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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

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 

MSC

94A08 35F20 42B37 68U10 

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Copyright information

© 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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