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Asymptotic Analysis of the SVD for the Truncated Hilbert Transform with Overlap
- Publication Year :
- 2015
- Publisher :
- Society for Industrial and Applied Mathematics Publications, 2015.
-
Abstract
- The truncated Hilbert transform with overlap $H_T$ is an operator that arises in tomographic reconstruction from limited data, more precisely in the method of Differentiated Back-Projection (DBP). Recent work [1] has shown that the singular values of this operator accumulate at both zero and one. To better understand the properties of the operator and, in particular, the ill-posedness of the inverse problem associated with it, it is of interest to know the rates at which the singular values approach zero and one. In this paper, we exploit the property that $H_T$ commutes with a second-order differential operator $L_S$ and the global asymptotic behavior of its eigenfunctions to find the asymptotics of the singular values and singular functions of $H_T$.<br />Comment: 25 pages
- Subjects :
- Asymptotic analysis
Mathematics(all)
tomography
01 natural sciences
030218 nuclear medicine & medical imaging
Mathematics - Spectral Theory
03 medical and health sciences
symbols.namesake
0302 clinical medicine
Singular value decomposition
Classical Analysis and ODEs (math.CA)
FOS: Mathematics
0101 mathematics
Spectral Theory (math.SP)
Mathematics
Applied Mathematics
Operator (physics)
010102 general mathematics
Spectrum (functional analysis)
Mathematical analysis
Eigenfunction
Differential operator
Computational Mathematics
Singular value
Mathematics - Classical Analysis and ODEs
symbols
Hilbert transform
Hilbert Transform
Analysis
Limited data
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....c9a979e6eea12bb8eb044e09d8cba388