Stability estimates for the regularized inversion of the truncated Hilbert transform

TL;DR

This paper investigates the stability of regularized inversion of the truncated Hilbert transform in limited data tomography, demonstrating improved stability estimates under certain prior assumptions on the function.

Contribution

It provides new stability estimates for the truncated Hilbert transform inversion, showing H"older continuity when reconstructing on the overlap region with prior knowledge.

Findings

Better stability with H"older continuity for restricted reconstruction

Improved understanding of ill-posedness in limited data tomography

Quantitative stability estimates under prior assumptions

Abstract

In limited data computerized tomography, the 2D or 3D problem can be reduced to a family of 1D problems using the differentiated backprojection (DBP) method. Each 1D problem consists of recovering a compactly supported function $f \in L^2(\mathcal F)$, where $\mathcal F$ is a finite interval, from its partial Hilbert transform data. When the Hilbert transform is measured on a finite interval $\mathcal G$ that only overlaps but does not cover $\mathcal F$ this inversion problem is known to be severely ill-posed [1]. In this paper, we study the reconstruction of $f$ restricted to the overlap region $\mathcal F \cap \mathcal G$. 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 H\"older continuity (typical for mildly ill-posed problems) can be obtained.