NOWNUNM: Nonlocal Weighted Nuclear Norm Minimization for Sparse-Sampling CT Reconstruction

Computed tomography (CT) image reconstruction using classical total variation (TV)-based methods or its variations inevitably suffers from a blocky effect when the sampling number is low, because of the piecewise assumption. A low-rank based method is an effective way to circumvent this side effect. Normally, a nuclear norm is used to impose the low rank constraint, and its numerical computation depends on the sum of singular values, which are calculated by singular value decomposition. However, as larger singular values mainly deliver the structural information, treating all the singular values equally may lead to imperfect preservation of edges and textures. To deal with this problem, we here propose to reconstruct the CT image by explicitly exploring the nonlocal similarity in the target image with nonlocal weighted nuclear norm minimization. First, a matrix is constructed by grouping nonlocal patches similar to the current patch. Then, the original nuclear norm minimization is replaced by a weighted version, which treats the singular values differently according to their magnitudes. By doing this, we can eliminate noise and streak artifacts without introducing any side effects. The corresponding numerical algorithm is given by an alternating optimization strategy. Experimental results demonstrate that our method outperforms several existing reconstruction methods in both qualitative and quantitative aspects, including filtered back projection, TV, and total generalized variation methods.