Part I. Inverse and Semi-discrete Problems
1. III-posed problems and regularization methods ............... 5
2. Approximate inverse in L2-spaces ........................... 11
2.1. The idea of approximate inverse ...................... 11
2.2. A first example: The Radon transform ................. 17
3. Approximate inverse in Hilbert spaces ...................... 25
3.1. Semi-discrete operator equations ..................... 25
3.2. Convergence and stability ............................ 32
4. Approximate inverse in distribution spaces ................. 39
4.1. Mollifier and reconstruction kernels in dual
spaces of smooth functions ........................... 40
4.2. Dealing with semi-discrete equations ................. 44
5. Conclusion and perspectives ................................ 49
Part II. Application to 3D Doppler Tomography
6. A semi-discrete setup for Doppler tomography ............... 55
7. Solving the semi-discrete problem .......................... 63
7.1. Definition of the operators Пp,q,r and Ed ............. 63
7.2. Computation of reconstruction kernels for Dj ......... 70
7.3. The method of approximate inverse for
Ωр,q,rD ............................................... 76
8. Convergence and stability .................................. 81
9. Approaches for defect correction ........................... 89
9.1. Potentials as solutions of elliptic boundary
value problems ....................................... 90
9.2. The Neumann problem .................................. 93
9.2.1. A boundary element method for the Neumann
problem ...................................... 93
9.2.2. The computation of the Newton potentials ..... 94
9.2.3. Numerical results ........................... 100
9.3. The Dirichlet problem ............................... 101
10. Conclusion and perspectives ............................... 105
Part III. Application to the spherical mean operator
11. The spherical mean operator ............................... 111
11.1. Spherical means in SONAR and SAR .................... 111
11.2. Properties of the spherical mean operator ........... 113
11.3. Approximate inverse for M ........................... 118
12. Design of a mollifier ..................................... 123
13. Computation of reconstruction kernels ..................... 133
14. Numerical experiments ..................................... 139
15. Conclusion and perspectives ............................... 145
Part IV. Further Applications
16. Approximate inverse and X-ray diffractometry .............. 151
16.1. X-ray diffractometry ................................ 151
16.2. Approximate inverse for the Laplace transform ....... 153
16.3. A solution scheme for the X-ray diffractometry
problem ............................................. 161
17. A filtered backprojection algorithm for thermoacoustic
computerized tomography (TCT) ............................. 165
17.1. Thermoacoustic computerized tomography (TCT) ........ 165
17.2. An inversion method for the spherical geometry ...... 168
17.3. Numerical results ................................... 175
18. Computation of reconstruction kernels in 3D computerized
tomography ................................................ 181
19. Conclusion and perspectives ............................... 187
References .................................................... 189
Index ......................................................... 197
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