Title

ASYMPTOTIC ANALYSIS OF THE SVD FOR THE TRUNCATED HILBERT TRANSFORM WITH OVERLAP

Authors

Authors

R. Alaifari; M. Defrise;A. Katsevich

Comments

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Abbreviated Journal Title

SIAM J. Math. Anal.

Keywords

limited data; computerized tomography; spectrum; asymptotic analysis; Hilbert transform; ill-posedness; CONE-BEAM CT; IMAGE-RECONSTRUCTION; FAN-BEAM; BACKPROJECTION; Mathematics, Applied

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. Recent work [R. Al-Aifari and A. Katsevich, SIAM J. Math. Anal., 46 (2014), pp. 192213] 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.

Journal Title

Siam Journal on Mathematical Analysis

Volume

47

Issue/Number

1

Publication Date

1-1-2015

Document Type

Article

Language

English

First Page

797

Last Page

824

WOS Identifier

WOS:000353952400029

ISSN

0036-1410

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