Shapes as empirical distributions

We address the problem of shape based classification. We interpret the shape of an object as a probability distribution governing the location of the points of the object. An image of the object, represented as an arbitrary set of unlabeled points, corresponds to a random drawing from the shape probability distribution and can thus be analyzed as an empirical distribution. Using this framework, classification of shapes is robust to the number of points in the image and there is no need to solve the correspondence problem when comparing two images. The framework allows us to estimate geometrical transformations between images in a statistically meaningful way using maximum likelihood. We formulate the decision problem associated with shape classification as a hypothesis test for which we can characterize the performance. We particularize this framework to two-dimensional shapes related by an affine transformation. Under this assumption, we develop a descriptor invariant to affine movement, permutations, and sampling density, and robust to noise, occlusion, and reasonable non-linear deformations. Experimental results demonstrate the quality of our approach.