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
Abstract
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
Abstract
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
Abstract
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
Abstract
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).