Inverse problems in the space of Radon measures

In this talk, regularization of illposed inverse problems involving Radon measures is studied. The main motivation is to nd a sparse reconstruction for an unknown Radon measure based on indirect measurements. Similar questions has been previously examined in [3].

In this talk, it is shown that the minimization of the Tikhonov functional with the total
variation norm penalty is a well-posed problem for Radon measures. The minimizer of the Tikhonov functional is characterized by using the corresponding Fenchel predual problem. Convergence and convergence rate properties of the minimizer are examined under common source conditions, like in [1, 2]. A numerical algorithm promoting sparsity is proposed. The algorithm is illustrated by a numerical example.

References
[1] M. Burger and S. Osher. Convergence rates of convex variational regularization.

Inverse Problems, 20(5):1411{1421, 2004.
[2] B. Hofmann, B. Kaltenbacher, C. Poeschl, and O. Scherzer. A convergence rates result
for Tikhonov regularization in Banach spaces with non-smooth operators. Inverse
Problems, 23(3):987{1010, 2007.

[3] O. Scherzer and B. Walch. Sparsity regularization for Radon measures. In X.-C. Tai,
K. Morken, M. Lysaker, and K.-A. Lie, editors, Scale Space and Variational Meth-
ods in Computer Vision, volume 5567 of Lecture Notes in Computer Science, pages
452{463. Springer-Verlag, 2009. Proceedings of the Second International Conference,
SSVM 2009.