In this paper, we present the latest results of the development of a novel non-rigid image registration method
(NiRuDeGG) using a well-established mathematical framework known as the deformation based grid generation. The
deformation based grid generation method is able to generate a grid with desired grid density distribution which is free
from grid folding. This is achieved by devising a positive monitor function describing the anticipated grid density in the
computational domain. Based on it, we have successfully developed a new non-rigid image registration method, which
has many advantages. Firstly, the functional to be optimized consists of only one term, a similarity measure. Thus, no
regularization functional is required in this method. In particular, there is no weight to balance the regularization
functional and the similarity functional as commonly required in many non-rigid image registration methods.
Nevertheless, the regularity (no mesh folding) of the resultant deformation is theoretically guaranteed by controlling the
Jacobian determinant of the transformation. Secondly, since no regularization term is introduced in the functional to be
optimized, the resultant deformation field is highly flexible that large deformation frequently experienced in inter-patient
or image-atlas registration tasks can be accurately estimated. Detailed description of the deformation based grid
generation, a least square finite element (LSFEM) solver for the underlying div-curl system, and a fast div-curl solver
approximating the LSFEM solution using inverse filtering, along with several 2D and 3D experimental results are
presented.
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 1 and thus has the potential to achieve higher registration accuracy. Detailed comparison
among the three algorithms is described in the paper.
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