Variational regularization theory of ill-posed inverse problems with known forward models has a long tradition spanning all the way back to the seminal contributions of Tikhonov in the 1940s. It studies questions like consistency as the regularization parameter and noise level converge to zero simultaneously, generally from a functional-analytic point of view.
Often, and in particular when dealing with imaging applications like deblurring or Radon transform inversion for tomography, the regularization energies used in such approaches contain spatial derivatives. As such, they also have rich analytical backgrounds in terms of properties and regularity of minimizers.
In this talk, I will present some recent work bridging these two areas together. In the regime of vanishing noise and regularization parameter, we obtain results of convergence in Hausdorff distance of level sets of minimizers (which can be interpreted as objects to be recovered in an imaging context) and uniform L∞ bounds. These hold not only with total variation regularization, but also when regularizing with the fractional Laplacian.
Based on joint works with Gwenael Mercier, Kristian Bredies, and Otmar Scherzer.