Similarity Metric For Curved Shapes In Euclidean Space

In this paper, we introduce a similarity metric for curved shapes that can be described, distinctively, by ordered points. The proposed method represents a given curve as a point in the deformation space, the direct product of rigid transformation matrices, such that the successive action of the matrices on a fixed starting point reconstructs the full curve. In general, both open and closed curves are represented in the deformation space modulo shape orientation and orientation preserving diffeomorphisms. The use of direct product Lie groups to represent curved shapes led to an explicit formula for geodesic curves and the formulation of a similarity metric between shapes by the $L^{2}$-norm on the Lie algebra. Additionally, invariance to reparametrization or estimation of point correspondence between shapes is performed as an intermediate step for computing geodesics. Furthermore, since there is no computation of differential quantities on the curves, our representation is more robust to local perturbations and needs no pre-smoothing. We compare our method with the elastic shape metric defined through the square root velocity (SRV) mapping, and other shape matching approaches