Approximation of weighted local mean operators

Abstract

Recently, weighted local mean operators are widely used in image processing, compressive sensing and other areas. A weighted local mean operator changes its characteristics depending on a function content within a local area in order to preserve the function features. The directional diffusion filter and Yaroslavsky neighbourhood filter (also called the sigma filter) are discrete versions of such operators. Although these operators are not convolution ones, due to their sparsity, the corresponding numerical algorithms have simple structure and fast performance. In this paper, we study the approximate properties of the weighted local mean operators, particularly focus on their asymptotic expansions, which are related to non-linear diffusion equations.

Keywords:

• weighted local mean
• anisotropic diffusion
• approximation
• asymptotic expansion
• neighbourhood filters

AMS Subject Classifications:

• 41A35
• 41A60
• 47N20
• 94A08

Acknowledgments

The author would like to thank the reviewers of this paper for their valuable comments.

Notes

The research was supported by SHSU-ERG-290029.