The x-ray transform: singular value decomposition and resolution
- 1 November 1987
- journal article
- Published by IOP Publishing in Inverse Problems
- Vol. 3 (4) , 729-741
- https://doi.org/10.1088/0266-5611/3/4/016
Abstract
The x-ray transform maps a compactly supported function in Rn to its integrals over all the straight lines in Rn. A singular value decomposition (SVD) for this operator is given in arbitrary dimensions. The proof uses results from the representation theory of Lie groups. This paper addresses questions concerning the stability of the inversion problem. The SVD shows which parts of a reconstructed function are affected by data errors and by how much. The resolution in the reconstruction is determined if only a finite set of data is available.Keywords
This publication has 12 references indexed in Scilit:
- Reconstruction from Restricted Radon Transform Data: Resolution and Ill-ConditionednessSIAM Journal on Mathematical Analysis, 1986
- Incomplete data problems in x-ray computerized tomographyNumerische Mathematik, 1986
- Nonuniqueness in inverse radon problems: The frequency distribution of the ghostsMathematische Zeitschrift, 1984
- Orthogonal Function Series Expansions and the Null Space of the Radon TransformSIAM Journal on Mathematical Analysis, 1984
- The Ill-Conditioned Nature of the Limited Angle Tomography ProblemSIAM Journal on Applied Mathematics, 1983
- Mathematical problems of computerized tomographyProceedings of the IEEE, 1983
- A singular value decomposition for the radon transform inn-dimensional euclidean spaceNumerical Functional Analysis and Optimization, 1981
- The uncertainty principle in reconstructing functions from projectionsDuke Mathematical Journal, 1975
- On the reconstruction of a function on a circular domain from a sampling of its line integralsJournal of Mathematical Analysis and Applications, 1974
- Representation of a Function by Its Line Integrals, with Some Radiological Applications. IIJournal of Applied Physics, 1964