Advertisement

Filtering on Graphs

Chapter
  • 2.8k Downloads

Abstract

Measured data often includes noise. A data point measured in isolation offers little opportunity to tease signal apart from noise. However, this separation of noise from the signal becomes more possible when multiple data points are acquired which have a relationship with each other. A spatial relationship, such as the edge set of a graph, permits the use of the collective data acquisition to make better decisions about the true data underlying each measurement. This process whereby the spatial relationships of the data are used to provide better estimates of the noiseless data is called a filtering or a denoising process. In this chapter, we outline the assumptions used to justify spatial filtering, describe the equivalent of Fourier analysis on a general graph and discuss how different parameter settings of a small number of variational approaches to filtering lead to a large number of commonly used filters. Although our focus in this chapter is on the filtering of node data (0-cochains), we also discuss how these techniques may be applied to the filtering of edge data (i.e., flows, or 1-cochains) and to the filtering of data associated with higher-dimensional cells.

Keywords

Discrete Fourier Transform Edge Weight Laplacian Matrix Circulant Matrix Smoothness Term 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 2.
    Agrawal, A., Raskar, R., Chellappa, R.: What is the range of surface reconstructions from a gradient field? In: Proc. of ECCV. Lecture Notes in Computer Science, vol. 3954, pp. 578–591. Springer, Berlin (2006) Google Scholar
  2. 11.
    Appleton, B., Talbot, H.: Globally optimal surfaces by continuous maximal flows. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(1), 106–118 (2006) CrossRefGoogle Scholar
  3. 15.
    Awate, S.P., Whitaker, R.T.: Unsupervised, information-theoretic, adaptive image filtering for image restoration. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(3), 364–376 (2006) CrossRefGoogle Scholar
  4. 38.
    Black, M.J., Sapiro, G., Marimont, D.H., Heeger, D.: Robust anisotropic diffusion. IEEE Transactions on Image Processing 7(3), 421–432 (1998) CrossRefGoogle Scholar
  5. 39.
    Blake, A., Zisserman, A.: Visual Reconstruction. MIT Press, Cambridge (1987) Google Scholar
  6. 49.
    Bougleux, S., Elmoataz, A., Melkemi, M.: Discrete regularization on weighted graphs for image and mesh filtering. In: Proc. of SSVM. Lecture Notes in Computer Science, vol. 4485, pp. 128–139. Springer, Berlin (2007) Google Scholar
  7. 50.
    Bouman, C., Sauer, K.: A generalized Gaussian image model for edge-preserving MAP estimation. IEEE Transactions on Image Processing 2(3), 296–310 (1993) CrossRefGoogle Scholar
  8. 54.
    Boykov, Y., Kolmogorov, V.: Computing geodesics and minimal surfaces via graph cuts. In: Proceedings of International Conference on Computer Vision, vol. 1 (2003) Google Scholar
  9. 56.
    Boykov, Y., Veksler, O., Zabih, R.: Fast approximate energy minimization via graph cuts. IEEE Transactions on Pattern Analysis and Machine Intelligence 23(11), 1222–1239 (2001) CrossRefGoogle Scholar
  10. 63.
    Buades, A., Coll, B., Morel, J.M.: A review of image denoising algorithms, with a new one. Multiscale Modeling and Simulation 4(2), 490–530 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 68.
    Chambolle, A.: An algorithm for total variation minimization and applications. Journal of Mathematical Imaging and Vision 20(1–2), 89–97 (2004) MathSciNetGoogle Scholar
  12. 69.
    Chambolle, A., Lions, P.L.: Image recovery via total variation minimization and related problems. Numerische Mathematik 76(2), 167–188 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 81.
    Chung, F.R.K.: Spectral Graph Theory. Regional Conference Series in Mathematics, vol. 92. Am. Math. Soc., Providence (1997) zbMATHGoogle Scholar
  14. 82.
    Cliff, A.D., Ord, J.K.: Spatial Processes: Models and Applications. Pion, London (1981) zbMATHGoogle Scholar
  15. 88.
    Cosgriff, R.L.: Identification of shape. Technical Report 820-11 ASTIA AD 254 792, Ohio State Univ. Res. Foundation (1960) Google Scholar
  16. 96.
    Dale, A.M., Fischl, B., Sereno, M.I.: Cortical surface-based analysis. I. Segmentation and surface reconstruction. NeuroImage 9(2), 179–194 (1999) CrossRefGoogle Scholar
  17. 98.
    Darbon, J., Sigelle, M.: Image restoration with discrete constrained total variation part I: Fast and exact optimization. Journal of Mathematical Imaging and Vision 26(3), 261–276 (2006) MathSciNetCrossRefGoogle Scholar
  18. 104.
    Desbrun, M., Meyer, M., Schröder, P., Barr, A.H.: Implicit fairing of irregular meshes using diffusion and curvature flow. In: Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques, pp. 317–324. ACM Press/Addison-Wesley, New York (1999) Google Scholar
  19. 116.
    DuBois, E.: The sampling and reconstruction of time varying imagery with application in video systems. Proceedings of the IEEE 73(4), 502–522 (1985) CrossRefGoogle Scholar
  20. 123.
    Elmoataz, A., Lézoray, O., Bougleux, S.: Nonlocal discrete regularization on weighted graphs: A framework for image and manifold processing. IEEE Transactions on Image Processing 17(7), 1047–1060 (2008) MathSciNetCrossRefGoogle Scholar
  21. 131.
    Fattal, R., Lischinski, D., Werman, M.: Gradient domain high dynamic range compression. In: Proc. of SIGGRAPH (2002) Google Scholar
  22. 136.
    Fischl, B., Sereno, M.I., Dale, A.M.: Cortical surface-based analysis. II: Inflation, flattening, and a surface-based coordinate system. NeuroImage 9(2), 195–207 (1999) CrossRefGoogle Scholar
  23. 144.
    Geman, D., Reynolds, G.: Constrained restoration and the discovery of discontinuities. IEEE Transactions on Pattern Analysis and Machine Intelligence 14(3), 367–383 (1992) CrossRefGoogle Scholar
  24. 145.
    Geman, S., Geman, D.: Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence 6(6), 721–741 (1984) zbMATHCrossRefGoogle Scholar
  25. 146.
    Geman, S., McClure, D.: Statistical methods for tomographic image reconstruction. In: Proc. 46th Sess. Int. Stat. Inst. Bulletin ISI, vol. 52, pp. 4–21 (1987) Google Scholar
  26. 158.
    Grady, L.: Space-variant computer vision: A graph-theoretic approach. Ph.D. thesis, Boston University, Boston, MA (2004) Google Scholar
  27. 163.
    Grady, L., Alvino, C.: The piecewise smooth Mumford-Shah functional on an arbitrary graph. IEEE Transactions on Image Processing 18(11), 2547–2561 (2009) MathSciNetCrossRefGoogle Scholar
  28. 166.
    Grady, L., Schwartz, E.L.: The Graph Analysis Toolbox: Image processing on arbitrary graphs. Technical Report TR-03-021, Boston University, Boston, MA (2003) Google Scholar
  29. 172.
    Gray, R.M.: Toeplitz and Circulant Matrices: A Review. Now Publishers, Hanover (2006) zbMATHGoogle Scholar
  30. 209.
    Hughes, A.: The topography of vision in mammals of contrasting life style: Comparative optics and retinal organization. In: Crescitelli, F. (ed.) The Visual System in Vertebrates. The Handbook of Sensory Physiology, vol. 7, pp. 613–756. Springer, Berlin (1977). Chap. 11 CrossRefGoogle Scholar
  31. 256.
    Lestrel, P.E. (ed.): Fourier Descriptors and Their Applications in Biology. Cambridge University Press, Cambridge (1997) zbMATHGoogle Scholar
  32. 277.
    Mead, C.: Analog VLSI and Neural Systems. Addison-Wesley, Reading (1989) zbMATHCrossRefGoogle Scholar
  33. 289.
    Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Communications on Pure and Applied Mathematics 42, 577–685 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  34. 297.
    Oppenheim, A.V., Schafer, R.W.: Discrete-Time Signal Processing. Prentice-Hall, New York (1989) zbMATHGoogle Scholar
  35. 300.
    Osher, S., Shen, J.: Digitized PDE method for data restoration. In: Anastassiou, G.A. (ed.) Handbook of Analytic Computational Methods in Applied Mathematics, pp. 751–771. CRC Press, Boca Raton (2000). Chap. 16 Google Scholar
  36. 306.
    Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence 12(7), 629–639 (1990) CrossRefGoogle Scholar
  37. 312.
    Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes: The Art of Scientific Computing, 3rd edn. Cambridge University Press, Cambridge (2007) Google Scholar
  38. 317.
    Ripley, B.D.: Spatial Statistics. Wiley-Interscience, New York (2004) Google Scholar
  39. 321.
    Rojer, A.S., Schwartz, E.L.: Design considerations for a space-variant visual sensor with complex-logarithmic geometry. In: Proc. ICPR, vol. 2, pp. 278–285. IEEE Comput. Soc., Los Alamitos (1990) Google Scholar
  40. 326.
    Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60(1–4), 259–268 (1992) zbMATHCrossRefGoogle Scholar
  41. 328.
    Sandini, G., Questa, P., Scheffer, D., Mannucci, A.: A retina-like CMOS sensor and its applications. In: IEEE Sensor Array and Multichannel Signal Processing Workshop, IEEE Comput. Soc., Cambridge (2000) Google Scholar
  42. 329.
    Scannell, J.W., Burns, G.A.P.C., Hilgetag, C.C., O’Neil, M.A., Young, M.P.: The connectional organization of the cortico-thalamic system of the cat. Cerebral Cortex 9, 277–299 (1999) CrossRefGoogle Scholar
  43. 336.
    Schwartz, E.L.: Computational anatomy and functional architecture of striate cortex: a spatial mapping approach to perceptual coding. Vision Research 20(8), 645–669 (1980) CrossRefGoogle Scholar
  44. 342.
    Shen, J.: The Mumford–Shah digital filter pair (MS-DFP) and applications. In: Proc. of ICIP, vol. 2, pp. 849–852 (2002) Google Scholar
  45. 354.
    Sporns, O., Kotter, R.: Motifs in brain networks. PLoS Biology 2, 1910–1918 (2004) CrossRefGoogle Scholar
  46. 371.
    Taubin, G.: A signal processing approach to fair surface design. In: Cook, R. (ed.) Computer Graphics Proceedings. Special Interest Group in Computer Graphics (SIGGRAPH) 95, pp. 351–358. ACM, Los Angeles (1995) Google Scholar
  47. 372.
    Taubin, G., Zhang, T., Golub, G.: Optimal surface smoothing as filter design. In: Proc. of ECCV 1996, pp. 283–292 (1996) Google Scholar
  48. 377.
    Tobler, W.R.: A computer movie simulating urban growth in the Detroit region. Economic Geography 46, 234–240 (1970) CrossRefGoogle Scholar
  49. 384.
    Tsai, A., Yezzi, A., Willsky, A.: Curve evolution implementation of the Mumford–Shah functional for image segmentation, denoising, interpolation, and magnification. IEEE Transactions on Image Processing 10(8), 1169–1186 (2001) zbMATHCrossRefGoogle Scholar
  50. 386.
    Unger, M., Pock, T., Trobin, W., Cremers, D., Bischof, H.: TVSeg—Interactive total variation based image segmentation. In: Proc. of British Machine Vision Conference (2008) Google Scholar
  51. 389.
    Vese, L., Chan, T.: A multiphase level set framework for image segmentation using the Mumford and Shah model. International Journal of Computer Vision 50(3), 271–293 (2002) zbMATHCrossRefGoogle Scholar
  52. 391.
    Wallace, R., Ong, P.W., Bederson, B., Schwartz, E.: Space variant image processing. International Journal of Computer Vision 13(1), 71–90 (1994) CrossRefGoogle Scholar
  53. 393.
    Wang, H., Chen, Y., Fang, T., Tyan, J., Ahuja, N.: Gradient adaptive image restoration and enhancement. In: Proc. of Int. Conf. on Image Procession, pp. 2893–2896. IEEE Press, New York (2006) Google Scholar
  54. 406.
    Worsley, K., Friston, K.: Analysis of fMRI time-series revisited—again. NeuroImage 2(3), 173–181 (1995) CrossRefGoogle Scholar
  55. 410.
    Xu, W., Zhou, K., Yu, Y., Tan, Q., Peng, Q., Guo, B.: Gradient domain editing of deforming mesh sequences. In: Proc. of SIGGRAPH, vol. 26 (2007) Google Scholar
  56. 417.
    Zahn, C.T., Roskies, R.Z.: Fourier descriptors for plane closed curves. IEEE Transactions on Computers C-21(3), 269–281 (1972) MathSciNetCrossRefGoogle Scholar
  57. 423.
    Zhou, D., Schölkopf, B.: Regularization on discrete spaces. In: Proc. of the 27th DAGM Symp. Lecture Notes in Computer Science, vol. 3663, pp. 361–368. Springer, Berlin (2005) Google Scholar

Copyright information

© Springer-Verlag London Limited 2010

Authors and Affiliations

  1. 1.Siemens Corporate ResearchPrincetonUSA
  2. 2.Athinoula A. Martinos Center for Biomedical Imaging, Department of RadiologyMassachusetts General Hospital, Harvard Medical SchoolCharlestownUSA

Personalised recommendations