A fractional diffusion-wave equation with non-local regularization for image denoising

•The fractional PDE interpolates between a diffusion equation and a wave equation.•The anti-diffusion effect of the non-local term results in edge enhancement.•We have proved that the proposed model is well-posed.•The stable and convergent numerical scheme is given.•We have discussed the choice of the parameters in our model.

This paper introduces a novel fractional diffusion-wave equation with non-local regularization for noise removal. Using the fractional time derivative, the model interpolates between the heat diffusion equation and the wave equation, which leads to a mixed behavior of diffusion and wave propagation and thus it can preserve edges in a highly oscillatory region. On the other hand, the usual diffusion is used to reduce the noise whereas the non-local term which exhibits an anti-diffusion effect is used to enhance the image structure. We prove that the proposed model is well-posed, and the stable and convergent numerical scheme is also given in this paper. The experimental results indicate superiority of the proposed model over the baseline diffusion models.