![Preface to the Classics Edition
We are pleased that SIAM isrepublishing this book on the principlesof
computerized tomography. We enjoyed writingit and have heard from many
people who found the book valuable. We're glad that the book isnow back
in print.
The worlds of tomography and medical imaging have not stood still in
the twelve years since this book was first published, yet the basic algorithms
presented in this book are just as important today as when they were first
applied to tomographic imaging. The reconstruction formalism that under-
lies these algorithms helps us create efficient implementations for medical
scanners and provides us with a theoretical framework to think about issues
such as noise, sampling, resolution and artifacts. We hope that the new
readers of our book will also appreciate the beauty of these algorithms.
The primary strength of this book consists of derivations that are
appropriate for engineers and scientists who want to understand the princi-
ples of computerized tomographic imaging. We hope that these derivations
will provide readers with insights to create their own implementations. For
our readers more interested in the medical applications of computerized
tomography we suggest they also consult Lee's [Lee98] or Haaga's [Haa94]
books. Forreadersinterested in more mathematical rigoror adiscussionof the
mathematical foundations ofcross-sectional imagingwesuggesttheysupple-
ment our book with the book byNatterer and Wubbeling [Nat0l].
The yearssince our book was first published have witnessed an increased
interest in helical acquisition and reconstruction methods. We cover the
basics of this topic in Section 3.6. Interested readers should also consult the
paper by Crawford and King [Cra90] and the special issue of Transactionson
Medical Imaging [Wan00] for an overview of current approaches.
Perhaps the most surprising development in the tomography field has
been the application of these algorithms to geophysical phenomena. Re-
searchers have used tomography to study the ocean [Mun95] at scales weare
sure were not imagined by Hounsfield when he created his first scans.
Likewise, tomographic algorithms have been used for whole-earth imaging
[Iye93].
PREFACE TO THE CLASSICS EDITION xi](https://image.slidesharecdn.com/2222090-250613173804-a44e63fe/85/Principles-Of-Computerized-Tomographic-Imaging-Aninash-C-Kak-16-320.jpg)
![Lately, much work has used tomographic data (and MRI data) to generate
volume renderings of internal organs. This work can allow physicians to "fly"
through internal organs and provideinterestingways forradiologists to understand
the state of a human body.
Before wewrote this book, it taxed our laboratory's abilities to collect the data
and create 64 X 64 reconstructions. Today, high-end medical scanners do helical
cone-beam tomography, collect four rowsofdata (1000 detectors per row and 1000
views per rotation) in one-half second, and produce 512 x 512 images in near-real
time. We hope our book will help people understand both ends of this spectrum.
The authors would like to thank the IEEEfor their many years ofsupport and
for allowing us to republish our book. We would also like to thank our colleague
Carl Crawford for his support over the years and his creative means
of encouraging usto publish a new edition. One ofthe authors (MS) would like to
publicly thank Barb Booth, Jim Ziegler,Gordon Speer, and Anne (Atkins) Burch
for their earlyencouragement in the areas ofmathematics, science, and publishing.
A number ofcorrections and some sample codes are available on the Web at
http://www.slaney.org/pct.
[Cra90] C. R. Crawford,K. F. King,Computed tomography scanning withsimulta-
neous patienttranslation, Medical Physics, Vol. 17,No. 6, pp. 967-982, November/
December 1990.
[Haa94] J.R. Haaga,C. F.Lanzieri, D.J. Sartoris (Eds.),Computed Tomography and
Magnetic Resonance Imaging of the Whole Body. 3rd edition, St. Louis, MO: Mosby,
1994.
[Iye93] H.M. Iyer, K. Hirahara (Eds.),SeismicTomography: Theory and Practice.
London; New York:Chapman & Hall, 1993.
[Lee98] J. K.T. Lee,S.S. Sagel.R. J. Stanley,J. P.Heiken (Eds.),Computed Body
Tomography with MRI Correlation. 3rd edition, Philadelphia, PA:
Lippincott Williams & Wilkins, 1998.
[Mun95] W. H. Munk, P. Worchester, C. Wunsch, Ocean Acoustic Tomography.
Cambridge: Cambridge University Press, 1995.
[Nat0l] F. Natterer, F. Wiibbeling, Mathematical Methods in Image
Reconstruction. Philadelphia, PA: SIAM, 2001.
[Wan00] G. Wang, C. Crawford, W. Kalender (Eds.), Special Issue on Multirow
Detector and Cone-Beam Spiral/Helical Computed Tomography, IEEETransactions
on Medical Imaging,Vol. 19, No. 9, September 2000.
xii PREFACE TO THE CLASSICS EDITION](https://image.slidesharecdn.com/2222090-250613173804-a44e63fe/85/Principles-Of-Computerized-Tomographic-Imaging-Aninash-C-Kak-17-320.jpg)
![It is perhaps fair to say that the breakneck pace at which x-ray computed
tomography images improved after Hounsfield's invention was in large
measure owing to the developments that were made in reconstruction
algorithms. Hounsfield used algebraic techniques, described in Chapter 7,
and was able to reconstruct noisy looking 80 x 80 images with an accuracy
of one part in a hundred. This was followed by the application of convolution-
backprojection algorithms, first developed by Ramachandran and Lak-
shminarayanan [Ram71] and later popularized by Shepp and Logan [She74],
to this type of imaging. These later algorithms considerably reduced the
processing time for reconstruction, and the image produced was numerically
more accurate. As a result, commercial manufacturersof x-ray tomographic
scanners started building systems capable of reconstructing 256 x 256 and
512 X 512 images that were almost photographically perfect (in the sense
that the morphological detail produced was unambiguous and in perfect
agreement with the anatomical features). The convolution-backprojection
algorithms are discussed in Chapter 3.
Given the enormous success of x-ray computed tomography, it is not
surprising that in recent years much attention has been focused on extending
this image formation technique to nuclear medicine and magnetic resonance
on the one hand; and ultrasound and microwaves on the other. In nuclear
medicine, our interest is in reconstructing a cross-sectional image of
radioactive isotope distributions within the human body; and in imagingwith
magnetic resonance we wish to reconstruct the magnetic properties of the
object. In both these areas, the problem can be set up as reconstructing an
image from its projections of the type shown in Fig. 1.1. This is not the case
when ultrasound and microwaves are used as energy sources; although the
Fig. 1.1: Two projections are
shown of an object consisting of
a pair of cylinders.
2 COMPUTERIZED TOMOGRAPHIC IMAGING](https://image.slidesharecdn.com/2222090-250613173804-a44e63fe/85/Principles-Of-Computerized-Tomographic-Imaging-Aninash-C-Kak-21-320.jpg)
![References
aim is the same as with x-rays, viz., to reconstruct the cross-sectional image
of, say, the attenuation coefficient. X-raysare nondiffracting, i.e., they travel
in straight lines, whereas microwaves and ultrasound are diffracting. When
an object is illuminated with a diffracting source, the wave field is scattered in
practically all directions, althoughunder certain conditions one might be able
to get away with the assumption of straight line propagation; these conditions
being satisfied when the inhomogeneities are much larger than the wave-
length and when the imaging parameter is the refractive index. For situations
when one must take diffraction effects (inhomogeneity caused scattering of
the wave field) into account, tomographic imaging can in principle be
accomplished with the algorithms described in Chapter 6.
This book covers three aspects of tomography: Chapters 2 and 3 describe
the mathematical principles and the theory. Chapters 4 and 5 describe how to
apply the theory to actual problems in medical imaging and other fields.
Finally, Chapters 6, 7, and 8 introduce several variations of tomography that
are currently being researched.
During the last decade, there has been an avalanche of publications on
different aspects of computed tomography. No attempt will be made to
present a comprehensive bibliography on the subject, since that was recently
accomplished in a book by Dean [Dea83]. We will only give selected
references at the end of each chapter, their purpose only being to cite material
that provides further details on the main ideas discussed in the chapter.
The principal textbooks that have appeared on the subject of tomographic
imaging are [Her80], [Dea83], [Mac83], [Bar8l]. The reader is also referred
to the review articles in the field [Gor74], [Bro76], [Kak79] and the two
special issues of IEEE journals [Kak8l], [Her83]. Reviews of the more
popular algorithms also appeared in [Ros82], [Kak84], [Kak85], [Kak86].
[Bar8l] H. H. Barrett and W. Swindell, Radiological Imaging: The Theory of Image
Formation, Detection and Processing. New York, NY: Academic Press, 1981.
[Bro76] R. A. Brooks and G. DiChiro, "Principles of computer assisted tomography (CAT)
in radiographic and radioisotopicimaging," Phys. Med. Biol., vol. 21, pp. 689-
732, 1976.
[Dea83] S. R. Dean, The Radon Transform and Some of Its Applications. New York,
NY: John Wiley and Sons, 1983.
[Gor74] R. Gordon and G. T. Herman, "Three-dimensional reconstructions from projections:
A review of algorithms," in International Review of Cytology, G. H. Bourne and
J. F. Danielli, Eds. New York, NY: Academic Press, 1974, pp. 111-151.
[Her80] G. T. Herman, Image Reconstructions from Projections. New York, NY:
Academic Press, 1980.
[Her83] ----, Guest Editor, Special Issue on Computerized Tomography, Proceedings of
the IEEE, vol. 71, Mar. 1983.
[Kak79] A. C. Kak, "Computerized tomography with x-ray emission and ultrasound
sources," Proc. IEEE, vol. 67, pp. 1245-1272, 1979.
[Kak8l] , Guest Editor, Special Issue on Computerized Medical Imaging, IEEE
Transactions on Biomedical Engineering, vol. BME-28, Feb. 1981.
INTRODUCTION 3](https://image.slidesharecdn.com/2222090-250613173804-a44e63fe/85/Principles-Of-Computerized-Tomographic-Imaging-Aninash-C-Kak-22-320.jpg)
![[Kak84] ____, "Image reconstructions from projections," in Digital Image Processing
Techniques, M. P. Ekstrom, Ed. New York, NY: Academic Press, 1984.
[Kak85] ____, "Tomographic imaging with diffracting and non-diffracting sources," in
Array Signal Processing, S. Haykin, Ed. Englewood Cliffs, NJ: Prentice-Hall,
1985.
[Kak86] A. C. Kak and B. Roberts, "Image reconstruction from projections," in Handbook
of Pattern Recognition and Image Processing, T. Y. Young and K. S. Fu, Eds.
New York, NY: Academic Press, 1986.
[Mac83] A. Macovski, Medical Imaging Systems. Englewood Cliffs, NJ: Prentice-Hall,
1983.
[Ram71] G. N. Ramachandranand A. V. Lakshminarayanan, "Three dimensional reconstruc-
tions from radiographs and electron micrographs: Application of convolution instead
of Fourier transforms," Proc. Nat. Acad. Sci., vol. 68, pp. 2236-2240, 1971.
[Ros82] A. Rosenfeld and A. C. Kak, Digital Picture Processing, 2nded. New York, NY:
Academic Press, 1982.
[She74] L. A. Shepp and B. F. Logan, "The Fourier reconstruction of a head section," IEEE
Trans. Nucl. Sci., vol. NS-21, pp. 21-43, 1974.
4 COMPUTERIZED TOMOGRAPHIC IMAGING](https://image.slidesharecdn.com/2222090-250613173804-a44e63fe/85/Principles-Of-Computerized-Tomographic-Imaging-Aninash-C-Kak-23-320.jpg)
![2Signal Processing Fundamentals
We can't hope to cover all the important details of one- and two-
dimensional signal processing in one chapter. For those who have already
seen this material, we hope this chapter will serve as a refresher. For those
readers who haven't had prior exposure to signal and image processing, we
hope that this chapter will provide enough of an introductionso that the rest of
the book will make sense.
All readers are referred to a number of excellent textbooks that cover one-
and two-dimensional signal processing in more detail. For information on
1-D processing the reader is referred to [McG74], [Sch75], [Opp75], [Rab75].
The theory and practice of image processing havebeen described in [Ros82],
[Gon77], [Pra78]. The more general case of multidimensional signal
processing has been described in [Dud84].
2.1 One-Dimensional Signal Processing
2.1.1 Continuous and Discrete One-Dimensional Functions
One-dimensional continuous functions, such as in Fig. 2.1(a), will be
represented in this book by the notation
where x(t) denotes the value as a function at t. This function may be given a
discrete representation by sampling its value over a set of points as illustrated
in Fig. 2.1(b). Thus the discrete representation can be expressed as the list
As an example of this, the discrete representation of the data in Fig. 2. l(c) is
It is also possible to represent the samples as a single vector in a
multidimensional space. For example, the set of seven samples could also be
represented as a vector in a 7-dimensional space, with the first element of the
vector equal to 1, the second equal to 3, and so on.
There is a special function that is often useful for explaining operations on
functions. It is called the Dirac delta or impulse function. It can't be defined
SIGNAL PROCESSINGFUNDAMENTALS 5
2Signal Processing Fundamentals
We can't hope to cover all the important details of one- and two-
dimensional signal processing in one chapter. For those who have already
seen this material, we hope this chapter will serve as a refresher. For those
readers who haven't had prior exposure to signal and image processing, we
hope that this chapter will provide enough of an introductionso that the rest of
the book will make sense.
All readers are referred to a number of excellent textbooks that cover one-
and two-dimensional signal processing in more detail. For information on
1-D processing the reader is referred to [McG74], [Sch75], [Opp75], [Rab75].
The theory and practice of image processing havebeen described in [Ros82],
[Gon77], [Pra78]. The more general case of multidimensional signal
processing has been described in [Dud84].
2.1 One-Dimensional Signal Processing
2.1.1 Continuous and Discrete One-Dimensional Functions
One-dimensional continuous functions, such as in Fig. 2.1(a), will be
represented in this book by the notation
where x(t) denotes the value as a function at t. This function may be given a
discrete representation by sampling its value over a set of points as illustrated
in Fig. 2.1(b). Thus the discrete representation can be expressed as the list
As an example of this, the discrete representation of the data in Fig. 2. l(c) is
It is also possible to represent the samples as a single vector in a
multidimensional space. For example, the set of seven samples could also be
represented as a vector in a 7-dimensional space, with the first element of the
vector equal to 1, the second equal to 3, and so on.
There is a special function that is often useful for explaining operations on
functions. It is called the Dirac delta or impulse function. It can't be defined
SIGNAL PROCESSINGFUNDAMENTALS 5](https://image.slidesharecdn.com/2222090-250613173804-a44e63fe/85/Principles-Of-Computerized-Tomographic-Imaging-Aninash-C-Kak-24-320.jpg)
![Fig. 2.5: Thefirst three
components of a Fourier series
are shown. The cosine waves
represent the real part of the
signal while the sine waves
represent the imaginary.
called the frequency of the sinusoids. Note that the sinusoids in (24) are at
multiples of the frequency f0, which is called the fundamental frequency.
The coefficients Zk in (22) are called the complex amplitude of the kth
component, and can be obtained by using the following formula
The representation in (22) is called the Fourier Series. To illustrate pictorially
the representation in (22), we have shown in Fig. 2.6, a triangular function
and some of the components from the expansion.
A continuous signal x(t) defined for t between - and also possesses
another Fourier representation called the continuous Fourier transform and
defined by
One can show that this relationship may be inverted to yield
Comparing (22) and (27), we see that in both representations, x(t) has been
expressed as a sum of sinusoids, ejwt;
the difference being that in the former,
the frequencies of the sinusoids are at multiples of w0, whereas in the latter we
have all frequencies between — to . The two representations are not
independent of each other. In fact, the series representation is contained in the
continuous transform representation since Zk 's in (25) are similar to X(w) in
(26) for w = kwo = k(2 /T), especially if we assume that x(t) is zero
outside [T1 T2], in which case the range of integration in (27) can be cut
SIGNAL PROCESSING FUNDAMENTALS 11](https://image.slidesharecdn.com/2222090-250613173804-a44e63fe/85/Principles-Of-Computerized-Tomographic-Imaging-Aninash-C-Kak-30-320.jpg)
![Fig. 2.6: This illustrates the
Fourier seriesfor a simple
waveform. A triangle wave is
shown in (a) with the magnitude
(b) and phase (c) of the first few
terms of the Fourier series.
down to [T1 T]. For the case whenx(t) is zero outside [T1, T2], the reader
might ask that since one can recover x(t) from Zk using (22), why use (27)
since we require X(w) at frequencies in addition to kw0's. The informationin
X(w) for w kw0 is necessary to constrain the values of x(t) outside the
interval [T1, T2].
If we compute ZK'S using (25), and then reconstruct x(t) from zk's using
(22), we will of course obtain the correct values of x(t) within [T1, 7"2];
however, if we insist on carrying out this reconstruction outside [T1 T2], we
will obtain periodic replications of the original x(t) (see Fig. 2.7). On the
other hand, if X(w) is used for reconstructing the signal, we will obtain x(t)
within [T1, T2] and zero everywhere outside.
The continuous Fourier transform defined in (26) may not exist unless x(t)
satisfies certain conditions, of which the following are typical [Goo68]:
2) g(t) must have only a finite number of discontinuities and a finite
number of maxima and minima in any finite interval.
3) g(t) must have no infinite discontinuities.
Some useful mathematicalfunctions, like the Dirac function, do not obey
the preceding conditions. But if it is possible to represent these functions as
limits of a sequence of well-behaved functions that do obey these conditions
then the Fourier transforms of the members of this sequence will also form a
12 COMPUTERIZED TOMOGRAPHIC IMAGING](https://image.slidesharecdn.com/2222090-250613173804-a44e63fe/85/Principles-Of-Computerized-Tomographic-Imaging-Aninash-C-Kak-31-320.jpg)
![Fig. 2.7: The signal represented
by a Fourierseries is actually a
periodic version of the original
signal defined betweenT1,and T2.
Here the original function is
shown in (a) and the replications
caused by the Fourier series
representation are shown in (b).
sequence. Now if this sequence of Fourier transformspossesses a limit, then
this limit is called the "generalized Fourier transform" of the original
function. Generalized transformscan be manipulated in the same manner as
the conventionaltransforms, and the distinction between the two is generally
ignored; it being understood that when a function fails to satisfy the existence
conditions and yet is said to have a transform, then the generalized transform
is actually meant [Goo68], [Lig60].
Various transforms described in this section obey many useful properties;
these will be shown for the two-dimensional case in Section 2.2.4. Given a
relationship for a function of two variables, it is rather easy to suppress one
and visualize the one-dimensional case; the opposite is usually not the case.
2.1.4 Discrete Fourier Transform (DFT)
As in the continuous case, a discrete function may also be given a
frequency domain representation:
where x(nT) are the samples of some continuous function x(t), and X(w) the
frequency domain representation for the sampled data. (In this book we will
generally use lowercase letters to represent functions of time or space and
the uppercase lettersfor functions in the frequency domain.)
Note that our strategy for introducing the frequency domain representation
is opposite of that in the preceding subsection. In describing Fourier series we
defined the inverse transform (22), and then described how to compute its
coefficients. Now for the DFT we have first described the transform from
time into the frequency domain. Later in this section we will describe the
inverse transform.
SIGNAL PROCESSINGFUNDAMENTALS 13](https://image.slidesharecdn.com/2222090-250613173804-a44e63fe/85/Principles-Of-Computerized-Tomographic-Imaging-Aninash-C-Kak-32-320.jpg)
![Fig. 2.9: The effect of circular
convolution is shown in'(a), (b)
shows how the data can be
zero-padded so that when an FFT
convolution isperformed the
result represents samples of an
aperiodic convolution.
summation is equal to xoy0 but the second term is equal tox-1y1.Althoughin
the original formulation of the finite Fourier transform, the x sequence was
only specified for indices from 0 through N - 1, the periodicity property in
(48) implies that x-1 be equal to xN-1. This leads to the name circular
convolution since the undefined portions of the original sequence are replaced
by a circular replication of the original data.
The effect of circular convolution is shown in Fig. 2.9(a). Here we have
shown an exponential sequence convolved with an impulse. The result
represents a circular convolution and not samples of the continuous
convolution.
A circular convolution can be turned into an aperiodic convolutionby zero-
padding the data. As shown in Fig. 2.9(b) if the original sequences are
doubled in length by adding zeros then the original N samples of the product
sequence will represent an aperiodic convolution of the two sequences.
Efficient procedures for computing the finite Fourier transform are known
as fast Fourier transform (FFT) algorithms. To calculate each of the N points
of the summation shown in (43) requires on the order of N2
operations. In a
fast Fourier transform algorithm the summation is rearranged to take
advantage of common subexpressions and the computational expense is
reduced to N log N. For a 1024 point signal this represents an improvement
by a factor of approximately 100. The fast Fourier transform algorithm has
revolutionized digital signal processing and is described in more detail in
[Bri74].
18 COMPUTERIZED TOMOGRAPHIC IMAGING](https://image.slidesharecdn.com/2222090-250613173804-a44e63fe/85/Principles-Of-Computerized-Tomographic-Imaging-Aninash-C-Kak-37-320.jpg)
![Fig. 2.14: Truncating a sequence
of data is equivalent to
multiplying it by arectangular
window. The result in the
frequency domain is to convolve
the Fourier transform of the
signal with the window shown
above.
where
with N = 16. This function is displayed in Fig. 2.14, and illustrates the
nature of distortion introduced by data truncation.
2.2 Image Processing
The signal processing concepts described in the first half of this chapter are
easily extended to two dimensions.As was done before, we will describe how
to represent an image with delta functions, linear operations on images and
the use of the Fourier transform.
2.2.1 Point Sources and Delta Functions
Let O be an operation that takes pictures into pictures; given the input
picture /, the result of applying O to / is denoted by O[f]. Like the 1-
dimensional case discussed earlier in this chapter, we call O linear if
for all pictures, /, g and all constants a, b.
In the analysis of linear operations on pictures, the concept of a point
28 COMPUTERIZED TOMOGRAPHIC IMAGING](https://image.slidesharecdn.com/2222090-250613173804-a44e63fe/85/Principles-Of-Computerized-Tomographic-Imaging-Aninash-C-Kak-47-320.jpg)
![called a Dirac deltafunction. Evidently, we have (x, y) = 0 for all (x, y)
other than (0, 0) where it is infinite. It follows that (-x, -y) = (x, y).
A number of the properties of the one-dimensional delta function described
earlier extend easily to the two-dimensional case. For example, in light of
(103), we can write
More generally, consider the integral g(x, y) n(x, y) dx dy. This
isjust the average of g(x, y) over a 1/n x 1/n square centered at the origin.
Thus in the limit we retain just the value at the origin itself, so that we can
conclude that the area under the delta function is one and write
If weshift 6bytheamount (a, ), i.e., we use (x - a, y - ) insteadof
(x, y), we similarly obtain the value of g at the point (a, ), i.e.,
The same is true for any region of integration containing (a, ). Equation
(106) is called the "sifting" property of the 5 function.
As a final useful property of 6, we have
For a discussion of this property, see Papoulis [Pap62].
2.2.2 Linear Shift InvariantOperations
Again let us consider a linear operation on images. The point spread
function, which is the output image for an input point source at the origin of
the xy-plane, is denoted by h(x, y).
A linear operation is said to be shift invariant (or space invariant, or
position invariant) iftheresponse to (x - a, y - ), which isapoint source
located at (a, ) in the xy-plane, is given by h(x - a, y - ). In other
words, the output is merely shifted by and in the x and y directions,
respectively.
30 COMPUTERIZED TOMOGRAPHIC IMAGING](https://image.slidesharecdn.com/2222090-250613173804-a44e63fe/85/Principles-Of-Computerized-Tomographic-Imaging-Aninash-C-Kak-49-320.jpg)
![Now let us consider an arbitrary inputpicture f(x, y). By (106) thispicture
can be considered to be a linear sumof point sources. We can writef(x, y) as
In other words, the image f(x, y) is a linear sum of point sources located at
( , ) inthe xy-plane witha and ranging from - to + . Inthis sum the
point source at a particular value of (a , ) has "strength" f(a, ). Letthe
response of the operation to the input f(x, y) be denoted by O[f]. If we
assume the operation to be shift invariant, then by the interpretationjustgiven
to the right-hand side of (108), we obtain
by the linearity of the operation, which means that the response to a sum of
excitations is equal to the sum of responses to each excitation. As stated
earlier, the response to (a - x, - y) [ = (x - a, y — )], which is a
point source located at (a, ), is given by h(x - a, y - ) andif O[f] is
denoted by g, we obtain
The right-hand side is called the convolution of f and h, and is often denoted
by f * h. The integrand is a product of two functionsf(a, ) and h(a, ) with
the latter rotated about the origin by 180° and shifted by x and y along the x
and y directions, respectively. A simple change of variables shows that (111)
can also be written as
so that f* h = h *f.
Fig. 2.16 shows the effect of a simple blurring operation on two different
images. In this case the point response, h, is given by
As can be seen in Fig. 2.16 one effect of this convolution is to smooth out the
edges of each image.
SIGNAL PROCESSING FUNDAMENTALS 31](https://image.slidesharecdn.com/2222090-250613173804-a44e63fe/85/Principles-Of-Computerized-Tomographic-Imaging-Aninash-C-Kak-50-320.jpg)
![This integral is called the inverse Fourier transform of F(u, v). By (114)
and (127), f(x, y) and F(u, v) form a Fourier transform pair.
If x and y represent spatial coordinates, (127) can be used to give a physical
interpretation to the Fourier transform F(ut u)and to the coordinates u and v.
Let us first examine the function
The real and imaginary parts of this function are cos 2 (ux + vy) and sin
2 (ux + vy), respectively. In Fig. 2.18(a), we have shown cos 2 (ux +
vy). It is clear that if one took a section of this two-dimensional pattern
parallel to the x-axis, it goes through u cycles per unit distance, while a
section parallel to the .y-axis goes through v cycles per unit distance. This is
the reason why u and v are called the spatialfrequencies along the x- and y-
axes, respectively. Also, from the figure it can be seen that the spatial period
of the pattern is (u2
+ u2
)-1/2
. The plot for sin 2 (ux + vy) looks similar to
the one in Fig. 2.18(a) except that it is displaced by a quarter period in the
direction of maximum rate of change.
From the preceding discussion it is clear that eJ2 (ux+vy)
is a two-
dimensional pattern, the sections of which, parallel to the x- and.y-axes, are
spatially periodic with frequencies u and u, respectively. The pattern itself
has a spatial period of (u2
+ v2
)-1/2
along a direction that subtends an angle
tan-1
(v/u) with the x-axis. By changing u and v, one can generate patterns
with spatial periods ranging from 0 to in any direction in the xy-plane.
Equation (127) can, therefore, be interpreted to mean that/(x,y) is a linear
combination of elementary periodic patterns of the form ej2 (ux+vy)
.
Evidently, the function, F(u, v), is simply a weighting factor that is a
measure of the relative contribution of the elementary pattern to the total sum.
Since u and v are the spatial frequency of the pattern in the x andy directions,
F(u, u) is called thefrequency spectrum of f(x, y).
2.2.4 Properties of Fourier Transforms
Several properties of the two-dimensional Fourier transform follow easily
from the defining integrals equation. Let F{f} denote the Fourier transform
of a function f(x, y). Then F{f(x, y)} - F(u, v). We will nowpresent
without proof some of the more common properties of Fourier transforms.
The proofs are, for the most part, left for the reader (see the books by
Goodman [Goo68] and Papoulis [Pap62]).
1) Linearity:
This follows from the linearity of the integration operation.
SIGNAL PROCESSING FUNDAMENTALS 35](https://image.slidesharecdn.com/2222090-250613173804-a44e63fe/85/Principles-Of-Computerized-Tomographic-Imaging-Aninash-C-Kak-54-320.jpg)
Principles Of Computerized Tomographic Imaging Aninash C Kak
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- 7. SIAM's Classics in Applied Mathematics series consists of books that were previously allowed to go out of print. These books are republished by SLAM as a professional service because they continue to be important resources for mathematical scientists. Editor-in-Chief Robert E. O'Malley, Jr., University of Washington Editorial Board Richard A. Brualdi, University ofWisconsin-Madison Herbert B. Keller, California Institute of Technology Andrzej Z. Manitius, George Mason University Ingram Olkin, Stanford University Stanley Richardson, University of Edinburgh Ferdinand Verhulst, Mathematisch Instituut, Universityof Utrecht Classics in Applied Mathematics C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences Johan G. F.Belinfante and Bernard Kolman, A Survey of Lie Groups and Lie Algebras with Applications and Computational Methods James M. Ortega, Numerical Analysis: A Second Course Anthony V. Fiacco and Garth P.McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques F. H. Clarke, Optimization and Nonsmooth Analysis George F.Carrier and Carl E. Pearson, Ordinary Differential Equations Leo Breiman, Probability R. Bellman and G. M. Wing, An Introduction to Invariant Imbedding Abraham Berman and Robert J. Plemmons, Nonnegative Matrices in the Mathematical Sciences Olvi L. Mangasarian, Nonlinear Programming *Carl Friedrich Gauss, Theory of the Combination of Observations Least Subject to Errors: Part One, Part Two, Supplement. Translated by G. W. Stewart Richard Bellman, Introduction to Matrix Analysis U. M. Ascher, R. M. M. Mattheij, and R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations K. E. Brenan, S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial- Value Problems in Differential-Algebraic Equations Charles L. Lawson and Richard J. Hanson, Solving Least Squares Problems J. E. Dennis, Jr. and Robert B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations Richard E. Barlow and Frank Proschan, Mathematical Theory of Reliability *First time in print.
- 8. Classics in Applied Mathematics (continued) Cornelius Lanczos, Linear Differential Operators Richard Bellman, Introduction to Matrix Analysis, Second Edition Beresford N. Parlett, The SymmetricEigenvalue Problem Richard Haberman, Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow Peter W. M. John, Statistical Design and Analysis ofExperiments Tamer Basar and Geert Jan Olsder, Dynamic NoncooperativeGame Theory, Second Edition Emanuel Parzen, StochasticProcesses Petar Kokotovic", Hassan K. Khalil, and John O'Reilly, Singular Perturbation Methods in Control: Analysis and Design Jean Dickinson Gibbons, Ingram Olkin, and Milton Sobel, Selecting and Ordering Populations: A New Statistical Methodology James A. Murdock, Perturbations: Theory and Methods Ivar Ekeland and Roger Temam, ConvexAnalysis and Variational Problems Ivar Stakgold, Boundary Value Problemsof Mathematical Physics, Volumes I and II J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables David Kinderlehrer and Guido Stampacchia, An Introduction to Variational Inequalities and Their Applications F. Natterer, The Mathematics of Computerized Tomography Avinash C. Kak and Malcolm Slaney, Principlesof Computerized Tomographic Imaging R. Wong, AsymptoticApproximations of Integrals O. Axelsson and V. A. Barker, Finite ElementSolutionof Boundary Value Problems: Theory and Computation David R. Brillinger, Time Series: Data Analysis and Theory Joel N. Franklin, Methods of Mathematical Economics:Linear and Nonlinear Programming, Fixed-PointTheorems Philip Hartman, Ordinary Differential Equations, Second Edition Michael D. Intriligator, Mathematical Optimization and Economic Theory Philippe G. Ciarlet, The Finite ElementMethod for Elliptic Problems Jane K. Cullum and Ralph A. Willoughby, Lanczos Algorithmsfor LargeSymmetric Eigenvalue Computations, Vol. I: Theory M. Vidyasagar, Nonlinear Systems Analysis, Second Edition Robert Mattheij and Jaap Molenaar, Ordinary Differential Equations in Theory and Practice Shanti S. Gupta and S. Panchapakesan, Multiple Decision Procedures: Theory and Methodology of Selecting and RankingPopulations Eugene L. Allgower and Kurt Georg, Introduction to Numerical Continuation Methods
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- 10. Principles of Computerized Tomographic Imaging Avinash C.Kak Purdue University West Lafayette, Indiana Malcolm Slaney IBM Almaden Research Center San Jose, California siam. Society for Industrial and Applied Mathematics Philadelphia
- 11. Copyright © 2001 by the Society for Industrialand Applied Mathematics. This SIAM edition is an unabridged republicationof the work first published by IEEE Press, New York, 1988. 1 0 9 8 7 6 5 4 3 2 All rights reserved. Printed in the United States ofAmerica. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrialand Applied Mathematics, 3600 UniversityCity Science Center, Philadelphia, PA 19104-2688. Library of Congress Cataloging-in-Publication Data Kak, Avinash C. Principles of computerized tomographic imaging / Avinash C. Kak, Malcolm Slaney. p. cm. —(Classics in applied mathematics; 33) "This SIAM edition is an abridgedrepublicationof the work first published by IEEE Press, New York, 1988." Includes bibliographical references and index. ISBN 0-8987l-494-X(pbk.) 1. Tomography. I. Slaney, Malcolm. II. Title. III. Series. RC78.7.T6 K35 2001 616.07'57--dc21 2001020475 Siam is a registered trademark.
- 12. Contents 1 2 3 Preface to the Classics Edition Preface Introduction References 3 Signal Processing Fundamentals 2.1 One-Dimensional Signal Processing 5 Continuous and Discrete One-Dimensional Functions • Linear Operations • Fourier Representation • Discrete Fourier Transform (DFT) • Finite Fourier Transform • Just How Much Data Is Needed? • Interpretation of the FFT Output • How to Increase the Display Resolution in the Frequency Domain • Howto Deal with Data Defined for Negative Time • Howto Increase Frequency Domain Display Resolution of Signals Defined for Negative Time • Data Truncation Effects 2.2 Image Processing 28 Point Sources andDelta Functions • Linear Shift Invariant Operations • Fourier Analysis • Properties of Fourier Transforms •The Two-Dimensional Finite Fourier Transform • NumericalImplementation of the Two-Dimensional FFT 2.3 References 47 Algorithms for Reconstruction with Nondiffracting Sources 3.1 Line Integrals and Projections 49 3.2 The Fourier Slice Theorem 56 3.3 Reconstruction Algorithms for Parallel Projections 60 The Idea • Theory • Computer Implementation oftheAlgorithm 3.4 Reconstruction from Fan Projections 75 Equiangular Rays • Equally Spaced Collinear Detectors • ARe-sorting Algorithm 3.5 Fan Beam Reconstruction from a Limited Number of Views 93 3.6 Three-Dimensional Reconstructions 99 Three-Dimensional Projections • Three-Dimensional Filtered Backprojection xi xiii 1 5 49 VII
- 13. 4 5 6 3.7 Bibliographic Notes 107 3.8 References 110 Measurement of Projection Data— The Nondiffracting Case 4.1 X-Ray Tomography 114 Monochromatic X-Ray Projections • Measurement of Projection Data with Polychromatic Sources • Polychromaticity Artifacts in X-Ray CT • Scatter • Different Methods for Scanning • Applications 4.2 Emission Computed Tomography 134 Single Photon Emission Tomography • Attenuation Compensation for Single Photon Emission CT • Positron Emission Tomography • Attenuation Compensation for Positron Tomography 4.3 Ultrasonic Computed Tomography 147 Fundamental Considerations • Ultrasonic Refractive Index Tomography • Ultrasonic Attenuation Tomography • Applications 4.4 Magnetic Resonance Imaging 158 4.5 Bibliographic Notes 168 4.6 References 169 Aliasing Artifacts and Noise in CT Images 5.1 Aliasing Artifacts 177 What Does Aliasing Look Like? • Sampling in a Real System 5.2 Noise in Reconstructed Images 190 The Continuous Case • The Discrete Case 5.3 Bibliographic Notes 200 5.4 References 200 Tomographic Imaging with Diffracting Sources 6.1 Diffracted Projections 204 Homogeneous Wave Equation • Inhomogeneous Wave Equation 6.2 Approximations to the Wave Equation 211 The First Born Approximation • The First Rytov Approximation 6.3 The Fourier Diffraction Theorem 218 Decomposing theGreen's Function • Fourier Transform Approach • Short Wavelength Limit of the Fourier Diffraction Theorem • The Data Collection Process 6.4 Interpolation and a Filtered Backpropagation Algorithm for Diffracting Sources 234 Frequency Domain Interpolation • Backpropagation Algorithms 113 177 203 viii
- 14. 7 8 6.5 Limitations 247 Mathematical Limitations • Evaluation of the Born Approximation • Evaluation of the Rytov Approximation • Comparison of the Born and Rytov Approximations 6.6 Evaluation of Reconstruction Algorithms 252 6.7 Experimental Limitations 261 Evanescent Waves • Sampling the Received Wave • The Effects of a Finite Receiver Length • Evaluation of the Experimental Effects • Optimization • Limited Views 6.8 Bibliographic Notes 268 6.9 References 270 Algebraic Reconstruction Algorithms 7.1 Image and Projection Representation 276 7.2 ART (Algebraic Reconstruction Techniques) 283 7.3 SIRT (Simultaneous Iterative Reconstructive Technique) 284 7.4 SART (Simultaneous Algebraic Reconstruction Technique) 285 Modeling the Forward Projection Process • Implementation of the Reconstruction Algorithm 7.5 Bibliographic Notes 292 7.6 References 295 Reflection Tomography 8.1 Introduction 297 8.2 B-Scan Imaging 298 8.3 Reflection Tomography 303 Plane Wave Reflection Transducers • Reflection Tomography vs. Diffraction Tomography • Reflection Tomography Limits 8.4 Reflection Tomography with Point Transmitter/Receivers 313 Reconstruction Algorithms • Experimental Results 8.5 Bibliographic Notes 321 8.6 References 321 Index 275 297 323 ix
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- 16. Preface to the Classics Edition We are pleased that SIAM isrepublishing this book on the principlesof computerized tomography. We enjoyed writingit and have heard from many people who found the book valuable. We're glad that the book isnow back in print. The worlds of tomography and medical imaging have not stood still in the twelve years since this book was first published, yet the basic algorithms presented in this book are just as important today as when they were first applied to tomographic imaging. The reconstruction formalism that under- lies these algorithms helps us create efficient implementations for medical scanners and provides us with a theoretical framework to think about issues such as noise, sampling, resolution and artifacts. We hope that the new readers of our book will also appreciate the beauty of these algorithms. The primary strength of this book consists of derivations that are appropriate for engineers and scientists who want to understand the princi- ples of computerized tomographic imaging. We hope that these derivations will provide readers with insights to create their own implementations. For our readers more interested in the medical applications of computerized tomography we suggest they also consult Lee's [Lee98] or Haaga's [Haa94] books. Forreadersinterested in more mathematical rigoror adiscussionof the mathematical foundations ofcross-sectional imagingwesuggesttheysupple- ment our book with the book byNatterer and Wubbeling [Nat0l]. The yearssince our book was first published have witnessed an increased interest in helical acquisition and reconstruction methods. We cover the basics of this topic in Section 3.6. Interested readers should also consult the paper by Crawford and King [Cra90] and the special issue of Transactionson Medical Imaging [Wan00] for an overview of current approaches. Perhaps the most surprising development in the tomography field has been the application of these algorithms to geophysical phenomena. Re- searchers have used tomography to study the ocean [Mun95] at scales weare sure were not imagined by Hounsfield when he created his first scans. Likewise, tomographic algorithms have been used for whole-earth imaging [Iye93]. PREFACE TO THE CLASSICS EDITION xi
- 17. Lately, much work has used tomographic data (and MRI data) to generate volume renderings of internal organs. This work can allow physicians to "fly" through internal organs and provideinterestingways forradiologists to understand the state of a human body. Before wewrote this book, it taxed our laboratory's abilities to collect the data and create 64 X 64 reconstructions. Today, high-end medical scanners do helical cone-beam tomography, collect four rowsofdata (1000 detectors per row and 1000 views per rotation) in one-half second, and produce 512 x 512 images in near-real time. We hope our book will help people understand both ends of this spectrum. The authors would like to thank the IEEEfor their many years ofsupport and for allowing us to republish our book. We would also like to thank our colleague Carl Crawford for his support over the years and his creative means of encouraging usto publish a new edition. One ofthe authors (MS) would like to publicly thank Barb Booth, Jim Ziegler,Gordon Speer, and Anne (Atkins) Burch for their earlyencouragement in the areas ofmathematics, science, and publishing. A number ofcorrections and some sample codes are available on the Web at http://www.slaney.org/pct. [Cra90] C. R. Crawford,K. F. King,Computed tomography scanning withsimulta- neous patienttranslation, Medical Physics, Vol. 17,No. 6, pp. 967-982, November/ December 1990. [Haa94] J.R. Haaga,C. F.Lanzieri, D.J. Sartoris (Eds.),Computed Tomography and Magnetic Resonance Imaging of the Whole Body. 3rd edition, St. Louis, MO: Mosby, 1994. [Iye93] H.M. Iyer, K. Hirahara (Eds.),SeismicTomography: Theory and Practice. London; New York:Chapman & Hall, 1993. [Lee98] J. K.T. Lee,S.S. Sagel.R. J. Stanley,J. P.Heiken (Eds.),Computed Body Tomography with MRI Correlation. 3rd edition, Philadelphia, PA: Lippincott Williams & Wilkins, 1998. [Mun95] W. H. Munk, P. Worchester, C. Wunsch, Ocean Acoustic Tomography. Cambridge: Cambridge University Press, 1995. [Nat0l] F. Natterer, F. Wiibbeling, Mathematical Methods in Image Reconstruction. Philadelphia, PA: SIAM, 2001. [Wan00] G. Wang, C. Crawford, W. Kalender (Eds.), Special Issue on Multirow Detector and Cone-Beam Spiral/Helical Computed Tomography, IEEETransactions on Medical Imaging,Vol. 19, No. 9, September 2000. xii PREFACE TO THE CLASSICS EDITION
- 18. Preface The purpose of this book is to provide a tutorial overview on the subject of computerized tomographic imaging. We expect the book to be useful for practicing engineers and scientists for gaming an understanding of what can and cannot be done with tomographic imaging. Toward this end, we have tried to strike a balance among purely algorithmic issues, topics dealing with how to generate data for reconstruction in different domains, and artifacts inherent to different data collection strategies. Our hope is that the style of presentation used will also make the book useful for a beginning graduate course on the subject. The desired prerequisites for taking such a course will depend upon the aims of the instructor. If the instructor wishes to teach the course primarily at a theoretical level, with not much emphasis on computer implementations of the reconstruction algorithms, the book is mostly self-contained for graduate students in engineering, the sciences, and mathematics. On the other hand, if the instructor wishes to impart proficiency in the implementations, it would be desirable for the students to have had some prior experience with writing computer programs for digital signal or image processing. The introductory material we have included in Chapter 2 should help the reader review the relevant practical details in digital signal and image processing. There are no homework problems in the book, the reason being that in our own lecturing on the subject, we have tended to emphasize the implementation aspects and, therefore, the homework has consisted of writing computer programs for reconstruction algorithms. The lists of references by no means constitute a complete bibliography on the subject. Basically, we have included those references that we have found useful in our own research over the years. Whenever possible, we have referenced books and review articles to provide the reader with entry points for more exhaustive literature citations. Except in isolated cases, we have not made any attempts to establish historical priorities. No value judgments should be implied by our including or excluding a particular work. Many of our friends and colleagues deserve much credit for helping bring this book to fruition. This book draws heavily from research done at Purdue by our past and present colleagues and collaborators: Carl Crawford, Mani Azimi, David Nahamoo, Anders Andersen, S. X. Pan, Kris Dines, andBarry Roberts. A number of people, Carl Crawford, Rich Kulawiec, Gary S. Peterson, and the anonymous reviewers, helped us proofread the manuscript; PREFACE xiii
- 19. we are grateful for the errors they caught and we acknowledge that any errors that remain are our own fault. We are also grateful to Carl Crawford and Kevin King at GE Medical Systems Division, Greg Kirk at Resonex, Dennis Parker at the University of Utah, and Kris Dines of XDATA, for sharing their knowledge with us about many newly emerging aspects of medical imaging. Our editor, RandiScholnick, at the IEEE PRESS was most patient with us; her critical eye did much to improve the quality of this work. Sharon Katz, technical illustrator for the School of Electrical Engineering at Purdue University, was absolutely wonderful. She produced most of the illustrations in this book and always did it with the utmost professionalism and a smile. Also, Pat Kerkhoff (Purdue), and Tammy Duarte, Amy Atkinson, and Robin Wallace (SPAR) provided excellent secretarial support, even in the face of deadlines and garbled instructions. Finally, one of the authors (M.S.) would like to acknowledge the support of his friend Kris Meade during the long time it took to finish this project. AVINASH C. KAK MALCOLM SLANEY XIV PREFACE
- 20. Tomography refers to the cross-sectional imaging of an object from either transmission or reflection data collected by illuminating the object from many different directions. The impact of this technique in diagnostic medicine has been revolutionary, since it has enabled doctors to view internal organs with unprecedented precision and safety to the patient. The first medical application utilized x-rays for forming images of tissues based on their x-ray attenuation coefficient. More recently, however, medical imaging has also been successfully accomplished with radioisotopes, ultrasound, and magnetic resonance; the imaged parameter being different in each case. There are numerous nonmedical imaging applications which lend them- selves to the methods of computerized tomography. Researchers have already applied this methodology to the mapping of underground resources via cross- borehole imaging, some specialized cases of cross-sectional imaging for nondestructive testing, the determination of the brightness distribution over a celestial sphere, and three-dimensional imaging with electron microscopy. Fundamentally, tomographic imaging deals with reconstructing an image from its projections. In the strict sense of the word, a projection at a given angle is the integral of the image in the direction specified by that angle, as illustrated in Fig. 1.1. However, in a loose sense, projection means the information derived from the transmitted energies, when an object is illuminated from a particular angle; the phrase "diffracted projection" may be used when energy sources are diffracting, as is the case with ultrasound and microwaves. Although, from a purely mathematical standpoint, the solution to the problem of how to reconstruct a function from its projections dates back to the paper by Radon in 1917, the current excitement in tomographic imaging originated with Hounsfield's invention of the x-ray computed tomographic scanner for whichhe received a Nobel prize in 1972. He shared the prize with Allan Cormack who independently discovered some of the algorithms. His invention showed that it is possible to compute high-quality cross-sectional images with an accuracy now reaching one part in a thousand in spite of the fact that the projection data do not strictly satisfy the theoretical models underlying the efficiently implementable reconstruction algorithms. His invention also showed that it is possible to process a very large number of measurements (now approaching a million for the case of x-raytomography) with fairly complex mathematical operations, and still get an image that is incredibly accurate. INTRODUCTION 1 Introduction 1
- 21. It is perhaps fair to say that the breakneck pace at which x-ray computed tomography images improved after Hounsfield's invention was in large measure owing to the developments that were made in reconstruction algorithms. Hounsfield used algebraic techniques, described in Chapter 7, and was able to reconstruct noisy looking 80 x 80 images with an accuracy of one part in a hundred. This was followed by the application of convolution- backprojection algorithms, first developed by Ramachandran and Lak- shminarayanan [Ram71] and later popularized by Shepp and Logan [She74], to this type of imaging. These later algorithms considerably reduced the processing time for reconstruction, and the image produced was numerically more accurate. As a result, commercial manufacturersof x-ray tomographic scanners started building systems capable of reconstructing 256 x 256 and 512 X 512 images that were almost photographically perfect (in the sense that the morphological detail produced was unambiguous and in perfect agreement with the anatomical features). The convolution-backprojection algorithms are discussed in Chapter 3. Given the enormous success of x-ray computed tomography, it is not surprising that in recent years much attention has been focused on extending this image formation technique to nuclear medicine and magnetic resonance on the one hand; and ultrasound and microwaves on the other. In nuclear medicine, our interest is in reconstructing a cross-sectional image of radioactive isotope distributions within the human body; and in imagingwith magnetic resonance we wish to reconstruct the magnetic properties of the object. In both these areas, the problem can be set up as reconstructing an image from its projections of the type shown in Fig. 1.1. This is not the case when ultrasound and microwaves are used as energy sources; although the Fig. 1.1: Two projections are shown of an object consisting of a pair of cylinders. 2 COMPUTERIZED TOMOGRAPHIC IMAGING
- 22. References aim is the same as with x-rays, viz., to reconstruct the cross-sectional image of, say, the attenuation coefficient. X-raysare nondiffracting, i.e., they travel in straight lines, whereas microwaves and ultrasound are diffracting. When an object is illuminated with a diffracting source, the wave field is scattered in practically all directions, althoughunder certain conditions one might be able to get away with the assumption of straight line propagation; these conditions being satisfied when the inhomogeneities are much larger than the wave- length and when the imaging parameter is the refractive index. For situations when one must take diffraction effects (inhomogeneity caused scattering of the wave field) into account, tomographic imaging can in principle be accomplished with the algorithms described in Chapter 6. This book covers three aspects of tomography: Chapters 2 and 3 describe the mathematical principles and the theory. Chapters 4 and 5 describe how to apply the theory to actual problems in medical imaging and other fields. Finally, Chapters 6, 7, and 8 introduce several variations of tomography that are currently being researched. During the last decade, there has been an avalanche of publications on different aspects of computed tomography. No attempt will be made to present a comprehensive bibliography on the subject, since that was recently accomplished in a book by Dean [Dea83]. We will only give selected references at the end of each chapter, their purpose only being to cite material that provides further details on the main ideas discussed in the chapter. The principal textbooks that have appeared on the subject of tomographic imaging are [Her80], [Dea83], [Mac83], [Bar8l]. The reader is also referred to the review articles in the field [Gor74], [Bro76], [Kak79] and the two special issues of IEEE journals [Kak8l], [Her83]. Reviews of the more popular algorithms also appeared in [Ros82], [Kak84], [Kak85], [Kak86]. [Bar8l] H. H. Barrett and W. Swindell, Radiological Imaging: The Theory of Image Formation, Detection and Processing. New York, NY: Academic Press, 1981. [Bro76] R. A. Brooks and G. DiChiro, "Principles of computer assisted tomography (CAT) in radiographic and radioisotopicimaging," Phys. Med. Biol., vol. 21, pp. 689- 732, 1976. [Dea83] S. R. Dean, The Radon Transform and Some of Its Applications. New York, NY: John Wiley and Sons, 1983. [Gor74] R. Gordon and G. T. Herman, "Three-dimensional reconstructions from projections: A review of algorithms," in International Review of Cytology, G. H. Bourne and J. F. Danielli, Eds. New York, NY: Academic Press, 1974, pp. 111-151. [Her80] G. T. Herman, Image Reconstructions from Projections. New York, NY: Academic Press, 1980. [Her83] ----, Guest Editor, Special Issue on Computerized Tomography, Proceedings of the IEEE, vol. 71, Mar. 1983. [Kak79] A. C. Kak, "Computerized tomography with x-ray emission and ultrasound sources," Proc. IEEE, vol. 67, pp. 1245-1272, 1979. [Kak8l] , Guest Editor, Special Issue on Computerized Medical Imaging, IEEE Transactions on Biomedical Engineering, vol. BME-28, Feb. 1981. INTRODUCTION 3
- 23. [Kak84] ____, "Image reconstructions from projections," in Digital Image Processing Techniques, M. P. Ekstrom, Ed. New York, NY: Academic Press, 1984. [Kak85] ____, "Tomographic imaging with diffracting and non-diffracting sources," in Array Signal Processing, S. Haykin, Ed. Englewood Cliffs, NJ: Prentice-Hall, 1985. [Kak86] A. C. Kak and B. Roberts, "Image reconstruction from projections," in Handbook of Pattern Recognition and Image Processing, T. Y. Young and K. S. Fu, Eds. New York, NY: Academic Press, 1986. [Mac83] A. Macovski, Medical Imaging Systems. Englewood Cliffs, NJ: Prentice-Hall, 1983. [Ram71] G. N. Ramachandranand A. V. Lakshminarayanan, "Three dimensional reconstruc- tions from radiographs and electron micrographs: Application of convolution instead of Fourier transforms," Proc. Nat. Acad. Sci., vol. 68, pp. 2236-2240, 1971. [Ros82] A. Rosenfeld and A. C. Kak, Digital Picture Processing, 2nded. New York, NY: Academic Press, 1982. [She74] L. A. Shepp and B. F. Logan, "The Fourier reconstruction of a head section," IEEE Trans. Nucl. Sci., vol. NS-21, pp. 21-43, 1974. 4 COMPUTERIZED TOMOGRAPHIC IMAGING
- 24. 2Signal Processing Fundamentals We can't hope to cover all the important details of one- and two- dimensional signal processing in one chapter. For those who have already seen this material, we hope this chapter will serve as a refresher. For those readers who haven't had prior exposure to signal and image processing, we hope that this chapter will provide enough of an introductionso that the rest of the book will make sense. All readers are referred to a number of excellent textbooks that cover one- and two-dimensional signal processing in more detail. For information on 1-D processing the reader is referred to [McG74], [Sch75], [Opp75], [Rab75]. The theory and practice of image processing havebeen described in [Ros82], [Gon77], [Pra78]. The more general case of multidimensional signal processing has been described in [Dud84]. 2.1 One-Dimensional Signal Processing 2.1.1 Continuous and Discrete One-Dimensional Functions One-dimensional continuous functions, such as in Fig. 2.1(a), will be represented in this book by the notation where x(t) denotes the value as a function at t. This function may be given a discrete representation by sampling its value over a set of points as illustrated in Fig. 2.1(b). Thus the discrete representation can be expressed as the list As an example of this, the discrete representation of the data in Fig. 2. l(c) is It is also possible to represent the samples as a single vector in a multidimensional space. For example, the set of seven samples could also be represented as a vector in a 7-dimensional space, with the first element of the vector equal to 1, the second equal to 3, and so on. There is a special function that is often useful for explaining operations on functions. It is called the Dirac delta or impulse function. It can't be defined SIGNAL PROCESSINGFUNDAMENTALS 5 2Signal Processing Fundamentals We can't hope to cover all the important details of one- and two- dimensional signal processing in one chapter. For those who have already seen this material, we hope this chapter will serve as a refresher. For those readers who haven't had prior exposure to signal and image processing, we hope that this chapter will provide enough of an introductionso that the rest of the book will make sense. All readers are referred to a number of excellent textbooks that cover one- and two-dimensional signal processing in more detail. For information on 1-D processing the reader is referred to [McG74], [Sch75], [Opp75], [Rab75]. The theory and practice of image processing havebeen described in [Ros82], [Gon77], [Pra78]. The more general case of multidimensional signal processing has been described in [Dud84]. 2.1 One-Dimensional Signal Processing 2.1.1 Continuous and Discrete One-Dimensional Functions One-dimensional continuous functions, such as in Fig. 2.1(a), will be represented in this book by the notation where x(t) denotes the value as a function at t. This function may be given a discrete representation by sampling its value over a set of points as illustrated in Fig. 2.1(b). Thus the discrete representation can be expressed as the list As an example of this, the discrete representation of the data in Fig. 2. l(c) is It is also possible to represent the samples as a single vector in a multidimensional space. For example, the set of seven samples could also be represented as a vector in a 7-dimensional space, with the first element of the vector equal to 1, the second equal to 3, and so on. There is a special function that is often useful for explaining operations on functions. It is called the Dirac delta or impulse function. It can't be defined SIGNAL PROCESSINGFUNDAMENTALS 5
- 25. Fig. 2.1: A one-dimensional signal is shown in (a) with its sampled version in (b). The discrete version of the signal is illustrated in (c). directly; instead it must be expressed as the limit of a sequence of functions. First we define a new function called rect (short for rectangle) as follows This is illustrated in Fig. 2.2(a). Consider a sequence of functions of ever decreasing support on the t-axis as described by and illustrated in Fig. 2.2(b). Each function in this sequence has the same area but is of ever increasing height, which tends to infinity as n . The limit of this sequence of functions is of infinite height but zero width in such a manner that the area is still unity.This limit is often pictorially represented as shown in Fig. 2.2(c) and denoted by (t). Our explanation leads to the definition of the Dirac delta function that follows The delta function has the following "sampling" property 6 COMPUTERIZED TOMOGRAPHIC IMAGING
- 26. Fig. 2.2: A rectanglefunction as shown in (a) is scaled in both width and height (b). In the limit the result is the delta function illustrated in (c). where 5(t - t') isanimpulse shifted tothelocation t = t'. When animpulse enters into a product with an arbitrary x(t), all the values of x(t) outside the location t = t' are disregarded. Then by the integral property of the delta function weobtain (7); sowecansaythat (t - t') samples thefunction x(t) at t'. 2.1.2 Linear Operations Functions may be operated on for purposes such as filtering, smoothing, etc. The application of an operator O to a function x(t) will be denoted by The operator is linear provided for any pair of constants a and and for any pair of functions x(t) and y(t). An interesting class of linear operations is defined by the followingintegral form where h is called the impulse response. It is easily shown that h is the system response of the operator applied to a delta function. Assume that the input SIGNAL PROCESSING FUNDAMENTALS 7
- 27. Fig. 2.3: The impulse response of a shift invariant filter is shown convolved with three impulses. function is an impulse at t = t0 or Substituting into (10), we obtain Therefore h(t, t') can be called the impulseresponse for the impulse applied at t'. A linear operation is called shift invariantwhen implies or equivalently This implies that when the impulse is shifted by t', so is the response, as is further illustrated in Fig. 2.3. In other words, the response produced by the linear operation does not vary with the location of the impulse; it is merely shifted by the sameamount. For shift invariant operations, the integral form in (10) becomes This is now called a convolutionand is represented by The process of convolution can be viewed as flipping one of the two functions, shifting one with respect to the other, multiplying the two and integrating the product for every shift as illustrated by Fig. 2.4. 8 COMPUTERIZED TOMOGRAPHIC IMAGING
- 28. Fig. 2.4: The results of convolving an impulse response with an impulse (top) and a square pulse (bottom) are shown here. Convolution can also be defined for discrete sequences. If and then the convolution of xi with yi can be written as This is a discrete approximation to the integral of (17). 2.1.3 Fourier Representation For many purposes it is useful to represent functions in the frequency domain. Certainly the most common reason is because it gives a new perspective to an otherwise difficult problem. This is certainly true with the SIGNAL PROCESSING FUNDAMENTALS 9
- 29. convolution integral; in the time domain convolution is an integral while in the frequency domain it is expressed as a simple multiplication. In the sections to follow we will describe four different varieties of the Fourier transform. The continuous Fourier transform is mostly used in theoretical analysis. Given that with real world signals it is necessary to periodically sample the data, we are led to three other Fourier transformsthat approximate either the time or frequency data as samples of thecontinuous functions. The four typesof Fourier transforms are summarized in Table2.1. Assume that we have a continuousfunction x(t) defined for T1 t T2. This function can be expressed in the following form: wherej = andw0 = 2 f0 = 2 /T, T = T2 - T1 andzk arecomplex coefficients to be discussed shortly. What is being said here is that x(t) is the sum of a number of functions of the form This function represents The two functions on the right-hand side, commonly referred to as sinusoids, are oscillatory with kf0 cycles per unit of t as illustrated by Fig.2.5. kf0 is Table 2.1: Four different Fourier transforms can be defined by sampling the time and frequency domains.* Continuous Frequency Discrete Frequency Continuous Time Name: Fourier Transform Forward: X(a) = x(t)e-Jwt dt Inverse: x(t)= 1/2 X(w)ejet dw Periodicity: None Name: Fourier Series Forward: Xn=1/Tx(t)e-jn(2/T)t Inerse: x(t) = XneJn(2 /T) ' Periodicity: x(t) =x(t + iT) Discrete Time Name: Discrete Fourier Transform Forward: X(w) = x(nT)e-jwnT Inverse: x(nT) = T/2 X(w)eJumT dw Periodicity : X(w) =X(w+i(2 /T)) Name: Finite Fourier Transform Forward: Xk=/N xne-j(2 /N)kn Inverse: xk = Xnej(2 /N)kn Periodicity: xk=xk+iN andXk = Xk+iN * In the above table time domain functions are indicated by x and frequency domain functions are X. The time domain sampling interval is indicated byT. 10 COMPUTERIZED TOMOGRAPHIC IMAGING
- 30. Fig. 2.5: Thefirst three components of a Fourier series are shown. The cosine waves represent the real part of the signal while the sine waves represent the imaginary. called the frequency of the sinusoids. Note that the sinusoids in (24) are at multiples of the frequency f0, which is called the fundamental frequency. The coefficients Zk in (22) are called the complex amplitude of the kth component, and can be obtained by using the following formula The representation in (22) is called the Fourier Series. To illustrate pictorially the representation in (22), we have shown in Fig. 2.6, a triangular function and some of the components from the expansion. A continuous signal x(t) defined for t between - and also possesses another Fourier representation called the continuous Fourier transform and defined by One can show that this relationship may be inverted to yield Comparing (22) and (27), we see that in both representations, x(t) has been expressed as a sum of sinusoids, ejwt; the difference being that in the former, the frequencies of the sinusoids are at multiples of w0, whereas in the latter we have all frequencies between — to . The two representations are not independent of each other. In fact, the series representation is contained in the continuous transform representation since Zk 's in (25) are similar to X(w) in (26) for w = kwo = k(2 /T), especially if we assume that x(t) is zero outside [T1 T2], in which case the range of integration in (27) can be cut SIGNAL PROCESSING FUNDAMENTALS 11
- 31. Fig. 2.6: This illustrates the Fourier seriesfor a simple waveform. A triangle wave is shown in (a) with the magnitude (b) and phase (c) of the first few terms of the Fourier series. down to [T1 T]. For the case whenx(t) is zero outside [T1, T2], the reader might ask that since one can recover x(t) from Zk using (22), why use (27) since we require X(w) at frequencies in addition to kw0's. The informationin X(w) for w kw0 is necessary to constrain the values of x(t) outside the interval [T1, T2]. If we compute ZK'S using (25), and then reconstruct x(t) from zk's using (22), we will of course obtain the correct values of x(t) within [T1, 7"2]; however, if we insist on carrying out this reconstruction outside [T1 T2], we will obtain periodic replications of the original x(t) (see Fig. 2.7). On the other hand, if X(w) is used for reconstructing the signal, we will obtain x(t) within [T1, T2] and zero everywhere outside. The continuous Fourier transform defined in (26) may not exist unless x(t) satisfies certain conditions, of which the following are typical [Goo68]: 2) g(t) must have only a finite number of discontinuities and a finite number of maxima and minima in any finite interval. 3) g(t) must have no infinite discontinuities. Some useful mathematicalfunctions, like the Dirac function, do not obey the preceding conditions. But if it is possible to represent these functions as limits of a sequence of well-behaved functions that do obey these conditions then the Fourier transforms of the members of this sequence will also form a 12 COMPUTERIZED TOMOGRAPHIC IMAGING
- 32. Fig. 2.7: The signal represented by a Fourierseries is actually a periodic version of the original signal defined betweenT1,and T2. Here the original function is shown in (a) and the replications caused by the Fourier series representation are shown in (b). sequence. Now if this sequence of Fourier transformspossesses a limit, then this limit is called the "generalized Fourier transform" of the original function. Generalized transformscan be manipulated in the same manner as the conventionaltransforms, and the distinction between the two is generally ignored; it being understood that when a function fails to satisfy the existence conditions and yet is said to have a transform, then the generalized transform is actually meant [Goo68], [Lig60]. Various transforms described in this section obey many useful properties; these will be shown for the two-dimensional case in Section 2.2.4. Given a relationship for a function of two variables, it is rather easy to suppress one and visualize the one-dimensional case; the opposite is usually not the case. 2.1.4 Discrete Fourier Transform (DFT) As in the continuous case, a discrete function may also be given a frequency domain representation: where x(nT) are the samples of some continuous function x(t), and X(w) the frequency domain representation for the sampled data. (In this book we will generally use lowercase letters to represent functions of time or space and the uppercase lettersfor functions in the frequency domain.) Note that our strategy for introducing the frequency domain representation is opposite of that in the preceding subsection. In describing Fourier series we defined the inverse transform (22), and then described how to compute its coefficients. Now for the DFT we have first described the transform from time into the frequency domain. Later in this section we will describe the inverse transform. SIGNAL PROCESSINGFUNDAMENTALS 13
- 33. Fig. 2.8: The discrete Fourier transform (DFT) of a two element sequence is shown here. As will be evident shortly, X(w) represents the complex amplitude of the sinusoidal component ejwT of the discrete signal. Therefore, with one important difference, X(w) plays the same role here as Zk in the preceding subsection; the difference being that in the preceding subsection the frequency domain representation was discrete (since it only existed at multiples of the fundamental frequency), while the representation here is continuous as X(w) is defined for all w. For example, assume that For this signal Note that X(w) obeys the following periodicity which follows from (28) by simple substitution. In Fig. 2.8 we haveshown several periods of this X(w). X(w) is called the discrete Fourier transform of the function x(nT). From the DFT, the function x(nT) can be recovered by using 14 COMPUTERIZED TOMOGRAPHICIMAGING
- 34. which points to the discrete function x(nT) being a sum (an integral sum, to be more specific) of sinusoidal components like ejanT . An important property of the DFT is that it provides an alternate method for calculating the convolution in (21). Given a pair of sequences xI = x(iT) and hi = h(iT), their convolution as defined by can be calculated from This can be derived by noting that the DFT of the convolution is written as Rewriting the exponential we find The second summation now can be written as Note that the limits of the summation remain from — to . At this point it is easy to see that A dual to the above relationship can be stated as follows. Let's multiply two discrete functions, xn andyn, each obtained by sampling the correspond- ing continuous function with a sampling interval of T and call the resulting sequence zn Then the DFT of the new sequence is given by the following convolution in the frequency domain SIGNAL PROCESSINGFUNDAMENTALS 15
- 35. 2.1.5 Finite Fourier Transform Consider a discrete function that is N elements long. Let's represent this sequence with the following subscripted notation Although the DFT defined in Section 2.1.4 is useful for many theoretical discussions, for practical purposes it is the following transformation, called the finite Fourier transform (FFT),1 that is actually calculated with a computer: for u = 0, 1,2, , N - 1. To explain the meaning of the values Xu, rewrite (43) as Comparing (44) and (28), we see that the Xu's are the samples of the continuous function X(w) for. Therefore, we see that if (43) is used to compute the frequency domain representation of a discrete function, a sampling interval of Tin the t-domain implies a sampling interval of l/NT in the frequency domain. The inverse of the relationship shown in (43) is Both (43) and (46) define sequences that are periodically replicated. First consider (43). If the u = Nm + i term is calculated then by noting that ej(2 /N)Nm = 1 for all integer values of m, it is easy to see that 1 The acronym FFT also stands for fast Fourier transform, which is an efficient algorithm for the implementation of the finite Fourier transform. 16 COMPUTERIZED TOMOGRAPHIC IMAGING
- 36. A similar analysis can be made for the inverse case so that When the finite Fourier transforms of two sequences are multiplied the result is still a convolution, as it was for the discrete Fourier transform defined in Section 2.1.4, but now the convolution is with respect to replicated sequences. This is often known as circular convolution because of the effect discussed below. To see this effect consider the product of two finite Fourier transforms. First write the product of two finite Fourier transforms and then take the inverse finite Fourier transform to find Substituting the definition of Xu and Yu as given by (43) the product can now be written The order of summation can be rearranged and the exponential terms combined to find There are twocases to consider. When n —i —k 0 then as a functionof u the samples of the exponential eJ(2 /N)uni-ui-uk represent an integral number of cycles of a complex sinusoid and their sum is equal to zero. On the other hand, when i = n - k then each sample of the exponential is equal toone and thus the summation is equal to N. The summation in (52)over i and k represents asum ofallthepossible combinations ofxi,andyk.When i = n - k then the combination is multiplied by a factor of N while when i n — k then the term is ignored. This means that the original product of two finite Fourier transforms can be simplified to This expression is very similar to (21)except for the definitionofx -k and yk for negative indices. Consider the case when n = 0. The first term of the SIGNAL PROCESSING FUNDAMENTALS 17
- 37. Fig. 2.9: The effect of circular convolution is shown in'(a), (b) shows how the data can be zero-padded so that when an FFT convolution isperformed the result represents samples of an aperiodic convolution. summation is equal to xoy0 but the second term is equal tox-1y1.Althoughin the original formulation of the finite Fourier transform, the x sequence was only specified for indices from 0 through N - 1, the periodicity property in (48) implies that x-1 be equal to xN-1. This leads to the name circular convolution since the undefined portions of the original sequence are replaced by a circular replication of the original data. The effect of circular convolution is shown in Fig. 2.9(a). Here we have shown an exponential sequence convolved with an impulse. The result represents a circular convolution and not samples of the continuous convolution. A circular convolution can be turned into an aperiodic convolutionby zero- padding the data. As shown in Fig. 2.9(b) if the original sequences are doubled in length by adding zeros then the original N samples of the product sequence will represent an aperiodic convolution of the two sequences. Efficient procedures for computing the finite Fourier transform are known as fast Fourier transform (FFT) algorithms. To calculate each of the N points of the summation shown in (43) requires on the order of N2 operations. In a fast Fourier transform algorithm the summation is rearranged to take advantage of common subexpressions and the computational expense is reduced to N log N. For a 1024 point signal this represents an improvement by a factor of approximately 100. The fast Fourier transform algorithm has revolutionized digital signal processing and is described in more detail in [Bri74]. 18 COMPUTERIZED TOMOGRAPHIC IMAGING
- 38. 2.1.6 Just How Much Data Is Needed? In Section 2.1.1 we used a sequence of numbers xi to approximate a continuous function x(t). An important question is, how finely must the data be sampled for xi to accurately represent the original signal? This question was answered by Nyquist who observed that a signal must be sampled at least twice during each cycle of the highest frequency of the signal. More rigorously, if a signal x(t) has a Fourier transform such that then samples of x must be measured at a rate greater than wN. In other words, if T is the interval between consecutive samples, we want 2 /T wN. The frequency wN is known as the Nyquist rate and represents the minimum frequency at which the data can be sampled without introducing errors. Since most real world signals aren't limited to a small range of frequencies, it is important to know the consequences of sampling at below the Nyquist rate. We can consider the process of sampling to be equivalent to multiplication of the original continuous signal x(t) by a sampling function given by The Fourier transform of h(t) can be computed from (26) to be By (40) we can convert the multiplication to a convolution in the frequency domain. Thus the result of the sampling can be written This result is diagrammed in Fig. 2.10. It is important to realize that when sampling the original data (Fig. 2.10(a)) at a rate faster than that defined by the Nyquist rate, the sampled data are an exact replica of the original signal. This is shown in Fig. 2.10(b). If the sampled signal is filtered such that all frequencies above the Nyquist rate are removed, then the original signal will be recovered. On the other hand, as the sampling interval is increased the replicas of the signal in Fig. 2.10(c) move closer together. With a sampling interval greater SIGNAL PROCESSINGFUNDAMENTALS 19
- 39. Fig. 2.10: Sampling a waveform generates replications of the original Fourier transform of the object at periodic intervals. If the signal is sampled at a frequency of w then the Fourier transform of the object will be replicated at intervals of 2w. (a) shows the Fourier transform of the original signal, (b) shows the Fourier transform when x(t) is sampled at a ratefaster than the Nyquist rate, (c) when sampled at the Nyquist rate and finally (d) when the data are sampled at a rate less than the Nyquist rate. than that predicted by the Nyquist rate some of the information in the original data has been smeared by replications of the signal at other frequencies and the original signal is unrecoverable. (See Fig. 2.10(d).) The error caused by the sampling process is given by the inverse Fourier transform of the frequency information in the overlap as shown in Fig. 2.10(d). These errors are also known as aliasing. 2.1.7 Interpretation of the FFT Output Correct interpretation of the Xu's in (43) is obviously important. Toward that goal, it is immediately apparent that X0 stands for the average (or, what is more frequently called the dc) component of the discrete function, since from (43) Interpretation of X1 requires, perhaps, a bit more effort; it stands for 1 cycle per sequence length. This can be made obvious by setting X1 = 1, while all 20 COMPUTERIZED TOMOGRAPHIC IMAGING
- 40. other Ays are set equal to 0 in (46). We obtain for n = 0,1, 2, •••, N - 1. Aplot ofeither thecosine or the sinepart ofthis expression will showjust one cycle of the discrete function xn, which is why we consider X1 as representing one cycle per sequence length. One may similarly show that X2 represents two cycles per sequence length. Unfortu- nately, this straightforward approach for interpreting Xu breaks down for u > N/2. For these high values of the index u, we make use of the following periodicity property which is easily proved by substitution in (43). For further explanation, consider now a particular value for N, say 8. We already know that X0 represents dc X represents 1 cycle per sequence length X2 represents 2 cycles per sequence length X3 represents 3 cycles per sequence length X4 represents 4 cycles per sequence length. From the periodicity property we can now add the following X5 represents - 3 cycles persequence length X6 represents - 2 cycles persequence length X-7 represents - 1cycle per sequence length. Note that we could also have added "X4 represents —4cycles per sequence length." The fact is that for any N element sequence, XN/2 will always be equal to X-N/2, since from (43) The discussion is diagrammatically represented by Fig. 2.11, which shows that when an N element data sequence is fed into an FFT program, the output sequence, also N elements long, consists of the dc frequency term, followed by positive frequencies and then by negative frequencies. This type of an output where the negative axis information follows the positive axis information is somewhat unnatural to look at. To display the FFT output with a more naturalprogression of frequencies, we can, of course, rearrange the output sequence, although if the aim is SIGNAL PROCESSING FUNDAMENTALS 21
- 41. Fig. 2.11: The output of an 8 element FFT is shown here. merely to filter the data, it may not be necessary to do so. In that case the filter transfer function can be rearranged to correspond to the frequency assignments of the elements of the FFT output. It is also possible to produce normal-looking FFT outputs (with dc at the center between negative and positive frequencies) by "modulating" the data prior to taking the FFT. Suppose we multiplythe data with (— 1)" to produce a new sequence x'n Let X'u designate the FFT of this new sequence. Substituting(63) in (43), we obtain for u = 0, 1,2, •••, N - 1. This implies the followingequivalences 2.1.8 How to Increase the Display Resolution in the Frequency Domain The right column of Fig. 2.12 showsthe magnitudeof the FFT output(the dc is centered) of the sequence that represents a rectangular function asshown in the left column. As was mentioned before, the Fourier transform of a discrete sequence contains all frequencies, although it is periodic, and the FFT output represents the samples of one period. For many situations, the 22 COMPUTERIZED TOMOGRAPHIC IMAGING
- 42. Fig. 2.12: As shown here, padding a sequence of data with zeros increases the resolution in the frequency domain. The sequence in fa) has only 16 points, (b) has 32 points, while (c) has 64 points. frequency domain samples supplied by the FFT, although containing practically all the informationfor the reconstruction of the continuous Fourier transform, are hard to interpret visually. This is evidenced by Fig. 2.12(a), where for part of the display we have only one sample associated with an oscillation in the frequency domain. It is possible to produce smoother- looking outputs by what is called zero-padding the data before taking the FFT. For example, if the sequence of Fig. 2.12(a) is extended with zeros to SIGNAL PROCESSING FUNDAMENTALS 23
- 43. twice its length, the FFT of the resulting 32 element sequence will be as shown in Fig. 2.12(b), which is visually smoother looking than the pattern in Fig. 2.12(a). If we zero-pad the data to four times its original length, the output is as shown in Fig. 2.12(c). That zero-padding a data sequence yields frequency domain points that are more closely spaced can be shown by the following derivation. Again let x1, x2 xN- 1 represent the original data. By zero-padding the data we will define a new x' sequence: Let X 'u be the FFT of the new sequence x'n. Therefore, which in terms of the original data is equal to If we evaluate this expression at even values of u, that is when we get In Fig. 2.13 is illustrated the equality between the even-numbered elementsof the newtransformandthe original transform. That X1', X'3, , etc. are the interpolated values between X0 and X1; between X1 and X2; etc. can be seen from the summations in (43) and (74) written in the following form Comparing the two summations, we see that the upper one simply represents the sampled DFT with half the sampling interval. 24 COMPUTERIZED TOMOGRAPHIC IMAGING
- 44. Fig. 2.13: When a data sequence is padded with zeros the effect is to increase the resolution in the frequency domain. The points in (a) are also in the longer sequence shown in (b), but there are additional points, as indicated by circles, that provide interpolated values of the FFT. So we have the following conclusion: to increase the display resolution in the frequency domain, we must zero-extend the time domain signal. This also means that if we are comparing the transforms of sequences of different lengths, they must all be zero-extended to the same number, so that they are all plotted with the same display resolution. This is because the upper summation, (79), has a sampling interval in the frequency domain of 2 r/27W while the lower summation, (80), has a sampling interval that is twice as long or 2 /N . 2.1.9 How to Deal with Data Defined for Negative Time Since the forward and the inverse FFT relationships, (43) and (46), are symmetrical, the periodicity property described in (62) also applies in time domain. What is being said here is that if a time domain sequence and its transform obey (43) and (46), then an N element data sequence in the time domain must satisfy the following property To explain the implications of this property, consider the case of N= 8, for which the data sequence may be written down as By the property under discussion, this sequence should be interpreted as Then if our data are defined for negative indices (times), and, say, are of the following form SIGNAL PROCESSING FUNDAMENTALS 25
- 45. they should be fed into an FFT program as To further drive home the implications of the periodicity property in (62), consider the followingexample, which consists of taking an 8 element FFT of the data We insist for the sake of explaining a point, that only an 8 element FFT be taken. If the given data have no association with time, then the data should be fed into the program as they are presented. However, if it is definitely known that the data are ZERO before the first element, then the sequence presented to the FFT program should look like This sequence represents the given fact that att = -1, - 2and - 3thedata are supposed to be zero. Also, since the fifth element represents both x4 and x-4 (these two elements are supposed to be equal for ideal data), and since in the given data the element x_4 is zero, we simply replace the fifth element by the average of the two. Note that in the data fed into the FFT program, the sharp discontinuity at the origin, as represented by the transition from 0 to 0.9, has been retained. This discontinuitywill contribute primarily to thehigh frequency content of the transform of the signal. 2.1.10 How to Increase Frequency Domain Display Resolution of Signals Defined for Negative Time Let's say that we have an eight element sequence of data defined for both positive and negative times as follows: It can be fed into an FFT algorithm after it is rearranged to look like If x-4 was also defined in the original sequence, we have three options: we can either ignore x_4, or ignore x4 and retain x_4 for the fifth from left position in the above sequence, or, better yet, use (x-4 + x4)/2 for the fifth 26 COMPUTERIZED TOMOGRAPfflC IMAGING
- 46. position. Note we are making use of the property that due to the data periodicity properties assumed by the FFT algorithm, the fifth element corresponds to both x4 and x_4 and in the ideal case they are supposed to be equal to each other. Now suppose we wish to double the display resolution in the frequency domain; we must then zero-extend the data as follows Note that we have now given separate identities to x4 and x_4, since they don't have to be equal to each other anymore. So if they are separately available, they can be used as such. 2.1.11 Data Truncation Effects To see the data truncation effects, consider a signal defined for all indices n. If X(w) is the true DFT of this signal, we have Suppose we decide to take only a 16 element transform, meaning that of all the xn'S, we will retain only 16. Assuming that the most significant transitions of the signal occur in the base interval defined by n going from -7 to 8, we may write approximately More precisely, if X'(w) denotes the DFT of the truncated data, we may write where I16(n) is a function that is equal to 1for n between - 7 and 8, andzero outside. By the convolution theorem SIGNAL PROCESSING FUNDAMENTALS 27
- 47. Fig. 2.14: Truncating a sequence of data is equivalent to multiplying it by arectangular window. The result in the frequency domain is to convolve the Fourier transform of the signal with the window shown above. where with N = 16. This function is displayed in Fig. 2.14, and illustrates the nature of distortion introduced by data truncation. 2.2 Image Processing The signal processing concepts described in the first half of this chapter are easily extended to two dimensions.As was done before, we will describe how to represent an image with delta functions, linear operations on images and the use of the Fourier transform. 2.2.1 Point Sources and Delta Functions Let O be an operation that takes pictures into pictures; given the input picture /, the result of applying O to / is denoted by O[f]. Like the 1- dimensional case discussed earlier in this chapter, we call O linear if for all pictures, /, g and all constants a, b. In the analysis of linear operations on pictures, the concept of a point 28 COMPUTERIZED TOMOGRAPHIC IMAGING
- 48. Fig. 2.15: As in the one-dimensional case, the delta function ( ) is defined as the limit of the rectanglefunction shown here. source is very convenient. If any arbitrary picture/could be considered to be a sum of point sources, then a knowledge of the operation's output for a point source input could be used to determine the output for /. Whereas for one- dimensional signal processing the response due to a point source input is called the impulse response, in image processing it is usually referred to as thepoint spread function of O. If in addition the point spread function is not dependent on the location of the point source input then the operation is said to be space invariant. A point source can be regarded as the limit of a sequence of pictures whose nonzero values become more and more concentrated spatially. Note that in order for the total brightness to be the same for each of these pictures, their nonzero values must get larger and larger. As an example of such a sequence of pictures, let (see Fig. 2.15) and let Thus n is zero outside the l/n x l/n square described by x l/2n, y l/2n and has constant value n2 inside that square. It follows that for any n. As n the sequence does not have a limit in the usual sense, but it is convenient to treat it as though its limit existed. This limit, denoted by 6, is SIGNAL PROCESSING FUNDAMENTALS 29
- 49. called a Dirac deltafunction. Evidently, we have (x, y) = 0 for all (x, y) other than (0, 0) where it is infinite. It follows that (-x, -y) = (x, y). A number of the properties of the one-dimensional delta function described earlier extend easily to the two-dimensional case. For example, in light of (103), we can write More generally, consider the integral g(x, y) n(x, y) dx dy. This isjust the average of g(x, y) over a 1/n x 1/n square centered at the origin. Thus in the limit we retain just the value at the origin itself, so that we can conclude that the area under the delta function is one and write If weshift 6bytheamount (a, ), i.e., we use (x - a, y - ) insteadof (x, y), we similarly obtain the value of g at the point (a, ), i.e., The same is true for any region of integration containing (a, ). Equation (106) is called the "sifting" property of the 5 function. As a final useful property of 6, we have For a discussion of this property, see Papoulis [Pap62]. 2.2.2 Linear Shift InvariantOperations Again let us consider a linear operation on images. The point spread function, which is the output image for an input point source at the origin of the xy-plane, is denoted by h(x, y). A linear operation is said to be shift invariant (or space invariant, or position invariant) iftheresponse to (x - a, y - ), which isapoint source located at (a, ) in the xy-plane, is given by h(x - a, y - ). In other words, the output is merely shifted by and in the x and y directions, respectively. 30 COMPUTERIZED TOMOGRAPHIC IMAGING
- 50. Now let us consider an arbitrary inputpicture f(x, y). By (106) thispicture can be considered to be a linear sumof point sources. We can writef(x, y) as In other words, the image f(x, y) is a linear sum of point sources located at ( , ) inthe xy-plane witha and ranging from - to + . Inthis sum the point source at a particular value of (a , ) has "strength" f(a, ). Letthe response of the operation to the input f(x, y) be denoted by O[f]. If we assume the operation to be shift invariant, then by the interpretationjustgiven to the right-hand side of (108), we obtain by the linearity of the operation, which means that the response to a sum of excitations is equal to the sum of responses to each excitation. As stated earlier, the response to (a - x, - y) [ = (x - a, y — )], which is a point source located at (a, ), is given by h(x - a, y - ) andif O[f] is denoted by g, we obtain The right-hand side is called the convolution of f and h, and is often denoted by f * h. The integrand is a product of two functionsf(a, ) and h(a, ) with the latter rotated about the origin by 180° and shifted by x and y along the x and y directions, respectively. A simple change of variables shows that (111) can also be written as so that f* h = h *f. Fig. 2.16 shows the effect of a simple blurring operation on two different images. In this case the point response, h, is given by As can be seen in Fig. 2.16 one effect of this convolution is to smooth out the edges of each image. SIGNAL PROCESSING FUNDAMENTALS 31
- 51. Fig. 2.16: The two-dimensional convolutions of a circular point spread function and a square (a) and a binary image (b) are shown.
- 52. 2.2.3 Fourier Analysis Representing two-dimensional images in the Fourier domain is as useful as it is in the one-dimensional case. Let f(x, y) be a function of two independent variables x and y; then its Fourier transform F(u, v) is defined by In the definition of the one- and two-dimensional Fourier transforms we have used slightly different notations. Equation (26)represents the frequency in terms of radians per unit length while the above equation represents frequency in terms of cycles per unit length. The two forms are identical except for a scaling and either form can be converted to the other using the relation By splitting the exponential into two halves it is easy to see that thetwo- dimensional Fourier transform can be considered as two one-dimensional transforms; first with respect to x and then y In general, F isa complex-valued function of u andv.Asanexample, let f(x, y) = rect (x, y). Carrying out the integration indicated in (114) we find This last function is usually denoted by sinc (u, v) and is illustrated in Fig. 2.17. More generally, using thechange ofvariables x' = nx andy' = ny, it is easy to show that the Fourier transform of rect (nx, ny) is Given the definition of the Dirac delta function as a limit of a sequence of the functions n2 rect (nx, ny); by the arguments in Section 2.1.3, the Fourier transform of the Dirac delta function is the limit of the sequence of Fourier SIGNAL PROCESSING FUNDAMENTALS 33
- 53. Fig. 2.17: The two-dimensional Fourier transform of the rectanglefunction is shown here. transforms sinc (u/n, v/n). In other words, when then The inverse Fourier transform of F(u, v) is found by multiplying both sides of (114) by eJ2 (ux+v ) and integrating with respect to u and v to find or Making use of (107) it is easily shown that or equivalently 34 COMPUTERIZED TOMOGRAPHIC IMAGING
- 54. This integral is called the inverse Fourier transform of F(u, v). By (114) and (127), f(x, y) and F(u, v) form a Fourier transform pair. If x and y represent spatial coordinates, (127) can be used to give a physical interpretation to the Fourier transform F(ut u)and to the coordinates u and v. Let us first examine the function The real and imaginary parts of this function are cos 2 (ux + vy) and sin 2 (ux + vy), respectively. In Fig. 2.18(a), we have shown cos 2 (ux + vy). It is clear that if one took a section of this two-dimensional pattern parallel to the x-axis, it goes through u cycles per unit distance, while a section parallel to the .y-axis goes through v cycles per unit distance. This is the reason why u and v are called the spatialfrequencies along the x- and y- axes, respectively. Also, from the figure it can be seen that the spatial period of the pattern is (u2 + u2 )-1/2 . The plot for sin 2 (ux + vy) looks similar to the one in Fig. 2.18(a) except that it is displaced by a quarter period in the direction of maximum rate of change. From the preceding discussion it is clear that eJ2 (ux+vy) is a two- dimensional pattern, the sections of which, parallel to the x- and.y-axes, are spatially periodic with frequencies u and u, respectively. The pattern itself has a spatial period of (u2 + v2 )-1/2 along a direction that subtends an angle tan-1 (v/u) with the x-axis. By changing u and v, one can generate patterns with spatial periods ranging from 0 to in any direction in the xy-plane. Equation (127) can, therefore, be interpreted to mean that/(x,y) is a linear combination of elementary periodic patterns of the form ej2 (ux+vy) . Evidently, the function, F(u, v), is simply a weighting factor that is a measure of the relative contribution of the elementary pattern to the total sum. Since u and v are the spatial frequency of the pattern in the x andy directions, F(u, u) is called thefrequency spectrum of f(x, y). 2.2.4 Properties of Fourier Transforms Several properties of the two-dimensional Fourier transform follow easily from the defining integrals equation. Let F{f} denote the Fourier transform of a function f(x, y). Then F{f(x, y)} - F(u, v). We will nowpresent without proof some of the more common properties of Fourier transforms. The proofs are, for the most part, left for the reader (see the books by Goodman [Goo68] and Papoulis [Pap62]). 1) Linearity: This follows from the linearity of the integration operation. SIGNAL PROCESSING FUNDAMENTALS 35
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- 56. Nearly all that remained of the ancient palace was the prison or “conciergerie,” where Montgomery, who by mishap had slain his king in a tournament, and, at a later period, Damiens of the Four Horses had been confined. The tower of the conciergerie was for a long time called the Montgomery Tower. Besides the conciergerie, the hall known as the Salle des Pas Perdus and the so-called “Kitchen of Saint-Louis,” with an immense chimney-piece in each of the four corners, formed part of the ancient building. In 1776 the Palais de Justice again took fire, and again was in great part reconstructed. In 1835, under Louis Philippe, the Town of Paris decided to enlarge it, and the plan by M. Huyot, the architect, was adopted by the Municipal Council in 1840. The royal sanction was then obtained; but Louis Philippe did not remain long enough on the throne to see the work of construction terminated. The Republican Government of 1848 stopped the building; and it was only under the Second Empire in 1854 that it was resumed, to be completed in 1868. More important by far than the re- alterations, additions, and reconstructions of which the Palais de Justice has in successive centuries been made the subject have been the changes in the French law, and in various matters connected with its administration. Up to the time of the Revolution citizens were {254} arrested in the most arbitrary manner on mere suspicion, and imprisoned for an indefinite time without being able to demand justice in any form. Some half a dozen years before the uprising of 1789 the king had decreed that no one should be arrested except on a definite accusation; but the order was habitually set at nought. The Palais de Justice of the present day occupies about one third of the total surface of the Cité. Enclosed on the east by the Boulevard du Palais, on the west by the Rue de Harlay, on the north by the Quai de l’Horloge, and on the south by the Quai des Orfèvres, it forms a quadrilateral mass in which all styles are opposed and confused, from the feudal towers of the Quai de l’Horloge to the new buildings begun in Napoleon III.’s reign, but never completed. To the left of this strange agglomeration the air is pierced by the graceful spire of the Sainte-Chapelle, admirable monument of the piety and of the art of the middle ages. Some portions of the ancient Palace of Justice are preserved in the modern edifice, but only the substructures, as, for instance, in the northern buildings facing the Seine. The principal gate, and the central pavilion with
- 57. its admirable façade at the bottom of the courtyard opening on to the Boulevard du Palais, were constructed under the reign of Louis XVI. The northern portion, from the clock tower, at the corner of the quay, to the third tower behind, has been restored or rebuilt in the course of the last thirty years. All the rest of the building is absolutely new. The clock tower, a fine specimen of the military architecture of the fourteenth century, was furnished in 1370 by order of Charles V. with the first large clock that had been seen in Paris, the work of a German, called in France Henri de Vic. To this clock the northern quay owes its name of “Quai de l’Horloge du Palais” or “Quai de l’Horloge.” The bell suspended in the upper part of the tower is said to have sounded the signal for the massacre of the Protestants on the eve of St. Bartholomew’s Day, August 24, 1572; a doubtful honour, which is also claimed for the bell of Saint- Germain-l’Auxerrois. The Palais de Justice, as it now exists, possesses a threefold character— legal, administrative, and punitive. Here cases are tried, here the Prefect of Police performs the multifarious duties of his office, and here criminals are imprisoned. Of the various law courts the Palais de Justice contains five: the Court of Cassation, in which appeal cases are finally heard on questions of form, but of form only; the Court of Appeal, the Court of Assizes, the Tribunal of First Instance, and the Tribunal of Police. These fill the halls of the immense building. The Court of Cassation, divided into three chambers, counts forty-eight counsellors, a first president, three presidents of chamber, a procurator- general, six advocates-general, a registrar-in-chief, four ordinary registrars, three secretaries of the court, a librarian, eight ushers, and a receiver of registrations and fines; altogether seventy-seven persons. The Court of Appeal, divided into seven chambers, is composed of a first president, seven presidents of chamber, sixty-four counsellors, a procurator-general, seven advocates-general, eleven substitutes attached to the court, a registrar-in-chief, and fourteen ordinary registrars; altogether 106 persons. The number of officials and clerks employed in the Tribunal of First Instance is still greater. Divided into eleven chambers, the tribunal comprises one president, eleven vice-presidents, sixty-two judges, and fifteen supplementary judges, a public prosecutor, twenty-six substitutes, a registrar-in-chief, and forty-five clerks of registration. As for the Police Court, it is presided over in turn by each of the twenty magistrates of Paris,
- 58. two Commissaries of Police doing duty as assessors. With the addition of two registrars and a secretary the entire establishment consists of six persons. The entire number of judges, magistrates, registrars, and secretaries employed at the Palais de Justice amounts to 351; without counting a floating body of some hundreds of barristers, solicitors, ushers, and clerks, thronging like a swarm of black ants a labyrinth of staircases, corridors, and passages. Yet the Palais de Justice, constantly growing, is still insufficient for the multiplicity of demands made upon it. The history of the Palais de Justice is marked by the fires in which it has from time to time been burned down. The first of these broke out on the night of the 5th of March, 1618, when the principal hall and most of the buildings adjoining it were destroyed. The second, which took place on the 27th of October, 1737, consumed the buildings forming the Chamber of Accounts, situated at the bottom of the courtyard of the Sainte-Chapelle— an edifice of surpassing beauty, constructed in the fifteenth century by Jean Joconde, a monk of the Order of Saint Dominic. {255} The third fire declared itself during the night of January 10, 1776, in the hall known as the Prisoners’ Gallery, from which it spread to all the central buildings. In this conflagration perished the old Montgomery Tower. The last of the fires in which so many portions of the Palais de Justice have turn by turn succumbed, was lighted by order of the insurgent Commune on the 24th of May, 1871, when the troops from Versailles were entering Paris. The principal hall, the prison, the old towers with all the civil and criminal archives (in the destruction of the latter the insurgents may have been specially interested) were all consumed. These repeated catastrophes, together with numerous restorations, have left standing but very little of the ancient Palais de Justice. The central pavilion, reconstructed under Louis XVI. in accordance with the plans of the architect Desmaisons, is connected with two galleries of historical interest, on one side with the Galerie Mercière, on the other with the Galerie Marchande. The names of “Mercière” and “Marchande” recall the time when the galleries so named, as well as the principal hall and the outer walls of the palace, were occupied by stalls and booths in which young and pretty shop-girls sold all sorts of fashionable and frivolous trifles, such as ribbons, bows, and embroideries. Here, too, new books were offered for sale. Here Claude Barbin and his rivals sold to the patrons and patronesses
- 59. THE CLOCK OF THE PALAIS DE JUSTICE. of the stage the latest works of Corneille, Molière, and Racine. Here appointments of various kinds were made, but especially of one kind. The Palace Gallery, or Galerie du Palais, was the great meeting-place for the fashionable world until only a few years before the great Revolution, when it was deserted for the Palais Royal. Some of its little shops continued to live a meagre life until the reign of Louis Philippe. Now everything of the kind has disappeared, with the exception of two privileged establishments where “toques” and togas—in plain English, caps and gowns—can be bought, or even hired, by barristers attending the “palace.” The entrance to the central building is from the Galerie Mercière, through a portico supported by Ionic columns, and surmounted by the arms of France. The visitor reaches a broad, well-lighted staircase, where, half- way up, stands in a niche an impressive statue of Law, the work of Gois, bearing in one hand a sceptre, and in the other the Book of the Law, inscribed with the legend “In legibus salus.” The grand staircase of the Palais leads through a waiting-room, which serves also as a library, to the three first chambers of the Court of Appeal. The rooms are of a becomingly severe aspect. The walls are painted a greenish grey, of one uniform tint. The tribunal is sometimes oblong, sometimes in horse- shoe form. On the right sits the assessor representing the Minister of Justice, on the left the registrar on duty. In the “parquet,” or enclosure beneath the tribunal, is the table of the usher, who calls the next case, executes the president’s behests, and maintains order in the court, exclaiming “Silence, gentlemen,” with the traditional voice and accent. The “parquet” is shut in by a balustrade technically known as the bar, on which lean the advocates as they deliver their speeches. The space {256}
- 60. furnished with benches which is reserved for them, and where plaintiff and defendant may also sit, is enclosed by a second bar, designed to keep off the public properly so-called, and prevent it from pressing too closely upon the court. There is no witness-box in a French court. The witness stands in the middle of the court and recites, often in a speech that has evidently been prepared beforehand, all he knows about the case under trial. ENTRANCE TO THE COURT OF ASSIZE. Such is the general disposition of all the assize chambers in the Palais de Justice. Some, however, present features of their own. The first chamber, for instance, contains a magnificent Calvary, by Van Eyck; one of the rare objects of art which survive from the ancient ornamentation of the palace. On the centre of the picture, rising like a dome between two side panels, is the Saviour on the Cross. On His right is the Virgin supported by two holy women, by Saint John the Baptist and by Saint Louis, graced with the exact features of King Charles VII., under whose reign this masterpiece was executed. On the left are Saint John the Evangelist, Saint Denis, and Saint Charlemagne. Above the head of our Lord are the Holy Ghost and the Eternal Father surrounded by angels, while the background is occupied by a landscape less real than curious; for it represents the City of Jerusalem, the
- 61. Tower of Nesle, the Louvre, and the Gothic buildings of the Palais de Justice. This work, by the great painter of Bruges, executed in the early part of the fifteenth century, was formerly in the Principal Hall of the Parliament, beneath the portrait of Louis XII., which the people (whose “father” he claimed to be) destroyed in 1793. The portion of the building which contains the three first chambers of the court—behind the portico opening on to the Galerie Mercière—escaped the fire of 1776. Its lateral and southern {257} façade, turned towards the courtyard of the Sainte- Chapelle, is pierced with lofty windows, sculptured in the Renaissance style. It must have been constructed under the Valois, or under the reign of Henri IV. But it is difficult to ascertain its early history, for but few writers have given much attention to the subject. THE PALAIS DE JUSTICE.
- 62. THE PALAIS DE JUSTICE AND SAINTE-CHAPELLE. {258} The fifth, sixth, and seventh chambers of the Court of Appeal are all entered from the Galerie Marchande; while the fourth chamber stands in the north-east corner of the said gallery. On the left of the Galerie Mercière is the famous Salle des Pas Perdus, seventy-four metres long and twenty-eight broad. This is the great entrance hall to the courts generally. Why it should be called “Salle des Pas Perdus” is not evident, though the name may be due either to the “lost steps” of litigants bringing or defending actions without result, or, more probably, to the “lost steps” of those who walk
- 63. wearily to and fro for an indefinite time, vainly expecting their case to be called on. Whatever the derivation of its name, the Salle des Pas Perdus is considered one of the finest halls in Europe. Twice has it been destroyed by fire and twice rebuilt. The first large hall of the palace, as it was at that time called, was built under Philip the Fair and finished towards 1313. It was adorned successively with the statues of the kings of France from Pharamond to Francis I.; the successful ones being represented with their hands raised to heaven in token of thanksgiving, the unfortunate ones with head and hands lowered towards the ground. The most celebrated ornament of the large hall was the immense marble table of which ample mention has already been made. After the fire of 1618 (in which the table split into several pieces, still preserved in the vaults of the palace) a new hall on the same site, and of the same dimensions as the old one, was built by Jacques Desbrosses, which was burnt in 1871 by the Commune, to be promptly rebuilt by MM. Duc Dommey and Daumet. The seven civil chambers of the tribunal are entered through the Salle des Pas Perdus, either from the ground floor or from the upper storey, which is reached by two staircases. This portion of the palace was partly reconstructed in 1853 under the reign of Napoleon III., Baron Haussmann being Prefect of the Seine. The fact is recorded on a marble slab let into one of the walls. In the middle of the south part of the Salle des Pas Perdus, a marble monument was raised in 1821 to Malesherbes, the courageous advocate who defended Louis XVI. at the bar of the Convention. The monument comprises the statue of Malesherbes with figures of France and Fidelity by his side. On the pedestal are low reliefs, representing the different phases of the memorable trial. The statues are by Cortot, the illustrative details by Bosio. The Latin inscription engraved on the pedestal was composed by Louis XVIII., in whose reign the monument was executed and placed in its present position. This king, who translated Horace and otherwise distinguished himself as a Latinist, is the author of more than one historical inscription in the Latin language, and he commemorated by this means, not only the heroism of Malesherbes, who defended Louis XVI. at the trial, but also the piety of the Abbé Edgeworth, who accompanied him to the scaffold. Towards the end of the hall, on the other side, is the statue of Berryer, which, according to M. Vitu, is “the homage paid to eloquence considered
- 64. as the auxiliary of justice.” In the north-east corner of the Hall of Lost Steps, to the left of Berryer’s monument, is the entrance to the first chamber, once the bed-chamber of Saint Louis, and which, reconstructed with great magnificence by Louis XII. for his marriage with Mary of England, daughter of King Henry VII., took the name of the Golden Room. It afterwards played an important part in the annals of the Parliament of Paris. Here Marshal de Biron was condemned to death on the 28th of July, 1602. Here a like sentence was pronounced against Marshal d’Ancre on the 8th of July, 1617. Here the kings of France held their Bed of Justice, solidly built up at the bottom of the hall in the right corner, and composed of a lofty pile of cushions, covered with blue velvet, in which golden fleurs de lis were worked. Here, finally, on the 3rd of May, 1788, the Marquis d’Agoult, commanding three detachments of French Guards, Swiss Guards, Sappers, and Cavalry, entered to arrest Counsellors d’Épréménil and Goislard, when the president, surrounded by 150 magistrates and seventeen peers of France, every one wearing the insignia of his dignity, called upon him to point out the two inculpated members, and exclaimed: “We are all d’Épréménil and Goislard! What crime have they committed?” A resolution had been obtained from the Parliament declaring that the nation alone had the right to impose taxes through the States-General. This resolution and the scene which followed were the prelude to the French Revolution. Four years later there was no longer either monarch or parliament, French Guards or Swiss Guards. The great chamber of the palace had become the “Hall of Equality,” where, on the 17th of April, 1792, was established the first Revolutionary Tribunal, to be replaced {259} on the 10th of May, 1793, by the criminal tribunal extraordinary; which was reorganised on the 26th of September by a decree which contained this phrase, still more extraordinary than the tribunal itself: “A defender is granted by law to calumniated patriots, but refused to conspirators.” Here were arraigned—one cannot say tried—that same d’Épréménil who had proclaimed the rights of the nation, and Barnave, the Girondists, the Queen of France, Mme. Élizabeth, Danton, Camille Desmoulins, Chaumette, Hébert, and Fabre d’Églantine; then, one after the other, the Robespierres, with Couthon, Collot d’Herbois, Saint-Just, Henriot, and Fouquier-Tinville —altogether 2,742 victims, whose 2,742 heads fell into the red basket either on the former Place Louis XV., which had become the Place de la Révolution and was afterwards to be known as the Place de la Concorde, or
- 65. on the Place du Trône. The numbered list, which used to be sent out, like a newspaper, to subscribers, has been preserved. It began with the slaughter of the 26th of August, 1792, in which La Porte, intendant of the civil list, the journalist Durozoi, and the venerable Jacques Cazotte, author of “Le Diable Amoureux,” lost their heads. Cazotte had kept up a long correspondence with Ponteaux, secretary of the civil list, and had sent him several plans for the escape of the Royal Family, together with suggestions, from his point of view invaluable, for crushing the revolution. The letters were seized at the house of the intendant of the civil list, the before-mentioned La Porte; and thereupon Cazotte was arrested. His daughter Elizabeth followed him to prison; and they were both at the Abbaye during the atrocious massacres of September. The unhappy young girl had been separated from her father since the beginning of the executions, and she now thought only of rejoining him either to save his life or to die with him. Suddenly she heard him call out, and then hurried down a staircase in the midst of a jingle of arms. Before there was time to arrest him she rushed towards him, reached him, threw her arms around him, and so moved the terrible judges by her daughterly affection that they were completely disarmed. Not only was the old man spared, but he and his heroic daughter were sent back with a guard of honour to their home. Soon afterwards, however, the father was again arrested, and brought before the revolutionary tribunal. On the advice of the counsel defending him, he denied the competence of the court on the plea of autrefois acquit. It was ruled, however, that the court was dealing with new facts, and the judges had indeed simply to apply the decree pronounced against those who had taken part in preparing the repression of the 10th of August. The evidence against Cazotte was only too clear, and he was condemned to death; which suggested the epigram that “Judges struck where executioners had spared.” But these very judges, bound by inflexible laws, could not refuse the expression of their pity and esteem to the unhappy old man. While condemning him to death they rendered homage to his honesty and his courage. “Why,” exclaimed the public accuser, “after a virtuous life of seventy-two years, must you now be declared guilty? Because it is not sufficient to be a good husband and a good father; because one must also be a good citizen.” The President of the Court, in pronouncing sentence, said with gravity and emotion: “Old man, regard the approach of death without
- 66. fear. It has no power to alarm you. It can have no terrors for such a man as you.” Cazotte ascended with fortitude the steps of the scaffold, and exclaimed, before lowering his head: “I die as I have lived, faithful to my God and to my king.” The last victim of the 2,472 was Coffinhal, vice-president of the Revolutionary Tribunal, and member of the Council-General of the Paris Commune. No show of equity, no imitation even of judicial forms, gave colour to these bloody sacrifices. Most of the victims, condemned beforehand, were brought to the prison of the Conciergerie at eight in the morning, led before the tribunal at two, and executed at four. A printing office established in a room adjoining the court was connected with the latter by an opening in the wall, through which notes and documents relating to the case before the tribunal were passed; and often the sentence was composed, printed, and hawked for sale in the streets before being read to the victims. “You disgrace the guillotine!” said Robespierre one day to Fouquier- Tinville, the public accuser. Of this historic hall nothing now remains but the four walls. Still, however, may be seen the little door of the staircase which Marie Antoinette ascended to appear before the revolutionary jury, and which she afterwards descended on the way to her dungeon. The Galerie Saint-Louis is the name given to the ancient gallery {260} connected with the Galerie Marchande, its name being justified by the various forms in which incidents from the life of Saint Louis are represented on its walls. Here, in sculptured and coloured wood, is the effigy of Saint Louis, close to the open space where, when centuries ago it was a garden, the pious king was wont to imitate, and sometimes to render, justice beneath the spreading trees. One of the bureaux in the Palais de Justice contains an alphabetical list of all the sentences passed, by no matter what court, against any person born in one of the districts of Paris or of the department of the Seine. This record, contemplated by Napoleon I., was established in 1851 by M. Rouher, at that time Minister of Justice. The list is kept strictly secret; nor is any extract permitted except on the requisition of a magistrate, or on the application of one of the persons sentenced, requiring it in his own interest.
- 67. THE FAÇADE OF THE OLD PALAIS DE JUSTICE. The Bureau of “Judicial Assistance,” dating from 1851, enables any indigent person to plead in formâ pauperis, whether as plaintiff or defendant. Nor is he obliged to plead in person. Not only stamped paper, but solicitors, barristers, and every legal luxury are supplied to him gratuitously. It is at the expense of the lawyers that the pauper litigant is relieved. Two curious bureaux connected with the Palais de Justice are those in which are kept, sealed up and divided into series indicated by different colours, objects of special value taken from persons brought before the court, or voluntarily deposited by them; together with sums of money which, in like manner, have passed into the hands of legal authorities. Still more curious is the collection of articles of all kinds stored in a sort of museum, which presents the aspect at once of a bazaar and of a pawnbroker’s shop. Here, in striking confusion, are seen boots and shoes, clothes, wigs, rags, and a variety of things seized and condemned as fraudulent imitations; likewise instruments of fraud, such as false scales. Here, too, in abundance are murderous arms—knives, daggers, and revolvers. Singularly interesting is the collection of burglarious instruments of the most different patterns, from the enormous lump of iron, which might be used as a battering ram, to the most delicately-made skeleton key, feeble enough in appearance, but sufficiently strong to force the lock of an iron safe. There is now scarcely room for the constantly increasing collection of objects at the service of fraud and crime. Beneath this strange exhibition, rendered still more sinister by the method and order with which it is arranged, are disposed in two storeys the
- 68. four chambers which together constitute the civil tribunal. {261} Connected with the criminal tribunal, their duty is to try offences punishable by a scale of sentences, with five years’ imprisonment as the maximum. According to one of the last legislative enactments of the Second Empire, persons brought before a police-court remained provisionally at liberty except under grave circumstances. Cases, moreover, in which the offender has been taken in flagrante delicto are decided in three days. “This is a sign of progress,” says M. Vitu; “but Paris still needs an institution of which London is justly proud, that of district magistrates, something like our juges de paix, deciding police cases forthwith. The principal merit of this institution is that it prevents arbitrary detention and serious mistakes such as unfortunately are only too frequent with us. Instances have occurred, and will occur again, in which an inoffensive man, arrested by mistake, in virtue of a regular warrant intended for another of the same name, is sent straight to the criminal prison of Mazas. It will then take him a week to get set at liberty. In London he would have been taken at once to the magistrate of the district, who would have proceeded without delay to the verification of his identity. It would have been the affair of two hours at most, thanks to the service of constables at the disposal, day and night, of the English magistrate.” THE SALLE DES PAS PERDUS.
- 69. The police-courts have sometimes to deal with remarkable cases, but as a rule their duties are of a somewhat trivial character. Adventurers of a low order, swindlers on a petty scale, and street thieves who have been caught with their hands in the pocket of a gentleman or the muff of a lady, are the sort of persons they usually deal with. To these may be added vendors of pretended theatrical admissions, hawkers of forbidden books, and a few drunkards. From morning till night the police are constantly bringing in poor wretches of both sexes; the men for the most part in blouses, the women in rags. They arrive in “cellular” {262} carriages, vulgarly called “salad baskets”; and leaving the vehicle they are kept together by a long cord attached to the wrist of each prisoner. The place of confinement where they remain pending the trial is called the “mouse-trap”: two rows, placed one above the other, each of twenty-five cells, containing one prisoner apiece. Every cell is closed in front by an iron grating, in the centre of which is a small aperture—a little square window looking into the corridor. Through this window, which can be opened and shut, but which is almost invariably kept open, the prisoner sees all that takes place in the passage, and the occasional arrival of privileged visitors helps to break the monotony of his day. The wire cages in which the prisoners are detained suggest those of the Zoological Gardens; and the character of the wild beast is too often imprinted on the vicious criminal features of the incarcerated ones. Disputes with cab-drivers and hackney coachmen generally are, as a rule, settled by the commissary of the district or the quartier. But serious complaints have now and then to be brought before the Tribunal of Police. In former times the hackney coaches of Paris were at once the disgrace and the terror of the town. “Nothing,” writes Mercier, “can more offend the eye of a stranger than the shabby appearance of these vehicles, especially if he has ever seen the hackney coaches of London and Brussels. Yet the aspect of the drivers is still more shocking than that of the carriages, or of the skinny hacks that drag those frightful machines. Some have but half a coat on, others none at all; they are uniform in one point only, that is extreme wretchedness and insolence. You may observe the following gradation in the conduct of these brutes in human shape. Before breakfast they are pretty tractable, they grow restive towards noon, but in the evening they are not to be borne. The commissaries or justices of the peace are the only umpires between the driver and the drivee; and, right or wrong, their award is in favour of the former, who are generally taken from the honourable body of
- 70. police greyhounds, and are of course allied to the formidable phalanx of justices of the peace. However, if you would roll on at a reasonable pace, be sure you take a hackney coachman half-seas-over. Nothing is more common than to see the traces giving way, or the wheels flying off at a tangent. You find yourself with a broken shin or a bloody nose; but then, for your comfort, you have nothing to pay for the fare. Some years ago a report prevailed that some alterations were to take place in the regulation of hackney coaches; the Parisian phaetons took the alarm and drove to Choisy, where the King was at that time. The least appearance of a commotion strikes terror to the heart of a despot. The sight of 1,800 empty coaches frightened the monarch; but his apprehensions were soon removed by the vigilance of his guard and courtiers. Four representatives of the phaetonic body were clapped into prison and the speaker sent to Bicêtre, to deliver his harangue before the motley inhabitants of that dreary mansion. The safety of the inhabitants doubtless requires the attention of the Government, in providing carriages hung on better springs and generally more cleanly; but the scarcity of hay and straw, not to mention the heavy impost of twenty sols per day for the privilege of rattling over the pavement of Paris, when for the value of an English shilling you may go from one end of the town to the other, prevents the introduction of so desirable a reformation.” In another part of his always interesting “Picture of Paris,” Mercier becomes quite tragic on the subject of Paris coaches and Paris coachmen. “Look to the right,” he says, “and see the end of all public rejoicings in Paris; see that score of unfortunate men, some of them with broken legs and arms, some already dead or expiring. Most of them are parents of families, who by this catastrophe must be reduced to the most horrible misery. I had foretold this accident as the consequence of that file of coaches which passed us before. The police take so little notice of these chance medleys that it is simply a wonder such accidents, already too frequent, are not still more numerous. The threatening wheel which runs along with such rapidity carries an obdurate man in power, who has not leisure, or indeed cares not, to observe that the blood of his fellow-subjects is yet fresh on the stones over which his magnificent chariot rattles so swiftly. They talk of a reformation, but when is it to take place? All those who have any share in the administration keep carriages, and what care they for the pedestrian traveller? Jean Jacques Rousseau, in the year 1776, on the road to Mesnil- Montant, was knocked down by a large Lapland dog and remained on the
- 71. spot, whilst the master, secure in his berline, passed him by with that stoic indifference which amounts to savage barbarity. Rousseau, lame and bruised, was taken up and conducted to his house by some {263} charitable peasants. The gentleman, or rather savage, learning the identity of the person whom the dog had knocked down, sent a servant to know what he could do for him. ‘Tell him,’ said Rousseau, ‘to keep his dog chained,’ and dismissed the messenger. When a coachman has crushed or crippled a passenger, he may be carried before a commissaire, who gravely inquires whether the accident was occasioned by the fore wheels or the hind wheels. If one should die under the latter, no pecuniary damage can be recovered by the heirs-at-law, because the coachman is answerable only for the former; and even in this case there is a police standard by which he is merely judged at so much an arm and so much a leg! After this we boast of being a civilised nation!” In addition to the place of detention already described, the Palais de Justice contains a permanent prison known historically as the Conciergerie, and, by its official name, as the House of Justice. Here are received, on the one hand, prisoners about to be tried before the Assize Court or the Appeal Court of Police; on the other, certain prisoners who are the object of special favour and who consider themselves fortunate to be confined in this rather than any other prison. The list of celebrated persons who have been detained in the Conciergerie would be a long one, from the Constable of Armagnac (1440) to Prince Napoleon (1883). Here may still be seen the dungeons of Damiens, of Ravaillac, of Lacenaire the murderer, of André Chenier the poet, of Mme. Roland, and of Robespierre. The name whose memory, in connection with this fatal place, extinguishes all others is that of the unhappy Marie Antoinette. After a captivity of nearly a year in the Temple the queen was conducted on the 5th of August, 1792, to the Conciergerie, and there shut up in a dark narrow cell called the Council Hall, lighted from the courtyard by a little window crossed with iron bars. This Council Hall was previously divided into two by a partition, which had now been removed; and in place of it a screen was fixed which, during her sleep, shut the queen off from the two gendarmes ordered to watch her day and night. The daughter of the Cæsars left her dungeon on the 15th of October, 1793, dressed in black, to appear before the Revolutionary Tribunal, and the next day, dressed in white, to step into the cart which conveyed her to the guillotine erected on the Place Louis XV.
- 72. POLICE CARRIAGES. This historical dungeon, which, says M. Vitu, could not contain the tears which it has caused to be shed, and ought to have been walled up in order to bury the memory of a crime unworthy of the French nation, was transformed into a chapel by order of Louis XVIII. in 1816. The altar bears a Latin inscription which, like others previously referred to, was composed by the king himself. Close to the queen’s dungeon is the so-called Hall of the Girondists (formerly a chapel), in which the most enlightened and the most heroic of the Revolutionists are said, by a not too trustworthy legend, to have passed their last night. Locally and even architecturally connected with the Palace of Justice is the Holy Chapel, one of the most perfect sacred buildings that {264} Paris possesses. The courtyard of the Holy Chapel, mentioned more than once in connection with the Palace of Justice, stands at the south-east corner of the principal building, and is shut in by the Tribunal of Police and a portion of the Court of Appeal. It can be entered from five different points: from the Boulevard of the Palace of Justice; by two different openings from the Police Tribunal; from the so-called depôt of the Prefecture of Police; and from the Cour du Mai on the north-east. No more admirable specimen of the religious architecture of the middle ages is to be found; nor is any church or chapel more venerable by its origin and its antiquity. Founded by Robert I. in 921, the year of his accession to the throne, it replaced, in the royal palace of which it had formed part, a chapel dedicated to Saint Bartholomew, which dated from the kings of the first dynasty.
- 73. THE CONCIERGERIE, PALAIS DE JUSTICE. The royal palace contained, moreover, several private oratories, including in particular one dedicated to the Holy Virgin. In 1237 Baudouin II., Emperor of Constantinople, exhausted by the wars he had been sustaining against the Greeks, came to France to beg assistance from King Saint Louis. Baudouin was of the House of Flanders, and in consideration of a large sum of money, he pledged to the French king his county of Namur, and allowed him to redeem certain holy relics—the crown of thorns, the sponge which had wiped away the blood and sweat of the Saviour, and the lance with which his side had been pierced—on which the Venetians, the Genoese, the Abbess of Perceul, Pietro Cornaro, and Peter Zauni had lent 13,000 gold pieces. The relics arrived in France the year afterwards, and crossed the country in the midst of pious demonstrations from the whole population. The king himself, and the Count of Artois, went to receive them at Sens and bore on their {265} shoulders the case containing the crown of thorns. Thus, in formal procession, they passed through the streets of Sens and of Paris; and the holy king deposited the relics in the oratory of the Virgin until a building should be erected specially for their reception. This was the Holy Chapel, of which the first stone was laid in
- 74. 1245. The work had been entrusted to the architect Pierre de Montreuil or de Montereau. In three years it was finished, the chapel being inaugurated on the 25th of April, 1248. “Only three years for the construction of such an edifice,” exclaims a French writer, “when the nineteenth century cannot manage to restore it in thirty years!” THE SAINTE-CHAPELLE. The Holy Chapel is composed of two chapels one above the other, having a single nave without transept, each chapel possessing a separate entrance. The upper chapel, approached through the Galerie Mercière, was reserved for the king and his family, who, from the royal palace, entered it on foot. The lower chapel, intended for the inferior officers attached to the
- 75. court, became later on, in virtue of a papal bull, the parish church of all who lives in the immediate neighbourhood of the palace. If the Holy Chapel is admirable by its design and proportions, it is a marvel of construction from a technical point of view. It rests on slender columns, which seem incapable of supporting it. The roof, in pointed vaulting, is very lofty; and for the last six centuries it has {266} resisted every cause of destruction, including the fire which, in 1630, threatened the entire building. No more beautiful specimens of stained glass are to be seen than in the Holy Chapel, with its immense windows resplendent in rich and varied colours. A remarkable statue of the Virgin bowing her head as if in token of assent, now at the Hôtel Cluny, belonged originally to the Holy Chapel. According to a pious legend, the figure bent forward to show approval of the doctrine of the Immaculate Conception as formulated by Duns Scotus, who was teaching theology at Paris in 1304, and from the time of the miracle until now maintains the same gesture of inclination. More than one mediæval tradition makes statues, and especially statues of the Virgin, perform similar actions. There is, for example, in the Contes Dévots a story of a statue of the Virgin to which a certain bourgeois qui aimait une dame prayed that she would either make the lady return his love or cause that love to cease. Some time previously a Hebrew magician had offered to secure the lady’s affections for the infatuated bourgeois provided he would renounce God, the saints, and especially the Blessed Virgin; to which the despondent lover replied that though, in his grief and despair, he might abandon everything else, yet nothing could make him relinquish his allegiance and devotion to the Blessed Virgin. This fidelity, under all temptations, gave him some right, he hoped, to implore the influence of the merciful Virgin towards softening the heart of the woman he so passionately loved; and the statue of the Virgin, before which he prostrated himself, showed by a gentle inclination of the head that his prayer was heard. Fortunately, the lady whose cold demeanour had so vexed the heart of her lover was in the church at the very moment of the miracle, and, seeing the Virgin bow her head to the unhappy bourgeois, felt convinced that he must be an excellent man. Thereupon she went up to him, asked him why he looked so sad, reproached him gently with not having visited her of late, and ended by assuring him that if he still loved her she fully returned his affection. Somewhat analogous to this legend, though in a different order of
- 76. ideas, is that of the Commander whose statue Don Juan invited to supper, with consequences too familiar to be worth repeating. The ancient statue of the Virgin, once in the Holy Chapel, venerated now in the Hôtel Cluny, regarded simply as a curiosity, has been replaced by a modern statue. The sacred relics which the Holy Chapel at one time possessed are still preserved at Notre Dame. The gold case which enclosed them was, at the beginning of the Revolution, sent to the Mint to be converted into coin. The spire which now surmounts the Holy Chapel is the fourth since the erection of the building. The first one, by Pierre de Montreuil, was crumbling away from age under the reign of Charles V., who thereupon had it restored by a master-carpenter, Robert Foucher. Burnt in the great fire of 1630, this second spire was re-constructed by order of Louis XIII., and destroyed during the Revolution. The fourth edition of it, which still exists, was built by M. Lassus in the florid style of the first years of the fifteenth century. The one thing which strikes the visitor to the Holy Chapel above everything else, and which cannot but make a lasting impression on him, is the wonderful beauty of the stained glass windows already referred to. They date, for the most part, from the reign of Saint Louis, and were put in on the day the building was consecrated in 1248. In their present condition and form, however, they take us back only to the year 1837. During forty-six years (1791 to 1837) the Holy Chapel was given up to all kinds of uses. First it was a club-house, then a flour magazine, and finally a bureau for official documents. This last was the least injurious of the purposes to which it was turned. Nevertheless the incomparable stained glass windows were interfered with by the construction of various boxes and cupboards along the sides of the building, no less than three metres of the lower part of each window being thus sacrificed. Certain glaziers, moreover, employed to take down the windows, clean them, and put them back, had made serious mistakes, restoring portions of windows to the wrong frames. The subjects of the stained art-work are all from the Holy Scriptures, and on a thousand glass panels figure a thousand different personages. The restoration of the windows had been entrusted, after a public competition, to M. Henri Gérante, a French artist who, more than any other, has contributed to the resurrection of the seemingly lost art of painting on
- 77. glass. But, unhappily, M. Gérante died before beginning his work, which, thereupon, was divided between M. Steintheil, for the {267} drawing and painting, and M. Lusson for the material preparation. Their labours were crowned with the most complete success. Entering the Holy Chapel one is literally dazzled by the bright rich colours from the windows on all sides, blending together in the most harmonious manner. {268} THE LOWER CHAPEL OF THE SAINTE-CHAPELLE. Right and left of the nave the place is shown where Saint Louis and Blanche de Castille were accustomed to sit opposite one another to hear mass and other religious services. A corner, moreover, is pointed out, with an iron network before it, where, according to a doubtful tradition, the suspicious Louis XI. used to retire in order to hear mass without being seen; perhaps also to watch the faithful at their prayers. In many an old French church corners and passages may be met with, protected by a network or simply by rails, which served, it is said, to shut off lepers from the general congregation. Closely associated with the Palais de Justice is the Tribunal of Commerce, which has its own code, its own judges and functionaries. Three centuries ago the necessity was recognised in France of leaving commercial
- 78. and industrial cases to the decision of men competent, from their occupation, to deal with such matters. Paris owes its Tribunal of Commerce to King Charles IX.; but the code under which issues are now decided dates only from September, 1807—from the First Empire, that is to say. The commercial judges are named for two years by the merchants and tradesmen domiciled in the department of the Seine. Formerly the Tribunal of Commerce, or Consular Tribunal, held its sittings at the back of the Church of Saint-Méry in the Hôtel des Consuls, the gate of which used to support a statue of Louis XIV., by Simon Guilain. THE UPPER CHAPEL OF THE SAINTE-CHAPELLE.
- 79. This mercantile court consists of five merchants, the first bearing the title of judge, and the four others that of consuls. The Tribunal of Commerce was removed from the old house in the Rue Saint-Méry in 1826, to be installed on the first storey of the newly constructed Bourse. Soon, however, the place assigned to it became inadequate for the constantly increasing number of cases brought before the court; and a special edifice was erected for the Tribunal of Commerce in {269} the immediate vicinity of the Palais de Justice. This structure, quadrilateral in form, is bounded on the north by the Quai aux Fleurs, on the east by the Rue Aubé, on the south by the Rue de Lutèce, and on the west by the Boulevard du Palais. To build a new Palais de Justice it was necessary to destroy all that existed of the ancient Cité. One curious building, which, after undergoing every kind of modification, ultimately, in order to make room for the Court of Commerce, disappeared altogether, was the ancient Church of Saint Bartholomew. This sacred edifice during the early days of the Revolution, when churches had gone very much out of fashion, became the Théâtre Henri IV., to be afterwards called Palais Variété, Théâtre de la Cité, Cité Variété, and Théâtre Mozart. Here was represented, in 1795, “The Interior of the Revolutionary Committees,” the most cutting satire ever directed against the tyranny of the Jacobins; and, in another style, “The Perilous Forest, or the Brigands of Calabria,” a true type of the ancient melodrama. Suppressed in 1807, this theatre underwent a number of transformations, to serve at last as a dancing saloon, known to everyone and beloved by students under the title of The Prado.
- 80. THE TRIBUNAL OF COMMERCE. The cupola of the Tribunal of Commerce is a reproduction, as to form, of the cupola of a little church which attracted the attention of Napoleon III. on the borders of the Lake of Garda while he was awaiting the result of the attack on the Solferino Tower. The Audience Chamber of the Tribunal is adorned with paintings by Robert Fleury, representing incidents in the commercial history of France from Charles IX. to Napoleon III. {270}
- 81. T CHAPTER XXIV. THE FIRE BRIGADE AND THE POLICE. The Sapeurs-pompiers—The Prefect of Police—The Garde Républicaine—The Spy System. HE Tribunal of Commerce, standing north of the Rue de Lutèce, has for pendant on its south side (that is to say, between the Rue de Lutèce and the quay) the barrack of the Republican Guard and two houses adjoining it, one of which is the private residence of the Prefect of Police: where, moreover, he has his private office; while the second contains the station of the firemen of the town of Paris. The Fire Brigade, or corps of Sapeurs-pompiers, is partly under the direction of the Prefect of Police, partly under that of the Minister of War, who takes charge of its organisation, its recruitment, and its internal administration. Much was said at the time of the terrible fire at the Opéra Comique in 1887 of the evils of this dual system; the chief of the corps, an officer appointed by the War Minister, being often an experienced soldier, but never before his appointment a skilled fireman. There is a reason, however, for placing the Sapeurs-pompiers under the orders of the Minister of War. During the campaign of 1870 and 1871 the Germans refused to recognise the military character of corps not holding their commission from this minister. Thus the National Guards, as a purely civic body, were not looked upon as soldiers, and were threatened with the penalties inflicted on persons taking up arms without authority from the central military power. In the next war against Germany the French propose to call out the whole of their available forces; and to be recognised as regular troops the Sapeurs- pompiers must have a military organisation and act under military chiefs formally appointed and responsible to a superior officer. All this, however, could surely be accomplished without rendering the corps unfit for the special duties assigned to it. The Sapeurs-pompiers are organised in twelve companies, forming two battalions, and are distributed among the 150 barracks, stations, and watch-
- 82. houses comprised in the twenty districts, or arrondissements, of Paris. The Magistracy of the Prefect of Police was created under the Consulate of the 1st of July, 1800, when the Central Power took over the general police duties entrusted under the Monarchy to the Lieutenant-General of Police, and which had been transferred by the Revolution to the Commune of Paris. The Prefect is specially empowered to take, personally, every step necessary for the discovery and repression of crime and for the punishment of criminals. He is charged, moreover, under the authority of the Minister of the Interior, with all that relates to the administrative and economic government of the prisons and houses of detention and correction, not only in Paris, but throughout the department of the Seine, as well as in the communes of Saint-Cloud, Sèvres, Meudon, and Enghien, suburbs of Paris belonging to the department of Seine-et-Oise. The Prefect of Police has beneath his orders all the police of the capital, or rather of the department to which the capital belongs. This service is divided into two special organisations: Municipal Police and Agents of Security. The “Security” force consists of three hundred agents with the title of inspector, commanded by five chief inspectors, ten brigadiers, and twenty sub-brigadiers. These agents are employed in arresting malefactors, and are viewed with intense hatred by the criminal class generally. The Municipal Police counts an effective of about 8,000 men, commanded by 38 peace officers, 25 chief inspectors, 100 brigadiers, and 700 sub- brigadiers. The entire expenditure of the Prefecture of the Police Service amounts to twenty-five million francs a year, of which eleven millions are put down for pay and the remainder for uniforms, office expenses, and all kinds of extras. “If,” says a French writer who knows London as well as Paris, “our police is not always so clear-sighted and so clever as it might be, it is, at any rate, more tolerant than vexatious. Our ‘keepers of the peace’ do not impose on the Paris population all the respect that the English people feels for its policemen; nor have they the same rigid bearing or the same herculean aspect. But, on the other hand, they are without their brutality—quite incredible to anyone who has not lived in London. Nearly all have been in the army, and they preserve the familiar aspect of the French soldier; while of the rules laid down by the Prefecture, the one they least observe is that which forbids {271} them to talk in the street with servant maids and cooks. But they are intelligent, ingenious, possessed of a certain tact, and brave to
- 83. the point of self-sacrifice. They are at present more appreciated and more popular, with their tunic, their military cap, their high boots, and their little cloak, which give them the look of troops on a campaign, than were the Sergents de Ville whose swallow-tail coat and black cocked hat were so much feared by rioters under the reign of Louis Philippe.” The Barracks of the Prefecture are occupied by the Garde Républicaine, which succeeds the Garde de Paris, the latter having itself succeeded the Garde Municipale, which was simply the Gendarmerie Royale of the Town of Paris, created under the Restoration. After the Revolution of 1848 the name of the Garde Municipale was changed, as after the Revolution of 1830 the title of Gendarmerie Royale was abolished. Notwithstanding alterations of name and certain slight modifications of uniform, the Republican Guard is a legion of gendarmerie like the different corps that preceded it. Commanded by a colonel, the Republican Guard is divided into two detachments or brigades, each under a lieutenant-colonel; the first consisting of three battalions of infantry, the second of three squadrons of cavalry. The whole force comprises 118 officers, with 2,800 men beneath their orders—2,200 infantry, and 600 cavalry. The Republican Guard, one of the finest corps that can be seen, belongs to the cadres of the regular army; and it served brilliantly in the war of 1870 and 1871. Its special duties, however, are to keep order in the City of Paris; though, in consideration of its mixed character, the pay assigned to it is furnished, half by the State, half by the Town of Paris. Among other merits it possesses an admirable band, in which may be found some of the finest orchestral players in a capital possessing an abundance of fine orchestras. The evidence of a Garde Républicaine, or gendarme, is accepted at the police courts as unimpeachable. The written statement drawn up by a gendarme may be denied by the accused, but it cannot be set aside. “As a matter of fact,” says M. Auguste Vitu, in his work on “Paris,” “very few evil results are caused by this rule; for the gendarme is honest. But he may make a mistake. In London, the magistrate, having generally to deal only with policemen of his own district, knows them personally, can judge of their intelligence and disposition, and is able in certain cases to see whether they are obscuring or altering the truth. He exercises over them, in case of negligence or error, accidental or intentional, the right of reprimanding and of suspending them. In Paris the ‘judges of correction,’ before whom, at one time or another, every one of the ‘keepers of the
- 84. peace’ or of the Republican Guards (altogether about 10,000 men) may appear, can only accept their evidence. It is doubtless sincere, but there is no way of testing it.” Of the spy system in connection with police administration it is difficult to speak with accurate knowledge, for the simple reason that it is not until long afterwards that secret arrangements of this kind are divulged. But in principle the system described by Mercier more than a hundred years ago still exists. “This,” writes that faithful chronicler, “may be termed the second part of Parisian grievances. Yet, like even the most poisonous reptile, these bloodhounds are of some service to the community: they form a mass of corruption which the police distil, as it were, with equal art and judgment, and, by mixing it with a few salutary ingredients, soften its baneful nature, and turn it to public advantage. The dregs that remain at the bottom of the still are the spies of whom I have just spoken; for these also belong to the police. The distilled matter itself consists of the thief-catchers, etc. They, like other spies, have persons to watch over them; each is foremost to impeach the other, and a base lucre is the bone of contention amongst those wretches, who are, of all evils, the most necessary. Such are the admirable regulations of the Paris police that a man, if suspected, is so closely watched that the most minute transaction in which he is concerned is treasured up till it is fit time to arrest him. The police does not confine its care to the capital only. Droves of its runners are sent to the principal towns and cities in this kingdom, where, by mixing with those whose character is suspicious, they insinuate themselves into their confidence, and by pretending to join in their mischievous schemes, get sufficient information to prevent their being carried into execution. The mere narrative of the following fact, which happened when M. de Sartine was at the head of this department, will give the reader an idea of the watchfulness of the police. A gentleman travelling from Bordeaux to Paris with only one servant in his company was stopped at the turnpike {272} by the Custom House officer, who, having inquired his name, told him he must go directly to M. de Sartine. The traveller was both astonished and frightened at this peremptory command, which, however, it would have been imprudent to disobey. He went, and his fears soon subsided at the civil reception he met with; but his surprise was greatly increased when the magistrate, whom, to his knowledge, he had never seen before, calling him by his name, gave him an
- 85. account of every transaction that had taken place previous to the gentleman’s departure from Bordeaux, and even minutely described the full contents of his portmanteau. ‘Now, sir,’ continued the Lieutenant de Police, ‘to show that I am well informed I have a trifle more to disclose to you. You are going to such and such an hotel, and a scheme is laid by your servant to murder you by ten o’clock.’ ‘Then, my lord, I must shift my quarters to defeat his wicked intention.’ ‘By no means, sir; you must not even take notice of what I have said. Retire to bed at your usual hour, and leave the rest to me.’ The gentleman followed the advice of the magistrate and went to the hotel. About an hour after he had lain down, when, no doubt, he was but little inclined to compose himself to rest, the servant, armed with a clasp-knife, entered the room on tip-toe, drew near the bed, and was about to fulfil his murderous intention. Then four men, rushing from behind the hangings, seized the wretch, who confessed all, and soon afterwards paid to the injured laws of humanity the forfeit of his life.” A POMPIER.
- 86. Since the Revolution the number of spies employed in France has doubtless diminished. But they have existed in that country, as in others, from time immemorial. A French writer, dealing with this subject, traces the history of espionage to the remotest antiquity; the first spies being, according to his view, the brothers of Joseph, who were for that reason detained when they visited him in Egypt as Pharaoh’s minister. The Romans employed spies in their armies, and both Nero and Caligula had an immense number of secret agents. Alfred the Great was a spy of the chivalrous, self- sacrificing kind; for, risking his life on behalf of his own people he would assuredly, had he been recognised in the Danish camp, have been put to death. The spy system was first established in France on a large, widely organised scale by Richelieu, under whose orders the notorious Father Joseph became the director of a network of spies which included not only all the religious orders of France, but many persons belonging to the nobility and middle classes. This sort of conspiracy had, moreover, its correspondents abroad. The Police, strongly organised under Louis XIV., included a numerous body of spies. But all that had before been known in the way of espionage was eclipsed in Louis XV.’s reign, when the too famous De Sartine, Lieutenant of Police, gave to his spy system a prodigious extension. Under the administration of De Sartine spies were employed to follow the Court; and the Minister of Foreign Affairs maintained a subdivision of spies to watch the doings of all foreigners arriving in Paris, and to ascertain, in particular, the object of their visit. This course of action is followed to the present day in Russia, not only secretly, but in the first instance openly. Thus the chief of a bureau connected with the Foreign Office questions the stranger in the politest manner as to his motive in coming to Russia, the friends, if any, that {273} he has there, his occupation, and his pecuniary resources. A report is attributed to the above-named Lieutenant of Police in which it is set forth that to watch thoroughly a family of twenty persons forty spies would be necessary. This, however, was an ideal calculation, for, in reality, the cost of the spy system under Louis XV., as set down in the official registers of the police, did not amount annually to more than 20,000 francs. The Government had, however, at its disposal much larger sums received for licences from the gambling houses, and as fines and ransoms from evil- doers of all kinds. Berryer, the successor of De Sartine—bearer of a name
- 87. which, in the nineteenth century, was to be rendered honourable— conceived the idea, inspired, perhaps, by a familiar proverb, of employing as spies criminals of various kinds, principally thieves who had escaped from prison or from the pursuit of the police. These wretches, banded together in a secret army of observation, were only too zealous in the performance of the work assigned to them; for, on the slightest negligence or prevarication, they were sent back to the hulks or to gaol, where a hot reception awaited them from their former comrades in crime. Hackney- coachmen, innkeepers, and lodging-house keepers were also engaged as spies, not to speak of domestic servants, who, through secret agencies, were sometimes supplied to householders by the police themselves. Many a person was sent to the Bastille in virtue of a lettre de cachet issued on the representation of some valet before whom his master had uttered an imprudent word. Mercier’s picture of the spy system in Paris a few years before the Revolution is, to judge from other contemporary accounts, in no way exaggerated. The Revolution did not think even of suppressing espionage, but it endeavoured to moralise this essentially immoral, if sometimes necessary, institution. In a report on this subject dated November 30, 1789, only a few months after the taking of the Bastille, the following significant passage occurs:—“We have been deprived of a sufficient number of observers, a sort of army operating under the orders of the {274} old police, which made considerable use of it. If all the districts were well organised, if their committees were wisely chosen and not too numerous, we should apparently have no reason to regret the suppression of that odious institution which our oppressors employed so long against us.” The writer of the report was, in fact, recommending, without being apparently aware of it, a system of open denunciation necessitating previously that secret espionage which he found so hateful; for before denouncing it would be necessary to observe and watch. Nevertheless, the Police of the Revolution employed no regular spies, registered, organised, and paid, until 1793; though this did not prevent wholesale denunciation on the part of officious volunteers. Robespierre, however, maintained a spy system more or less on the ancient pattern; and when the Empire was established, Napoleon’s famous Prefect of Police, Fouché, made of espionage a perfect science. Fouché had at his service spies of all classes and kinds; and the ingenious Mme. de Bawr has, in one of her best tales, imagined the case of a poor
- 88. A GUARDIAN OF THE PEACE. curé, who, after the suppression of churches and religious services, calls upon Fouché, an old schoolfellow of his, to ask for some employment; when the crafty police minister assigns a certain salary to his simple- minded friend and tells him not to do any serious work for the present, but to go about Paris amusing himself in various cafés and places of entertainment, after which he can look in from time to time and say what has chiefly struck him in the persons he has seen and the conversations he has heard. At last the innocent curé finds that he has been doing the work of a spy. Fortunately, when he discovers to what a base purpose he has been turned, Napoleon has just restored public worship; whereupon, by way of amends, Fouché uses his influence with the Emperor to get the poor man re-appointed to his old parish.
- 89. AN ORDERLY OF THE GARDE DE PARIS. Under the Restoration the spy system was maintained as under the Empire, but with additional intricacies. Fouché had been replaced by Vidocq, who, among other strange devices for getting at the thoughts of the public, obtained from the Government permission to establish a public bowling alley, which collected crowds of people, whose conversations were listened to and reported by agents employed for the purpose. The bowling alley brought in some 4,000 to 5,000 francs a year, which was spent on additional spies. The Prefect Delavau, with Vidocq as his lieutenant, went back to the system of Berryer under the ancient régime, taking into the State service escaped criminals, who for the {275} slightest fault were sent back to gaol. An attempt was made by the same Delavau, in humble imitation of Berryer, to get into his service all the domestics of Paris; and in this way he renewed an old regulation by which each servant was to keep a book and bring it to the Prefecture of Police on entering or leaving a situation. To their credit, be it recorded, most of the servants abstained from obeying this discreditable order. Finding that his plan for watching private families through their servants did not answer, Delavau multiplied the number of agents charged with attending places of public entertainment. “The Police,” writes M. Peuchet in his “Mémoires tirés des Archives de la Police,” “will never learn to respect an order so long as its superintendents are taken from the hulks and feel that they have their revenge to take on the society which has punished them.” The justice of this remark has since been recognised. The first care of Delavau’s successor, the honourable and much regretted M. de Belleyme, was to dismiss, and even to send back to their
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