Scale-space Properties of Nonstationary Iterative Regularization Methods Scale-space Properties of Nonstationary Iterative Regularization Methods

The technical reports of the CVGPR Group are listed under Abstract Most scale-space concepts have been expressed as parabolic or hyperbolic partial diierential equations (PDEs). In this paper we extend our work on scale-space properties of elliptic PDEs arising from regularization methods: we study linear and nonlinear regularization methods that are applied iteratively and with different regularization parameters. For these so-called nonstationary iterative reg-ularization techniques we clarify their relations to both isotropic diiusion lters with a scalar-valued diiusivity and anisotropic diiusion lters with a diiusion tensor. We establish scale-space properties for iterative regularization methods that are in complete accordance with those for diiusion ltering. In particular, we show that nonstationary iterative regularization satisses a causality property in terms of a maximum{minimum principle, possesses a large class of Lyapunov functionals, and converges to a constant image as the regularization parameters tend to innnity. We also establish continuous dependence of the result with respect to the sequence of regularization parameters. Numerical experiments in two and three space dimensions are presented that illustrate the scale-space behavior of regularization methods.