Adapted Basis for Non-Local Reconstruction of Missing Spectrum
  The object of this work is to design an adequate regularization for the problem of recovering missing Fourier coefficients, particularly in some non standard situations were low frequency coefficients are lost. In the framework of non-local regularization, we propose a technique to build an original patchwise similarity measure that is adapted to the missing spectrum. Then, a simple Non-Local quadratic energy is minimized. By construction, the similarity criterion is invariant under the corruption process so that the distance between two patches of the corrupted image is almost exactly equal to the one computed on the clean image. We illustrate our method with experiments which show its efficiency, both in terms of speed and quality of the results, with respect to other common approaches. We show that the method is practical on synthetic examples which are built upon models of inverse scattering problems, synthetic aperture mirrors for spatial imaging or also medical imaging. 
Antonin Chambolle, Khalid Jalalzai
math.NA