A Nonlocal Denoising Algorithm for Manifold-Valued Images Using Second Order Statistics

Nonlocal patch-based methods, in particular the Bayesian approach of Lebrun, Buades, and Morel [SIAM J. Imaging Sci., 6 (2013), pp. 1665-1688], are considered to be state-of-the-art methods for denoising (color) images corrupted by white Gaussian noise of moderate variance. This paper is the first attempt to generalize this technique to manifold-valued images. Such images, for example, images with phase or directional entries or with values in the manifold of symmetric positive definite matrices, are frequently encountered in real-world applications. Generalizing the normal law to manifolds is not canonical, and different attempts have been considered. Here, we focus on a straightforward intrinsic model and discuss the relation to other approaches for specific manifolds. We reinterpret the Bayesian approach of Lebrun, Buades, and Morel [SIAM J. Imaging Sci., 6 (2013), pp. 1665-1688] in terms of minimum mean squared error estimation, which motivates our definition of a corresponding estimator on the manifold. With this estimator at hand we present a nonlocal patch-based method for the restoration of manifold-valued images. Various proof -of-concept examples demonstrate the potential of the proposed algorithm.