Journal of Convex Analysis 33 (2026), No. 3&4, 925--952
Copyright Heldermann Verlag 2026
High Order Symmetrised Fractional Variation for Signal and Image Analysis
Alessandro Lanza
Department of Mathematics, University of Bologna, Bologna, Italy
alessandro.lanza2@unibo.it
Antonio Leaci
Dip. di Matematica e Fisica "Ennio De Giorgi", Università del Salento, Lecce, Italy
antonio.leaci@unisalento.it
Serena Morigi
Department of Mathematics, University of Bologna, Bologna, Italy
serena.morigi@unibo.it
Franco Tomarelli
Dipartimento di Matematica, Politecnico di Milano, Milano, Italy
franco.tomarelli@polimi.it
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 L1 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.
Keywords: Symmetrized fractional variation, Riemann-Liouville fractional derivatives, Grünwald-Letnikov formulas, discretization of fractional derivatives, calculus of variations, image denoising, Abel equation, bounded variation functions.
MSC: 26A33, 26A45, 65K10.
[ Fulltext-pdf (874 KB)] open access