期刊
IEEE ACCESS
卷 4, 期 -, 页码 10110-10135出版社
IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/ACCESS.2016.2636159
关键词
Fractional calculus; fractional-order Green formula; fractional-order steepest descent approach; fractional-order extreme point; fractional-order image restoration
资金
- Foundation Franco-Chinoise Pour La Science Et Ses Applications
- National Natural Science Foundation of China [61571312]
- Science and Technology Support Project of Sichuan Province of China [2013SZ0071]
- Science and Technology Support Project of Chengdu PU Chip Science and Technology Company, Ltd.
This paper discusses a novel conceptual formulation of the fractional-order Euler Lagrange equation for the fractional-order variational method, which is based on the fractional-order extremum method. In particular, the reverse incremental optimal search of the fractional-order variational method is based on the fractional-order steepest descent approach. Fractional calculus has been applied to the solution of a necessary condition for the fractional-order fixed boundary optimization problems in signal processing and image processing mainly because of its inherent strengths in terms of long-term memory, non locality, and weak singularity. At first, for the convenience of comparison, the first-order Euler Lagrange equation for the first-order variational method is derived based on the first-order Green formula. Second, the fractional-order Euler Lagrange equation for the fractional-order variational method is derived based on Wiener Khintchine theorem. Third, in order to directly and easily achieve the fractional-order variational method in the spatial domain or the time domain, the fractional-order Green formula and the fractional-order Euler Lagrange equation based on the fractional-order Green formula are derived, respectively. Fourth, the solution procedure of the fractional-order Euler Lagrange equation is derived. Finally, a fractional-order inpainting algorithm and a fractional-order denoising algorithm based on the fractional-order variational method are illustrated, respectively. The capability of restoring and maintaining the edges and textural details of the fractional-order image restoration algorithm based on the fractional-order variational method is superior to that of the integer-order image restoration algorithm based on the classical first-order variational method, especially for images rich in textural details. The fractional-order Euler Lagrange equation for the fractional-order variational method proposed by this paper is a necessary condition for the fractional-order fixed boundary optimization problems, which is a basic mathematical method in the fractional-order optimization and can be widely applied to the fractional-order field of signal analysis, signal processing, image processing, machine intelligence, automatic control, biomedical engineering, intelligent transportation, computational finance and so on.
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