Frontiers of Mathematics in China

, Volume 8, Issue 4, pp 761–782 | Cite as

Approximations for modulus of gradients and their applications to neighborhood filters

  • Yan Chen
  • Zhuangji Wang
  • Kewei ZhangEmail author
Research Article


We define an integral approximation for the modulus of the gradient |∇u(x)| for functions f: Ω ⊂ ℝ n → ℝ by modifying a classical result due to Calderon and Zygmund. Our integral approximations are more stable than the pointwise defined derivatives when applied to numerical differentiation for discrete data. We apply our results to design and analyse neighborhood filters. These filters correspond to well-behaved nonlinear heat equations with the conductivity decreasing with respect to the modulus of gradient |∇u(x)|. We also show some numerical experiments and evaluate the effectiveness of our filters.


BMO modulus of gradient harmonic analysis and PDE neighborhood filter image processing 


94A08 35F20 42B37 68U10 


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  1. 1.
    Ambrosio L, Fusco N, Pallara D. Function of Bounded Variation and Free Discontinuity Problems. Oxford: Oxford Univ Press, 2000Google Scholar
  2. 2.
    Aubert G, Kornprobst P. Mathematical Problems in Image Processing. Berlin: Springer, 2002zbMATHGoogle Scholar
  3. 3.
    Barash D. Fundamental relationship between bilateral filtering, adaptive smoothing, and the nonlinear diffusion equation. IEEE Trans Pattern Anal Mach Intell, 2002, 24: 844–847CrossRefGoogle Scholar
  4. 4.
    Buades A, Coll B, Lisani J L, Sbert C. Conditional image diffusion. Inverse Probl Imaging, 2007, 1: 593–608MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Buades A, Coll B, Morel J -M. Neighborhood filters and PDE’s. Numer Math, 2006, 105: 1–34MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Buades A, Coll B, Morel J -M. The staircasing effect in neighborhood filters and its solution. IEEE Trans Image Process, 2006, 15: 1499–1505CrossRefGoogle Scholar
  7. 7.
    Calderon A P, Zygmund A. Local properties of solutions of elliptic differential equations. Studia Math, 1961, 20: 171–225MathSciNetzbMATHGoogle Scholar
  8. 8.
    Campbell J B. Introduction to Remote Sensing. New York: The Guilford Press, 1996Google Scholar
  9. 9.
    Canny J. A computational approach to edge detection. IEEE Trans Pattern Anal Mach Intell, 1986, 8: 679–714CrossRefGoogle Scholar
  10. 10.
    Chen Y, Zhang K. Young measure solutions of the two-dimensional Perona-Malik equation in image processing. Commun Pure Appl Anal, 2006, 5: 615–635MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Evans L C, R. F. Gariepy R F. Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. Boca Raton: CRC Press, 1992Google Scholar
  12. 12.
    Lindenbaum M, Fischer M, Bruckstein A M. On Gabor’s contribution to image enhancement. Pattern Recognition, 1994, 27: 1–8CrossRefGoogle Scholar
  13. 13.
    Mumford D, Shah J. Boundary detection by minimizing functionals. In: Proc International Conference on Computer Vision and Pattern Recognition, San Francisco, CA, 1985. 1985, 22–26Google Scholar
  14. 14.
    Perona P, Malik J. Scale space and edge detection using anisotropic diffusion. IEEE Trans Pattern Anal Mach Intell, 1990, 12: 629–639CrossRefGoogle Scholar
  15. 15.
    Sapiro G. Geometric Partial Differential Equations and Image Analysis. Cambridge: Cambridge University Press, 2001zbMATHCrossRefGoogle Scholar
  16. 16.
    Smith S M, Brady J M. SUSAN—A new approach to low level image processing. Int J Comput Vis, 1997, 23: 45–78CrossRefGoogle Scholar
  17. 17.
    Stein E M. Singular integrals and differentiability properties of functions. Princeton Mathematical Series, 30. Princeton: Princeton University Press, 1970Google Scholar
  18. 18.
    Stein E M. Harmonic Analysis. Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton: Princeton University Press, 1993Google Scholar
  19. 19.
    Taheri S, Tang Q, Zhang K. Young measure solutions and instability of the one-dimensional Perona-Malik equation. J Math Anal Appl, 2005, 308: 467–490MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Tomasi C, Manduchi R. Bilateral filtering for gray and color images. In: Sixth International Conference on Computer Vision. 1998, 839-846Google Scholar
  21. 21.
    Wang J. Construction of local nonlinear filter without staircase effect in image restoration. Appl Anal, 2011, 90: 1257–1273MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Weickert J. Anisotropic Diffusion in Image Processing. ECMI Series. Stuttgart: Teubner, 1998zbMATHGoogle Scholar
  23. 23.
    Yaroslavsky L P. Digital Picture Processing—An Introduction. Berlin: Springer-Verlag, 1985zbMATHCrossRefGoogle Scholar
  24. 24.
    Zhang K. On the coercivity of elliptic systems in two dimensional spaces. Bull Aust Math Soc, 1996, 54: 423–430zbMATHCrossRefGoogle Scholar
  25. 25.
    Zhang K. Existence of infinitely many solutions for the one-dimensional Perona-Malik model. Calc Var Partial Differential Equations, 2006, 26: 171–199MathSciNetCrossRefGoogle Scholar
  26. 26.
    Ziou D, Tabbone S. Edge detection techniques—An overview. Int J Pattern Recognition Image Anal, 1998, 8: 537–559Google Scholar

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© Higher Education Press and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.College of Resource and EnvironmentChina Agricultural UniversityBeijingChina
  2. 2.Department of AgronomyIowa State UniversityAmesUSA
  3. 3.School of Mathematical SciencesUniversity of NottinghamNottinghamUK

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