Principal component analysis of random particles

Particles are random sets whose position and orientation are irrelevant. They are traditionally handled by calculating summaries of shape, such as compactness and elongation, or by defining landmarks, whose positions are then subject to statistical analysis. It would be advantageous in many applications if shape variability could be addressed without the need for landmarks. This paper proposes a way to do this. The first step is to define similarity/distance between shapes. This is done in terms of the area of non-overlap between them, when they have been brought into the closest possible alignment. The resulting distance matrix can then be treated by the methods of principal coordinate analysis. It is shown that this is equivalent to principal component analysis on the binary sets in R-2 defined as the regions within the shape outlines. The method is illustrated by application to a set of carrot outlines.