Approximations for modulus of gradients and their applications to neighborhood filters

Abstract

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.

Keywords

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

References

[1] Ambrosio L, Fusco N, Pallara D. Function of Bounded Variation and Free Discontinuity Problems, 2000, Oxford: Oxford Univ Press

[2] Aubert G, Kornprobst P. Mathematical Problems in Image Processing, 2002, Berlin: Springer

[3] Barash D. Fundamental relationship between bilateral filtering, adaptive smoothing, and the nonlinear diffusion equation. IEEE Trans Pattern Anal Mach Intell, 2002, 24: 844-847

[4] Buades A, Coll B, Lisani J L, Sbert C. Conditional image diffusion. Inverse Probl Imaging, 2007, 1: 593-608

[5] Buades A, Coll B, Morel J -M. Neighborhood filters and PDE’s. Numer Math, 2006, 105: 1-34

[6] Buades A, Coll B, Morel J -M. The staircasing effect in neighborhood filters and its solution. IEEE Trans Image Process, 2006, 15: 1499-1505

[7] Calderon A P, Zygmund A. Local properties of solutions of elliptic differential equations. Studia Math, 1961, 20: 171-225

[8] Campbell J B. Introduction to Remote Sensing, 1996, New York: The Guilford Press

[9] Canny J. A computational approach to edge detection. IEEE Trans Pattern Anal Mach Intell, 1986, 8: 679-714

[10] Chen Y, Zhang K. Young measure solutions of the two-dimensional Perona-Malik equation in image processing. Commun Pure Appl Anal, 2006, 5: 615-635

[11] Evans L C, Gariepy R F, Measure R F. Theory and Fine Properties of Functions. Studies in Advanced Mathematics, 1992, Boca Raton: CRC Press

[12] Lindenbaum M, Fischer M, Bruckstein A M. On Gabor’s contribution to image enhancement. Pattern Recognition, 1994, 27: 1-8

[13] Mumford D, Shah J. Boundary detection by minimizing functionals. Proc International Conference on Computer Vision and Pattern Recognition, San Francisco, CA, 1985, 1985 22 26

[14] Perona P, Malik J. Scale space and edge detection using anisotropic diffusion. IEEE Trans Pattern Anal Mach Intell, 1990, 12: 629-639

[15] Sapiro G. Geometric Partial Differential Equations and Image Analysis, 2001, Cambridge: Cambridge University Press

[16] Smith S M, Brady J M. SUSAN—A new approach to low level image processing. Int J Comput Vis, 1997, 23: 45-78

[17] Stein E M. Singular integrals and differentiability properties of functions. Princeton Mathematical Series, 30, 1970, Princeton: Princeton University Press

[18] Stein E M. Harmonic Analysis. Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III, 1993, Princeton: Princeton University Press

[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: 467-490

[20] Tomasi C, Manduchi R. Bilateral filtering for gray and color images. In: Sixth International Conference on Computer Vision. 1998, 839-846

[21] Wang J. Construction of local nonlinear filter without staircase effect in image restoration. Appl Anal, 2011, 90: 1257-1273

[22] Weickert J. Anisotropic Diffusion in Image Processing, 1998, Stuttgart: Teubner

[23] Yaroslavsky L P. Digital Picture Processing—An Introduction, 1985, Berlin: Springer-Verlag

[24] Zhang K. On the coercivity of elliptic systems in two dimensional spaces. Bull Aust Math Soc, 1996, 54: 423-430

[25] Zhang K. Existence of infinitely many solutions for the one-dimensional Perona-Malik model. Calc Var Partial Differential Equations, 2006, 26: 171-199

[26] Ziou D, Tabbone S. Edge detection techniques—An overview. Int J Pattern Recognition Image Anal, 1998, 8: 537-559