Anisotropic regularization for inverse problems with application to the wiener filter with gaussian and impulse noise

Most inverse problems require a regularization term on the data. The classic approach for the variational formulation is to use the L ₂ norm on the data gradient as a penalty term. This however acts as a low pass filter and thus is not good at preserving edges in the reconstructed data. In this paper we propose a novel approach whereby an anisotropic regularization is used to preserve object edges. This is achieved by calculating the data gradient over a Riemannian manifold instead of the standard Euclidean space using the Laplace-Beltrami approach. We also employ a modified fidelity term to handle impulse noise. This approach is applicable to both scalar and vector valued images. The result is demonstrate via the Wiener filter with several approaches for minimizing the functional including a novel GSVD based spectral approach applicable to functionals containing gradient based features.