Image analysis using p ‐Laplacian and geometrical PDEs

Abstract

AbstractMinimizing the integral ∫Ω1/p |∇L |p d Ω for an image L under suitable boundary conditions gives PDEs that are well‐known for p = 1, 2, namely Total Variation evolution and Laplacian diffusion (also known as Gaussian scale space and heat equation), respectively. Without fixing p, one obtains a framework related to the p ‐Laplace equation. The partial differential equation describing the evolution can be simplified using gauge coordinates (directional derivatives), yielding an expression in the two second order gauge derivatives and the norm of the gradient. Ignoring the latter, one obtains a series of PDEs that form a weighted average of the second order derivatives, with Mean Curvature Motion as a specific case. Both methods have the Gaussian scale space in common. Using singularity theory, one can use properties of the heat equation (namely. the role of scale) in the full L (x, t) space and obtain a framework for topological image segmentation. In order to be able to extend image analysis aspects of Gaussian scale space in future work, relations between these methods are investigated, and general numerical schemes are developed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)