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Signal II and optimization

Presentation

Part I : Wavelets

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

  • Elements for convex analysis. Notion of convex subdifferential, optimality necessary condition, Lagrangian duality
  • Algorithms. Descent methods (steepest descent and stabilization), subgradient methods, cutting planes methods and bundle methods.
  • Some examples (minimax problems, proximal algorithms, how to manage constraints)
  • 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.

Objectives

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

  •  Wavelet transform
  • Filter Banks with exact reconstruction
  • Properties of wavelets (localisation in space and frequency) and applications to the approximation of functions.

 

The student will be able to:

  •  Provide examples of wavelets
  • Carry out numerical approximation of images with wavelets.

Needed prerequisite

Signal 1 [I4MMMF71]

Optimization 1 & 2 [I4MMMO71] [I3MIMT11]

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...