Abstract
In this paper, a primal-dual algorithm for total bounded variation (TV)--type image restoration is analyzed and tested. Analytically it turns out that employing a global $\boldsymbol{L}^s$-regularization, with $1 < s \leq 2$, in the dual problem results in a local smoothing of the TV-regularization term in the primal problem. The local smoothing can alternatively be obtained as the infimal convolution of the $\ell_r$-norm, with $r^{-1} + s^{-1} = 1$, and a smooth function. In the case $r = s = 2$, this results in Gauss-TV--type image restoration. The globalized primal-dual algorithm introduced in this paper works with generalized derivatives, converges locally at a superlinear rate, and is stable with respect to noise in the data. In addition, it utilizes a projection technique which reduces the size of the linear system that has to be solved per iteration. A comprehensive numerical study ends the paper.
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