Smoothing techniques and difference of convex functions algorithms for image reconstructions

Abstract

In this paper, we study characterizations of differentiability for real-valued functions based on generalized differentiation. These characterizations provide the mathematical foundation for Nesterov's smoothing techniques in infinite dimensions. As an application, we provide a simple approach to image reconstructions based on Nesterov's smoothing and algorithms for minimizing differences of convex (DC) functions that involve the ${\ell }_1-{\ell }_2$ regularization.

Keywords:

• Generalized differentiation
• Nesterov's smoothing techniques
• DC algorithm
• image reconstruction

AMS subject classifications:

• 49J52
• 49J53
• 90C31

Acknowledgements

Part of this work was done during the first author's visit to the Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thank the VIASM and Prof. Nguyen Dong Yen for the hospitality and support.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Available at https://www2.eecs.berkeley.edu/Research/Projects/CS/vision/bsds/

Additional information

Funding

Research of Nguyen Mau Nam was partly supported by the National Science Foundation (Division of Mathematical Sciences) under grant DMS-1716057. Research of Nguyen Thai An was supported by the China Postdoctoral Science Foundation under grant No. 2017M622991.