Stability Estimates For The Regularized Inversion Of The Truncated Hilbert Transform

Keywords

limited data computerized tomography; regularized inversion; severely ill-posed; stability estimate; truncated Hilbert transform

Abstract

In limited data computerized tomography, the 2D or 3D problem can be reduced to a family of 1D problems using the differentiated backprojection method. Each 1D problem consists of recovering a compactly supported function f ϵ L2 (F), where F is a finite interval from its partial Hilbert transform data. When the Hilbert transform is measured on a finite interval G that only overlaps but does not cover F , this inversion problem is known to be severely ill-posed (Alaifari et al 2015 SIAM J. Math. Anal. 47 797824). In this paper, we study the reconstruction of f restricted to the overlap region F ∩ G. We show that with this restriction and by assuming prior knowledge on the L2 norm or on the variation of f, better stability with Hlder continuity (typical for mildly ill-posed problems) can be obtained.

Publication Date

4-28-2016

Publication Title

Inverse Problems

Volume

32

Issue

6

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1088/0266-5611/32/6/065005

Socpus ID

84969916538 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84969916538

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