Principles of Computer Tomography Imaging

331 Pages·2005·27.51 MB·English

Computerized
Tomographic
Imaging
Avinash C. Kak
School of Electrical Engineering
P undue University
Malcolm Slaney
Schfumberger Palo Alto Research
Electronic Copy (c) 1999 by A. C. Kak and Malcolm Slaney
Copies can be made for personal use only.
IEEE
e PRESS
+
The lnstbte of Electrical and Electronics Engineers, Inc., New York
IEEE PRESS
1987 Editorial Board
R. F. Cotellessa, Editor in Chief
J. K. Aggarwal. Editor, Selected Reprint Series
Glen Wade, Editor, Special Issue Series
James Aylor J. F. Hayes A. C. Schell
F. S. Barnes W. K. Jenkins L. G. Shaw
J. E. Brittain A. E. Joel, Jr. M. I. Skolnik
B. D. Carrol Shlomo Karni P. W. Smith
Aileen Cavanagh R. W. Lucky M. A. Soderstrand
D. G. Childers R. G. Meyer M. E. Van Valkenburg
H. W. Colborn Seinosuke Narita John Zaborsky
J. D. Ryder
W. R. Crone, Managing Editor
Hans P. Leander, Technical Editor
Laura J. Kelly, Administrative Assistant
Randi E. Scholnick and David G. Boulanger, Associate Editors
This book is set in Times Roman.
Copyright 0 1988 by
THE INSTITUTE OF ELECTRICAL AND ELECTRONICS ENGINEERS, INC.
345 East 47th Street, New York, NY 10017-2394
All rights reserved.
PRINTED IN THE UNITED STATES OF AMERICA
IEEE Order Number: PC02071
Library of Congress Cataloging-in-Publication Data
Kak, Avinash C.
Principles of computerized tomographic imaging.
“Published under the sponsorship of the IEEE Engineering in Medicine and
Biology Society.”
Includes bibliographies and index.
1. Tomography. I. Slaney, Malcolm. II. IEEE Engineering in Medicine and
Biology Society. Ill. Title.
RC78.7.T6K35 1987 616.07’572 87-22645
ISBN O-87942-1 98-3
Contents
Preface ix
I Introduction
References 3
2 Signal Processing Fundamentals 5
2.1 One-Dimensional Signal Processing 5
Continuous and Discrete One-Dimensional Functions Linear
l
Operations Fourier Representation Discrete Fourier Transform
l l
(DFT) Finite Fourier Transform Just How Much Data Is Needed?
l l l
Interpretation of the FFT Output How to Increase the Display
l
Resolution in the Frequency Domain How to Deal with Data Defined
l
for Negative Time How to Increase Frequency Domain Display
l
Resolution of Signals Defined for Negative Time Data Truncation
l
Effects
2.2 Image Processing 28
Point Sources and Delta Functions Linear Shift Invariant Operations
l l
Fourier Analysis Properties of Fourier Transforms The
l l
Two-Dimensional Finite Fourier Transform Numerical Implementation
l
of the Two-Dimensional FFT
2.3 References 47
3 Algorithms for Reconstruction with Nondiffracting Sources 49
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 of the Algorithm
l l
3.4 Reconstruction from Fan Projections 75
Equiangular Rays Equally Spaced Collinear Detectors A Re-sorting
l l
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
l
Backprojection
3.7 Bibliographic Notes 107
3.8 References 110
4 Measurement of Projection Data- The Nondiffracting Case 113
4.1 X-Ray Tomography 114
Monochromatic X-Ray Projections Measuremento f Projection Data
l
with Polychromatic Sources Polychromaticity Artifacts in X-Ray CT
l l
Scatter Different Methods for Scanning Applications
l l
4.2 Emission Computed Tomography 134
Single Photon Emission Tomography Attenuation Compensationf or
l
Single Photon Emission CT Positron Emission Tomography
l l
Attenuation Compensationf or Positron Tomography
4.3 Ultrasonic Computed Tomography 147
FundamentalC onsiderations Ultrasonic Refractive Index Tomography
l
Ultrasonic Attenuation Tomography Applications
l l
4.4 Magnetic Resonance Imaging 158
4.5 Bibliographic Notes 168
4.6 References 169
5 Abasing Artifacts and Noise in CT Images 177
5.1 Aliasing Artifacts 177
What Does Aliasing Look Like? Sampling in a Real System
l
5.2 Noise in Reconstructed Images 190
The ContinuousC ase The Discrete Case
l
5.3 Bibliographic Notes 200
5.4 References 200
6 Tomographic Imaging with Diffracting Sources 203
6.1 Diffracted Projections 204
HomogeneousW ave Equation InhomogeneousW ave Equation
l
6.2 Approximations to the Wave Equation 211
The First Born Approximation The First Rytov Approximation
l
6.3 The Fourier Diffraction Theorem 218
Decomposingt he Green’sF unction Fourier Transform Approach
l l
Short WavelengthL imit of the Fourier Diffraction Theorem The Data
l
Collection Process
6.4 Interpolation and a Filtered Backpropagation Algorithm for Diffracting
Sources 234
FrequencyD omain Interpolation BackpropagationA lgorithms
l
6.5 Limitations 247
Mathematical Limitations Evaluation of the Born Approximation
l l
Evaluation of the Rytov Approximation Comparison of the Born and
l
Rytov Approximations
6.6 Evaluation of Reconstruction Algorithms 252
6.7 Experimental Limitations 261
Evanescent Waves Sampling the Received Wave The Effects of a
l l
Finite Receiver Length Evaluation of the Experimental Effects
l l
Optimization Limited Views
l
6.8 Bibliographic Notes 268
6.9 References 270
7 Algebraic Reconstruction Algorithms 275
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
l
Reconstruction Algorithm
7.5 Bibliographic Notes 292
7.6 References 295
8 Reflection Tomography 297
8.1 Introduction 297
8.2 B-Scan Imaging 298
8.3 Reflection Tomography 303
Plane Wave Reflection Transducers Reflection Tomography vs.
l
Diffraction Tomography Reflection Tomography Limits
l
8.4 Reflection Tomography with Point Transmitter/Receivers 313
Reconstruction Algorithms Experimental Results
l
8.5 Bibliographic Notes 321
8.6 References 321
Index 323
About the Authors 329
vii
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 gaining 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, and Barry
Roberts. A number of people, Carl Crawford, Rich Kulawiec, Gary S.
Peterson, and the anonymous reviewers, helped us proofread the manuscript;
PREFACE ix
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, Randi Scholnick, 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
X PREFACE
1 Introduction
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 which he 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-ray tomography)
with fairly complex mathematical operations, and still get an image that is
incredibly accurate.
INTRODUCTION 1
Fig. 1.1: Two projections are It is perhaps fair to say that the breakneck pace at which x-ray computed
shown of an object consisting of tomography images improved after Hounsfield’s invention was in large
a pair of cylinders.
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 [Ram711 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 manufacturers of 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 imaging with
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
2 COMPUTERIZED TOMOGRAPHIC IMAGING
aim is the same as with x-rays, viz., to reconstruct the cross-sectional image
of, say, the attenuation coefficient. X-rays are 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, although under 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], [Bar8 11. The reader is also referred
to the review articles in the field [Gor74], [Bro76], [Kak79] and the two
special issues of IEEE journals [Kak81], [Her83]. Reviews of the more
popular algorithms also appeared in [Ros82], [Kak84], [Kak85], [Kak86].
References
[Bar811 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 radioisotopic imaging,” 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. Boume and
J. F. Danielli, Eds. New York, NY: Academic Press, 1974, pp. 111-151.
[Her801 G. T. Herman, Image Reconstructions from Projections. New York, NY:
Academic Press, 1980.
[Her831 -, 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.
[Kak81] -, Guest Editor, Special Issue on Computerized Medical Imaging, IEEE
Transactions on Biomedical Engineering, vol. BME-28, Feb. 1981.
INTRODUCTION 3