A SPECIALIZED NEURAL NETWORK BASED ON LIE GROUP THEORY FOR EXTRACTING 3D MOTION AND 3D LAYOUT OF VISIBLE SURFACES

SIMILARITY DEFINED THROUGH ABSTRACTION AND GENERALIZATION IS DIFFERENT FROM SIMILARITY UNDER TRANSFORMATION GROUPS. IN THEIR EARLY WORK TITLE "HOW TO KNOW UNIVERSALS", MCCULLOCH AND PITTS STUDIED NEURAL NETWORKS WHICH RECOGNIZED UNIVERSALS, I.E., FEATURES INVARIANT UNDER GEOMETRIC TRANSFORMATIONS. MUCH WORK HAS BEEN DONE IN RECENT YEARS ON NEURAL NETWORKS FOR ABSTRACTION AND GENERALIZATION BUT LITTLE ATTENTION HAS BEEN PAID TO INVARIANCE UNDER TRANSFORMATIONS, AS ASPECT WHICH IS IMPORTANT FOR DYNAMIC SCENE ANALYSIS. PITTS AND MCCULLOCH EXPLORED THE MECHANISM OF PERFORMING TRANSOFRMATIONS FROM CERTAIN TRANSFORMATION GROUPS IN ORDER TO EXTRACT "UNIVERSALS". HOWEVER, THEY DID NOT USE THE CONCEPT OF "ORIMITIVE OPERATION". ALL THE TRANSFORMATIONS FROM THE GROUP ARE EQUALLY IMPLEMENTED, AND THE NUMBER OF GEOMETRIC TRANSFORMATIONS IS VERY LARGE. USING LIE GROUP AND LIE ALGEBRA THEORY, WE PROPOSE A NEURAL NETWORK WHICH IMPLEMENTS THE LIE TRANSFORMATION GROUP VIA A FEW PRIMITIVE OPERATIONS CALLED ATOMIC TRANSFORMATIONS (I.E., GENERATORS OF THE DISCRETIZED LIE GROUP) THUS AVOIDING THE INTRACTABILITY OF PITTS AND MCCULLOCH'S NEURAL NETWORKS. DUE TO THE ONE-TO-ONE CORRESPONDENCE BETWEEN 3D RIGID MOTION AND THE INDUCED COHERENT IMAGE TRANSFORMATIONS, THIS NEURAL NETWORK IS CAPABLE OF PICKING UP 3D MOTION AND ED LOCATION PARAMETERS.