Elastic Shape Analysis of Functions, Curves and Trajectories

Abstract

International audienceWe present a Riemannian framework for geometric shape analysis of curves, functions, and trajectories on nonlinear manifolds. Since scalar functions and trajectories can also have important geometric features, we use shape as an all-encompassing term for the descriptors of curves, scalar functions and trajectories. Our framework relies on functional representation and analysis of curves and scalar functions, by square-root velocity fields (SRVF) under the Fisher–Rao metric, and of trajectories by transported square-root vector fields (TSRVF). SRVFs are general functional representations that jointly capture both the shape (geometry) and the reparameterization (sampling speed) of curves, whereas TSRVFs also capture temporal reparameterizations of time-indexed shapes. The space of SRVFs for shapes of curves becomes a subset of a spherical Riemannian manifold under certain special constraints. A fundamental tool in shape analysis is the construction and implementation of geodesic paths between shapes. This is used to accomplish a variety of tasks, including the definition of a metric to compare shapes, the computation of intrinsic statistics for a set of shapes, and the definition of probability models on shape spaces. We demonstrate our approach using several applications from computer vision and medical imaging including the analysis of (1) curves, (2) human growth, (3) bird migration patterns, and (4) human actions from video surveillance images and skeletons from depth images

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“…Shape analysis is a research area that spans a variety of different scientific fields, including mathematics, computer vision, image processing, machine vision, geographic information systems and computer graphics. There is a vast body of literature related to statistical shape analysis, and applications of these techniques are seen in a wide array of fields ranging anywhere from medical imaging, optical character recognition and interpretation of hand gestures to bird migration (Srivastava et al 2011;Belongie, Malik, & Puzicha 2002;Joshi et al 2016;Pavlovic, Sharma, and Huang 1997;Su et al 2014). Loncaric (1998) provides a comprehensive review of shape analysis techniques available nearly two decades ago, motivating the exposition in the context of grayscale image processing systems.…”

“…There are many ways to compare and analyze 2D shapes. For example, shapes can be characterized as smooth curves, and techniques from differential and Riemannian geometry can both smoothly deform shapes into each other and measure the amount of "deformation energy" needed to transmogrify one into the other (Joshi, Su, Zhang, & Amor, 2016). Or, the boundary of the shape can be sampled at regular intervals, shape descriptors calculated at each sampled point and the descriptors matched using optimization algorithms (Belongie, Malik, & Puzicha, 2002).…”