  In a recent paper by A. Chambolle et al. [Geometric properties of solutions
to the total variation denoising problem. Inverse Problems 33, 2017] it was
proven that if the subgradient of the total variation at the noise free data is
not empty, the level-sets of the total variation denoised solutions converge to
the level-sets of the noise free data with respect to the Hausdorff distance.
The condition on the subgradient corresponds to the source condition introduced
by Burger and Osher [Convergence rates of convex variational regularization.
Inverse Problems 20, 2004], who proved convergence rates results with respect
to the Bregman distance under this condition. We generalize the result of
Chambolle et al. to total variation regularization of general linear inverse
problems under such a source condition. As particular applications we present
denoising in bounded and unbounded, convex and non convex domains, deblurring
and inversion of the circular Radon transform. In all these examples the
convergence result applies. Moreover, we illustrate the convergence behavior
through numerical examples.
