Published Paper

Inserted: 25 aug 2026
Last Updated: 25 aug 2026

Journal: Nonlinear Analysis TMA
Volume: 274
Number: 114260
Year: 2027
Doi: 10.1016/j.na.2026.114260
Links: journal site

Abstract:

We collect several results related to the Symmetrized Fractional Variation model for signal and image denoising (shortly denoted SFV): a variational approach based on $L^1$ fitting data term together with regularizing terms exploiting a distributional version of Riemann-Liouville fractional derivatives. We enhance the analysis of the one-dimensional case through the study of the space $BV^s_*$ of admissible signals on a bounded interval, say the functions with bounded variation of both sides fractional derivatives for a prescribed real positive order s. We show that the embedding in $BV^s_*$ of the Sobolev space of the same fractional order is strict. We exhibit some nontrivial borderline examples of admissible or non admissible functions in the space $BV^s_*$. We prove several relationships between related fractional calculus and the integral transforms. The SFV model is discretized based on a second-order consistent Grunwald-Letnikov scheme and coupled with an automatic selection procedure of all model parameters relying on the whiteness principle: some numerical simulations are presented to show the efficacy of the proposed approach in denoising one-dimensional signals corrupted by impulsive noise modelled by the Laplace distribution.

Keywords: calculus of variations, bounded variation functions, Image denoising, Riemann-Liouville fractional derivatives, Symmetrized fractional variation, Grünwald-Letnikov formulas, discretization of fractional derivatives, Abel integral equation