Discrete Tomography

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ISBN: 9780817641016 | 0817641017
Cover: Hardcover
Copyright: 9/1/1999

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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.

Preface

xiii

Contributors

xvii

I Foundations

(188)

Discrete Tomography: A Historical Overview

(32)

Attila Kuba

Gabor T. Herman

Introduction

(3)

Foundations and algorithms

(20)

Definitions, notations, and basic problems

(4)

Reconstruction on binary matrices from two projections

(16)

Applications

(3)

Data compression, data coding, and image processing

(1)

Electron microscopy

(1)

Biplane angiography

(1)

Computerized tomography

(1)

Conclusion

(6)

References

(6)

Sets of Uniqueness and Additivity in Integer Lattices

(24)

Peter C. Fishburn

Lawrence A. Shepp

Introduction

(2)

Uniqueness and additivity

(6)

Bad rectangles

(2)

No bad rectangles

(1)

Nonadditive sets of uniqueness

(4)

Planar sets and lines

(5)

Linear programming

(1)

Discussion

(3)

References

(2)

Tomographic Equivalence and Switching Operations

(26)

T. Yung Kong

Gabor T. Herman

Ryser's theorem for binary pictures on the square grid

(11)

Binary and ternary pictures on the square grid

(1)

Four statements of Ryser's theorem

(2)

Metropolis algorithms based on Ryser graphs

(5)

A proof of Ryser's theorem

(2)

Nonexistence of analogs on m-grids with m > 2

(2)

m-grids

(1)

Validity of Ryser's theorem for arbitrary 2-grids

(1)

The main result

(1)

A proof of the main result

(10)

Derivation of the main result from the Main Claim

(2)

Justification of the Main Claim

(7)

Concluding remarks

(3)

References

(2)

Uniqueness and Complexity in Discrete Tomography

(78)

Richard J. Gardner

Peter Gritzmann

Introduction

(1)

Definitions and preliminaries

(2)

Uniqueness results

(9)

Complexity results

(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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Discrete Tomography

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Table of Contents

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