The investigators and their colleagues study the problem of automatically clustering sets of points that lie on smooth manifolds of arbitrary co-dimension. They use the geodesic snake approach in which they set up a Partial Differential Equation (PDE) that drives the evolution of a manifold in some potential field created by the data points, such that the solution of the PDE converges to a manifold that is very close to the data points (ideally contains them). The "shape" of this manifold (including its co-dimension) characterizes the set of points. They investigate computationally efficient implementations for evolving manifolds of arbitrary co-dimension by level sets methods, because of their robustness and flexibility. These implementations are being used to study the Euclidean and affine mean-curvature motion of curves in three-dimensional spaces. Two applications are being pursued by the investigators. The first application is the detection of blood vessels in Magnetic Resonance Angiography (MRA) images. In MRA images, blood vessels appear as bright and noisy curve-like patterns, possibly with gaps. It is important to detect those patterns independently of the noise and to bridge the gaps. The detection of blood vessels can be considered, as a first approximation, as an instance of the three-dimensional case, co-dimension two. The second application is the clustering of feature points in higher dimensional spaces with an eye on two specific problems, the description of the "shape" of a cluster and the design of automated content-based image retrieval systems. A given pattern, e.g. an image, can be represented by a vector of features in some n-dimensional space. Patterns in the same class usually fall into clusters that may have some very tricky "shapes". The investigators and their collegues believe that their methods can provide ways of automatically obtaining shape descriptions from sets of sample feature vectors thereby facilitating the design of recognition algorithms and improving their performances. A target application is the design of automated content-based image retrieval systems. The investigators and their colleagues study the problem of automatically fitting a set of points in a possibly high-dimensional space to a set of smooth manifolds. This is important because in many practical applications data can be represented as points in high-dimensional spaces. They are currently pursuing two applications of these techniques. The first one is the detection of blood vessels in Magnetic Resonance Angiography (MRA) images. In MRA images, blood vessels appear as bright and noisy curve-like patterns, possibly with gaps. It is important to detect those patterns independently of the noise and to bridge the gaps. The results of this detection can be used in the diagnosis of diseases and in planning surgery. The second application is the description of the "shape" of a cluster of points in a high-dimensional space. The investigators and their colleagues are investigating means of automatically obtaining such descriptions from a set of samples. They believe that this will facilitate the design of recognition algorithms and increase their performances. A target application that is particularly relevant to the average citizen is the design of automated content-based image retrieval systems. In such systems each image can be represented by a point in some high-dimensional (10 to 20) feature space, and images in the same class, e.g. images of cars, cluster on some manifold, e.g a "car-manifold". The description of the shape of this manifold (which is simpler than the real shapes of the objects in the class) can then be used to retrieve all images of, e.g. cars, in the data base because their representations in the high-dimensional space will be closer to the corresponding manifold than to any other.
