J Math Imaging Vis (2009) 34: 61–88 DOI 10.1007/s10851-008-0129-7 Symmetric Non-rigid Registration: A Geometric Theory and Some Numerical Techniques

Abstract This paper proposes L2- and information-theory-based (IT) non-rigid registration algorithms that are exactly symmetric. Such algorithms pair the same points of two im-ages after the images are swapped. Many commonly-used L2 and IT non-rigid registration algorithms are only approx-imately symmetric. The asymmetry is due to the objective function as well as due to the numerical techniques used in discretizing and minimizing the objective function. This paper analyzes and provides techniques to eliminate both sources of asymmetry. This paper has five parts. The first part shows that objec-tive function asymmetry is due to the use of standard differ-ential volume forms on the domain of the images. The sec-ond part proposes alternate volume forms that completely eliminate objective function asymmetry. These forms, called graph-based volume forms, are naturally defined on the graph of the registration diffeomorphism f, rather than on the domain of f. When pulled back to the domain of f they involve the Jacobian Jf and therefore appear “non-standard”. In the third part of the paper, graph-based volume forms are analyzed in terms of four key objective-function properties: symmetry, positive-definiteness, invariance, and lack of bias. Graph-based volume forms whose associated L2 objective functions have the first three properties ar