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Poster
Monday
    9h00-9h30 : Welcome 
  9h30-11h00 : Level sets method 1 (J.D. Benamou)
11h00-11h30 : coffee break
11h30-13h00 : Level sets method 2 (J.D. Benamou) 

13h00-14h30 : Lunch

14h30-16h00 : Graph cuts method 1 (A. Chambolle)
16h00-16h20 : coffee break
16h20-17h20 : Short communications
    16h20-16h40 : E. Cristiani : "A new Fast and efficient implementation of narrow band methods for Hamilton-Jacobi-Bellman equations" Slides
    16h40-17h00 : N. Komodakis : "Fast and accurate energy minimization for static or time-varying MRFs" Slides
    17h00-17h20 : G. Peyré : "Geodesic Computations for Image
    and Mesh Processing
    " Slides


Tuesday

  9h30-11h00 : Level sets method 3 (J.D. Benamou)
11h00-11h30 : coffee break
11h30-13h00 : Fast marching method 1 (M. Falcone) Slides

13h00-14h30 : Lunch

14h30-16h00 : Graph cuts method 2 (A. Chambolle)
16h00-16h20 : coffee break
16h20-17h20 : Short communications
    16h20-16h40 : E. Carlini : "A single-pass scheme for the Mean Curvature Motion of convex curve"  Slides
    16h40-17h00 : H. Talbot : "Surfaces minimales par flots maximaux continus" Slides
    17h00-17h20 : P.Cumsille : "A simple mathematical model of biofilm formation"     Slides


Wednesday
  9h30-11h00 : Fast marching method 2 (M. Falcone) Slides
11h00-11h30 : coffee break
11h30-13h00 : Fast marching method 3 (N. Forcadel)

13h00-14h00 : Lunch

14h00-15h30 : Graph cut method 3 (D. Cremers) Slides




Abstract:

Level Sets method
Lecture 1
Models  : Lagrangian versus Eulerian

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.

Lecture 2
Theory (minimal) :  Multivalued solutions versus viscosity solutions.

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.

Lecture 3
Numerical Methods :  The need for "upwinding".

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.


Fast marching method
Lecture 1 (Slides)
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.


Graph Cuts method
Lecture 1 and 2
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.

Lecture 3 (Slides)
Continuous versus Discrete Shape Optimization in Computer Vision 

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.