Summary
Wherever anisotropic behavior in physical measurements or models is encountered matrices provide adequate means to describe this anisotropy. Prominent examples are the diffusion 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 offers 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 applied to minimize various energy functionals defined for matrix fields. These functionals couple the different matrix channels taking into account the relations between them. Furthermore, new functionals for the regularization of matrix data are proposed and the corresponding Euler–Lagrange equations are derived by means of matrix differential calculus. Numerical experiments substantiate the usefulness of the proposed methods for the restoration of matrix fields.
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Steidl, G., Setzer, S., Popilka, B. et al. Restoration of matrix fields by second-order cone programming. Computing 81, 161–178 (2007). https://doi.org/10.1007/s00607-007-0247-x
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DOI: https://doi.org/10.1007/s00607-007-0247-x