A fractional-order derivative based variational framework for image denoising

Fangfang  Dong; Yunmei Chen

doi:10.3934/ipi.2016.10.27

Article Contents

2016, Volume 10, Issue 1: 27-50. Doi: 10.3934/ipi.2016.10.27

This issue Previous Article On the choice of the Tikhonov regularization parameter and the discretization level: A discrepancy-based strategy Next Article The topological gradient method for semi-linear problems and application to edge detection and noise removal

A fractional-order derivative based variational framework for image denoising

Fangfang Dong^1, and
Yunmei Chen^2,

1.
School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018
2.
Department of Mathematics, 358 Little Hall, PO Box 118105, Gainesville, FL 32611

Received: November 30, 2013

Revised: May 31, 2015

Published: January 2016

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Abstract

In this paper, we propose a unified variational framework for noise removal, which uses a combination of different orders of fractional derivatives in the regularization term of the objective function. The principle of the combination is taking the order two or higher derivatives for smoothing the homogeneous regions, and a fractional order less than or equal to one to smooth the locations near the edges. We also introduce a novel edge detector to better detect edges and textures. A main advantage of this framework is the superiority in dealing with textures and repetitive structures as well as eliminating the staircase effect. To effectively solve the proposed model, we extend the first-order primal dual algorithm to minimize a functional involving fractional-order derivatives. A set of experiments demonstrates that the proposed method is able to avoid the staircase effect and preserve accurately edges and structural details of the image while removing the noise.

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Mathematics Subject Classification: Primary: 49M29, 65K10, 68U10; Secondary: 49K20.

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