Symmetric Non-rigid Registration: A Geometric Theory and Some Numerical Techniques

This paper proposes ℒ 2 - and information-theory-based (IT) non-rigid registration algorithms that are exactly symmetric . Such algorithms pair the same points of two images after the images are swapped. Many commonly-used ℒ 2 and IT non-rigid registration algorithms are only approximately 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 objective function asymmetry is due to the use of standard differential volume forms on the domain of the images. The second 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 J f 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 ℒ 2 objective functions have the first three properties are completely classified. There is an infinite-dimensional space of such graph-based forms. But within this space, up to scalar multiple, there is a unique volume form whose associated ℒ 2 objective function is unbiased. This volume form, which when pulled back to the domain of f is (1+det( J f )) times the standard volume form on Euclidean space, is exactly the differential-geometrically natural volume form on the graph of f . The fourth part of the paper shows how the same volume form also makes the IT objective functions symmetric, positive semi-definite, invariant, and unbiased. The fifth part of the paper introduces a method for removing asymmetry in numerical computations and presents results of numerical experiments. The new objective functions and numerical method are tested on a coronal slice of a 3-D MRI brain image. Numerical experiments show that, even in the presence of noise, the new volume form and numerical techniques reduces asymmetry practically down to machine precision without compromising registration accuracy.