This course focuses on formulating signal and image processing problems as convex optimization problems, analyzing their properties (e.g. existence and uniqueness of solutions) and designing efficient algorithms to solve them numerically. We aim at giving students the background and skills to formulate problems and use appropriate algorithms on their own applications. Syllabus:?* Convex optimization: existence and uniqueness of solutions, subdifferential and gradient, constraints and indicator functions, monotone inclusions, nonexpansive operators and fixed point algorithms, duality, proximal operator, splitting algorithms.?* From estimation to optimization: formulating priors and constraints, regularity and parsimony, Bayesian interpretation.
Inverse problems: well- and ill-posed problems, data fidelity and regularization, study of signal and image recovery problems.
H. H. Bauschke and P. L. Combettes, « Convex Analysis and Monotone Operator Theory in Hilbert Spaces », 2011 N. Parikh and S. Boyd, « Proximal Algorithms », Foundations and Trends in Optimization Vol. 1, No. 3 (2013) 123–231 P. L. Combettes and J.-C. Pesquet, « Proximal splitting methods in Signal Processing », chapitre de « Fixed-point algorithms for inverse problems in science and engineering », p. 185-212, 2011.
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