Figure 1. From two real blurred frames (left), we automatically and simultaneously estimate the motion region, the motion vector, and the image intensity of the foreground (middle). Based on this and the background intensity we reconstruct the two frames (right).

Abstract. 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 ill-posed 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.

We present and study a family of metrics on the space of compact subsets of � N (that we call “shapes”). These metrics are “geometric”, that is, they are independent of rotation and translation; and these metrics enjoy many interesting properties, as, for example, the existence of minimal geodesics. We view our space of shapes as a subset of Banach (or Hilbert) manifolds: so we can define a “tangent manifold ” to shapes, and (in a very weak form) talk of a “Riemannian Geometry” of shapes. Some of the metrics that we propose are topologically equivalent to the Hausdorff metric; but at the same time, they are more “regular”, since we can hope for a local uniqueness of minimal geodesics. We also study general properties of the metrics obtained by isometrically identifying a generic metric space with a subset of a Banach space and we obtain a rigidity result.

Median averaging is a powerful averaging concept on sets of vector data in finite dimensions. A generalization of the median for shapes in the plane is introduced. The underlying distance measure for shapes takes into account the area of the symmetric difference of shapes, where shapes are considered to be invariant with respect to different classes of affine transformations. To obtain a well–posed problem the perimeter is introduced as a geometric prior. Based on this model, an existence result can be established in the class of sets of finite perimeter. As alternative invariance classes other classical transformation groups such as pure translation, rotation, scaling, and shear are investigated. The numerical approximation of median shapes uses a level set approach to describe the shape contour. The level set function and the parameter sets of the group action on every given shape are incorporated in a joint variational functional, which is minimized based on step size controlled, regularized gradient descent. Various applications show in detail the qualitative properties of the median.

Median averaging is a powerful averaging concept on sets of vector data in finite dimensions. A generalization of the median for shapes in the plane is introduced. The underlying distance measure for shapes is based on the area of the symmetric difference of shapes and takes into account different invariance classes. These classes are generated by classical transformation groups such as translation, rotation, anisotropic scaling, and shear. As in the finite dimensional case, non-uniqueness of the median is observed. The numerical approximation of shape medians is based on a level set approach for the description of the shape contour. The level set function and the parameter sets of the group action on every given shape are incorporated in a joint variational functional, which is minimized based on step size controlled, regularized gradient descent. Various applications show in detail the qualitative behavior of the method. 1

of curves in shape optimization and analysis