Adaptive Approximation Methods for Signal Data Processing
This application-oriented tutorial introduces adaptive approximation methods for signal data processing. Special emphasis is placed on
(a) the analysis of high-dimensional signals by manifold learning;
(b) geometrical methods for adaptive approximation of image and video data.

Lecture I: Monday 6, 15:00-16:30
Analysis of High-Dimensional Signal Data: Objectives, Problems, and Basic Tools
This first lecture unit discusses basic objectives and problems concerning the analysis of high-dimensional signal data, where motivating application examples from neuro and bioscience are given for illustration. We then explain relevant mathematical tools from signal data analysis, such as Fourier and wavelet transforms. This is followed by a more comprehensive discussion on linear and nonlinear projection methods for dimensionality reduction.

Lecture II: Tuesday 7, 9:30-11:00
Analysis of High-Dimensional Signal Data: Manifold Learning and Curvature Analysis
Further to our discussion of the previous lecture unit, we now focus on the construction and analysis of dimensionality reduction methods by using recent concepts from manifold learning. The purpose of our construction is to obtain suitable low-dimensional parameterizations of high-dimensional signal data. This requires analyzing the geometrical distortion of manifolds, as incurred by their corresponding (nonlinear) embedding maps. To this end, a more detailed discussion concerning the curvature analysis of manifolds is provided, where the computation of their related metric and curvature tensors is explained. Finally, numerical examples concerning low-dimensional parameterizations of scale- and frequency-modulated manifolds are presented for illustration.

Lecture III: Wednesday 8, 9:30-11:00
Adaptive Thinning Algorithms
Adaptive thinning algorithms are geometrical approximation methods for computing sparse representations of digital image and video data. The nonlinear approximation scheme of adaptive thinning works with a greedy removal of (image or video) pixels to extract geometrical features from the given signal data. Moreover, adaptive thinning relies on least squares approximation using linear splines over anisotropic Delaunay triangulations (tetrahedralizations). Adaptive thinning algorithms are fast, stable and accurate. In this lecture unit, we first introduce the basic ingredients of adaptive thinning algorithms, before we then analyze their computational complexity and the asymptotic behaviour of related N-term approximations.

Lecture IV: Thursday 9, 9:30-11:00
Nonlinear Approximation and Contextual Compression of Image and Video Data
Further to our discussion of the previous lecture unit, we now use adaptive thinning algorithms in combination with customized contextual encoding schemes to obtain a complete compression method for image and video data. Further details concerning the contruction of the proposed compression method are explained and, moreover, alternative wavelet-based schemes are discussed. In order to demonstrate the good performance of the proposed contextual compression method, we finally provide several numerical simulations and comparisons with the relevant competitors JPEG2000 (for images) and MPEG4 (for videos).