Optimization Methods for Manifold-Valued Image Processing

Recent image acquisition techniques, e. g., diffusion tensor magnetic resonance tomography and electron backscatter diffraction, allow to obtain images having pixel values in manifolds. However, the acquisition process often leads to corrupted images. In this thesis, we are facing similar tasks as in real-valued image processing, namely image restoration, i. e., noise removal and inpainting, segmentation, and interpolation between images. In the beginning we generalize a state-of-the-art method for color image denoising, namely the nonlocal patch-based method of Lebrun et al., which is equivalent to a minimum mean squared error estimator when considering a Gaussian model. We generalize the minimum mean squared error estimation to the manifold-valued setting. Next, we consider variational models for denoising and inpainting. For the manifold total variation model we develop two minimization algorithms. First, half-quadratic minimization is used to solve a relaxed problem by alternating minimization. Second, we generalize the parallel Douglas-Rachford algorithm to functions on symmetric Hadamard manifolds and show its convergence for certain functions. Similarly to the classical total variation, its counterpart for manifold-valued images suffers from stair-casing. To avoid this effect we introduce different priors based on the infimal convolution of first and second order differences, and the total generalized variation. The next topic of this thesis is the labeling of manifold-valued images. Based on an idea of Aström et al. we propose a multiplicative filtering method on the open probability simplex. We prove convergence of the method and give an interpretation as gradient ascent reprojection algorithm on positive numbers. In the last part of this thesis we investigate the morphing of manifold-valued images, i. e., smooth interpolation between a template and a target image. We generalize the time discrete geodesic path model of Berkels et al. to nonlinear Lebesgue spaces on Hadamard manifolds. After proving the existence of a minimizer of the variational problem, an alternating minimization scheme is proposed. Finally, we apply our morphing for the colorization of face images in the YUV color space. While state-of-the-art methods for exemplar-based colorization employ texture matching, which does not yield reasonable results for face images, our method focuses on the geometry and leads to convincing results.