Gems - Geometric Median Shapes

Abstract

We present an algorithm to compute the geometric median of shapes which is based on the extension of median to high dimensions. The median finding problem is formulated as an optimization over distances and it is solved directly using the watershed method as an optimizer. We show that the geometric median shape faithfully represents the true central tendency of the data, contaminated or not. It is superior to the mean shape which can be negatively affected by the presence of outliers. Our approach can be applied to manifold and non manifold shapes, with single or multiple connected components. The application of distance transform and watershed algorithm, two well established constructs of image processing, lead to an algorithm that can be quickly implemented to generate fast solutions with linear storage requirement. We demonstrate our methods in synthetic and natural shapes and compare median and mean results under increasing outlier contamination.