Optimization on Manifolds for Models using Second Order Differences

In many real life scenarios, measured data appears as values on a Riemannian manifold. For example in interferometric Synthetic Aperture Radar (InSAR) data is given as a phase, in electron backscattered diffraction (EBSD) as data items being from a quotient of the orientation group SO(3), and in diffusion tensor magnetic resonance imaging (DT-MRI) the measured data are symmetric positive definite matrices. These data items are often measured on an equispaced grid like usual signals and image but they also suffer from the same measurement errors like resence of noise or incompleteness. Hence there is a need to perform data processing tasks like denoising, inpainting or interpolation on these manifold-valued data.
In this talk we present variational models for these tasks involving discrete second order differences for manifold-valued data. To compute a minimizer of such a the model, we obtain a high-dimensional, possibly non-smooth, optimization problem defined on a Riemannian manifold. We present algorithms to efficiently solve these problems and illustrate their performance.