Correspondence between frame shrinkage and high-order nonlinear diffusion

Authors:
Qingtang Jiang
Affiliations:
Department of Mathematics and Computer Science, University of Missouri-St. Louis, St. Louis, MO 63121, USA
Venue:
Applied Numerical Mathematics
Year:
2012

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Abstract

Nonlinear diffusion filtering and wavelet/frame shrinkage are two popular methods for signal and image denoising. The relationship between these two methods has been studied recently. In this paper we investigate the correspondence between frame shrinkage and nonlinear diffusion. We show that the frame shrinkage of Ron-Shen@?s continuous-linear-spline-based tight frame is associated with a fourth-order nonlinear diffusion equation. We derive high-order nonlinear diffusion equations associated with general tight frame shrinkages. These high-order nonlinear diffusion equations are different from the high-order diffusion equations studied in the literature. We also construct two sets of tight frame filter banks which result in the sixth- and eighth-order nonlinear diffusion equations. The correspondence between frame shrinkage and diffusion filtering is useful to design diffusion-inspired shrinkage functions with competitive performance. On the other hand, the study of such a correspondence leads to a new type of diffusion equations and helps to design frame-inspired diffusivity functions. The denoising results with diffusion-inspired shrinkages provided in this paper are promising.