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Nonisometric Surface Registration via Conformal Laplace–Beltrami Basis Pursuit

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Abstract

Surface registration is one of the most fundamental problems in geometry processing. Many approaches have been developed to tackle this problem in cases where the surfaces are nearly isometric. However, it is much more challenging to compute correspondence between surfaces which are intrinsically less similar. In this paper, we propose a variational model to align the Laplace-Beltrami (LB) eigensytems of two non-isometric genus zero shapes via conformal deformations. This method enables us to compute geometrically meaningful point-to-point maps between non-isometric shapes. Our model is based on a novel basis pursuit scheme whereby we simultaneously compute a conformal deformation of a ’target shape’ and its deformed LB eigensystem. We solve the model using a proximal alternating minimization algorithm hybridized with the augmented Lagrangian method which produces accurate correspondences given only a few landmark points. We also propose a re-initialization scheme to overcome some of the difficulties caused by the non-convexity of the variational problem. Intensive numerical experiments illustrate the effectiveness and robustness of the proposed method to handle non-isometric surfaces with large deformation with respect to both noises on the underlying manifolds and errors within the given landmarks or feature functions.

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Availability of data and material

The datasets generated during and/or analysed during the current study are available in the FAUST repository (http://faust.is.tue.mpg.de).

Notes

  1. In fact, we do not need to require that the surfaces have the same number of points, but doing so for now will allow for more convenient notation.

References

  1. Attouch, H., Bolte, J., Redont, P., Soubeyran, A.: Proximal alternating minimization and projection methods for nonconvex problems: an approach based on the kurdyka-łojasiewicz inequality. Math. Oper. Res. 35(2), 438–457 (2010)

    Article  MathSciNet  Google Scholar 

  2. Attouch, H., Bolte, J., Svaiter, B.F.: Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized gauss-seidel methods. Math. Program. 137(1–2), 91–129 (2013)

    Article  MathSciNet  Google Scholar 

  3. Aubry, M., Schlickewei, U., Cremers, D.: The wave kernel signature: a quantum mechanical approach to shape analysis. In: 2011 IEEE International Conference on Computer Vision Workshops (ICCV Workshops), pp. 1626–1633 . IEEE(2011)

  4. Barzilai, J., Borwein, J.M.: Two-point step size gradient methods. IMA J. Numer. Anal. 8(1), 141–148 (1988)

    Article  MathSciNet  Google Scholar 

  5. Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms. Wiley, Hoboken (2013)

    MATH  Google Scholar 

  6. Bérard, P., Besson, G., Gallot, S.: Embedding riemannian manifolds by their heat kernel. Geom. Funct. Anal 4(4), 373–398 (1994)

    Article  MathSciNet  Google Scholar 

  7. Bogo, F., Romero, J., Loper, M., Black, M.J.: Faust: Dataset and evaluation for 3d mesh registration. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 3794–3801 (2014)

  8. Bolte, J., Sabach, S., Teboulle, M.: Proximal alternating linearized minimization for nonconvex and nonsmooth problems. Math. Program. 146(1–2), 459–494 (2014)

    Article  MathSciNet  Google Scholar 

  9. Bronstein, A.M., Bronstein, M.M., Kimmel, R.: Efficient computation of isometry-invariant distances between surfaces. SIAM J. Sci. Comput. 28(5), 1812–1836 (2006)

    Article  MathSciNet  Google Scholar 

  10. Bronstein, M.M., Kokkinos, I.: Scale-invariant heat kernel signatures for non-rigid shape recognition. In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 1704–1711 (2010)

  11. Chavel, I.: Eigenvalues in Riemannian Geometry, vol. 115. Academic press, New York (1984)

    MATH  Google Scholar 

  12. Chen, W., Ji, H., You, Y.: An augmented lagrangian method for \(\ell _1\)-regularized problems with orthogonality constrains. SIAM J. Sci. Comput. 38(4), B570–B592 (2016)

    Article  Google Scholar 

  13. Dziuk, G., Elliott, C.M.: Finite element methods for surface pdes. Acta Numer. 22, 289–396 (2013)

    Article  MathSciNet  Google Scholar 

  14. Elad, A., Kimmel, R.: On bending invariant signatures for surfaces. IEEE Trans. Pattern Anal. Mach. Intell. 25(10), 1285–1295 (2003)

    Article  Google Scholar 

  15. Glowinski, R., Le Tallec, P.: Augmented Lagrangian and operator-splitting methods in nonlinear mechanics. SIAM (1989)

  16. Gu, X., Wang, Y., Chan, T.F., Thompson, P.M., Yau, S.T.: Genus zero surface conformal mapping and its application to brain surface mapping. IEEE Trans. Med. Imaging 23(8), 949–958 (2004)

    Article  Google Scholar 

  17. Haker, S., Angenent, S., Tannenbaum, A., Kikinis, R., Sapiro, G., Halle, M.: Conformal surface parameterization for texture mapping. IEEE Trans. Vis. Comput. Graph. 6(2), 181–189 (2000)

    Article  Google Scholar 

  18. Heimann, T., Meinzer, H.P.: Statistical shape models for 3d medical image segmentation: a review. Med. Image Anal. 13(4), 543–563 (2009)

    Article  Google Scholar 

  19. Hurdal, M.K., Stephenson, K., Bowers, P., Sumners, D., Rottenberg, D.: Coordinate systems for conformal cerebellar flat maps. NeuroImage 11, S467 (2000)

    Article  Google Scholar 

  20. Jost, J.: Riemannian Geometry and Geometric Analysis. Springer, Berlin (2008)

    MATH  Google Scholar 

  21. Kao, C.Y., Lai, R., Osting, B.: Maximization of laplace- beltrami eigenvalues on closed riemannian surfaces. ESAIM: Control Optim. Calc. Var. 23(2), 685–720 (2017)

    MathSciNet  MATH  Google Scholar 

  22. Kim, V.G., Lipman, Y., Funkhouser, T.: Blended intrinsic maps. In: ACM Transactions on Graphics (TOG), vol 30, p 79. ACM (2011)

  23. Kovnatsky, A., Bronstein, M.M., Bronstein, A.M., Glashoff, K., Kimmel, R.: Coupled quasi-harmonic bases. Comput. Grap. Forum 32, 439–448 (2013)

    Article  Google Scholar 

  24. Kovnatsky, A., Glashoff, K., Bronstein, M.M.: Madmm: a generic algorithm for non-smooth optimization on manifolds. In: European Conference on Computer Vision, pp. 680–696. Springer (2016)

  25. Kraevoy, V., Sheffer, A.: Cross-parameterization and compatible remeshing of 3d models. ACM Trans. Graph. (TOG) 23(3), 861–869 (2004)

    Article  Google Scholar 

  26. Lai, R., Osher, S.: A splitting method for orthogonality constrained problems. J. Sci. Comput. 58(2), 431–449 (2014)

    Article  MathSciNet  Google Scholar 

  27. Lai, R., Zhao, H.: Multiscale nonrigid point cloud registration using robust sliced-wasserstein distance via laplace-beltrami eigenmap. SIAM J. Imaging Sci. 10(2), 449–483 (2017)

    Article  MathSciNet  Google Scholar 

  28. Lai, R., Shi, Y., Scheibel, K., Fears, S., Woods, R., Toga, A.W., Chan, T.F.: Metric-induced optimal embedding for intrinsic 3D shape analysis. In: Computer Vision and Pattern Recognition (CVPR), pp. 2871–2878 (2010)

  29. Lai, R., Shi, Y., Sicotte, N., Toga, A.W.: Automated corpus callosum extraction via laplace-beltrami nodal parcellation and intrinsic geodesic curvature flows on surfaces. In: 2011 IEEE International Conference on Computer Vision (ICCV), pp. 2034–2040. IEEE (2011)

  30. Lai, R., Liang, J., Zhao, H.: A local mesh method for solving pdes on point clouds. Inverse Prob. Imaging 7(3), 737–755 (2013)

    Article  MathSciNet  Google Scholar 

  31. Levy, B.: Laplace-beltrami eigenfunctions: Towards an algorithm that understands geometry. In: IEEE International Conference on Shape Modeling and Applications (invited talk) (2006)

  32. Litman, R., Bronstein, A.M.: Learning spectral descriptors for deformable shape correspondence. IEEE Trans. Pattern Anal. Mach. Intell. 36(1), 171–180 (2014)

    Article  Google Scholar 

  33. Ovsjanikov, M., Ben-Chen, M., Solomon, J., Butscher, A., Guibas, L.: Functional maps: a flexible representation of maps between shapes. ACM Trans. Graph. (TOG) 31(4), 30 (2012)

    Article  Google Scholar 

  34. Raviv, D., Kimmel, R.: Affine invariant non-rigid shape analysis. Int. J. Comput. Vis. 111, 1–11 (2015)

    Article  MathSciNet  Google Scholar 

  35. Raviv, D., Bronstein, A.M., Bronstein, M.M., Kimmel, R., Sochen, N.: Affine-invariant diffusion geometry for the analysis of deformable 3d shapes. In: CVPR, pp. 2361–2367 (2011)

  36. Reuter, M., Wolter, F.E., Peinecke, N.: Laplace-spectra as fingerprints for shape matching. In: Proceedings of the 2005 ACM Symposium on Solid and Physical Modeling, pp. 101–106. ACM (2005)

  37. Reuter, M., Wolter, F.E., Peinecke, N.: Laplace-beltrami spectra as ‘shape-dna’of surfaces and solids. Comput. Aided Des. 38(4), 342–366 (2006)

    Article  Google Scholar 

  38. Rodola, E., Moeller, M., Cremers, D.: Point-wise map recovery and refinement from functional correspondence (2015). arXiv:1506.05603

  39. Rustamov, R.M.: Laplace-beltrami eigenfunctions for deformation invariant shape representation. In: Eurographics Symposium on Geometry Processing (2007)

  40. Shi, Y., Lai, R., Krishna, S., Sicotte, N., Dinov, I., Toga, A.W.: Anisotropic Laplace-Beltrami eigenmaps: bridging Reeb graphs and skeletons. In: Computer Vision and Pattern Recognition Workshops, pp. 1–7 (2008)

  41. Shi, Y., Lai, R., Gill, R., Pelletier, D., Mohr, D., Sicotte, N., Toga, A.W.: Conformal metric optimization on surface (cmos) for deformation and mapping in laplace-beltrami embedding space. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 327–334. Springer (2011)

  42. Shi, Y., Lai, R., Toga, A.W.: Cortical surface reconstruction via unified reeb analysis of geometric and topological outliers in magnetic resonance images. IEEE Trans. Med. Imaging 32(3), 511–530 (2013)

    Article  Google Scholar 

  43. Shi, Y., Lai, R., Wang, D.J., Pelletier, D., Mohr, D., Sicotte, N., Toga, A.W.: Metric optimization for surface analysis in the laplace-beltrami embedding space. IEEE Trans. Med. Imaging 33(7), 1447–1463 (2014)

    Article  Google Scholar 

  44. Shtern, A., Kimmel, R.: Spectral gradient fields embedding for nonrigid shape matching. Comput. Vis. Image Underst. 140, 21–29 (2015)

    Article  Google Scholar 

  45. Shtern, A., Sela, M., Kimmel, R.: Fast blended transformations for partial shape registration. J. Math. Imaging Vis. 60, 1–16 (2016)

    MathSciNet  MATH  Google Scholar 

  46. Siddiqi, K., Lauziere, Y.B., Tannenbaum, A., Zucker, S.W.: Area and length minimizing flows for shape segmentation. IEEE Trans. Image Process. 7(3), 433–443 (1998)

    Article  Google Scholar 

  47. Springborn, B., Schröder, P., Pinkall, U.: Conformal equivalence of triangle meshes. In: ACM Transactions on Graphics (TOG)—Proceedings of ACM SIGGRAPH 2008, 27(3) (2008)

  48. Sun, J., Ovsjanikov, M., Guibas, L.: A concise and provably informative multi-scale signature based on heat diffusion. Comput. Graph. Forum 28, 1383–1392 (2009)

    Article  Google Scholar 

  49. Tombari, F., Salti, S., Di Stefano, L.: Unique signatures of histograms for local surface description. In: European Conference on Computer Vision, pp. 356–369. Springer (2010)

  50. Vallet, B., Levy, B.: Spectral geometry processing with manifold harmonics. In: Computer Graphics Forum (Proceedings Eurographics) (2008)

  51. Van Kaick, O., Zhang, H., Hamarneh, G., Cohen-Or, D.: A survey on shape correspondence. Comput. Graph. Forum 30, 1681–1707 (2011)

    Article  Google Scholar 

  52. Vestner, M., Lähner, Z., Boyarski, A., Litany, O., Slossberg, R., Remez, T., Rodola, E., Bronstein, A., Bronstein, M., Kimmel, R., et al.: Efficient deformable shape correspondence via kernel matching. In: 2017 International Conference on 3D Vision (3DV), pp. 517–526. IEEE (2017)

  53. Wang, Y., Yin, W., Zeng, J.: Global convergence of admm in nonconvex nonsmooth optimization (2015). arXiv:1511.06324

  54. Wen, Z., Yin, W.: A feasible method for optimization with orthogonality constraints. Math. Program. 142(1–2), 397–434 (2013)

    Article  MathSciNet  Google Scholar 

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Funding

The research of S. Schonsheck and R. Lai is supported in part by NSF DMS–1522645 and an NSF Career Award DMS–1752934. M. Bronstein is partially supported by ERC Consolidator grant No. 724228 (LEMAN).

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Authors and Affiliations

  1. Department of Mathematics, Rensselaer Polytechnic Institute, Troy, NY, 12180, USA

    Stefan C. Schonsheck & Rongjie Lai

  2. Department of Computing, Imperial College London, London, UK

    Michael M. Bronstein

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  1. Stefan C. Schonsheck
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R. Lai and M. Bronstein proposed the research. R. Lai and S. Schonsheck performed the research and wrote the paper.

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Correspondence to Rongjie Lai.

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Schonsheck, S.C., Bronstein, M.M. & Lai, R. Nonisometric Surface Registration via Conformal Laplace–Beltrami Basis Pursuit. J Sci Comput 86, 30 (2021). https://doi.org/10.1007/s10915-020-01390-y

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  • DOI: https://doi.org/10.1007/s10915-020-01390-y

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