We design a new class of shearlet-based functionals resembling the classical Ginzburg-Landau energy, and prove the variational convergence of those to the weighted TV functionals as the diffuse interface parameter approaches zero.

We use the essential features of the differential operator representations in the Fourier domain to create a new class of anisotropic diffusive operators based on sparse representations. While preserving the convenient diffusive features, new operators bring in the combined advantages of sparsity and non-locality.

The anisotropic shearlet energies and associated operators are (by design) highly tunable and very effective in the signal recovery: we illustrate it with examples of directional-sensitive image inpainting.