A total fractional-order variation model for image restoration with nonhomogeneous boundary conditions and its numerical solution. (English) Zbl 1327.62388
Summary: To overcome the weakness of a total variation based model for image restoration, various high order (typically second order) regularization models have been proposed and studied recently. In this paper we analyze and test a fractional-order derivative based total \(\alpha\)-order variation model which can outperform the currently popular high order regularization models. There exist several previous works using total \(\alpha\)-order variations for image restoration; however, first, no analysis has been done yet, and second, all tested formulations, differing from each other, utilize the zero Dirichlet boundary conditions which are not realistic (while nonzero boundary conditions violate definitions of fractional-order derivatives). This paper first reviews some results of fractional-order derivatives and then analyzes the theoretical properties of the proposed total \(\alpha\)-order variational model rigorously. It then develops four algorithms for solving the variational problem – one based on the variational Split-Bregman idea and three based on direct solution of the discretize-optimization problem. Numerical experiments show that, in terms of restoration quality and solution efficiency, the proposed model can produce highly competitive results, for smooth images, to two established high order models: the mean curvature and the total generalized variation.
MSC:
| 62H35 | Image analysis in multivariate analysis |
| 65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |
| 65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |
| 74G65 | Energy minimization in equilibrium problems in solid mechanics |
| 74G75 | Inverse problems in equilibrium solid mechanics |
Keywords:
fractional-order derivatives; total \(\alpha\)-order variation; PDE; image denoising; image inverse problems; optimization methodsSoftware:
ma2dfcReferences:
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