High Order Symmetrised Fractional Variation for Signal and Image Analysis

We introduce and study a variational model for signal and image denoising based on Riemann-Liouville fractional derivatives of every positive order higher than zero. Both the one-dimensional and two-dimensional cases are studied. The model exploits an $L^1$ fitting data term together with both right and left Riemann-Liouville fractional derivatives as regularizing terms, with the aim of achieving an orientation independent analysis. To provide evidence of effectiveness for the proposed model we introduce a discretisation based on a second-order consistent Grünwald Letnikov scheme and show some numerical simulations aiming to denoise images corrupted by impulsive noise, which can be well modeled by the Laplace distribution.