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.
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.
A class of implementations of mutual information (MI) based image registration estimate MI from the joint histogram of
the overlap of two images. The consequence of this approach is that the MI estimate thus obtained is not overlap
invariant: its value tends to increase when the overlapped region is getting smaller. When the two images are very noisy
or are so different that the correct MI peak is very weak, it may lead to incorrect registration results using the
maximization of mutual information (MMI) criterion. In this paper, we present a new joint histogram estimation scheme
for overlap invariant MI estimation. The idea is to keep it a constant the number of samples used for joint histogram
estimation. When one image is completely within another, this condition is automatically satisfied. When one image
(floating image) partially overlaps another image (reference image) after applying a certain geometric transformation, it
is possible that, for a pixel from the floating image, there is no corresponding point in the reference image. In this case,
we generate its corresponding point by assuming that its value is a random variable following the distribution of the
reference image. In this way, the number of samples utilized for joint histogram estimation is always the same as that of
the floating image. The efficacy of this joint histogram estimation scheme is demonstrated by using several pairs of
remote sensing images. Our results show that the proposed method is able to produce a mutual information measure that
is less sensitive to the size of overlap and the peak found is more reliable for image registration.
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