Accurate Inversion of 3-D Transformation Fields

This correspondence addresses the inversion of 3-D transformation fields, which is a problem that typically arises in image warping problems. A topology preserving parametric B-spline-based representation of the deformation field is considered. Topology preservation ensures that the transformation is a one-to-one mapping and consequently that it is invertible. Inverting such transformation fields amounts to solving a system of nonlinear equations. To tackle this problem, we rely on interval analysis techniques. The proposed algorithm yields a solution whose accuracy is user-controlled. This method may be extended to any dense transformation field and also to deformations defined on a grid of points, by considering a projection in the space of topology preserving B-spline-based deformation fields. The performance of the algorithm is illustrated on transformation fields coming from intersubject brain registration.