Mathematical Methods in Tomography Proceedings of a Conference held
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 developme
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G. T. Herman
A. K. Louis
F. Natterer R. Applying Proposition 1 (or the analogous statement for the Radon transform Rp on Rn) to all tangentplanes to Sr0 we find that N*(Sr0 ) n WFA(f) must be empty. Lemma 2 now shows that f must vanish in some neighbourhood of Sro· This contradicts the definition of r 0 , hence the proof is complete.
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REFERENCES
[B1]. Uniqueness theorems for generalized Radon transforma, in "Constructive Theory of Functions '84," Sofia, 1984, pp. 173-176. [B2]. J. Boman, An ezample of non-uniqueness for a generalized Radon transform, Dept. of Math., University of Stockholm 1984:13. (Bj]. J.-E. Bjiirk, "Rings of Dilferential Operators," North-Holland Publishing Comp., Amsterdam, 1979. (BQ1]. J. Boman and E. T. Quinto, Support tl&eorems for real-analytic Radon transforma, Duke Math. J. 55 (1987), 943-948. (BQ2]. J. Boman and E. T. Quinto, Support theorems for real-analytic Radon transforma online comple:z:es in three-space, to appear in Trans. Amer. Math. Soc. (GS]. V. Guillemin and S. Sternberg, "Geometric Asymptotics," Amer. Math. Soc., Providence, RI, 1977. (GU]. A. Greenleaf and G. Uhlmann, Nonlocal inversion formulas for the X-ray transform, Duke Math. J. 58 (1989), 205-240. (H1]. L. Hiirmander, Fourier integral operators I, Acta Math. 127 (1971), 79-183. [H2]. L. Hiirmander, "The analysis of linear partial dilferential operators, voi. 1," Springer-Verlag, Berlin, Heidelberg, and New York, 1983. (He1]. S. Helgason, The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds, Acta Math. 113 (1965), 153-180. . (He2]. S. Helgason, "The Radon transform," Birkhii.user, Boston, 1980. (He3]. S. Helgason, Support of Radon transforma, Adv. Math. 38 (1980), 91-100. [Q1]. E. T: Quinto, On the locality and invertibility of Radon transforma, Thesis, M.I.T., Cambridge, Mass. (1978). [Q2]. E. T. Quinto, The dependence of the generalized Radon transforma on the defining measureş, Trans. Amer. Math. Soc. 257 (1980), 331-346. (Q3]. E. T. Quinto, The invertibility ofrotation invariant Radon transforma, J. Math. Anal. Appl. 91 (1983), 510-522.
Singular Value Decompositions for Radon Transforms Peter Maass Depa.rtment of Ma.thema.tics, Tufts University Medford, MA 02155,
U. S. A.
1 Introd uction In this paper simple techniques are developed for the construction of singular value decompositions (SVD) for rotationally invariant Radon transforms in euclidean spaces. First we introduce the definition of a SVD. Definition 1. Let A be a linear operator between (separable) Hilbert spaces X,Y A:X-+Y
The triple {un, Vn, O'n}n>O is called a Singular Value Decomposition (SVD) · of the operator A if {un}n>l is a complete orthonormal system in X, {vn}n;l is an orthonormal system in Y, {O'n} fS a set of non-negative real numbers,
The singular values O'n are usually ordered such that
0'1
2:
0'2
2: .. 2: O .
Sometimes one refers to the singular functions {Un} as generalized Eigenfunctions, since (A* A)un = O'~ Un The importance of a SVD comes form it