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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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Fractional-Order Euler-Lagrange Equation for Fractional-Order Variational Method: A Necessary Condition for Fractional-Order Fixed Boundary Optimization Problems in Signal Processing and Image Processing