Time-Frequency Analysis

First Lecture: Tuesday, 21.02.2012, HG F 26.5.

Synopsis

Many classes of signals and operators can be characterized by their time-frequency localization properties. A case in point is given by functions with point singularities and Calderon-Zygmund operators which are optimally adapted to the time-frequency localization pattern of wavelets. This fact lies at the very foundation of the success of wavelet methods for the solution of elliptic PDEs.

In the last decade several new developments of representation systems besides wavelets have been made with the ability to handle more general signal classes and operators. As an example we mention the curvelet transform which is optimally adapted to functions with curve singularities (think of images) and wave propagation operators. These constructions have lead to a paradigm shift in the fields of signal processing and operator compression, with new applications being developed as we speak.

This course will discuss these developments from a unified view based on partitions of the time-frequency space (phase space). I will lead to the current frontier of research in the area and also discuss various research problems and algorithmic aspects, if time permits.

The goals of this course are

-Familiarity with recent developments in Geometric Multiscale Analysis.

-Obtaining an intuition on which representation system should be used for the compression of a given class of signals or operators.

-Knowledge of the elements of nonlinear approximation theory.

-Knowledge of common algorithms in computational harmonic analysis

Literature

• Emmanuel Candes and Laurent Demanet, The Curvelet representation of WavePropagators is optimally sparse, Communications in Pure and Applied  Mathematics 58 (2004), 1472--1528.
• Emmanuel Candes and David Donoho, New tight Frames of Curvelets and Optimal Representations of Objects with Piecewise C^2 Singularities, Communications in Pure and Applied Mathematics 57 (2002), 219--266.
• David Donoho, Ingrid Daubechies, Ronald DeVore and Martin Vetterli, Data Compression and Harmonic Analysis, IEEE Transactions on Information Theory 44/6 (1998), 2435--2476
• Ronald DeVore, Nonlinear Approximation, Acta Numerica (1998), 51--150.
• Minh Do and Martin Vetterli, The Contourlet Transform: An Efficient Directional Multiresolution Image Representation, IEEE Transactions on Image Processing 14 (2005), 2091--2106.
• David Donoho, Sparse Components of Images and Optimal Atomic Decompositions, Constructive Approximation 17 (1999), 353--382.
• Emmanuel LePennec and Stephane Mallat, Sparse Geometric Image Representation with Bandelets, IEEE Transactions on Image Processing 14 (2005), 423--438.
• E. Cordero and F. Nicola, Sparsity of Gabor representations of Schrödinger propagators, Applied and Computational Harmonic Analysis 26 (2009), 357--370.