Microlocal sequential regularization in imaging

In this article, we consider imaging problems in which the dataconsist of noisy observations of the true image through a linearfilter such as blurring, sparse sampling or tomographic projections.The image restoration problem is ill-posed and in order to obtain ameaningful result, the problem needs to be regularized or augmentedby additional information. In this article, we consider Tikhonovregularization by a class of non-linear smoothness filters that arecapable of detecting and restoring edges in the image. Theregularization function is microlocal in the sense that it issensitive to the location and the direction of the non-smoothness ofthe image. The implementation of the algorithm leads to a sequenceof simple linear least squares problems, the penalty term beingcalculated as a direction-sensitive weighted finite differenceapproximation of the Laplacian. The algorithm is applied to twoclassical imaging problems, image zooming and limited angletomography.