Chapter 1. Major Fields of Application .......................... 1
1.1. Introduction .............................................. 1
1.2. Medicine; X-ray Computed Tomography ...................... 11
1.3. Medicine; Emission Computed Tomography ................... 16
1.4. Medicine; Ultrasound CT .................................. 21
1.5. Astronomy ................................................ 24
1.6. Electron Microscopy ...................................... 31
1.7. Nuclear Magnetic Resonance ............................... 35
1.8. Optics ................................................... 42
1.9. Stress Analysis, Geophysics, and Other Areas ............. 48
1.10. References ............................................... 50
Chapter 2. Definition of the Radon Transform ................... 55
2.1. Introduction ............................................. 55
2.2. Two Dimensions ........................................... 56
2.3. Three Dimensions ......................................... 59
2.4. Extension to Higher Dimensions ........................... 60
2.5. Some Important Examples .................................. 61
Chapter 3. Basic Properties .................................... 66
3.1. Introduction ............................................. 66
3.2. Homogeneity .............................................. 66
3.3. Linearity ................................................ 67
3.4. Transform of a Linear Transformation ..................... 68
3.5. Shifting Property ........................................ 71
3.6. Transform of Derivatives ................................. 77
3.7. Transforms Involving Hermite Polynomials ................. 80
3.8. Transforms Involving Laguerre Polynomials ................ 86
3.9. Derivatives of the Transform ............................. 91
3.10. Transform of Convolution ................................. 94
Chapter 4. Relation to Other Transforms ........................ 96
4.1. Introduction ............................................. 96
4.2. Relation to the Fourier Transform ........................ 96
4.3. Relation to the Gegenbauer Transform .................... 100
4.4. Relation to the Hough Transform ......................... 106
4.5. Relation to the Hankel Transform ........................ 107
Chapter 5. Inversion .......................................... 108
5.1. Introduction ............................................ 108
5.2. Odd Dimension ........................................... 108
5.3. Even Dimension .......................................... 112
5.4. Unification and the Adjoint ............................. 115
5.5. Fourier Methods ......................................... 121
Chapter 6. Recent Development of Inversion Methods ............ 125
6.1. Introduction ............................................ 125
6.2. Projection-Slice Theorem ................................ 128
6.3. Backprojection .......................................... 131
6.4. Backprojection of Filtered Projections .................. 136
6.5. Filter of Backprojections ............................... 140
6.6. Iterative Methods ....................................... 142
6.7. Three-dimensional Methods ............................... 147
6.8. Categorized References .................................. 147
Chapter 7. Series Methods ..................................... 151
7.1. Introduction ............................................ 151
7.2. Gegenbauer Transform Pair ............................... 151
7.3. Circular Harmonic Expansion (n = 2) ..................... 152
7.4. Spherical Harmonic Expansion (n = 3) .................... 160
7.5. A Tchebycheff Transform Pair of the Second
Kind .................................................... 163
7.6. Orthogonal Function Expansions on the Unit
Disk .................................................... 163
7.7. Orthogonal Function Expansions Over the Entire
Plane ................................................... 180
7.8. Other Approaches ........................................ 182
Chapter 8. More Properties, Applications, and
Generalizations .................................... 184
8.1. Introduction ............................................ 184
8.2. Characterization of the Transform ....................... 184
8.3. A Discrete Version ...................................... 187
8.4. Picture Restoration ..................................... 191
8.5. Transformations in Geophysics ........................... 192
8.6. The Integral Equation of Potential Scattering ........... 196
8.7. Partial Differential Equations .......................... 198
8.8. Generalizations and Other Uses .......................... 201
Appendix A. Translation of Radon's 1917 Paper ................. 204
Appendix B. Generalized Functions ............................. 218
Appendix C. Special Functions ................................. 240
References .................................................... 247
Additional and Recent References .............................. 282
Index ......................................................... 289
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