Article

Restoration of matrix fields by second-order cone programming.

Computing (impact factor: 0.7). 01/2007; 81:161-178. DOI:10.1007/s00607-007-0247-x
Source: DBLP

ABSTRACT 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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    ABSTRACT: The restoration of scalar-valued images via minimization of an energy functional is a well-established technique in image processing. Recently, higher-order methods have proved their advantages in edge preserving image denoising. In this chapter, we transfer successful techniques like the minimization of the Rudin-Osher-Fatemi functional and the infimal convolution to matrix fields, where our functionals couple with different matrix channels. For the numerical computation, we use second-order cone programming. Moreover, taking the operator structure of matrices into account, we consider a new operator-based regularization term. This is the first variational approach for denoising tensor-valued data that takes the operator structure of matrices, in particular the operation of matrix multiplication into account. Using matrix differential calculus, we deduce the corresponding Euler-Lagrange equation and apply it for the numerical solution by a steepest descent method.
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