Dual algorithm for truncated fractional variation based image denoising

Abstract

Fractional-order derivative is attracting more and more attention of researchers in image processing because of its better property in restoring more texture than the total variation. To improve the performance of fractional-order variation model in image restoration, a truncated fractional-order variation model was proposed in Chan and Liang [Truncated fractional-order variation model for image restoration, J. Oper. Res. Soc. China]. In this paper, we propose a dual approach to solve this truncated fractional-order variation model on noise removal. The proposed algorithm is based on the dual approach proposed by Chambolle [An algorithm for total variation minimisation and applications, J. Math Imaging Vis. 20 (2004), pp. 89–97]. Conversely, the Chambolle's dual approach can be treated as a special case of the proposed algorithm with fractional order $\alpha =1$. The work of this paper modifies the result in Zhang et al. [Adaptive fractional-order multi-scale method for image denoising, J. Math. Imaging Vis. 43(1) (2012), pp. 39–49. Springer Netherlands 0924–9907, Computer Science, pp. 1–11, 2011], where the convergence is not analysed. Based on the truncation, the convergence of the proposed dual method can be analysed and the convergence criteria can be provided. In addition, the accuracy of the reconstruction is improved after the truncation is taken.

Keywords:

• Truncated fractional-order derivative
• truncated fractional-order variation model
• dual algorithm
• image denoising
• texture preserving

2010 Mathematics Subject Classification:

• 65K10

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research is supported partly by Jiangsu National Natural Science Foundation of China (Grant No. BK20150373), partly by Jiangsu National Natural Science Foundation of China (Grant No. BK20171237), partly by XJTLU Research Enhancement Fund (Grant Nos. REF-17-01-08 and REF-18-01-04), partly by the Key Programme Special Fund in XJTLU (Grant Nos. KSF-E-32 and KSF-E-21). The research is supported by National Natural Science Foundation of China (Grant No. 11801362).