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Cover image for Image processing : tensor transform and discrete tomography with MATLAB
Title:
Image processing : tensor transform and discrete tomography with MATLAB
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Publication Information:
Boca Raton, FL : CRC Press/Taylor & Francis, 2013.
Physical Description:
xv, 442 p. : ill. ; 24 cm.
ISBN:
9781466509948
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33000000000744 TA1637 G75 2013 Open Access Book Book
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Summary

Summary

Focusing on mathematical methods in computer tomography, Image Processing: Tensor Transform and Discrete Tomography with MATLAB® introduces novel approaches to help in solving the problem of image reconstruction on the Cartesian lattice. Specifically, it discusses methods of image processing along parallel rays to more quickly and accurately reconstruct images from a finite number of projections, thereby avoiding overradiation of the body during a computed tomography (CT) scan.

The book presents several new ideas, concepts, and methods, many of which have not been published elsewhere. New concepts include methods of transferring the geometry of rays from the plane to the Cartesian lattice, the point map of projections, the particle and its field function, and the statistical model of averaging. The authors supply numerous examples, MATLAB®-based programs, end-of-chapter problems, and experimental results of implementation.

The main approach for image reconstruction proposed by the authors differs from existing methods of back-projection, iterative reconstruction, and Fourier and Radon filtering. In this book, the authors explain how to process each projection by a system of linear equations, or linear convolutions, to calculate the corresponding part of the 2-D tensor or paired transform of the discrete image. They then describe how to calculate the inverse transform to obtain the reconstruction. The proposed models for image reconstruction from projections are simple and result in more accurate reconstructions.

Introducing a new theory and methods of image reconstruction, this book provides a solid grounding for those interested in further research and in obtaining new results. It encourages readers to develop effective applications of these methods in CT.


Author Notes

Artyom M. Grigoryan, Ph.D., is currently an associate professor at the Department of Electrical Engineering, University of Texas at San Antonio. He has authored or co-authored three books, including Brief Notes in Advanced DSP: Fourier Analysis with MATLAB® (2009) and Multidimensional Discrete Unitary Transforms: Representation: Partitioning, and Algorithms (2003) as well as two book chapters and many journal papers. He specializes in the theory and application of fast one- and multi-dimensional Fourier transforms, elliptic Fourier transforms, tensor and paired transforms, integer unitary heap transforms, design of robust linear and nonlinear filters, image encryption, computerized 2-D and 3-D tomography, and processing of biomedical images.

Merughan M. Grigoryan is currently conducting research on the theory and application of quantum mechanics in signal processing, differential equations, Fourier analysis, elliptic Fourier transforms, Hadamard matrices, fast integer unitary transformations, the theory and methods of the fast unitary transforms generated by signals, and methods of encoding in cryptography. He is the coauthor of the book Brief Notes in Advanced DSP: Fourier Analysis with MATLAB® (2009).


Table of Contents

Author Biosp. xi
Prefacep. xiii
1 Discrete 2-D Fourier Transformp. 1
1.1 Separable 2-D transformsp. 2
1.2 Vector forms of representationp. 4
1.3 Partitioning of 2-D transformsp. 5
1.3.1 Tensor representationp. 8
1.3.2 Covering with cyclic groupsp. 9
1.4 Tensor representation of the 2-D DFTp. 12
1.4.0.1 Code: Splitting-signal calculationp. 13
1.4.1 Tensor algorithm of the 2-D DFTp. 13
1.4.2 N is primep. 14
1.4.2.1 Code: 2-D DFT by tensor transformp. 20
1.4.3 N is a power of twop. 21
1.4.4 N is a power of an odd primep. 27
1.4.5 Case N = L 1 L 2 (L 1 ≠ L 2 > 1)p. 29
1.4.6 General casep. 29
1.4.7 Other orders N 1 × N 2p. 30
1.5 Discrete Fourier transform and its geometryp. 32
1.5.1 Inverse DFTp. 35
Problemsp. 39
2 Direction Imagesp. 41
2.1 2-D direction images on the latticep. 41
2.1.1 Superposition of directionsp. 44
2.2 The inverse tensor transform: Case N is primep. 51
2.2.1 Inverse tensor transformp. 51
2.2.2 Formula of the inverse tensor transformp. 57
2.2.2.1 Code for inverse tensor transformp. 58
2.3 3-D paired representationp. 60
2.3.1 2D-to-3D paired transformp. 62
2.3.2 The splitting of the 2-D DFTp. 66
2.4 Complete system of 2-D paired functionsp. 75
2.4.0.1 Code: System of basic paired functionsp. 80
2.4.1 1-D DFT and paired transformp. 81
2.5 Paired transform direction imagesp. 83
2.6 L-paired representation of the imagep. 87
2.6.1 Principle of superposition: General casep. 90
Problemsp. 94
3 Image Sampling Along Directionsp. 97
3.1 Image reconstruction: Model Ip. 98
3.1.1 Coordinate systems and raysp. 100
3.2 Inverse paired transformp. 101
3.3 Example: Image 4×4p. 103
3.3.1 Horizontal and vertical projectionsp. 103
3.3.2 Diagonal projectionsp. 107
3.3.3 Other projectionsp. 109
3.3.3.1 Generator (1,3)p. 109
3.3.3.2 Generator (1,2)p. 111
3.3.3.3 Generator (2,1)p. 115
3.4 Property of the directed multiresolutionp. 120
3.5 Example: Image 8×8p. 121
3.5.1 Horizontal projectionp. 121
3.5.2 Vertical projectionp. 124
3.5.3 Diagonal projectionp. 125
3.5.4 (2,1)- and (1,2)-projectionsp. 129
3.5.4.1 (2,1)-projectionp. 129
3.5.4.2 (1,2)-projectionp. 137
3.5.5 (1,3)-projectionp. 143
3.5.6 (1,4)- and (4,1)-projectionsp. 158
3.5.7 (1,5)-projectionp. 172
3.5.8 (1,6)-projectionp. 189
3.5.9 (6,1)-projectionp. 196
3.5.10 (1,7)-projectionp. 202
3.6 Summary of resultsp. 208
3.6.1 Equations of raysp. 210
3.6.2 Equations for line-integralsp. 213
3.7 Equations in the coordinate system (X, 1 - Y)p. 214
3.7.1 Convolution equationsp. 219
Problemsp. 224
4 Main Program of Image Reconstructionp. 227
4.1 The main diagram of the reconstructionp. 227
4.2 Part 1: Image modelp. 229
4.3 The coordinate system and raysp. 231
4.4 Part 2: Projection datap. 232
4.5 Part 3: Transformation of geometryp. 237
4.6 Part 4: Linear transformation of projectionsp. 241
4.7 Part 5: Calculation the 2-D paired transformp. 245
4.7.1 Method of incomplete 1-D DPTp. 246
4.7.2 Fast 1-D paired transformp. 247
4.7.3 Inverse 2-D DPTp. 250
4.7.4 Preliminary resultsp. 252
4.8 Fast projection integrals by squaresp. 254
4.9 Selection of projectionsp. 265
Problemsp. 268
5 Reconstruction for Prime Size Imagep. 271
5.1 Image reconstruction: Model IIp. 271
5.2 Example with image 7×7p. 272
5.2.1 Horizontal projectionp. 273
5.2.2 Vertical projectionp. 274
5.2.3 Diagonal projectionp. 275
5.2.4 (1,2)-Projectionp. 279
5.2.5 (1,3)-projectionp. 285
5.2.6 (1,4)-projectionp. 291
5.2.7 (1,5)-projectionp. 299
5.2.8 (1,6)-projectionp. 306
5.2.9 Reconstructed image 7×7p. 311
5.3 General algorithm of image reconstructionp. 313
5.4 Program description and image modelp. 315
5.5 System of equationsp. 318
5.6 Solutions of convolution equationsp. 319
5.6.1 Splitting-signal compositionp. 321
5.6.2 Inverse 2-D tensor transformp. 322
5.7 MATLAB®-based code (N prime)p. 324
Problemsp. 327
6 Method of Particlesp. 329
6.1 Point-map of projectionsp. 329
6.1.1 A-particle and the fieldp. 332
6.1.2 Representation by field functionsp. 337
6.2 Method of G-raysp. 343
6.2.1 G-rays for the first set of generatorsp. 343
6.2.2 G-rays for the second set of generatorsp. 348
6.2.3 G-rays for the third set of generatorsp. 351
6.2.4 G-rays for the fourth set of generatorsp. 354
6.2.5 Map of projections for one squarep. 355
6.2.5.1 Codes for particlesp. 360
6.3 Reconstruction by field transformp. 365
6.4 Method of circular convolutionp. 374
6.4.1 Uniform framesp. 379
Problemsp. 380
7 Methods of Averaging Projectionsp. 383
7.1 Filtered backprojectionp. 384
7.2 BP and method of splitting-signalsp. 386
7.2.1 Tensor method of summation of projectionsp. 390
7.3 Method of summation of Line-integralsp. 397
7.4 Models with averagingp. 398
7.4.1 Method of proportionp. 399
7.4.2 Method with probability modelp. 402
7.4.3 Reconstruction of the shifted imagep. 404
7.4.4 Method of minimization of errorp. 407
7.4.5 Corpuscular approachp. 409
7.5 General case: Probability modelp. 411
7.5.0.1 Code of the reconstructionp. 414
Problemsp. 417
Bibliographyp. 423
Appendix Ap. 427
Appendix Bp. 433
Indexp. 441
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