To overcome the weakness of a total variation based model for imagerestoration, various high order (typically second order) regularization modelshave been proposed and studied recently. In this paper we analyze and test afractional-order derivative based total $\alpha$-order variation model, whichcan outperform the currently popular high order regularization models. Thereexist several previous works using total $\alpha$-order variations for imagerestoration; however first no analysis is done yet and second all testedformulations, differing from each other, utilize the zero Dirichlet boundaryconditions which are not realistic (while non-zero boundary conditions violatedefinitions of fractional-order derivatives). This paper first reviews someresults of fractional-order derivatives and then analyzes the theoreticalproperties of the proposed total $\alpha$-order variational model rigorously.It then develops four algorithms for solving the variational problem, one basedon the variational Split-Bregman idea and three based on direct solution of thediscretise-optimization problem. Numerical experiments show that, in terms ofrestoration quality and solution efficiency, the proposed model can producehighly competitive results, for smooth images, to two established high ordermodels: the mean curvature and the total generalized variation.
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