Stability estimates for the regularized inversion of the truncated Hilbert transform

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.