Almost all disciplines in information sciences and engineering use and need to compute the distance between events and geodesics or optimal paths between them. Geodesics are used for example for path planning in robotics, brain analysis in computational neuroscience, image segmentation, and finding fundamental geometric objects on surfaces. In high dimensions, they are the building blocks for problems in data mining and information discovery, with applications ranging from finance to biology to image and video processing. In general spaces, generalized geodesics also have numerous applications, from brain imaging to color image enhancement to computer graphics
Since distance functions and geodesics are fundamental for the interpretation and utilization of data and the knowledge and information embedded in it, it is important to have a good theoretical understanding of generalized distance functions and geodesics, and to develop efficient techniques for their accurate and fast computation. The work is needed in high dimensions and for diverse types of data representations, including implicit hyper-surfaces, sets of unorganized points such as manifold samples obtained when learning high dimensional information from examples, and point clouds from popular 3D range scanners. This project addresses these theoretical and computational issues, and concentrates on specific important applications in the areas mentioned above. These applications can be efficiently approached only once the theoretical and computational frameworks here studied have been derived.
In the educational arena, the goal is to continue with current efforts in the interdisciplinary aspects of advanced education and in the distribution and popularization of the developed techniques, including making software publicly available.
