Regularity and Scale-Space Properties of Fractional High Order Linear Filtering

We investigate the use of fractional powers of the Laplacian for signal and image simplification. We focus both on their corresponding variational techniques and parabolic pseudodifferential equations. We perform a detailed study of the regularisation properties of energy functionals, where the smoothness term consists of various linear combinations of fractional derivatives. The associated parabolic pseudodifferential equations with constant coefficients are providing the link to linear scale-space theory. These encompass the well-known α-scale-spaces, even those with parameter values α > 1 known to violate common maximumminimum principles. Nevertheless, we show that it is possible to construct positivity-preserving combinations of high and low-order filters. Numerical experiments in this direction indicate that non-integral orders play an essential role in this construction. The paper reveals the close relation between continuous and semi-discrete filters, and by that helps...

• Type: Conference
• Year: 2005
• Where: SCALESPACE
• Authors: Stephan Didas, Bernhard Burgeth, Atsushi Imiya, Joachim Weickert
• Keywords: Corresponding Variational Techniques, Parabolic Pseudodifferential Equations, SCALESPACE 2005, Various Linear Combinations