Author Bios .................................................... xi
Preface ...................................................... xiii
1 Discrete 2-D Fourier Transform ............................. 1
1.1 Separable 2-D transforms ................................... 2
1.2 Vector forms of representation ............................. 4
1.3 Partitioning of 2-D transforms ............................. 5
1.3.1 Tensor representation ............................... 8
1.3.2 Covering with cyclic groups ......................... 9
1.4 Tensor representation of the 2-D DFT ...................... 12
1.4.0.1 Code: Splitting-signal calculation ............... 13
1.4.1 Tensor algorithm of the 2-D DFT .................... 13
1.4.2 N is prime ......................................... 14
1.4.2.1 Code: 2-D DFT by tensor transform ......... 20
1.4.3 N is a power of two ................................ 21
1.4.4 N is a power of an odd prime ....................... 27
1.4.5 Case N = L1L2 (L1 ≠ L2 > 1) ........................ 29
1.4.6 General case ....................................... 29
1.4.7 Other orders N1 × N2 ............................... 30
1.5 Discrete Fourier transform and its geometry ............... 32
1.5.1 Inverse DFT ........................................ 35
Problems .................................................. 39
2 Direction Images .......................................... 41
2.1 2-D direction images on the lattice ....................... 41
2.1.1 Superposition of directions ........................ 44
2.2 The inverse tensor transform: Case N is prime ............. 51
2.2.1 Inverse tensor transform ........................... 51
2.2.2 Formula of the inverse tensor transform ............ 57
2.2.2.1 Code for inverse tensor transform ......... 58
2.3 3-D paired representation ................................. 60
2.3.1 2D-to-3D paired transform .......................... 62
2.3.2 The splitting of the 2-D DFT ....................... 66
2.4 Complete system of 2-D paired functions ................... 75
2.4.0.1 Code: System of basic paired functions .... 80
2.4.1 1-D DFT and paired transform ....................... 81
2.5 Paired transform direction images ......................... 83
2.6 L-paired representation of the image ...................... 87
2.6.1 Principle of superposition: General case .......... 90
Problems .................................................. 94
3 Image Sampling Along Directions ........................... 97
3.1 Image reconstruction: Model I ............................. 98
3.1.1 Coordinate systems and rays ...................... 100
3.2 Inverse paired transform ................................. 101
3.3 Example: Image 4x4 ....................................... 103
3.3.1 Horizontal and vertical projections ............... 103
3.3.2 Diagonal projections .............................. 107
3.3.3 Other projections ................................. 109
3.3.3.1 Generator (1,3) .......................... 109
3.3.3.2 Generator (1,2) .......................... 111
3.3.3.3 Generator (2,1) .......................... 115
3.4 Property of the directed multiresolution ................. 120
3.5 Example: Image 8x8 ....................................... 121
3.5.1 Horizontal projection ............................. 121
3.5.2 Vertical projection ............................... 124
3.5.3 Diagonal projection ............................... 125
3.5.4 (2,1)- and (1,2)-projections ...................... 129
3.5.4.1 (2,1)-projection ......................... 129
3.5.4.2 (l,2)-projection ......................... 137
3.5.5 (1,3)-projection .................................. 143
3.5.6 (1,4)-and (4, l)-projections ...................... 158
3.5.7 (l,5)-projection .................................. 172
3.5.8 (l,6)-projection .................................. 189
3.5.9 (6,l)-projection .................................. 196
3.5.10 (l,7)-projection .................................. 202
3.6 Summary of results ....................................... 208
3.6.1 Equations of rays ................................. 210
3.6.2 Equations for line-integrals ...................... 213
3.7 Equations in the coordinate system (X, 1 - Y) ............ 214
3.7.1 Convolution equations ............................. 219
Problems ................................................. 224
4 Main Program of Image Reconstruction ..................... 227
4.1 The main diagram of the reconstruction ................... 227
4.2 Part 1: Image model ...................................... 229
4.3 The coordinate system and rays ........................... 231
4.4 Part 2: Projection data .................................. 232
4.5 Part 3: Transformation of geometry ....................... 237
4.6 Part 4: Linear transformation of projections ............. 241
4.7 Part 5: Calculation the 2-D paired transform ............. 245
4.7.1 Method of incomplete 1-D DPT ...................... 246
4.7.2 Fast 1-D paired transform ......................... 247
4.7.3 Inverse 2-D DPT ................................... 250
4.7.4 Preliminary results ............................... 252
4.8 Fast projection integrals by squares ..................... 254
4.9 Selection of projections ................................. 265
Problems ................................................. 268
5 RECONSTRUCTION FOR PRIME SIZE IMAGE ...................... 271
5.1 Image reconstruction: Model II ........................... 271
5.2 Example with image 7 × 7 ................................. 272
5.2.1 Horizontal projection ............................. 273
5.2.2 Vertical projection ............................... 274
5.2.3 Diagonal projection ............................... 275
5.2.4 (l,2)-Projection .................................. 279
5.2.5 (l,3)-projection .................................. 285
5.2.6 (l,4)-projection .................................. 291
5.2.7 (l,5)-projection .................................. 299
5.2.8 (l,6)-projection .................................. 306
5.2.9 Reconstructed image 7 × 7 ......................... 311
5.3 General algorithm of image reconstruction ................ 313
5.4 Program description and image model ...................... 315
5.5 System of equations ...................................... 318
5.6 Solutions of convolution equations ....................... 319
5.6.1 Splitting-signal composition ...................... 321
5.6.2 Inverse 2-D tensor transform ...................... 322
5.7 MATLAB®-based code (N prime) ............................. 324
Problems ................................................. 327
6 Method of Particles ...................................... 329
6.1 Point-map of projections ................................. 329
6.1.1 A-particle and the field .......................... 332
6.1.2 Representation by field functions ................. 337
6.2 Method of G-rays ......................................... 343
6.2.1 G-rays for the first set of generators ............ 343
6.2.2 G-rays for the second set of generators ........... 348
6.2.3 G-rays for the third set of generators ............ 351
6.2.4 G-rays for the fourth set of generators ........... 354
6.2.5 Map of projections for one square ................. 355
6.2.5.1 Codes for particles ...................... 360
6.3 Reconstruction by field transform ........................ 365
6.4 Method of circular convolution ........................... 374
6.4.1 Uniform frames .................................... 379
Problems ................................................. 380
7 Methods of Averaging Projections ......................... 383
7.1 Filtered backprojection .................................. 384
7.2 BP and method of splitting-signals ....................... 386
7.2.1 Tensor method of summation of projections ......... 390
7.3 Method of summation of line-integrals .................... 397
7.4 Models with averaging .................................... 398
7.4.1 Method of proportion .............................. 399
7.4.2 Method with probability model ..................... 402
7.4.3 Reconstruction of the shifted image ............... 404
7.4.4 Method of minimization of error ................... 407
7.4.5 Corpuscular approach .............................. 409
7.5 General case: Probability model .......................... 411
7.5.0.1 Code of the reconstruction ...................... 414
Problems ................................................. 417
Bibliography .................................................. 423
Appendix A .................................................... 427
Appendix В .................................................... 433
Index ......................................................... 441
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