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.

介绍了一种新的非光滑变分模型，用于恢复具有二阶差分项的流形值数据的噪声，其中需要数值分析，凸优化和微分几何技术的结合，证明了循环近端点算法的收敛性并实现了在实践中的效果。