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Informatique et Mathématiques appliquées  # Optimal Transport, levelset: applications to image - 5MM2531C

• #### Number of hours

• Lectures : 18.0
ECTS : 1.5

### Goals

An image may be considered as an application from a domain of R2 (or R3 in the case of three-dimensional images ) with values ??in a subset of R (or R3 for color images). Many algorithms of edge detection, pattern recognition, denoising, are based on partial differential equations formulations.
A first part of this course is to introduce an Eulerian representation of regions of interest of the image. We begin by giving the main principles of the Eulerian description of the interfaces and what we can extract from it geometrically . Then we will detail some active contour algorithms developed in the past years. We will explicit some numerical tools used to implement those algorithms.
Optimal transport is an old theme introduced in 1781 by Gaspard Monge to optimize the transport of construction materials. It has applications in economics and has been formulated in an original form by Brenier in the 2000s , in the form of an equation of conservation of mass acting as the constraint of kinetic energy .
The use of this tool in image analysis was proposed by Benamou - Brenier and then Haker et al. Their work has helped to develop a method to interpolate between two images (or more ) in a manner taking into account displacement of an image on the other. This determines intermediate frames between two images (morphing) or calculate distances between images.
This course will also consider conjoint use of these two topics, namely the study of optimal transport for more general energy in the context of image processing, obtained from level set representations .

Contact Emmanuel MAITRE

Content

1 Introduction
2 Mathematical Tools
2.1 Notations
2.2 Characteristic curves
2.3 Formulas Reynolds
2.4 Transport equation
2.6 Rearrangements of sets and functions
2.7 Rearrangements and Angenent Haker Tannenbaum model

3 Level Set in image analysis
3.1 Introduction
3.2 Level Set Representation
3.3 Sets and Operations level functions
3.4 General principle of a method of active contour ( snakes )
3.5 Calculation of volume and surface integrals using level-set representation.
3.6 Deformations of curves and Level Set
3.7 Application methods of active contours and generalizations
3.8 Numerical aspects of the level-set method

4 Introduction to optimal transport and application to image analysis
4.1 Problems and Monge Kantorovich
4.2 Minimizers functional gradients are convex
4.3 Alternative Formulation
4.4 Numerical Algorithm
4.4.1 Computation of an optimal path: Benamou - Brenier algorithm
4.4.2 Computation of the Monge problem: Angenent , Haker - Tanenbaum
4.4.3 Construction of a non-optimal application
4.4.4 Canceling the rotational
4.4.5 Other algorithms
4.5 Applications to Image
4.5.1 Elastic distance of Younes
4.5.2 Examples of application of the algorithm Benamou - Brenier
4.5.3 Registration / deformation algorithm of Haker - Tannenbaum - Angenent
4.5.4 Other Applications

5 To go further ...

Prerequisites

Classical tools from optimization and applied analysis. Those will be recalled if necessary.

Tests

Semester 9 - The exam is given in english only Reading of a scientific article and short presentation talk with slides, plus a short written note (2 pages) on that article.

N1=P (résumé article + soutenance)
N2=P

Semester 9 - This course is given in english only Curriculum->Math. Modelling, Image & Simulation->Semester 9

Bibliography

C. Villani, Topics in optimal transportation, Graduate Studies in Mathematics, Vol. 50, AMS (2003)
J. Sethian : Level Set & Fast Marching Methods, Cambridge
S. Osher & R. Fedkiw : Level Set Methods and Dynamics Implicit Surfaces, Springer
S. Osher & N. Paragios (2003) : Geometric Level Set in Imaging Vision and Graphics, Springer
S. Agenent, S. Haker and A. Tannenbaum, Minimizing flows for the Monge-Kantorovich pro- blem, SIAM J. Math. Analysis, 35 :61–97 (2003)
J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution of the Monge- Katorovich mass transfer problem, Numer. Math. 84 :375–393 (2000)
L.C. Evans, Partial Differential Equations and Monge-Kantorovich Mass Transfer, Notes de
cours sur http://math.berkeley.edu/~evans/
M. Fortin and R. Glowinski, Augmented Lagrangian Methods : Applications to the Solution of Boundary Value Problems, Studies in Mathematics and its Applications 15, North-Holland, Amsterdam, 1983.
V. Caselles, R. Kimmel and G. Sapiro , "Geodesic Active Contours ", International Journal of Computer Vision (1997)
T. Chan and L. Vese , "Active Contours without Edges", IEEE Transactions on Image Processing (2001)
G. Aubert et. al., "Image Segmentation Using Active Contours: Calculus Of Variations Or Shape Gradients", SIAM (2003)