After a classification of data this coarse presents fundamental concepts, like time and frequency representation, analysis tools (Fourier, wavelets, multiresolution, SVD, matching pursuit, …) and then the compression and denoising of data. Different applications (signals, images and vector fields) will illustrate the properties of the different tools.


Introduction and signal classification

Complex numbers and norms

Some elements of linear algebra and Hilbert analysis in finite dimensions

A primer on standard notations, linear applications (discrete Fourier transform in finite dimension)

Matrix calculus, diagonalization and SVD

Possible extensions to infinite dimensions (Hilbert bases)

Approximation, projection, decomposition, orthogonal and non orthogonal projection.

Application of the SVD

Bases and approximation by projection on subspaces generated by a sub-family of a basis. Example of the DCT.

Discrete wavelets (example Haar basis), matching pursuit.

Computational aspects (complexity, conditioning)