Discrete curve model for non-elastic shape analysis on shape manifold

In this paper, we construct a novel finite dimensional shape manifold for shape analyses. Elements of the shape manifold are a set of discrete, planar, and closed curves, which stand for object boundaries and are represented by direction function. On this manifold, we use a set of N-dimensional Fourier basis to construct the tangent space of the shape manifold as a finite dimensional space. Furthermore, we construct the shape manifold as a Riemannian manifold, in which the Riemannian metric is interpreted as an l² metric. Our method improves the performance of bending-only models in the issues of shape analysis including the shape synthesis, comparison, and statistic analysis. We evaluate the performance of the manifold via the following applications: 1) shape interpolation and extrapolation between curves, 2) shape retrieval on the Flavia leaf database, 3) shape synthesis using an estimated probability distribution on the manifold, and 4) a novel application named shape arithmetic. All the above experiments clearly demonstrate our approach achieves superior performance to state-of-the-art methods.