Stability estimates for the regularized inversion of the 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 (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 Hölder continuity (typical for mildly ill-posed problems) can be obtained.

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Cited by in corpus (1)

• Stability Estimates for Truncated Fourier and Laplace Transforms