A new parametric non-rigid image registration method based on Helmholtz's theorem

Helmholtz's theorem states that, with suitable boundary condition, a vector field is completely determined if both of its divergence and curl are specified everywhere. Based on this, we developed a new parametric non-rigid image registration algorithm. Instead of the displacements of regular control grid points, the curl and divergence at each grid point are employed as the parameters. The closest related work was done by Kybic where the parameters are the Bspline coefficients of the displacement field at each control grid point. However, in Kybic's work, it is very likely to result in grid folding in the final deformation field if the distance between adjacent control grid points (knot spacing) is less than 8. This implies that the high frequency components in the deformation field can not be accurately estimated. Another relevant work is the NiRuDeGG method where by solving a div-curl system, an intermediate vector field is generated and, in turn, a well-regularized deformation field can be obtained. Though the present work does not guarantee the regularity (no mesh folding) of the resulting deformation field, which is also suffered by Kybic's work, it allows for a more efficient optimization scheme over the NiRuDeGG method. Our experimental results showed that the proposed method is less prone to grid folding than Kybic's work and that in many cases, in a multi-resolution fashion; the knot spacing can be reduced down to I and thus has the potential to achieve higher registration accuracy. Detailed comparison among the three algorithms is described in the paper.