Total variation wavelet inpainting

We consider the problem of filling in missing or damaged wavelet coefficients due to lossy image transmission or communication. The task is closely related to classical inpainting problems, but also remarkably differs in that the inpainting regions are in the wavelet domain. New challenges include that the resulting inpainting regions in the pixel domain are usually not geometrically well defined, as well as that degradation is often spatially inhomogeneous. We propose two related variational models to meet such challenges, which combine the total variation (TV) minimization technique with wavelet representations. The associated Euler-Lagrange equations lead to nonlinear partial differential equations (PDE's) in the wavelet domain, and proper numerical algorithms and schemes are designed to handle their computation. The proposed models can have effective and automatic control over geometric features of the inpainted images including sharp edges, even in the presence of substantial loss of wavelet coefficients, including in the low frequencies. Existence and uniqueness of the optimal inpaintings are also carefully investigated.