Geometric Median Shapes

TL;DR

This paper introduces a robust algorithm for computing the geometric median of shapes using distance transforms and watershed methods, effectively handling outliers and applicable to various shape types.

Contribution

The paper presents a novel shape median computation method based on extending median to high dimensions and using watershed optimization, improving robustness over mean shapes.

Findings

The geometric median shape is more robust to outliers than the mean shape.

The proposed algorithm is fast, with linear storage requirements.

It performs well on both synthetic and natural shapes, outperforming mean-based methods.

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 computing the geometric median of shapes is robust in the presence of outliers and it is superior to the mean shape which can easily be affected by the presence of outliers. The geometric median shape thus faithfully represents the true central tendency of the data, contaminated or not. Our approach can be applied to manifold and non manifold shapes, with connected or disconnected shapes. The application of distance transforms 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 requirements. We demonstrate our methods in synthetic and natural shapes and compare median and mean results under increasing contamination by strong outliers.