Discrete Tomography,9780817641016

Discrete Tomography

by ;
ISBN13: 9780817641016
ISBN10: 0817641017
Format: Hardcover
Pub. Date: 1999-09-01
Publisher(s): Birkhauser
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Summary

Discrete Tomography provides a critical survey of new methods, algorithms, analytical and computational tools, and select applications that are the foundations of multidimensional image analysis, construction, and reconstruction. Professionals and researchers in multidimensional image processing will find the book an essential resource and guide to the latest developments in this field, which has important applications in medical imaging, industrial inspection and graphics.

Table of Contents

Preface xiii
Contributors xvii
I Foundations 1(188)
Discrete Tomography: A Historical Overview
3(32)
Attila Kuba
Gabor T. Herman
Introduction
3(3)
Foundations and algorithms
6(20)
Definitions, notations, and basic problems
6(4)
Reconstruction on binary matrices from two projections
10(16)
Applications
26(3)
Data compression, data coding, and image processing
26(1)
Electron microscopy
27(1)
Biplane angiography
27(1)
Computerized tomography
28(1)
Conclusion
29(6)
References
29(6)
Sets of Uniqueness and Additivity in Integer Lattices
35(24)
Peter C. Fishburn
Lawrence A. Shepp
Introduction
35(2)
Uniqueness and additivity
37(6)
Bad rectangles
43(2)
No bad rectangles
45(1)
Nonadditive sets of uniqueness
46(4)
Planar sets and lines
50(5)
Linear programming
55(1)
Discussion
56(3)
References
57(2)
Tomographic Equivalence and Switching Operations
59(26)
T. Yung Kong
Gabor T. Herman
Ryser's theorem for binary pictures on the square grid
59(11)
Binary and ternary pictures on the square grid
60(1)
Four statements of Ryser's theorem
61(2)
Metropolis algorithms based on Ryser graphs
63(5)
A proof of Ryser's theorem
68(2)
Nonexistence of analogs on m-grids with m > 2
70(2)
m-grids
70(1)
Validity of Ryser's theorem for arbitrary 2-grids
71(1)
The main result
72(1)
A proof of the main result
72(10)
Derivation of the main result from the Main Claim
73(2)
Justification of the Main Claim
75(7)
Concluding remarks
82(3)
References
83(2)
Uniqueness and Complexity in Discrete Tomography
85(78)
Richard J. Gardner
Peter Gritzmann
Introduction
85(1)
Definitions and preliminaries
86(2)
Uniqueness results
88(9)
Complexity results
97(11)
Further extensions and variations
108(7)
Higher-dimensional X-rays
108(1)
Successive determination
109(1)
The polyatomic case
110(1)
References
111(4)
Reconstructionm of Plane Figures from Two Projections
Akira Kaneko
115(1)
Lei Huang
Review of the problem and known results
115(8)
Choice of good reconstruction for the discrete case
123(5)
Passage to continuous limit
128(9)
References
134(3)
Reconstruction of Two-Valued Functions and Matrices
Attila Kuba
137(1)
Introduction
137(1)
Characteristic functions
138(1)
Two-valued functions
139(4)
Reconstruction of two-value functions
143(4)
Two-valued matrices
147(6)
Reconstruction of two-valued matrices
153(4)
Uniqueness in the case of two-valued matrices
157(6)
References
160(3)
Reconstruction of Connected Sets from Two Projections
163(26)
Alberto Del Lungo
Maurice Nivat
Introduction
163(1)
Preliminaries
164(3)
Intractability results
167(5)
The complexity of RSHVP on (h), (p,h), (v), and (p,v)
167(2)
The complexity of RSHVP on (p) and (h,v)
169(3)
Reconstruction of convex polyominoes
172(17)
The medians
173(2)
The spine and its reconstruction
175(4)
The reconstruction algorithm
179(8)
References
187(2)
II Algorithms 189(154)
Binary Tomography Using Gibbs Priors
191(22)
Samuel Matej
Avi Vardi
Gabor T. Herman
Eilat Vardi
Introduction
192(1)
Gibbs distributions on the hexagonal grid
193(2)
Reconstruction algorithms
195(3)
Experimental study
198(6)
Finding additional invariant elements
204(3)
Gibbs prior definition using a look-up table
207(2)
Conclusions
209(4)
References
211(2)
Probabilistic Modeling of Discrete Images
213(24)
Michael T. Chan
Gabor T. Herman
Emanuel Levitan
Introduction
213(2)
Image-modeling distributions for discrete images
215(2)
Parameter estimation
217(2)
Isotropic considerations
217(1)
Combining global and local characteristics
217(2)
Recovering images corrupted by additive noice
219(7)
Adaptation to discrete tomographic reconstruction
226(6)
The posterior model and optimization criteria
226(2)
An approximate two-step reconstruction approach
228(1)
Experimental results
229(3)
Conclusions
232(5)
References
232(5)
Multiscale Bayesian Methods for Discrete Tomogaphy
237(28)
Thomas Frese
Charles A. Bouman
Ken Souer
Introduction
237(3)
Stochastic data models for tomography
240(3)
Markov random field prior models
243(2)
Optimization techniques
245(2)
Estimation of discrete levels
247(3)
Multiscale approaches
250(1)
Multiscale MRF
251(1)
Computational complexity
252(2)
Results
254(6)
Conclusion
260(5)
References
261(4)
An Algebraic Solution for Discrete Tomography
265(20)
Andrew E. Yagle
Introduction
265(2)
Applications
265(1)
Discussion
266(1)
Problem formulation and nonuniqueness
267(1)
Problem formulation
267(1)
Nonuniqueness
267(1)
Reformulation and insights using the DFT
268(2)
Reformulation using the DFT
268(1)
Ambiguity insights using the DFT
269(1)
Relation to phase retrieval
269(1)
Agarwal-Cooley fast convolution
269(1)
Solution using DFT reformulation
270(2)
Solving for X (k1, k2, k3)
270(1)
Linear systems of equations for X (k1, k2, k3)
271(1)
Solving for x (i1, i2, i3)
271(1)
Illustrative examples
272(2)
2 x 2 x 2 general problem
272(1)
2 x 2 x 2 general solution
272(1)
3 x 3 x 3 example
273(1)
Closed-form solution to limited-angle discrete tomography
274(2)
Intoduction
274(1)
Limited-angle and discrete tomography
275(1)
An explicit formula for bandwidth extrapolation
276(1)
Basic idea
276(1)
Application to tomography
277(2)
Discussion of application
277(1)
Comments
278(1)
Summary of procedure
278(1)
A simple, illustrative numerical example
279(3)
Problem statement
279(1)
Extrapolation equations
280(1)
Extrapolation of unknown F(m,n)
281(1)
Application to large images
282(1)
Conclusion
282(3)
References
283(2)
Binary Steering of Nonbinary Iterative Algorithms
285(12)
Yair Censor
Samuel Matej
Introduction: Problem definition, approach, and motivation
285(1)
The steering mechanism
286(3)
Experimental study
289(5)
Conclusions
294(3)
References
295(2)
Reconstuction of Binary Images via the EM Algorithm
297(46)
Yehuda Vardi
Cun-Hui Zhang
Introduction
297(4)
The EM/ML algorithm
301(2)
Properties of the algorithm
303(1)
Implementations and experiments
304(5)
Further comments
309(1)
Proofs
310(3)
Characterization of local MLEs
310(2)
Convergence of the algorithm
312(1)
Appendix
313(4)
References
315(2)
Compact Object Reconstruction
317(1)
Ali Mohammad-Djafari
Charles Soussen
Introduction
317(1)
Review of methods
318(4)
Classical pixel representation approach
319(1)
Level-set approach
320(1)
Shape reconstruction approach
321(1)
Polygonal and polyhedral shape modeling
322(8)
Exact reconstruction of polygonal shapes in 2D
323(2)
Exact reconstruction of polyhedral shapes in 3D
325(5)
Desciption of the proposed method
330(2)
Simulation results
332(7)
2D reconstructions
332(2)
3D case
334(5)
Conclusions
339(4)
References
339(4)
III Applications 343(129)
CT-Assisted Engineering and Manufacturing
345(18)
Jolyon A. Browne
Mathew Koshy
Introduction
345(2)
Agile manufacturing: A new manufacturing paradigm
347(2)
Obtaining the digital model
349(2)
The role of computed tomography: Selected examples
351(7)
The role of discrete tomography in improving the digital model
358(1)
Technical challenges
359(1)
Conclusions
360(3)
References
360(3)
3D Reconstruction from Sparse Radiographic Data
363(22)
James Sachs
Jr., Ken Sauer
Introduction
363(2)
Radiographic/tomographic data
365(4)
Idealized photon counting model
365(1)
Compton scattering effects
366(3)
3D maximum a posteriori reconstruction
369(7)
Bayesian formulation
369(3)
Reconstruction from simulated radiographs
372(2)
Optimization considerations
374(2)
Physical radiographic experiments
376(5)
Conclusion
381(4)
References
382(3)
Heart Chamber Reconstruction from Biplane Angiography
385(20)
Dietrich G. W. Onnasch
Guido P.M. Prause
Introduction
385(1)
Cardiac X-ray angiography
386(3)
Image recording and processing
386(1)
Quantitative biplane angiography
387(2)
Binary tomography
389(3)
The problem
389(1)
Probability-driven reconstruction
390(2)
Input data restoration
392(2)
Implemented algorithm
394(3)
Pre-processing
394(1)
Ventricular reconstruction
395(1)
Post-processing
396(1)
Results
397(2)
Patient data
397(2)
Ventricular phantoms
399(1)
Discussion
399(6)
References
401(4)
Discrete Tomography in Electron Microscopy
405(12)
J. M. Carazo
C. O. Sorzano
E. Rietzel
R. Schroder
R. Marabini
Introduction
406(1)
3D electron microscopy
407(3)
Discrete tomography in electron microscopy
410(1)
Considering experimental applications: The problems
410(3)
The Contrast Transfer Function (CTF)
411(1)
Discretization of the electron density of specimens
412(1)
A practical application
413(1)
Conclusions
414(3)
References
414(3)
Tomography on the 3D-Torus and Crystals
417(18)
Pablo M. Salzberg
Raul Figueroa
Introduction
417(1)
A fast inverse transform
418(4)
The continuous case
422(3)
Binary tomography on lattices
425(3)
Binary tomography and crystals
425(1)
Stating the binary tomography problem
426(2)
The reconstruction algorithm
428(1)
Final remarks and examples
429(6)
References
432(3)
A Recursive Algorithm for Diffuse Planar Tomography
435(20)
Sarah K. Patch
Introduction
435(2)
Diffuse tomography model
437(4)
Forward problem
440(1)
Consistency conditions
440(1)
Recursive solution
441(2)
Elimination of parameters
443(4)
Conclusion
447(1)
Appendix I --- Matrix identities
448(2)
Elementary identities
448(1)
Graβmann identities
449(1)
Range conditions
449(1)
Appendix II --- Enforcing range conditions
450(5)
References
453(2)
From Orthogonal Projections to Symbolic Projections
455(17)
Shi-Kuo Chang
Introduction
455(2)
2D string representations of symbolic pictures
457(2)
Picture matching
459(2)
Computer-aided design database
461(3)
Geographical information systems
464(3)
Retrieval of similar Chinese characters
467(1)
Three-dimensional image database querying
468(1)
Medical image database system
469(3)
References
470(2)
Index 472

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