Jean-Marie Mirebeau

 


Research


I develop numerical schemes for Partial Differential Equations (PDEs), focusing on issues related to Anisotropy. I also investigate their applications, in particular to medical image processing.


Keywords : Anisotropic Partial Differential Equations,

Anisotropic Diffusion, Anisotropic Eikonal equations, Monge-Ampere equations, Convexity Constraint.


Curriculum Vitae, List of publications, Seminars and conferences, Google scholar page

Habilitation a diriger les recherches (Post graduate level dissertation), 2018


Open source codes and reproducible research (Github page)


Main developer of the Hamiltonian-Fast-Marching C++ library, a numerical solver of a variety of generalized eikonal equations, based on the author's research and used by several teams in the context of image processing and motion planning. It has interfaces for Python, Matlab, Mathematica


Developer of the Adaptive Grid Discretizations book of Python notebooks (view online), intended as material for summer schools, and  for making my research reproducible. Topics include anisotropic fast marching methods, Hamilton-Jacobi-Bellman PDEs, and adequate algorithmic tools.


Anisotropic Eikonal Equations


The eikonal equation is a Partial Differential Equation (PDE) which characterizes the geodesic distance on a (Finsler or Riemannian) manifold. It has numerous applications, including medical image processing in collaboration with L. Cohen. I have developed solvers for this PDE which are particularly fast and accurate in the context of large anisotropies. Slides.


These methods have been used to find the optimal solutions to control problems, including the Reed-Shepp car and the Euler/Mumford elasticae, which be approximated by shortest path problems with respect to strongly anisotropic metrics. The approach allows for data-driven terms in the PDE, and is mostly applied medical image segmentation. Slides


Journal papers:

-M., J. Portegies, Hamiltonian fast marching: A numerical solver for anisotropic and non-holonomic eikonal PDEs, Image Processing on Line, 9, 47–93, 2019.

- R. Duits, S. Meesters, M, J. Portegies, Optimal paths for variants of the 2D and 3D Reeds-Shepp Car with Applications in Image Analysis, JMIV, 2018

-M., Riemannian fast-marching on cartesian grids using Voronoi's first reduction of quadratic forms, SINUM, 2019

-M., Fast Marching Methods for Curvature Penalized Shortest Paths, JMIV, 2017

-M., Efficient Fast Marching with Finsler Metrics, Numerische Math, 2014

-M., Anisotropic Fast-Marching on cartesian grids using Lattice Basis Reduction, SINUM, 2014


Conference proceedings:

- M, J. Dreo, Automatic differentiation of non-holonomic fast marching for computing most threatening trajectories under sensors surveillance, GSI 2017

- D. Chen, M, L. Cohen, Finsler Geodesic Evolution Model for Region based Active Contours, BMVC 2016

- D. Chen, L. Cohen, M., A New Finsler Minimal Path Model with Curvature Penalization for Image Segmentation and Closed Contour Detection, CVPR 2016

  1. -D. Chen, L. Cohen, M., Global Minimum for Curvature Penalized Minimal Path Method, BMVC 2015

- G. Sanguinetti, E. Bekkers, R. Duits, M. Janssen, A. Mashtakov, M., Sub-Riemannian Fast Marching in SE2, Preprint, 2015

- D. Chen, L. Cohen, M.,Vessel Extraction using Anisotropic Minimal Paths and Path Score, ICIP 2014


Monge-Ampere Equations and the Constraint of Convexity


The Monge-Ampere equation is a fully non-linear PDE, which requires specific discretizations. I study two classes of numerical schemes, based on the general method of monotone wide stencil schemes, and the more specific approach of semi-discrete optimal transport based on weighted Voronoi diagrams, implemented in CGAL.


Slides on semi discrete optimal transport, and volume preserving maps (by Q. Merigot)

Slides on the constraint of convexity.


PDE discretizations based on wide stencil schemes:

  1. -M., Adaptive, Anisotropic and Hierarchical cones of Discrete Convex Functions, Numerische Math, 2015

  2. -Benamou, Collino, M., Monotone and Consistent Discretization of the Monge Ampere Operator, Math of Comp, 2015

  3. -M., Discretization of the 3D Monge-Ampere Operator, between Wide Stencils and Power Diagrams, M2AN, 2015


Applications of semi-discrete optimal transport:

-M., Numerical resolution of Euler equations, through semi-discrete optimal transport, Proceedings des journées EDP de Roscoff 2015

-Merigot, M., Minimal geodesics along volume-preserving maps, through semi-discrete optimal transport, Preprint



Anisotropic Diffusion


Nonlinear Anisotropic Diffusion is involved in state of the art image processing methods, as well as in physical phenomena. We developed a discretization for this PDE, on grids, which combines low numerical cost and strong mathematical guarantees.


Scheme description and analysis:

-Fehrenbach, M., Sparse Non-Negative Stencils for Anisotropic Diffusion, JMIV, 2014

-M., Minimal Stencils for Monotony or Causality Preserving Discretizations of Anisotropic PDEs, Preprint


Open source code available as an Insight Toolkit module :

Anisotropic diffusion in ITK, 2015 (module description), Interactive Figures



Adaptive and Anisotropic Finite Element Approximation 


Video réalisée par EADS sur mes travaux 


  Mesh adaption procedures for finite elements approximation allow to adapt locally the resolution, by local refinement around the places of strong variation of the function. The use of anisotropic triangles allows to improve the efficiency of the procedure by introducing long and thin triangles fitting in particular the directions of the possible curves of discontinuity.


   It is therefore natural to try to characterize, an produce optimal meshes, in the sense of the compromise between approximation error and complexity of the mesh, for a given function.


   In the case of smooth functions, we have derived optimal error estimates and propose a caracterisation of the corresponding meshes (Article). We have also elaborated a hierarchical algorithm that produces such anisotropic triangulations, and which optimality has been proved for certain classes of functions.



PhD Dissertation


Sharp asymptotic interpolation error:

-M., Optimally Adapted Meshes for Finite Elements of Arbitrary Order and W1p Norms, Numer Math, 2011

-Y. Babenko, T. Leskevich, M., Sharp asymptotics of the L p approximation error for interpolation on block partitions, Numerische Mathematik, 2011

-M., Optimal meshes for finite elements of arbitrary order, Constructive Approximation, 2010



Greedy refinement algorithms:

-A. Cohen, N. Dyn, F. Hecht, M., Adaptive multiresolution analysis based on anisotropic triangulations, Mathematics of Computation, 2011

-A. Cohen, M., Greedy bisection generates optimally adapted triangulations, Mathematics of Computation, 2011

-A. Cohen, M., Adaptive and anisotropic piecewise polynomial approximation, Chapter of the book Multiscale, Nonlinear and Adaptive Approximation , 2009


Link with image analysis:

-A. Cohen, M., Anisotropic smoothness classes: from finite element approximation to image models, JMIV, 2010


Design of Riemannian metrics for mesh generation:

  1. -M., The optimal aspect ratio for piecewise quadratic anisotropic finite element approximation, proceedings of the conference SampTA 2011



Other subjects in numerical analysis


-J. Bleyer, G. Carlier, V. Duval, M., G. Peyré, A Gamma-Convergence Result for the Upper Bound Limit Analysis of Plates, M2AN, 2015


-M., Non conforming vector finite elements for H (curl) intersected with H (div), Applied Mathematic Letters, 2011


PhD students


François Desquilbet, in co-direction with Ludovic Metivier, started fall 2019. Preliminary title: Fast marching methods for the computation of seismic wave traveltimes


Guillaume Bonnet, in co-direction with Frederic Bonnans, started fall 2018. Preliminary title:

Efficient numerical schemes for Hamilton-Jacobi-Bellman PDEs


Da Chen, in co-direction with Laurent Cohen, 2013-2016. Title: New Minimal Paths Models for Tubular Structure Extraction and Image Segmentation


Grants


Member of the ANR project MAGA, Monge Ampere and Algorithmic Geometry, coordinated by Quentin Merigot. 2016-


Coordinator of the ANR project NS-LBR, Numerical Schemes using Lattice Basis Reduction. 2013-2017.


Awards


Winner of the 8th Popov Prize, for Outstanding Research Achievements in Approximation Theory

Prize received at the 15th International Conference on Approximation Theory, May 2016.


Prix de la meilleure thèse de la fondation d'entreprise EADS (2011), en Mathématiques et leurs interactions.


Prix solennel de la chancellerie des universités de Paris (2011), Perrissin-Pirasset/Schneider, en Mathématiques Fondamentales et Appliquées, pour ma thèse.

CNRS researcher in Applied Mathematics

Université Paris-Sud, CNRS, Université Paris-Saclay,

Laboratoire de Mathématiques d’Orsay, bureau 2F1,

jean-marie.mirebeau «at» math.u-psud.fr


Address :

Département de Mathématiques Bâtiment 307

Faculté des Sciences d'Orsay Université Paris-Sud

F-91405 Orsay Cedex

Determination of the optimal aspect ratio, for anisotropic mesh generation

Anisotropic refinement by bisection

(1) Geometry of numerical scheme stencils for anisotropic eikonal equations. (2) An example of front propagation.

Three kinds of consumers in a principal agent model.

(Left) Original image. (Right) Effect of anisotropic diffusion.

Receiving the 8th Popov Prize, with Albert Cohen (left, former PhD advisor), Pencho Petrushev (right, chairman of the selection comittee).