A Second Order Non-Smooth Variational Model for Restoring Manifold-Valued Images

We introduce a new non-smooth variational model for the restoration of manifold-valued data which includes second order differences in the regularization term. While such models were successfully applied for real-valued images, we introduce the second order difference and the corresponding variational models for manifold data, which up to now only existed for cyclic data. The approach requires a combination of techniques from numerical analysis, convex optimization and differential geometry. First, we establish a suitable definition of absolute second order differences for signals and images with values in a manifold. Employing this definition, we introduce a variational denoising model based on first and second order differences in the manifold setup. In order to minimize the corresponding functional, we develop an algorithm using an inexact cyclic proximal point algorithm. We propose an efficient strategy for the computation of the corresponding proximal mappings in symmetric spaces utilizing the machinery of Jacobi fields. For the n-sphere and the manifold of symmetric positive definite matrices, we demonstrate the performance of our algorithm in practice. We prove the convergence of the proposed exact and inexact variant of the cyclic proximal point algorithm in Hadamard spaces. These results which are of interest on its own include, e.g., the manifold of symmetric positive definite matrices

Smoothing algorithms for nonsmooth and nonconvex minimization over the stiefel manifold

We consider a class of nonsmooth and nonconvex optimization problems over the Stiefel manifold where the objective function is the summation of a nonconvex smooth function and a nonsmooth Lipschitz continuous convex function composed with an linear mapping. We propose three numerical algorithms for solving this problem, by combining smoothing methods and some existing algorithms for smooth optimization over the Stiefel manifold. In particular, we approximate the aforementioned nonsmooth convex function by its Moreau envelope in our smoothing methods, and prove that the Moreau envelope has many favorable properties. Thanks to this and the scheme for updating the smoothing parameter, we show that any accumulation point of the solution sequence generated by the proposed algorithms is a stationary point of the original optimization problem. Numerical experiments on building graph Fourier basis are conducted to demonstrate the efficiency of the proposed algorithms.Comment: 22 page

Inpainting of Cyclic Data using First and Second Order Differences

Cyclic data arise in various image and signal processing applications such as interferometric synthetic aperture radar, electroencephalogram data analysis, and color image restoration in HSV or LCh spaces. In this paper we introduce a variational inpainting model for cyclic data which utilizes our definition of absolute cyclic second order differences. Based on analytical expressions for the proximal mappings of these differences we propose a cyclic proximal point algorithm (CPPA) for minimizing the corresponding functional. We choose appropriate cycles to implement this algorithm in an efficient way. We further introduce a simple strategy to initialize the unknown inpainting region. Numerical results both for synthetic and real-world data demonstrate the performance of our algorithm.Comment: accepted Converence Paper at EMMCVPR'1