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arXiv · 1507.01141

Stability estimates for the regularized inversion of the truncated Hilbert transform

Rima Alaifari·Michel Defrise·Alexander Katsevich
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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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Rima Alaifari, Michel Defrise, Alexander Katsevich. 2016-01-28. Stability estimates for the regularized inversion of the truncated Hilbert transform. https://doi.org/10.1088/0266-5611%2F32%2F6%2F065005

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