Stability estimates for the regularized inversion of the truncated Hilbert transform. (28th April 2016)

• Record Type: Journal Article
• Authors: Alaifari, Rima; Defrise, Michel; Katsevich, Alexander
• Is Part Of: Inverse problems. Volume 32:Number 6(2016:Jun.)
• Publication Date: 2016-04-28
• Subjects: limited data computerized tomography -- truncated Hilbert transform -- severely ill-posed -- regularized inversion -- stability estimate; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/0266-5611/32/6/065005
• Languages: English
• ISSNs: 0266-5611

Abstract

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 ∊ L 2 (  ), where  is a finite interval from its partial Hilbert transform data. When the Hilbert transform is measured on a finite interval  that only overlaps but does not cover , 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  ∩  . We show that with this restriction and by assuming prior knowledge on the L 2 norm or on the variation of f, better stability with Hölder continuity (typical for mildly ill-posed problems) can be obtained.