An Algorithm for Optimal Transport between a Simplex Soup and a Point Cloud

We propose a numerical method to find the optimal transport map between a measure supported on a lower-dimensional subset of $\mathbb{R}^d$ and a finitely supported measure. More precisely, the source measure is assumed to be supported on a simplex soup, i.e., on a union of simplices of arbitrary dimension between 2 and $d$. As in [F. Aurenhammer, F. Hoffman, and B. Aronov, Algorithmica, 20 (1998), pp. 61--76] we recast this optimal transport problem as the resolution of a nonlinear system where one wants to prescribe the quantity of mass in each cell of the so-called Laguerre diagram. We prove the convergence with linear speed of a damped Newton's algorithm to solve this nonlinear system. The convergence relies on two conditions: (i) a genericity condition on the point cloud with respect to the simplex soup, and (ii) a (strong) connectedness condition on the support of the source measure defined on the simplex soup. Finally, we apply our algorithm in ${R}^3$ to compute optimal transport plans between a measure supported on a triangulation and a discrete measure. We also detail some applications such as optimal quantization of a probability density over a surface, remeshing, or rigid point set registration on a mesh.

  • 1.  F. Aurenhammer F. Hoffmann and  B. Aronov , Minkowski-type theorems and least-squares clustering , Algorithmica , 20 ( 1998 ), pp. 61 -- 76 . CrossrefISIGoogle Scholar

  • 2.  J.-D. Benamou B. D. Froese and  A. M. Oberman , Numerical solution of the optimal transportation problem using the Monge--Ampere equation , J. Comput. Phys. , 260 ( 2014 ), pp. 107 -- 126 . CrossrefISIGoogle Scholar

  • 3.  D. P. Bertsekas , The auction algorithm: A distributed relaxation method for the assignment problem , Ann. Oper. Res. , 14 ( 1988 ), pp. 105 -- 123 . CrossrefGoogle Scholar

  • 4.  P. J. Besl and  N. D. McKay , Method for registration of $3$-D shapes, in Robotics-DL Tentative , International Society for Optics and Photonics , 1992 , pp. 586 -- 606 . Google Scholar

  • 5.  L. A. Caffarelli S. A. Kochengin and  V. I. Oliker , On the numerical solution of the problem of reflector design with given far-field scattering data , in Monge Ampère Equation: Applications to Geometry and Optimization: NSF-CBMS Conference on the Monge Ampère Equation, Applications to Geometry and Optimization , 1997 , Florida Atlantic University , Contemp . Math. 226, American Mathematical Society, 1999, pp. 13 -- 32 . Google Scholar

  • 6.  G. Carlier A. Galichon and  F. Santambrogio , From Knothe's transport to Brenier's map and a continuation method for optimal transport , SIAM J. Math. Anal. , 41 ( 2010 ), pp. 2554 -- 2576 , https://doi.org/10.1137/080740647. LinkISIGoogle Scholar

  • 7.  P. M. M. Castro Q. Mérigot and  B. Thibert , Far-field reflector problem and intersection of paraboloids , Numer. Math. , 2 ( 2016 ), pp. 389 -- 411 . CrossrefISIGoogle Scholar

  • 8.  F. de Goes K. Breeden V. Ostromoukhov and  M. Desbrun , Blue noise through optimal transport , ACM Trans. Graphics , 31 ( 2012 ), 171 . CrossrefISIGoogle Scholar

  • 9.  J. Digne D. Cohen-Steiner P. Alliez F. De Goes and  M. Desbrun , Feature-preserving surface reconstruction and simplification from defect-laden point sets , J. Math. Imaging Vision , 48 ( 2014 ), pp. 369 -- 382 . CrossrefISIGoogle Scholar

  • 10.  B. Engquist and  B. D. Froese , Application of the Wasserstein metric to seismic signals , Commun. Math. Sci. , 12 ( 2014 ), pp. 979 -- 988 . CrossrefISIGoogle Scholar

  • 11.  W. Gangbo and  R. J. McCann , The geometry of optimal transportation , Acta Math. , 177 ( 1996 ), pp. 113 -- 161 . CrossrefISIGoogle Scholar

  • 12.  J. Kitagawa Q. Mérigot and  B. Thibert , Convergence of a Newton Algorithm for Semi-discrete Optimal Transport, preprint, https://arxiv.org/abs/1603.05579 , 2016 . , https://arxiv.org/abs/1603.05579. Google Scholar

  • 13.  B. Lévy , A numerical algorithm for $L^2$ semi-discrete optimal transport in $3$D , ESAIM Math. Model. Numer. Anal. , 49 ( 2015 ), pp. 1693 -- 1715 . CrossrefISIGoogle Scholar

  • 14.  D. C. Liu and  J. Nocedal , On the limited memory BFGS method for large scale optimization , Math. Programming , 45 ( 1989 ), pp. 503 -- 528 . CrossrefISIGoogle Scholar

  • 15.  G. Loeper , On the regularity of solutions of optimal transportation problems , Acta Math. , 202 ( 2009 ), pp. 241 -- 283 . CrossrefISIGoogle Scholar

  • 16.  X.-N. Ma and  N. S. Trudinger . Wang, Regularity of potential functions of the optimal transportation problem , Arch. Ration. Mech. Anal. , 177 ( 2005 ), pp. 151 -- 183 . CrossrefISIGoogle Scholar

  • 17.  M. Mandad D. Cohen-Steiner L. Kobbelt P. Alliez and  M. Desbrun , Variance-minimizing transport plans for inter-surface mapping , ACM Trans. Graphics , 36 ( 2017 ), 39 . CrossrefISIGoogle Scholar

  • 18.  Q. Mérigot , A multiscale approach to optimal transport , Computer Graphics Forum , 30 ( 2011 ), pp. 1583 -- 1592 . CrossrefISIGoogle Scholar

  • 19.  J.-M. Mirebeau , Discretization of the 3D Monge-Ampère operator, between wide stencils and power diagrams , ESAIM Math. Model. Numer. Anal., 49 ( 2015 ), pp. 1511 -- 1523 . Google Scholar

  • 20.  V. Oliker and  L. Prussner , On the numerical solution of the equation $(\partial^2 z/\partial x^2) + (\partial^2z/\partial y^2) - (\partial^2z/\partial x\partial y)$ and its discretizations, I , Numer. Math. , 54 ( 1989 ), pp. 271 -- 293 . CrossrefISIGoogle Scholar

  • 21.  F. Santambrogio , Optimal Transport for Applied Mathematicians , Birkhäuser , 2015 . Google Scholar

  • 22.  B. Schmitzer and  C. Schnörr , A hierarchical approach to optimal transport , in International Conference on Scale Space and Variational Methods in Computer Vision , Springer , 2013 , pp. 452 -- 464 . Google Scholar

  • 23.  J. Solomon F. De Goes G. Peyré M. Cuturi A. Butscher A. Nguyen T. Du and  L. Guibas , Convolutional Wasserstein distances: Efficient optimal transportation on geometric domains , ACM Trans. Graphics , 34 ( 2015 ), 66 . CrossrefISIGoogle Scholar

  • 24.  Z. Su W. Zeng R. Shi Y. Wang J. Sun and  X. Gu , Area preserving brain mapping , in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition , 2013 , pp. 2235 -- 2242 . CrossrefGoogle Scholar

  • 25.  The CGAL Project , CGAL User and Reference Manual , CGAL Editor ial Board , 4.9 ed., 2016 , http://doc.cgal.org/4.9/Manual/packages.html. , http://doc.cgal.org/4.9/Manual/packages.html. Google Scholar

  • 26.  M. Thorpe S. Park S. Kolouri G. K. Rohde and  D. Slepčev , A transportation $L^p$ distance for signal analysis , J. Math. Imaging Vision , 59 ( 2017 ), pp. 187 -- 210 . CrossrefISIGoogle Scholar

  • 27.  C. Villani , Optimal Transport: Old and New , Springer-Verlag , 2009 . Google Scholar