Abstract
This work presents a novel surface matching and registration method based on the landmark curve-driven canonical surface quasiconformal mapping, where an open genus zero surface decorated with landmark curves is mapped to a canonical domain with horizontal or vertical straight segments and the local shapes are preserved as much as possible. The key idea of the canonical mapping is to minimize the harmonic energy with the landmark curve straightening constraints and generate a quasi-holomorphic 1-form which is zero in one parameter along landmark and results in a quasiconformal mapping. The mapping exists and is unique and intrinsic to surface and landmark geometry. The novel shape representation provides a conformal invariant shape signature. We use it as Teichmüller coordinates to construct a subspace of the conventional Teichmüller space which considers geometry feature details and therefore increases the discriminative ability for matching. Furthermore, we present a novel and efficient registration method for surfaces with landmark curve constraints by computing an optimal mapping over the canonical domains with straight segments, where the curve constraints become linear forms. Due to the linearity of 1-form and harmonic map, the algorithms are easy to compute, efficient and practical. Experiments on human face and brain surfaces demonstrate the efficiency and efficacy and the potential for broader shape analysis applications.
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Ahlfors, L.: Lectures in Quasiconformal Mappings. Van Nostrand Reinhold, New York (1966)
Boris, S., Schröder, P., Pinkall, U.: Conformal equivalence of triangle meshes. ACM TOG 27(3), 1–11 (2008)
Desbrun, M., Meyer, M., Alliez, P.: Intrinsic parameterizations of surface meshes. In: Eurographics 2002, pp. 209–218 (2002)
Farkas, H.M., Kra, I.: Riemann Surfaces (Graduate Texts in Mathematics). Springer (1991)
Floater, M.S., Hormann, K.: Surface parameterization: a tutorial and survey, pp. 157–186. Springer
Funkhouser, T., Min, P., Kazhdan, M., Chen, J., Halderman, A., Dobkin, D., Jacobs, D.: A search engine for 3D models. ACM TOG 22(1), 83–105 (2003)
Gu, D.X., Zeng, W., Luo, F., Yau, S.T.: Numerical computation of surface conformal mappings. Computational Methods and Functional Theory 11(2), 747–787 (2011)
Jin, M., Zeng, W., Luo, F., Gu, X.: Computing Teichmüller shape space. IEEE TVCG 15(3), 504–517 (2009)
Kazhdan, M., Funkhouser, T., Rusinkiewicz, S.: Rotation invariant spherical harmonic representation of 3D shape descriptors. In: SGP 2003, pp. 156–164 (2003)
Kurtek, S., Srivastava, A., Klassen, E., Laga, H.: Landmark-guided elastic shape analysis of spherically-parameterized surfaces. Computer Graphics Forum (Proceedings of Eurographics 2013) 32(2), 429–438 (2013)
Levy, B., Petitjean, S., Ray, N., Maillot, J.: Least squares conformal maps for automatic texture atlas generation. In: SIGGRAPH 2002 (2002)
Lui, L.M., Wong, T.W., Zeng, W., Gu, X., Thompson, P.M., Chan, T.F., Yau, S.T.: Optimization of surface registrations using Beltrami holomorphic flow. J. of Scie. Comp. 50(3), 557–585 (2012)
Lui, L.M., Zeng, W., Yau, S.T., Gu, X.: Shape analysis of planar multiply-connected objects using conformal welding. IEEE TPAMI 36(7), 1384–1401 (2014)
Lui, L., Wang, Y., Chan, T., Thompson, P.: Automatic landmark tracking and its application to the optimization of brain conformal mapping, pp. II:1784–II:1792 (2006)
Schoen, R., Yau, S.T.: Lecture on Harmonic Maps, vol. 2. International Press Incorporated, Boston (1997)
Sharon, E., Mumford, D.: 2D-shape analysis using conformal mapping. IJCV 70, 55–75 (2006)
Sheffer, A., Praun, E., Rose, K.: Mesh parameterization methods and their applications, vol. 2 (2006)
Shi, R., Zeng, W., Su, Z., Damasio, H., Lu, Z., Wang, Y., Yau, S.T., Gu, X.: Hyperbolic harmonic mapping for constrained brain surface registration. In: IEEE CVPR 2013 (2013)
Weber, O., Myles, A., Zorin, D.: Computing extremal quasiconformal maps. Comp. Graph. Forum 31(5), 1679–1689 (2012)
Yin, L., Wei, X., Sun, Y., Wang, J., Rosato, M.J.: A 3D facial expression database for facial behavior research. In: IEEE FG 2006, pp. 211–216 (2006)
Zeng, W., Zeng, Y., Wang, Y., Yin, X., Gu, X., Samaras, D.: 3D non-rigid surface matching and registration based on holomorphic differentials. In: Forsyth, D., Torr, P., Zisserman, A. (eds.) ECCV 2008, Part III. LNCS, vol. 5304, pp. 1–14. Springer, Heidelberg (2008)
Zeng, W., Gu, X.: Registration for 3D surfaces with large deformations using quasi-conformal curvature flow. In: IEEE CVPR 2011 (2011)
Zeng, W., Samaras, D., Gu, X.D.: Ricci flow for 3D shape analysis. IEEE TPAMI 32(4), 662–677 (2010)
Zeng, W., Shi, R., Wang, Y., Yau, S.T., Gu, X.: Teichmüller shape descriptor and its application to Alzheimer’s disease study. IJCV 105(2), 155–170 (2013)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Zeng, W., Yang, YJ. (2014). Surface Matching and Registration by Landmark Curve-Driven Canonical Quasiconformal Mapping. In: Fleet, D., Pajdla, T., Schiele, B., Tuytelaars, T. (eds) Computer Vision – ECCV 2014. ECCV 2014. Lecture Notes in Computer Science, vol 8689. Springer, Cham. https://doi.org/10.1007/978-3-319-10590-1_46
Download citation
DOI: https://doi.org/10.1007/978-3-319-10590-1_46
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-10589-5
Online ISBN: 978-3-319-10590-1
eBook Packages: Computer ScienceComputer Science (R0)