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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 Bios | p. xi |
Preface | p. xiii |
1 Discrete 2-D Fourier Transform | p. 1 |
1.1 Separable 2-D transforms | p. 2 |
1.2 Vector forms of representation | p. 4 |
1.3 Partitioning of 2-D transforms | p. 5 |
1.3.1 Tensor representation | p. 8 |
1.3.2 Covering with cyclic groups | p. 9 |
1.4 Tensor representation of the 2-D DFT | p. 12 |
1.4.0.1 Code: Splitting-signal calculation | p. 13 |
1.4.1 Tensor algorithm of the 2-D DFT | p. 13 |
1.4.2 N is prime | p. 14 |
1.4.2.1 Code: 2-D DFT by tensor transform | p. 20 |
1.4.3 N is a power of two | p. 21 |
1.4.4 N is a power of an odd prime | p. 27 |
1.4.5 Case N = L 1 L 2 (L 1 ≠ L 2 > 1) | p. 29 |
1.4.6 General case | p. 29 |
1.4.7 Other orders N 1 × N 2 | p. 30 |
1.5 Discrete Fourier transform and its geometry | p. 32 |
1.5.1 Inverse DFT | p. 35 |
Problems | p. 39 |
2 Direction Images | p. 41 |
2.1 2-D direction images on the lattice | p. 41 |
2.1.1 Superposition of directions | p. 44 |
2.2 The inverse tensor transform: Case N is prime | p. 51 |
2.2.1 Inverse tensor transform | p. 51 |
2.2.2 Formula of the inverse tensor transform | p. 57 |
2.2.2.1 Code for inverse tensor transform | p. 58 |
2.3 3-D paired representation | p. 60 |
2.3.1 2D-to-3D paired transform | p. 62 |
2.3.2 The splitting of the 2-D DFT | p. 66 |
2.4 Complete system of 2-D paired functions | p. 75 |
2.4.0.1 Code: System of basic paired functions | p. 80 |
2.4.1 1-D DFT and paired transform | p. 81 |
2.5 Paired transform direction images | p. 83 |
2.6 L-paired representation of the image | p. 87 |
2.6.1 Principle of superposition: General case | p. 90 |
Problems | p. 94 |
3 Image Sampling Along Directions | p. 97 |
3.1 Image reconstruction: Model I | p. 98 |
3.1.1 Coordinate systems and rays | p. 100 |
3.2 Inverse paired transform | p. 101 |
3.3 Example: Image 4×4 | p. 103 |
3.3.1 Horizontal and vertical projections | p. 103 |
3.3.2 Diagonal projections | p. 107 |
3.3.3 Other projections | p. 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 multiresolution | p. 120 |
3.5 Example: Image 8×8 | p. 121 |
3.5.1 Horizontal projection | p. 121 |
3.5.2 Vertical projection | p. 124 |
3.5.3 Diagonal projection | p. 125 |
3.5.4 (2,1)- and (1,2)-projections | p. 129 |
3.5.4.1 (2,1)-projection | p. 129 |
3.5.4.2 (1,2)-projection | p. 137 |
3.5.5 (1,3)-projection | p. 143 |
3.5.6 (1,4)- and (4,1)-projections | p. 158 |
3.5.7 (1,5)-projection | p. 172 |
3.5.8 (1,6)-projection | p. 189 |
3.5.9 (6,1)-projection | p. 196 |
3.5.10 (1,7)-projection | p. 202 |
3.6 Summary of results | p. 208 |
3.6.1 Equations of rays | p. 210 |
3.6.2 Equations for line-integrals | p. 213 |
3.7 Equations in the coordinate system (X, 1 - Y) | p. 214 |
3.7.1 Convolution equations | p. 219 |
Problems | p. 224 |
4 Main Program of Image Reconstruction | p. 227 |
4.1 The main diagram of the reconstruction | p. 227 |
4.2 Part 1: Image model | p. 229 |
4.3 The coordinate system and rays | p. 231 |
4.4 Part 2: Projection data | p. 232 |
4.5 Part 3: Transformation of geometry | p. 237 |
4.6 Part 4: Linear transformation of projections | p. 241 |
4.7 Part 5: Calculation the 2-D paired transform | p. 245 |
4.7.1 Method of incomplete 1-D DPT | p. 246 |
4.7.2 Fast 1-D paired transform | p. 247 |
4.7.3 Inverse 2-D DPT | p. 250 |
4.7.4 Preliminary results | p. 252 |
4.8 Fast projection integrals by squares | p. 254 |
4.9 Selection of projections | p. 265 |
Problems | p. 268 |
5 Reconstruction for Prime Size Image | p. 271 |
5.1 Image reconstruction: Model II | p. 271 |
5.2 Example with image 7×7 | p. 272 |
5.2.1 Horizontal projection | p. 273 |
5.2.2 Vertical projection | p. 274 |
5.2.3 Diagonal projection | p. 275 |
5.2.4 (1,2)-Projection | p. 279 |
5.2.5 (1,3)-projection | p. 285 |
5.2.6 (1,4)-projection | p. 291 |
5.2.7 (1,5)-projection | p. 299 |
5.2.8 (1,6)-projection | p. 306 |
5.2.9 Reconstructed image 7×7 | p. 311 |
5.3 General algorithm of image reconstruction | p. 313 |
5.4 Program description and image model | p. 315 |
5.5 System of equations | p. 318 |
5.6 Solutions of convolution equations | p. 319 |
5.6.1 Splitting-signal composition | p. 321 |
5.6.2 Inverse 2-D tensor transform | p. 322 |
5.7 MATLAB®-based code (N prime) | p. 324 |
Problems | p. 327 |
6 Method of Particles | p. 329 |
6.1 Point-map of projections | p. 329 |
6.1.1 A-particle and the field | p. 332 |
6.1.2 Representation by field functions | p. 337 |
6.2 Method of G-rays | p. 343 |
6.2.1 G-rays for the first set of generators | p. 343 |
6.2.2 G-rays for the second set of generators | p. 348 |
6.2.3 G-rays for the third set of generators | p. 351 |
6.2.4 G-rays for the fourth set of generators | p. 354 |
6.2.5 Map of projections for one square | p. 355 |
6.2.5.1 Codes for particles | p. 360 |
6.3 Reconstruction by field transform | p. 365 |
6.4 Method of circular convolution | p. 374 |
6.4.1 Uniform frames | p. 379 |
Problems | p. 380 |
7 Methods of Averaging Projections | p. 383 |
7.1 Filtered backprojection | p. 384 |
7.2 BP and method of splitting-signals | p. 386 |
7.2.1 Tensor method of summation of projections | p. 390 |
7.3 Method of summation of Line-integrals | p. 397 |
7.4 Models with averaging | p. 398 |
7.4.1 Method of proportion | p. 399 |
7.4.2 Method with probability model | p. 402 |
7.4.3 Reconstruction of the shifted image | p. 404 |
7.4.4 Method of minimization of error | p. 407 |
7.4.5 Corpuscular approach | p. 409 |
7.5 General case: Probability model | p. 411 |
7.5.0.1 Code of the reconstruction | p. 414 |
Problems | p. 417 |
Bibliography | p. 423 |
Appendix A | p. 427 |
Appendix B | p. 433 |
Index | p. 441 |