modulation transfer function (MTF)

Exposure Systems
Contact Proximity Projection

AJAL.A.J
Introduction: Historical and cultural relevance of MTS
Motivations: Why MTF supported also by incompetents (like me)
Technical issues: How do we achieve MTF and analyse its data (a few
examples for MTF)
Scientific rationale: What can we do reasonably and first with MTF ?
SLIDES PREPARED FOR S8 ECE . METS SCHOOL OF ENGINEERING , MALA THRISSUR, KERALA , INDIA ON 10-12-09
modulation transfer function (MTF)

Instrument properties: (1) Linearity
• Linearity tested using high-quality test target
Set of squares of precisely calibrated optical absorbance
Note that the measured absorbance is greater than the real one.
This is due to the extra absorbing elements through which the beam passes
between the reference and imaging detectors.
Measured absorbance
Realabsorbance
Line of
unity

Instrument properties: (2) MTF
• MTF measured using alternating bands .
MTF limited by diameter of laser beam.
It is the beam size as it passes through the sample that matters:
Laser scanner design needs to be modified if resolution is to approach that of
pixelated (CCD) scanners.
0.5 mm resolution

Instrument properties: (3) SNR
• Experimental test using different dye concentrations
Projection SNR defined by A / σ(A): direct measurement for squares on HQTT
Note that SNR → 0 as A → 0
Reconstruction SNR calculated via measurement of uniform dye in matching
tank and then using data to simulate a scanned cylinder.
Peak SNR of almost 400 ⇒ possibility of dose precision to < 0.5% !

Instrument properties: (4) Speed
• To date
Acquisition rate up to 65536 samples / s demonstrated.
Equivalent to a 256 × 256 projection image in 1 s.
Gives 2563
3-D image in 400 s ≈ 7 mins. At full Nyquist sampling
Images shown are acquired at around 20 mins. for a 2563
3-D image
This is comparable with the best performance of our CCD scanner
• The future ...
This architecture is very similar to that used in 3-D confocal microscopy.
In that field images are acquired at video frame rates.
We confidently expect 3-D scans in a minute or less.

DEFINE MTF ?
• The resolution and performance of an optical
microscope can be characterized by a quantity
known as the modulation transfer function (MTF),
which is a measurement of the microscope's ability
to transfer contrast from the specimen to the
intermediate image plane at a specific resolution.
• Computation of the modulation transfer function is
a mechanism that is often utilized by optical
manufacturers to incorporate resolution and
contrast data into a single specification.

• The modulation transfer function (MTF) indicates the ability of an
optical system to reproduce (transfer) various levels of detail (spatial
frequencies) from the object to the image.
• Its units are the ratio of image contrast over the object contrast as a
function of spatial frequency.
• It is the optical contribution to the contrast sensitivity function (CSF).

MTF: Cutoff FrequencyMTF: Cutoff Frequency
0
0.5
1
0 50 100 150 200 250 300
1 mm
2 mm
4 mm
6 mm
8 mm
modulationtransfer
spatial frequency (c/deg)
cut-off frequency
57.3
cutoff
a
f
λ
=
⋅
Rule of thumb: cutoff
frequency increases by
~30 c/d for each mm
increase in pupil size

Another useful concept is the
modulation transfer function or
MTF, defined as shown below
MTF is the ratio between image
intensity modulation over the
object intensity modulation
This parameter qualifies the
capability of an optical system
Photolithography- MTF

Function describes
contrast as a function of
size of features on the
mask
Generally, MTF needs to
be > 0.5 for the resist to
resolve features
Photolithography- MTF

Modulation transfer function
i(x,y)=o(x,y)⊗PSF(x,y) +noise
c c c c
I(kx,ky)=O(kx,ky)⋅MTF(kx,ky)+ℑnoise{ }

Information Processing by Human
Observer
• Visual perception
– Concerns how an image is perceived by a human
observer
• preliminary processing by eye  this lecture
• further processing by brains
– Important for developing image fidelity measures
• needed for design and evaluate DIP/DVP algorithms &
systems
imageimage eyeeye perceived imageperceived image
understanding of content

The Eye
– Cross section illustration
Figure is from slides at
Gonzalez/ Woods DIP book
website (Chapter 2)

Two Types of Photoreceptors at
Retina
• Rods
– Long and thin
– Large quantity (~ 100 million)
– Provide scotopic vision (i.e., dim light vision or at low illumination)
– Only extract luminance information and provide a general overall
picture
• Cones
– Short and thick, densely packed in fovea (center of retina)
– Much fewer (~ 6.5 million) and less sensitive to light than rods
– Provide photopic vision (i.e., bright light vision or at high illumination)
– Help resolve fine details as each cone is connected to its own nerve end
– Responsible for color vision
our interest
(well-lighted display)

Light
• Light is an electromagnetic wave
– with wavelength of 350nm to 780nm stimulating human visual response
• Expressed as spectral energy distribution I(λ)
– The range of light intensity levels that human visual system can adapt is
huge: ~ on 10 orders of magnitude (1010
) but not simultaneously

Luminance vs. Brightness
• Luminance (or intensity)
– Independent of the luminance of surroundings
I(x,y,λ) -- spatial light distribution
V(λ) -- relative luminous efficiency func. of visual system ~ bell shape
(different for scotopic vs. photopic vision;
highest for green wavelength, second for red, and least for blue )
• Brightness
– Perceived luminance
– Depends on surrounding luminance
Same lum.
Different
brightness
Different lum.
Similar
brightness

Luminance vs. Brightness
• Example: visible digital watermark
– How to make the watermark
appears the same gray level
all over the image?

Testing MethodologyTesting Methodology
 Study Objective
• To compare the image quality of different designs using
the modulation transfer function (MTF) testing method
 Modulation Transfer Function (MTF) Testing
• Objective method of measuring image contrast
degradation at different spatial frequencies
 Study Objective
• To compare the image quality of different designs using
the modulation transfer function (MTF) testing method
 Modulation Transfer Function (MTF) Testing
• Objective method of measuring image contrast
degradation at different spatial frequencies

The optical bench setup
Pinhole Target
IOL
Light
Source
CCD Camera
MTF measurement systemMTF measurement systemMTF measurement systemMTF measurement system

paradigmsparadigms
 The IQ ReSTORThe IQ ReSTOR®®
SN6AD3 asphericSN6AD3 aspheric
intraocular lensintraocular lens (IOL)(IOL) produced the highestproduced the highest
overall image quality for MTFoverall image quality for MTF
 Clinical investigation is needed to determine whetherClinical investigation is needed to determine whether
superior IOL optical quality demonstrated in opticalsuperior IOL optical quality demonstrated in optical
bench testing results in measurable visualbench testing results in measurable visual
improvements in clinical practiceimprovements in clinical practice
 The IQ ReSTORThe IQ ReSTOR®®
SN6AD3 asphericSN6AD3 aspheric
intraocular lensintraocular lens (IOL)(IOL) produced the highestproduced the highest
overall image quality for MTFoverall image quality for MTF
 Clinical investigation is needed to determine whetherClinical investigation is needed to determine whether
superior IOL optical quality demonstrated in opticalsuperior IOL optical quality demonstrated in optical
bench testing results in measurable visualbench testing results in measurable visual
improvements in clinical practiceimprovements in clinical practice

Image quality
• Spatial resolution can be best described by
• The limiting resolution of an imaging
system is where the MTF approaches zero
• Higher magnification modes (smaller fields
of view) are capable of better resolution
• Video imaging system degrades the MTF
substantially

Color of LightColor of Light
 Perceived color depends on spectral content
(wavelength composition)
– e.g., 700nm ~ red.
– “spectral color”
A light with very narrow bandwidth
 A light with equal energy in all visible bands
appears white
“Spectrum” from http://www.physics.sfasu.edu/astro/color.html

Perceptual Attributes of ColorPerceptual Attributes of Color
 Value of Brightness
(perceived luminance)
 Chrominance
– Hue
specify color tone (redness, greenness,
etc.)
depend on peak wavelength
– Saturation
describe how pure the color is
depend on the spread (bandwidth) of
light spectrum
reflect how much white light is added
 RGB  HSV Conversion ~ nonlinear
HSV circular cone is from
online documentation of Matlab
image processing toolbox
http://www.mathworks.com/acc
ess/helpdesk/help/toolbox/imag
es/color10.shtml

• Any color can be
reproduced by mixing an
appropriate set of three
primary colors
(Thomas Young, 1802)
Representation by Three Primary Colors

Example: Seeing Yellow
Without Yellow
mix green and red light to obtain perception of yellow, without
shining a single yellow photon
520nm 630nm
570nm
=

Fourier Transform
The co-ordinate (ω) in Fourier space is often
referred to as spatial frequency or just
frequency

Graphical Representation Of The Fourier
Transform

Transfer Functions
• In Fourier Space this representation is simplified
)()()( sTsXsX inputoutput ⋅=
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20 25
x =
0
0.5
1
1.5
2
2.5
3
0 5 10 15 20 25
0
0.5
1
1.5
2
2.5
3
0 5 10 15 20 25

Point Spread Function (PSF)
• The blurring of an imaginary point as it passes
through an optical system
• Convolution of the input function with a
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20 25 0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20 25

∫ ∫
∫ ∫
∞
∞−
∞
∞−
+−
=
dudvvuh
dudvvfufjvuh
ff
YX
YX 2
2
),(
)](2exp[),(
),(
π
H
By international agreement, the function is known as the optical
transfer function (OTF) of the system and it is also the normalized
autocorrelation of the amplitude transfer function.
Its modulus H is known as the modulation transfer function (MTF).
)(
)(
PTFi
e
MTFOTF =
Where PTF is phase transfer function.

• MTF is the Fourier Transform Of the PSF
• MTF is a Transfer Function
CONCLUSION
)(
)(
PTFi
e
MTFOTF =

Modulation Transfer Function (MTF)
• A representation of the point spread function
in Fourier space
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20 25
x =
0
0.5
1
1.5
2
2.5
3
0 5 10 15 20 25
0
0.5
1
1.5
2
2.5
3
0 5 10 15 20 25

42
)},({)},({)},({
),(),(
),(),(),(
0
0
0
zySFzyIFZYIF
zySzyI
dydzzyIzZyYSZYI
i
i
⋅=
⊗=
−−= ∫ ∫
+∞
∞−
+∞
∞−
[ ]),(exp),(
),()},({
ZYZY
ZY
kkikkM
kkTzySF
Φ=
=
Transfer functions:
Optical transfer function T (OTF)
Modulation transfer function M (MTF)
Phase transfer function Φ (PTF)
Frequency convolution theorem:
}{}{
2
1
}{ hFfFhfF ⊗=⋅
π
Example: Transform of a Gaussian wave packet.
akkxikax
ekFee
a
xE 4/)( 2
00
2
)(,)( −−−−
=×=
π
Please prove
it.

MTF Definition
• MTF is a measure of intensity contrast transfer per unit
resolution of an image or signal.
• It is used in optics, electronics, and related signal
processing applications.

Imaging Task
As spatial separation
decreases, the “good” system
maintains clear separation of
point source images, while the
“poor” system eventually can
no longer distinguish them.
MTF quantifies this
phenomenon in terms of
contrast between the center
peak intensities versus
intensity at their midpoint
across a scale of separation
distances.
At large separations, even a poor system
can completely resolve the two images. As
separation decreases, only the good
systems can still recognize separate
sources.
Good Poor

Contrast Modulation: A Basic
MTF
Contrast Modulation is defined simply by
averaging the difference of maximum and
minimum transmitted intensities:
“Spatial Frequency” typically implies an array of
sine or bar targets at a given spacing,
expressed in line-pairs-per-millimeter (lp/mm)
or cylces -per-milliradian (cy/mrad)
Contrast
I I
I I
=
−
+
max min
max min
0 200 400 600 800 1000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
One line pair
Imax
Imin
Original
Signal
Output
Signal

Nyquist Sampling Theory
Nyquist Theory: In order to achieve perfect
reconstruction of an input signal which has a maximum
spatial frequency "f" (the cutoff), sampling must occur at
a rate of at least "2f". (Note: Phase is still an issue!)
Sample Intervals in Phase
InputWaveformSampledOutput
Samples Out of Phase
InputWaveformSampledOutput

Optical MTF
Imaging optical systems
perform sampling, with the
maximum sample frequency
determined by the “spot size”
image of a perfect point source
object (e.g., “Impulse
Response”).
A “perfect” optical system is
limited in resolution by
wavelength (λ) dependent
diffraction effects. Lens
aberrations can only worsen
performance.
The MTF of an optical system
is found by Fourier operations
on the “spot size”, or Point
Spread Function.
ImageObject

System MTF
The MTF of cascaded optical assemblies is NOT
equal to the product of component MTF’s!
Why? Lenses transmit not just intensity, but
wavefront phase as well, and hence aberrations in
one lens can cancel those in another.
MTF of cascaded objective lenses, detector, and
displays may be multiplied for composite “System
MTF”, with a component MTF measured at each
intensity transfer point.
Focal Plane Image Focal Plane Image
MTFsystem = MTFobjective*MTFdetector*MTFdisplay*MTFeyepiece*MTFeye
MTFsystem ≠ MTFLens1*MTFLens2

System MTF
Example System MTF calculation:
Freq. ObjLens FPA Display Eyepiece System
cy/mrad MTF MTF MTF MTF MTF
1.0 0.9999 0.9999 0.9999 0.9999 0.99
2.0 0.90 0.99 0.99 0.90 0.79
3.0 0.85 0.98 0.92 0.88 0.67
For average human observers, MTF values around 0.05 are considered barely
resolvable. If the above system MTF reached 0.05 at 10 cy/mrad, for example, then
you can predict that a human observer could identify (6 cylce criteria) a 2.4 meter taget
through this sensor at a maximum range (this is a coarse estimate!) of about
Range m
m cycles
mrad
km( )
[2. ] [6 ]
tan[ / ]
≈
÷
=
4
1 10
4

Measurement: Knife Edge
Knife Edge Measure of the Line Spread Function
(LSF):
1. Drag a knife-edge across the focal
plane of the optic to be tested and record
the intensity
2. Calculating the derivative of this data
gives us the LSF we are looking for so we
can continue with MTF.
Edge
Response
Line Spread
Differe
-ntiate

Measurements of Image Quality
• PSF = Point Spread Function
• LSF = Line Spread Function
• CTF = Contrast Transfer Function
• MTF = Modulation Traffic Function

Point Spread Function
PSF
• “Point” object imaged as circle due to
blurring
• Causes
– finite focal spot size
– finite detector size
– finite matrix size
– Finite separation between object and detector
• Ideally zero
– Finite distance to focal spot
• Ideally infinite

Quantifying Blurring
• Object point becomes image circle
• Difficult to quantify total image circle size
– difficult to identify beginning & end of object
Intensity
?

Quantifying Blurring
Full Width at Half Maximum (FWHM)
• width of point spread
function at half its
maximum value
• Maximum value easy
to identify
• Half maximum value
easy to identify
• Easy to quantify width
at half maximum
FWHM
Maximum
Half
Maximum

Line Spread Function
LSF
• Line object image blurred
• Image width larger than object width
Intensity
?

Contrast Response Function
CTF or CRF
• Measures contrast response of imaging
system as function of spatial frequency
Lower
Frequency
Higher
Frequency
Loss of contrast between light and dark areas as
bars & spaces get narrower. Bars & spaces blur into
one another.

Contrast Response Function
CTF or CRF
• Blurring causes loss of contrast
– darks get lighter
– lights get darker
Lower
Frequency
Higher
Frequency
Higher
Contrast
Lower
Contrast

Modulation Transfer Function
MTF
• Fraction of contrast reproduced as a function
of frequency
Recorded
Contrast
(reduced by blur)
frequency
MTF
1
0
Contrast provided
to film
Freq. =
line pairs / cm
50%

MTF
• Can be derived from
– point spread function
– line spread function
• MTF = 1 means
– all contrast reproduced at this frequency
• MTF = 0 means
– no contrast reproduced at this frequency

MTF
• If MTF = 1
– all contrast reproduced at this frequency
Recorded
Contrast
Contrast provided
to film

MTF
• If MTF = 0.5
– half of contrast reproduced at this frequency
Recorded
Contrast
Contrast provided
to film

MTF
• If MTF = 0
– no contrast reproduced at this frequency
Recorded
Contrast
Contrast provided
to film

Modulation Transfer Function (MTF)
MTF = Imax- Imin
Imax + Imin
-MTF is a measure of the contrast of
an aerial pattern,
-For well-separated images,
MTF ~ 1,
-For smaller images, MTF<1
-In general, MTF should be >0.5.
Intensity
Intensity
DisplacementDisplacement

Component MTF
• Each component in an imaging system has its own
MTF
– each component retains a fraction of contrast as function
of frequency
• System MTF is product of MTF’s for each
component.
• Since MTF is between 0 and 1,
• composite MTF <= MTF of poorest component

Modulation Transfer Function
• MTF is a measure of an imaging system’s
ability to recreate the spatial frequency
content of scene
MTF is the magnitude of the
Fourier Transform of
the Point Spread Function / Line
Spread Function.
1.0
Cut-off
Spatial frequency

3 steps for
MTF Measurement of IJ Printer
MTF of an IJ printer
1. Print a test target page 2. Scan the printed target
3. Analyze scanned data
Test Target Page

MTS : for noise reduction example
Low frequency noise artifacts are visible in the
sky, which has been enhanced (darkened and
boosted in contrast) for aesthetic purposes.
This is clear evidence of noise reduction– in a
camera has an excellent reputation for low
noise. The CMOS sensor evidently requires
noise reduction.

MTF measurement methods:
1] Point source METHOD
Xe lamp: 3kW Xe lamp: 1kW

MTF measurement methods: step edge
Step edge method
– Image of a target (artificial or natural) with a sharp transition
between dark and bright area
– With a slight edge inclination, we can interleave successive rows
(or columns) to rebuild a sufficiently sampled response to
Heaviside function
• Again, this is not necessary with THR mode
– Modulus of ratio of FT (edge response) to FT (edge) = in-flight
MTF
Two kinds of edge
– Natural edge: agricultural fields
• Difficulty to find a good one and to validate it
– Artificial edge
• A checkerboard target has been laid out (Salon-de-Provence in south of
France)
• 60 x 60 m

Method Description
• Edge Method (MTF estimation method)
– Sub-pixel edge locations were found by Fermi function fit.
– A least-square error line was calculated through the edge locations.
– Savitzky-Golay Helder-Choi filtering was applied on each line
– The filtered profile was differentiated to obtain LSF
– MTF calculated by applying Fourier transform to LSF.
Fig 1. Edge Method

• Pulse method
– A pulse input is given to an imaging system.
– Output of the system is the resulting image.
– Edge detection and SGHC filtering was applied to get
output profile.
– Take Fourier transform of the input and output.
– MTF is calculated by dividing output by input.
Figure 2. Pulse method
10 12 14 16 18 20 22 24
2
4
6
8
10
12
14
Edge detection
Pixels
Pixels
Curve inflection point
Least square fit line

MTF measurement methods: Bi-resolution
Principle
– Same landscape acquired with two spatial resolutions (same
spectral band)
• High resolution image = reference
• Low resolution image = sensor under assessment
– In-flight MTF = Modulus of ratio of FT (LR image) to FT (HR
image)
Two situations
– Satellite image versus aerial image
• Attempt with SPOT4 HRVIR
– Both sensors on the same satellite
• Attempt with SPOT4: VGT1 versus HRVIR

MTF measurement methods: Periodic target
Opportunity to acquire Center radial target
THRESHOLD (5m) THRESHOLD (2.5m)

APPLICATIONSAPPLICATIONS
The MTF curves are used to characterize photographicThe MTF curves are used to characterize photographic
objectives and are determined by the comparison of inputobjectives and are determined by the comparison of input
images with their photographic reproductions.images with their photographic reproductions.
With this study the procedure is also extended to casesWith this study the procedure is also extended to cases
where the characterization of the acquisition system is notwhere the characterization of the acquisition system is not
possible.possible.
?ACQUISITION
SYSTEM
OUTPUT
IMAGE
INPUT
IMAGE
?
MTF(f)
ω, f
ACQUISITION
SYSTEM
OUTPUT
IMAGE
INPUT
IMAGE
MTF(f)
ω, f

AIM:AIM:
extension of the traditional MTF technique to evaluate theextension of the traditional MTF technique to evaluate the
debated resolution of the body image of the Turin Shroud.debated resolution of the body image of the Turin Shroud.

"“Relic certainly it is…”"“Relic certainly it is…” (John Paul II, April 28(John Paul II, April 28thth
1989)1989)
““The Shroud is provocation to the intelligence …The Shroud is provocation to the intelligence …
The Church submits to the scientists the assignment toThe Church submits to the scientists the assignment to
keep on investigating”keep on investigating” (John Paul II, Turin 1998(John Paul II, Turin 1998

Purdue UniversityHP-Purdue Confidential 77
MTF (Modulation Transfer Function)MTF (Modulation Transfer Function)
• The MTF of an imaging system
 The MTF is the magnitude of the OTF (optical transfer function) as
.
)0(
)(
)()(
H
fH
fOTFfMTF ==
 If the input signal x(·) is an impulse, then the MTF can be obtained by
calculating |Y(f)| / |Y(0)|, where Y(f) is the impulse response of the system.
 The MTF describes how much the system attenuates the input modulation
signal as a function of frequency.
• MTF measurement by using a set of sinusoidal signals
 Measure the output response for a set of sinusoidal signals.
 Extract the output magnitude at the given frequency for each sinusoidal signal.
 Normalize the extracted output magnitudes and estimate the MTF curve.

Questions & Contacts
AJAL.A.J --- 4u.ajal@gmail.com
Phone: (9633 – 910 911)
www.ajal4u.0catch.com
http://www.metsengg.org/dept_home.php?
dept_name=ece&opt=faculty

References
• Modulation Transfer Function (MTF)
– Implement and compare different methods for
calculating/measuring the MTF of an imaging lens
– Reference:
• Backmann et al., “Random target method for fast MTF inspection,”
Optics Express, vol. 12, no. 12, pp. 2610 (2004)
• MTF with defocus
– Implement and evaluate the MTF for a defocusing error
– Reference:
• C. S. Williams, O.A. Becklund, “Introduction to the Optical Transfer
Function,” (1989)

Thanks so much. It has been a
pleasure.
Enjoy the meeting!
AJAL.AJ, ME .
Assistant Professor
METS School of Engineering
University of Calicut
Mala , Thrissur ( Dt )
Email: 4u.ajal@gmail.com

modulation transfer function (MTF)

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