# Differential Calculus, Wavelets and Applications

## Public

M2 MSIAM (DS, MSCI)

6 ECTS

## Instructors

Sylvain Meignen and Kevin Polisano

## Description

The course is structured in two parts, treated respectively and independently by Sylvein Meignen and Kévin Polisano. The first part is devoted to differential calculus and its applications in image restoration and edge detection. The second part is dedicated to the construction and practical use of the wavelet transform. Wavelets are basis functions widely used in a large variety of fields: signal and image processing, data compression, smoothing/denoising data, numerical schemes for partial differential equations, scientific visualization, etc. Connections between the two parts will be made on the aspects of denoising, edge detection and graph analysis.

## Course outline

Part I: Differential Calculus

1. Differentiability on normed vector spaces
2. Image restoration
3. Edge detection

Part II: Wavelets and Applications

1. From Fourier to the 1D Continuous Wavelet Transform
2. Wavelet zoom, a local characterization of functions
3. The 2D Continuous Wavelet Transform
4. The 1D and 2D Discrete Wavelet Transform
5. Linear and nonlinear approximations in wavelet bases
6. The graph Fourier and wavelets transforms

## Assessments

### First session

• Part I – Differential calculus: a written exam (3h) [N1]
• Part II – Wavelets and application: one project and two lab sessions [N2]

### Second session

The student will have the choice of retaking only one or both parts:

• Part I: a new written exam (3h)
• Part II: continuation of the project after feedbacks from the teacher

Details for N1:

• The lab sessions are each one graded out of 2,5 points
• The project is graded out of 15 points
• Choice of the article (difficulty, length, …): 1 point
• Summary and outline (in line with the targets announced): 2 points
• Report redaction (including statement of the method, novelty of the paper, …): 3 points
• Codes (from scratch or existing librairies): 4 points
• Numerical results (replication results or extended): 3 points
• Interpretations of the results: 2 points