A new framework for sparse regularization in limited angle x-ray tomography

TL;DR: In this paper, the authors proposed the use of the sparse regularization technique in combination with curvelets, which gives rise to an edge-preserving reconstruction, and showed that the dimension of the problem can be significantly reduced in the curvelet domain.

TL;DR: This work proposes the use of the sparse regularization technique in combination with curvelets and argues that this technique gives rise to an edge-preserving reconstruction and shows that the dimension of the problem can be significantly reduced in the curvelet domain.

TL;DR: Computer simulations show that ART + TV3D method substantially enhances the reconstructed image with fewer artifacts and smaller error rates than the other two algorithms under the same configuration and parameters and it provides faster convergence rate.

TL;DR: In this paper, a characterization of filtered backprojection reconstructions from limited angle data is presented, and a strategy for artifact reduction and stabilization is developed for tomographic reconstruction at limited angular range.

TL;DR: It is proved that replacing the usual quadratic regularizing penalties by weighted 𝓁p‐penalized penalties on the coefficients of such expansions, with 1 ≤ p ≤ 2, still regularizes the problem.

TL;DR: Candes et al. as discussed by the authors established new results about the accuracy of the reconstruction from undersampled measurements, which improved on earlier estimates, and have the advantage of being more elegant. But they did not consider the restricted isometry property of the sensing matrix.

TL;DR: In this paper, the Radon transform and related transforms have been studied for stability, sampling, resolution, and accuracy, and quite a bit of attention is given to the derivation, analysis, and practical examination of reconstruction algorithm, for both standard problems and problems with incomplete data.

Abstract: We consider linear inverse problems where the solution is assumed to have a sparse expansion on an arbitrary pre-assigned orthonormal basis. We prove that replacing the usual quadratic regularizing penalties by weighted l^p-penalties on the coefficients of such expansions, with 1 < or = p < or =2, still regularizes the problem. If p < 2, regularized solutions of such l^p-penalized problems will have sparser expansions, with respect to the basis under consideration. To compute the corresponding regularized solutions we propose an iterative algorithm that amounts to a Landweber iteration with thresholding (or nonlinear shrinkage) applied at each iteration step. We prove that this algorithm converges in norm. We also review some potential applications of this method.