Multiscale decomposition of manifold-valued data.

摘要

Science and engineering have long experienced a 'Data deluge'. But now, inundation by data is taking a new turn. While analysis of ever-increasing volume of data still remains a challenge, much richer challenges are posed by novel data types of highly geometric nature arising in all branches of science and engineering. In this thesis, we focus on one particular new type of data, 'Riemannian Symmetric Space-Valued Data'. More specifically, we are interested in data which are indexed by an equispaced cartesian grid in time or space and which take values in a smooth Riemannian manifold. We call such data 'Manifold-Valued Data'. Inspired by the success of wavelets, we develop a multiscale, 'wavelet-like' decomposition of M-Valued data. This is achieved by deploying linear interpolating refinement schemes in the tangent space of these manifolds. Our decomposition has many properties in common with traditional wavelet transform for real-valued data. Wavelet coefficients of our transform can be manipulated much like the traditional wavelet transform e.g. thresholding, quantization, scaling, etc. Many data analysis tasks, for which wavelet decomposition is used in traditional settings like denoising, compression, feature extraction, and pattern analysis, can easily be realized for M-Valued data. We present many examples of M-Valued data and processes facilitated by our decomposition. A Matlab toolbox, Symmlab, is available online that can reproduce all the results in this thesis.