The Geometry of Orthogonal-Series, Square-Root Density Estimators: Applications in Computer Vision and Model Selection

Abstract

We consider the geometry and model order specification of a class of density models where the square-root of the distribution is expanded in an orthogonal series. The simplicity of the resulting spherical geometry makes this framework ideal for many applications that rely on information geometric concepts like distances and manifold statistics. Specifically, we demonstrate applications of these models in the computer vision field of object recognition and retrieval. We illustrate how invariant shape representations can be used in conjunction with these probabilistic models to yield state-of-the-art classifiers. Moreover, the viability of formulating classification models that take into account shape deformation in an optimal transport context are investigated, yielding insight into the practicalities of working with the parameter space of the densities versus the Wasserstein measure space approach. The free parameters associated with these square-root estimators can be rigorously selected using the Minimum Description Length (MDL) criterion for model selection. Under these models, it is shown that the MDL has a closed-form representation, atypical for most applications of MDL in density estimation. Experimental evaluation of our techniques are conducted on one, two, and three dimensional density estimation problems in shape analysis, with comparative analysis demonstrating our approach to be state-of-the-art in object recognition and model selection.