Reconstruction of 3D localized objects from limited angle X-ray projections: an approach based on sparsity and multigrid image representation

Abstract

This paper is about 3D image reconstruction from a limited set of computed tomography (CT) projections. We focus on configurations with very limited angle of view and on applications in which the image to be reconstructed is composed of one or several localized objects laying in a known background. We propose an original method based on the detection of the localized object voxels and on a sparse modeling of the image. Reconstruction is done by computing the maximum a posteriori estimator of the image parameters. To implement image reconstruction, we adopt a multigrid strategy in which coarse-to-fine resoluted images are successively reconstructed. This strategy provides detection of localized object voxels as well as accurate initial solutions at each resolution level. Each optimization stage is carried out by using an iterative deterministic descent algorithm. We propose a convergent single site update algorithm which consists of successive constrained optimizations with respect to one voxel at a time. We show the performance of the multigrid method on simulated data corresponding to a set of limited angle cone-beam projections of a synthetic image. The results are accurate, while both memory storage and numerical time of computation are dramatically reduced compared with the monogrid method.