We will first present the stationnary Eikonal partial differential equation as the local "Level Set" (or Eulerian) model for normal front propagation with prescribed speed. We then introduce reduced time dependent models. We will also discuss briefly the relationship with High frequency wave propagation (Ray tracing) and distance functions. Finally we will reframe everything in the more general first order Hamilton Jacobi equation setting.

On the advantages and disadvantages of the "level set" method and its numerical implications. We explain how Eulerian "level set" viscosity solutions can devellop singularities and how they relate to Multivalued Lagrangian solutions.

We first explain the need for "upwinding"  when discretizing a simple linear transport equation with Finite Differences. We show how to extend this idea to time dependent and stationnary Eikonal equations. The same ideas apply to first order HJ equation with convex Hamiltonian function. Time permitting we will discuss Higher order Finite difference methods and implied arising difficulties.

Front propagation in the normal direction driven by a given velocity c(x) with constant sign. The eikonal equation and the minimum time problem. The "classical" Fast Marching (FM) method based on finite differences. Convergence and complexity.

Lecture 2 (Slides)
Other fast methods for the eikonal equation (Group marching, Sweeping, FM-semilagrangian). Extensions of the FM method to more general first order Hamilton-Jacobi equations. The Narrow Band algorithm for the evolutive Hamilton-Jacobi equation when c(x) changes sign.

Lecture 3
A generalized FM method for c(x,t) changing sign.  Discontinuous representation of the front. Properties of the scheme. Convergence. Some applications.

This course will describe the links between total variation minimization problems and minimal perimeter problems, and introduce some algorithms, and in particular combinatorial optimization techniques that have been widely used in image processing for the past ten years.
We will first show how minimal and evolving (by mean curvature + other terms) surfaces can be practically computed with simple techniques, based on algorithms for total variation minimization.
Then, we will describe optimization techniques for solving discrete minimal perimeter problems, based on the max-flow/min-cut duality. We also will shortly review efficient algorithms, and discuss the main advantages and drawbacks of  these approaches.

A multitude of computer vision challenges can be cast as problems of energy minimization. In my presentation I will introduce shape optimization methods which allow to segment moving objects in image sequences, to detect obstacles in traffic videos, to reconstruct 3D shapes from a collection of 2D images and to track familiar shapes (for example walking people) in videos. I will detail how respective cost functionals can be minimized both in a spatially continuous (level set methods or TV-L1 minimization) and in a discrete (graph theoretic) formulation.