Stability estimates for the regularized inversion of the truncated Hilbert transform. (28th April 2016) - Alaifari, Rima; Defrise, Michel; Katsevich, Alexander
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.