Morphological Amoebas Are Self-snakes

Abstract

This paper is concerned with amoeba median filtering, a structureadaptive morphological image filter. It has been introduced by Lerallut et al. in a discrete formulation. Experimental evidence shows that iterated amoeba median filtering leads to segmentation-like results that are similar to those obtained by self-snakes, an image filter based on a partial differential equation. We establish this correspondence by analysing a space-continuous formulation of iterated median filtering. We prove that in the limit of vanishing radius of the structuring elements, iterated amoeba median filtering indeed approximates the partial differential equation of self-snakes. This result holds true under very general assumptions on the metric used to construct the amoebas. We present experiments with discrete iterated amoeba median filtering that confirm qualitative and quantitative predictions of our analysis.

Why Is the Census Transform Good for Robust Optic Flow Computation?

“…Other examples include e.g. continuous reinterpretations of median filters in terms of mean curvature motion [20] and morphological amoebae as self-snakes [21].…”

Mathematical Foundations and Generalisations of the Census Transform for Robust Optic Flow Computation

“…Other examples include e.g. continuous reinterpretations of median filters in terms of mean curvature motion [12] and morphological amoebae as self-snakes [46].…”

Analysis and Application of a Nonlocal Hessian

“…We draw our motivation for choosing the weights partly from a recent discussion of non-local "Amoeba" filters [LDM07,WBV11,Wel12]. Amoeba filters use classical techniques such as iterated median filtering, but the structuring element is chosen in a highly adaptive local way that can follow the image structures, instead of being restricted to a small parametrized set of shapes.…”

“…We show an example where the weights are used to construct a regularizer that both favours piecewise affine functions but also allows for jumps in the data. It is motivated by the recent discussion of "Amoeba" filters in [LDM07,WBV11,Wel12] which combine standard filters such as median filters with non-parametric structuring elements that are based on the data -that is in long thin objects they would extend along the structure and thus prevent smoothing perpendicular to the structure. In Amoeba filtering, the shape of a structuring element at a point is defined as a unit circle with respect to the geodesic distance on a manifold defined by the image itself.…”