FIGURE 9 Scheme to match cortical regions with high-dimensional transformations and color-coded spherical maps. Highresolution surface models of the cerebral cortex are extracted in parametric form, which produces a continuous, invertible one-to-one mapping between cortical surface points, (a), and their counterparts on a sphere. To find matches between cortical regions in different subjects [(a)/(d)j, a Dirichlet problem is framed in the parametric space [26, 111, 113, 115]. Each point in the spherical map, (b), is color-coded at 16 bits per channel with a color value that represents the location of its counterpart on the convoluted surface. When spherical maps are made from two different cortical surfaces, the respective sulci will be in different positions in each spherical map [(b),(c)], reflecting their different locations on the folded brain surface. Using a vector-valued flow field defined on the sphere (c), the system of sulcal curves in one spherical map is driven into exact correspondence with their counterparts in the target spherical map, guiding the transformation of the adjacent regions. The effect of the transformation is illustrated in (c) by its effect on a uniform grid, ruled over the starting spherical map and passively carried along in the resultant deformation. Complex nonlinear flow is observed in superior temporal regions, as the superior temporal sulcus (STS) extends further posteriorly in the target brain, and the posterior upswing of the Sylvian fissure (SYLV) is more pronounced in the reference brain (a) than in the target (d). Outlines are also shown for the superior frontal sulcus (SFS) and central sulcus (CENT), which is less convoluted in the reference brain than in the target. Because the color-coded spherical maps index cortical surface locations in 3D, the transformation is recovered in 3D stereotaxic space as a displacement of points in one subject's cortex onto their counterparts in another subject (Fig. 12). See also Plate 86.

L(u(r))+ F (r — u(r)) = 0, Vr e fl, u(r) = u0(r), Vr e M0 U M1. (22)

Here M0, M1 are sets of points and (sulcal and gyral) curves where vectors u(r) = u0(r) matching regions of one anatomy with their counterparts in the other are known, and L and F are 2D equivalents of the differential operators and body forces defined earlier. Unfortunately, the recovered solution x^Dq(Fpq(Dp—1(x))) will in general be prone to variations in the metric tensors gjk (rp) and gjk (rq) of the mappings Dp and Dq (Fig. 10). Since the cortex is not a developable surface [26], it cannot be given a parameterization whose metric tensor is uniform. As in fluid dynamics or general relativity applications, the intrinsic curvature of the solution domain should be taken into account when computing flow vector fields in the cortical parameter space and mapping one mesh surface onto another.

To counteract this problem, we developed a covariant formalism [115] that makes cortical mappings independent of how each cortical model is parameterized. Although spherical, or planar, maps involve different amounts of local dilation or contraction of the surface metric, this metric tensor field is stored and used later, to adjust the flow that describes the mapping of one cortex onto another. The result is a covariant regularization approach that makes it immaterial whether a spherical or planar map is used to perform calculations. Flows defined on the computational domain are adjusted for variations in the metric tensor of the mapping, and the results become independent of the underlying parameterization (i.e., spherical or planar).

The covariant approach was introduced by Einstein [38] to allow the solution of physical field equations defined by elliptic operators on manifolds with intrinsic curvature. Similarly, the problem of deforming one cortex onto another involves solving a similar system of elliptic partial differential equations [26, 35, 115], defined on an intrinsically curved computational mesh (in the shape of the cortex). In the covariant formalism, the differential operators governing the mapping of one cortex to another are adaptively modified to reflect changes in the underlying metric tensor of the surface parameterizations (Fig. 11).

Cortical surfaces are matched as follows. We first establish the cortical parameterization, in each subject, as the solution of a time-dependent partial differential equation (PDE) with a spherical computational mesh (Eq. (19); Fig. 8; [26, 11]). This procedure sets up an invertible parameterization of each surface in deformable spherical coordinates (Fig. 9), from which the metric tensors gjk (rp) and gjk (r?) of the mappings are computed. The solution to this PDE defines a Riemannian manifold [10]. In contrast to prior approaches, this Riemannian manifold is then not flattened (as in [35,123]), but is used directly as a computational mesh on which a second PDE is defined (see Fig. 11). The second PDE matches sulcal networks from subject to subject. Dependencies between the metric tensors of the underlying surface parameterizations and the matching field itself are eliminated using generalized coordinates and Christoffel symbols [115]. In the PDE formulation, we replace L by the covariant differential operator L1. In L1, all L's partial derivatives are replaced with covariant derivatives. These covariant derivatives3 are defined with respect to the metric tensor of the surface domain where calculations are performed.

Using this method, surface matching can be driven by anatomically significant features, and the mappings are independent of the chosen parameterization for the surfaces being matched. High spatial accuracy of the match is guaranteed in regions of functional significance or structural complexity, such as sulcal curves and cortical landmarks (Fig. 11). Consequently, the transformation of one cortical surface model onto another is parameterized by one translation vector for each mesh point in the surface model, or 3 x 65536«

3The covariant derivative of a (contravariant) vector field, u' (x), is defined as u' k = 8uj/8xk + T^ iku' [38] where the P ik are Christoffel symbols of the second kind. This expression involves not only the rate of change of the vector field itself, as we move along the cortical model, but also the rate of change of the local basis, which itself varies due to the intrinsic curvature of the cortex (cf [65]). On a surface with no intrinsic curvature, the extra terms (Christoffel symbols) vanish. The Christoffel symbols are expressed in terms of derivatives of components of the metric tensor gjk (x), which are calculated from the cortical model, with T^ = (l/2)g'l (8gj/8xk + 8glk/8xj — 8gjk/8x'). Scalar, vector, and tensor quantities, in addition to the Christoffel symbols required to implement the diffusion operators on a curved manifold, are evaluated by finite differences. These correction terms are then used in the solution of the Dirichlet problem for matching one cortex with another. A final complication is that different metric tensors gjk (rp) and gjk (rq) relate (1) the physical domain of the input data to the computation mesh (via mapping Dp—1) and (2) the solution on the computation mesh to the output data domain (via mapping Dq). To address this problem (Fig. 9), the PDE u(rq ) = —F is solved first, to find a flow field T^: r^-r — u(r) on the target spherical map with anatomically-driven boundary conditions u(r^) = u0(r^), Vr^ e M0 U M1. Here L11 is the covariant adjustment of the differential operator L with respect to the tensor field gjk (r^) induced by Dq. Next, the PDE L^u(rp) = —F is solved, to find a reparameterization Tp: r^-r — u(r) of the initial spherical map with boundary conditions u(rp) = 0, Vrp e M0 U M1. Here L1 is the covariant adjustment of L with respect to the tensor field gjk (rp) induced by Dp. The full cortical matching field (Fig. 12) is then defined as x^Dq(Fpq(Dp—1(x))) with Fpq = (Tq)—1 o(Tp)—1.

FIGURE 10 High-dimensional matching of cortical surfaces and sulcal networks. 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. A cascade of mathematical mappings is required to achieve this. 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 (Dp-1, Dq-1) to connected networks of curved sulci in each subject transforms the problem into one of computing an angular flow vector field Fpq, in spherical coordinates, which drives the network elements into register on the sphere [111]. To greatly accelerate computation of the overall mappings DqFpqDp-1, the forward mapping Dq is preencoded via the mapping Iq-lKq as a three-channel floating-point array (shown in color) defined on the codomain of Fpq. The full mapping DqFpqDp-1 can be expressed as a displacement vector field that drives cortical points and regions in brain P into precise structural registration with their counterparts in brain Q. See also Plate 87.

FIGURE 10 High-dimensional matching of cortical surfaces and sulcal networks. 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. A cascade of mathematical mappings is required to achieve this. 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 (Dp-1, Dq-1) to connected networks of curved sulci in each subject transforms the problem into one of computing an angular flow vector field Fpq, in spherical coordinates, which drives the network elements into register on the sphere [111]. To greatly accelerate computation of the overall mappings DqFpqDp-1, the forward mapping Dq is preencoded via the mapping Iq-lKq as a three-channel floating-point array (shown in color) defined on the codomain of Fpq. The full mapping DqFpqDp-1 can be expressed as a displacement vector field that drives cortical points and regions in brain P into precise structural registration with their counterparts in brain Q. See also Plate 87.

Covariant Matching

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