Advertisement

International Journal of Computer Vision

, Volume 84, Issue 2, pp 220–236 | Cite as

Local and Nonlocal Discrete Regularization on Weighted Graphs for Image and Mesh Processing

  • Sébastien BougleuxEmail author
  • Abderrahim Elmoataz
  • Mahmoud Melkemi
Article

Abstract

We propose a discrete regularization framework on weighted graphs of arbitrary topology, which unifies local and nonlocal processing of images, meshes, and more generally discrete data. The approach considers the problem as a variational one, which consists in minimizing a weighted sum of two energy terms: a regularization one that uses the discrete p-Dirichlet form, and an approximation one. The proposed model is parametrized by the degree p of regularity, by the graph structure and by the weight function. The minimization solution leads to a family of simple linear and nonlinear processing methods. In particular, this family includes the exact expression or the discrete version of several neighborhood filters, such as the bilateral and the nonlocal means filter. In the context of images, local and nonlocal regularizations, based on the total variation models, are the continuous analog of the proposed model. Indirectly and naturally, it provides a discrete extension of these regularization methods for any discrete data or functions.

Keywords

Discrete variational problems on graphs Discrete diffusion processes Smoothing Denoising Simplification 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alvarez, L., Guichard, F., Lions, P.-L., & Morel, J.-M. (1993). Axioms and fundamental equations of image processing. Archive for Rational Mechanics and Analysis, 123(3), 199–257. zbMATHCrossRefMathSciNetGoogle Scholar
  2. Arbeláez, P. A., & Cohen, L. D. (2004). Energy partitions and image segmentation. Journal of Mathematical Imaging and Vision, 20(1–2), 43–57. CrossRefMathSciNetGoogle Scholar
  3. Aubert, G., & Kornprobst, P. (2006). Applied mathematical sciences : Vol. 147. Mathematical problems in image processing, partial differential equations and the calculus of variations (2nd ed.). Berlin: Springer. zbMATHGoogle Scholar
  4. Bajaj, C. L., & Xu, G. (2003). Anisotropic diffusion of surfaces and functions on surfaces. ACM Transactions on Graphics, 22(1), 4–32. CrossRefGoogle Scholar
  5. Barash, D. (2002). A fundamental relationship between bilateral filtering, adaptive smoothing, and the nonlinear diffusion equation. IEEE Transactions Pattern Analysis and Machine Intelligence, 24(6), 844–847. CrossRefGoogle Scholar
  6. Bensoussan, A., & Menaldi, J.-L. (2005). Difference equations on weighted graphs. Journal of Convex Analysis, 12(1), 13–44. zbMATHMathSciNetGoogle Scholar
  7. Bobenko, A. I., & Schröder, P. (2005). Discrete Willmore flow. In M. Desbrun & H. Pottmann (Eds.), Eurographics symposium on geometry processing (pp. 101–110). Google Scholar
  8. Bougleux, S., & Elmoataz, A. (2005). Image smoothing and segmentation by graph regularization. In LNCS : Vol. 3656. Proc. int. symp. on visual computing (ISVC) (pp. 745–752). Berlin: Springer. Google Scholar
  9. Bougleux, S., Elmoataz, A., & Melkemi, M. (2007a). Discrete regularization on weighted graphs for image and mesh filtering. In F. Sgallari, A. Murli, & N. Paragios (Eds.), LNCS : Vol. 4485. Proc. of the 1st int. conf. on scale space and variational methods in computer vision (SSVM) (pp. 128–139). Berlin: Springer. CrossRefGoogle Scholar
  10. Bougleux, S., Melkemi, M., & Elmoataz, A. (2007b). Local Beta-Crusts for simple curves reconstruction. In C. M. Gold (Ed.), 4th international symposium on Voronoi diagrams in science and engineering (ISVD’07) (pp. 48–57). IEEE Computer Society. Google Scholar
  11. Brox, T., & Cremers, D. (2007). Iterated nonlocal means for texture restoration. In F. Sgallari, A. Murli, & N. Paragios (Eds.), LNCS : Vol. 4485. Proc. of the 1st int. conf. on scale space and variational methods in computer vision (SSVM) (pp. 12–24). Berlin: Springer. Google Scholar
  12. Buades, A., Coll, B., & Morel, J.-M. (2005). A review of image denoising algorithms, with a new one. Multiscale Modeling and Simulation, 4(2), 490–530. zbMATHCrossRefMathSciNetGoogle Scholar
  13. Chambolle, A. (2005). Total variation minimization and a class of binary MRF models. In LNCS : Vol. 3757. Proc. of the 5th int. work. EMMCVPR (pp. 136–152). Berlin: Springer. Google Scholar
  14. Chan, T., & Shen, J. (2005). Image processing and analysis—variational, PDE, wavelets, and stochastic methods. SIAM. Google Scholar
  15. Chan, T., Osher, S., & Shen, J. (2001). The digital TV filter and nonlinear denoising. IEEE Transactions on Image Processing, 10(2), 231–241. zbMATHCrossRefGoogle Scholar
  16. Chung, F. (1997). Spectral graph theory. CBMS Regional Conference Series in Mathematics, 92, 1–212. Google Scholar
  17. Clarenz, U., Diewald, U., & Rumpf, M. (2000). Anisotropic geometric diffusion in surface processing. In VIS’00: Proc. of the conf. on visualization (pp. 397–405). IEEE Computer Society Press. Google Scholar
  18. Coifman, R., Lafon, S., Lee, A., Maggioni, M., Nadler, B., Warner, F., & Zucker, S. (2005). Geometric diffusions as a tool for harmonic analysis and structure definition of data. Proc. of the National Academy of Sciences, 102(21). Google Scholar
  19. Coifman, R., Lafon, S., Maggioni, M., Keller, Y., Szlam, A., Warner, F., & Zucker, S. (2006). Geometries of sensor outputs, inference, and information processing. In Proc. of the SPIE: intelligent integrated microsystems (Vol. 6232). Google Scholar
  20. Cvetković, D. M., Doob, M., & Sachs, H. (1980). Spectra of graphs, theory and application. Pure and applied mathematics. San Diego: Academic Press. Google Scholar
  21. Darbon, J., & Sigelle, M. (2004). Exact optimization of discrete constrained total variation minimization problems. In R. Klette & J. D. Zunic (Eds.), LNCS : Vol. 3322. Proc. of the 10th int. workshop on combinatorial image analysis (pp. 548–557). Berlin: Springer. Google Scholar
  22. Desbrun, M., Meyer, M., Schröder, P., & Barr, A. (2000). Anisotropic feature-preserving denoising of height fields and bivariate data. Graphics Interface, 145–152. Google Scholar
  23. Fleishman, S., Drori, I., & Cohen-Or, D. (2003). Bilateral mesh denoising. ACM Transactions on Graphics, 22(3), 950–953. CrossRefGoogle Scholar
  24. Gilboa, G., & Osher, S. (2007a). Nonlocal linear image regularization and supervised segmentation. SIAM Multiscale Modeling and Simulation, 6(2), 595–630. zbMATHCrossRefMathSciNetGoogle Scholar
  25. Gilboa, G., & Osher, S. (2007b). Nonlocal operators with applications to image processing (Technical Report 07-23). UCLA, Los Angeles, USA. Google Scholar
  26. Griffin, L. D. (2000). Mean, median and mode filtering of images. Proceedings: Mathematical, Physical and Engineering Sciences, 456(2004), 2995–3004. zbMATHCrossRefMathSciNetGoogle Scholar
  27. Hein, M., Audibert, J.-Y., & von Luxburg, U. (2007). Graph Laplacians and their convergence on random neighborhood graphs. Journal of Machine Learning Research, 8, 1325–1368. MathSciNetGoogle Scholar
  28. Hildebrandt, K., & Polthier, K. (2004). Anisotropic filtering of non-linear surface features. Eurographics 2004: Comput. Graph. Forum, 23(3), 391–400. CrossRefGoogle Scholar
  29. Jones, T. R., Durand, F., & Desbrun, M. (2003). Non-iterative, feature-preserving mesh smoothing. ACM Transactions on Graphics, 22(3), 943–949. CrossRefGoogle Scholar
  30. Kervrann, C., Boulanger, J., & Coupé, P. (2007). Bayesian non-local means filter, image redundancy and adaptive dictionaries for noise removal. In F. Sgallari, A. Murli, & N. Paragios (Eds.), LNCS : Vol. 4485. Proc. of the 1st int. conf. on scale space and variational methods in computer vision (SSVM) (pp. 520–532). Berlin: Springer. CrossRefGoogle Scholar
  31. Kimmel, R., Malladi, R., & Sochen, N. (2000). Images as embedding maps and minimal surfaces: Movies, color, texture, and volumetric medical images. International Journal of Computer Vision, 39(2), 111–129. zbMATHCrossRefGoogle Scholar
  32. Kinderman, S., Osher, S., & Jones, S. (2005). Deblurring and denoising of images by nonlocal functionals. SIAM Multiscale Modeling and Simulation, 4(4), 1091–1115. CrossRefGoogle Scholar
  33. Lee, J. S. (1983). Digital image smoothing and the sigma filter. Computer Vision, Graphics, and Image Processing, 24(2), 255–269. CrossRefGoogle Scholar
  34. Lezoray, O., Elmoataz, A., & Bougleux, S. (2007). Graph regularization for color image processing. Computer Vision and Image Understanding, 107(1–2), 38–55. CrossRefGoogle Scholar
  35. Lopez-Perez, L., Deriche, R., & Sochen, N. (2004). The Beltrami flow over triangulated manifolds. In LNCS : Vol. 3117. ECCV 2004 workshops CVAMIA and MMBIA (pp. 135–144). Berlin: Springer. Google Scholar
  36. Meila, M., & Shi, J. (2000). Learning segmentation by random walks. Advances in Neural Information Processing Systems, 13, 873–879. Google Scholar
  37. Meyer, F. (2001). An overview of morphological segmentation. International Journal of Pattern Recognition and Artificial Intelligence, 15(7), 1089–1118. CrossRefGoogle Scholar
  38. Mrázek, P., Weickert, J., & Bruhn, A. (2006). On robust estimation and smoothing with spatial and tonal kernels. In Computational imaging and vision : Vol. 31. Geometric properties from incomplete data (pp. 335–352). Berlin: Springer. CrossRefGoogle Scholar
  39. Osher, S., & Shen, J. (2000). Digitized PDE method for data restoration. In E. G. A. Anastassiou (Ed.), In analytical-computational methods in applied mathematics (pp. 751–771). London: Chapman & Hall/CRC. Google Scholar
  40. Paragios, N., Chen, Y., & Faugeras, O. (Eds.) (2005). Handbook of mathematical models in computer vision. Berlin: Springer. zbMATHGoogle Scholar
  41. Paris, S., Kornprobst, P., Tumblin, J., & Durand, F. (2007). A gentle introduction to bilateral filtering and its applications. In SIGGRAPH ’07: ACM SIGGRAPH 2007 courses. Washington: ACM. Google Scholar
  42. Peyré, G. (2008). Image processing with non-local spectral bases. SIAM Multiscale Modeling and Simulation, 7(2), 731–773. CrossRefGoogle Scholar
  43. Requardt, M. (1997). A new approach to functional analysis on graphs, the connes-spectral triple and its distance function. Google Scholar
  44. Requardt, M. (1998). Cellular networks as models for Planck-scale physics. Journal of Physics A: Mathematical and General, 31, 7997–8021. zbMATHCrossRefMathSciNetGoogle Scholar
  45. Rudin, L. I., Osher, S., & Fatemi, E. (1992). Nonlinear total variation based noise removal algorithms. Physica D, 60(1–4), 259–268. zbMATHCrossRefGoogle Scholar
  46. Shi, J., & Malik, J. (2000). Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(8), 888–905. CrossRefGoogle Scholar
  47. Smith, S., & Brady, J. (1997). SUSAN—a new approach to low level image processing. International Journal of Computer Vision (IJCV), 23, 45–78. CrossRefGoogle Scholar
  48. Sochen, N., Kimmel, R., & Bruckstein, A. M. (2001). Diffusions and confusions in signal and image processing. Journal of Mathematical Imaging and Vision, 14(3), 195–209. zbMATHCrossRefMathSciNetGoogle Scholar
  49. Szlam, A. D., Maggioni, M., & Coifman, R. R. (2006). A general framework for adaptive regularization based on diffusion processes on graphs (Technical Report YALE/DCS/TR1365). YALE. Google Scholar
  50. Tasdizen, T., Whitaker, R., Burchard, P., & Osher, S. (2003). Geometric surface processing via normal maps. ACM Transactions on Graphics, 22(4), 1012–1033. CrossRefGoogle Scholar
  51. Taubin, G. (1995). A signal processing approach to fair surface design. In SIGGRAPH’95: Proc. of the 22nd annual conference on computer graphics and interactive techniques (pp. 351–358). New York: ACM Press. CrossRefGoogle Scholar
  52. Tomasi, C., & Manduchi, R. (1998). Bilateral filtering for gray and color images. In ICCV’98: Proc. of the 6th int. conf. on computer vision (pp. 839–846). IEEE Computer Society. Google Scholar
  53. Weickert, J. (1998). Anisotropic diffusion in image processing. Leipzig: Teubner. zbMATHGoogle Scholar
  54. Xu, G. (2004). Discrete Laplace-Beltrami operators and their convergence. Computer Aided Geometric Design, 21, 767–784. zbMATHCrossRefMathSciNetGoogle Scholar
  55. Yagou, H., Ohtake, Y., & Belyaev, A. (2002). Mesh smoothing via mean and median filtering applied to face normals. In Proc. of the geometric modeling and processing (GMP’02)—theory and applications (pp. 195–204). IEEE Computer Society. Google Scholar
  56. Yoshizawa, S., Belyaev, A., & Seidel, H.-P. (2006). Smoothing by example: mesh denoising by averaging with similarity-based weights. In Proc. international conference on shape modeling and applications (pp. 38–44). Google Scholar
  57. Zhou, D., & Schölkopf, B. (2005). Regularization on discrete spaces. In LNCS : Vol. 3663. Proc. of the 27th DAGM symp. (pp. 361–368). Berlin: Springer. Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Sébastien Bougleux
    • 1
    Email author
  • Abderrahim Elmoataz
    • 2
  • Mahmoud Melkemi
    • 3
  1. 1.GREYC CNRS UMR 6072, Équipe ImageENSICAENCaen CedexFrance
  2. 2.GREYC CNRS UMR 6072, Équipe ImageUniversité de CaenCaen CedexFrance
  3. 3.LMIA, Équipe MAGEUniversité de Haute-AlsaceMulhouse CedexFrance

Personalised recommendations