Reconstruction from truncated Hilbert transform data - a wavelet approach

This paper presents a study of the stability of the inverse problem of tomography with incomplete data. It was commonly assumed, since the early years of Computerized Tomography, that the exact reconstruction of a section from measurements of its Radon transform $R f (s,\theta)$ is possible only if all lines crossing the section are measured, even if one is interested in the reconstruction of only a sub-region (''region of interest'') of the section. This belief had implications on data acquisition and on the radiation dose. It is now known that exact and stable reconstruction from incomplete data is possible for specific subsets of line integrals.

One class of already identified configurations relies on the reduction of the 2D or 3D reconstruction problem to the inversion of the Hilbert transform (obtained by ''Differentiated Back-Projection'') along a family of lines covering the region of interest. Thus, we consider the problem of reconstructing a 1D signal from its (partially known) Hilbert transform.



Suppose $D \subset \mathbb{R}^2$ is compact, $f \in L^2(D)$, $\mathcal{L}$ a line that intersects $D$ and the Hilbert transform $\mh f_{\mathcal{L}}$ is known on a segment that covers $D \cap \mathcal{L}$. Then, the inverse finite Hilbert transform [1] allows for the unique and stable reconstruction of $f_{\mathcal{L}}$. However, if the segment on which $\mh f_{\mathcal{L}}$ is known does not cover $D \cap \mathcal{L}$, the problem is more delicate. This situation can occur when the field of view of the scanner does not cover the object support $D$. Given real numbers $a_1
X_{[a_2,a_4]} \mathcal{H} X_{[a_1,a_3]} f_{\mathcal{L}} = g,

where $g:[a_2,a_4] \rightarrow \mathbb{R}$, $g \in L^2(\mathbb{R})$ is the measured Hilbert transform and $X$ is the characteristic function.

A remaining open question is whether it is possible to obtain improved stability estimates for the corresponding ill-posed inverse problem. Prior estimates, based on analyticity arguments [2], demonstrate that the problem is not severely ill-posed (unlike for instance an analytic continuation). However, they are pessimistic point-wise error estimates. Better estimates could be obtained by analyzing the truncated Hilbert transform in (\ref{tht}) in wavelet bases. The rationale for using wavelets is that their Hilbert transform decays as $1/|x|^{n+1}$, $n$ being their number of vanishing moments. As a consequence, the Hilbert operator is diagonally dominated in this basis with entries decaying faster than in the canonical basis.

We introduce an approach in which the problem is formulated with wavelets that form bases of finite intervals rather than of the whole line [3]. We show that this allows to exploit the properties of the classical Hilbert transform (without truncations) and reformulate the problem in the sense that we only aim at recovering $f_{\mathcal{L}}$ on the overlap region $[a_2,a_3]$. It will be necessary to use, as part of the data, an a priori bound on the function $f_{\mathcal{L}}$ on $[a_1,a_2]$. This bound could be formulated in terms of a weighted $\ell^p$-norm on the wavelet coefficients, which would lead to a stability estimate that at the same time is not overly restrictive on $f_{\mathcal{L}}$.

We will conclude by showing some results on the numerical realization of the wavelet decomposition of the system operator and its SVD.