Our aim was to develop an intensity-based non-rigid matcher that would be fast enough to use during image guided surgery. In particular, our aim was to develop methods whose dominant computational complexity do not scale like the number of voxels in an image. We accomplished those goals. We developed methods based on an adaptive tetrahedral mesh that is used to represent a displacement field. Those methods were able to non-rigidly warp three dimensional images into each other in roughly 5 minutes. The algorithm made errors that generally less than 2 millimeters, which is comparable to other methods (FNM+01).
Chapter 7 Conclusion
In this thesis, we argue that by making compact representations of anatomical fea-tures, we can make more efficient and more effective algorithms. We began by noting that voxel based representations inefficiently describe uniform regions found in seg-mented anatomical structures. Furthermore those representations describe the surface of anatomical structures as unrealistic jagged edges. We therefore developed a com-pact tetrahedra-based representation of an anatomical structure. The representations we created used fewer tetrahedra than the voxel-based representations used voxels.
Furthermore, the tetrahedra-based representation could straightforwardly describe a smooth surface.
We then used the compact representations we developed to create free-form non-rigid shape matcher using a linear elastic regularization term. We overcame the ill-conditioned equations that resulted to create a non-rigid matcher that typically converged in thirty seconds to a minute on a desktop machine.
We used the results of that non-rigid registration algorithm to perform morpho-logical studies of shape. We showed that the correspondences found by the non-rigid registration algorithm led to classifiers with improved performance over classifiers based on distance maps. We furthermore showed that the correspondences we found led to effective linear classifiers, while distance maps did not.
Not only did the non-rigid shape registration algorithm lead to effective morpho-logical studies, it also led to an effective deformable model for segmentation. We
cre-ated a deformable model based on the non-rigid registration of a group of anatomical structures and merged that deformable model with an intensity based segmentation method. The result was a more accurate final method than intensity based methods alone. In particular, the segmentations found by intensity based method alone in-cluded disconnected regions of tissue, jagged edges, and unrealistic protrusions. Our combined method produced smooth surfaces and one connected region of tissue. Fur-thermore, our method achieved greater agreement with manual segmentations than the intensity based method alone.
Finally, we created a novel, adaptive method to non-rigidly register medical im-ages. That technique uses a tetrahedral representation of a displacement field that adapts both to anatomy and to the displacement field between images. The result-ing method had a computational cost that was dominated by the number of nodes in the mesh, rather than the number of voxels in the image. The resulting method converged in about five minutes, making it fast enough for use during surgery.
Clinically, we validated other’s results on finding the correlation of shape of the amygdala-hippocampus with Schizophrenia. Furthermore, we presented new results showing a correlation between the shape of the thalamus and first episode schizophre-nia. We developed a full brain segmentation algorithm and validated the accuracy of that method in segmenting the thalamus. Finally, we developed a non-rigid registra-tion tool for use during surgery.
7.1 Future Directions of Research
In this thesis, we discussed representational issues in medical images. We plan to continue pursuing that direction of research. In particular, we have pointed out that segmented anatomical shapes have uniform intensity regions. Medical images have nearly uniform textures or uniform intensity over large regions; it seems inefficient to describe such regions using a voxel representation. We plan to investigate representing a gray scale image using volumetric elements – it should be possible to represent an image as low order polynomials over tetrahedra. The large regions of uniform intensity
could be represented very efficiently. Though, it will be very challenging to find an efficient representation in regions of the image where the intensity varies a lot; in these regions volumetric elements may not be an efficient representation.
A second direction of research we would like to pursue concerns hierarchical vol-umetric representations of anatomical objects. Such a representation could be very powerful for non-rigidly matching complicated shapes between subjects such as the cortical folds. In particular, the non-rigid shape matcher developed in Chapter 3 has very little chance of succeeding in matching the cortical fold because of the strong likelihood that a cortical fold in the deforming structure would be attracted to the wrong cortical fold in the target structure. A hierarchical method, where an align-ment is done one level at a time, may be able to overcome that problem. Furthermore, much of the technology needed to create such a representation was already developed in Chapter 6.
A hierarchical representations may also be particularly useful for comparing anatom-ical structures. Nodes in a hierarchanatom-ical mesh in one level of the mesh can be introduced based on local coordinate system relative to the coarser level of the mesh. Such a representation has the advantage of having a dependence on global orientation only in its coarsest level. As discussed in Chapter 4, choosing the “correct” orientation is often a problem when comparing shapes. A hierarchical description using this type of relative coordinate system could strongly mitigate these problems; on the coarsest level would have any dependence on orientation.
There is a second direction of research we intend to pursue. In our view, choosing a good representation is only part of the challenge of developing efficient and effective methods. Another part of the challenge is using a good metric to compare those representations. In Chapter 4 and Chapter 5, we used Euclidean metrics to compare displacement fields or node locations. As displacement fields form a vector space (Section 4.1.3), a Euclidean metric is probably appropriate.
However, we are also interested in comparing tensors. In particular, we desire to compare local strain or stress tensors between subjects, rather than comparing the vector field. Furthermore, researchers at Brigham and Women’s hospital are
collect-ing diffusion tensor data across many subjects which they would like to compare.
For tensor comparisons, we believe that the Amari metric (Ama98) or the metric recently introduced by Miller and Chefd’hotel (MC03) are likely appropriate choices to compare strain matrices. Both of these choices have been designed to account for the group structure of tensors. We expect improved performance using metrics that account for group structure over metrics that do not.