Geometric Median Shapes
This work addresses the need for robust shape representation in image processing, particularly for data with outliers, though it is incremental as it builds on existing methods like distance transforms and watershed algorithms.
The authors tackled the problem of computing a robust central tendency for shape data by introducing an algorithm for the geometric median of shapes, which outperforms the mean shape in the presence of outliers, as demonstrated in synthetic and natural datasets under increasing contamination.
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.