| A new algorithm is presented for tracking the contours of objects as they move through a video-stream. Two sets are identified: one which contains all relevant curves as found in the image, and a second which contains all curves which characterize the object of interest. The former is constructed out of edge-points in the image, while the latter is learned prior to running. The tracked curve is taken to be the element of the first set which is nearest the second set. Setting up the problem in this manner divides tracking into two subproblems: matching, that is the solution of the aforementioned optimization problem, and learning. The matching problem is difficult because it is of a mixed continuous-discrete nature, and the discrete set is very large. Furthermore, it is of crucial importance that the global optimum be found. Results are established which show that an approximate solution may be computed; which achieve bounds on the accuracy of the approximation; and which show how to attain the solution efficiently. In the realm of learning, an algorithm is proposed which constructs a finite-dimensional manifold from a set of its samples. This algorithm, whose output is a simplicial complex, does not require any prior knowledge of the geometry or topology of the manifold; these are learned from the samples.;The efficacy of the algorithm is demonstrated in the experimental context. Several video-streams are presented, in which there are a variety of moving objects. The tracker is quite successful in following the objects as they move, even against heavily cluttered backgrounds. The advantages of the proposed tracker over a condensation tracker is demonstrated in a particular experiment. |
