Abstract
This paper introduces a novel method to combine total variation and ℓ2 regularizations to reconstruct piecewise smooth signals. The main idea is to consider the signal as a sum of two components: a piecewise constant component and a smooth component. For the solution of ill-posed problems, the Tikhonov method with a special stabilizer in the form of a sum of two different stabilizers is used: the total variation for the first component and the Sobolev norm for the second one. An iteratively re-weighted least squares technique is used as a fast and an efficient algorithm for minimization of the Tikhonov functional. A method is also presented for determining the regularization parameters. Numerical experiments, among the many performed, in denoising, deblurring, and compressed sensing demonstrate high performance of the new regularization for reconstruction of piecewise-smooth solutions with sharp discontinuities.
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