Shearlet-based regularization in statistical inverse learning with an application to X-ray tomography

Statistical inverse learning theory, a field that lies at the intersection of inverse problems and statistical learning, has lately gained more and more attention. In an effort to steer this interplay more towards the variational regularization framework, convergence rates have recently been proved for a class of convex, $p$-homogeneous regularizers with $p \in (1,2]$, in the symmetric Bregman distance. Following this path, we take a further step towards the study of sparsity-promoting regularization and extend the aforementioned convergence rates to work with $\ell^p$-norm regularization, with $p \in (1,2)$, for a special class of non-tight Banach frames, called shearlets, and possibly constrained to some convex set. The $p = 1$ case is approached as the limit case $(1,2) \ni p \rightarrow 1$, by complementing numerical evidence with a (partial) theoretical analysis, based on arguments from $\Gamma$-convergence theory. We numerically demonstrate our theoretical results in the context of X-ray tomography, under random sampling of the imaging angles, using both simulated and measured data

Shearlet-based regularization in statistical inverse learning with an application to x-ray tomography

Statistical inverse learning theory, a field that lies at the intersection of inverse problems and statistical learning, has lately gained more and more attention. In an effort to steer this interplay more towards the variational regularization framework, convergence rates have recently been proved for a class of convex, p-homogeneous regularizers with p (1, 2], in the symmetric Bregman distance. Following this path, we take a further step towards the study of sparsity-promoting regularization and extend the aforementioned convergence rates to work with .," p -norm regularization, with p (1, 2), for a special class of non-tight Banach frames, called shearlets, and possibly constrained to some convex set. The p = 1 case is approached as the limit case (1, 2) p → 1, by complementing numerical evidence with a (partial) theoretical analysis, based on arguments from "-convergence theory. We numerically validate our theoretical results in the context of x-ray tomography, under random sampling of the imaging angles, using both simulated and measured data. This application allows to effectively verify the theoretical decay, in addition to providing a motivation for the extension to shearlet-based regularization

A nonsmooth regularization approach based on shearlets for Poisson noise removal in ROI tomography

Due to its potential to lower exposure to X-ray radiation and reduce the scanning time, region-of-interest (ROI) computed tomography (CT) is particularly appealing for a wide range of biomedical applications. To overcome the severe ill-posedness caused by the truncation of projection measurements, ad hoc strategies are required, since traditional CT reconstruction algorithms result in instability to noise, and may give inaccurate results for small ROI. To handle this difficulty, we propose a nonsmooth convex optimization model based on ℓ1 shearlet regularization, whose solution is addressed by means of the variable metric inexact line search algorithm (VMILA), a proximal-gradient method that enables the inexact computation of the proximal point defining the descent direction. We compare the reconstruction performance of our strategy against a smooth total variation (sTV) approach, by using both Poisson noisy simulated data and real data from fan-beam CT geometry. The results show that, while for synthetic data both shearets and sTV perform well, for real data, the proposed nonsmooth shearlet-based approach outperforms sTV, since the localization and directional properties of shearlets allow to detect finer structures of a textured image. Finally, our approach appears to be insensitive to the ROI size and location