The Community for Technology Leaders
RSS Icon
Issue No.09 - Sept. (2013 vol.35)
pp: 2284-2297
E. Konukoglu , Med. Sch., Athinoula A. Martinos Center for Biomed. Imaging, Harvard Univ., Cambridge, MA, USA
B. Glocker , Microsoft Res. Cambridge, Cambridge, UK
A. Criminisi , Microsoft Res. Cambridge, Cambridge, UK
K. M. Pohl , Univ. of Pennsylvania, Philadelphia, PA, USA
This paper presents a new distance for measuring shape dissimilarity between objects. Recent publications introduced the use of eigenvalues of the Laplace operator as compact shape descriptors. Here, we revisit the eigenvalues to define a proper distance, called Weighted Spectral Distance (WESD), for quantifying shape dissimilarity. The definition of WESD is derived through analyzing the heat trace. This analysis provides the proposed distance with an intuitive meaning and mathematically links it to the intrinsic geometry of objects. We analyze the resulting distance definition, present and prove its important theoretical properties. Some of these properties include: 1) WESD is defined over the entire sequence of eigenvalues yet it is guaranteed to converge, 2) it is a pseudometric, 3) it is accurately approximated with a finite number of eigenvalues, and 4) it can be mapped to the ([0,1)) interval. Last, experiments conducted on synthetic and real objects are presented. These experiments highlight the practical benefits of WESD for applications in vision and medical image analysis.
Shape, Eigenvalues and eigenfunctions, Geometry, Laplace equations, Heating, Equations, Global Positioning System,medical images, Shape distance, spectral distance, Laplace operator, Laplace spectrum, segmentations, label maps
E. Konukoglu, B. Glocker, A. Criminisi, K. M. Pohl, "WESD--Weighted Spectral Distance for Measuring Shape Dissimilarity", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.35, no. 9, pp. 2284-2297, Sept. 2013, doi:10.1109/TPAMI.2012.275
[1] D. Zhang and G. Lu, "Review of Shape Representation and Description Techniques," Pattern Recognition, vol. 37, no. 1, pp. 1-19, 2004.
[2] N. Iyer, S. Jayanti, K. Lou, Y. Kalyanaraman, and K. Ramani, "Three-Dimensional Shape Searching: State-of-the-Art Review and Future Trends," Computer-Aided Design, vol. 37, pp. 509-30, 2005.
[3] M. Reuter, F.-E. Wolter, and N. Peinecke, "Laplace-Beltrami Spectra as 'Shape-DNA' of Surfaces and Solids," Computer-Aided Design, vol. 38, pp. 342-366, 2006.
[4] B. Lévy, "Laplace-Beltrami Eigenfunctions towards an Algorithm that 'Understands' Geometry," Proc. IEEE Conf. Shape Modeling and Applications, 2006.
[5] R.M. Rustamov, "Laplace-Beltrami Eigenfunctions for Deformation Invariant Shape Representations," Proc. Eurographics Symp. Geometry Processing, 2007.
[6] J. Sun, M. Ovsjanikov, and L. Guibas, "A Concise and Provably Informative Multi-Scale Signature Based on Heat Diffusion," Proc. Eurographics Symp. Geometry Processing, 2009.
[7] M. Reuter, F.-E. Wolter, M. Shenton, and M. Niethammer, "Laplace-Beltrami Eigenvalues and Topological Features of Eigenfunctions for Statistical Shape Analysis," Computer-Aided Design, vol. 41, pp. 739-755, 2009.
[8] M.M. Bronstein and A.M. Bronstein, "Shape Recognition with Spectral Distances," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 33, no. 5, pp. 1065-1071, May 2011.
[9] H. Weyl, "Das Asymptotische Verteilungsgesetz der Eigenwerte Linearer Partieller Differentialgleichungen," Mathematische Annalen, vol. 71, pp. 441-479, 1912.
[10] M. Kac, "Can One Hear the Shape of a Drum?" The Am. Math. Monthly, vol. 73, no. 7, pp. 1-23, 1966.
[11] R. Courant and D. Hilbert, Method of Mathematical Physics, vol. 1. Interscience Publishers, 1966.
[12] V. Jain and H. Zhang, "Robust 3D Shape Correspondence in the Spectral Domain," Proc. Int'l Conf. Shape Modeling and Applications, 2006.
[13] A.M. Bronstein, M. Bronstein, M. Mahmoudi, R. Kimmel, and G. Sapiro, "A Gromov-Hausdorff Framework with Diffusion Geometry for Topologically-Robust Non-Rigid Shape Matching," Int'l J. Computer Vision, vol. 89, no. 2/3, pp. 266-286, 2010.
[14] F. Mémoli, "A Spectral Notion of Gromov-Wasserstein Distance and Related Methods," Applied and Computational Harmonic Analysis, vol. 30, no. 3, pp. 363-401, 2010.
[15] M. Ovsjanikov, A.M. Bronstein, M.M. Bronstein, and L.J. Guibas, "Shape Google: A Computer Vision Approach to Invariant Shape Retrieval," Proc. IEEE Int'l Conf. Computer Vision Workshops, pp. 320-327, 2009.
[16] M. Bronstein, "Scale-Invariant Heat Kernel Signatures for Non-Rigid Shape Recognition," Proc. IEEE Conf. Computer Vision and Pattern Recognition, 2010.
[17] R. Lai, Y. Shi, K. Scheibel, S. Fears, R. Woods, A. Toga, and T. Chan, "Metric-Induced Optimal Embedding for Intrinsic 3D Shape Analysis," Proc. IEEE Conf. Computer Vision and Pattern Recognition, 2010.
[18] S. Gnutzmann, U. Smilansky, and N. Sondergaard, "Resolving Isospectral 'Drums' by Counting Nodal Domains," J. Physics A: Math. and General, vol. 38, no. 41, pp. 8921-8933, 2005.
[19] S. Gnutzmann, P. Karageorge, and U. Smilansky, "Can One Count the Shape of a Drum?" Physical Rev. Letters, vol. 97, no. 9, 2006.
[20] R. Lai, Y. Shi, I. Dinov, T. Chan, and A. Toga, "Laplace-Beltrami Nodal Counts: A New Signature for 3D Shape Analysis," Proc. Int'l Symp. Biomedical Imaging, 2009.
[21] A. Pleijel, "A Study of Certain Green's Functions with Applications in the Theory of Vibrating Membranes," Arkiv för Matematik, vol. 2, no. 6, pp. 553-569, 1954.
[22] H. McKean and I. Singer, "Curvature and the Eigenvalues of the Laplacian," J. Differential Geometry, vol. 1, pp. 43-69, 1967.
[23] L. Smith, "The Asymptotics of the Heat Equation for a Boundary Value Problem," Inventiones Math., vol. 63, pp. 467-493, 1981.
[24] M. Protter, "Can One Hear the Shape of a Drum? Revisited," SIAM Rev., vol. 29, no. 2, pp. 185-197, June 1987.
[25] D. Vassilevich, "Heat Kernel Expansion: User's Manual," Physics Reports, vol. 388, no. 5/6, pp. 279-360, 2003.
[26] G. Jurman, R. Visintainer, and C. Furlanello, "An Introduction to Spectral Distances in Networks (Extended Version)," preprint in ArXiv, Oct. 2010.
[27] Z. Lian, A. Godil, B. Bustos, M. Daoudi, J. Hermans, S. Kawamura, Y. Kurita, G. Lavoué, H.V. Nguyen, R. Ohbuchi, Y. Ohkita, Y. Ohishi, F. Porikli, M. Reuter, I. Sipiran, D. Smeets, P. Suetens, H. Tabia, and D. Vandermeulen, "SHREC '11 Track: Shape Retrieval on Non-Rigid 3D Watertight Meshes," Proc. Eurographics/ACM Siggraph Symp. 3D Object Retrieval, 2011.
[28] C. Gordon, D. Webb, and S. Wolpert, "Isospectral Plane Domains and Surfaces via Riemannian Orbifolds," Inventiones Math., vol. 110, no. 1, pp. 1-22, 1992.
[29] M. Niethammer, M. Reuter, F.-E. Wolter, S. Bouix, N. Peinecke, M.-S. Koo, and M.E. Shenton, "Global Medical Shape Analysis Using the Laplace-Beltrami Spectrum," Proc. 10th Int'l Conf. Medical Image Computing and Computer Assisted Intervention, 2007.
[30] E. Whittaker and G. Watson, A Course of Modern Analysis. Cambridge Math. Library, 1996.
[31] L. Evans, Partial Differential Equations. Am. Math. Soc., 1998.
[32] W.F. Ames, Numerical Methods for Partial Differential Equations. Academic Press, 1977.
[33] W.E. Arnoldi, "The Principle of Minimized Iterations in the Solution of the Matrix Eigenvalue Problem," Quarterly of Applied Math., vol. 9, no. 17, pp. 17-29, 1954.
[34] A. Bronstein, A.B.M.M. Bronstein, and R. Kimmel, "Analysis of Two-Dimensional Non-Rigid Shapes," Int'l J. Computer Vision, vol. 78, no. 1, pp. 67-77, 2008.
[35] A. Bronstein, M. Bronstein, and R. Kimmel, Numerical Geometry of Non-Rigid Shapes. Springer, 2008.
[36] J. Tenenbaum, V. de Silva, and J. Langford, "A Global Geometric Framework for Nonlinear Dimensionality Reduction," Science, vol. 290, no. 5500, pp. 2319-2323, 2000.
[37] Z. Lian, A. Godil, X. Sun, and H. Zhang, "Non-Rigid 3D Shape Retrieval Using Multidimensional Scaling and Bag-of-Features," Proc. 17th IEEE Int'l Conf. Image Processing, pp. 3181-3184, 2010.
[38] C. Maes, T. Fabry, J. Keustermans, D. Smeets, P. Suetens, and D. Vandermuelen, "Feature Detection on 3D Face Surfaces for Pose Normalisation and Recognition," Proc. Fourth IEEE Int'l Conf. Biometrics: Theory, Applications and Systems, 2010.
[39] D. Smeets, T. Fabry, J. Hermans, D. Vandermuelen, and P. Suetens, "Isometric Deformation Modelling for Object Recognition," Proc. 13th Int'l Conf. Computer Analysis of Images and Patterns, pp. 757-765, 2009.
[40] F.L. Bookstein, P.D. Sampson, A.P. Streissguth, and P.D. Connor, "Geometric Morphometrics of Corpus Callosum and Subcortical Structures in the Fetal-Alcohol-Affected Brain," Teratology, vol. 64, pp. 4-32, July 2001.
[41] D. Shattuck, M. Mirza, V. Adisetiyo, C. Hojatkashani, G. Salamon, K. Narr, R. Poldrack, R. Bilder, and A. Toga, "Construction of a 3D Probabilistic Atlas of Human Cortical Structures," NeuroImage, vol. 39, no. 3, pp. 1064-80, 2008.
[42] T. Mansi, I. Voigt, B. Leonardi, X. Pennec, S. Durrleman, M. Sermesant, H. Delingette, A. Taylor, Y. Boudjemline, G. Pongiglione, and N. Ayache, "A Statistical Model for Quantification and Prediction of Cardiac Remodeling: Application to Tetralogy of Fallot," IEEE Trans. Medical Imaging, vol. 30, no. 9, pp. 1605-1616, Sept. 2011.
[43] E. Bernardis, E. Konukoglu, Y. Ou, D.N. Metaxas, B. Desjardins, and K.M. Pohl, "Temporal Shape Analysis via the Spectral Signature," Proc. 15th Int'l Conf. Medical Image Computing and Computer Assisted Intervention, 2012.
31 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool