Restoration of Matrix Fields by SOCP

Wherever anisotropic behaviour in physical measurements or models is encountered matrices provide adequate means to describe this anisotropy. Prominent examples are the di usion tensor magnetic resonance imaging in medical imaging or the stress tensor in civil engineering. As most measured data these matrix-valued data are also polluted by noise and require restoration. The restoration of scalar images corrupted by noise via minimization of an energy functional is a well-established technique that o ers many advantages. A convenient way to achieve this minimization is second order cone programming (SOCP). The goal of this article is to transfer this method to the matrix-valued setting. It is shown how SOCP can be extended to minimize energy functionals de ned for matrix elds. Furthermore, new functionals for the regularization of matrix data are proposed and the corresponding Euler-Lagrange equations derived by means of matrix di erential calculus. Numerous experiments substantiate the usefulness of the proposed methods for the restoration of matrix elds.