Analysis of Amoeba Active Contours

Subject of this paper is the theoretical analysis of structure-adaptive median filter algorithms that approximate curvature-based PDEs for image filtering and segmentation. These so-called morphological amoeba filters are based on a concept introduced by Lerallut et al. They achieve similar results as the well-known geodesic active contour and self-snakes PDEs. In the present work, the PDE approximated by amoeba active contours is derived for a general geometric situation and general amoeba metric. This PDE is structurally similar but not identical to the geodesic active contour equation. It reproduces the previous PDE approximation results for amoeba median filters as special cases. Furthermore, modifications of the basic amoeba active contour algorithm are analysed that are related to the morphological force terms frequently used with geodesic active contours. Experiments demonstrate the basic behaviour of amoeba active contours and its similarity to geodesic active contours.Comment: Revised version with several improvements for clarity, slightly extended experiments and discussion. Accepted for publication in Journal of Mathematical Imaging and Visio

A conservative shock filter model for the numerical approximation of conservation laws

A new shock filter model designed to sharpen numerically diffused discontinuities in a conservative fashion is presented. Besides the description of the modeling, the discussion includes a mathematically rigorous validation with respect to the meaning of the model as well as a presentation of some numerical results

Theoretical foundations for 1-D shock filtering

While shock filters are popular morphological image enhancement methods, no well-posedness theory is available for their corresponding partial differential equations (PDEs). By analysing the dynamical system of ordinary differential equations that results from a space discretisation of a PDE for 1-D shock filtering, we derive an analytical solution and prove well-posedness. We show that the results carry over to the fully discrete case when an explicit time discretisation is applied. Finally we establish an equivalence result between discrete shock filtering and local mode filtering