CSDL Home IEEE Transactions on Pattern Analysis & Machine Intelligence 2014 vol.36 Issue No.03 - March

Subscribe

Issue No.03 - March (2014 vol.36)

pp: 466-478

Patrice Koehl , Dept. of Comput. Sci., Univ. of California, Davis, Davis, CA, USA

Joel Hass , Dept. of Math., Univ. of California, Davis, Davis, CA, USA

ABSTRACT

A new algorithm is presented that provides a constructive way to conformally warp a triangular mesh of genus zero to a destination surface with minimal metric deformation, as well as a means to compute automatically a measure of the geometric difference between two surfaces of genus zero. The algorithm takes as input a pair of surfaces that are topological 2-spheres, each surface given by a distinct triangulation. The algorithm then constructs a map f between the two surfaces. First, each of the two triangular meshes is mapped to the unit sphere using a discrete conformal mapping algorithm. The two mappings are then composed with a Mobius transformation to generate the function f. The Mobius transformation is chosen by minimizing an energy that measures the distance of f from an isometry. We illustrate our approach using several “real life” data sets. We show first that the algorithm allows for accurate, automatic, and landmark-free nonrigid registration of brain surfaces. We then validate our approach by comparing shapes of proteins. We provide numerical experiments to demonstrate that the distances computed with our algorithm between low-resolution, surface-based representations of proteins are highly correlated with the corresponding distances computed between high-resolution, atomistic models for the same proteins.

INDEX TERMS

Shape, Conformal mapping, Proteins, Shape measurement, Geometry, Equations,nonrigid registration, Conformal mapping, mesh warping, Möbius transformation

CITATION

Patrice Koehl, Joel Hass, "Automatic Alignment of Genus-Zero Surfaces",

*IEEE Transactions on Pattern Analysis & Machine Intelligence*, vol.36, no. 3, pp. 466-478, March 2014, doi:10.1109/TPAMI.2013.139REFERENCES

- [1] R. Kötter and E. Wanke, "Mapping Brains without Coordinates,"
Philosophical Trans. Royal Soc. of London, Series B, Biological Sciences, vol. 360, pp. 751-766, 2005.- [2] A. Otte and U. Halsband, "Brain Imaging Tools in Neurosciences,"
J. Physiology Paris, vol. 99, pp. 281-292, 2006.- [3] A. Gholipour, N. Kehtarnavaz, R. Briggs, M. Devous, and K. Gonipath, "Brain Functional Localization: A Survey of Image Registration Techniques,"
IEEE Trans. Medical Imaging, vol. 26, no. 4, pp. 427-451, Apr. 2007.- [4] N. Max and E. Getzoff, "Spherical Harmonic Molecular Surfaces,"
IEEE Computer Graphic and Applications, vol. 8, no. 4, pp. 42-50, July 1988.- [5] P. Koehl, "Protein Structure Classification,"
Reviews in Computational Chemistry, K. Lipkowitz, T. Cundari, V. Gillet, and B. Boyd, eds., vol. 22, pp. 1-56, John Wiley & Sons, 2006.- [6] R. Kolodny, D. Petrey, and B. Honig, "Protein Structure Comparison: Implications for the Nature of Fold Space, and Structure and Function Prediction,"
Current Opinion in Structural Biology, vol. 16, pp. 393-398, 2006.- [7] V. Venkatraman, L. Sael, and D. Kihara, "Potential for Protein Surface Shape Analysis Using Spherical Harmonics and 3D Zernike Descriptors,"
Cell Biochemistry and Biophysics, vol. 54, pp. 23-32, 2009.- [8] M. Eck, T. DeRose, T. Duchamp, H. Hoppe, M. Lounsbery, and W. Stuetzle, "Multiresolution Analysis of Arbitrary Meshes,"
Proc. ACM SIGGRAPH '95, pp. 175-182, 1995.- [9] P. Alliez, M. Meyer, and M. Desbrun, "Interactive Geometry Remeshing,"
Proc. ACM SIGGRAPH '02, pp. 347-354, 2002.- [10] Y. Lipman, R. Al-Aifari, and I. Daubechies, "The Continuous Procrustes Distance between Two Surfaces,"
Communications in Pure and Applied Math., http://arxiv.org/absarXiv:1106.4588v2 [math.DG] , 2011.- [11] A. Bronstein, M. Bronstein, and R. Kimmel, "Generalized Multidimensional Scaling: A Framework for Isometry-Invariant Partial Surface Matching,"
Proc. Nat'l Academy of Sciences USA, vol. 103, pp. 1168-1172, 2006.- [12] F. Mémoli, "On the Use of Gromov-Hausdorff Distances for Shape Comparison,"
Proc. Point Based Graphics, pp. 81-90, 2007.- [13] Y. Lipman and I. Daubechies, "Conformal Wasserstein Distances: Comparing Surfaces in Polynomial Time,"
Advances in Math., vol. 227, pp. 1047-1077, 2011.- [14] Y. Lipman, J. Puente, and I. Daubechies, "Conformal Wasserstein Distance: II. Computational Aspects and Extensions,"
Math. of Computation, http://arxiv.org/absarXiv:1103.4681v2[math.NA] , 2011.- [15] D. Boyer, Y. Lipman, E. StClair, J. Puente, B. Patel, T. Funkhouser, J. Jernvall, and I. Daubechies, "Algorithms to Automatically Quantify the Geometric Similarity of Anatomical Surfaces,"
Proc. Nat'l Academy of Sciences USA, vol. 108, pp. 18221-18226, 2011.- [16] M. Zelditch, D. Swiderski, D. Sheets, and W. Fink,
Geometric Morphometrics for Biologists. Elsevier Academic, 2004.- [17] H. Lu, L.-P. Nolte, and M. Reyes, "Interest Points Location for Brain Image Using Landmark-Annotated Atlas,"
Int'l J. Imaging Systems Technology, vol. 22, pp. 145-152, 2012.- [18] Q. Huang, B. Adams, M. Wicke, and L. Guibas, "Non-Rigid Registration under Isometric Deformations,"
Proc. Symp. Geometry Processing, pp. 1149-1458, 2008.- [19] R. Lasowski, A. Tevs, H.-P. Seidel, and M. Wand, "A Probabilistic Framework for Partial Intrinsic Symmetries in Geometric Data,"
Proc. IEEE Int'l Conf. Computer Vision, pp. 963-970, 2009.- [20] R. Rustamov, "Laplace-Beltrami Eigenfunctions for Deformation Invariant Shape Representation,"
Proc. Fifth Eurographics Symp. Geometry Processing, pp. 225-233, 2007.- [21] J. Sun, M. Ovsjanikov, and L. Guibas, "A Concise and Provably Informative Multi-Scale Signature Based on Heat Diffusion,"
Proc. Symp. Geometry Processing, pp. 1383-1392, 2009.- [22] B. Fischl, M. Sereno, and A. Dale, "Cortical Surface-Based Analysis. II: Inflation, Flattening, and a Surface-Based Coordinate System,"
Neuroimage, vol. 9, pp. 195-207, 1999.- [23] B. Fischl, M. Sereno, R. Tootell, and A. Dale, "High-Resolution Inter-Subject Averaging and a Coordinate System for the Cortical Surface,"
Human Brain Mapping, vol. 8, pp. 272-284, 1999.- [24] M. Valliant and J. Glaunès, "Surface Matching via Currents,"
Lecture Notes in Computer Science, vol. 3565, pp. 381-392, 2005.- [25] L. Bers, "Uniformization, Moduli, and Kleinian Groups,"
Bull. London Math. Soc., vol. 4, pp. 257-300, 1972.- [26] J. Gray, "On the History of the Riemann Mapping Theorem,"
Rendiconti del Circolo Matematico di Palermo, ser. II, Supplemento 34, pp. 47-94, 1994.- [27] S. Angenent, S. Haker, A. Tannenbaum, and R. Kikinis, "Conformal Geometry and Brain Flattening,"
Proc. Medical Image Computing and Computer-Assisted Intervention (MICCAI '99), pp. 271-278, 1999.- [28] M. Hurdal, P. Bowers, K. Stephenson, D. Sumners, K. Rehm, K. Shaper, and D. Rotenberg, "Quasiconformally Flat Mapping the Human Cerebellum,"
Proc. Medical Image Computing and Computer-Assisted Intervention (MICCAI '99), pp. 279-286, 1999.- [29] X. Gu and S.-T. Yau, "Global Conformal Surface Parametrization,"
Proc. Eurographics Symp. Geometry Processing, pp. 127-137, 2003.- [30] X. Gu, Y. Wang, T. Chan, P. Thompson, and S.-T. Yau, "Genus Zero Surface Conformal Mapping and Its Application to Brain Surface Mapping,"
IEEE Trans. Medical Imaging, vol. 23, no. 8, pp. 949-958, Aug. 2004.- [31] B. Springborn, P. Schröder, and U. Pinkall, "Conformal Equivalence of Triangle Meshes,"
Proc. SIGGRAPH Asia, pp. 79-89, 2008.- [32] Y. Lipman and T. Funkhouser, "Möbius Voting for Surface Correspondence,"
ACM Trans. Graphics, vol. 28, pp. 72-83, 2009.- [33] D. Tosun, M. Rettmann, and J. Prince, "Mapping Techniques for Aligning Sulci across Multiple Brains,"
Medical Image Analysis, vol. 8, pp. 295-309, 2004.- [34] Y. Wang, L. Lui, T. Chan, and P. Thompson, "Optimization of Brain Conformal Mapping with Landmarks,"
Proc. Medical Image Computing and Computer-Assisted Intervention (MICCAI '05), pp. 675-683, 2005.- [35] A. Joshi, D. Shattuck, P. Thompson, and R. Leahy, "Surface-Constrained Volumetric Brain Registration Using Harmonic Mappings,"
IEEE Trans. Medical Imaging, vol. 26, no. 12, pp. 1657-1669, Dec. 2004.- [36] A. Bobenko, U. Pinkall, and B. Springborn, "Discrete Conformal Maps and Ideal Hyperbolic Polyhedra," http://arxiv.org/absarXiv:1005.2698 [math.GT] , 2010.
- [37] B. Springborn, "A Unique Representation of Polyhedral Types. Centering via Möbius Transformations,"
Math. Z., vol. 249, pp. 513-517, 2005.- [38] C.-J. Lin and J. Moré, "Newton's Method for Large Bound-Constrained Optimization Problems,"
SIAM J. Optimization, vol. 9, pp. 1100-1127, 1999.- [39] Y. Wu, Y. He, and H. Tian, "A Spherical Point Location Algorithm Based on Spherical Coordinates,"
Proc. Int'l Conf. Computational Science and Its Applications, vol. 3482, pp. 1099-1108, 2005.- [40] A. Dale, B. Fischl, and M. Sereno, "Cortical Surface-Based Analysis. I: Segmentation and Surface Reconstruction,"
Neuroimage, vol. 9, pp. 179-194, 1999.- [41] B. Yeo, M. Sabuncu, T. Vercauteren, N. Ayache, B. Fischl, and P. Golland, "Spherical Demons: Fast Diffeomorphic Landmark-Free Surface Registration,"
IEEE Trans. Medical Imaging, vol. 29, no. 3, pp. 650-668, Mar. 2010.- [42] R. Desikan, F. Ségonne, B. Fischl, B. Quinn, B. Dickerson, D. Blacker, R. Buckner, A. Dale, R. Maguire, B. Hyman, M. Albert, and R. Killiany, "An Automated Labeling System for Subdividing the Human Cerebral Cortex on MRI Scans into Gyral Based Regions of Interest,"
Neuroimage, vol. 31, pp. 968-980, 2006.- [43] M.-P. Dubuisson and A. Jain, "A Modified Hausdorff Distance for Object Matching,"
Proc. 12th IAPR Int'l Conf. Computer Vision and Image Processing, pp. 566-568, 1994.- [44] J. Chou, S. Li, C. Klee, and A. Bax, "Solution Structure of Ca(2)-Calmodulin Reveals Flexible Hand-Like Properties of Its Domains,"
Nature Structural Biology, vol. 8, pp. 990-997, 2001.- [45] S.D.J. Franklin, P. Koehl, and M. Delarue, "MinActionPath: Maximum Likelihood Trajectory for Large-Scale Structural Transitions in a Coarse-Grained Locally Harmonic Energy Landscape,"
Nucleic Acids Research, vol. 35, pp. W477-W482, 2007.- [46] H. Edelsbrunner, "Deformable Smooth Surface Design,"
Discrete and Computational Geometry, vol. 21, pp. 87-115, 1999.- [47] H. Cheng and X. Shi, "Guaranteed Quality Triangulation of Molecular Skin Surfaces,"
Proc. IEEE Visualization, pp. 481-488, 2004.- [48] H. Cheng and X. Shi, "Quality Mesh Generation for Molecular Skin Surfaces Using Restricted Union of Balls,"
Proc. IEEE Visualization, pp. 399-405, 2005.- [49] X. Shi and P. Koehl, "Adaptive Surface Meshes Coarsening with Guaranteed Quality and Topology,"
Proc. Int'l Conf. Computer Graphics, pp. 53-61, 2009.- [50] B. Lee and F.M. Richards, "Interpretation of Protein Structures: Estimation of Static Accessibility,"
J. Molecular Biology, vol. 55, pp. 379-400, 1971. |