We presented in this chapter a method to extract reliable differential geometry features (crest lines and extremal points) from 3D images and several rigid registration algorithms to put into correspondence these features in two different images and to compute the rigid transformation between them. We also presented an analysis of the robustness with the computation of the probability (or mean number) of false positives and an analysis of the accuracy of the transformation.

This method proves to be very powerful for monomodal rigid registration of the same patient imaged at different times, as we show that an accuracy of less than a tenth of voxel can be achieved. In the last experiment, we showed that this uncertainty estimation technique is precise enough to put into evidence systematic biases on the order of 0.1 voxel between features in echo-1 and echo-2 images. Once corrected, multiple experiments on several patients show that our uncertainty prediction is validated on real data.

This registration technique is currently used in many medical applications, such as the registration of a preoperative MR used for planning and MR with a stereotactic frame for neurosurgery navigation (European Community project Roboscope), or the registration of a series of acquisitions over time of images of multiple sclerosis patients to study the disease's evolution (European Community project Biomorph).

Several tracks have been followed to generalize this work to

TABLE 1 Theoretical and observed values of the real validation index with bias for different patients"

Theoretical values 6 VT2 = 3.46 0.01-1 n < 24 n *(n - 1)/2

Patient 1 6.29 4.58 0.14 15 105

Patient 2 5.42 3.49 0.12 18 153

Patient 3 6.50 3.68 0.25 14 91

Patient 4 6.21 3.67 0.78 21 210

"The number of registrations (which is also the number of values used to compute the statistics on the validation index) is directly linked to the number of images used. Results indicate a very good validation of the registration accuracy prediction: The mean validation index is within 10% of its theoretical value and the K-S test exhibits impressively high values.

nonrigid registration. Feldmar [11] used the principal curvatures to register surfaces with rigid, affine, and locally affine transformations. Subsol designed an algorithm for nonrigid matching of crest lines. In [36], he used this method to warp 3D MR images of different patients' brains in the same coordinate system, and even to warp an atlas onto a given 3D MR image of a brain in order to segment it. In [35], he used the same method to construct automatically a morphometric atlas of the skull crest lines from several acquisitions of different patients' CT images, with applications in craniofacial surgery.

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