Finite-dimensional approximation of convex regularization via hexagonal pixel grids

Abstract

This work extends the existing convergence analysis for discrete approximations of minimizers of convex regularization functionals. In particular, some solution concepts are generalized, namely the standard minimum norm solutions for squared-norm regularizers and the -minimizing solutions for general convex regularizers, respectively. A central part of the manuscript addresses finite-dimensional approximations of solutions of ill-posed operator equations with basis functions defined on hexagonal grids, which require the novel solution concept.

Keywords:

• ill-posed problem
• convex regularization
• total variation
• hexagonal pixel

AMS Subject Classifications:

• 65J20
• 47J06 (65M60)

Acknowledgements

The list of helpful references provided by Luminita Vese is gratefully acknowledged. C.P. and E.R. acknowledge the support by Austrian Science Fund, project J2970 (Schrödinger scholarship), project T644-N26 (Hertha Firnberg fellowship), and project FWF V82-N118 (Elise Richter fellowship), respectively. The work of E.R. has been partly done while affiliated with the Johannes Kepler University, Linz. The work of C.K. and O.S. is supported through the Austrian Science Fund (FWF) via projects S11704 and S10505.

Notes

The work of C.K. has also been supported by the Vienna Graduate School in Computational Science (IK I059-N) funded by the University of Vienna.