This project is devoted to innovative applications of the geometry of moving frames and the mathematical theory of differential invariants to problems arising in computer vision, with emphasis on medical imaging. Since symmetry forms a basic component of the human visual system, it should be naturally incorporated into mathematical methods for image processing and object recognition. The image processing applications to be considered will include denoising and smoothing, segmentation and edge detection, as well as object recognition, including symmetry detection and partial occlusions. Practical implementations will be based on the signature curve paradigm, that is based on the construction of suitable differential invariants for the symmetry group under consideration. Metrics that distinguish signature curves of different objects will be compared, with the goal being to recognize and extract objects from both two and three-dimensional images. Practical implementations of the image processing systems will rely on a new method of constructing symmetry-preserving numerical approximations to differential invariants that is to be developed as a part of this project. The applications will be based on a new, practical theory of moving frames, that can be applied to general group actions, both finite and infinite-dimensional. Part of the project will be devoted to the further development of this general theory. Various additional applications, to problems arising in physics, geometry, invariant theory, and the calculus of variations, will also be investigated.
