Formation M2 mathematics for data sciences
Semester1
BlocSafety and protection of data
Teaching staffLecturer : Julien Barral, TA :
Credits 4 ECTS
Teaching hours 21h of lectures + 21h of TA sessions
Validation schemeContinuous examination+final exam

Presentation

This course is intended to be an introduction to Fourier analysis as a tool for processing information contained in signals such as images , sound signals , gravitational waves , or DNA sequences . This tool brings into play the concepts of Fourier transform, of reconstruction and approximation of a signal , which contributed in an essential way to the revolution that constituted the discovery of wavelets, the development of their theory, and their implementation in applications. The points that can be covered in this course are:

  • Fourier and solving the heat equation. Elementary and less elementary properties.
    • integrable functions and their Fourier series. The Gibbs phenomenon. Direct applications of the Fourier series in geometry and dynamics.
  • The Fourier transform of functions and measures. Poisson summation formula.
  • Shannon's Sampling Theorem. Fast Fourier transform. Heisenberg's principle of uncertainty.
  • From the Fourier transform with sliding window to the wavelet transform continuous.
  • The transform into discrete wavelets. Multi-resolution analyzes. Construction and properties of wavelets. Wavelets with compact support.
  • Applications: edge detection, denoising, compression . Detection of gravitational waves.