Flat Tensor Maps

3D Models

3D Matching Field

FIGURE 6 Maps of the human cerebral cortex: flat maps, spherical maps, and tensor maps. Extreme variations in cortical anatomy (3D models; top left) present challenges in brain mapping, because of the need to compare and integrate cortically derived brain maps from many subjects. Comparisons of cortical geometry can be based on the warped mapping of one subject's cortex onto another (top right, [109]). These warps can also be used to transfer functional maps from one subject to another, or onto a common anatomic template for comparison. Accurate and comprehensive matching of cortical surfaces requires more than the matching of overall cortical geometry. Connected systems of curved sulcal landmarks, distributed over the cortical surface, must also be driven into correspondence with their counterparts in each target brain. Current approaches for deforming one cortex into the shape of another typically simplify the problem by first representing cortical features on a 2D plane, sphere or ellipsoid, where the matching procedure (i.e., finding u(r 2), shown in figure) is subsequently performed [24, 26, 108]. In one approach [109] active surface extraction of the cortex provides a continuous inverse mapping from the cortex of each subject to the spherical template used to extract it. Application of these inverse maps to connected networks of curved sulci in each subject transforms the problem into one of computing an angular flow vector field u(r2), in spherical coordinates, which drives the network elements into register on the sphere (middle panel, [108]). The full mapping (top right) can be recovered in 3D space as a displacement vector field which drives cortical points and regions in one brain into precise structural registration with their counterparts in the other brain. Tensor maps (middle and lower left): Although these simple two-parameter surfaces can serve as proxies for the cortex, different amounts of local dilation and contraction (encoded in the metric tensor if the mapping, g^(r)) are required to transform the cortex into a simpler two-parameter surface. These variations complicate the direct application of 2D regularization equations for matching their features. A covariant tensor approach is introduced in ([112]; see red box) to address this difficulty. The regularization operator L is replaced by its covariant form L*, in which correction terms (Christoffel symbols, rjt) compensate for fluctuations in the metric tensor of the flattening procedure. A covariant tensor approach [112] allows either flat or spherical maps to support cross-subject comparisons and registrations of cortical data by eliminating the confounding effects of metric distortions that necessarily occur during the flattening procedure. See also Plate 109.

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