On approximation in total variation penalization for image reconstruction and inverse problems

In this paper, we examine some theoretical issues associated with the use of total variation based image reconstruction. Our investigations are motivated by problems of inverse interferome-try, in which laser light phase shifts are used to reconstruct medium density profiles in flow field sensing. The reconstruction problem is posed as a residual minimization with total variation reg-ularization applied to handle the inherent ill-posedness. We consider numerical approximations of these penalized minimal residual problems, and analyze some approximation strategies and their properties. The standard definition of total variation leads to inconsistent approximations, with piecewise constant basis functions, so we consider alternative definitions, which preserve the needed compactness and produce convergent approximations.