Topological data analysis is becoming a popular way to study high dimensional feature spaces without any contextual clues or assumptions. This paper concerns itself with one popular topological feature, which is the number of @math dimensional holes in the dataset, also known as the Betti @math number. The persistence of the Betti numbers over various scales is encoded into a persistence diagram (PD), which indicates the birth and death times of these holes as scale varies. A common way to compare PDs is by a point-to-point matching, which is given by the @math -Wasserstein metric. However, a big drawback of this approach is the need to solve correspondence between points before computing the distance; for @math points, the complexity grows according to @math n @math . Instead, we propose to use an entirely new framework built on Riemannian geometry, that models PDs as 2D probability density functions that are represented in the square-root framework on a Hilbert Sphere. The resulting space is much more intuitive with closed form expressions for common operations. The distance metric is 1) correspondence-free and also 2) independent of the number of points in the dataset. The complexity of computing distance between PDs now grows according to @math , for a @math discretization of @math . This also enables the use of existing machinery in differential geometry towards statistical analysis of PDs such as computing the mean, geodesics, classification etc. We report competitive results with the Wasserstein metric, at a much lower computational load, indicating the favorable properties of the proposed approach.
The square-root representation for 1D probability density functions (pdfs) was proposed in @cite_4 and was used for shape classification. It has since been used in several applications, including activity recognition @cite_1 - where the square-root velocity representation is used to model the space of time warping functions. This representation extends quite easily to arbitrary high-dimensional pdfs as well.
@cite_1 @cite_4
2160303236 2169678505
Pattern recognition in video is a challenging task because of the multitude of spatio-temporal variations that occur in different videos capturing the exact same event. While traditional pattern-theoretic approaches account for the spatial changes that occur due to lighting and pose, very little has been done to address the effect of temporal rate changes in the executions of an event. In this paper, we provide a systematic model-based approach to learn the nature of such temporal variations (time warps) while simultaneously allowing for the spatial variations in the descriptors. We illustrate our approach for the problem of action recognition and provide experimental justification for the importance of accounting for rate variations in action recognition. The model is composed of a nominal activity trajectory and a function space capturing the probability distribution of activity-specific time warping transformations. We use the square-root parameterization of time warps to derive geodesics, distance measures, and probability distributions on the space of time warping functions. We then design a Bayesian algorithm which treats the execution rate function as a nuisance variable and integrates it out using Monte Carlo sampling, to generate estimates of class posteriors. This approach allows us to learn the space of time warps for each activity while simultaneously capturing other intra- and interclass variations. Next, we discuss a special case of this approach which assumes a uniform distribution on the space of time warping functions and show how computationally efficient inference algorithms may be derived for this special case. We discuss the relative advantages and disadvantages of both approaches and show their efficacy using experiments on gait-based person identification and activity recognition. Applications in computer vision involve statistically analyzing an important class of constrained, non-negative functions, including probability density functions (in texture analysis), dynamic time-warping functions (in activity analysis), and re-parametrization or non-rigid registration functions (in shape analysis of curves). For this one needs to impose a Riemannian structure on the spaces formed by these functions. We propose a "spherical" version of the Fisher-Rao metric that provides closed-form expressions for geodesies and distances, and allows fast computation of sample statistics. To demonstrate this approach, we present an application in planar shape classification.