Avoiding symmetry-breaking spatial non-uniformity in deformable image registration via a quasi-volume-preserving constraint

The choice of a reference image typically influences the results of deformable image registration, thereby making it asymmetric. This is a consequence of a spatially non-uniform weighting in the cost function integral that leads to general registration inaccuracy. The inhomogeneous integral measure--which is the local volume change in the transformation, thus varying through the course of the registration--causes image regions to contribute differently to the objective function. More importantly, the optimization algorithm is allowed to minimize the cost function by manipulating the volume change, instead of aligning the images. The approaches that restore symmetry to deformable registration successfully achieve inverse-consistency, but do not eliminate the regional bias that is the source of the error. In this work, we address the root of the problem: the non-uniformity of the cost function integral. We introduce a new quasi-volume-preserving constraint that allows for volume change only in areas with well-matching image intensities, and show that such a constraint puts a bound on the error arising from spatial non-uniformity. We demonstrate the advantages of adding the proposed constraint to standard (asymmetric and symmetrized) demons and diffeomorphic demons algorithms through experiments on synthetic images, and real X-ray and 2D/3D brain MRI data. Specifically, the results show that our approach leads to image alignment with more accurate matching of manually defined neuroanatomical structures, better tradeoff between image intensity matching and registration-induced distortion, improved native symmetry, and lower susceptibility to local optima. In summary, the inclusion of this space- and time-varying constraint leads to better image registration along every dimension that we have measured it.