Symmetrized Fractional Variation for signal denoising

We define, study and implement the model SFV (Symmetrized Fractional Variation): a variational approach to signal analysis exploiting the Riemann-Liouville fractional derivatives of every positive real order higher than zero. The model exploites an L_1 fitting data term together with both right and left fractional derivatives as regularizing terms: this approach aims to achieve an orientation-independent protocol. We express the model as an energy minimization and introduce a functional framework where existence of minimizers is proved: namely the space BVs∗. We show that the embedding in BVs∗ of the Sobolev space of the same fractional order is strict, and we exhibit some nontrivial borderline examples of admissible or non admissible functions in the space BVs∗. To provide evidence of effectiveness for the proposed model, a discretisation based on a second-order consistent Gr¨unwald-Letnikov scheme is introduced. A multi-parameter whiteness criterion is proposed which provides an unsupervised, automatic and simultaneous selection of the two free parameters in the model, namely the fractional order of differentiation and the regularization parameter. Numerical experiments on 1-d and 2-d signals are performed which show how the proposed model holds the potential to achieve good quality results for denoising signals corrupted by additive Laplace noise. This is a joint research with Alessandro Lanza(1), Antonio Leaci(2) and Serena Morigi(1). (1) Dipartimento di Matematica, Universit´a di Bologna (2) Universit´a del Salento, Dipartimento di Matematica e Fisica “Ennio De Giorgi”, Lecce