Comparison to other methods

Other investigators have proposed methods for enforcing pairwise consistent transformations. For example, Woods et al. [27] computes all pairwise registrations of a population of image volumes using a linear transformation model, i.e., a $3 \times 3$ matrix transformation. They then average the transformation from $T$ to $S$ with all the transformations from $T$ to $X$ to $S$. The original transformation from $T$ to $S$ is replaced with average transformation. The procedure is repeated for all the image pairs until convergence. This technique is limited by the fact that it can not be applied to two data sets. Also, there is no guarantee that the generated set of consistent transformations are valid. For example, a poorly registered pair of images can adversely effect all of the pairwise transformations.

The method proposed in this paper is most similar to the heuristic approach described by Thirion [6]. Thirion's idea was to iteratively estimate the forward $h$, reverse $g$, and residual $r=h \circ g$ transformations in order to register the images $T$ and $S$. At each iteration, half of the residual $r$ is added to $h$ and half of the residual $r$ is mapped through $h$ and added to $g$. After performing this operation, $h \circ g$ is close to the identity transformation. The advantage of Thirion's method is that it enforces the inverse consistency constraint without having to explicitly compute the inverse transformations as in Eq. 3. The residual method is an approximation to the inverse consistency method in that the residual method approximates the correspondences between the forward and reverse transformations while the inverse consistency method computes those correspondences. Thus, the residual approach only works under a small deformation assumption since the residual is computed between points that do not correspond to one another. This drawback limits the residual approach to small deformations and it therefore can not be extended to nonlinear transformation models. On the other hand, the approach presented in this paper can be extended to the nonlinear case by modifying the procedure used to calculate the inverse transformation to include nonlinear transformations.

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