# 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...