Skip to main content
SpringerLink
Log in

The Square Root Normal Field Distance and Unbalanced Optimal Transport

  • Published:
Applied Mathematics & Optimization Submit manuscript

Abstract

This paper explores a novel connection between two areas: shape analysis of surfaces and unbalanced optimal transport. Specifically, we characterize the square root normal field (SRNF) shape distance as the pullback of the Wasserstein–Fisher–Rao (WFR) unbalanced optimal transport distance. In addition we propose a new algorithm for computing the WFR distance and present numerical results that highlight the effectiveness of this algorithm. As a consequence of our results we obtain a precise method for computing the SRNF shape distance directly on piecewise linear surfaces and gain new insights about the degeneracy of this distance.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Notes

  1. Our code is available at https://github.com/emmanuel-hartman/WassersteinFisherRaoDistance.

References

  1. Alexandrov, A.D.: Zur theorie der gemischten volumina von konvexen körpern i. Mat. Sbornik NS 1, 227–251 (1938)

    MATH  Google Scholar 

  2. Bauer, M., Bruveris, M., Michor, P.W.: Overview of the geometries of shape spaces and diffeomorphism groups. J. Math. Imaging Vis. 50(1), 60–97 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bauer, M., Charon, N., Harms, P., Hsieh, H.-W.: A numerical framework for elastic surface matching, comparison, and interpolation. Int. J. Comput. Vis. 129, 2425–2444 (2021)

    Article  MATH  Google Scholar 

  4. Bauer, M., Harms, P., Michor, P.W.: Sobolev metrics on shape space of surfaces. J. Geom. Mech. 3(4) (2011)

  5. Benamou, J.-D., Brenier, Y.: A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84(3), 375–393 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bertsekas, D.P.: Nonlinear programming. J. Oper. Res. Soc. 48(3), 334–334 (1997)

    Article  Google Scholar 

  7. Bethuel, F., Zheng, X.: Density of smooth functions between two manifolds in Sobolev spaces. J. Funct. Anal. 80(1), 60–75 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bruveris, M.: Optimal reparametrizations in the square root velocity framework. SIAM J. Math. Anal. 48(6), 4335–4354 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bruveris, M., Michor, P.W., Mumford, D.: Geodesic completeness for Sobolev metrics on the space of immersed plane curves. In: Forum of Mathematics, Sigma, vol. Cambridge University Press (2014)

  10. Burkard, R., Dell’Amico, M., Martello, S.: Assignment Problems: Revised Reprint. SIAM (2012)

  11. Charon, N., Pierron, T.: On length measures of planar closed curves and the comparison of convex shapes. Ann. Glob. Anal. Geom. 60(4), 863–901 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chizat, L., Peyré, G., Schmitzer, B., Vialard, F.-X.: An interpolating distance between optimal transport and Fisher-Rao metrics. Found. Comput. Math. 18(1), 1–44 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chizat, L., Peyré, G., Schmitzer, B., Vialard, F.-X.: Scaling algorithms for unbalanced optimal transport problems. Math. Comput. 87(314), 2563–2609 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chizat, L., Peyré, G., Schmitzer, B., Vialard, F.-X.: Unbalanced optimal transport: dynamic and Kantorovich formulations. J. Funct. Anal. 274(11), 3090–3123 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cuturi, M.: Sinkhorn distances: lightspeed computation of optimal transport. In: Burges, C.J.C., Bottou, L., Welling, M., Ghahramani, Z., Weinberger, K.Q. (eds.) Advances in Neural Information Processing Systems, vol. 26. Curran Associates Inc, Red Hook (2013)

    Google Scholar 

  16. Dryden, I.L., Mardia, K.V.: Statistical Shape Analysis: With Applications in R, vol. 995. Wiley, Chichester (2016)

    Book  MATH  Google Scholar 

  17. Flamary, R., Courty, N., Gramfort, A., Alaya, M.Z., Boisbunon, A., Chambon, S., Chapel, L., Corenflos, A., Fatras, K., Fournier, N., Gautheron, L., Gayraud, N.T.H., Janati, H., Rakotomamonjy, A., Redko, I., Rolet, A., Schutz, A., Seguy, V., Sutherland, D.J., Tavenard, R., Tong, A., Vayer, T.: Pot: Python optimal transport. J. Mach. Learn. Res. 22(78), 1–8 (2021)

    MATH  Google Scholar 

  18. Gallouët, T., Ghezzi, R., Vialard, F.-X.: Regularity theory and geometry of unbalanced optimal transport. arXiv preprint arXiv:2112.11056 (2021)

  19. Haker, S., Zhu, L., Tannenbaum, A., Angenent, S.: Optimal mass transport for registration and warping. Int. J. Comput. Vis. 60(3), 225–240 (2004)

    Article  MATH  Google Scholar 

  20. Hartman, E., Sukurdeep, Y., Charon, N., Klassen, E., Bauer, M.: Supervised deep learning of elastic SRV distances on the shape space of curves. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (2021)

  21. Jermyn, I.H., Kurtek, S., Klassen, E., Srivastava, A.: Elastic shape matching of parameterized surfaces using square root normal fields. In European Conference on Computer Vision, pp. 804–817. Springer (2012)

  22. Jermyn, I.H., Kurtek, S., Laga, H., Srivastava, A.: Elastic shape analysis of three-dimensional objects. Synth. Lect. Comput. Vis. 12(1), 1–185 (2017)

    Article  Google Scholar 

  23. Joshi, S.H., Xie, Q., Kurtek, S., Srivastava, A., Laga, H.: Surface shape morphometry for hippocampal modeling in Alzheimer’s disease. In: 2016 International Conference on Digital Image Computing: Techniques and Applications (DICTA), pp. 1–8. IEEE (2016)

  24. Klassen, E., Michor, P.W.: Closed surfaces with different shapes that are indistinguishable by the SRNF. Arch. Math. 56(2), 107–114 (2020)

    MathSciNet  MATH  Google Scholar 

  25. Kondratyev, S., Monsaingeon, L., Vorotnikov, D., et al.: A new optimal transport distance on the space of finite radon measures. Adv. Differ. Equ. 21(11/12), 1117–1164 (2016)

    MathSciNet  MATH  Google Scholar 

  26. Kurtek, S., Klassen, E., Ding, Z., Srivastava, A.: A novel Riemannian framework for shape analysis of 3D objects. In: 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp. 1625–1632. IEEE (2010)

  27. Kurtek, S., Klassen, E., Gore, J.C., Ding, Z., Srivastava, A.: Elastic geodesic paths in shape space of parameterized surfaces. IEEE Trans. Pattern Anal. Mach. Intell. 34(9), 1717–1730 (2011)

    Article  Google Scholar 

  28. Kurtek, S., Samir, C., Ouchchane, L.: Statistical shape model for simulation of realistic endometrial tissue. In: ICPRAM, pp. 421–428 (2014)

  29. Kurtek, S., Srivastava, A., Klassen, E., Laga, H.: Landmark-guided elastic shape analysis of spherically-parameterized surfaces. In: Computer Graphics Forum, vol. 32, pp. 429–438. Wiley Online Library (2013)

  30. Laga, H., Guo, Y., Tabia, H., Fisher, R.B., Bennamoun, M.: 3D Shape Analysis: Fundamentals, Theory, and Applications. Wiley, New York (2018)

    Google Scholar 

  31. Laga, H., Padilla, M., Jermyn, I. H., Kurtek, S., Bennamoun, M., Srivastava, A.: 4d atlas: Statistical analysis of the spatiotemporal variability in longitudinal 3D shape data. arXiv preprint arXiv:2101.09403 (2021)

  32. Laga, H., Xie, Q., Jermyn, I.H., Srivastava, A.: Numerical inversion of SRNF maps for elastic shape analysis of genus-zero surfaces. IEEE Trans. Pattern Anal. Mach. Intell. 39(12), 2451–2464 (2017)

    Article  Google Scholar 

  33. Lahiri, S., Robinson, D., Klassen, E.: Precise matching of PL curves in \(\mathbb{R}^N\) in the square root velocity framework. Geom. Imaging Comput. 2(3), 133–186 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  34. Liero, M., Mielke, A., Savaré, G.: Optimal transport in competition with reaction: the Hellinger-Kantorovich distance and geodesic curves. SIAM J. Math. Anal. 48(4), 2869–2911 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  35. Liero, M., Mielke, A., Savaré, G.: Optimal entropy-transport problems and a new Hellinger-Kantorovich distance between positive measures. Invent. Math. 211(3), 969–1117 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  36. Maas, J., Rumpf, M., Schönlieb, C., Simon, S.: A generalized model for optimal transport of images including dissipation and density modulation. ESAIM: Math. Model. Numer. Anal. 49(6), 1745–1769 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  37. Matuk, J., Mohammed, S., Kurtek, S., Bharath, K.: Biomedical applications of geometric functional data analysis. In: Handbook of Variational Methods for Nonlinear Geometric Data, pp. 675–701. Springer (2020)

  38. Mérigot, Q.: A multiscale approach to optimal transport. In: Computer Graphics Forum, vol. 30, pp. 1583–1592. Wiley Online Library (2011)

  39. Michor, P.W., Mumford, D.: Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Doc. Math. 10, 217–245 (2005)

    MathSciNet  MATH  Google Scholar 

  40. Monge, G.: Mémoire sur la théorie des déblais et des remblais. Histoire de l’Académie Royale des Sciences de Paris (1781)

  41. Pennec, X.: Intrinsic statistics on riemannian manifolds: basic tools for geometric measurements. J. Math. Imaging Vis. 25(1), 127–154 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  42. Peyré, G., Cuturi, M., et al. Computational optimal transport: With applications to data science. Found. Trends® Mach. Learn. 11(5-6):355–607 (2019)

  43. Piccoli, B., Rossi, F.: Generalized Wasserstein distance and its application to transport equations with source. Arch. Ration. Mech. Anal. 211(1), 335–358 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  44. Praun, E., Hoppe, H.: Spherical parametrization and remeshing. ACM Trans. Graph. 22(3), 340–349 (2003)

    Article  Google Scholar 

  45. Rubner, Y., Tomasi, C., Guibas, L.J.: The earth mover’s distance as a metric for image retrieval. Int. J. Comput. Vis. 40(2), 99–121 (2000)

    Article  MATH  Google Scholar 

  46. Rumpf, M., Wardetzky, M.: Geometry processing from an elastic perspective. GAMM-Mitteilungen 37(2), 184–216 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  47. Rumpf, M., Wirth, B.: Variational methods in shape analysis. Handb. Math. Methods Imaging 2, 1819–1858 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  48. Schneider, R.: Convex surfaces, curvature and surface area measures. In: Handbook of Convex Geometry, pp. 273–299. Elsevier (1993)

  49. Schneider, R.: Convex Bodies: The Brunn-Minkowski theory, vol. 151. Cambridge University Press, Cambridge (2014)

    MATH  Google Scholar 

  50. Sellaroli, G.: An algorithm to reconstruct convex polyhedra from their face normals and areas. arXiv preprint arXiv:1712.00825 (2017)

  51. Sheffer, A., Praun, E., Rose, K., et al.: Mesh parameterization methods and their applications. Found. Trends Comput. Graph. Vis. 2(2), 105–171 (2007)

    Article  MATH  Google Scholar 

  52. Sinkhorn, R.: A relationship between arbitrary positive matrices and doubly stochastic matrices. Ann. Math. Stat. 35(2), 876–879 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  53. Solomon, J.: Transportation Techniques for Geometric Data Processing. Stanford University, Stanford (2015)

    Google Scholar 

  54. Solomon, J., Rustamov, R., Guibas, L., Butscher, A.: Wasserstein propagation for semi-supervised learning. In: International Conference on Machine Learning, pp. 306–314. PMLR (2014)

  55. Srivastava, A., Klassen, E.P.: Functional and Shape Data Analysis, vol. 1. Springer, New York (2016)

    Book  MATH  Google Scholar 

  56. Su, Z., Bauer, M., Preston, S.C., Laga, H., Klassen, E.: Shape analysis of surfaces using general elastic metrics. J. Math. Imaging Vis. 62, 1087–1106 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  57. Van Schaftingen, J.: Approximation in Sobolev spaces by piecewise affine interpolation. J. Math. Anal. Appl. 420(1), 40–47 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  58. Villani, C.: Topics in Optimal Transportation, vol. 58. American Mathematical Soc, Providence, RI (2003)

    MATH  Google Scholar 

  59. Villani, C.: Optimal Transport: Old and New, vol. 338. Springer, New York (2008)

    MATH  Google Scholar 

  60. Whitehead, J.H.C.: On C1-complexes. Ann. Math. 41, 809–824 (1940)

    Article  MathSciNet  MATH  Google Scholar 

  61. Younes, L.: Computable elastic distances between shapes. SIAM J. Appl. Math. 58(2), 565–586 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  62. Younes, L.: Shapes and Diffeomorphisms, vol. 171. Springer, New York (2010)

    MATH  Google Scholar 

Download references

Acknowledgements

We thank Nicolas Charon, Ian Jermyn, Cy Maor, Zhe Su, François-Xavier Vialard, and the statistical shape analysis group at FSU for helpful discussions during the preparation of this manuscript.

Funding

Funding was provided by National Science Foundation (Grant Nos. 49Q10 and 49Q22).

Author information

Authors and Affiliations

  1. Department of Mathematics, Florida State University, Tallahassee, USA

    Martin Bauer, Emmanuel Hartman & Eric Klassen

Authors
  1. Martin Bauer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Emmanuel Hartman
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Eric Klassen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Emmanuel Hartman.

Ethics declarations

Conflict of interest

The authors have not disclosed any competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Martin Bauer and Emmanuel Hartman were partially supported by NSF-Grants 1912037 and 1953244.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bauer, M., Hartman, E. & Klassen, E. The Square Root Normal Field Distance and Unbalanced Optimal Transport. Appl Math Optim 85, 35 (2022). https://doi.org/10.1007/s00245-022-09867-y

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00245-022-09867-y

Keywords

Mathematics Subject Classification

Search

Navigation