Advances in Computer Vision and Pattern Recognition

Image Reconstruction from Projections

By: Gabor T. Herman

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Hardcover | 24 September 2009 | Edition Number 2

At a Glance

Hardcover
312 Pages

Edition Type
Revised

Dimensions(cm)
23.9 x 16.3 x 1.8

Hardcover

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This revised and updated text presents the computational and mathematical procedures underlying data collection, image reconstruction, and image display in computerized tomography. New topics: the fast calculation of a ray sum for a digitized picture, the task-oriented comparison of reconstruction algorithm performance, blob basis functions and the linogram method for image reconstruction. Features: Describes how projection data are obtained and the resulting reconstructions are used; Presents a comparative evaluation of reconstruction methods; Investigates reconstruction algorithms; Explores basis functions, functions to be optimized, norms, generalized inverses, least squares solutions, maximum entropy solutions, and most likely estimates; Discusses SNARK09, a large programming system for image reconstruction; Concludes each chapter with helpful Notes and References sections. An excellent guide for practitioners, it can also serve as a textbook for an introductory graduate course.

Industry Reviews

From the reviews of the first edition.

"... the author is a giant in the field. One expects an exciting book and one is not disappointed."
S. Webb, Physics in Medicine and Biology, v. 26. p. 721, 1981.

"... the book is an excellent introduction to the subject and is well written, thoughtfully organized, and a pleasure to read."
M.W. Vannier, The Journal of Nuclear Medicine, v. 23, p. 369, 1982.

"On the one hand the book is well suited for a course on computerized tomography at the graduate level, on the other hand I found it a well balanced review on the subject for somebody outside the field."
W.N Brouw, Space Science Reviews, v.32. p. 467, 1982.

"This book will be valuable to the engineer, the physicist, the mathematician, and the physician (with a sufficient mathematical background) interested in the field of CT."
M.M. Ter-Pogossian, Medical Physics, v. 11, p. 90, 1984.

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You Can Find This Book In

Non-Fiction Medicine Other Branches of Medicine Medical Imaging Engineering & Technology Other Technologies & Applied Sciences Applied Optics Computing & I.T.Computer Science Image Processing Artificial Intelligence Computer Vision

Mathematical Theory of Computation Graphical & Digital Media Applications Pre-Clinical Medicine & Basic Sciences Databases Data Capture & Analysis

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Paperback

Published: 14th March 2012

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Introduction	p. 1
Image Reconstruction from Projections	p. 1
Probability and Random Variables	p. 18
An Overview of the Process of CT	p. 27
What Are We Trying to Do?	p. 27
Traditional Tomography	p. 27
Data Collection for CT	p. 30
Voxels, Pixels, and CT Numbers	p. 31
The Problem of Polychromaticity	p. 32
Reconstruction Algorithms	p. 34
Physical Problems Associated with Data Collection in CT	p. 37
Photon Statistics	p. 37
Beam Hardening	p. 42
Other Sources of Error	p. 44
Scanning Modes	p. 47
Computer Simulation of Data Collection in CT	p. 53
Pictures and Digitization	p. 53
Creation of a Phantom	p. 54
A Piecewise-Homogeneous Head Phantom	p. 56
Head Phantom with a Large Tumor and Local Inhomogeneities	p. 59
Creation of the Ray Sums	p. 60
Fast Calculation of a Ray Sum for a Digitized Picture	p. 63
Data Collection and Reconstruction of the Head Phantom	p. 67
Methods of Picture Comparison	p. 67
Task-Oriented Comparison of Algorithm Performance	p. 69
An Illustration Using Selective Smoothing	p. 73
Reconstruction from Perfect Data	p. 78
Effects of Photons Statistics	p. 83
Effect of Beam Hardening	p. 86
The Effects of Detector Width and Scatter	p. 91
Simulation of Different Scanning Modes	p. 95
Basic Concepts of Reconstruction Algorithms	p. 101
Problem Statement	p. 101
Transform Methods	p. 106
Series Expansion Methods	p. 108
Optimization Criteria	p. 111
Blob Basis Functions	p. 119
Computational Efficiency	p. 122
Backprojection	p. 125
Continuous Backprojection	p. 125
Implementation of the Backprojection Operator	p. 127
Discrete Backprojection	p. 131
Filtered Backprojection for Parallel Beams	p. 135
Convolutions, Hilbert Transforms, Regularization	p. 135
Derivation of the FBP Method	p. 139
Implementation of the FBP Method	p. 140
Fourier Transforms	p. 143
Sampling and Interpolation	p. 146
The Choice of Convolving and Interpolating Functions	p. 147
Why So Popular?	p. 157
Other Transform Methods for Parallel Beams	p. 159
Two-Dimensional Fourier Transforms	p. 159
The Fourier Method of Reconstruction	p. 161
Linograms	p. 166
Rho-Filtered Layergram	p. I73
Filtered Backprojection for Divergent Beams	p. 177
The Divergent Beam FBP Algorithm	p. 177
Choice of the Window Function	p. 181
Point Response Function	p. 182
Noise Reconstruction	p. 186
Comparison of Algorithms Based on Reconstructions	p. 188
Algebraic Reconstruction Techniques	p. 193
What is Art?	p. 193
Relaxation Methods for Solving Systems of Inequalities and Equalities	p. 196
Additive Art	p. 201
Tricks	p. 205
Efficacy of Art	p. 210
Quadratic Optimization Methods	p. 217
Mathematical Background to Quadratic Optimization	p. 217
Richardson's Method for Solving Systems of Equations	p. 221
Smoothing Matrices	p. 224
Implementation of Richardson's Methods for Image Reconstruction	p. 226
A Demonstration of Quadratic Optimization	p. 227
Truly Three-Dimensional Reconstruction	p. 235
Three-Dimensional Series Expansion	p. 236
Dynamically Changing 3D Phantoms and Their Projections	p. 237
Three-Dimensional Reconstructions of the Dynamic Phantom	p. 240
Three-Dimensional Display of Organs	p. 243
The Basic Approach	p. 243
Boundary Detection	p. 246
Hidden Surface Removal	p. 251
Shading	p. 253
Mathematical Background	p. 259
The Dimensionality of the Linear Attenuation Coefficient	p. 259
The Line Integral of the Relative Linear Attenuation	p. 260
The Radon Inversion Formula	p. 261
A Picture is Not Uniquely Determined by a Finite Number of Its Views	p. 265
Analysis of the Photon Statistics	p. 267
The Integral Expression for Polychromatic Ray Sums	p. 269
Proof of the Regularization Theorem	p. 270
Convergence of the Relaxation Method for Inequalities	p. 273
References	p. 277
Index	p. 293
Table of Contents provided by Ingram. All Rights Reserved.

Advances in Computer Vision and Pattern Recognition

Image Reconstruction from Projections

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

You Can Find This Book In

Other Editions and Formats

Paperback

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