Stability estimates for the regularized inversion of the truncated Hilbert transform

Rima Alaifari, Michel Defrise, Alexander Katsevich

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4 Citations (Scopus)

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 in 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. 47 797–824).

In this paper, we study the reconstruction of f restricted to the overlap region between G anf F. We show that with this restriction and by assuming prior knowledge on the L2 norm or on the variation of f, better stability with Hölder continuity (typical for mildly ill-posed problems) can be obtained.
Original languageEnglish
Article number065005
Number of pages17
JournalInverse Problems
Volume32
Issue number6
DOIs
Publication statusPublished - Apr 2016

Keywords

  • hilbert transform
  • TOMOGRAPHY

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