Mathematical Methods in Tomography: Proceedings of a Conference held in Oberwolfach, Germany, 5–11 June, 1990

Jan Boman (auth.), Gabor T. Herman, Alfred K. Louis, Frank Natterer (eds.)3540549706, 9783540549703, 0387549706

The conference was devoted to the discussion of present and future techniques in medical imaging, including 3D x-ray CT, ultrasound and diffraction tomography, and biomagnetic ima- ging. The mathematical models, their theoretical aspects and the development of algorithms were treated. The proceedings contains surveys on reconstruction in inverse obstacle scat- tering, inversion in 3D, and constrained least squares pro- blems.Research papers include besides the mentioned imaging techniques presentations on image reconstruction in Hilbert spaces, singular value decompositions, 3D cone beam recon- struction, diffuse tomography, regularization of ill-posed problems, evaluation reconstruction algorithms and applica- tions in non-medical fields. Contents: Theoretical Aspects: J.Boman: Helgason’ s support theorem for Radon transforms-a newproof and a generalization -P.Maass: Singular value de- compositions for Radon transforms- W.R.Madych: Image recon- struction in Hilbert space -R.G.Mukhometov: A problem of in- tegral geometry for a family of rays with multiple reflec- tions -V.P.Palamodov: Inversion formulas for the three-di- mensional ray transform – Medical Imaging Techniques: V.Friedrich: Backscattered Photons – are they useful for a surface – near tomography – P.Grangeat: Mathematical frame- work of cone beam 3D reconstruction via the first derivative of the Radon transform -P.Grassin,B.Duchene,W.Tabbara: Dif- fraction tomography: some applications and extension to 3D ultrasound imaging -F.A.Gr}nbaum: Diffuse tomography: a re- fined model -R.Kress,A.Zinn: Three dimensional reconstruc- tions in inverse obstacle scattering -A.K.Louis: Mathemati- cal questions of a biomagnetic imaging problem – Inverse Problems and Optimization: Y.Censor: On variable block algebraic reconstruction techniques -P.P.Eggermont: On Volterra-Lotka differential equations and multiplicative algorithms for monotone complementary problems

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Table of contents :
Helgason’s support theorem for Radon transforms — A new proof and a generalization….Pages 1-5
Singular value decompositions for Radon transforms….Pages 6-14
Image reconstruction in Hilbert space….Pages 15-45
A problem of integral geometry for a family of rays with multiple reflections….Pages 46-52
Inversion formulas for the three-dimensional ray transform….Pages 53-62
Backscattered photons — Are they useful for a surface-near tomography?….Pages 63-65
Mathematical framework of cone beam 3D reconstruction via the first derivative of the radon transform….Pages 66-97
Diffraction tomography some applications and extension to 3-D ultrasound imaging….Pages 98-105
Diffuse tomography: A refined model….Pages 106-111
Three dimensional reconstructions in inverse obstacle scattering….Pages 112-125
Mathematical questions of a biomagnetic imaging problem….Pages 126-132
On variable block algebraic reconstruction techniques….Pages 133-140
On Volterra-Lotka differential equations and multiplicative algorithms for monotone complementarity problems….Pages 141-152
Constrained regularized least squares problems….Pages 153-166
Multiplicative iterative methods in computed tomography….Pages 167-186
Remark on the informative content of few measurements….Pages 187-193
Theorems for the number of zeros of the projection radial modulators of the 2D exponential radon transform….Pages 194-214
Evaluation of reconstruction algorithms….Pages 215-228
Radon transform and analog coding….Pages 229-241
Determination of the specific density of an aerosol through tomography….Pages 242-260
Computed tomography and rockets….Pages 261-268