Abstract
The ubiquity of the Laplace-Beltrami operator in shape analysis can be seen by observing the wide variety of applications where it has been found to be useful. Here we demonstrate a small subset of such uses with their latest developments including a scale invariant transform for general triangulated meshes, an effective and efficient method for denoising meshes using Beltrami flows via high dimensional embeddings of 2D manifolds and finally the possibility of viewing the framework of geodesic active contours as a surface minimization having the Laplace-Beltrami operator as its main ingredient.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aflalo, Y., Raviv, D., Kimmel, R.: Scale invariant geometry for non-rigid shapes. Technical report, Technion University (2011)
Belkin, M., Niyogi, P.: Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput. 15(6), 1373–1396 (2003)
Besl, P.J., McKay, N.D.: A method for registration of 3-D shapes. IEEE Transactions on Pattern Analysis and Machine Intelligence 14(2), 239–256 (1992)
Bogdanova, I., Bresson, X., Thiran, J.P., Vandergheynst, P.: Scale space analysis and active contours for omnidirectional images. IEEE Transactions on Image Processing 16(7), 1888–1901 (2007)
Bresson, X., Vandergheynst, P., Thiran, J.-P.: Multiscale active contours. In: Kimmel, R., Sochen, N.A., Weickert, J. (eds.) Scale-Space 2005. LNCS, vol. 3459, pp. 167–178. Springer, Heidelberg (2005)
Bronstein, A.M., Bronstein, M., Guibas, L.J., Ovsjanikov, M.: Shape google: Geometric words and expressions for invariant shape retrieval. ACM Transactions on Graphics 30(1), Article 1 (2011)
Bronstein, A.M., Bronstein, M.M., Kimmel, R.: Generalized multidimensional scaling: A framework for isometry-invariant partial surface matching. Proceedings of the National Academy of Science, pp. 1168–1172 (2006)
Bronstein, A.M., Bronstein, M.M., Kimmel, R., Mahmoudi, M., Sapiro, G.: A Gromov-Hausdorff framework with diffusion geometry for topologically-robust non-rigid shape matching. International Journal of Computer Vision 89(2-3), 266–286 (2010)
Bronstein, M., Kokkinos, I.: Scale-invariant heat kernel signatures for non-rigid shape recognition. In: Proc. Computer Vision and Pattern Recognition (CVPR), San Francisco, USA, December 13-18, pp. 1704–1711. IEEE Computer Society (2010)
Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. International Journal of Computer Vision 22(1), 61–79 (1997)
Chen, Y., Medioni, G.: Object modeling by registration of multiple range images. In: Proceedings of IEEE International Conference on Robotics and Automation, vol. 3, pp. 2724–2729 (1991)
Coifman, R.R., Lafon, S.: Diffusion maps. Applied and Computational Harmonic Analysis 21(1), 5–30 (2006); Special Issue: Diffusion Maps and Wavelets
Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: Image denoising by sparse 3-d transform-domain collaborative filtering. IEEE Transactions on Image Processing, 2080–2095 (2007)
Elad (Elbaz), A., Kimmel, R.: On bending invariant signatures for surfaces. IEEE Trans. on Pattern Analysis and Machine Intelligence (PAMI) 25(10), 1285–1295 (2003)
Hilaga, M., Shinagawa, Y., Kohmura, T., Kunii, T.L.: Topology matching for fully automatic similarity estimation of 3D shapes. In: ACM SIGGRAPH 2001, Los Angeles, CA, August 12-17 (2001)
Karni, Z., Gotsman, C.: Spectral compression of mesh geometry. In: Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH 2000), pp. 279–286. ACM Press/Addison-Wesley Publishing Co., New York (2000)
Levy, B.: Laplace-beltrami eigenfunctions towards an algorithm that ”understands” geometry. In: IEEE International Conference on Shape Modeling and Applications (SMI 2006), p. 13 (2006)
Mémoli, F., Sapiro, G.: A theoretical and computational framework for isometry invariant recognition of point cloud data. Foundations of Computational Mathematics 5(3), 313–347 (2005)
Osada, R., Funkhouser, T., Chazelle, B., Dobkin, D.: Shape distributions. ACM Transactions on Graphics 21(4), 807–832 (2002)
Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. Journal of Computational Physics 79(1), 12–49 (1988)
Raviv, D., Bronstein, A.M., Bronstein, M.M., Kimmel, R., Sochen, N.: Affine-invariant geodesic geometry of deformable 3D shapes. In: Computers and Graphics, Herzliya, Israel, June 22-24. Proceedings of Shape Modelling International (SMI 2011), vol. 35(3). Elsevier (2011)
Raviv, D., Bronstein, A.M., Bronstein, M.M., Kimmel, R., Sochen, N.: Affine-invariant geometry of deformable 3D shapes. In: Proc. of Computer vision and Pattern Recognition (CVPR). IEEE Computer Society (June 2011)
Roussos, A., Maragos, P.: Tensor-based image diffusions derived from generalizations of the total variation and Beltrami functionals. In: ICIP (September 2010)
Sochen, N., Kimmel, R., Bruckstein, A.M.: Diffusions and confusions in signal and image processing. Journal of Mathematical Imaging and Vision 14(3), 195–209 (2001)
Sochen, N., Kimmel, R., Malladi, R.: A general framework for low level vision. IEEE Trans. on Image Processing, 310–318 (1998)
Sochen, N.A.: Stochastic processes in vision: From langevin to beltrami. In: IEEE International Conference on Computer Vision, vol. 1, p. 288 (2001)
Sun, J., Ovsjanikov, M., Guibas, L.J.: A concise and provably informative multi-scale signature based on heat diffusion. Computer Graphics Forum 28(5), 1383–1392 (2009)
Wetzler, A., Kimmel, R.: Efficient beltrami flow in patch-space. In: Bruckstein, A.M., ter Haar Romeny, B.M., Bronstein, A.M., Bronstein, M.M. (eds.) SSVM 2011. LNCS, vol. 6667, pp. 134–143. Springer, Heidelberg (2012)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Wetzler, A., Aflalo, Y., Dubrovina, A., Kimmel, R. (2013). The Laplace-Beltrami Operator: A Ubiquitous Tool for Image and Shape Processing. In: Hendriks, C.L.L., Borgefors, G., Strand, R. (eds) Mathematical Morphology and Its Applications to Signal and Image Processing. ISMM 2013. Lecture Notes in Computer Science, vol 7883. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38294-9_26
Download citation
DOI: https://doi.org/10.1007/978-3-642-38294-9_26
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-38293-2
Online ISBN: 978-3-642-38294-9
eBook Packages: Computer ScienceComputer Science (R0)