A New Geometric Metric in the Space of Curves, and Applications to Tracking Deforming Objects by Prediction and Filtering

Abstract

We define a novel metric on the space of closed planar curves which decomposes into three intuitive components. According to this metric centroid translations, scale changes and deformations are orthogonal, and the metric is also invariant with respect to reparameterizations of the curve. While earlier related Sobolev metrics for curves exhibit some general similarities to the novel metric proposed in this work, they lacked this important three-way orthogonal decomposition which has particular relevance for tracking in computer vision. Another positive property of this new metric is that the Riemannian structure that is induced on the space of curves is a smooth Riemannian manifold, which is isometric to a classical well-known manifold. As a consequence, geodesics and gradients of energies defined on the space can be computed using fast closed-form formulas, and this has obvious benefits in numerical applications.The obtained Riemannian manifold of curves is ideal to address complex problems in computer vision; one such example is the tracking of highly deforming objects. Previous works have assumed that the object deformation is smooth, which is realistic for the tracking problem, but most have restricted the deformation to belong to a finite-dimensional group -such as affine motions -or to finitely-parameterized models. This is too restrictive for highly deforming objects such as the contour of a beating heart. We adopt the smoothness assumption implicit in previous work, but we lift the restriction to finite-dimensional motions/deformations. We define a dynamical model in this Riemannian manifold of curves, and use it to perform filtering and prediction to infer and extrapolate not just the pose (a finitely parameterized quantity) of an object, but its deformation (an infinite-dimensional quantity) as well. We illustrate these ideas using a simple first-order dynamical model, and show that it can be effective even on image sequences where existing methods fail.

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

“…See [123][124][125] for more details on Sobolev active contours and applications to segmentation and tracking. The same idea has been employed for gradient flows of surfaces in [140].…”

“…More general and higher order Sobolev metrics on plane curves have been studied in [84,96], and they have been applied to the field of active contours in [27,125]. Other Sobolev type metrics on curves that have been studied include a metric for which translations, scale changes and deformations of the curve are orthogonal [123] and an H 2 -type (semi)-metric whose kernel is generated by translations, scalings and rotations. [116].…”

Geodesics in infinite dimensional Stiefel and Grassmann manifolds

“…Note that point (5) in the above theorem implies that minimal geodesics can be numerically computed using a finite dimensional algorithm; see Sec. 3.3.4 in [6].…”

“…Thus Gr(2, V ) with V = L 2 ([0, 1]) is isometric to the space of planar closed curves up to translations, scalings and rotations. See [7], [8], [5] and [6]. Any result that is proven about the Stiefel or Grassmannian immediately carries over to the corresponding space of curves.…”

“…[6]. Any result that is proven about the Stiefel or Grassmannian immediately carries over to the corresponding space of curves.…”