Scale-Space Properties of Nonstationary Iterative Regularization Methods

Most seale-spaee eoneepts have been expressed as parabolie or hyperbolie partial differential equations (PDEs). In this paper we extend our work on seale-spaee properties of elliptie PDEs arising from regularization methods: we study linear and nonlinear regularization methods that are applied iteratively and with different regularization parameters. For these so-ealled nonstationary iterative regularization teehniques we darify their relations to both isotropie diffusion filters with a sealar-valued diffusivity and anisotropie diffusion filters with a diffusion tensor. We establish seale-spaee properties for iterative regularization methods that are in eomplete aeeordance with those for diffusion filtering. In particular, we show that nonstationary iterative regularization satisfies a causCl,lityproperty in terms of a maximum-minimum principle, possesses a large dass of Lyapunov functionals, and converges to a constant image as the regularization parameters tend to infinity. 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 seale-space behavior of regularization methods.