Abstract: Computational procedures for rigid and non-rigid registration are widely used, but a conceptual and theoretical framework for understanding these procedures has been scarce. Such a framework is critical for designing a new generation of registration algorithms - algorithms with guaranteed (i.e. provable) properties. Research presented in this talk suggests that viewing registration algorithms from a geometric point-of-view provides a basis for understanding registration. In this framework, intra- and inter-modality registration appear on common ground and the fundamental role of the geometric volume form becomes clear. Different volume forms give registration algorithms with different properties. Four properties of registration objective functions are identified as useful, and a unique volume form is shown to impart these properties. Experimental results confirm that the theoretical results hold in practice even in the presence of noise in the images.