Image structure preserving denoising using generalized fractional time integrals

A generalization of the linear fractional integral equation u(t)=u0+∂−αAu(t)u(t)=u0+∂−αAu(t), 1<α<21<α<2, which is written as a Volterra matrix-valued equation when applied as a pixel-by-pixel technique is proposed in this paper for image denoising (restoration, smoothing, etc.). Since the fractional integral equation interpolates a linear parabolic equation and a hyperbolic equation, the solution enjoys intermediate properties. The Volterra equation we propose is well-posed for all t>0t>0, and allows us to handle the diffusion by means of a viscosity parameter instead of introducing nonlinearities in the equation as in the Perona–Malik and alike approaches. Several experiments showing the improvements achieved by our approach are provided.

► A parabolic type PDE based model is considered, now with fractional time derivatives.
► The order derivative αα allows to control the diffusion avoiding nonlinearities.
► A pixel-by-pixel application of this idea leads to a linear Volterra type equation.
► The model fits into a closed mathematical framework, i.e., of well posed problems.
► The numerical part is closed as well, i.e., this has been closely studied.