227-0434-00L  Harmonic Analysis: Theory and Applications in Advanced Signal Processing

SemesterSpring Semester 2015
LecturersH. Bölcskei
Periodicitytwo-yearly recurring course
Language of instructionEnglish


AbstractThis course is an introduction to the field of applied harmonic analysis with emphasis on applications in signal processing such as transform coding, inverse problems, imaging, signal recovery, and inpainting. We will consider theoretical, applied, and algorithmic aspects.
ObjectiveThis course is an introduction to the field of applied harmonic analysis with emphasis on applications in signal processing such as transform coding, inverse problems, imaging, signal recovery, and inpainting. We will consider theoretical, applied, and algorithmic aspects.
ContentFrame theory: Frames in finite-dimensional spaces, frames for Hilbert spaces, sampling theorems as frame expansions

Spectrum-blind sampling: Sampling of multi-band signals with known support set, density results by Beurling and Landau, unknown support sets, multi-coset sampling, the modulated wideband converter, reconstruction algorithms

Sparse signals and compressed sensing: Uncertainty principles, recovery of sparse signals with unknown support set, recovery of sparsely corrupted signals, orthogonal matching pursuit, basis pursuit, the multiple measurement vector problem

High-dimensional data and dimension reduction: Random projections, the Johnson-Lindenstrauss Lemma, the Restricted Isometry Property, concentration inequalities, covering numbers, Kashin widths
Lecture notesLecture notes, problem sets with documented solutions.
LiteratureS. Mallat, ''A wavelet tour of signal processing: The sparse way'', 3rd ed., Elsevier, 2009

I. Daubechies, ''Ten lectures on wavelets'', SIAM, 1992

O. Christensen, ''An introduction to frames and Riesz bases'', Birkhäuser, 2003

K. Gröchenig, ''Foundations of time-frequency analysis'', Springer, 2001

M. Elad, ''Sparse and redundant representations -- From theory to applications in signal and image processing'', Springer, 2010
Prerequisites / NoticeThe course is heavy on linear algebra, operator theory, and functional analysis. A solid background in these areas is beneficial. We will, however, try to bring everybody on the same page in terms of the mathematical background required, mostly through reviews of the mathematical basics in the discussion sessions. Moreover, the lecture notes contain detailed material on the advanced mathematical concepts used in the course. If you are unsure about the prerequisites, please contact C. Aubel or H. Bölcskei.