An Inexact Majorized Proximal Alternating Direction Method of Multipliers for Diffusion Tensors\ast

This paper focuses on studying the denoising problem for positive semidefinite fourth-order tensor field estimation from noisy observations. The positive semidefiniteness of the tensor is preserved by mapping the tensor to a 6-by-6 symmetric positive semidefinite matrix where its matrix rank is less than or equal to three. For denoising, we propose to use an anisotropic discrete total variation function over the tensor field as the regularization term. We propose an inexact majorized proximal alternating direction method of multipliers for such a nonconvex and nonsmooth optimization problem. We show that an \varepsilon -stationary solution of the resulting optimization problem can be found in no more than O(\varepsilon - 4) iterations. The effectiveness of the proposed model and algorithm is tested using multifiber diffusion weighted imaging data, and our numerical results demonstrate that our method outperforms existing methods in terms of denoising performance.