A novel approach to grouping symmetrical planar curves under a projective transform is described. Symmetric curves are important as a generic model for object recognition where an object class is defined by the set of symmetries that any object in the class obeys. In this paper, a new algorithm is presented for grouping curves based on their correspondence under a plane projectivity. The correspondence between curves is established from an initial correspondence between two pairs of distinguished lines, such as lines tangent to inflection points. This initial correspondence leads to a reduced dimensional form for the projective mapping between the curves and a natural method for establishing correspondence between all points on the curves. A saliency measure is introduced which permits grouping results to be ordered in terms of the degree of symmetry supported by each curve pair. This saliency measure provides a basis for recognition in the case of approximate symmetry.

Magnetic resonance imaging (MRI) is unique in its ability to noninvasively and selectively alter tissue magnetization and create tagged patterns within a deforming body such as the heart muscle. The resulting patterns define a time-varying curvilinear coordinate system on the tissue, which we track with coupled B-snake grids. B-spline bases provide local control of shape, compact representation, and parametric continuity. Efficient spline warps are proposed which warp an area in the plane such that two embedded snake grids obtained from two tagged frames are brought into registration, interpolating a dense displacement vector field. The reconstructed vector field adheres to the known displacement information at the intersections, forces corresponding snakes to be warped into one another, and for all other points in the plane, where no information is available, a C1 continuous vector field is interpolated. The implementation proposed in this paper improves on our previous variational-based implementation and generalizes warp methods to include biologically relevant contiguous open curves, in addition to standard landmark points. The methods are validated with a cardiac motion simulator, in addition to in-vivo tagging data sets.

A framework for visual tracking of contours is presented, based on a synthesis of elastic models, stochastic filtering, and geometric invariance. Flexibly coupled curve templates implement soft prior assumptions about shape. Affine invariance, built into the flexible coupling, ensures that the affine deformations that arise naturally from image projection are favored. The stochastic basis of the framework is shown to result naturally in an automatic mechanism for the control of spatiotemporal scale.>