Sparse regularization in limited angle tomography

TL;DR

This paper explores sparse regularization with curvelets for improved edge-preserving reconstruction in limited angle tomography, addressing severe ill-posedness due to incomplete data.

Contribution

It introduces a novel approach combining sparse regularization with curvelets, providing a characterization of solutions and data dimension reduction in limited angle tomography.

Findings

Curvelet-based regularization enhances edge preservation.

Significant reduction in problem dimension in the curvelet domain.

Numerical experiments demonstrate practical effectiveness.

Abstract

We investigate the reconstruction problem of limited angle tomography. Such problems arise naturally in applications like digital breast tomosynthesis, dental tomography, electron microscopy etc. Since the acquired tomographic data is highly incomplete, the reconstruction problem is severely ill-posed and the traditional reconstruction methods, such as filtered backprojection (FBP), do not perform well in such situations. To stabilize the reconstruction procedure additional prior knowledge about the unknown object has to be integrated into the reconstruction process. In this work, we propose the use of the sparse regularization technique in combination with curvelets. We argue that this technique gives rise to an edge-preserving reconstruction. Moreover, we show that the dimension of the problem can be significantly reduced in the curvelet domain. To this end, we give a characterization of the kernel of limited angle Radon transform in terms of curvelets and derive a characterization of solutions obtained through curvelet sparse regularization. In numerical experiments, we will present the practical relevance of these results.