1. Some applications of 'a contrario' methods in computer vision
The aim of this course is to give a large introduction to a general detection methodology used in computer vision and called the 'a contrario' detection theory. This theory relies on a basic principle of visual perception, called the Helmholtz principle, which states that perception are large deviations from a uniform stochastic distribution. It can be surprising that such a general statement can lead to effective computations in order to detect structures in digital images. After a basic introduction to visual perception, we will tackle several applications of this approach: low-level applications, such as the detection of alignments in digital images, or more sophisticated ones, such as object recognition.
2. The Lax-Oleinik syndrome
The Lax-Oleinik formula (or, more precisely, formulae) is a kind of explicit representation of solutions to a class of nonlinear equations comprising hyperbolic conservation laws, the inviscid Burgers equation, the Hamilton-Jacobi equation, and some other models of mathematical physics. Roughly speaking, a Lax-Oleinik formula reflects a number of structural features of the equation or a system of equations, such as the existence of a variational principle, locality and irreversibility of the evolution, min-plus linearity, Hamiltonian structure, and convexity. These features form a "syndrome" exploited in a number of recent and actively developed theories. In this course we will discuss main problems and results of these theories, aged between 10-20 and 1-2 years. The prospective audience are graduate students specializing in mathematics and theoretical physics.