No 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.

We consider regions of images that exhibit smooth statistics, and pose the question of characterizing the "essence" of these regions that matters for recognition. Ideally, this would be a statistic (a function of the image) that does not depend on viewpoint and illumination, and yet is sufficient for the task. In this manuscript, we show that such statistics exist. That is, one can compute deterministic functions of the image that contain all the "information" present in the original image, except for the effects of viewpoint and illumination. We also show that such statistics are supported on a "thin" (zero-measure) subset of the image domain, and thus the "information" in an image that is relevant for recognition is sparse. Yet, from this thin set one can reconstruct an image that is equivalent to the original up to a change of viewpoint and local illumination (contrast). Finally, we formalize the notion of "information" an image contains for the purpose of viewpoint-and illuminationinvariant tasks, which we call "actionable information" following ideas of J. J. Gibson.

Recently, the Sobolev metric was introduced to define gradient flows of various geometric active contour energies. It was shown that the Sobolev metric outperforms the traditional metric for the same energy in many cases such as for tracking where the coarse scale changes of the contour are important. Some interesting properties of Sobolev gradient flows include that they stabilize certain unstable traditional flows, and the order of the evolution PDEs are reduced when compared with traditional gradient flows of the same energies. In this paper, we explore new possibilities for active contours made possible by Sobolev metrics. The Sobolev method allows one to implement new energy-based active contour models that were not otherwise considered because the traditional minimizing method render them illposed or numerically infeasible. In particular, we exploit the stabilizing and the order reducing properties of Sobolev gradients to implement the gradient descent of these new energies. We give examples of this class of energies, which include some simple geometric priors and new edge-based energies. We also show that these energies can be quite useful for segmentation and tracking. We also show that the gradient flows using the traditional metric are either ill-posed or numerically difficult to implement, and then show that the flows can be implemented in a stable and numerically feasible manner using the Sobolev gradient.

Recently proposed Sobolev active contours introduced a new paradigm for minimizing energies defined on curves by changing the traditional cost of perturbing a curve and thereby redefining their gradients. Sobolev active contours evolve more globally and are less attracted to certain intermediate local minima than traditional active contours, and it is based on a well-structured Riemannian metric. In this paper, we analyze Sobolev active contours using scale-space analysis in order to understand their evolution across different scales. This analysis shows an extremely important and useful behavior of Sobolev contours, namely, that they move successively from coarse to increasingly finer scale motions in a continuous manner. This property illustrates that one justification for using the Sobolev technique is for applications where coarse-scale deformations are preferred over fine scale deformations. Along with other properties to be discussed, the coarse-to-fine observation reveals that Sobolev active contours are, in particular, ideally suited for tracking algorithms that use active contours. We will also justify our assertion that the Sobolev metric should be used over the traditional metric for active contours in tracking problems by experimentally showing how a variety of active contour based tracking methods can be significantly improved merely by evolving the active contour according to the Sobolev method.