Curvature in image and shape processing

Yonathan Aflalo[1]; Anastasia Dubrovina[1]; Ron Kimmel[1]; Aaron Wetzler[1]

  • [1] GIP Lab, Technion, Haifa 32000, Israel

Actes des rencontres du CIRM (2013)

  • Volume: 3, Issue: 1, page 131-139
  • ISSN: 2105-0597

Abstract

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The laplacian operator applied to the coordinates of a manifold provides the mean curvature vector. Manipulating the metric of the manifold or interpreting its coordinates in various ways provide useful tools for shape and image processing and representation. We will review some of these tools focusing on scale invariant geometry, curvature flow with respect to an embedding of the image manifold in a high dimensional space, and object segmentation by active contours defined via the shape laplacian operator. Such generalizations of the curvature vector and its numerical approximation as part of an image flow or triangulated shape representation, demonstrate the omnipresence of this operator and its usefulness in imaging sciences.

How to cite

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Aflalo, Yonathan, et al. "Curvature in image and shape processing." Actes des rencontres du CIRM 3.1 (2013): 131-139. <http://eudml.org/doc/275392>.

@article{Aflalo2013,
abstract = {The laplacian operator applied to the coordinates of a manifold provides the mean curvature vector. Manipulating the metric of the manifold or interpreting its coordinates in various ways provide useful tools for shape and image processing and representation. We will review some of these tools focusing on scale invariant geometry, curvature flow with respect to an embedding of the image manifold in a high dimensional space, and object segmentation by active contours defined via the shape laplacian operator. Such generalizations of the curvature vector and its numerical approximation as part of an image flow or triangulated shape representation, demonstrate the omnipresence of this operator and its usefulness in imaging sciences.},
affiliation = {GIP Lab, Technion, Haifa 32000, Israel; GIP Lab, Technion, Haifa 32000, Israel; GIP Lab, Technion, Haifa 32000, Israel; GIP Lab, Technion, Haifa 32000, Israel},
author = {Aflalo, Yonathan, Dubrovina, Anastasia, Kimmel, Ron, Wetzler, Aaron},
journal = {Actes des rencontres du CIRM},
keywords = {Image denoising; scale invariant; active contours; segmentation; Laplace-Beltrami; denoising},
language = {eng},
month = {11},
number = {1},
pages = {131-139},
publisher = {CIRM},
title = {Curvature in image and shape processing},
url = {http://eudml.org/doc/275392},
volume = {3},
year = {2013},
}

TY - JOUR
AU - Aflalo, Yonathan
AU - Dubrovina, Anastasia
AU - Kimmel, Ron
AU - Wetzler, Aaron
TI - Curvature in image and shape processing
JO - Actes des rencontres du CIRM
DA - 2013/11//
PB - CIRM
VL - 3
IS - 1
SP - 131
EP - 139
AB - The laplacian operator applied to the coordinates of a manifold provides the mean curvature vector. Manipulating the metric of the manifold or interpreting its coordinates in various ways provide useful tools for shape and image processing and representation. We will review some of these tools focusing on scale invariant geometry, curvature flow with respect to an embedding of the image manifold in a high dimensional space, and object segmentation by active contours defined via the shape laplacian operator. Such generalizations of the curvature vector and its numerical approximation as part of an image flow or triangulated shape representation, demonstrate the omnipresence of this operator and its usefulness in imaging sciences.
LA - eng
KW - Image denoising; scale invariant; active contours; segmentation; Laplace-Beltrami; denoising
UR - http://eudml.org/doc/275392
ER -

References

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  1. Yonathan Aflalo, Ron Kimmel, Spectral multidimensional scaling, Proceedings of the National Academy of Sciences 110 (2013), 18052-18057 Zbl1292.62078MR3151774
  2. Yonathan. Aflalo, Ron Kimmel, Dan Raviv, Scale Invariant Geometry for Nonrigid Shapes, SIAM Journal on Imaging Sciences 6 (2013), 1579-1597 Zbl1282.53014MR3092743
  3. Luis Alvarez, Frédéric Guichard, Pierre-Louis Lions, Jean-Michel Morel, Axioms and fundamental equations of image processing, Archive for Rational Mechanics and Analysis 123 (1993), 199-257 Zbl0788.68153MR1225209
  4. M. Belkin, P. Niyogi, Laplacian Eigenmaps for Dimensionality Reduction and Data Representation, Neural Comput. 15 (2003), 1373-1396 Zbl1085.68119
  5. P.J. Besl, N.D. McKay, A Method for Registration of 3-D Shapes, IEEE Transactions on Pattern Analysis and Machine Intelligence 14 (1992), 239-256 
  6. I. Bogdanova, X. Bresson, J. P. Thiran, P. Vandergheynst, Scale space analysis and active contours for omnidirectional images, Image Processing, IEEE Transactions on 16 (2007), 1888-1901 MR2463092
  7. X. Bresson, P. Vandergheynst, J. P. Thiran, Multiscale active contours, Scale Space and PDE Methods in Computer Vision (2005), 167-178 Zbl1119.68458
  8. A. M. Bronstein, M. M. Bronstein, R. Kimmel, Generalized multidimensional scaling: A framework for isometry-invariant partial surface matching, Proceedings of the National Academy of Science (2006), 1168-1172 Zbl1160.65306MR2204074
  9. A. M. Bronstein, M. M. Bronstein, R. Kimmel, M. Mahmoudi, G. Sapiro, A Gromov-Hausdorff framework with diffusion geometry for topologically-robust non-rigid shape matching, International Journal of Computer Vision 89 (2010), 266-286 
  10. A.M. Bruckstein, G. Sapiro, D. Shaked, Affine-invariant Evolutions of Planar Polygons, (1992) 
  11. V. Caselles, R. Kimmel, G. Sapiro, Geodesic active contours, International journal of computer vision 22 (1997), 61-79 Zbl0894.68131
  12. Y. Chen, G. Medioni, Object modeling by registration of multiple range images, Proceedings of IEEE International Conference on Robotics and Automation 3 (1991), 2724-2729 
  13. L. D. Cohen, R. Kimmel, Global minimum for active contour models: A minimal path approach, International Journal of Computer Vision 24 (1997), 57-78 
  14. R. R. Coifman, S. Lafon, Diffusion maps, Applied and Computational Harmonic Analysis 21 (2006), 5-30 Zbl1095.68094MR2238665
  15. K. Dabov, A. Foi, V. Katkovnik, K. Egiazarian, Image denoising by sparse 3-D transform-domain collaborative filtering, IEEE Transactions on Image Processing (2007), 2080-2095 MR2460626
  16. A. Elad (Elbaz), R. Kimmel, On bending invariant signatures for surfaces, IEEE Trans. on Pattern Analysis and Machine Intelligence (PAMI) 25 (2003), 1285-1295 
  17. M. Hilaga, Y. Shinagawa, T. Kohmura, T.L. Kunii, Topology Matching for Fully Automatic Similarity Estimation of 3D Shapes, ACM SIGGRAPH 2001 (2001), Los Angeles, CA 
  18. Z. Karni, C. Gotsman, Spectral compression of mesh geometry, Proceedings of the 27th annual conference on Computer graphics and interactive techniques (2000), 279-286, ACM Press/Addison-Wesley Publishing Co., New York, NY, USA 
  19. B. Levy, Laplace-Beltrami Eigenfunctions Towards an Algorithm That “Understands” Geometry, Conference on Shape Modeling and Applications, 2006. SMI 2006. IEEE International (2006), 13-13 
  20. F. Mémoli, G. Sapiro, A Theoretical and Computational Framework for Isometry Invariant Recognition of Point Cloud Data, Foundations of Computational Mathematics 5 (2005), 313-347 Zbl1101.53022MR2168679
  21. R. Osada, T. Funkhouser, B. Chazelle, D. Dobkin, Shape Distributions, ACM Transactions on Graphics 21 (2002), 807-832 Zbl1331.68256
  22. S. Osher, J. A. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, Journal of computational physics 79 (1988), 12-49 Zbl0659.65132MR965860
  23. D. Raviv, A. M. Bronstein, M. M. Bronstein, R. Kimmel, N. Sochen, Affine-invariant geodesic geometry of deformable 3D shapes, Computers & Graphics 35 (2011), 692-697 
  24. A. Roussos, P. Maragos, Tensor-based image diffusions derived from generalizations of the total variation and Beltrami functionals, ICIP (2010) 
  25. Leonid I. Rudin, Stanley Osher, Emad Fatemi, Nonlinear Total Variation Based Noise Removal Algorithms, Phys. D 60 (1992), 259-268 Zbl0780.49028
  26. N. Sochen, R. Kimmel, A. M. Bruckstein, Diffusions and confusions in signal and image processing, Journal of Mathematical Imaging and Vision 14 (2001), 195-209 Zbl0994.94011MR1847811
  27. N. Sochen, R. Kimmel, A.M. Bruckstein, Diffusions and Confusions in Signal and Image Processing, Journal of Mathematical Imaging and Vision 14 (2001), 195-209 Zbl0994.94011MR1847811
  28. N. Sochen, R. Kimmel, R. Malladi, A general framework for low level vision, IEEE Trans. on Image Processing (1998), 310-318 Zbl0973.94502MR1669540
  29. Nir A. Sochen, Stochastic Processes in Vision: From Langevin to Beltrami, Computer Vision, IEEE International Conference on 1 (2001) Zbl0985.68084
  30. C. Tomasi, R. Manduchi, Bilateral filtering for gray and color images, Proc. IEEE ICCV (1998), 836-846 
  31. C. Tomasi, R. Manduchi, Bilateral Filtering for Gray and Color Images, Proceedings of the Sixth International Conference on Computer Vision (1998), IEEE Computer Society, Washington, DC, USA 
  32. A. Wetzler, R. Kimmel, Efficient Beltrami Flow in Patch-Space, Proceedings of the 3rd International Conference on Scale Space and Variational Methods in Computer Vision 2011 6667 (2011), 134-143, Springer Zbl1137.68483

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