| Course label : |
Representation of signals and inverse problems |
| Teaching departement : |
MIN / Applied Mathematics and General Computing |
| Teaching manager : |
Mister PIERRE-ANTOINE THOUVENIN / Mister PIERRE CHAINAIS |
| Education language : |
French |
| Potential ects : |
4 |
| Results grid : |
|
| Code and label (hp) : |
G1G2_ED_MIN_RSP - Repr. signaux et pblm inverses |
Education team
Teachers : Mister PIERRE-ANTOINE THOUVENIN / Mister PIERRE CHAINAIS / Mister Pierre PALUD
External contributors (business, research, secondary education): various temporary teachers
Summary
This course is an introduction to inverse problems arising in signal and image processing, building on the notion of signal representation. The content of the course covers essential mathematical concepts in this field (time-frequency transform, time-scale transform, multi-resolution analysis and convex optimization), as well as the associated algorithms allowing a numerical application. Applications to sound, images, videos or more abstract data (graphs) are considered, ranging from image denoising and image deconvolution to musical source separation. This course serves as an introduction to a wide and active multi-disciplinary research field at the interface of machine learning and signal processing. These topics have endured a lasting interest over the past 15 years, following the introduction of the fundamental role of sparsity in signal processing.
The first part of the course is devoted to signal representation, and the considerations underlying the choice of a domain transform for data analysis (compression, restoration, parameter estimation, feature extraction). In particular, time-frequency and scale frequency analysis are introduced to highlight characteristic features of different signals.
These two representations naturally find applications in the second part of the course devoted to inverse problems. An inverse problem consists in estimating a collection of characteristic parameters from corrupted, incomplete observations of a phenomenon based on a predefined model. The notion of an ill-posed problem is first introduced, leading to the concept of regularization, considered either as a penalty within an optimization problem or resulting from a statistical bayesian interpretation of the model. On the one hand, the model is aimed at reducing the number of mathematically acceptable solutions using physical considerations specific to the data to be analyzed. This is typically the case when considering indirect observations, e.g., in foetal echography or astronomical imaging. The problem consists in reconstructing/estimating the underlying signal as precisely as possible. On the other hand, the model needs to be sufficiently simple to be numerically addressed by efficient inference algorithms. Several modelling aspects are covered (bayesian approach, sparsity prior, ...), with two fundamental descent algorithms (gradient descent, forward-backward algorithm). Illustrative applications to signal denoising and restoration are considered.
The concepts introduced in the lectures are applied during Python programming sessions based on Jupyter notebooks. These sessions illustrate the different stages necessary to address an inverse problem, from the most theoretical aspects to their numerical application.
Keywords: signal representation, time-frequency/time-scale analysis, multi-
resolution analysis, wavelets, inverse problem, Bayesian inference, convex
optimization.
Content of the lectures:
Part 1: Signal representations
- Chapter 1: Signal and image representations
- Keywords: representation, Fourier transform (Parseval, Plancherel), discrete Fourier transform, Gabor-Heisenberg theorem, Hilbert space orthonormal basis;
- Chapter 2: Time-frequency analysis
- Keywords: Short-time Fourier transform (continuous and discrete), time-frequency atoms, inversion and energy conservation theorems;
- Chapter 3: Continuous wavelet transform
- Keywords: real wavelets, continuous wavelet transform, scaling function, reconstruction theorem, scalogram;
- Chapter 4: Orthogonal wavelet basis
- Keywords: Multi-resolution analysis, dyadic wavelet transform, orthogonal wavelet basis, scaling equation, conjugate mirror filters, Mallat's ￜ trou algorithm, 2D wavelets
Part 2: Introduction to inverse problems
- Chapter 1: Introduction
- Keywords: Ill-posed problem (Hadamard), least-squares, pseudo-inverse (Penrose-Moore), SVD, matrix conditioning number, regularizarion (Thikhonov, sparsity, TV);
- Chapter 2: Statistical interpretation
- Keywords: noise (Gaussian, Poisson, Laplace), maximum likelihood estimator, bayesian regularization, Bayes' theorem, maximum a posteriori estimator;
- Chapter 3: Notions of convex optimization
- Keywords: local/global minimizer, convex set/function, lower semi-continuity, optimization problem, existence and unicity of solution(s), sub-differential, 1st order optimality criterion, proximal operator, gradient descent, proximal gradient descent (forward-backward algorithm).
Educational goals
At the end of the course, students will be able to
- Understand and solve an elementary inverse problem;
- Explain and understand the considerations at stake related to the notions of: signal representation; time-frequency analysis; time-scale analysis; wavelet transform (continuous, orthogonal); ill-posed inverse problem; maximum likelihood / a posteriori estimator; sparsity; convex optimization; proximal operator; gradient descent; proximal gradient descent.
Contribution of the course to the reference framework of competences. By the end of the course, the students will have made progress in the ability to:
- C2 (Represent and model): Exploit a time-frequency or a wavelet representations to extract meaningful information from a signal. Model a multi-dimensional linear inverse in a statistical (Bayesian) framework, formulate the problem as an optimization problem.
- C2 (Solve and decide): Apply an elementary algorithm to obtain a time-frequency / time-scale representation, interpret the information revealed by these transforms. Apply an elementary algorithm to solve basic linear inverse problems, identify and justify the limits of the approach.
Sustainable development goals
Knowledge control procedures
Continuous Assessment
Comments: Continuous assessment / final examination:
- 2 exams (1h for each part of the course, middle and end)
- practical session reports (4 reports in total).
Online resources
Lecture slides, Python notebooks for the practical sessions, 2 exercise sheets, additional articles made available on Moodle (discrete Fourier transform, article on source separation).
Reference book: Mallat, S. G. (2009) A wavelet tour of signal processing: the sparse way. 3rd ed. Amsterdam?; Boston: Elsevier/Academic Press.
Pedagogy
Lecture, practical sessions conducted in pairs, tutorial sessions.
Sequencing / learning methods
| Number of hours - Lectures : |
22 |
| Number of hours - Tutorial : |
12 |
| Number of hours - Practical work : |
0 |
| Number of hours - Seminar : |
12 |
| Number of hours - Half-group seminar : |
0 |
| Number of student hours in TEA (Autonomous learning) : |
23 |
| Number of student hours in TNE (Non-supervised activities) : |
0 |
| Number of hours in CB (Fixed exams) : |
0 |
| Number of student hours in PER (Personal work) : |
0 |
| Number of hours - Projects : |
0 |
Prerequisites
- Lecture: good understanding of the courses on Probability & Statistics, Signal processing, notions of optimization and functional analysis, interest in physical applications.
- Practicals: basic Python programming skills.
Maximum number of registrants
64
Remarks
Remark: course primarily oriented towards G2 students.