Fractional bounded variation and signal analysis

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.