![The first step in a super-resolution image reconstruction algorithm is to estimate
the supixel shifts of each frame with respect to a reference frame. The second step is
to increase the effective spatial sampling by operating on a sequence of low-resolu-
tion subpixel-shifted images. There are also spatial and spatial frequency domain
methods for the subpixel shift estimation and the generation of the high-resolution
output images. These algorithms, examples, and the image performance are
discussed in Chapter 6.
1.6 Image Restoration—Deblurring
An imaging system’s blurring function, also called the point spread function (PSF), is
another common factor in the reduction of high-frequency components in the
image. Image restoration tries to inverse this blurring degradation phenomenon, but
within the bandlimit of the imager (i.e., it enhances the spatial frequencies within the
imager band). This includes deblurring images that are degraded by the limitations
of a sensor or environment. The estimate or knowledge of the blurring function is
essential to the application of these algorithms. One of the most important consider-
ations of designing a deblurring filter is to control noise, since the noise is likely
amplified at high frequencies. The amplification of noise results in undesired arti-
facts in the output image. Figure 1.2 shows examples of image deblurring. One input
image [Figure 1.2(a)] contains the blur, while the deblurred version of it [Figure
1.2(b)] removes the most blur. Another input image [Figure 1.2(c)] contains the blur
and noise; the noise effect illustrates on the deblurred version of it [Figure 1.2(d)].
Image restoration tries to recover the high-frequency information below the diffrac-
tion limit while limiting the noise artifacts. The designs of deblurring filters, the
6 Introduction
(a) (b)
(c) (d)
Figure 1.2 Examples of image deblurring: (a) blurred bar image; (b) deblurred version of (a); (c)
blurred bar image with noise added; and (d) deblurred version of (c).](https://image.slidesharecdn.com/810291-250522124357-52de8d91/85/Signal-Processing-And-Performance-Analysis-For-Imaging-Systems-S-Susan-Young-27-320.jpg)
![noise control mechanisms, examples, and image performance are discussed in
Chapter 7.
1.7 Image Contrast Enhancement
Image details can also be enhanced by image contrast enhancement techniques in
which certain image edges are emphasized as desired. For an example of a medical
application in diagnosing breast cancer from mammograms, radiologists follow the
ductal networks to look for abnormalities. However, the number of ducts and the
shape of ductal branches vary with individuals, which make the visual process of
locating the ducts difficult. The image contrast enhancement provides the ability to
enhance the appearance of the ductal elements relative to the fatty-tissue surround-
ings, which helps radiologists to visualize abnormalities in mammograms.
Image contrast enhancement methods can be divided into single-scale approach
and multiscale approach. In the single-scale approach, the image is processed in the
original image domain, such as a simple look-up table. In the multiscale approach,
the image is decomposed into multiple resolution scales, and processing is per-
formed in the multiscale domain. Because the information at each scale is adjusted
before the image is reconstructed back to the original image intensity domain, the
output image contains the desired detail information. The multiscale approach can
also be coupled with the dynamic range reduction. Therefore, the detail information
in different scales can be displayed in one output image. Localized contrast
enhancement (LCE) is the process in which these techniques are applied on a local
scale for the management of dynamic range in the image. For example, the
sky-to-ground interface in infrared imaging can include a huge apparent tempera-
ture difference that occupies most of the image dynamic range. Small targets with
smaller signals can be lost, while LCE can reduce the large sky-to-ground interface
signal and enhance small target signals (see Figure 8.10 later in this book). Details of
the algorithms, examples, and image performance are discussed in Chapter 8.
1.8 Nonuniformity Correction (NUC)
Focal plan array (FPA) sensors have been used in many commercial and military
applications, including both visible and infrared imaging systems, since they have
wide spectral responses, compact structures, and cost-effective production. How-
ever, each individual photodetector in the FPA has a different photoresponse, due to
detector-to-detector variability in the FPA fabrication process [1]. Images that are
acquired by an FPA sensor suffer from a common problem known as fixed-pattern
noise, or spatial nonuniformity. The technique to compensate for this distortion is
called nonuniformity correction (NUC). Figure 1.3 shows an example of a
nonuniformity corrected image from an original input image with the fixed-pattern
noise.
There are two main categories of NUC algorithms, namely, calibration-based
and scene-adaptive algorithms. A conventional, calibration-based NUC is the stan-
dard two-point calibration, which is also called linear NUC. This algorithm esti-
1.7 Image Contrast Enhancement 7](https://image.slidesharecdn.com/810291-250522124357-52de8d91/85/Signal-Processing-And-Performance-Analysis-For-Imaging-Systems-S-Susan-Young-28-320.jpg)
![mates the gain and offset parameters by exposing the FPA to two distinct and
uniform irradiance levels. The scene-adaptive NUC uses the data acquired in the
video sequence and a motion estimation algorithm to register each point in the scene
across all of the image frames. This way, continuous compensation can be applied
adaptively for individual detector responses and background changes. These algo-
rithms, examples, and imaging system performance are discussed in Chapter 9.
1.9 Tone Scale
Tone scale is a technique that improves the image presentation on an output display
medium (softcopy display or hardcopy print). Tone scale is also a mathematical
mapping of the image pixel values from the sensor to a region of interest on an out-
put medium. Note that tone scale transforms improve only the appearance of the
image, but not the image quality itself. The gray value resolution is still the same.
However, a proper tone scale allows the characteristic curve of a display system to
match the sensitivity of the human eye to enhance the image interpretation task per-
formance. There are various tone scale techniques, including piece-wise linear tone
scale, nonlinear tone scale, and perceptual linearization tone scale. These techniques
and a tone scale performance example are discussed in Chapter 10.
1.10 Image Fusion
Because researchers realize that different sensors provide different signature cues of
the scene, image fusion has been receiving additional attention in signal processing.
Some of those applications are shown to benefit from fusing the images of multiple
sensors.
Imaging sensor characteristics are determined by the wavebands that they
respond to in the electromagnetic spectrum. Figure 1.4 is a diagram of the electro-
magnetic spectrum with wavelength indicated in metric length units [2]. The most
familiar classifications of wavebands are the radiowave, microwave, infrared, visi-
ble, ultraviolet, X-ray, and gamma-ray wavebands. Figure 1.5 shows further subdi-
vided wavebands for broadband sensors [3]. For example, the infrared waveband is
8 Introduction
(a) (b)
Figure 1.3 Example of nonuniformity correction: (a) input image with the fixed-pattern noise
shown in the image; and (b) nonuniformity corrected image in which the helicopter in the center
is clearly illustrated.](https://image.slidesharecdn.com/810291-250522124357-52de8d91/85/Signal-Processing-And-Performance-Analysis-For-Imaging-Systems-S-Susan-Young-29-320.jpg)
![Many questions of image fusion remain unanswered and open to new research
opportunities. Some of the questions involve how to select different sensors to pro-
vide better image information from the scene; whether different imaging informa-
tion can be effectively combined to provide a better cue in the scene; and how to best
combine the information. These issues are presented and examples and imaging sys-
tem performance are provided in Chapter 11.
References
[1] Milton, A. F., F. B. Barone, and M. R. Kruer, “Influence of Nonuniformity on Infrared
Focal Plan Array Performance,” Optical Engineering, Vol. 24, No. 5, 1985, pp. 855–862.
[2] Richards, A., Alien Vision—Exploring the Electromagnetic Spectrum with Imaging Tech-
nology, Bellingham, WA: SPIE Press, 2001.
[3] Driggers, R. G., P. Cox, and T. Edwards, Introduction to Infrared and Electro-Optical Sys-
tems, Norwood, MA: Artech House, 1999.
10 Introduction
Road cracks
MW LW
Road cracks
clearly visible in
LW but not MW
Sun glint
Fused image
Figure 1.6 Example of fusing MWIR and LWIR images. The road cracks are visible in LWIR, but
not in MWIR. Similarly, the Sun glint is visible in MWIR image, but not in LWIR. The fused image
shows both Sun glint and road cracks.](https://image.slidesharecdn.com/810291-250522124357-52de8d91/85/Signal-Processing-And-Performance-Analysis-For-Imaging-Systems-S-Susan-Young-31-320.jpg)
![been adjusted. Dynamic range may be fully utilized in a visible sensor. For the case
of an infrared sensor, a portion of the dynamic range is used by the large back-
ground flux radiated by everything in the scene. This flux is never a small value;
hence, sensitivity and dynamic range requirements are much more difficult to satisfy
in infrared sensors than in visible sensors.
2.2 Resolution and Sensitivity
There are three general categories of infrared sensor performance characterizations.
The first is sensitivity and the second is resolution. When end-to-end, or
human-in-the-loop (HITL), performance is required, the third type of performance
characterization describes the visual acuity of an observer through a sensor, which
will be discussed in Chapter 3. The former two are both related to the hardware and
software that comprises the system, while the latter includes both the sensor and the
observer. The first type of measure, sensitivity, is determined through radiometric
analysis of the scene/environment and the quantum electronic properties of the
detectors. Resolution is determined by analysis of the physical optical properties,
the detector array geometry, and other degrading components of the system in
much the same manner as complex electronic circuit/signals analysis.
Sensitivity describes how the sensor performs with respect to input signal level.
It relates noise characteristics, responsivity of the detector, light gathering of the
optics, and the dynamic range/quantization of the sensor. Radiometry describes
how much light leaves the object and background and is collected by the detector.
Optical design and detector characteristics are of considerable importance in sensor
sensitivity analysis. In infrared systems, noise equivalent temperature difference
(NETD) is often a first-order description of the system sensitivity. The three-dimen-
sional (3-D) noise model [1] describes more detailed representations of sensitivity
parameters. In visible systems, the noise equivalent irradiance (NEI) is a similar
term that is used to determine the sensitivity of the system.
The second type of measure is resolution. Resolution is the ability of the sensor
to image small targets and to resolve fine detail in large targets. Modulation transfer
function (MTF) is the most widely used resolution descriptor in infrared systems.
Alternatively, it may be specified by a number of descriptive metrics, such as the
optical Rayleigh Criterion or the instantaneous field-of-view (FOV), of the detector.
While these metrics are component-level descriptions, the system MTF is an
all-encompassing function that describes the system resolution.
Sensitivity and resolution can be competing system characteristics, and they are
the most important issues in initial studies for a design. For example, given a fixed
sensor aperture diameter, an increase in focal length can provide an increase in reso-
lution, but it may decrease sensitivity [1]. Typically, visible band systems have
plenty of sensitivity and are resolution-limited, while infrared imagers have been
more sensitivity-limited. With staring infrared sensors, the sensitivity has seen
significant improvements.
Quite often metrics, such as NETD and MTF, are considered to be separable.
However, in an actual sensor, sensitivity and resolution performance are not inde-
pendent. As a result, minimum resolvable temperature difference (MRT or MRTD)
2.2 Resolution and Sensitivity 15](https://image.slidesharecdn.com/810291-250522124357-52de8d91/85/Signal-Processing-And-Performance-Analysis-For-Imaging-Systems-S-Susan-Young-36-320.jpg)
![or the sensor contrast threshold function (CTF) has become the primary perfor-
mance metrics for infrared systems. MRT and MRC (minimum resolvable contrast)
are a quantitative performance measure in terms of both sensitivity and resolution.
A simple MRT curve is shown in Figure 2.5. The performance is bounded by the sen-
sor’s limits and the observer’s limits. The temperature difference, or thermal con-
trast, required to image smaller details in a scene increases with detail size. The
inclusion of observer performance yields a single-sensor performance characteriza-
tion. It describes sensitivity as a function of resolution and includes the human visual
system.
2.3 Linear Shift-Invariant (LSI) Imaging Systems
A linear imaging system requires two properties [1, 2]: superposition and scaling.
Consider an input scene, i(x, y) and an output image, o(x, y). Given that a linear sys-
tem is described by L{}, then
( ) ( )
{ }
o x y L i x y
, ,
= (2.3)
The superposition and scaling properties are satisfied if
( ) ( )
{ } ( )
{ } ( )
{ }
L ai x y bi x y aL i x y bL i x y
1 2 1 2
, , , ,
+ = + (2.4)
where i1(x, y) and i2(x, y) are input scenes and a and b are constants. Superposition,
simply described, is that the image of two scenes, such as a target scene and a back-
ground scene, is the sum of individual scenes imaged separately. The simplest exam-
ple here is that of a point source as shown in Figure 2.6. The left side of the figure
shows the case where a single point source is imaged, then a second point source is
imaged, and the two results are summed to give an image of the two point sources.
16 Imaging Systems
Spatial frequency
Minimum
resolvable
temperature
Visual
sensitivity
limit
System
resolution
limit
System
response
Figure 2.5 Sensor resolution and sensitivity.](https://image.slidesharecdn.com/810291-250522124357-52de8d91/85/Signal-Processing-And-Performance-Analysis-For-Imaging-Systems-S-Susan-Young-37-320.jpg)
![filter that is convolved with the input scene to obtain an output image. The simpli-
fied LSI imaging system model is shown in Figure 2.7.
The system described here is valid for LSI systems only. This analysis technique
is a reasonable description for continuous and well-sampled imaging systems. It is
not a good description for an undersampled or a well-designed sampled imaging
system. These sampled imaging systems do satisfy the requirements of a linear sys-
tem, but they do not follow the shift invariance property. The sampling nature of
these systems is described later in this chapter. The representation of sampled imag-
ing systems is a modification to this approach.
For completeness, we take the spatial domain linear systems model and convert
it to the spatial frequency domain. Spatial filtering can be accomplished in both
domains. Given that x and y are spatial coordinates in units of milliradians, the
spatial frequency domain has independent variables of fx and fy, cycles per
milliradian. A spatial input or output function is related to its spectrum by the Fou-
rier transform
( ) ( ) ( )
F f f f x y e dxdy
x y
j f x f y
x y
, ,
=
− +
−∞
∞
−∞
∞
∫
∫
2 π
(2.12)
where the inverse Fourier transform converts an image spectrum to a spatial
function
( ) ( ) ( )
f x y F f f e df df
x y
j f x f y
x y
x y
, ,
=
+
−∞
∞
−∞
∞
∫
∫
2 π
(2.13)
The properties and characteristics of the Fourier transform are provided in
[2–4]. A function and its spectrum are collectively described as a Fourier transform
pair. We will use the notation of the Fourier transform operator
( ) ( )
[ ] ( ) ( )
[ ]
G f f g x y g x y G f f
x y x y
, , , ,
= = −
ᑤ ᑤ
and 1
(2.14)
in order to simplify analyses descriptions. One of the very important properties of
the Fourier transform is that the Fourier transform of a convolution results in a
product. Therefore, the spatial convolution described in (2.11) results in a spectrum
of
( ) ( ) ( )
O f f I f f H f f
x y x y x y
, , ,
= (2.15)
Here, the output spectrum is related to the input spectrum by the product of the
Fourier transform of the system impulse response. Therefore, the Fourier transform
of the system impulse response is called the transfer function of the system. Multi-
plication of the input scene spectrum by the transfer function of an imaging system
provides the same filtering action as the convolution of the input scene with the
imaging system PSF. In imaging systems, the magnitude of the Fourier transform of
the system PSF is the modulation transfer function (MTF).
2.3 Linear Shift-Invariant (LSI) Imaging Systems 19](https://image.slidesharecdn.com/810291-250522124357-52de8d91/85/Signal-Processing-And-Performance-Analysis-For-Imaging-Systems-S-Susan-Young-40-320.jpg)
![beginning with the optical effects. Also, the transfer function of a system, as given in
(2.17), is frequently described without the eye transfer function.
2.4.1 Optical Filtering
Two filters account for the optical effects in an imaging system: diffraction and
aberrations. The diffraction filter accounts for the spreading of the light as it passes
an obstruction or an aperture. The diffraction impulse response for an incoherent
imaging system with a circular aperture of diameter D is
( )
h x y
D
somb
Dr
diff , =
λ λ
2
2
(2.18)
where λ is the average band wavelength and r x y
= +
2 2
. The somb (for som-
brero) function is given by Gaskill [3] to be
( )
( )
somb r
J r
r
=
1 π
π
(2.19)
where J1 is the first-order Bessel function of the first kind. The filtering associated
with the optical aberrations is sometimes called the geometric blur. There are many
ways to model this blur and there are numerous commercial programs for calculat-
ing geometric blur at different locations on the image. However, a convenient
method is to consider the geometric blur collectively as a Gaussian function
( )
h x y Gaus
r
geom
gb gb
, =
1
2
σ σ
(2.20)
where σgb is an amplitude that best describes the blur associated with the
aberrations.
The Gaussian function, Gaus, is
( )
Gaus r e r
= −π 2
(2.21)
Note that the scaling values in front of the somb and the Gaus functions are
intended to provide a functional area (under the curve) of unity so that no gain
is applied to the scene. Examples of the optical impulse responses are given in
Figure 2.9 corresponding to a wavelength of 10 µm, an optical diameter of 10 cen-
timeters, and a geometric blur of 0.1 milliradian.
The overall impulse response of the optics is the combined blur of both the dif-
fraction and aberration effects
( ) ( ) ( )
h x y h x y h x y
optics diff geom
, , ** ,
= (2.22)
The transfer functions corresponding to these impulse responses are obtained
by taking the Fourier transform of the functions given in (2.18) and (2.20). The Fou-
rier transform of the somb is given by Gaskill [3] so that the transfer function is
2.4 Imaging System Point Spread Function and Modulation Transfer Function 21](https://image.slidesharecdn.com/810291-250522124357-52de8d91/85/Signal-Processing-And-Performance-Analysis-For-Imaging-Systems-S-Susan-Young-42-320.jpg)
![( )
H f f
D D D
diff x y
, cos
=
− −
−
2
1
1
2
π
ρλ ρλ ρλ
(2.23)
where ρ = +
f f
x y
2 2
and is plotted in cycles per milliradian and D is the entrance
aperture diameter. The Fourier transform of the Gaus function is simply the Gaus
function [4], with care taken on the scaling property of the transform. The transfer
function corresponding to the aberration effects is
( ) ( )
H f f Gaus
geom x y gb
, = σ ρ (2.24)
For the example described here, the transfer functions are shown in Figure 2.10.
Note that the overall optical transfer function is the product of the two functions.
2.4.2 Detector Spatial Filters
The detector spatial filter is also comprised of a number of different effects, includ-
ing spatial integration, sample-and-hold, crosstalk, and responsivity, among others.
The two most common effects are spatial integration and sample-and-hold; that is,
( ) ( ) ( )
h x y h x y h x y
sp sh
det det_ det_
, , ** ,
= (2.25)
22 Imaging Systems
−0.2
−0.1
0
0.1
0.2
−0.2
−0.1
0
0.1
0.2
0
0.2
0.4
0.6
0.8
1
−0.2
−0.1
0 0.1
0.2
−0.1
0
0.1
0.2
0
0.2
0.4
0.6
0.8
1
y milliradians
hdiff
( )
x,y hgeom
( )
x,y
m
x illiradians
y milliradians m
x illiradians
Figure 2.9 Spatial representations of optical blur.
10
0.2
0.4
0.6
0.8
1
0
5
0
−5
0.2
0.4
0.6
0.8
1
0
cycles per
milliradians
H f f
( , )
geom
−10
10
5
0
−5
−10
10
f
5
0
−5
−10
10
5
0
−5
−10
cycles per
milliradians
cycles per
milliradians
cycles per
milliradians
x fx
fy fy
x y
H f f
diff x y
( , )
Figure 2.10 Optical transfer functions of optical blur.](https://image.slidesharecdn.com/810291-250522124357-52de8d91/85/Signal-Processing-And-Performance-Analysis-For-Imaging-Systems-S-Susan-Young-43-320.jpg)
![The other effects can be included, but they are usually considered negligible
unless there is good reason to believe otherwise (i.e., the detector responsivity varies
dramatically over the detector).
The detector spatial impulse response is due to the spatial integration of the
light over the detector. Since most detectors are rectangular in shape, the rectangle
function is used as the spatial model of the detector
( )
h x y
DAS DAS
rect
x
DAS
y
DAS
DAS
r
sp
x y x y
x
det_ , ,
=
=
1
1
ect
x
DAS DAS
rect
y
DAS
x y y
1
(2.26)
where DASx and DASy are the horizontal and vertical detector angular subtenses in
milliradians. The detector angular subtense is the detector width (or height) divided
by the sensor focal length. The transfer function corresponding to the detector spa-
tial integration is determined by taking the Fourier transform of (2.26)
( ) ( ) ( )
H f f DAS f DAS f DAS f DAS f
sp x y x x y y x x y
det_ , ,
= =
sinc sinc sinc( )
y (2.27)
where the sinc function is defined as [2]
( )
( )
sinc x
x
x
=
sin π
π
(2.28)
The impulse response and the transfer function for a detector with a 0.1 by 0.1
milliradian detector angular subtense is shown in Figure 2.11.
The detector sample-and-hold function is an integration of the light as the
detector scans across the image. This sample-and-hold function is not present in
staring arrays, but it is present in most scanning systems where the output of the
integrated signal is sampled. The sampling direction is assumed to be the horizontal,
or x, direction. Usually, the distance, in milliradians, between samples is smaller
than the detector angular subtense by a factor called samples per IFOV or samples
per DAS, spdas. The sample-and-hold function can be considered a rectangular
2.4 Imaging System Point Spread Function and Modulation Transfer Function 23
40
0
0.5
1
h x y
( , )
det_sp
−0.5
20
0
−20
0.2
0.4
0.6
0.8
1
0
y milliradians
H f
( , )
f
det_sp
−40
1.0
0
−0.1
1.0
0
−0.1
x milliradians
cycles per
milliradians
cycles per
milliradians
fy fx
40
20
0
−20
−40
x y
Figure 2.11 Detector spatial impulse response and transfer function.](https://image.slidesharecdn.com/810291-250522124357-52de8d91/85/Signal-Processing-And-Performance-Analysis-For-Imaging-Systems-S-Susan-Young-44-320.jpg)
![function in x where the size of the rectangle corresponds to the distance between
samples. In the spatial domain y direction, the function is an impulse function.
Therefore, the impulse response of the sample-and-hold function is
( ) ( )
h x y
spdas
DAS
rect
x spdas
DAS
y
sh
x x
det_ , =
δ (2.29)
The Fourier transform of the impulse response gives the transfer function of the
sample- and-hold operation
( )
H f f
DAS f
spdas
sh x y
x x
det_ , =
sinc (2.30)
Note that the Fourier transform of the impulse function in the y direction is 1.
The impulse response and the transfer function for sample-and-hold associated with
the detector given in Figure 2.11 with a two-sample-per-DAS sample-and-hold are
shown in Figure 2.12.
2.4.3 Electronics Filtering
The electronics filtering function is one of the more difficult to characterize and one
of the more loosely applied functions. First, it involves the conversion of temporal
frequencies to spatial frequencies. Usually, this involves some scan rate or readout
rate. Second, most impulse response functions in space are even functions. With
electronic filtering, the impulse function can be a one-sided function. Finally, most
engineers apply a two-sided impulse response that violates the rules of causality.
This gross approximation does not usually have a heavy impact on sensor perfor-
mance estimates since the electronics are not typically the limiting component of the
sensor. Holst [5] and Vollmerhausen and Driggers [6] provide electronic filter (digi-
tal and analog) approximations that can be used in transfer function estimates.
Digital filters also provide a spatial blur and a corresponding transfer function.
Finite impulse response (FIR) filters are common in electro-optical and infrared sys-
24 Imaging Systems
40
0
0.5
1
h x y
( , )
det_sh
−0.5
20
0
−20
0.2
0.4
0.6
0.8
1
0
y milliradians
H f
( , )
f
det_sh
−40
1.0
0
−0.1
1.0
0
−0.1
x milliradians
cycles per
milliradians
cycles per
milliradians
fy fx
40
20
0
−20
−40
x y
−0.05
0.05
−0.05
0.05
Figure 2.12 Detector sample-and-hold impulse response and transfer function.](https://image.slidesharecdn.com/810291-250522124357-52de8d91/85/Signal-Processing-And-Performance-Analysis-For-Imaging-Systems-S-Susan-Young-45-320.jpg)
![( ) ( )
H f f Gaus
disp x y disp angle
, _
= σ ρ Gaussian display (2.34)
or
( ) ( )
H f f W f H f
disp x y disp angle h x disp angle v y
, ,
_ _ _ _
= sinc Flat-panel display (2.35)
Again, these transfer functions are shown in Figures 2.9 and 2.11.
2.4.5 Human Eye
Note that the human eye is not part of the system performance MTF as shown in
Figure 2.8, and the eye MTF should not be included in the PSF of the system. In
Chapter 3, the eye CTF is used to include the eye sensitivity and resolution limita-
tions in performance calculations. It is, however, useful to understand the PSF and
MTF of the eye such that the eye blur can be compared to sensor blur. A system with
much higher resolution than the eye is a waste of money, and a system with much
lower resolution than the eye is a poorly performing system.
The human eye certainly has a PSF that is a combination of three physical com-
ponents: optics, retina, and tremor [7, 8]. In terms of these components, the PSF is
( ) ( ) ( ) ( )
h x y h x y h x y h x y
eye optics retina tremor
, , ** , ** ,
_
= (2.36)
Therefore, the transfer function of the eye is
( ) ( ) ( ) ( )
H f f H f f H f f H f f
eye x y eye optics x y retina x y tremor x y
, , , ,
_
= (2.37)
The transfer function associated with the eye optics is a function of display light
level. This is because the pupil diameter changes with light level. The number of
foot-Lamberts (fL) at the eye from the display is Ld/0.929, where Ld is the display
luminance in millilamberts. The pupil diameter is then
( )
{ } [ ]
D fL
pupil = − + −
9011 1323 21082
10
. . exp log . mm (2.38)
This equation is valid, if one eye is used as in some targeting applications. If both
eyes view the display, the pupil diameter is reduced by 0.5 millimeter. Two parame-
ters, io and fo, are required for the eye optics transfer function. The first parameter is
( )
io Dpupil
= +
0 7155 0277
2
. . (2.39)
and the second is
( )
{ }
fo D D
pupil pupil
= −
exp . . * log
3663 00216 2
(2.40)
Now, the eye optics transfer function can be written as
( ) ( )
[ ]
{ }
H M fo
eye optics
io
_ exp .
ρ ρ
= − 4369 (2.41)
26 Imaging Systems](https://image.slidesharecdn.com/810291-250522124357-52de8d91/85/Signal-Processing-And-Performance-Analysis-For-Imaging-Systems-S-Susan-Young-47-320.jpg)
![where o1(x, y) is the presampled blur image or the output of the presample blur pro-
cess. The convolution is denoted by the *, so ** denotes the two-dimensional con-
volution. The sampling process can be modeled with the multiplication of the
presample blur image with the sampling function. For convention, we use Gaskill’s
comb function [9]
( ) ( )
comb
x
a
y
b
a b x ma y nb
n
m
,
= − −
=−∞
∞
=−∞
∞
∑
∑ δ δ (2.47)
which is a two-dimensional separable function. Now the output of the sampling
process can be written as the product of the presample blurred image with the sam-
pling function (note that a and b are the distances in milliradians or millimeters
between samples)
( ) ( ) ( ) ( )
[ ]
o x y o x y
ab
comb
x
a
y
b
i x y h x y
ab
comb
x
a
2 1
1 1
, , , , ** , ,
=
=
y
b
(2.48)
At this point, all that is present is a set of discrete values that represent the
presample blurred image at discrete locations. This output can be thought of as a
weighted “bed of nails” that is meaningless to look at unless the “image” is recon-
structed. The display and the eye, if applied properly, reconstruct the image to a
function that is interpretable. This reconstruction is modeled as the convolution of
the display and eye blur (and any other spatial postsample blur) and the output of
the sampling process; that is,
( ) ( ) ( ) ( ) ( )
[ ]
{
o x y o x y d x y i x y h x y
ab
comb
x
a
y
b
, , ** , , ** ,
, *
= =
×
2
1
( )
* ,
d x y
(2.49)
While (2.49) appears to be a simple spatial process, there is a great deal that is
inherent in the calculation. We have simplified the equation with the aggregate
presample blur effects and the aggregate postsample reconstruction effects.
The frequency analysis of the three-step process shown in Figure 2.15 can be
presented simply by taking the Fourier transform of each process step. Consider the
first step in the process, the presample blur. The transform of the convolution in
space is equivalent to a product in spatial frequency
2.5 Sampled Imaging Systems 29
Presample
blur
Image sample Reconstruction
x x x
i x,y
( )
h x,y
( ) s x,y
( ) d x,y
( )
o x,y
( )
o1( )
x,y o2( )
x,y
Figure 2.15 Three-step imaging process.](https://image.slidesharecdn.com/810291-250522124357-52de8d91/85/Signal-Processing-And-Performance-Analysis-For-Imaging-Systems-S-Susan-Young-50-320.jpg)
![( ) ( ) ( )
O f f I f f H f f
x y x y pre x y
1 , , ,
= (2.50)
where fx and fy are the horizontal and vertical spatial frequencies. If x and y are in
milliradians, then the spatial frequencies are in cycles per milliradian. Hpre(fx, fy) is
the Fourier transform of the presample blur spot. Note that the output spectrum can
be normalized to the input spectrum so that Hpre(fx, fy) is a transfer function that fol-
lows the linear systems principles. Consider the presample blur spectrum (i.e., the
presample blur transfer function given in Figure 2.16). Note that this is the image
spectrum on the output of the blur that would occur if an impulse were input to the
system.
Next, we address the sampling process. The Fourier transform of (2.48) gives
( ) ( ) ( )
[ ] ( )
O f f I f f H f f comb af bf
x y x y pre x y x y
2 , , , ** ,
= (2.51)
where
( ) ( ) ( )
comb af bf f kf f lf f a f b
x y x xs y ys xs ys
l
, ,
= − − = =
=−∞
δ δ 1 1
and
∞
=−∞
∞
∑
∑
k
(2.52)
If an impulse were input to the system, the response would be
( ) ( ) ( )
O f f H f f comb af bf
x y pre x y x y
2 , , ** ,
= (2.53)
which is a replication of the presample blur at sample spacings of 1/a and 1/b. Con-
sider the case shown in Figure 2.17. The sampled spectrum shown corresponds to a
Gaussian blur of 0.5-milliradian radius (to the 0.043 cutoff) and a sample spacing of
0.5 milliradian.
Note that the reproduction in frequency of the presample blur is at 2 cycles per
milliradian. The so-called Nyquist rate of the sensor (the sensor half-sample rate) is
at 1 cycle per milliradian. Any frequency from the presample blur baseband that is
greater than the half-sample rate is also present as a mirror signal, or classical
aliasing, under the half-sample rate by the first-order reproduction. The amount of
classical aliasing is easily computed as the area of this mirrored signal. However,
this is not the aliased signal seen on the output of the display as the display transfer
30 Imaging Systems
0
0.2
0.4
0.6
0.8
1
1.2
−3 −2 −1 0 1 2 3
Cycles/mrad
H (fx,fy)
pre
Figure 2.16 Presample blur transfer function.](https://image.slidesharecdn.com/810291-250522124357-52de8d91/85/Signal-Processing-And-Performance-Analysis-For-Imaging-Systems-S-Susan-Young-51-320.jpg)
![has not been applied to the signal. The higher-order replications of the baseband are
real-frequency components. The curves to the left and the right of the central curve
are the first- and second-order replications at the positive and negative positions.
The current state of the sample signal is tiny infinitesimal points weighted with the
presample blurred image values. These points have spectra that extend in the fre-
quency domain to very high frequencies. The higher-order replications are typically
filtered with a reconstruction function involving the display and the eye. There is no
practical way to implement the perfect reconstruction filter; however, the perfect
rectangular filter would eliminate these higher-order replications and result only in
the classical aliased signal. The reconstruction filter usually degrades the baseband
and allows some of the signal of the higher-order terms through to the observer.
The final step in the process corresponds to the reconstruction of the sampled
information. This is accomplished simply by blurring the infinitesimal points so that
the function looks nearly like that of the continuous input imagery. The blur is
convolved in space, so it is multiplied in frequency
( ) ( ) ( )
[ ] ( )
O f f H f f comb af bf D f f
x y pre x y x y x y
, , ** , ,
= (2.54)
where this output corresponds to a point source input. Note that the postsampling
transfer function is multiplied by the sampled spectrum to give the output of the
whole system. Consider the sampled spectrum and the dashed display transfer func-
tion shown in Figure 2.18. The postsampling transfer function is shown in the graph
as the display, passes part of the first-order replications. However, this display
degrades the baseband signal relatively little. The tradeoff here is baseband resolu-
tion versus spurious response content.
Given that all of the signals traverse through the postsample blur transfer func-
tion, the output is shown in Figure 2.19. There is classical aliasing on the output of
the system, but a large signal corresponds to higher-order replication signals that
passed through the display. Aliasing and the higher-order signals are collectively the
spurious response of the sensor. These signals were not present on the input imag-
ery, but there are artifacts on the output imagery. Without sampling, these spurious
2.5 Sampled Imaging Systems 31
Cycles/mrad
0
0.2
0.4
0.6
0.8
1
−5 −3 −1 1 3 5
2
O (fx ,fy)
Figure 2.17 Output of sampling.](https://image.slidesharecdn.com/810291-250522124357-52de8d91/85/Signal-Processing-And-Performance-Analysis-For-Imaging-Systems-S-Susan-Young-52-320.jpg)
![signals would not be present. The higher-order replicated signals are combined at
each spatial frequency in terms of a vector sum, so it is convenient to represent the
magnitude of the spurious signals as the root-sum-squared (RSS) of the spurious
orders that make it through the reconstruction process.
Three aggregate quantities have proven useful in describing the spurious
response of a sampled imaging system: total integrated spurious response as defined
by (2.55), in-band spurious response as defined by (2.56), and out-of-band spurious
response as defined by (2.57) [10]; that is,
( )
( )
SR
Spurious df
df
x
x
= −∞
∞
−∞
∞
∫
∫
Response
BasebandSignal
(2.55)
( )
SR
Spurious df
in band
x
f
f
s
s
−
−
=
∫ Response
BasebandSigna
2
2
( )
l dfx
−∞
∞
∫
(2.56)
32 Imaging Systems
Cycles/mrad
0
0.2
0.4
0.6
0.8
1
−5 −3 −1 1 3 5
Sampled signal and display transfer
Figure 2.18 Sampled signal and display transfer.
Cycles/mrad
System output spectrum
0
0.2
0.4
0.6
0.8
1
−5 −3 −1 1 3 5
Figure 2.19 System output signal.](https://image.slidesharecdn.com/810291-250522124357-52de8d91/85/Signal-Processing-And-Performance-Analysis-For-Imaging-Systems-S-Susan-Young-53-320.jpg)
![St. Paul as he appeared in public. They may have known very little
of the inner history of his life as he reveals it in his Epistle to the
Galatians when vindicating his apostolic authority and mission.[55]
Let us now see whether we cannot harmonise St. Paul's
autobiographical narrative in the Epistle with the Evangelist's
narrative in the Acts; always remembering, however, that an
imperfect knowledge is never more completely felt than in such
cases. When we try to harmonise an account written from the
subjective side by one individual with an objective and exterior
narrative written by some one else, we are like a man looking at a
globe and trying to take it all in at one glance. One side must be
hidden from him; and so in this case, many circumstances are
necessarily concealed from us which would solve difficulties that now
completely puzzle us. But let us to our task, in which we have
derived much assistance from the commentary of Bishop Lightfoot
upon Galatians. St. Paul, we are told in ch. ix. 19, received meat
after the visit of Ananias and was strengthened. St. Paul was never
one of those high-wrought fanatics who despise food and the care of
the body. There was nothing of the Gnostic or the Manichean about
him, leading him to despise and neglect the body which the Lord has
given to be the soul's instrument. He recognised under all
circumstances that if the human spirit is to do its work, and if God's
glory is to be promoted, the human body must be sustained in force
and vigour. When he was on board ship and in imminent peril of
shipwreck and death, and men thought they should be at their
prayers, thinking of the next world alone, he took bread and blessed
and set the crew and passengers alike the healthy example of eating
a hearty meal, and thus keeping his body in due preparation for
whatever deliverances the Lord might work for them; and so, too, at
Damascus, his spiritual joy and hallowed peace and deep gratitude
for his restoration to sight did not prevent him paying due attention
to the wants of his body. "He took food, and was strengthened." And
now comes the first note of time. "Then was Saul certain days with
the disciples which were at Damascus. And straightway (εὐθέως) he
preached Christ in the synagogues, that He is the Son of God." The](https://image.slidesharecdn.com/810291-250522124357-52de8d91/85/Signal-Processing-And-Performance-Analysis-For-Imaging-Systems-S-Susan-Young-56-320.jpg)
![very same expression is used by St. Paul in Galatians, where, after
speaking of his conversion, he says, "Immediately (εὐθέως) I
conferred not with flesh and blood, but went away into Arabia, and
again returned unto Damascus." Now my explanation, and not mine
alone, but that of Bishop Lightfoot, is this. After the new convert had
rested for a short time at Damascus, he retired into the Sinaitic
desert, where he remained for several months, perhaps for a whole
year. During this period he disappeared from the sight and
knowledge of men as if the earth had opened its mouth and
swallowed him. Then he returned to Damascus and preached with
such power that the Jews formed a plot against his life, enlisting the
help of the governor on their side, so that even the gates were
watched that he might be arrested. He escaped their hands,
however, through the assistance of his converts, and went up to
Jerusalem.[56]
But here another difficulty arises. The Acts tells us that "when Saul
was come to Jerusalem, he assayed to join himself to the disciples;
but they were all afraid of him, and believed not that he was a
disciple," whereupon Barnabas, fulfilling his office of mediation,
explanation, and consolation, took him and introduced him to the
Apostles; while on the other hand in the first chapter of Galatians St.
Paul himself speaks of his first visit to the Jerusalem Church thus:
"Then after three years I went up to Jerusalem to visit Cephas, and
tarried with him fifteen days. But other of the Apostles saw I none,
save James the Lord's brother." Now the difficulty consists in this.
First, how could the disciples at Jerusalem have been suspicious of
St. Paul, if at least a year and a half had elapsed since his
conversion? for the Jewish method of counting time would not
require three whole years to have elapsed since that event.
Secondly, how could Barnabas have brought him to the Apostles as
the Acts states, if St. Paul himself says he saw none of them save
Peter and James? As to the first difficulty, we acknowledge at once
that it seems at first sight a very considerable one, and yet a little
reflection will show that there are many explanations of it. If St. Paul
kept quiet, as we believe he did, after his conversion and baptism,](https://image.slidesharecdn.com/810291-250522124357-52de8d91/85/Signal-Processing-And-Performance-Analysis-For-Imaging-Systems-S-Susan-Young-57-320.jpg)
![and departed into the solitudes of Arabia, and then upon his return
to Damascus, perhaps after a year's retirement, began his
aggressive work, there may not have been time for the Church at
large to get knowledge of the facts. Communication, again, may
have been interrupted because of the contest between Herod and
Aretas, in which Damascus played no small part. Communication
may not have been possible between the two Churches.[57]
Then,
again, the persecution raised by Saul himself seems to have
practically extirpated the Jerusalem Church for a time. "They were
all scattered abroad except the Apostles," is the account given of the
Christian community at Jerusalem. The terror of that persecution
may have lasted many a long month. Numbers of the original
members may never have ventured back again to the Holy City. The
Jerusalem Church may have been a new formation largely composed
of new converts who never had heard of a wondrous circumstance
which had happened a year or two before to the high priest's
delegate, which the Sanhedrin would doubtless desire to keep
secret.[58]
These and many other considerations offer themselves when we
strive to throw ourselves back into the circumstances of the time and
help to a solution of the first difficulty which we have indicated.
Human life is such a complex thing that the strangest combinations
may easily find place therein. In this particular case we are so
ignorant of the facts, so many hypotheses offer themselves to
account for the seeming inconsistencies, that we hesitate not to
identify the visit to Jerusalem mentioned in the Acts with that
recorded by St. Paul in the Epistle to the Galatians. The second
difficulty to which we have alluded is this, How could Barnabas have
brought him to the Apostles, if St. Paul himself states that he saw
none of the Apostles save Peter and James the Lord's brother? We
must remember, however, that St. Luke and St. Paul wrote with two
distinct objects. St. Paul, in the Galatians, wished to show the
independence of his revelations as regards the Apostles of the
circumcision, the Twelve technically so called. Of these Apostles he
saw not one, save St. Peter. St. Luke is giving a broad external](https://image.slidesharecdn.com/810291-250522124357-52de8d91/85/Signal-Processing-And-Performance-Analysis-For-Imaging-Systems-S-Susan-Young-58-320.jpg)
![account of the new convert's earliest religious history, and he tells us
that on his first visit to the Holy City his conversion was
acknowledged and guaranteed by the apostles,—not the Twelve
merely, but the apostles, that is, the senior members of the Christian
community, embracing not merely the original company chosen by
Christ, but all the senior members of the Church, like Barnabas,
James, and others who may have formed a supreme council to guide
the affairs of the infant society. The word apostle, in fact, is used
very variously in the New Testament; sometimes in a limited sense
as confined to the Twelve, sometimes in a wider and more general
sense, embracing men like Barnabas, as in Acts xiv. 4, 14; St. James,
the Lord's brother, as in 1 Cor. xv. 7; Andronicus and Junias, as in
Rom. xvi. 7, and many others. It is quite possible, then, that
Barnabas may have brought Saul to the Apostolic council, and told
there the tale of his conversion though not one of the original
Twelve was present save St. Peter.[59]
We have now endeavoured to explain some of the difficulties which
a comparison of St. Paul's own autobiographical narrative with the
Acts discloses. Let us look again at the retirement into Arabia. This
retirement seems to us full of instruction and pregnant with meaning
for the hidden as well as the practical life of the soul. St. Paul as
soon as he was baptized retired into Arabia; and why, it may be
asked, did he retire thither? Some of the ancient expositors, as St.
Chrysostom and St. Jerome, both of whom wrote about the same
period, A.D. 400, thought that St. Paul retired into Arabia in order
that he might preach to the Arabians. St. Chrysostom, for instance,
comments thus: "See how fervent was his soul, he was eager to
occupy lands yet untilled. He forthwith attacked a barbarous and
savage people, choosing a life of conflict and of much toil." And the
explanations of Hilary, Theodore of Mopsuestia, Theodoret, and
Œcumenius, all of them ancient and acute expositors, are of exactly
the same character. Now this would have been a reversal of the
Divine order in one important aspect. The power of the keys, the
office of opening the kingdom of heaven to the Gentiles had been
committed to St. Peter by Jesus Christ. He had not as yet baptized](https://image.slidesharecdn.com/810291-250522124357-52de8d91/85/Signal-Processing-And-Performance-Analysis-For-Imaging-Systems-S-Susan-Young-59-320.jpg)
![men are tempted to think so much of practical work which appears
at once in evidence, making its presence felt in tangible results, and
so very little of devotional work and spiritual retirement which
cannot be estimated by any earthly standard or tabulated according
to our modern methods. Men are now inclined to think laborare est
orare, and that active external work faithfully and vigorously
rendered can take the place and supply the want of prayer and
thought, of quiet study and devout meditation. Against such a
tendency the Lord's dealings with St. Paul, yea more, the Divine
dealings with and leadings of the eternal Son Himself, form a loud
and speaking protest. The world was perishing and men were going
down to the grave in darkness and Satan and sin were triumphing,
and yet Jesus was led up of the Spirit into the wilderness for forty
days, and Saul was brought out into the deserts of Arabia from amid
the teeming crowds of Damascus that he might learn those secrets
of the Divine life which are best communicated to those who wait
upon God in patient prayer and holy retirement. This is a lesson very
necessary for this hot and fitful and feverish age of ours, when men
are in such a hurry to have everything set right and every abuse
destroyed all at once. Their haste is not after the Divine model, and
their work cannot expect the stability and solidity we find in God's.
The nineteenth-century extreme is reproved by St. Paul's retirement
into Arabia.[60]
Man is, however, such a creature that if he avoids one
extreme he generally tumbles into another. And so it is in this matter.
Men have been ready to push this matter of retirement into an
extreme, and have considered that they were following St. Paul's
example in retiring into the Arabian and similar deserts and
remaining there. But they have made a great mistake. St. Paul
retired into Arabia for a while, and then "returned again unto
Damascus." They have retired into the deserts and have remained
there engaged in the one selfish task of saving their own souls, as
they thought, by the exercises of prayer and meditation, apart from
that life of active good works for the sake of others which
constitutes another department of Christianity equally vital to the
health of the soul.](https://image.slidesharecdn.com/810291-250522124357-52de8d91/85/Signal-Processing-And-Performance-Analysis-For-Imaging-Systems-S-Susan-Young-61-320.jpg)
![The history of Eastern monasticism is marked from its earliest days
by an eager desire to follow St. Paul in his retirement into Arabia,
and an equal disinclination to return with him unto Damascus. And
this characteristic, this intense devotion to a life of solitude strangely
enough passed over to our own Western islands, and is a dominant
feature of the monasticism which prevailed in Great Britain and
Ireland in the days of Celtic Christianity. The Syrian and Egyptian
monks passed over to Lerins and Southern Gaul, whence their
disciples came to England and Ireland, where they established
themselves, bringing with them all their Eastern love of solitary
deserts. This taste they perpetuated, as may be seen especially on
the western coast of Ireland, where the ruins of extensive monastic
settlements still exist, testifying to this craving. The last islands, for
instance, which a traveller sees as he steams away from Cork to
America, are called the Skelligs. They are ten miles west of the Kerry
coast, and yet there on these rocks where a boat cannot land
sometimes for months together the early monks of the fifth and six
centuries established themselves as in a desert in the ocean. The
topography of Ireland is full of evidences and witnesses of this
desire to imitate the Apostle of the Gentiles in his Arabian
retirement. There are dozens of town lands—subdivisions of the
parishes—which are called deserts or diserts,[61]
because they
constituted solitudes set apart for hermit life after the example of St.
Paul in Arabia and John the Baptist in the deserts of Judæa. While,
again, when we turn northwards along the western seaboard of
Ireland, we shall find numerous islands like the Skelligs, Ardoilen or
the High Island, off the coast of Connemara, and Innismurry off the
Sligo coast, where hermit cells in the regular Egyptian and Syrian
fashion were built, and still exist as they did a thousand years ago,
testifying to the longing of the human mind for such complete
solitude and close communion with God as Saul enjoyed when he
departed from Damascus.[62]
The monks of ancient times may have
run into one extreme: well would it be for us if we could avoid the
other, and learn to cultivate self-communion, meditation, self-
examination, and that realisation of the eternal world which God](https://image.slidesharecdn.com/810291-250522124357-52de8d91/85/Signal-Processing-And-Performance-Analysis-For-Imaging-Systems-S-Susan-Young-62-320.jpg)
![of the impious decree of Caligula, which they at last attained
through their own determination and by the intervention of Herod
Agrippa, who was then at Rome.[63]
It was during this awful period
of uncertainty and opposition that the infant Church enjoyed a brief
period of repose and quiet growth, because the whole nation from
the high priest to the lowest beggar had something else to think of
than how to persecute a new sect that was as yet rigorously
scrupulous in observing the law of Moses. During this period of
repose from persecution St. Peter made his tour of inspection
"throughout all parts," Samaria, Galilee, Judæa, terminating with
Lydda, where he healed, or at least prayed for the healing of,
Æneas,[64]
and with Joppa, where his prayer was followed by the
restoration of Tabitha or Dorcas, who has given a designation now
widely applied to the assistance which devout women can give to
their poorer sisters in Christ.
We thus see how God by the secret guidance of His Spirit, shaping
his course by ways and roads known only to Himself, led St. Peter to
the house of Simon the tanner, where he abode many days waiting
in patience to know God's mind and will which were soon to be
opened out to him. We have now traced the line of events which
connect the conversion of Saul of Tarsus with that of Cornelius the
centurion of Cæsarea. Let us apply ourselves to the circumstances
surrounding the latter event, which is of such vital importance to us
Gentile Christians as having been the formal Divine proclamation to
the Church and to the world that the mystery which had been hid for
ages was now made manifest, and that the Gentiles were spiritually
on an equality with the Jews. The Church was now about to burst
the bonds which had restrained it for five years at least. We stand by
the birth of European Christendom and of modern civilisation. It is
well, then, that we should learn and inwardly digest every, even the
slightest, detail concerning such a transcendent and notable crisis.
Let us take them briefly one by one as the sacred narrative reports
them.](https://image.slidesharecdn.com/810291-250522124357-52de8d91/85/Signal-Processing-And-Performance-Analysis-For-Imaging-Systems-S-Susan-Young-68-320.jpg)
![I. I note, then, in the first place that the time of this conversion was
wisely and providentially chosen. The time was just about eight
years after the Ascension and the foundation of the Church. Time
enough therefore had elapsed for Christianity to take root among the
Jews. This was most important. The gospel was first planted among
the Jews, took form and life and shape, gained its initial impulse and
direction among God's ancient people in order that the constitution,
the discipline, and the worship of the Church might be framed on
the ancient Jewish model and might be built up by men whose
minds were cast in a conservative mould. Not that we have the old
law with its wearisome and burdensome ritual perpetuated in the
Christian Church. That law was a yoke too heavy for man to bear.
But, then, the highest and best elements of the old Jewish system
have been perpetuated in the Church. There was in Judaism by
God's own appointment a public ministry, a threefold public ministry
too, exercised by the high priest, the priests, and the Levites. There
is in Christianity a threefold ministry exercised by bishops, presbyters
or elders, and deacons.[65]
There were in Judaism public and
consecrated sanctuaries, fixed liturgies, public reading of God's
Word, a service of choral worship, hymns of joy and thanksgiving,
the sacraments of Holy Communion and baptism in a rudimentary
shape; all these were transferred from the old system that was
passing away into the new system that was taking its place. Had the
Gentiles been admitted much earlier all this might not have so easily
happened. Men do not easily change their habits. Habits, indeed, are
chains which rivet themselves year by year with ever-increasing
power round our natures; and the Jewish converts brought their
habits of thought and worship into the Church of Christ, establishing
there those institutions of prayer and worship, of sacramental
communion and preaching which we still enjoy. But we must
observe, on the other hand, that, had the Gentiles been admitted a
little later, the Church might have assumed too Jewish and Levitical
an aspect. This pause of eight years, during which Jews alone
formed the Church, is another instance of those delays of the Lord[66]
which, whether they happen in public or in private life, are always](https://image.slidesharecdn.com/810291-250522124357-52de8d91/85/Signal-Processing-And-Performance-Analysis-For-Imaging-Systems-S-Susan-Young-69-320.jpg)
![found in the long run to be wise, blessed, and providential things,
though for a time they may seem dark and mysterious, according to
that ancient strain of the Psalmist, "Wait on the Lord, ... and He shall
strengthen thine heart: wait, I say, upon the Lord."[67]
II. Again, the place where the Church burst its Jewish shell and
emerged into full gospel freedom is noteworthy. It was at Cæsarea.
It is a great pity that people do not make more use of maps in their
study of Holy Scripture. Sunday evenings are often a dull time in
Christian households, and the bare mechanical reading of Scripture
and of good books often only makes them duller. How much livelier,
interesting, and instructive they would be were an attempt made to
trace the journeys of the apostles with a map, or to study the scenes
where they laboured—Jerusalem, Cæsarea, Damascus, Ephesus,
Athens, and Rome—with some of the helps which modern
scholarship and commercial enterprise now place within easy reach.
I can speak thus with the force of personal experience, for my own
keen interest in this book which I am expounding dates from the
Sunday evenings of boyhood thus spent, though without many of
the aids which now lie within the reach of all. This is essentially the
modern method of study, especially in matters historical. A modern
investigator and explorer of Bible sites and lands has well expressed
this truth when he said, "Topography is the foundation of history. If
we are ever to understand history, we must understand the places
where that history was transacted."[68]
The celebrated historians the
late Mr. Freeman and Mr. Green worked a revolution in English
historical methods by teaching people that an indefatigable use of
maps and a careful study of the physical features of any country are
absolutely needful for a true conception of its history. In this respect
at least secular history and sacred history are alike. Without a
careful study of the map we cannot understand God's dealings with
the Church of Christ, as is manifest from the case of Cæsarea at
which we have arrived. The narratives of the Gospels and of the Acts
will be confused, unintelligible, unless we understand that there
were two Cæsareas in Palestine, one never mentioned in the
Gospels, the other never mentioned in the Acts. Cæsarea Philippi](https://image.slidesharecdn.com/810291-250522124357-52de8d91/85/Signal-Processing-And-Performance-Analysis-For-Imaging-Systems-S-Susan-Young-70-320.jpg)
![was a celebrated city of North-eastern Palestine. It was when our
Lord was within its borders that St. Peter made his celebrated
confession, "Thou art the Christ, the Son of the living God," told of in
St. Matthew xvi. 13-16. This is the only Cæsarea of which we hear in
the Gospels. It was an inland town, built by the Herods in joint
honour of themselves and of their patrons the Emperors of Rome,
and bore all the traces of its origin. It was decorated with a splendid
pagan temple, was a thoroughly pagan town, and was therefore
abhorred by every true Jew. There was another Cæsarea, the great
Roman port of Palestine and the capital, where the Roman governors
resided. It was situated in the borders of Phœnicia, in a north-
westerly direction from Jerusalem, with which it was connected by a
fine military road.[69]
This Cæsarea had been originally built by Herod
the Great. He spent twelve years at this undertaking, and succeeded
in making it a splendid monument of the magnificence of his
conceptions. The seaboard of Palestine is totally devoid to this day
of safe harbours. Herod constructed a harbour at vast expense. Let
us hear the story of its foundation in the very words of the Jewish
historian. Josephus tells us that Herod, observing "that Joppa and
Dora are not fit for havens on account of the impetuous south winds
which beat upon them, which, rolling the sands which come from
the sea against the shores, do not admit of ships lying in their
station; but the merchants are generally there forced to ride at their
anchors in the sea itself. So Herod endeavoured to rectify this
inconvenience, and laid out such a compass toward the land as
might be sufficient for a haven, wherein the great ships might lie in
safety; and this he effected by letting down vast stones of above
fifty feet in length, not less than eighteen in breadth and nine in
depth, into twenty fathoms deep."[70]
The Romans, when they took
possession of Palestine, adopted and developed Herod's plans, and
established Cæsarea on the coast as the permanent residence of the
procurator of Palestine. And it was a wise policy. The Romans, like
the English, had a genius for government. They fixed their provincial
capitals upon or near the sea-coast that their communications might
be ever kept open. Thus in our own case Calcutta, Bombay, Madras,](https://image.slidesharecdn.com/810291-250522124357-52de8d91/85/Signal-Processing-And-Performance-Analysis-For-Imaging-Systems-S-Susan-Young-71-320.jpg)
![Capetown, Quebec, and Dublin are all seaport towns. And so in
ancient times Antioch, Alexandria, Tarsus, Ephesus, Marseilles,
Corinth, London, were all seaports and provincial Roman capitals as
Cæsarea was in Palestine. And it was a very wise policy. The Jews
were a fierce, bold, determined people when they revolted. If the
seat of Roman rule had been fixed at Jerusalem, a rebellion might
completely cut off all effective relief from the besieged garrison,
which would never happen at Cæsarea so long as the command of
the sea was vested in the vast navies which the Roman State
possessed. Cæsarea was to a large extent a Gentile city, though
within some seventy miles of Jerusalem. It had a considerable
Jewish population with their attendant synagogues, but the most
prominent features were pagan temples, one of them serving for a
lighthouse and beacon for the ships which crowded its harbour,
together with a theatre and an amphitheatre, where scenes were
daily enacted from which every sincere Jew must have shrunk with
horror. Such was the place—a most fitting place, Gentile, pagan,
idolatrous to the very core and centre—where God chose to reveal
Himself as Father of the Gentiles as well as of the Jews, and showed
Christ's gospel as a light to lighten the Gentiles as well as the glory
of His people Israel.
III. Then, again, the person chosen as the channel of this revelation
is a striking character. He was "Cornelius by name, a centurion of the
band called the Italian band."[71]
Here, then, we note first of all that
Cornelius was a Roman soldier. Let us pause and reflect upon this. In
no respect does the New Testament display more clearly its Divine
origin than in the manner in which it rises superior to mere
provincialism. There are no narrow national prejudices about it like
those which nowadays lead Englishmen to despise other nations, or
those which in ancient times led a thorough-going Jew to look down
with sovereign contempt on the Gentile world as mere dogs and
outcasts. The New Testament taught that all men were equal and
were brothers in blood, and thus laid the foundations of those
modern conceptions which have well-nigh swept slavery from the
face of civilised Christendom. The New Testament and its teaching is](https://image.slidesharecdn.com/810291-250522124357-52de8d91/85/Signal-Processing-And-Performance-Analysis-For-Imaging-Systems-S-Susan-Young-72-320.jpg)
![Another reason, however, may be found for this fact. The Scriptures
never make light of discipline or training. "Train up a child in the way
he should go" is a Divine precept. St. Paul, in his Pastoral Epistles,
lays down as one great qualification for a bishop that he should have
this power of exercising discipline and rule at home as well as
abroad: "For if he knoweth not how to rule his own house, how shall
he take care of the Church of God?" (1 Tim. iii. 5). By discipline, the
discipline of Egypt and the wilderness, did God prepare His people
for Canaan. By the discipline of captivity and dispersion, by the
discipline of Greek philosophy spreading novel intellectual ideas, by
the discipline of Roman dominion executing mighty public works,
carrying roads and intercommunication to the remotest and most
barbarous nations, did God prepare the world for the revelation of
His Son. By the discipline of life, by joy and sorrow, by strife and
suffering, by parting and by loss, does God still prepare His faithful
ones for the beatific vision of eternal beauty, for the rest and joy of
everlasting peace. And discipline worked out its usual results on
these military men, even though it was only an imperfect and pagan
discipline which these Roman soldiers received. Let us note carefully
how this was. The world of unregenerate man at the time of our
Lord's appearance had become utterly selfish. Discipline of every
kind had been flung off. Self-restraint was practically unknown, and
the devil and his works flourished in every circle, bringing forth the
fruits of wickedness, uncleanness, and impurity in every direction.
The army was the only place or region where in those times any
kind of discipline or self-restraint was practised. For no army can
permit—even if it be an army of atheists—profligacy and
drunkenness to rage, flaunting themselves beneath the very eye of
the sun. And as the spiritual result we find that this small measure of
pagan discipline acted as a preparation for Christianity, and became
under the Divine guidance the means of fitting men like Cornelius of
Cæsarea for the reception of the gospel message of purity and
peace.[72]
But we observe that Cornelius the centurion had one special feature
which made him peculiarly fitted to be God's instrument for opening](https://image.slidesharecdn.com/810291-250522124357-52de8d91/85/Signal-Processing-And-Performance-Analysis-For-Imaging-Systems-S-Susan-Young-75-320.jpg)
![Law was not imposed upon them because they were not
circumcised. They were merely bound to worship the true God and
observe certain moral precepts said to have been delivered to Noah.
[73]
Such was Cornelius whom the providence of God had led from
Italy to Cæsarea for this very purpose, to fulfil His purposes of
mercy towards the Gentile world. His residence there had taught him
the truth and beauty of the pure worship of Jehovah rendered by the
Jews. He had learned too, not only that God is, but that He is a
rewarder of them that diligently seek Him. Cornelius had set himself,
therefore, to the diligent discharge of all the duties of religion so far
as he knew them. He was earnest and diligent in prayer, for he
recognised himself as dependent upon an invisible God. He was
liberal in alms, for he desired to show forth his gratitude, for mercies
daily received. And acting thus he met with the divinely appointed
reward. Cornelius is favoured with a fuller revelation and a clearer
guidance by the angel's mouth, who tells him to send and summon
Peter from Joppa for this very purpose. What an eminently practical
lesson we may learn from God's dealings with this earliest Gentile
convert! We learn from the Divine dealings with Cornelius that
whosoever diligently improves the lower spiritual advantages which
he possesses shall soon be admitted to higher and fuller blessings.
It may well have been that God led him through successive stages
and rewarded him under each. In distant Italy, when residing amid
the abounding superstitions of that country, conscience was the only
preacher, but there the sermons of that monitor were heard with
reverence and obeyed with diligence. Then God ordered the course
of his life so that public duty summoned him to a distant land.
Cornelius may have at the time counted his lot a hard one when
despatched to Palestine as a centurion, for it was a province where,
from the nature of the warfare there prevalent, there were abundant
opportunities of death by assassination at the hands of the Zealots,
and but few opportunities of distinction such as might be gained in
border warfare with foreign enemies. But the Lord was shaping his
career, as He shapes all our careers, with reference to our highest
spiritual purposes. He led Cornelius, therefore, to a land and to a](https://image.slidesharecdn.com/810291-250522124357-52de8d91/85/Signal-Processing-And-Performance-Analysis-For-Imaging-Systems-S-Susan-Young-77-320.jpg)
![the perfect day," "In the way of righteousness is life, and in the
pathway thereof there is no death."
But the example of Cornelius is of still wider application. The position
of Cornelius was not a favourable one for the development of the
religious life, and yet he rose superior to all its difficulties, and
became thus an eminent example to all believers. Men may complain
that they have but few spiritual advantages, and that their station in
life is thickly strewn with difficulties, hindering the practices and
duties of religion. To such persons we would say, compare
yourselves with Cornelius and the difficulties external and internal he
had to overcome. Servants, for instance, may labour under great
apparent disadvantages. Perhaps, if living in an irreligious family,
they have few opportunities for prayer, public or private. Men of
business are compelled to spend days and nights in the
management of their affairs. Persons of commanding intellect or of
high station have their own disadvantages, their own peculiar
temptations, growing out of their very prosperity. The case of
Cornelius shows that each class can rise superior to their peculiar
difficulties and grow in the hidden life of the soul, if they but imitate
his example as he grew from grace to grace, improving his scanty
store till it grew into a fuller and ampler one, till it expanded into all
the glory of Christian privilege, when Cornelius, like Peter, was
enabled to rejoice in the knowledge and love of a risen and glorified
Redeemer.[74]](https://image.slidesharecdn.com/810291-250522124357-52de8d91/85/Signal-Processing-And-Performance-Analysis-For-Imaging-Systems-S-Susan-Young-79-320.jpg)
![I. Now, as in the case of Cornelius, so in the case of St. Peter, we
note the place where the chief actor in the scene abode. It was at
Joppa, and Joppa was associated with many memories for the Jews.
It has been from ancient times the port of Jerusalem, and is even
now rising into somewhat of its former commercial greatness,
specially owing to the late development of the orange trade, for the
production of which fruit Jaffa or Joppa has become famous. Three
thousand years ago Joppa was a favourite resort of the Phœnician
fleets, which brought the cedars of Lebanon to King Solomon for the
building of the temple (2 Chron. ii. 16). At a later period, when God
would send Jonah on a mission to Gentile Nineveh, and when Jonah
desired to thwart God's merciful designs towards the outer world,
the prophet fled to Joppa and there took ship in his vain effort to
escape from the presence of the Lord. And now again Joppa
becomes the refuge of another prophet, who feels the same natural
hesitation about admitting the Gentiles to God's mercy, but who,
unlike Jonah, yields immediate assent to the heavenly message, and
finds peace and blessing in the paths of loving obedience. The very
house where St. Peter abode is still pointed out.[75]
It is situated in
the south-western part of the town, and commands a view over the
bay of Joppa and the waters of that Mediterranean Sea which was
soon to be the channel of communication whereby the gospel
message should be borne to the nations of the distant West. We
remark, too, that it was with Simon the tanner of Joppa that St.
Peter was staying. When a great change is impending various little
circumstances occur all showing the tendencies of the age. By
themselves and taken one by one they do not express much. At the
time when they happen men do not regard them or understand their
meaning, but afterwards, and reading them in the light of
accomplished facts, men behold their significance. Thus it was with
Simon Peter and his visit to Simon the tanner of Joppa. Tanners as a
class were despised and comparatively outcast among the Jews.
Tanning was counted an unclean trade because of the necessary
contact with dead bodies which it involved. A tanyard must,
according to Jewish law, be separated by fifty yards at least from](https://image.slidesharecdn.com/810291-250522124357-52de8d91/85/Signal-Processing-And-Performance-Analysis-For-Imaging-Systems-S-Susan-Young-83-320.jpg)
Signal Processing And Performance Analysis For Imaging Systems S Susan Young
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- 6. Signal Processing and Performance Analysis for Imaging Systems
- 7. For a listing of recent titles in the Artech House Optoelectronics Series, turn to the back of this book.
- 8. Signal Processing and Performance Analysis for Imaging Systems S. Susan Young Ronald G. Driggers Eddie L. Jacobs artechhouse.com
- 9. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the U.S. Library of Congress. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. ISBN-13: 978-1-59693-287-6 Cover design by Igor Valdman © 2008 ARTECH HOUSE, INC. 685 Canton Street Norwood, MA 02062 All rights reserved. Printed and bound in the United States of America. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, includ- ing photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher. All terms mentioned in this book that are known to be trademarks or service marks have been appropriately capitalized. Artech House cannot attest to the accuracy of this informa- tion. Use of a term in this book should not be regarded as affecting the validity of any trade- mark or service mark. 10 9 8 7 6 5 4 3 2 1
- 10. To our families
- 12. Contents Preface xiii PART I Basic Principles of Imaging Systems and Performance 1 CHAPTER 1 Introduction 3 1.1 “Combined” Imaging System Performance 3 1.2 Imaging Performance 3 1.3 Signal Processing: Basic Principles and Advanced Applications 4 1.4 Image Resampling 4 1.5 Super-Resolution Image Reconstruction 5 1.6 Image Restoration—Deblurring 6 1.7 Image Contrast Enhancement 7 1.8 Nonuniformity Correction (NUC) 7 1.9 Tone Scale 8 1.10 Image Fusion 8 References 10 CHAPTER 2 Imaging Systems 11 2.1 Basic Imaging Systems 11 2.2 Resolution and Sensitivity 15 2.3 Linear Shift-Invariant (LSI) Imaging Systems 16 2.4 Imaging System Point Spread Function and Modulation Transfer Function 20 2.4.1 Optical Filtering 21 2.4.2 Detector Spatial Filters 22 2.4.3 Electronics Filtering 24 2.4.4 Display Filtering 25 2.4.5 Human Eye 26 2.4.6 Overall Image Transfer 27 2.5 Sampled Imaging Systems 28 2.6 Signal-to-Noise Ratio 34 2.7 Electro-Optical and Infrared Imaging Systems 38 2.8 Summary 39 References 39 vii
- 13. CHAPTER 3 Target Acquisition and Image Quality 41 3.1 Introduction 41 3.2 A Brief History of Target Acquisition Theory 41 3.3 Threshold Vision 43 3.3.1 Threshold Vision of the Unaided Eye 43 3.3.2 Threshold Vision of the Aided Eye 47 3.4 Image Quality Metric 50 3.5 Example 53 3.6 Summary 61 References 61 PART II Basic Principles of Signal Processing 63 CHAPTER 4 Basic Principles of Signal and Image Processing 65 4.1 Introduction 65 4.2 The Fourier Transform 65 4.2.1 One-Dimensional Fourier Transform 65 4.2.2 Two-Dimensional Fourier Transform 78 4.3 Finite Impulse Response Filters 83 4.3.1 Definition of Nonrecursive and Recursive Filters 83 4.3.2 Implementation of FIR Filters 84 4.3.3 Shortcomings of FIR Filters 85 4.4 Fourier-Based Filters 86 4.4.1 Radially Symmetric Filter with a Gaussian Window 87 4.4.2 Radially Symmetric Filter with a Hamming Window at a Transition Point 87 4.4.3 Radially Symmetric Filter with a Butterworth Window at a Transition Point 88 4.4.4 Radially Symmetric Filter with a Power Window 89 4.4.5 Performance Comparison of Fourier-Based Filters 90 4.5 The Wavelet Transform 90 4.5.1 Time-Frequency Wavelet Analysis 91 4.5.2 Dyadic and Discrete Wavelet Transform 96 4.5.3 Condition of Constructing a Wavelet Transform 97 4.5.4 Forward and Inverse Wavelet Transform 97 4.5.5 Two-Dimensional Wavelet Transform 98 4.5.6 Multiscale Edge Detection 98 4.6 Summary 102 References 102 PART III Advanced Applications 105 viii Contents
- 14. CHAPTER 5 Image Resampling 107 5.1 Introduction 107 5.2 Image Display, Reconstruction, and Resampling 107 5.3 Sampling Theory and Sampling Artifacts 109 5.3.1 Sampling Theory 109 5.3.2 Sampling Artifacts 110 5.4 Image Resampling Using Spatial Domain Methods 111 5.4.1 Image Resampling Model 111 5.4.2 Image Rescale Implementation 112 5.4.3 Resampling Filters 112 5.5 Antialias Image Resampling Using Fourier-Based Methods 114 5.5.1 Image Resampling Model 114 5.5.2 Image Rescale Implementation 115 5.5.3 Resampling System Design 117 5.5.4 Resampling Filters 118 5.5.5 Resampling Filters Performance Analysis 119 5.6 Image Resampling Performance Measurements 125 5.7 Summary 127 References 127 CHAPTER 6 Super-Resolution 129 6.1 Introduction 129 6.1.1 The Meaning of Super-Resolution 129 6.1.2 Super-Resolution for Diffraction and Sampling 129 6.1.3 Proposed Nomenclature by IEEE 130 6.2 Super-Resolution Image Restoration 130 6.3 Super-Resolution Image Reconstruction 131 6.3.1 Background 131 6.3.2 Overview of the Super-Resolution Reconstruction Algorithm 132 6.3.3 Image Acquisition—Microdither Scanner Versus Natural Jitter 132 6.3.4 Subpixel Shift Estimation 133 6.3.5 Motion Estimation 135 6.3.6 High-Resolution Output Image Reconstruction 143 6.4 Super-Resolution Imager Performance Measurements 158 6.4.1 Background 158 6.4.2 Experimental Approach 159 6.4.3 Measurement Results 166 6.5 Sensors That Benefit from Super-Resolution Reconstruction 167 6.5.1 Example and Performance Estimates 168 6.6 Performance Modeling and Prediction of Super-Resolution Reconstruction 172 6.7 Summary 173 References 174 Contents ix
- 15. CHAPTER 7 Image Deblurring 179 7.1 Introduction 179 7.2 Regularization Methods 181 7.3 Wiener Filter 181 7.4 Van Cittert Filter 182 7.5 CLEAN Algorithm 183 7.6 P-Deblurring Filter 184 7.6.1 Definition of the P-Deblurring Filter 185 7.6.2 Properties of the P-Deblurring Filter 186 7.6.3 P-Deblurring Filter Design 188 7.7 Image Deblurring Performance Measurements 199 7.7.1 Experimental Approach 200 7.7.2 Perception Experiment Result Analysis 203 7.8 Summary 204 References 204 CHAPTER 8 Image Contrast Enhancement 207 8.1 Introduction 207 8.2 Single-Scale Process 208 8.2.1 Contrast Stretching 208 8.2.2 Histogram Modification 209 8.2.3 Region-Growing Method 209 8.3 Multiscale Process 209 8.3.1 Multiresolution Analysis 210 8.3.2 Contrast Enhancement Based on Unsharp Masking 210 8.3.3 Contrast Enhancement Based on Wavelet Edges 211 8.4 Contrast Enhancement Image Performance Measurements 217 8.4.1 Background 217 8.4.2 Time Limited Search Model 218 8.4.3 Experimental Approach 219 8.4.4 Results 222 8.4.5 Analysis 223 8.4.6 Discussion 226 8.5 Summary 227 References 228 CHAPTER 9 Nonuniformity Correction 231 9.1 Detector Nonuniformity 231 9.2 Linear Correction and the Effects of Nonlinearity 232 9.2.1 Linear Correction Model 233 9.2.2 Effects of Nonlinearity 233 9.3 Adaptive NUC 238 9.3.1 Temporal Processing 238 9.3.2 Spatio-Temporal Processing 240 x Contents
- 16. 9.4 Imaging System Performance with Fixed-Pattern Noise 243 9.5 Summary 244 References 245 CHAPTER 10 Tone Scale 247 10.1 Introduction 247 10.2 Piece-Wise Linear Tone Scale 248 10.3 Nonlinear Tone Scale 250 10.3.1 Gamma Correction 250 10.3.2 Look-Up Tables 252 10.4 Perceptual Linearization Tone Scale 252 10.5 Application of Tone Scale to Enhanced Visualization in Radiation Treatment 255 10.5.1 Portal Image in Radiation Treatment 255 10.5.2 Locating and Labeling the Radiation and Collimation Fields 257 10.5.3 Design of the Tone Scale Curves 257 10.5.4 Contrast Enhancement 262 10.5.5 Producing the Output Image 264 10.6 Tone Scale Performance Example 264 10.7 Summary 266 References 267 CHAPTER 11 Image Fusion 269 11.1 Introduction 269 11.2 Objectives for Image Fusion 270 11.3 Image Fusion Algorithms 271 11.3.1 Superposition 272 11.3.2 Laplacian Pyramid 272 11.3.3 Ratio of a Lowpass Pyramid 275 11.3.4 Perceptual-Based Multiscale Decomposition 276 11.3.5 Discrete Wavelet Transform 278 11.4 Benefits of Multiple Image Modes 280 11.5 Image Fusion Quality Metrics 281 11.5.1 Mean Squared Error 282 11.5.2 Peak Signal-to-Noise Ratio 283 11.5.3 Mutual Information 283 11.5.4 Image Quality Index by Wang and Bovik 283 11.5.5 Image Fusion Quality Index by Piella and Heijmans 284 11.5.6 Xydeas and Petrovic Metric 285 11.6 Imaging System Performance with Image Fusion 286 11.7 Summary 290 References 290 About the Authors 293 Index 295 Contents xi
- 18. Preface In today’s consumer electronics market where a 5-megapixel camera is no longer considered state-of-the-art, signal and image processing algorithms are real-time and widely used. They stabilize images, provide super-resolution, adjust for detec- tor nonuniformities, reduce noise and blur, and generally improve camera perfor- mance for those of us who are not professional photographers. Most of these signal and image processing techniques are company proprietary and the details of these techniques are never revealed to outside scientists and engineers. In addition, it is not necessary for the performance of these systems (including the algorithms) to be determined since the metric of success is whether the consumer likes the product and buys the device. In other imaging communities such as military imaging systems (which, at a minimum, include visible, image intensifiers, and infrared) and medical imaging devices, it is extremely important to determine the performance of the imaging sys- tem, including the signal and image processing techniques. In military imaging sys- tems that involve target acquisition and surveillance/reconnaissance, the performance of an imaging system determines how effective the warfighter can accomplish his or her mission. In medical systems, the imaging system performance determines how accurately a diagnosis can be provided. Signal and image process- ing plays a key role in the performance of these imaging systems and, in the past 5 to 10 years, has become a key contributor to increased imaging system performance. There is a great deal of government funding in signal and image processing for imaging system performance and the literature is full of university and government laboratory developed algorithms. There are still a great number of industry algo- rithms that, overall, are considered company proprietary. We focus on those in the literature and those algorithms that can be generalized in a nonproprietary manner. There are numerous books in the literature on signal and image processing tech- niques, algorithms, and methods. The majority of these books emphasize the math- ematics of image processing and how they are applied to image information. Very few of the books address the overall imaging system performance when signal and image processing is considered a component of the imaging system. Likewise, there are many books in the area of imaging system performance that consider the optics, the detector, and the displays in the system and how the system performance behaves with changes or modifications of these components. There is very little book content where signal and imager processing is included as a component of the overall imaging system performance. This is the gap that we have attempted to fill with this book. While algorithm development has exploded in the past 5 to 10 years, xiii
- 19. the system performance aspects are relatively new and not quite fully understood. While the focus of this book is to help the scientist and engineer begin to understand that these algorithms are really an imaging system component and help in the system performance prediction of imaging systems with these algorithms, the performance material is new and will undergo dramatic improvements in the next 5 years. We have chosen to address signal and image processing techniques that are not new, but the real time implementation in military and medical systems are relatively new and the performance predication of systems with these algorithms are definitely new. There are some algorithms that are not addressed such as electronic stabiliza- tion and turbulence correction. There are current programs in algorithm develop- ment that will provide great advances in algorithm performance in the next few years, so we decided not to spend time on these particular areas. It is worth mentioning that there is a community called “computational imag- ing” where, instead of using signal/image processing to improve the performance of an existing imaging system approach, signal processing is an inherent part of the electro-optical design process for image formation. The field includes unconven- tional imaging systems and unconventional processing, where the performance of the collective system design is beyond any conventional system approach. In many cases, the resulting image is not important. The goal of the field is to maximize sys- tem task performance for a given electro-optical application using nonconventional design rules (with signal processing and electro-optical components) through the exploitation of various degrees of freedom (space, time, spectrum, polarization, dynamic range, and so forth). Leaders in this field include Dennis Healey at DARPA, Ravi Athale at MITRE, Joe Mait at the Army Research Laboratory, Mark Mirotznick at Catholic University, and Dave Brady at Duke University. These researchers and others are forging a new path for the rest of us and have provided some very stimulating experiments and demonstrations in the past 2 or 3 years. We do not address computational imaging in this book, as the design and approach methods are still a matter of research and, as always, it will be some time before sys- tem performance is addressed in a quantitative manner. We would like to thank a number of people for their thoughtful assistance in this work. Dr. Patti Gillespie at the Army Research Laboratory provided inspiration and encouragement for the project. Rich Vollmerhausen has contributed more to mili- tary imaging system performance modeling over the past 10 years than any other researcher, and his help was critical to the success of the project. Keith Krapels and Jonathan Fanning both assisted with the super-resolution work. Khoa Dang, Mike Prarie, Richard Moore, Chris Howell, Stephen Burks, and Carl Halford contributed material for the fusion chapter. There are many others who worked signal process- ing issues and with whom we collaborated through research papers to include: Nicole Devitt, Tana Maurer, Richard Espinola, Patrick O’Shea, Brian Teaney, Louis Larsen, Jim Waterman, Leslie Smith, Jerry Holst, Gene Tener, Jennifer Parks, Dean Scribner, Jonathan Schuler, Penny Warren, Alan Silver, Jim Howe, Jim Hilger, and Phil Perconti. We are grateful for the contributions that all of these people have pro- vided over the years. We (S. Susan Young and Eddie Jacobs) would like to thank our coauthor, Dr. Ronald G. Driggers for his suggestion of writing this book and encouragement in this venture. Our understanding and appreciation of system performance signifi- cance started from collaborating with him. S. Susan Young would like to thank Dr. xiv Preface
- 20. Hsien-Che Lee for his guidance and help early in her career in signal and image pro- cessing. On a personal side, we authors are very thankful to our families for their support and understanding. xv
- 22. P A R T I Basic Principles of Imaging Systems and Performance
- 24. C H A P T E R 1 Introduction 1.1 “Combined” Imaging System Performance The “combined” imaging system performance of both hardware (sensor) and soft- ware (signal processing) is extremely important. Imaging system hardware is designed primarily to form a high-quality image from source emissions under a large variety of environmental conditions. Signal processing is used to help highlight or extract information from the images that are generated from an imaging system. This processing can be automated for decision-making purposes or it can be utilized to enhance the visual acuity of a human looking through the imaging system. Performance measures of an imaging system have been excellent methods for better design and understanding of the imaging system. However, the imaging per- formance of an imaging system with the aid of signal processing has not been widely considered in the light of improving image quality from imaging systems and signal processing algorithms. Imaging systems can generate images with low-contrast, high-noise, blurring, or corrupted/lost high-frequency details, among others. How does the image performance of a low-cost imaging system with the aid of signal pro- cessing compare with the one of an expensive imaging system? Is it worth investing in higher image quality by improving the imaging system hardware or by develop- ing the signal processing software? The topic of this book is to relate the ability of extracting information from an imaging system with the aid of signal processing to evaluate the overall performance of imaging systems. 1.2 Imaging Performance Understanding the image formation and recording process helps in understanding the factors that affect image performance and therefore helps the design of imaging systems and signal processing algorithms. The image formation process and the sources of image degradation, such as loss of useful high-frequency details, noise, or low-contrast target environment, are discussed in Chapter 2. Methods of determining image performance are important tools in determining the merits of imaging systems and signal processing algorithms. Image performance determination can be performed via subjective human perception studies or image performance modeling. Image performance prediction and the role of image perfor- mance modeling are also discussed in Chapter 3. 3
- 25. 1.3 Signal Processing: Basic Principles and Advanced Applications The basic signal processing principles, including Fourier transform, wavelet trans- form, finite impulse response (FIR) filters, and Fourier-based filters, are discussed in Chapter 4. In an image formation and recording process, many factors affect sensor perfor- mance and image quality, and these can result in loss of high-frequency information or low contrast in an image. Several common causes of low image quality are the following: • Many low-cost visible and thermal sensors spatially or electronically undersample an image. Undersampling results in aliased imagery in which subtle/detailed information (high-frequency components) is lost in these images. • An imaging system’s blurring function (sometimes called the point spread function, or PSF) is another common factor in the reduction of high-frequency components in the acquired imagery and results in blurred images. • Low-cost sensors and environmental factors, such as lighting sources or back- ground complexities, result in low-contrast images. • Focal plan array (FPA) sensors have detector-to-detector variability in the FPA fabrication process and cause the fixed-pattern noise in the acquired imagery. There are many signal processing applications for the enhancement of imaging system performance. Most of them attempt to enhance the image quality or remove the degradation phenomena. Specifically, these applications try to recover the useful high-frequency components that are lost or corrupted in the image and attempt to suppress the undesired high-frequency components, which are noises. In Chapters 5 to 11, the following classes of signal processing applications are considered: 1. Image resampling; 2. Super-resolution image reconstruction; 3. Image restoration—deblurring; 4. Image contrast enhancement; 5. Nonuniformity correction (NUC); 6. Tone scale; 7. Image fusion. 1.4 Image Resampling The concept of image resampling originates from the sampled imager. The discus- sion in this chapter relates image resampling with image display and reconstruction from sampled points of one single image. These topics provide the reader with a fun- damental understanding that the way an image is processed and displayed is just as important as the blur and sampling characteristics of the sensor. It also provides a background for undersampled imaging for discussion on super-resolution image reconstruction in the following chapter. In signal processing, image resampling is 4 Introduction
- 26. also called image decimation, or image interpolation, according to whether the goal is to reduce or enlarge the size (or resolution) of a captured image. It can provide the image values that are not recorded by the imaging system, but are calculated from the neighboring pixels. Image resampling does not increase the inherent informa- tion content in the image, but poor image display reconstruction function can reduce the overall imaging system performance. The image resampling algorithms include spatial and spatial-frequency domain, or Fourier-based windowing, methods. The important considerations in image resampling include the image resampling model, image rescale implementation, and resampling filters, especially the anti-aliasing image resampling filter. These algo- rithms, examples, and image resampling performance measurements are discussed in Chapter 5. 1.5 Super-Resolution Image Reconstruction The loss of high-frequency information in an image could be due to many factors. Many low-cost visible and thermal sensors spatially or electronically undersample an image. Undersampling results in aliased imagery in which the high-frequency components are folded into the low-frequency components in the image. Conse- quently, subtle/detailed information (high-frequency components) is lost in these images. Super-resolution image reconstruction can produce high-resolution images by using the existing low-cost imaging devices from a sequence, or a few snapshots, of low-resolution images. Since undersampled images have subpixel shifts between successive frames, they represent different information from the same scene. Therefore, the informa- tion that is contained in an undersampled image sequence can be combined to obtain an alias-free (high-resolution) image. Super-resolution image reconstruction from multiple snapshots provides far more detail information than any interpolated image from a single snapshot. Figure 1.1 shows an example of a high-resolution (alias-free) infrared image that is obtained from a sequence of low-resolution (aliased) input images having subpixel shifts among them. 1.5 Super-Resolution Image Reconstruction 5 (b) (a) Figure 1.1 Example of super-resolution image reconstruction: (a) input sequence of aliased infra- red images having subpixel shifts among them; and (b) output alias-free (high-resolution) image in which the details of tree branches are revealed.
- 27. The first step in a super-resolution image reconstruction algorithm is to estimate the supixel shifts of each frame with respect to a reference frame. The second step is to increase the effective spatial sampling by operating on a sequence of low-resolu- tion subpixel-shifted images. There are also spatial and spatial frequency domain methods for the subpixel shift estimation and the generation of the high-resolution output images. These algorithms, examples, and the image performance are discussed in Chapter 6. 1.6 Image Restoration—Deblurring An imaging system’s blurring function, also called the point spread function (PSF), is another common factor in the reduction of high-frequency components in the image. Image restoration tries to inverse this blurring degradation phenomenon, but within the bandlimit of the imager (i.e., it enhances the spatial frequencies within the imager band). This includes deblurring images that are degraded by the limitations of a sensor or environment. The estimate or knowledge of the blurring function is essential to the application of these algorithms. One of the most important consider- ations of designing a deblurring filter is to control noise, since the noise is likely amplified at high frequencies. The amplification of noise results in undesired arti- facts in the output image. Figure 1.2 shows examples of image deblurring. One input image [Figure 1.2(a)] contains the blur, while the deblurred version of it [Figure 1.2(b)] removes the most blur. Another input image [Figure 1.2(c)] contains the blur and noise; the noise effect illustrates on the deblurred version of it [Figure 1.2(d)]. Image restoration tries to recover the high-frequency information below the diffrac- tion limit while limiting the noise artifacts. The designs of deblurring filters, the 6 Introduction (a) (b) (c) (d) Figure 1.2 Examples of image deblurring: (a) blurred bar image; (b) deblurred version of (a); (c) blurred bar image with noise added; and (d) deblurred version of (c).
- 28. noise control mechanisms, examples, and image performance are discussed in Chapter 7. 1.7 Image Contrast Enhancement Image details can also be enhanced by image contrast enhancement techniques in which certain image edges are emphasized as desired. For an example of a medical application in diagnosing breast cancer from mammograms, radiologists follow the ductal networks to look for abnormalities. However, the number of ducts and the shape of ductal branches vary with individuals, which make the visual process of locating the ducts difficult. The image contrast enhancement provides the ability to enhance the appearance of the ductal elements relative to the fatty-tissue surround- ings, which helps radiologists to visualize abnormalities in mammograms. Image contrast enhancement methods can be divided into single-scale approach and multiscale approach. In the single-scale approach, the image is processed in the original image domain, such as a simple look-up table. In the multiscale approach, the image is decomposed into multiple resolution scales, and processing is per- formed in the multiscale domain. Because the information at each scale is adjusted before the image is reconstructed back to the original image intensity domain, the output image contains the desired detail information. The multiscale approach can also be coupled with the dynamic range reduction. Therefore, the detail information in different scales can be displayed in one output image. Localized contrast enhancement (LCE) is the process in which these techniques are applied on a local scale for the management of dynamic range in the image. For example, the sky-to-ground interface in infrared imaging can include a huge apparent tempera- ture difference that occupies most of the image dynamic range. Small targets with smaller signals can be lost, while LCE can reduce the large sky-to-ground interface signal and enhance small target signals (see Figure 8.10 later in this book). Details of the algorithms, examples, and image performance are discussed in Chapter 8. 1.8 Nonuniformity Correction (NUC) Focal plan array (FPA) sensors have been used in many commercial and military applications, including both visible and infrared imaging systems, since they have wide spectral responses, compact structures, and cost-effective production. How- ever, each individual photodetector in the FPA has a different photoresponse, due to detector-to-detector variability in the FPA fabrication process [1]. Images that are acquired by an FPA sensor suffer from a common problem known as fixed-pattern noise, or spatial nonuniformity. The technique to compensate for this distortion is called nonuniformity correction (NUC). Figure 1.3 shows an example of a nonuniformity corrected image from an original input image with the fixed-pattern noise. There are two main categories of NUC algorithms, namely, calibration-based and scene-adaptive algorithms. A conventional, calibration-based NUC is the stan- dard two-point calibration, which is also called linear NUC. This algorithm esti- 1.7 Image Contrast Enhancement 7
- 29. mates the gain and offset parameters by exposing the FPA to two distinct and uniform irradiance levels. The scene-adaptive NUC uses the data acquired in the video sequence and a motion estimation algorithm to register each point in the scene across all of the image frames. This way, continuous compensation can be applied adaptively for individual detector responses and background changes. These algo- rithms, examples, and imaging system performance are discussed in Chapter 9. 1.9 Tone Scale Tone scale is a technique that improves the image presentation on an output display medium (softcopy display or hardcopy print). Tone scale is also a mathematical mapping of the image pixel values from the sensor to a region of interest on an out- put medium. Note that tone scale transforms improve only the appearance of the image, but not the image quality itself. The gray value resolution is still the same. However, a proper tone scale allows the characteristic curve of a display system to match the sensitivity of the human eye to enhance the image interpretation task per- formance. There are various tone scale techniques, including piece-wise linear tone scale, nonlinear tone scale, and perceptual linearization tone scale. These techniques and a tone scale performance example are discussed in Chapter 10. 1.10 Image Fusion Because researchers realize that different sensors provide different signature cues of the scene, image fusion has been receiving additional attention in signal processing. Some of those applications are shown to benefit from fusing the images of multiple sensors. Imaging sensor characteristics are determined by the wavebands that they respond to in the electromagnetic spectrum. Figure 1.4 is a diagram of the electro- magnetic spectrum with wavelength indicated in metric length units [2]. The most familiar classifications of wavebands are the radiowave, microwave, infrared, visi- ble, ultraviolet, X-ray, and gamma-ray wavebands. Figure 1.5 shows further subdi- vided wavebands for broadband sensors [3]. For example, the infrared waveband is 8 Introduction (a) (b) Figure 1.3 Example of nonuniformity correction: (a) input image with the fixed-pattern noise shown in the image; and (b) nonuniformity corrected image in which the helicopter in the center is clearly illustrated.
- 30. divided into near infrared (NIR), shortwave infrared (SWIR), midwave infrared (MWIR), longwave infrared (LWIR), and far infrared. The sensor types are driven by the type of image information that can be exploited within these bands. X-ray sensors can view human bones for disease diagnosis. Microwave and radiowave sensors have a good weather penetration in military applications. Infrared sensors detect both temperature and emissivity and are beneficial for night-vision applica- tions. Different subwaveband sensors in infrared wavebands can provide different information. For example, MWIR sensors respond better to hotter-than-terrestrial objects. LWIR sensors have better response to overall terrestrial object tempera- tures, which are around 300 Kelvins (K). Solar clutter is high in the MWIR in the daytime and is negligible in the LWIR. Figure 1.6 shows an example of fusing MWIR and LWIR images. The road cracks are visible in LWIR, but not in MWIR. Similarly, the Sun glint is visible in MWIR image, but not in LWIR. The fused image shows both Sun glint and road cracks. 1.10 Image Fusion 9 Microwave (and subbands) Radiowave Infrared Ultraviolet 400 nm Visible 750 nm Gamma rays X-rays −2 1 m 1 mm 1 cm 1 m µ 1 nm λ wavelength (meters)−13 −9 −6 −3 10 10 10 10 10 10 Figure 1.4 Electromagnetic spectrum. λ wavelength m (micrometers, ) µ Longwave infrared Midwave infrared Near - - and short wave infrared Visible 0.4 1.0 10 14 3.0 Figure 1.5 Subdivided infrared wavebands.
- 31. Many questions of image fusion remain unanswered and open to new research opportunities. Some of the questions involve how to select different sensors to pro- vide better image information from the scene; whether different imaging informa- tion can be effectively combined to provide a better cue in the scene; and how to best combine the information. These issues are presented and examples and imaging sys- tem performance are provided in Chapter 11. References [1] Milton, A. F., F. B. Barone, and M. R. Kruer, “Influence of Nonuniformity on Infrared Focal Plan Array Performance,” Optical Engineering, Vol. 24, No. 5, 1985, pp. 855–862. [2] Richards, A., Alien Vision—Exploring the Electromagnetic Spectrum with Imaging Tech- nology, Bellingham, WA: SPIE Press, 2001. [3] Driggers, R. G., P. Cox, and T. Edwards, Introduction to Infrared and Electro-Optical Sys- tems, Norwood, MA: Artech House, 1999. 10 Introduction Road cracks MW LW Road cracks clearly visible in LW but not MW Sun glint Fused image Figure 1.6 Example of fusing MWIR and LWIR images. The road cracks are visible in LWIR, but not in MWIR. Similarly, the Sun glint is visible in MWIR image, but not in LWIR. The fused image shows both Sun glint and road cracks.
- 32. C H A P T E R 2 Imaging Systems In this chapter, basic imaging systems are introduced and the concepts of resolution and sensitivity are explored. This introduction presents helpful background infor- mation that is necessary to understand imaging system performance, which is pre- sented in Chapter 3. It also provides a basis for later discussions on the implementation of advanced signal and image processing techniques. 2.1 Basic Imaging Systems A basic imaging system can be depicted as a cascaded system where the input signal is optical flux from a target and background and the output is an image presented for human consumption. A basic imaging system is shown in Figure 2.1. The system can begin with the flux leaving the target and the background. For electro-optical systems and more sophisticated treatments of infrared systems, the system can even begin with the illumination of the target with external sources. Regardless, the flux leaving the source traverses the atmosphere as shown. This path includes blur from turbulence and scattering and a reduction in the flux due to atmospheric extinction, such as scattering, and absorption, among others. The flux that makes it to the entrance of the optics is then blurred by optical diffraction and aberrations. The flux is also reduced by the optical transmission. The flux is imaged onto a detector array, either scanning or staring. Here, the flux is converted from photons to electrons. There is a quantum efficiency that reduces the signal, and the finite size of the detector imposes a blur on the image. The electronics further reduce, or in some cases enhance, the signal. The display also provides a signal reduction and a blur, due to the finite size of the display element. Finally, the eye consumes the image. The eye has its own inherent blur and noise, which are consid- ered in overall system performance. In some cases, the output of the electronics is processed by an automatic target recognizer (ATR), which is an automated process of detecting and recognizing targets. An even more common process is an aided tar- get recognizer (AiTR), which is more of a cueing process for a human to view the resultant cued image “chips” (a small area containing an object). All source and background objects above 0K emit electromagnetic radiation associated with the thermal activity on the surface of the object. For terrestrial tem- peratures (around 300K), objects emit a good portion of the electromagnetic flux in the infrared part of the electromagnetic spectrum. This emission of flux is some- times called blackbody thermal emission. The human eye views energy only in the visible portion of the electromagnetic spectrum, where the visible band spans wave- lengths from 0.4 to 0.7 micrometer (µm). Infrared imaging devices convert energy in 11
- 33. the infrared portion of the electromagnetic spectrum into displayable images in the visible band for human use. The infrared spectrum begins at the red end of the visible spectrum where the eye can no longer sense energy. It spans from 0.7 to 100 µm. The infrared spectrum is, by common convention, broken into five different bands (this may vary according to the application/community). The bands are typically defined in the following way: near-infrared (NIR) from 0.7 to 1.0 µm, shortwave infrared (SWIR) from 1.0 to 3.0 µm, midwave infrared (MWIR) from 3.0 to 5.0 µm, longwave infrared (LWIR) from 8.0 to 14.0 µm, and far infrared (FIR) from 14.0 to 100 µm. These bands are depicted graphically in Figure 2.2. Figure 2.2 shows the atmospheric transmission for a 1-kilometer horizontal ground path for a “standard” day in the United States. These types of transmission graphs can be tailored for any condition using sophisti- cated atmospheric models, such as MODTRAN (from http://www.ontar.com). Note that there are many atmospheric “windows” so that an imager designed with such a band selection can see through the atmosphere. 12 Imaging Systems Target and background Atmosphere Scanner Detector array and cooler Display Human vision Optics Electronics ATR Figure 2.1 Basic imaging system. Visible Ultra- violet Near- and short- wave infrared Midwave infrared Longwave infrared Far infrared Wavelength (micrometers) 0.4 Transmission 1 10 14 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 2.2 Atmospheric transmission for a 1-kilometer path on a standard U.S. atmosphere day.
- 34. The primary difference between a visible spectrum camera and an infrared imager is the physical phenomenology of the radiation from the scene being imaged. The energy used by a visible camera is predominantly reflected solar or some other illuminating energy in the visible spectrum. The energy imaged by infrared imagers, commonly known as forward looking infrareds (FLIRs) in the MWIR and LWIR bands, is primarily self-emitted radiation. From Figure 2.2, the MWIR band has an atmospheric window in the 3- to 5-µm region, and the LWIR band has an atmo- spheric window in the 8- to 12-µm region. The atmosphere is opaque in the 5- to 8-µm region, so it would be pointless to construct a camera that responds to this waveband. Figure 2.3 provides images to show the difference in the source of the radiation sensed by the two types of cameras. The visible image on the left side is all light that was provided by the Sun, propagated through Earth’s atmosphere, reflected off the objects in the scene, traversed through a second atmospheric path to the sensor, and then imaged with a lens and a visible band detector. A key here is that the objects in the scene are represented by their reflectivity characteristics. The image characteris- tics can also change by any change in atmospheric path or source characteristic change. The atmospheric path characteristics from the sun to the objects change fre- quently because the Sun’s angle changes throughout the day, plus the weather and cloud conditions change. The visible imager characterization model is a multipath problem that is extremely difficult. The LWIR image given on the right side of Figure 2.3 is obtained primarily by the emission of radiation by objects in the scene. The amount of electromagnetic flux depends on the temperature and emissivity of the objects. A higher temperature and a higher emissivity correspond to a higher flux. The image shown is white hot—a whiter point in the image corresponds to a higher flux leaving the object. It is interesting to note that trees have a natural self-cooling process, since a high temper- ature can damage foliage. Objects that have absorbed a large amount of solar energy are hot and are emitting large amounts of infrared radiation. This is some- times called solar loading. 2.1 Basic Imaging Systems 13 Figure 2.3 Visible image on the left side: reflected flux; LWIR infrared image on the right side: emitted flux. (Images courtesy of NRL Optical Sciences Division.)
- 35. The characteristics of the infrared radiation emitted by an object are described by Planck’s blackbody law in terms of spectral radiant emittance, in the following: ( ) ( ) M c ec T λ λ ε λ λ µ = − − 1 5 2 2 1 W cm m (2.1) where c1 and c2 are constants of 3.7418 × 10 4 W-µm 4 /cm 2 and 1.4388 × 10 4 µm-K. The wavelength, λ, is provided in micrometers and ε(λ) is the emissivity of the sur- face. A blackbody source is defined as an object with an emissivity of 1.0 and is con- sidered a perfect emitter. Source emissions of blackbodies at typical terrestrial temperatures are shown in Figure 2.4. Often, in modeling and system performance assessment, the terrestrial background temperature is assumed to be 300K. The source emittance curves are shown for other temperatures for comparison. One curve corresponds to an object colder than the background, and two curves corre- spond to temperatures hotter than the background. Planck’s equation describes the spectral shape of the source as a function of wavelength. It is readily apparent that the peak shifts to the left (shorter wave- lengths) as the body temperature increases. If the temperature of a blackbody were increased to that of the Sun (5,900K), the peak of the spectral shape would decrease to 0.55 µm or green light (note that this is in the visible band). This peak wavelength is described by Wien’s displacement law λ µ max , , = 2 898 T m (2.2) For a terrestrial temperature of 300K, the peak wavelength is around 10 µm. It is important to note that the difference between the blackbody curves is the “signal” in the infrared bands. For an infrared sensor, if the background is at 300K and the tar- get is at 302K, the signal is the difference in flux between the blackbody curves. Sig- nals in the infrared sensor are small riding on very large amounts of background flux. In the visible band, this is not the case. For example, consider the case of a white target on a black background. The black background is generating no signal, while the white target is generating a maximum signal, given the sensor gain has 14 Imaging Systems Blackbody curves for four temps from 290K to 320K Radiant emittance, W/cm - m µ 0 0.001 0.002 0.003 0.004 0.005 0 5 10 15 20 25 30 35 40 Wavelength (micrometers) 290 300 310 320 2 2 Figure 2.4 Planck’s blackbody radiation curves.
- 36. been adjusted. Dynamic range may be fully utilized in a visible sensor. For the case of an infrared sensor, a portion of the dynamic range is used by the large back- ground flux radiated by everything in the scene. This flux is never a small value; hence, sensitivity and dynamic range requirements are much more difficult to satisfy in infrared sensors than in visible sensors. 2.2 Resolution and Sensitivity There are three general categories of infrared sensor performance characterizations. The first is sensitivity and the second is resolution. When end-to-end, or human-in-the-loop (HITL), performance is required, the third type of performance characterization describes the visual acuity of an observer through a sensor, which will be discussed in Chapter 3. The former two are both related to the hardware and software that comprises the system, while the latter includes both the sensor and the observer. The first type of measure, sensitivity, is determined through radiometric analysis of the scene/environment and the quantum electronic properties of the detectors. Resolution is determined by analysis of the physical optical properties, the detector array geometry, and other degrading components of the system in much the same manner as complex electronic circuit/signals analysis. Sensitivity describes how the sensor performs with respect to input signal level. It relates noise characteristics, responsivity of the detector, light gathering of the optics, and the dynamic range/quantization of the sensor. Radiometry describes how much light leaves the object and background and is collected by the detector. Optical design and detector characteristics are of considerable importance in sensor sensitivity analysis. In infrared systems, noise equivalent temperature difference (NETD) is often a first-order description of the system sensitivity. The three-dimen- sional (3-D) noise model [1] describes more detailed representations of sensitivity parameters. In visible systems, the noise equivalent irradiance (NEI) is a similar term that is used to determine the sensitivity of the system. The second type of measure is resolution. Resolution is the ability of the sensor to image small targets and to resolve fine detail in large targets. Modulation transfer function (MTF) is the most widely used resolution descriptor in infrared systems. Alternatively, it may be specified by a number of descriptive metrics, such as the optical Rayleigh Criterion or the instantaneous field-of-view (FOV), of the detector. While these metrics are component-level descriptions, the system MTF is an all-encompassing function that describes the system resolution. Sensitivity and resolution can be competing system characteristics, and they are the most important issues in initial studies for a design. For example, given a fixed sensor aperture diameter, an increase in focal length can provide an increase in reso- lution, but it may decrease sensitivity [1]. Typically, visible band systems have plenty of sensitivity and are resolution-limited, while infrared imagers have been more sensitivity-limited. With staring infrared sensors, the sensitivity has seen significant improvements. Quite often metrics, such as NETD and MTF, are considered to be separable. However, in an actual sensor, sensitivity and resolution performance are not inde- pendent. As a result, minimum resolvable temperature difference (MRT or MRTD) 2.2 Resolution and Sensitivity 15
- 37. or the sensor contrast threshold function (CTF) has become the primary perfor- mance metrics for infrared systems. MRT and MRC (minimum resolvable contrast) are a quantitative performance measure in terms of both sensitivity and resolution. A simple MRT curve is shown in Figure 2.5. The performance is bounded by the sen- sor’s limits and the observer’s limits. The temperature difference, or thermal con- trast, required to image smaller details in a scene increases with detail size. The inclusion of observer performance yields a single-sensor performance characteriza- tion. It describes sensitivity as a function of resolution and includes the human visual system. 2.3 Linear Shift-Invariant (LSI) Imaging Systems A linear imaging system requires two properties [1, 2]: superposition and scaling. Consider an input scene, i(x, y) and an output image, o(x, y). Given that a linear sys- tem is described by L{}, then ( ) ( ) { } o x y L i x y , , = (2.3) The superposition and scaling properties are satisfied if ( ) ( ) { } ( ) { } ( ) { } L ai x y bi x y aL i x y bL i x y 1 2 1 2 , , , , + = + (2.4) where i1(x, y) and i2(x, y) are input scenes and a and b are constants. Superposition, simply described, is that the image of two scenes, such as a target scene and a back- ground scene, is the sum of individual scenes imaged separately. The simplest exam- ple here is that of a point source as shown in Figure 2.6. The left side of the figure shows the case where a single point source is imaged, then a second point source is imaged, and the two results are summed to give an image of the two point sources. 16 Imaging Systems Spatial frequency Minimum resolvable temperature Visual sensitivity limit System resolution limit System response Figure 2.5 Sensor resolution and sensitivity.
- 38. The superposition principle states that this sum of point source images would be identical to the resultant image if both point sources were included in the input scene. The second property simply states that an increase in input scene brightness increases the image brightness. Doubling a point source brightness would double the image brightness. The linear systems approach is extremely important with imaging systems, since any scene can be represented as a collection of weighted point sources. The output image is the collection of the imaging system responses to the point sources. In continuous (nonsampled) imaging systems, another property is typically assumed: shift-invariance. Sometimes a shift invariant system is called isoplanatic. Mathematically stated, the response of a shift invariant system to a shifted input, such as a point source, is a shifted output; that is, ( ) ( ) { } o x x x y L i x x y y o o o o − − = − − , , (2.5) where xo and yo are the coordinates of the point source. It does not matter where the point source is located in the scene, the image of the point source will appear to be the same, only shifted in space. The image of the point source does not change with position. If this property is satisfied, the shifting property of the point source, or delta function, can be used, ( ) ( ) ( ) i x y i x y x x y y dxdy o o o o x x y y , , , = − − ∫ ∫ δ 1 2 1 2 (2.6) 2.3 Linear Shift-Invariant (LSI) Imaging Systems 17 Imaging system Input scene Output image P1 Imaging system Input scene Output image P2 + = Output Image Imaging system Input scene Output image Imaged separately and added Imaged together Figure 2.6 Superposition principle.
- 39. where x1 ≤ xo ≤ x2 and y1 ≤ yo ≤ y2. The delta function, δ(x−xo, y−yo), is nonzero only at xo, yo and has an area of unity. The delta function is used frequently to describe infinitesimal sources of light. Equation (2.6) states that the value of the input scene at xo, yo can be written in terms of a weighted delta function. We can substitute i(x, y) in (2.6) ( ) ( ) ( ) i x y i x y d d , , , = − − −∞ ∞ −∞ ∞ ∫ ∫ α β δ α β α β (2.7) which states that the entire input scene can be represented as a collection of weighted point sources. The output of the linear system can then be written using (2.7) as the input, so that ( ) ( ) ( ) o x y L i x y d d , , , = − − −∞ ∞ −∞ ∞ ∫ ∫ α β δ α β α β (2.8) Since the linear operator, L{}, does not operate on α and β, (2.8) can be rewritten as ( ) ( ) ( ) { } o x y i L x y d d , , , = − − −∞ ∞ −∞ ∞ ∫ ∫ α β δ α β α β (2.9) If we call the point source response of the system the impulse response, defined as ( ) ( ) { } h x y L x y , , = δ (2.10) then the output of the system is the convolution of the input scene with the impulse response of the system; that is, ( ) ( ) ( ) ( ) ( ) o x y i h x y d d i x y h x y , , , , ** , = − − = −∞ ∞ −∞ ∞ ∫ ∫ α β α β α β (2.11) where ** denotes the two-dimensional (2-D) convolution. The impulse response of the system, h(x, y), is commonly called the point spread function (PSF) of the imag- ing system. The significance of (2.11) is that the system impulse response is a spatial 18 Imaging Systems i x,y ( ) x Point spread function o x,y ( ) ** h x,y ( ) Figure 2.7 Simplified LSI imaging system.
- 40. filter that is convolved with the input scene to obtain an output image. The simpli- fied LSI imaging system model is shown in Figure 2.7. The system described here is valid for LSI systems only. This analysis technique is a reasonable description for continuous and well-sampled imaging systems. It is not a good description for an undersampled or a well-designed sampled imaging system. These sampled imaging systems do satisfy the requirements of a linear sys- tem, but they do not follow the shift invariance property. The sampling nature of these systems is described later in this chapter. The representation of sampled imag- ing systems is a modification to this approach. For completeness, we take the spatial domain linear systems model and convert it to the spatial frequency domain. Spatial filtering can be accomplished in both domains. Given that x and y are spatial coordinates in units of milliradians, the spatial frequency domain has independent variables of fx and fy, cycles per milliradian. A spatial input or output function is related to its spectrum by the Fou- rier transform ( ) ( ) ( ) F f f f x y e dxdy x y j f x f y x y , , = − + −∞ ∞ −∞ ∞ ∫ ∫ 2 π (2.12) where the inverse Fourier transform converts an image spectrum to a spatial function ( ) ( ) ( ) f x y F f f e df df x y j f x f y x y x y , , = + −∞ ∞ −∞ ∞ ∫ ∫ 2 π (2.13) The properties and characteristics of the Fourier transform are provided in [2–4]. A function and its spectrum are collectively described as a Fourier transform pair. We will use the notation of the Fourier transform operator ( ) ( ) [ ] ( ) ( ) [ ] G f f g x y g x y G f f x y x y , , , , = = − ᑤ ᑤ and 1 (2.14) in order to simplify analyses descriptions. One of the very important properties of the Fourier transform is that the Fourier transform of a convolution results in a product. Therefore, the spatial convolution described in (2.11) results in a spectrum of ( ) ( ) ( ) O f f I f f H f f x y x y x y , , , = (2.15) Here, the output spectrum is related to the input spectrum by the product of the Fourier transform of the system impulse response. Therefore, the Fourier transform of the system impulse response is called the transfer function of the system. Multi- plication of the input scene spectrum by the transfer function of an imaging system provides the same filtering action as the convolution of the input scene with the imaging system PSF. In imaging systems, the magnitude of the Fourier transform of the system PSF is the modulation transfer function (MTF). 2.3 Linear Shift-Invariant (LSI) Imaging Systems 19
- 41. 2.4 Imaging System Point Spread Function and Modulation Transfer Function The system impulse response or point spread function (PSF) of an imaging system is comprised of component impulse responses as shown in Figure 2.8. Each of the com- ponents in the system contributes to the blurring of the scene. In fact, each of the components has an impulse response that can be applied in the same manner as the system impulse response. The blur attributed to a component may be comprised of a few different physical effects. For example, the optical blur is a combination of the diffraction and aberration effects of the optical system. The detector blur is a combi- nation of the detector shape and the finite time of detector integration as it traverses the scene. It can be shown that the PSF of the system is a combination of the individual impulse responses ( ) ( ) ( ) ( ) ( ) h x y h x y h x y h x y h x y system atm optics elec , , ** , ** , ** , * det = ( ) * , h x y disp (2.16) so that the total blur, or system PSF, is a combination of the component impulse responses. The Fourier transform of the system impulse response is called the transfer func- tion of the system. In fact, each of the component impulse responses given in (2.16) has a component transfer function that, when cascaded (multiplied), the resulting transfer function is the overall system transfer function; that is, ( ) ( ) ( ) ( ) ( ) O f f I f f H f f H f f H f f H f x y x y atm x y optics x y x y elec x , , , , , det = ( ) ( ) ( ) , , , f H f f H f f y disp x y eye x y (2.17) Note that the system transfer function is the product of the component transfer functions. A large number of imaging spatial filters are accounted for in the design and/or analysis of imaging system performance. These filters include effects from optics, detectors, electronics, displays, and the human eye. We use (2.16) and (2.17) as our spatial filtering guidelines, where we know that the treatment can be applied in either the spatial or frequency domain. We present the most common of these filters 20 Imaging Systems Input scene Atmosphere Optics Detectors Electronics Display hatm hoptics( ) x,y hdet helec hdisp i x,y ( ) o x,y ( ) Output scene ( ) x,y ( ) x,y ( ) x,y ( ) x,y Figure 2.8 Imaging system components.
- 42. beginning with the optical effects. Also, the transfer function of a system, as given in (2.17), is frequently described without the eye transfer function. 2.4.1 Optical Filtering Two filters account for the optical effects in an imaging system: diffraction and aberrations. The diffraction filter accounts for the spreading of the light as it passes an obstruction or an aperture. The diffraction impulse response for an incoherent imaging system with a circular aperture of diameter D is ( ) h x y D somb Dr diff , = λ λ 2 2 (2.18) where λ is the average band wavelength and r x y = + 2 2 . The somb (for som- brero) function is given by Gaskill [3] to be ( ) ( ) somb r J r r = 1 π π (2.19) where J1 is the first-order Bessel function of the first kind. The filtering associated with the optical aberrations is sometimes called the geometric blur. There are many ways to model this blur and there are numerous commercial programs for calculat- ing geometric blur at different locations on the image. However, a convenient method is to consider the geometric blur collectively as a Gaussian function ( ) h x y Gaus r geom gb gb , = 1 2 σ σ (2.20) where σgb is an amplitude that best describes the blur associated with the aberrations. The Gaussian function, Gaus, is ( ) Gaus r e r = −π 2 (2.21) Note that the scaling values in front of the somb and the Gaus functions are intended to provide a functional area (under the curve) of unity so that no gain is applied to the scene. Examples of the optical impulse responses are given in Figure 2.9 corresponding to a wavelength of 10 µm, an optical diameter of 10 cen- timeters, and a geometric blur of 0.1 milliradian. The overall impulse response of the optics is the combined blur of both the dif- fraction and aberration effects ( ) ( ) ( ) h x y h x y h x y optics diff geom , , ** , = (2.22) The transfer functions corresponding to these impulse responses are obtained by taking the Fourier transform of the functions given in (2.18) and (2.20). The Fou- rier transform of the somb is given by Gaskill [3] so that the transfer function is 2.4 Imaging System Point Spread Function and Modulation Transfer Function 21
- 43. ( ) H f f D D D diff x y , cos = − − − 2 1 1 2 π ρλ ρλ ρλ (2.23) where ρ = + f f x y 2 2 and is plotted in cycles per milliradian and D is the entrance aperture diameter. The Fourier transform of the Gaus function is simply the Gaus function [4], with care taken on the scaling property of the transform. The transfer function corresponding to the aberration effects is ( ) ( ) H f f Gaus geom x y gb , = σ ρ (2.24) For the example described here, the transfer functions are shown in Figure 2.10. Note that the overall optical transfer function is the product of the two functions. 2.4.2 Detector Spatial Filters The detector spatial filter is also comprised of a number of different effects, includ- ing spatial integration, sample-and-hold, crosstalk, and responsivity, among others. The two most common effects are spatial integration and sample-and-hold; that is, ( ) ( ) ( ) h x y h x y h x y sp sh det det_ det_ , , ** , = (2.25) 22 Imaging Systems −0.2 −0.1 0 0.1 0.2 −0.2 −0.1 0 0.1 0.2 0 0.2 0.4 0.6 0.8 1 −0.2 −0.1 0 0.1 0.2 −0.1 0 0.1 0.2 0 0.2 0.4 0.6 0.8 1 y milliradians hdiff ( ) x,y hgeom ( ) x,y m x illiradians y milliradians m x illiradians Figure 2.9 Spatial representations of optical blur. 10 0.2 0.4 0.6 0.8 1 0 5 0 −5 0.2 0.4 0.6 0.8 1 0 cycles per milliradians H f f ( , ) geom −10 10 5 0 −5 −10 10 f 5 0 −5 −10 10 5 0 −5 −10 cycles per milliradians cycles per milliradians cycles per milliradians x fx fy fy x y H f f diff x y ( , ) Figure 2.10 Optical transfer functions of optical blur.
- 44. The other effects can be included, but they are usually considered negligible unless there is good reason to believe otherwise (i.e., the detector responsivity varies dramatically over the detector). The detector spatial impulse response is due to the spatial integration of the light over the detector. Since most detectors are rectangular in shape, the rectangle function is used as the spatial model of the detector ( ) h x y DAS DAS rect x DAS y DAS DAS r sp x y x y x det_ , , = = 1 1 ect x DAS DAS rect y DAS x y y 1 (2.26) where DASx and DASy are the horizontal and vertical detector angular subtenses in milliradians. The detector angular subtense is the detector width (or height) divided by the sensor focal length. The transfer function corresponding to the detector spa- tial integration is determined by taking the Fourier transform of (2.26) ( ) ( ) ( ) H f f DAS f DAS f DAS f DAS f sp x y x x y y x x y det_ , , = = sinc sinc sinc( ) y (2.27) where the sinc function is defined as [2] ( ) ( ) sinc x x x = sin π π (2.28) The impulse response and the transfer function for a detector with a 0.1 by 0.1 milliradian detector angular subtense is shown in Figure 2.11. The detector sample-and-hold function is an integration of the light as the detector scans across the image. This sample-and-hold function is not present in staring arrays, but it is present in most scanning systems where the output of the integrated signal is sampled. The sampling direction is assumed to be the horizontal, or x, direction. Usually, the distance, in milliradians, between samples is smaller than the detector angular subtense by a factor called samples per IFOV or samples per DAS, spdas. The sample-and-hold function can be considered a rectangular 2.4 Imaging System Point Spread Function and Modulation Transfer Function 23 40 0 0.5 1 h x y ( , ) det_sp −0.5 20 0 −20 0.2 0.4 0.6 0.8 1 0 y milliradians H f ( , ) f det_sp −40 1.0 0 −0.1 1.0 0 −0.1 x milliradians cycles per milliradians cycles per milliradians fy fx 40 20 0 −20 −40 x y Figure 2.11 Detector spatial impulse response and transfer function.
- 45. function in x where the size of the rectangle corresponds to the distance between samples. In the spatial domain y direction, the function is an impulse function. Therefore, the impulse response of the sample-and-hold function is ( ) ( ) h x y spdas DAS rect x spdas DAS y sh x x det_ , = δ (2.29) The Fourier transform of the impulse response gives the transfer function of the sample- and-hold operation ( ) H f f DAS f spdas sh x y x x det_ , = sinc (2.30) Note that the Fourier transform of the impulse function in the y direction is 1. The impulse response and the transfer function for sample-and-hold associated with the detector given in Figure 2.11 with a two-sample-per-DAS sample-and-hold are shown in Figure 2.12. 2.4.3 Electronics Filtering The electronics filtering function is one of the more difficult to characterize and one of the more loosely applied functions. First, it involves the conversion of temporal frequencies to spatial frequencies. Usually, this involves some scan rate or readout rate. Second, most impulse response functions in space are even functions. With electronic filtering, the impulse function can be a one-sided function. Finally, most engineers apply a two-sided impulse response that violates the rules of causality. This gross approximation does not usually have a heavy impact on sensor perfor- mance estimates since the electronics are not typically the limiting component of the sensor. Holst [5] and Vollmerhausen and Driggers [6] provide electronic filter (digi- tal and analog) approximations that can be used in transfer function estimates. Digital filters also provide a spatial blur and a corresponding transfer function. Finite impulse response (FIR) filters are common in electro-optical and infrared sys- 24 Imaging Systems 40 0 0.5 1 h x y ( , ) det_sh −0.5 20 0 −20 0.2 0.4 0.6 0.8 1 0 y milliradians H f ( , ) f det_sh −40 1.0 0 −0.1 1.0 0 −0.1 x milliradians cycles per milliradians cycles per milliradians fy fx 40 20 0 −20 −40 x y −0.05 0.05 −0.05 0.05 Figure 2.12 Detector sample-and-hold impulse response and transfer function.
- 46. tems with such functions as interpolation, boost, and edge enhancements. These are filters that are convolved with a digital image and so they have a discrete “kernel” that is used to process the spatial image. The transfer function associated with these FIR filters is a summation of sines and cosines, where the filter is not band-limited. The combination of these filters with a display reconstruction provides for an over- all output filter (and corresponding transfer function). Chapter 4 discusses finite impulse response filters and the transfer function associated with these filters. 2.4.4 Display Filtering The finite size and shape of the display spot also corresponds to a spatial filtering of the image. Usually, the spot, or element, of a display is either Gaussian in shape like a cathode ray tube (CRT), or it is rectangular in shape, like a flat-panel display. Light emitting diode (LED) displays are also rectangular in shape. The PSF of the display is simply the size and shape of the display spot. The only difference is that the finite size and shape of the display spot must be converted from a physical dimension to the sensor angular space. For the Gaussian spot, the spot size dimen- sion in centimeters must be converted to an equivalent angular space in the sensor’s field of view (FOV) σ σ disp angle disp cm v disp v FOV L _ _ _ = (2.31) where Ldisp_v is the length in centimeters of the display vertical dimension and FOVv is FOV of the sensor in milliradians. For the rectangular display element, the height and width of the display element must also be converted to the sensor’s angular space. The vertical dimension of the rectangular shape is obtained using (2.31) and the horizontal dimension is similar with the horizontal display length and sensor FOV. Once these angular dimensions are obtained, the PSF of the display spot is simply the size and shape of the display element ( ) h x y Gaus r disp disp angle disp angle , _ _ = 1 2 σ σ for a Gaussian spot (2.32) or ( ) h x y W H rect x W disp disp angle h disp angle v disp ang , _ _ _ _ _ = 1 le h disp angle v y H _ _ _ , for flat panel (2.33) where the angular display element shapes are given in milliradians. These spatial shapes are shown in Figures 2.9 and 2.11. The transfer functions associated with these display spots are determined by taking the Fourier transform of the earlier PSF equations; that is, 2.4 Imaging System Point Spread Function and Modulation Transfer Function 25
- 47. ( ) ( ) H f f Gaus disp x y disp angle , _ = σ ρ Gaussian display (2.34) or ( ) ( ) H f f W f H f disp x y disp angle h x disp angle v y , , _ _ _ _ = sinc Flat-panel display (2.35) Again, these transfer functions are shown in Figures 2.9 and 2.11. 2.4.5 Human Eye Note that the human eye is not part of the system performance MTF as shown in Figure 2.8, and the eye MTF should not be included in the PSF of the system. In Chapter 3, the eye CTF is used to include the eye sensitivity and resolution limita- tions in performance calculations. It is, however, useful to understand the PSF and MTF of the eye such that the eye blur can be compared to sensor blur. A system with much higher resolution than the eye is a waste of money, and a system with much lower resolution than the eye is a poorly performing system. The human eye certainly has a PSF that is a combination of three physical com- ponents: optics, retina, and tremor [7, 8]. In terms of these components, the PSF is ( ) ( ) ( ) ( ) h x y h x y h x y h x y eye optics retina tremor , , ** , ** , _ = (2.36) Therefore, the transfer function of the eye is ( ) ( ) ( ) ( ) H f f H f f H f f H f f eye x y eye optics x y retina x y tremor x y , , , , _ = (2.37) The transfer function associated with the eye optics is a function of display light level. This is because the pupil diameter changes with light level. The number of foot-Lamberts (fL) at the eye from the display is Ld/0.929, where Ld is the display luminance in millilamberts. The pupil diameter is then ( ) { } [ ] D fL pupil = − + − 9011 1323 21082 10 . . exp log . mm (2.38) This equation is valid, if one eye is used as in some targeting applications. If both eyes view the display, the pupil diameter is reduced by 0.5 millimeter. Two parame- ters, io and fo, are required for the eye optics transfer function. The first parameter is ( ) io Dpupil = + 0 7155 0277 2 . . (2.39) and the second is ( ) { } fo D D pupil pupil = − exp . . * log 3663 00216 2 (2.40) Now, the eye optics transfer function can be written as ( ) ( ) [ ] { } H M fo eye optics io _ exp . ρ ρ = − 4369 (2.41) 26 Imaging Systems
- 48. where ρ is the radial spatial frequency, f f x y 2 2 + , in cycles per milliradian. M is the system magnification (angular subtense of the display to the eye divided by the sen- sor FOV). The retina transfer function is ( ) ( ) { } H M retina ρ ρ = − exp . . 0375 1 21 (2.42) Finally, the transfer function of the eye due to tremor is ( ) ( ) { } H M tremor ρ ρ = − exp . 04441 2 (2.43) which completes the eye model. For an example, let the magnification of the system equal 1. With a pupil diame- ter of 3.6 mm corresponding to a display brightness of 10 fL at the eye (with one viewing eye), the combined MTF of the eye is shown in Figure 2.13. The io and fo parameters were 0.742 and 27.2, respectively. All of the PSFs and transfer functions given in this section are used in the model- ing of infrared and electro-optical imaging systems. We covered only the more com- mon system components. There may be many more that must be considered when they are part of an imaging system. 2.4.6 Overall Image Transfer To quantify the overall system resolution, all of the spatial blurs are convolved and all of the transfer functions are multiplied. The system PSF is the combination of all the blurs, and the system MTF is the product of all the transfer functions. In the roll-up, the eye is typically not included to describe the resolution of the system. Also, the system can be described as “limited” by some aspect of the system. For example, a diffraction-limited system is one in which the diffraction cutoff fre- quency is smaller than all of the other components in the system (and spatial blur is larger). A detector-limited system would be one in which the detector blur is larger and the detector transfer cutoff frequency is smaller than the other system components. 2.4 Imaging System Point Spread Function and Modulation Transfer Function 27 fx fy 0 0.2 0.4 0.6 0.8 1 eye H 2 2 1 1 0 0 −1 −1 −2 −2 cyc/mrad cyc/mrad Figure 2.13 Eye transfer function.
- 49. The MTF for a typical MWIR system is shown in Figure 2.14. The pre-MTF shown is the rollup transfer function for the optics diffraction blur, aberrations, and the detector shape. The post-MTF is the rollup transfer for the electronics (many times negligible) and the display. The system MTF (system transfer) is the product of the pre- and post-MTFs as shown. In Figure 2.14, the horizontal MTF is shown. In most sensor performance mod- els, the horizontal and vertical blurs, and corresponding MTFs, are considered sepa- rable. That is, ( ) ( ) ( ) h x y h x h y , = (2.44) and the corresponding Fourier transform is ( ) ( ) ( ) H f f H f H f x y x y , = (2.45) This approach usually provides for small errors (a few percent) in performance calculations even when some of the components in the system are circularly symmetric. 2.5 Sampled Imaging Systems In the previous sections, we described the process of imaging for a continuous or well-sampled imager. In this case, the input scene is convolved with the imager PSF (i.e., the impulse response of the system). With sampled imaging systems, the process is different. As an image traverses through a sampled imaging system, the image undergoes a three-step process. Figure 2.15 shows this process as a presample blur, a sampling action, and a postsample blur (reconstruction). The image is blurred by the optics, the detector angular subtense, the spatial integration scan, if needed, and any other effects appropriate to presampling. This presample blur, h(x,y), is applied to the image in the manner of an impulse response, so the response is convolved with the input scene ( ) ( ) ( ) ( ) ( ) o x y i x y h d d i x y h x y 1 , , , , ** , = − − = −∞ ∞ −∞ ∞ ∫ ∫ α β α β α β (2.46) 28 Imaging Systems Horizontal system MTFs 0 0.2 0.4 0.6 0.8 1 0 5 10 15 Cycles/mrad Pre-MTF System transfer Post-MTF Figure 2.14 System transfer function.
- 50. where o1(x, y) is the presampled blur image or the output of the presample blur pro- cess. The convolution is denoted by the *, so ** denotes the two-dimensional con- volution. The sampling process can be modeled with the multiplication of the presample blur image with the sampling function. For convention, we use Gaskill’s comb function [9] ( ) ( ) comb x a y b a b x ma y nb n m , = − − =−∞ ∞ =−∞ ∞ ∑ ∑ δ δ (2.47) which is a two-dimensional separable function. Now the output of the sampling process can be written as the product of the presample blurred image with the sam- pling function (note that a and b are the distances in milliradians or millimeters between samples) ( ) ( ) ( ) ( ) [ ] o x y o x y ab comb x a y b i x y h x y ab comb x a 2 1 1 1 , , , , ** , , = = y b (2.48) At this point, all that is present is a set of discrete values that represent the presample blurred image at discrete locations. This output can be thought of as a weighted “bed of nails” that is meaningless to look at unless the “image” is recon- structed. The display and the eye, if applied properly, reconstruct the image to a function that is interpretable. This reconstruction is modeled as the convolution of the display and eye blur (and any other spatial postsample blur) and the output of the sampling process; that is, ( ) ( ) ( ) ( ) ( ) [ ] { o x y o x y d x y i x y h x y ab comb x a y b , , ** , , ** , , * = = × 2 1 ( ) * , d x y (2.49) While (2.49) appears to be a simple spatial process, there is a great deal that is inherent in the calculation. We have simplified the equation with the aggregate presample blur effects and the aggregate postsample reconstruction effects. The frequency analysis of the three-step process shown in Figure 2.15 can be presented simply by taking the Fourier transform of each process step. Consider the first step in the process, the presample blur. The transform of the convolution in space is equivalent to a product in spatial frequency 2.5 Sampled Imaging Systems 29 Presample blur Image sample Reconstruction x x x i x,y ( ) h x,y ( ) s x,y ( ) d x,y ( ) o x,y ( ) o1( ) x,y o2( ) x,y Figure 2.15 Three-step imaging process.
- 51. ( ) ( ) ( ) O f f I f f H f f x y x y pre x y 1 , , , = (2.50) where fx and fy are the horizontal and vertical spatial frequencies. If x and y are in milliradians, then the spatial frequencies are in cycles per milliradian. Hpre(fx, fy) is the Fourier transform of the presample blur spot. Note that the output spectrum can be normalized to the input spectrum so that Hpre(fx, fy) is a transfer function that fol- lows the linear systems principles. Consider the presample blur spectrum (i.e., the presample blur transfer function given in Figure 2.16). Note that this is the image spectrum on the output of the blur that would occur if an impulse were input to the system. Next, we address the sampling process. The Fourier transform of (2.48) gives ( ) ( ) ( ) [ ] ( ) O f f I f f H f f comb af bf x y x y pre x y x y 2 , , , ** , = (2.51) where ( ) ( ) ( ) comb af bf f kf f lf f a f b x y x xs y ys xs ys l , , = − − = = =−∞ δ δ 1 1 and ∞ =−∞ ∞ ∑ ∑ k (2.52) If an impulse were input to the system, the response would be ( ) ( ) ( ) O f f H f f comb af bf x y pre x y x y 2 , , ** , = (2.53) which is a replication of the presample blur at sample spacings of 1/a and 1/b. Con- sider the case shown in Figure 2.17. The sampled spectrum shown corresponds to a Gaussian blur of 0.5-milliradian radius (to the 0.043 cutoff) and a sample spacing of 0.5 milliradian. Note that the reproduction in frequency of the presample blur is at 2 cycles per milliradian. The so-called Nyquist rate of the sensor (the sensor half-sample rate) is at 1 cycle per milliradian. Any frequency from the presample blur baseband that is greater than the half-sample rate is also present as a mirror signal, or classical aliasing, under the half-sample rate by the first-order reproduction. The amount of classical aliasing is easily computed as the area of this mirrored signal. However, this is not the aliased signal seen on the output of the display as the display transfer 30 Imaging Systems 0 0.2 0.4 0.6 0.8 1 1.2 −3 −2 −1 0 1 2 3 Cycles/mrad H (fx,fy) pre Figure 2.16 Presample blur transfer function.
- 52. has not been applied to the signal. The higher-order replications of the baseband are real-frequency components. The curves to the left and the right of the central curve are the first- and second-order replications at the positive and negative positions. The current state of the sample signal is tiny infinitesimal points weighted with the presample blurred image values. These points have spectra that extend in the fre- quency domain to very high frequencies. The higher-order replications are typically filtered with a reconstruction function involving the display and the eye. There is no practical way to implement the perfect reconstruction filter; however, the perfect rectangular filter would eliminate these higher-order replications and result only in the classical aliased signal. The reconstruction filter usually degrades the baseband and allows some of the signal of the higher-order terms through to the observer. The final step in the process corresponds to the reconstruction of the sampled information. This is accomplished simply by blurring the infinitesimal points so that the function looks nearly like that of the continuous input imagery. The blur is convolved in space, so it is multiplied in frequency ( ) ( ) ( ) [ ] ( ) O f f H f f comb af bf D f f x y pre x y x y x y , , ** , , = (2.54) where this output corresponds to a point source input. Note that the postsampling transfer function is multiplied by the sampled spectrum to give the output of the whole system. Consider the sampled spectrum and the dashed display transfer func- tion shown in Figure 2.18. The postsampling transfer function is shown in the graph as the display, passes part of the first-order replications. However, this display degrades the baseband signal relatively little. The tradeoff here is baseband resolu- tion versus spurious response content. Given that all of the signals traverse through the postsample blur transfer func- tion, the output is shown in Figure 2.19. There is classical aliasing on the output of the system, but a large signal corresponds to higher-order replication signals that passed through the display. Aliasing and the higher-order signals are collectively the spurious response of the sensor. These signals were not present on the input imag- ery, but there are artifacts on the output imagery. Without sampling, these spurious 2.5 Sampled Imaging Systems 31 Cycles/mrad 0 0.2 0.4 0.6 0.8 1 −5 −3 −1 1 3 5 2 O (fx ,fy) Figure 2.17 Output of sampling.
- 53. signals would not be present. The higher-order replicated signals are combined at each spatial frequency in terms of a vector sum, so it is convenient to represent the magnitude of the spurious signals as the root-sum-squared (RSS) of the spurious orders that make it through the reconstruction process. Three aggregate quantities have proven useful in describing the spurious response of a sampled imaging system: total integrated spurious response as defined by (2.55), in-band spurious response as defined by (2.56), and out-of-band spurious response as defined by (2.57) [10]; that is, ( ) ( ) SR Spurious df df x x = −∞ ∞ −∞ ∞ ∫ ∫ Response BasebandSignal (2.55) ( ) SR Spurious df in band x f f s s − − = ∫ Response BasebandSigna 2 2 ( ) l dfx −∞ ∞ ∫ (2.56) 32 Imaging Systems Cycles/mrad 0 0.2 0.4 0.6 0.8 1 −5 −3 −1 1 3 5 Sampled signal and display transfer Figure 2.18 Sampled signal and display transfer. Cycles/mrad System output spectrum 0 0.2 0.4 0.6 0.8 1 −5 −3 −1 1 3 5 Figure 2.19 System output signal.
- 54. SR SR SR out of band in band − − − = − (2.57) where fs is the sampling frequency. Examples of total and in-band spurious response are illustrated in Figures 2.20 and 2.21, respectively. The spurious responses of the higher-order replications could be constructive or destructive in nature, depending on the phase and frequency content of the spurious signals. The combination in magnitude was identical to that of a vector sum. This magnitude, on average, was the quadrature sum of the signals. This integrated RSS spurious response value was normalized by the integral of the baseband area. This ratio was the spurious response ratio. There is good experimental and theoretical evidence to generalize the effects of in-band spurious response and out-of-band spurious response. An in-band spurious response is the same as classical aliasing in communication systems. These are sig- nals that are added to real signals and can corrupt an image by making the image look jagged, misplaced, (spatial registering), or even the wrong size (wider or thin- ner than the original object). The only way to decrease the amount of in-band spuri- ous response is to increase the sample rate or increase the amount of presample blur. Increasing presample blur, or reducing the MTF, only reduces the performance of the imaging system. Blur causes more severe degradation than aliased signals. The effects of out-of-band spurious response are manifested by display artifacts. Common out-of-band spurious response is raster where the display spot is small compared to the line spacing or pixelization where the display looks blocky. Pixelization occurs on flat-panel displays where the display elements are large or in cases where pixel replication is used as a reconstruction technique. 2.5 Sampled Imaging Systems 33 x f ) ( x f Gδ Transfer response Spurious response Figure 2.20 Example of total spurious response. x f ) ( x f Gδ Transfer response Spurious response Figure 2.21 Example of in-band spurious response.
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- 56. St. Paul as he appeared in public. They may have known very little of the inner history of his life as he reveals it in his Epistle to the Galatians when vindicating his apostolic authority and mission.[55] Let us now see whether we cannot harmonise St. Paul's autobiographical narrative in the Epistle with the Evangelist's narrative in the Acts; always remembering, however, that an imperfect knowledge is never more completely felt than in such cases. When we try to harmonise an account written from the subjective side by one individual with an objective and exterior narrative written by some one else, we are like a man looking at a globe and trying to take it all in at one glance. One side must be hidden from him; and so in this case, many circumstances are necessarily concealed from us which would solve difficulties that now completely puzzle us. But let us to our task, in which we have derived much assistance from the commentary of Bishop Lightfoot upon Galatians. St. Paul, we are told in ch. ix. 19, received meat after the visit of Ananias and was strengthened. St. Paul was never one of those high-wrought fanatics who despise food and the care of the body. There was nothing of the Gnostic or the Manichean about him, leading him to despise and neglect the body which the Lord has given to be the soul's instrument. He recognised under all circumstances that if the human spirit is to do its work, and if God's glory is to be promoted, the human body must be sustained in force and vigour. When he was on board ship and in imminent peril of shipwreck and death, and men thought they should be at their prayers, thinking of the next world alone, he took bread and blessed and set the crew and passengers alike the healthy example of eating a hearty meal, and thus keeping his body in due preparation for whatever deliverances the Lord might work for them; and so, too, at Damascus, his spiritual joy and hallowed peace and deep gratitude for his restoration to sight did not prevent him paying due attention to the wants of his body. "He took food, and was strengthened." And now comes the first note of time. "Then was Saul certain days with the disciples which were at Damascus. And straightway (εὐθέως) he preached Christ in the synagogues, that He is the Son of God." The
- 57. very same expression is used by St. Paul in Galatians, where, after speaking of his conversion, he says, "Immediately (εὐθέως) I conferred not with flesh and blood, but went away into Arabia, and again returned unto Damascus." Now my explanation, and not mine alone, but that of Bishop Lightfoot, is this. After the new convert had rested for a short time at Damascus, he retired into the Sinaitic desert, where he remained for several months, perhaps for a whole year. During this period he disappeared from the sight and knowledge of men as if the earth had opened its mouth and swallowed him. Then he returned to Damascus and preached with such power that the Jews formed a plot against his life, enlisting the help of the governor on their side, so that even the gates were watched that he might be arrested. He escaped their hands, however, through the assistance of his converts, and went up to Jerusalem.[56] But here another difficulty arises. The Acts tells us that "when Saul was come to Jerusalem, he assayed to join himself to the disciples; but they were all afraid of him, and believed not that he was a disciple," whereupon Barnabas, fulfilling his office of mediation, explanation, and consolation, took him and introduced him to the Apostles; while on the other hand in the first chapter of Galatians St. Paul himself speaks of his first visit to the Jerusalem Church thus: "Then after three years I went up to Jerusalem to visit Cephas, and tarried with him fifteen days. But other of the Apostles saw I none, save James the Lord's brother." Now the difficulty consists in this. First, how could the disciples at Jerusalem have been suspicious of St. Paul, if at least a year and a half had elapsed since his conversion? for the Jewish method of counting time would not require three whole years to have elapsed since that event. Secondly, how could Barnabas have brought him to the Apostles as the Acts states, if St. Paul himself says he saw none of them save Peter and James? As to the first difficulty, we acknowledge at once that it seems at first sight a very considerable one, and yet a little reflection will show that there are many explanations of it. If St. Paul kept quiet, as we believe he did, after his conversion and baptism,
- 58. and departed into the solitudes of Arabia, and then upon his return to Damascus, perhaps after a year's retirement, began his aggressive work, there may not have been time for the Church at large to get knowledge of the facts. Communication, again, may have been interrupted because of the contest between Herod and Aretas, in which Damascus played no small part. Communication may not have been possible between the two Churches.[57] Then, again, the persecution raised by Saul himself seems to have practically extirpated the Jerusalem Church for a time. "They were all scattered abroad except the Apostles," is the account given of the Christian community at Jerusalem. The terror of that persecution may have lasted many a long month. Numbers of the original members may never have ventured back again to the Holy City. The Jerusalem Church may have been a new formation largely composed of new converts who never had heard of a wondrous circumstance which had happened a year or two before to the high priest's delegate, which the Sanhedrin would doubtless desire to keep secret.[58] These and many other considerations offer themselves when we strive to throw ourselves back into the circumstances of the time and help to a solution of the first difficulty which we have indicated. Human life is such a complex thing that the strangest combinations may easily find place therein. In this particular case we are so ignorant of the facts, so many hypotheses offer themselves to account for the seeming inconsistencies, that we hesitate not to identify the visit to Jerusalem mentioned in the Acts with that recorded by St. Paul in the Epistle to the Galatians. The second difficulty to which we have alluded is this, How could Barnabas have brought him to the Apostles, if St. Paul himself states that he saw none of the Apostles save Peter and James the Lord's brother? We must remember, however, that St. Luke and St. Paul wrote with two distinct objects. St. Paul, in the Galatians, wished to show the independence of his revelations as regards the Apostles of the circumcision, the Twelve technically so called. Of these Apostles he saw not one, save St. Peter. St. Luke is giving a broad external
- 59. account of the new convert's earliest religious history, and he tells us that on his first visit to the Holy City his conversion was acknowledged and guaranteed by the apostles,—not the Twelve merely, but the apostles, that is, the senior members of the Christian community, embracing not merely the original company chosen by Christ, but all the senior members of the Church, like Barnabas, James, and others who may have formed a supreme council to guide the affairs of the infant society. The word apostle, in fact, is used very variously in the New Testament; sometimes in a limited sense as confined to the Twelve, sometimes in a wider and more general sense, embracing men like Barnabas, as in Acts xiv. 4, 14; St. James, the Lord's brother, as in 1 Cor. xv. 7; Andronicus and Junias, as in Rom. xvi. 7, and many others. It is quite possible, then, that Barnabas may have brought Saul to the Apostolic council, and told there the tale of his conversion though not one of the original Twelve was present save St. Peter.[59] We have now endeavoured to explain some of the difficulties which a comparison of St. Paul's own autobiographical narrative with the Acts discloses. Let us look again at the retirement into Arabia. This retirement seems to us full of instruction and pregnant with meaning for the hidden as well as the practical life of the soul. St. Paul as soon as he was baptized retired into Arabia; and why, it may be asked, did he retire thither? Some of the ancient expositors, as St. Chrysostom and St. Jerome, both of whom wrote about the same period, A.D. 400, thought that St. Paul retired into Arabia in order that he might preach to the Arabians. St. Chrysostom, for instance, comments thus: "See how fervent was his soul, he was eager to occupy lands yet untilled. He forthwith attacked a barbarous and savage people, choosing a life of conflict and of much toil." And the explanations of Hilary, Theodore of Mopsuestia, Theodoret, and Œcumenius, all of them ancient and acute expositors, are of exactly the same character. Now this would have been a reversal of the Divine order in one important aspect. The power of the keys, the office of opening the kingdom of heaven to the Gentiles had been committed to St. Peter by Jesus Christ. He had not as yet baptized
- 60. Cornelius, and thus formally opened the door of faith to the Gentiles. If St. Paul had preached to the Arabians, he would have usurped St. Peter's place and function. We believe, on the other hand, that God led the converted persecutor into the deserts of Arabia for very different purposes. Let us note a few of them. The Lord led Saul there for the purpose of quiet and retirement. The great commentators and expositors of the early Church, as we have already noted, used to call St. Paul by the special title of "Vas Electionis," the chosen vessel par excellence, chosen because surpassing in his gifts and graces and achievements all the other Apostles. Now it was with the "Vas Electionis" in the New Testament as with many of his types in the Old Testament. When God would prepare Moses for his life's work in shepherding, ruling, and guiding His people through the deserts of Arabia, He first called him for many a long day into retirement to the Mount of Horeb and the solitudes of the Sinaitic desert. When God would strengthen and console the spirit depressed, wounded and severely smitten, of his servant Elijah, He brought him to the same mysterious spot, and there restored his moral and spiritual tone, and equipped him with new strength for his warfare by the visions of the Almighty lovingly vouchsafed to him. The Founder or Former of the Jewish Dispensation and the Reformer of the same Dispensation were prepared and sustained for their work amid the solitudes of the Arabian deserts; and what more fitting place in which the "Vas Electionis," the chosen vessel of the New Dispensation, should be trained? What more suitable locality where the Lord Jesus should make those fuller and completer revelations of Christian doctrine and mystery which his soul needed, than there where lightning-blasted cliff and towering mountains all alike spoke of God and of His dealings with mankind in the mysterious ages of a long-departed past? The Lord thus taught St. Paul, and through him teaches the Church of every age, the need of seasons of retirement and communion with God preparatory to and in close connexion with any great work or scene of external activity, such as St. Paul was now entering upon. It is a lesson much needed by this age of ours when
- 61. men are tempted to think so much of practical work which appears at once in evidence, making its presence felt in tangible results, and so very little of devotional work and spiritual retirement which cannot be estimated by any earthly standard or tabulated according to our modern methods. Men are now inclined to think laborare est orare, and that active external work faithfully and vigorously rendered can take the place and supply the want of prayer and thought, of quiet study and devout meditation. Against such a tendency the Lord's dealings with St. Paul, yea more, the Divine dealings with and leadings of the eternal Son Himself, form a loud and speaking protest. The world was perishing and men were going down to the grave in darkness and Satan and sin were triumphing, and yet Jesus was led up of the Spirit into the wilderness for forty days, and Saul was brought out into the deserts of Arabia from amid the teeming crowds of Damascus that he might learn those secrets of the Divine life which are best communicated to those who wait upon God in patient prayer and holy retirement. This is a lesson very necessary for this hot and fitful and feverish age of ours, when men are in such a hurry to have everything set right and every abuse destroyed all at once. Their haste is not after the Divine model, and their work cannot expect the stability and solidity we find in God's. The nineteenth-century extreme is reproved by St. Paul's retirement into Arabia.[60] Man is, however, such a creature that if he avoids one extreme he generally tumbles into another. And so it is in this matter. Men have been ready to push this matter of retirement into an extreme, and have considered that they were following St. Paul's example in retiring into the Arabian and similar deserts and remaining there. But they have made a great mistake. St. Paul retired into Arabia for a while, and then "returned again unto Damascus." They have retired into the deserts and have remained there engaged in the one selfish task of saving their own souls, as they thought, by the exercises of prayer and meditation, apart from that life of active good works for the sake of others which constitutes another department of Christianity equally vital to the health of the soul.
- 62. The history of Eastern monasticism is marked from its earliest days by an eager desire to follow St. Paul in his retirement into Arabia, and an equal disinclination to return with him unto Damascus. And this characteristic, this intense devotion to a life of solitude strangely enough passed over to our own Western islands, and is a dominant feature of the monasticism which prevailed in Great Britain and Ireland in the days of Celtic Christianity. The Syrian and Egyptian monks passed over to Lerins and Southern Gaul, whence their disciples came to England and Ireland, where they established themselves, bringing with them all their Eastern love of solitary deserts. This taste they perpetuated, as may be seen especially on the western coast of Ireland, where the ruins of extensive monastic settlements still exist, testifying to this craving. The last islands, for instance, which a traveller sees as he steams away from Cork to America, are called the Skelligs. They are ten miles west of the Kerry coast, and yet there on these rocks where a boat cannot land sometimes for months together the early monks of the fifth and six centuries established themselves as in a desert in the ocean. The topography of Ireland is full of evidences and witnesses of this desire to imitate the Apostle of the Gentiles in his Arabian retirement. There are dozens of town lands—subdivisions of the parishes—which are called deserts or diserts,[61] because they constituted solitudes set apart for hermit life after the example of St. Paul in Arabia and John the Baptist in the deserts of Judæa. While, again, when we turn northwards along the western seaboard of Ireland, we shall find numerous islands like the Skelligs, Ardoilen or the High Island, off the coast of Connemara, and Innismurry off the Sligo coast, where hermit cells in the regular Egyptian and Syrian fashion were built, and still exist as they did a thousand years ago, testifying to the longing of the human mind for such complete solitude and close communion with God as Saul enjoyed when he departed from Damascus.[62] The monks of ancient times may have run into one extreme: well would it be for us if we could avoid the other, and learn to cultivate self-communion, meditation, self- examination, and that realisation of the eternal world which God
- 63. grants to those who wait upon Him apart from the bustle and din and dust of earth, which clog the spiritual senses and dim the heavenly vision. We can see many other reasons why Paul was led into Arabia. He was led there, for instance, that he might make a thorough scrutiny of his motives. Silence, separation, solitude, have a wondrous tendency to make a man honest with himself and humbly honest before his God. Saul might have been a hypocrite or a formalist elsewhere, where human eyes and jealous glances were bent upon him, but scarcely when there alone with Jehovah in the desert. Again, Saul was led there that his soul might be ennobled and enlarged by the power of magnificent scenery, of high and hallowed associations. Mountain and cliff and flood, specially those which have been magnified and made honourable by grand memories such as must have crowded upon Saul's mind, have a marvellous effect, enlarging, widening, developing, upon a soul like Saul's, long cribbed, cabined, and confined within the rigorous bonds of Pharisaic religionism. Saul, too, was led up into those mysterious regions away from the busy life and work, the pressing calls of Damascus, that he might speak a word in season to us all, and especially to those young in the Christian life, who think in the first burst of their zeal and faith as if they had nothing to do but go in and possess the whole land. Saul did not set out at once to evangelise the masses of Damascus, or to waste the first weak beginnings of his spiritual life in striving to benefit or awaken others. He was first led away into the deserts of Arabia, in order that there he might learn of the deep things of God and of the weak things of his own nature, and then, when God had developed his spiritual strength, He led him back to Damascus that he might testify out of the fulness of a heart which knew the secrets of the Most High. The teaching of Saul's example speaks loudly to us all. It was the same with Saul as with a greater than he. The Eternal Son Himself was trained amid years and years of darkness and secrecy, and even after His baptism the day of His manifestation unto Israel was delayed yet a little. Jesus Christ was no novice when He came preaching. And Saul of Tarsus was no
- 64. novice in the Christian life when he appeared as the Christian advocate in the synagogue of Damascus. Well would it have been for many a soul had this Divine example been more closely copied. Again and again have the young and ignorant and inexperienced been encouraged to stand up as public teachers immediately after they have been seriously impressed. They have yielded to the unwise solicitation. The vanity of the human heart has seconded the foolish advice given to them, and they have tried to declare the deep things of God when as yet they have need of learning the very first principles of the doctrine of Christ. Is it any wonder that such persons oftentimes make shipwreck of faith and a sound conscience? Truth is very large and wide and spacious, and requires much time and thought if it is to be assimilated; and even when truth is grasped in all its mighty fulness, then there are spiritual enemies within and without and spiritual pitfalls to be avoided which can be known only by experience. Woe is then to that man who is not assisted by grace and guided by Divine experience, and who knows not God and the powers of the world to come, and the devious paths of his own heart, as these things can only be known and learned as Saul of Tarsus knew and learned them in the deserts of Arabia. There was marvellous wisdom contained in the brief apostolic law enacted for candidates for holy orders in words gathered from St. Paul's own personal history, "Not a novice, lest being lifted up with pride he fall into the condemnation of the devil."
- 65. CHAPTER V. THE FIRST GENTILE CONVERT. "Now there was a certain man in Cæsarea, Cornelius by name, a centurion of the band called the Italian band, a devout man, and one that feared God with all his house, who gave much alms to the people, and prayed to God alway. He saw in a vision openly, as it were about the ninth hour of the day, an angel of God coming in unto him, and saying to him, Cornelius. And he, fastening his eyes upon him, and being affrighted, said, What is it, Lord? And he said unto him, Thy prayers and thine alms are gone up for a memorial before God. And now send men to Joppa, and fetch one Simon, who is surnamed Peter: he lodgeth with one Simon a tanner, whose house is by the sea side."—Acts x. 1-6. We have now arrived at another crisis in the history of the early Church of Christ. The Day of Pentecost, the conversion of Saul of Tarsus, the call of Cornelius, and the foundation of the Gentile Church of Antioch are, if we are to pick and choose amid the events related by St. Luke, the turning-points of the earliest ecclesiastical history. The conversion of St. Paul is placed by St. Luke before the conversion of Cornelius, and is closely connected with it. Let us then inquire by what events St. Luke unites the two. German commentators of the modern school, who are nothing unless they are original, have not been willing to allow that St. Luke's narrative is continuous. They have assigned various dates to the conversion of Cornelius. Some have made it precede the conversion of St. Paul, others have fixed it to the time of Paul's sojourn in Arabia, and so on, without any other solid reasons than what their own fancies suggest. I prefer, however, to think that St. Luke's narrative follows
- 66. the great broad outlines of the Christian story, and sets forth the events of the time in a divinely ordered sequence. At any rate I prefer to follow the course of events as the narrative suggests them, till I see some good reason to think otherwise. I do not think that the mere fact that the sacred writer states events in a certain order is a sufficient reason to think that the true order must have been quite a different one. Taking them in this light they yield themselves very naturally to the work of an expositor. Let us reflect then upon that sequence as here set forth for us. Saul of Tarsus went up to Jerusalem to confer with St. Peter, who had been hitherto the leading spirit of the apostolic conclave. He laboured in Jerusalem among the Hellenistic synagogues for some fifteen days. A conspiracy was then formed against his life. The Lord, ever watchful over His chosen servant, warned him to depart from Jerusalem, indicating to him as he prayed in the Temple the scope and sphere of his future work, saying, "Depart: for I will send thee forth far hence unto the Gentiles" (see Acts xxii. 21). The Christians of Jerusalem, having learned the designs of his enemies, conveyed Saul to Cæsarea, the chief Roman port of Palestine, whence they despatched him to Cilicia, his native province, where he laboured in obscurity and quietness for some time. St. Peter may have been one of the rescue party who saved Saul from the hands of his enemies escorting him to Cæsarea, and this circumstance may have led him to the western district of the country. At any rate we find him soon after labouring in Western Palestine at some distance from Jerusalem. Philip the Evangelist had been over the same ground a short time previously, and St. Peter may have been sent forth by the mother Church to supervise his work and confer that formal imposition of hands which from the beginning has formed the completion of baptism, and seems to have been reserved to the Apostles or their immediate delegates. Peter's visit to Western Palestine, to Lydda and Sharon and Joppa, may have been just like the visit he had paid some time previously, in company with St. John, to the city of Samaria, when he came for the first time in contact with Simon Magus. St. Luke gives us here a note of time
- 67. helping us to fix approximately the date of the formal admission of Cornelius and the Gentiles into the Church. He mentions that the Churches then enjoyed peace and quietness all through Palestine, enabling St. Peter to go upon his work of preaching and supervision. It may perhaps strike some persons that this temporary peace must have been attained through the conversion of Saul, the most active persecutor. But that event had happened more than two years before, in the spring of 37 A.D., and, far from diminishing, would probably have rather intensified the hostility of the Jewish hierarchy. It was now the autumn of the year 39, and a bitter spirit still lingered at Jerusalem, as Saul himself and the whole Church had just proved. External authorities, Jewish and Roman history, here step in to illustrate and confirm the sacred narrative. The Emperor Caius Caligula, who ascended the throne of the empire about the time of Stephen's martyrdom, was a strange character. He was wholly self-willed, madly impious, utterly careless of human life, as indeed unregenerate mankind ever is. Christianity alone has taught the precious value of the individual human soul the awful importance of human life as the probation time for eternity, and has thereby ameliorated the harshness of human laws, the sternness of human rulers, ready to inflict capital punishment on any pretence whatsoever. Caligula determined to establish the worship of himself throughout the world. He had no opposition to dread from the pagans, who were ready to adopt any creed or any cult, no matter how degrading, which their rulers prescribed. Caligula knew, however, that the Jews were more obstinate, because they alone were conscious that they possessed a Divine revelation. He issued orders, therefore, to Petronius, the Roman governor of Syria, Palestine, and the East, to erect his statue in Jerusalem and to compel the Jews to offer sacrifice thereto. Josephus tells us of the opposition which the Jews offered to Caligula; how they abandoned their agricultural operations and assembled in thousands at different points, desiring Petronius to slay them at once, as they could never live if the Divine laws were so violated. The whole energies of the nation were for months concentrated on this one object, the repeal
- 68. of the impious decree of Caligula, which they at last attained through their own determination and by the intervention of Herod Agrippa, who was then at Rome.[63] It was during this awful period of uncertainty and opposition that the infant Church enjoyed a brief period of repose and quiet growth, because the whole nation from the high priest to the lowest beggar had something else to think of than how to persecute a new sect that was as yet rigorously scrupulous in observing the law of Moses. During this period of repose from persecution St. Peter made his tour of inspection "throughout all parts," Samaria, Galilee, Judæa, terminating with Lydda, where he healed, or at least prayed for the healing of, Æneas,[64] and with Joppa, where his prayer was followed by the restoration of Tabitha or Dorcas, who has given a designation now widely applied to the assistance which devout women can give to their poorer sisters in Christ. We thus see how God by the secret guidance of His Spirit, shaping his course by ways and roads known only to Himself, led St. Peter to the house of Simon the tanner, where he abode many days waiting in patience to know God's mind and will which were soon to be opened out to him. We have now traced the line of events which connect the conversion of Saul of Tarsus with that of Cornelius the centurion of Cæsarea. Let us apply ourselves to the circumstances surrounding the latter event, which is of such vital importance to us Gentile Christians as having been the formal Divine proclamation to the Church and to the world that the mystery which had been hid for ages was now made manifest, and that the Gentiles were spiritually on an equality with the Jews. The Church was now about to burst the bonds which had restrained it for five years at least. We stand by the birth of European Christendom and of modern civilisation. It is well, then, that we should learn and inwardly digest every, even the slightest, detail concerning such a transcendent and notable crisis. Let us take them briefly one by one as the sacred narrative reports them.
- 69. I. I note, then, in the first place that the time of this conversion was wisely and providentially chosen. The time was just about eight years after the Ascension and the foundation of the Church. Time enough therefore had elapsed for Christianity to take root among the Jews. This was most important. The gospel was first planted among the Jews, took form and life and shape, gained its initial impulse and direction among God's ancient people in order that the constitution, the discipline, and the worship of the Church might be framed on the ancient Jewish model and might be built up by men whose minds were cast in a conservative mould. Not that we have the old law with its wearisome and burdensome ritual perpetuated in the Christian Church. That law was a yoke too heavy for man to bear. But, then, the highest and best elements of the old Jewish system have been perpetuated in the Church. There was in Judaism by God's own appointment a public ministry, a threefold public ministry too, exercised by the high priest, the priests, and the Levites. There is in Christianity a threefold ministry exercised by bishops, presbyters or elders, and deacons.[65] There were in Judaism public and consecrated sanctuaries, fixed liturgies, public reading of God's Word, a service of choral worship, hymns of joy and thanksgiving, the sacraments of Holy Communion and baptism in a rudimentary shape; all these were transferred from the old system that was passing away into the new system that was taking its place. Had the Gentiles been admitted much earlier all this might not have so easily happened. Men do not easily change their habits. Habits, indeed, are chains which rivet themselves year by year with ever-increasing power round our natures; and the Jewish converts brought their habits of thought and worship into the Church of Christ, establishing there those institutions of prayer and worship, of sacramental communion and preaching which we still enjoy. But we must observe, on the other hand, that, had the Gentiles been admitted a little later, the Church might have assumed too Jewish and Levitical an aspect. This pause of eight years, during which Jews alone formed the Church, is another instance of those delays of the Lord[66] which, whether they happen in public or in private life, are always
- 70. found in the long run to be wise, blessed, and providential things, though for a time they may seem dark and mysterious, according to that ancient strain of the Psalmist, "Wait on the Lord, ... and He shall strengthen thine heart: wait, I say, upon the Lord."[67] II. Again, the place where the Church burst its Jewish shell and emerged into full gospel freedom is noteworthy. It was at Cæsarea. It is a great pity that people do not make more use of maps in their study of Holy Scripture. Sunday evenings are often a dull time in Christian households, and the bare mechanical reading of Scripture and of good books often only makes them duller. How much livelier, interesting, and instructive they would be were an attempt made to trace the journeys of the apostles with a map, or to study the scenes where they laboured—Jerusalem, Cæsarea, Damascus, Ephesus, Athens, and Rome—with some of the helps which modern scholarship and commercial enterprise now place within easy reach. I can speak thus with the force of personal experience, for my own keen interest in this book which I am expounding dates from the Sunday evenings of boyhood thus spent, though without many of the aids which now lie within the reach of all. This is essentially the modern method of study, especially in matters historical. A modern investigator and explorer of Bible sites and lands has well expressed this truth when he said, "Topography is the foundation of history. If we are ever to understand history, we must understand the places where that history was transacted."[68] The celebrated historians the late Mr. Freeman and Mr. Green worked a revolution in English historical methods by teaching people that an indefatigable use of maps and a careful study of the physical features of any country are absolutely needful for a true conception of its history. In this respect at least secular history and sacred history are alike. Without a careful study of the map we cannot understand God's dealings with the Church of Christ, as is manifest from the case of Cæsarea at which we have arrived. The narratives of the Gospels and of the Acts will be confused, unintelligible, unless we understand that there were two Cæsareas in Palestine, one never mentioned in the Gospels, the other never mentioned in the Acts. Cæsarea Philippi
- 71. was a celebrated city of North-eastern Palestine. It was when our Lord was within its borders that St. Peter made his celebrated confession, "Thou art the Christ, the Son of the living God," told of in St. Matthew xvi. 13-16. This is the only Cæsarea of which we hear in the Gospels. It was an inland town, built by the Herods in joint honour of themselves and of their patrons the Emperors of Rome, and bore all the traces of its origin. It was decorated with a splendid pagan temple, was a thoroughly pagan town, and was therefore abhorred by every true Jew. There was another Cæsarea, the great Roman port of Palestine and the capital, where the Roman governors resided. It was situated in the borders of Phœnicia, in a north- westerly direction from Jerusalem, with which it was connected by a fine military road.[69] This Cæsarea had been originally built by Herod the Great. He spent twelve years at this undertaking, and succeeded in making it a splendid monument of the magnificence of his conceptions. The seaboard of Palestine is totally devoid to this day of safe harbours. Herod constructed a harbour at vast expense. Let us hear the story of its foundation in the very words of the Jewish historian. Josephus tells us that Herod, observing "that Joppa and Dora are not fit for havens on account of the impetuous south winds which beat upon them, which, rolling the sands which come from the sea against the shores, do not admit of ships lying in their station; but the merchants are generally there forced to ride at their anchors in the sea itself. So Herod endeavoured to rectify this inconvenience, and laid out such a compass toward the land as might be sufficient for a haven, wherein the great ships might lie in safety; and this he effected by letting down vast stones of above fifty feet in length, not less than eighteen in breadth and nine in depth, into twenty fathoms deep."[70] The Romans, when they took possession of Palestine, adopted and developed Herod's plans, and established Cæsarea on the coast as the permanent residence of the procurator of Palestine. And it was a wise policy. The Romans, like the English, had a genius for government. They fixed their provincial capitals upon or near the sea-coast that their communications might be ever kept open. Thus in our own case Calcutta, Bombay, Madras,
- 72. Capetown, Quebec, and Dublin are all seaport towns. And so in ancient times Antioch, Alexandria, Tarsus, Ephesus, Marseilles, Corinth, London, were all seaports and provincial Roman capitals as Cæsarea was in Palestine. And it was a very wise policy. The Jews were a fierce, bold, determined people when they revolted. If the seat of Roman rule had been fixed at Jerusalem, a rebellion might completely cut off all effective relief from the besieged garrison, which would never happen at Cæsarea so long as the command of the sea was vested in the vast navies which the Roman State possessed. Cæsarea was to a large extent a Gentile city, though within some seventy miles of Jerusalem. It had a considerable Jewish population with their attendant synagogues, but the most prominent features were pagan temples, one of them serving for a lighthouse and beacon for the ships which crowded its harbour, together with a theatre and an amphitheatre, where scenes were daily enacted from which every sincere Jew must have shrunk with horror. Such was the place—a most fitting place, Gentile, pagan, idolatrous to the very core and centre—where God chose to reveal Himself as Father of the Gentiles as well as of the Jews, and showed Christ's gospel as a light to lighten the Gentiles as well as the glory of His people Israel. III. Then, again, the person chosen as the channel of this revelation is a striking character. He was "Cornelius by name, a centurion of the band called the Italian band."[71] Here, then, we note first of all that Cornelius was a Roman soldier. Let us pause and reflect upon this. In no respect does the New Testament display more clearly its Divine origin than in the manner in which it rises superior to mere provincialism. There are no narrow national prejudices about it like those which nowadays lead Englishmen to despise other nations, or those which in ancient times led a thorough-going Jew to look down with sovereign contempt on the Gentile world as mere dogs and outcasts. The New Testament taught that all men were equal and were brothers in blood, and thus laid the foundations of those modern conceptions which have well-nigh swept slavery from the face of civilised Christendom. The New Testament and its teaching is
- 73. the parent of that modern liberalism which now rules every circle, no matter what its political designation. In no respect does this universal catholic feeling of the New Testament display itself more clearly than in the pictures it presents to us of Roman military men. They are uniformly most favourable. Without one single exception the pictures drawn for us of every centurion and soldier mentioned in the books of the New Testament are bright with some element of good shining out conspicuously by way of favourable contrast, when brought side by side with the Jewish people, upon whom more abundant and more blessed privileges had been in vain lavished. Let us just note a few instances which will illustrate our view. The soldiers sought John's baptism and humbly received John's penitential advice and direction when priests and scribes rejected the Lord's messenger (Luke iii. 14). A soldier and a centurion received Christ's commendation for the exercise of a faith surpassing in its range and spiritual perception any faith which the Master had found within the bounds and limits of Israel according to the flesh. "Verily I have not found so great faith, no, not in Israel," were Christ's almost wondering words as He heard the confession of His God-like nature, His Divine power involved in the centurion's prayer of humility, "I am not worthy that Thou shouldest come under my roof: but only say the word, and my servant shall be healed" (cf. Matt. viii. 5-13). So was it again with the centurion to whom the details of our Lord's execution were committed. He too is painted in a favourable light. He had an open mind, willing to receive evidence. He received that evidence under the most unfavourable conditions. His mind was convinced of our Lord's mission and character, not by His triumphs, but by His apparent defeat. As the victim of Jewish malice and prejudice yielded up the ghost and committed His pure, unspotted soul to the hands of His heavenly Father, then it was that, struck by the supernatural spirit of love and gentleness and forgiveness—those great forces of Christianity which never at any other time or in any other age have had their full and fair play—the centurion yielded the assent of his affections and of his intellect to the Divine mission of the suffering Saviour, and cried, "Truly this man was the Son of God" (Matt. xxvii. 54). So it was again with Julius the
- 74. centurion, who courteously entreated St. Paul on his voyage as a prisoner to Rome (Acts xxvii. 3); and so again it was with Cornelius the centurion, of the band called the Italian band. Now how comes this to pass? What a striking evidence of the workings and presence of the Divine Spirit in the writers of our sacred books we may find in this fact! The Roman soldiers were of course the symbols to a patriotic Jew of a hated foreign sway, of an idolatrous jurisdiction and rule. A Jew uninfluenced by supernatural grace and unguided by Divine inspiration would never have drawn such pictures of Roman centurions as the New Testament has handed down to us. The pictures, indeed, drawn by the opposition press of any country is not generally a favourable one when dealing with the persons and officials of the dominant party. But the apostles —Jews though they were of narrow, provincial, prejudiced Galilee— had drunk deep of the spirit of the new religion. They recognised that Jesus Christ, the King of the kingdom of heaven, cared nothing about what form of government men lived under. They knew that Christ ignored all differences of climate, age, sex, nationality, or employment. They felt that the only distinctions recognised in Christ's kingdom were spiritual distinctions, and therefore they recognised the soul of goodness wherever found. They welcomed the honest and true heart, no matter beneath what skin it beat, and found therefore in many of these Roman soldiers some of the ablest, the most devoted, and the most effective servants and teachers of the Cross of Jesus Christ. Verily the universal and catholic principles of the new religion which found their first formal proclamation in the age of Cornelius met with an ample vindication and a full reward in the trophies won and the converts gained from such an unpromising source as the ranks of the Roman army. This seems to me one reason for the favourable notices of the Roman soldiers in the New Testament. The Divine Spirit wished to impress upon mankind that birth, position, or employment have no influence upon a man's state in God's sight, and to prove by a number of typical examples that spiritual conditions and excellence alone avail to find favour with the Almighty.
- 75. Another reason, however, may be found for this fact. The Scriptures never make light of discipline or training. "Train up a child in the way he should go" is a Divine precept. St. Paul, in his Pastoral Epistles, lays down as one great qualification for a bishop that he should have this power of exercising discipline and rule at home as well as abroad: "For if he knoweth not how to rule his own house, how shall he take care of the Church of God?" (1 Tim. iii. 5). By discipline, the discipline of Egypt and the wilderness, did God prepare His people for Canaan. By the discipline of captivity and dispersion, by the discipline of Greek philosophy spreading novel intellectual ideas, by the discipline of Roman dominion executing mighty public works, carrying roads and intercommunication to the remotest and most barbarous nations, did God prepare the world for the revelation of His Son. By the discipline of life, by joy and sorrow, by strife and suffering, by parting and by loss, does God still prepare His faithful ones for the beatific vision of eternal beauty, for the rest and joy of everlasting peace. And discipline worked out its usual results on these military men, even though it was only an imperfect and pagan discipline which these Roman soldiers received. Let us note carefully how this was. The world of unregenerate man at the time of our Lord's appearance had become utterly selfish. Discipline of every kind had been flung off. Self-restraint was practically unknown, and the devil and his works flourished in every circle, bringing forth the fruits of wickedness, uncleanness, and impurity in every direction. The army was the only place or region where in those times any kind of discipline or self-restraint was practised. For no army can permit—even if it be an army of atheists—profligacy and drunkenness to rage, flaunting themselves beneath the very eye of the sun. And as the spiritual result we find that this small measure of pagan discipline acted as a preparation for Christianity, and became under the Divine guidance the means of fitting men like Cornelius of Cæsarea for the reception of the gospel message of purity and peace.[72] But we observe that Cornelius the centurion had one special feature which made him peculiarly fitted to be God's instrument for opening
- 76. the Christian faith to the Gentile world. The choice of Cornelius is marked by all that skill and prudence, that careful adaptation of means to ends which the Divine workmanship, whether in nature or in grace, ever displays. There were many Roman centurions stationed at Cæsarea, yet none was chosen save Cornelius, and that because he was "a devout man who feared God with all his house, praying to God always, and giving much alms to the people." He feared Jehovah, he fasted, prayed, observed Jewish hours of devotion. His habits were much more those of a devout Jew than of a pagan soldier. He was popular with the Jewish people therefore, like another centurion of whom it was said by the Jewish officials themselves "he loveth our nation and hath built us a synagogue." The selection of Cornelius as the leader and firstfruits of the Gentiles unto God was eminently prudent and wise. God when He is working out His plans chooses His instruments carefully and skilfully. He leaves nothing to chance. He does nothing imperfectly. Work done by God will repay the keenest scrutiny, the closest study, for it is the model of what every man's work in life ought as far as possible to be —earnest, wise, complete, perfect. IV. Again, looking at the whole passage we perceive therein illustrations of two important laws of the Divine life. We recognise in the case of Cornelius the working of that great principle of the kingdom of God often enunciated by the great Master: "To him that hath shall be given, and he shall have more abundantly," "If any man will do His will, he shall know of the doctrine"; or, to put it in other language, that God always bestows more grace upon the man who diligently uses and improves the grace which he already possesses; a principle which indeed we see constantly exemplified in things pertaining to this world as well as in matters belonging to the spiritual life. Thus it was with Cornelius. He was what was called among the Jews a proselyte of the gate. These proselytes were very numerous. They were a kind of fringe hanging upon the outskirts of the Jewish people. They were admirers of Jewish ideas, doctrines, and practices, but they were not incorporated with the Jewish nation nor bound by all their laws and ceremonial restraints. The Levitical
- 77. Law was not imposed upon them because they were not circumcised. They were merely bound to worship the true God and observe certain moral precepts said to have been delivered to Noah. [73] Such was Cornelius whom the providence of God had led from Italy to Cæsarea for this very purpose, to fulfil His purposes of mercy towards the Gentile world. His residence there had taught him the truth and beauty of the pure worship of Jehovah rendered by the Jews. He had learned too, not only that God is, but that He is a rewarder of them that diligently seek Him. Cornelius had set himself, therefore, to the diligent discharge of all the duties of religion so far as he knew them. He was earnest and diligent in prayer, for he recognised himself as dependent upon an invisible God. He was liberal in alms, for he desired to show forth his gratitude, for mercies daily received. And acting thus he met with the divinely appointed reward. Cornelius is favoured with a fuller revelation and a clearer guidance by the angel's mouth, who tells him to send and summon Peter from Joppa for this very purpose. What an eminently practical lesson we may learn from God's dealings with this earliest Gentile convert! We learn from the Divine dealings with Cornelius that whosoever diligently improves the lower spiritual advantages which he possesses shall soon be admitted to higher and fuller blessings. It may well have been that God led him through successive stages and rewarded him under each. In distant Italy, when residing amid the abounding superstitions of that country, conscience was the only preacher, but there the sermons of that monitor were heard with reverence and obeyed with diligence. Then God ordered the course of his life so that public duty summoned him to a distant land. Cornelius may have at the time counted his lot a hard one when despatched to Palestine as a centurion, for it was a province where, from the nature of the warfare there prevalent, there were abundant opportunities of death by assassination at the hands of the Zealots, and but few opportunities of distinction such as might be gained in border warfare with foreign enemies. But the Lord was shaping his career, as He shapes all our careers, with reference to our highest spiritual purposes. He led Cornelius, therefore, to a land and to a
- 78. town where the pure worship of Jehovah was practised and the elevated morality of Judaism prevailed. Here, then, were new opportunities placed within the centurion's reach. And again the same spiritual diligence is displayed, and again the same law of spiritual development and enlarging blessing finds a place. Cornelius is devout and liberal and God-fearing, and therefore a heavenly visitor directs his way to still fuller light and grander revelations, and Cornelius the centurion of the Italian band leads the Gentile hosts into the fulness of blessing, the true land flowing with milk and honey, found only in the dispensation of Jesus Christ and within the borders of the Church of God. This was God's course of dealing with the Roman centurion, and it is the course which the same loving dealings still pursues with human souls truly desirous of Divine guidance. The Lord imparts one degree of light and knowledge and grace, but withholds higher degrees till full use has been made of the lower. He speaks to us at first in a whisper; but if we reverently hearken, there is a gradual deepening of the voice, till it is as audible in the crowd as it is in the solitude, and we are continually visited with the messages of the Eternal King. Now cannot these ideas be easily applied to our own individual cases? A young man, for instance, may be troubled with doubts and questions concerning certain portions of the Christian faith. Some persons make such doubts an excuse for plunging into scenes of riot and dissipation, quenching the light which God has given them and making certain their own spiritual destruction. The case of Cornelius points out the true course which should in such a case be adopted. Men may be troubled with doubts concerning certain doctrines of revelation. But they have no doubt as to the dictates of conscience and the light which natural religion sheds upon the paths of morals and of life. Let them then use the light they have. Let them diligently practise the will of God as it has been revealed. Let them be earnest in prayer, pure and reverent in life, honest and upright in business, and then in God's own time the doubts will vanish, the darkness will clear away, and the ancient promises will be fulfilled, "Light is sown for the righteous," "The path of the just shineth more and more unto
- 79. the perfect day," "In the way of righteousness is life, and in the pathway thereof there is no death." But the example of Cornelius is of still wider application. The position of Cornelius was not a favourable one for the development of the religious life, and yet he rose superior to all its difficulties, and became thus an eminent example to all believers. Men may complain that they have but few spiritual advantages, and that their station in life is thickly strewn with difficulties, hindering the practices and duties of religion. To such persons we would say, compare yourselves with Cornelius and the difficulties external and internal he had to overcome. Servants, for instance, may labour under great apparent disadvantages. Perhaps, if living in an irreligious family, they have few opportunities for prayer, public or private. Men of business are compelled to spend days and nights in the management of their affairs. Persons of commanding intellect or of high station have their own disadvantages, their own peculiar temptations, growing out of their very prosperity. The case of Cornelius shows that each class can rise superior to their peculiar difficulties and grow in the hidden life of the soul, if they but imitate his example as he grew from grace to grace, improving his scanty store till it grew into a fuller and ampler one, till it expanded into all the glory of Christian privilege, when Cornelius, like Peter, was enabled to rejoice in the knowledge and love of a risen and glorified Redeemer.[74]
- 80. CHAPTER VI. THE PETRINE VISION AT JOPPA. "Now on the morrow, as they were on their journey, and drew nigh unto the city, Peter went up upon the housetop to pray, about the sixth hour: and he became hungry, and desired to eat: but while they made ready, he fell into a trance; and he beholdeth the heaven opened, and a certain vessel descending, as it were a great sheet, let down by four corners upon the earth: wherein were all manner of fourfooted beasts and creeping things of the earth and fowls of the heaven. And there came a voice to him, Rise, Peter; kill and eat. But Peter said, Not so, Lord; for I have never eaten anything that is common and unclean. And a voice came unto him again the second time, What God hath cleansed, make not thou common."—Acts x. 9- 15. There are two central figures in the conversion of Cornelius. The one is the centurion himself, the other is St. Peter, the selected and predestined agent in that great work. We have studied Cornelius in the last chapter, and have seen the typical character of all his circumstances. His time, his residence, his training, had all been providential, indicating to us the careful superintendence, the watchful oversight, which God bestows upon the history of individuals as well as of the Church at large. Let us now turn to the other figure, St. Peter, and see if the Lord's providence may not be traced with equal clearness in the circumstances of his case also. We have found Cornelius at Cæsarea, the great Roman port and garrison of Palestine, a very fitting and natural place for a Roman centurion to be located. We find Peter at this very same time at Joppa, a spot that was consecrated by many a memory and specially
- 81. associated with a mission to the Gentiles in the times of the Elder Dispensation. Here we trace the hand of the Lord providentially ruling the footsteps of Peter though he knew it not, and leading him, as Philip was led a short time before, to the spot where his intended work lay. The sickness and death of Tabitha or Dorcas led St. Peter to Joppa. The fame of his miracle upon that devout woman led to the conversion of many souls, and this naturally induced Peter to make a longer stay in Joppa at the house of Simon the tanner. How natural and unpremeditated, how very ordinary and unplanned to the natural eye seem the movements of St. Peter! So they would have seemed to us had we been living at Joppa, and yet now we can see with the light which the sacred narrative throws upon the story that the Lord was guiding St. Peter to the place where his work was cut out when the appointed time should come. Surely the history of Peter and his actions have abundant comfort and sustaining hope for ourselves! Our lives may be very ordinary and commonplace; the events may succeed one another in the most matter-of-fact style; there may seem in them nothing at all worthy the attention of a Divine Ruler; and yet those ordinary lives are just as much planned and guided by supernatural wisdom as the careers of men concerning whom all the world is talking. Only let us take care to follow St. Peter's example. He yielded himself completely to the Divine guidance, trusted himself entirely to Divine love and wisdom, and then found in such trust not only life and safety, but what is far better, perfect peace and sweetest calm. There is something very restful in the picture drawn for us of St. Peter at this crisis. There is none of that feverish hurry and restlessness which make some good men and their methods very trying to others. The notices of him have all an air of repose and Christian dignity. "As Peter went throughout all parts, he came down also to the saints which dwelt at Lydda"; "Peter put them all forth and prayed"; "Peter abode many days in Joppa"; "Peter went up upon the housetop to pray about the sixth hour." St. Peter, indeed, did not live in an age of telegrams and postcards and express trains, which all contribute more or less to that feverish activity and
- 82. restlessness so characteristic of this age. But even if he had lived in such a time, I am sure his faith in God would have saved him from that fussiness, that life of perpetual hurry, yet never bringing forth any abiding fruit, which we behold in so many moderns. This results a good deal, I believe, from the development—I was almost going to say the tyranny, the unwitting tyranny of modern journalism, which compels men to live so much in public and reports their every utterance. There are men never tired of running from one committee to another, and never weary of seeing their names in the morning papers. They count that they have been busily and usefully employed if their names are perpetually appearing in newspaper reports as speaking, or at any rate being present at innumerable meetings, leaving themselves no time for that quiet meditation whereby St. Peter gained closest communion with heaven. It is no wonder such men's fussiness should be fruitless, because their natures are poor, shallow, uncultivated, where the seed springs up rapidly but brings forth no fruit to perfection, because it has no deepness of earth. It is no wonder that St. Peter should have spoken with power at Cæsarea and been successful in opening the door of faith to the Gentiles, because he prepared himself for doing the Divine work by the discipline of meditation and thought and spiritual converse with his Risen Lord. And here we may remark, before we pass from this point, that the conversion of the first Gentile and the full and complete exercise of the power of the keys committed to St. Peter run on lines very parallel to those pertaining to the Day of Pentecost and the conversion of the earliest Jews in one respect at least. The Day of Pentecost was preceded by a period of ten days' waiting and spiritual repose. The conversion of Cornelius and the revelation of God's purposes to St. Peter were preceded by a season of meditation and prayer, when an apostle could find time amid all his pressing cares to seek the housetop for midday prayer and to abide many days in the house of one Simon a tanner. A period of pause, repose, and quietness preceded a new onward movement of development and of action.
- 83. I. Now, as in the case of Cornelius, so in the case of St. Peter, we note the place where the chief actor in the scene abode. It was at Joppa, and Joppa was associated with many memories for the Jews. It has been from ancient times the port of Jerusalem, and is even now rising into somewhat of its former commercial greatness, specially owing to the late development of the orange trade, for the production of which fruit Jaffa or Joppa has become famous. Three thousand years ago Joppa was a favourite resort of the Phœnician fleets, which brought the cedars of Lebanon to King Solomon for the building of the temple (2 Chron. ii. 16). At a later period, when God would send Jonah on a mission to Gentile Nineveh, and when Jonah desired to thwart God's merciful designs towards the outer world, the prophet fled to Joppa and there took ship in his vain effort to escape from the presence of the Lord. And now again Joppa becomes the refuge of another prophet, who feels the same natural hesitation about admitting the Gentiles to God's mercy, but who, unlike Jonah, yields immediate assent to the heavenly message, and finds peace and blessing in the paths of loving obedience. The very house where St. Peter abode is still pointed out.[75] It is situated in the south-western part of the town, and commands a view over the bay of Joppa and the waters of that Mediterranean Sea which was soon to be the channel of communication whereby the gospel message should be borne to the nations of the distant West. We remark, too, that it was with Simon the tanner of Joppa that St. Peter was staying. When a great change is impending various little circumstances occur all showing the tendencies of the age. By themselves and taken one by one they do not express much. At the time when they happen men do not regard them or understand their meaning, but afterwards, and reading them in the light of accomplished facts, men behold their significance. Thus it was with Simon Peter and his visit to Simon the tanner of Joppa. Tanners as a class were despised and comparatively outcast among the Jews. Tanning was counted an unclean trade because of the necessary contact with dead bodies which it involved. A tanyard must, according to Jewish law, be separated by fifty yards at least from
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