Research · Teaching · Software Quentin Mérigot

Optimal transport between two images

Numerical optimal transport and applications. The computation of solutions to the discrete version of this problem, that is when one or both measures are sum of Dirac masses, is of interest to me. I implemented a multiscale algorithm to solve the optimal transport problem on the plane and for the squared Euclidean cost based on Kantorovich duality. With J. André, D. Attali, B. Thibert and P. Machado, we are investigating the application of these techniques to solve inverse problems arising in non-imaging optics.

Geometric reconstruction from noisy data. With F. Chazal and David Cohen-Steiner, we introduced a notion of distance function to a probability measure which can replace the usual distance function in geometric inference. In order to perform global computations, such as computing topology of a certain sublevel set, one needs to approximate this function. This problem raises interesting question in connection with the classical k-set problem in computational geometry.

Boundary measure of a point cloud sampled on a mechanical shape Distance functions and curvature measures. The tube formulas assert that the volume of tubular neighborhoods of a manifold contains information about its curvature. Based on a generalization of this idea due to Federer, we proposed with F. Chazal and D. Cohen-Steiner a method to approximate the curvature measures of a set with positive reach from a discrete approximation. This has been applied to the detection of sharp edges with M. Ovsjanikov and L. Guibas.


  1. Handling convexity-like constraints in variational problems
    Q. Mérigot, É. Oudet
    SIAM Journal on Numerical Analysis, 52 (5), 2466–2487, 2014.
  2. Intersection of paraboloids and application to Minkowski-type problems
    P. Machado Manhães de Castro, Q. Mérigot, B. Thibert
    Proceedings of the 30th ACM Symposium on Computational Geometry, 2014
  3. On the reconstruction of convex sets from random normals measurements
    H. Abdallah, Q. Mérigot
    Discrete and Computational Geometry, to appear (also Proc SoCG 2014)
  4. Lower bounds for k-distance approximation.
    Q. Mérigot
    Proceedings of the 29th ACM Symposium on Computational Geometry, 2013
  5. Shape Matching via Quotient Spaces.
    M. Ovsjanikov, Q. Mérigot, V. Pătrăucean, L. Guibas
    Computer Graphics Forum 32 (5) 1–11, 2013 (also Proc SGP 2013).
  6. Witnessed k-distance.
    L. Guibas, Q. Mérigot, D. Morozov
    Discrete and Computational Geometry, 49 (1) 22–45, 2013 (also Proc SoCG 2011).
  7. A multiscale approach to optimal transport.
    Q. Mérigot
    Computer Graphics Forum 30 (5) 1583–1592, 2011 (also Proc SGP 2011).
  8. Size of the medial axis and stability of Federer’s curvature measures
    Q. Mérigot
    In Optimal Transportation: Theory and Applications, London Mathematical Society Lecture Note Series 413, pp 288–306
  9. Geometric inference for probability measures.
    F. Chazal, D. Cohen-Steiner, Q. Mérigot
    Foundations of Computational Mathematics 11, 733-751 (2011).
  10. Boundary measures for geometric inference.
    F. Chazal, D. Cohen-Steiner, Q. Mérigot
    Foundation of Computational Mathematics 10, 221-240 (2010).
  11. Feature Preserving Mesh Generation from 3D Point Clouds.
    N. Salman, M. Yvinec, Q. Mérigot
    Computer Graphics Forum 29 (5) 1623–1632, 2010 (also Proc. SGP 2010).
  12. One Point Isometric Matching with the Heat Kernel.
    M. Ovsjanikov, Q. Mérigot F. Mémoli, L. Guibas
    Computer Graphics Forum 29 (5) 1555–1564, 2010 (also Proc. SGP 2010).
  13. Voronoi-based Curvature and Feature Estimation from Point Clouds.
    Q. Mérigot, M. Ovsjanikov, L. Guibas
    IEEE Transactions on Visualization and Computer Graphics 17 (6) 743–756, 2011.
  14. Anosov AdS representations are quasi-Fuchsian.
    T. Barbot, Q. Mérigot
    Groups, Geometry and Dynamics 6(3), 441-483 (2012).

Notes, thesis, surveys

  1. A comparison of two dual methods for discrete optimal transport.
    Geometric Science of Information, LNCS 8085, 389-396, 2013
    Q. Mérigot
  2. Geometric structure detection in point clouds
    Thèse de doctorat, Université de Nice Sophia-Antipolis.


  1. Discretization of functionals involving the Monge-Ampère operator
    J.D. Benamou, G. Carlier, Q. Mérigot, É. Oudet
  2. Robust Geometry Estimation using the Generalized Voronoi Covariance Measure
    L. Cuel, J.O. Lachaud, Q. Mérigot, B. Thibert
  3. Discrete optimal transport: complexity, geometry and applications
    Q. Mérigot, É. Oudet