Abstract
In classical signal processing, it is common to analyze and process signals in the frequency domain, by representing the signal in the Fourier basis, and filtering it by applying a transfer function on the Fourier coefficients. In some applications, it is possible to design an optimal filter. A classical example is the Wiener filter that achieves a minimum mean squared error estimate for signal denoising. Here, we adopt similar concepts to construct optimal diffusion geometric shape descriptors. The analogy of Fourier basis are the eigenfunctions of the Laplace-Beltrami operator, in which many geometric constructions such as diffusion metrics, can be represented. By designing a filter of the Laplace-Beltrami eigenvalues, it is theoretically possible to achieve invariance to different shape transformations, like scaling. Given a set of shape classes with different transformations, we learn the optimal filter by minimizing the ratio between knowingly similar and knowingly dissimilar diffusion distances it induces. The output of the proposed framework is a filter that is optimally tuned to handle transformations that characterize the training set.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Bérard, P., Besson, G., Gallot, S.: Embedding riemannian manifolds by their heat kernel. Geometric and Functional Analysis 4(4), 373–398 (1994)
Bronstein, A.M., Bronstein, M.M., Bustos, B., Castellani, U., Crisani, M., Falcidieno, B., Guibas, L.J., Sipiran, I., Kokkinos, I., Murino, V., Ovsjanikov, M., Patané, G., Spagnuolo, M., Sun, J.: SHREC 2010: robust feature detection and description benchmark. In: Proc. 3DOR (2010)
Bronstein, A.M., Bronstein, M.M., Castellani, U., Falcidieno, B., Fusiello, A., Godil, A., Guibas, L.J., Kokkinos, I., Lian, Z., Ovsjanikov, M., Patané, G., Spagnuolo, M., Toldo, R.: Shrec 2010: robust large-scale shape retrieval benchmark. In: Proc. 3DOR (2010)
Bronstein, A.M., Bronstein, M.M., Ovsjanikov, M., Guibas, L.J.: Shape google: a computer vision approach to invariant shape retrieval. In: Proc. NORDIA (2009)
Bronstein, M.M., Bronstein, A.M.: Shape recognition with spectral distances. Trans. PAMI (2010) (to appear)
Bronstein, M.M., Kokkinos, I.: Scale-invariant heat kernel signatures for non-rigid shape recognition. In: Proc. CVPR (2010)
Coifman, R.R., Lafon, S.: Diffusion maps. Applied and Computational Harmonic Analysis 21, 5–30 (2006)
Floater, M.S., Hormann, K.: Surface parameterization: a tutorial and survey. In: Advances in Multiresolution for Geometric Modelling, vol. 1 (2005)
Lévy, B.: Laplace-Beltrami eigenfunctions towards an algorithm that “understands” geometry. In: Proc. Shape Modeling and Applications (2006)
Mahmoudi, M., Sapiro, G.: Three-dimensional point cloud recognition via distributions of geometric distances. Graphical Models 71(1), 22–31 (2009)
Mateus, D., Horaud, R.P., Knossow, D., Cuzzolin, F., Boyer, E.: Articulated shape matching using laplacian eigenfunctions and unsupervised point registration. In: Proc. CVPR (June 2008)
Meyer, M., Desbrun, M., Schroder, P., Barr, A.H.: Discrete differential-geometry operators for triangulated 2-manifolds. In: Visualization and Mathematics III, pp. 35–57 (2003)
Ovsjanikov, M., Sun, J., Guibas, L.J.: Global intrinsic symmetries of shapes. Computer Graphics Forum 27, 1341–1348 (2008)
Pinkall, U., Polthier, K.: Computing discrete minimal surfaces and their conjugates. Experimental Mathematics 2(1), 15–36 (1993)
Reuter, M., Wolter, F.-E., Peinecke N.: Laplace-spectra as fingerprints for shape matching. In: Proc. ACM Symp. Solid and Physical Modeling, pp. 101–106 (2005)
Rustamov, R.M.: Laplace-Beltrami eigenfunctions for deformation invariant shape representation. In: Proc. SGP, pp. 225–233 (2007)
Spira, A., Sochen, N., Kimmel, R.: Geometric filters, diffusion flows, and kernels in image processing. In: Handbook of Computational Geometry for Pattern Recognition, Computer Vision, Neurocomputing and Robotics. Springer, Heidelberg (2005)
Sun, J., Ovsjanikov, M., Guibas, L.J.: A concise and provably informative multi-scale signature based on heat diffusion. In: Proc. SGP (2009)
Wardetzky, M., Mathur, S., Kälberer, F., Grinspun, E.: Discrete Laplace operators: no free lunch. In: Conf. Computer Graphics and Interactive Techniques (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Aflalo, Y., Bronstein, A.M., Bronstein, M.M., Kimmel, R. (2012). Deformable Shape Retrieval by Learning Diffusion Kernels. In: Bruckstein, A.M., ter Haar Romeny, B.M., Bronstein, A.M., Bronstein, M.M. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2011. Lecture Notes in Computer Science, vol 6667. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24785-9_58
Download citation
DOI: https://doi.org/10.1007/978-3-642-24785-9_58
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-24784-2
Online ISBN: 978-3-642-24785-9
eBook Packages: Computer ScienceComputer Science (R0)