Differential Invariant Signatures and Flows in Computer Vision: A Symmetry Group Approach

Computer vision deals with image understanding at various levels. At the low level, it addresses issues such us planar shape recognition and analysis. Some classical results on differential invariants associated to planar curves are relevant to planar object recognition under different views and partial occlusion, and recent results concerning the evolution of planar shapes under curvature controlled diffusion have found applications in geometric shape decomposition, smoothing, and analysis, as well as in other image processing applications. In this work we first give a modern approach to the theory of differential invariants, describing concepts like Lie theory, jets, and prolongations. Based on this and the theory of symmetry groups, we present a high level way of defining invariant geometric flows for a given Lie group. We then analyze in detail different subgroups of the projective group, which are of special interest for computer vision. We classify the corresponding invariant flows and show that the geometric heat flow is the simplest possible one. This uniqueness result, together with previously reported results which we review in this paper, confirms the importance of this class of flows. *This work was supported in part by grants from the National Science Foundation DMS-9204192, DMS-8811084, and ECS-9122106, by the Air Force Office of Scientific Research AFOSR-90-0024, by the Army Research Office DAAL03-92-G-0115, DAAL03-91-G-0019, DAAH04-93-G-0332, by the Rothschild Foundation-Yad Hanadiv, and by Image Evolutions, Ltd. Report Documentation Page Form Approved OMB No. 0704-0188 Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. 1. REPORT DATE 10 DEC 1993 2. REPORT TYPE 3. DATES COVERED 00-12-1993 to 00-12-1993 4. TITLE AND SUBTITLE Differential Invariant Signatures and Flows in Computer Vision: A Symmetry Group Approach 5a. CONTRACT NUMBER