Mathematical methods for signal and image treatment

Signal processing is the discipline of developing and studying methods for analyzing, interpreting and transforming signals. A signal can be defined as any information carrier (such as a sequence of numbers, an electric current, a DNA sequence, an image or a video sequence...). Signal processing calls on many branches of applied mathematics (notably analysis, approximation theory, probability and statistics, information theory...) and now even pure mathematics (geometry, number theory...). Signals essentially take two forms: analog signals, which are the result of a physical (or other) measurement process, or obtained by "digital conversion $\mapsto$ analog", and digital signals stored on a computer or any other digital medium, or produced by "analog conversion $\mapsto$ digital". This last operation, one of the most fundamental in signal processing, is also known as sampling.

Signal processing covers a wide range of issues, from exploratory signal analysis to more complex tasks such as denoising and restoring degraded signals, encoding and compressing signals, images and video, estimating models and parameters, detecting specific events in signals and images... Moreover, the application framework in which these problems are posed often imposes severe constraints (causality, computational load, signal format...) which require processing to be adapted.

This dossier describes a wide range of signal and image processing methods and algorithms, focusing on mathematical foundations and algorithms. The first part focuses on the first essential point, namely the problem of signal representation. In this context, Fourier analysis and mathematical analysis in general play a central role. One of the essential tools of signal processing, convolution filtering, is also discussed, along with the problem of sampling. Since signals can be described as either deterministic or random objects, a number of probabilistic models are also discussed in detail, and concepts covered in the deterministic framework are revisited in the context of random signals.

The second part of this dossier is devoted to a number of specific signal analysis and processing problems, using the mathematical tools described in the first part. More specifically, the problems of analysis and estimation, coding and compression, and denoising are addressed. The final section is devoted to a brief discussion of very recent developments, based on a new paradigm, the notion of parsimony. Some more mathematical or technical aspects are...