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Signal II

Presentation

Programme (detailed contents):

 

Part I : Wavelets

1. Wavelet transform. Basis of wavelets: definition and properties. Multiresolution analysis.

2. Discrete wavelets: Haar basis, filter banks with exact reconstruction, 1d bi-orthogonal wavelets, wavelet decomposition, spatial and frequency localisation, 2d wavelets. Introduction to jpeg, denoising.

3. Discrete wavelet packets. Definition, localisation in space and time. Denoising, deconvolution.

 

Organisation: Lectures to introduce the concepts and Labwork for practical numerical implementation.

 

Part II – Algorithms for unconstrained nonsmooth nonsmooth optimization.

  1. Elements for convex analysis. Notion of convex subdifferential, optimality necessary condition, Lagrangian duality
  2. Algorithms. Descent methods (steepest descent and stabilization), subgradient methods, cutting planes methods and bundle methods.
  3. Some examples (minimax problems, proximal algorithms, how to manage constraints)
  4. Perspectives. Why BFGS is a good method for nonsmooth optimization ? Generalization to the nonconvex case.

 

Organisation : Lecture, Tutorials and Labwork. Some tutorials will be preliminaries to the Labworks.

 

 

Main difficulties for students:

Objectives

 

At the end of this module, the student will have understood and be able to explain (main concepts):

 

1)    Wavelet transform

2)    Filter Banks with exact reconstruction

3)    Properties of wavelets (localisation in space and frequency) and applications to the approximation of functions.

 

The student will be able to:

 

1)    Provide examples of wavelets

2)    Carry out numerical approximation of images with wavelets.

Needed prerequisite

Signal 1

Optimization 1 & 2

Form of assessment

The evaluation of outcome prior learning is made as a continuous training during the semester. According ot the teaching, the assessment will be different: as a written exam, an oral exam, a record, a written report, peers review...