Principles of Computerized Tomographic Imaging

Avinash C. Kak
School of Electrical Engineering
Purdue University

Malcolm Slaney
Originally: Schlumberger Palo Alto Research
Currently: Yahoo! Research

Buy the book

Our book has been republished by SIAM (Society of Industial and Applied Mathematics). Look for it in SIAM's catalog or your favorite bookstore (i.e. Amazon). The electronic copy available here is exactly the same as the original book. SIAM has been very gracious about letting us keep the electronic copy online. But if you are going to read more than a few pages we encourage you to buy the book from SIAM. It's not that expensive!!!

Original book

Originally published by IEEE Press.  (C) 1988 Institute for Electrical and Electronic Engineers.
Electronic copy (C) 1999 A. C. Kak and Malcolm Slaney. You are free to use this electronic version of Principles of Computerized Tomographic Imaging for your personal use. No commercial use is allowed without permission. Please reference this work as

A. C. Kak and Malcolm Slaney, Principles of Computerized Tomographic Imaging, IEEE Press, 1988.
or if you prefer
A. C. Kak and Malcolm Slaney, Principles of Computerized Tomographic Imaging, Society of Industrial and Applied Mathematics, 2001

Electronic copy

Each chapter of this book is available as an Adobe PDF File.  Free readers for most computer platforms are available from Adobe.

Preface  - PDF File (456K Bytes)

1 Introduction - PDF File (304K Bytes)

2 Signal Processing Fundamentals - PDF File (4.6M Bytes)
2.1 One-Dimensional Signal Processing
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 ° How to Deal with Data Defined for Negative Time ° How to Increase Frequency Domain Display ° Resolution of Signals Defined for Negative Time ° Data Truncation Effects
2.2 Image Processing
Point Sources and Delta Functions ° Linear Shift Invariant Operations ° Fourier Analysis ° Properties of Fourier Transforms ° The Two-Dimensional Finite Fourier Transform ° Numerical Implementation of the Two-Dimensional FFT
2.3 References

3 Algorithms for Reconstruction with Nondiffracting Sources - PDF File (5.5M Bytes)

3.1 Line Integrals and Projections

3.2 The Fourier Slice Theorem

3.3 Reconstruction Algorithms for Parallel Projections
The Idea ° Theory ° Computer Implementation of the Algorithm

3.4 Reconstruction from Fan Projections
Equiangular Rays ° Equally Spaced Collinear Detectors ° A Re-sorting Algorithm

3.5 Fan Beam Reconstruction from a Limited Number of Views

3.6 Three-Dimensional Reconstructions
Three-Dimensional Projections ° Three-Dimensional Filtered Backprojection

3.7 Bibliographic Notes

3.8 References

4 Measurement of Projection Data - The Nondiffracting Case  - PDF File (10.2M Bytes)
4.1 X-Ray Tomography
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
Single Photon Emission Tomography ° Attenuation Compensation for Single Photon Emission CT ° Positron Emission Tomography ° Attenuation Compensation for Positron Tomography

4.3 Ultrasonic Computed Tomography
Fundamental Considerations ° Ultrasonic Refractive Index Tomography ° Ultrasonic Attenuation Tomography ° Applications

4.4 Magnetic Resonance Imaging

4.5 Bibliographic Notes

4.6 References

(Individual sections of this large chapter are also available: 4.1, 4.2, 4.3, 4.4, 4.5& 4.6)

5 Aliasing Artifacts and Noise in CT Images - PDF File (2.3M Bytes)
5.1 Aliasing Artifacts
What Does Aliasing Look Like? ° Sampling in a Real System

5.2 Noise in Reconstructed Images
The Continuous Case ° The Discrete Case

5.3 Bibliographic Notes

5.4 References

6 Tomographic Imaging with Diffracting Sources - PDF File (6.9M Bytes)
6.1 Diffracted Projections
Homogeneous Wave Equation ° Inhomogeneous Wave Equation

6.2 Approximations to the Wave Equation
The First Born Approximation ° The First Rytov Approximation

6.3 The Fourier Diffraction Theorem
Decomposing the Green'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
Frequency Domain Interpolation ° Backpropagation Algorithms

6.5 Limitations
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

6.7 Experimental Limitations
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

6.9 References

7 Algebraic Reconstruction Algorithms - PDF File (2.3M Bytes)
7.1 Image and Projection Representation

7.2 ART (Algebraic Reconstruction Techniques)

7.3 SIRT (Simultaneous Iterative Reconstructive Technique)

7.4 SART (Simultaneous Algebraic Reconstruction Technique)
Modeling the Forward Projection Process ° Implementation of the Reconstruction Algorithm

7.5 Bibliographic Notes

7.6 References

8 Reflection Tomography - PDF File (3.4M Bytes)
8.1 Introduction

8.2 B-Scan Imaging

8.3 Reflection Tomography
Plane Wave Reflection Transducers ° Reflection Tomography vs. Diffraction Tomography ° Reflection Tomography Limits

8.4 Reflection Tomography with Point Transmitter/Receivers
Reconstruction Algorithms ° Experimental Results

8.5 Bibliographic Notes

8.6 References

Index -  HTML File (with links) - PDF File (494K Bytes)
About the Authors