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TILT: Topological Interface Recovery in Limited-Angle Tomography
(2024)
Elli Karvonen, Matti Lassas, Pekka Pankka, Samuli Siltanen
Abstract
A wavelet-based sparsity-promoting reconstruction method is studied in the context of tomography with severely limited projection data. Such imaging problems are ill-posed inverse problems, or very sensitive to measurement and modeling errors. The reconstruction method is based on minimizing a sum of a data discrepancy term based on an \$\ell\textasciicircum2\$-norm and another term containing an \$\ell\textasciicircum1\$-norm of a wavelet coefficient vector. Depending on the viewpoint, the method can be considered (i) as finding the Bayesian maximum a posteriori (MAP) estimate using a Besov-space \$B_\11\\textasciicircum\1\(\\mathbb T\\textasciicircum\2\)\$ prior, or (ii) as deterministic regularization with a Besov-norm penalty. The minimization is performed using a tailored primal-dual path following interior-point method, which is applicable to problems larger in scale than commercially available general-purpose optimization package algorithms. The choice of “regularization parameter” is done by a novel technique called the S-curve method, which can be used to incorporate a priori information on the sparsity of the unknown target to the reconstruction process. Numerical results are presented, focusing on uniformly sampled sparse-angle data. Both simulated and measured data are considered, and noise-robust and edge-preserving multiresolution reconstructions are achieved. In sparse-angle cases with simulated data the proposed method offers a significant improvement in reconstruction quality (measured in relative square norm error) over filtered back-projection (FBP) and Tikhonov regularization.