Cortical surfaces can be matched using a procedure that also matches large networks ofgyral and sulcal landmarks with their counterparts in the target brain [26,35,111,113,123]. Differences in the serial organization of cortical gyri prevent exact gyrus-by-gyrus matching of one cortex with another, but an important intermediate goal has been to match a comprehensive network of sulcal and gyral elements that are consistent in their incidence and topology across subjects. Elastic matching of primary cortical regions factors out a substantial component of confounding cortical variance in functional imaging studies, as it directly compensates for drastic variations in cortical patterns across subjects [100,114,129]. Quantitative comparison of cortical models can also be based on the mapping that drives one cortex with another [114,123]. Because of its role in pathology detection algorithms (Section 4), we focus on this mapping in some detail.
Our method  is conceptually similar to that ofDavatzikos . 3D active surfaces (Fig. 8; ) are used to automatically extract parametric representations of each subject's cortex, on which corresponding networks of anatomical curves are identified. The transformation relating these networks is expressed as a vector flow field in the parameter space of the cortex (Fig. 9). This vector flow field in parameter space indirectly specifies a correspondence field in three dimensions, which drives one cortical surface into the shape of another.
Several algorithms have been proposed to extract cortical surface models from 3D MR data [26,28,95,98,134]. In one algorithm [73,74], a spherical mesh surface is continuously deformed to match a target boundary defined by a threshold value in the continuous 3D MR image intensity field (Fig. 8). Evolution of the deformable surface is constrained by systems of partial differential equations. These equations have terms that attract the parametric model to regions with the preselected intensity value, while penalizing excessive local bending or stretching. If an initial estimate of the surface v0(s, r) is provided as a boundary condition (see Thompson and Toga  for details), the final position of the surface is given by the solution (as t— ro) of the Euler-Lagrange evolution equation:
+ 202/8s8r (wn|82 v/8s8r |) + 82/8s2(w20|82v/8s2|) + 82/8r2(w02|82v/8r2|) = F (v). (21)
Here F (v) is the sum of the external forces applied to the surface, and the w^ terms improve the regularity of the surface. The spherical parameterization of the deforming surface is maintained under the complex transformation, and the resulting model of the cortex consists of a high-resolution mesh of discrete triangular elements that tile the surface (Fig. 8).
Because the cortical model is obtained by deforming a spherical mesh, any point on the cortical surface (Fig. 9a) must map to exactly one point on the sphere (Fig. 9b), and vice versa. Each cortical surface is parameterized with an invertible mapping Dp, Dq: (r, s) — (x, y, z), so sulcal curves and landmarks in the folded brain surface can be reidentified in the spherical map (cf. Sereno et al.  for a similar approach; Fig. 9b). To retain relevant 3D information, cortical surface point position vectors in 3D stereotaxic space are color-coded, to form an image of the parameter space in color image format [Fig. 9(b)/(c)]. To find good matches between cortical regions in different subjects [Fig. 9(a)/(d)], we first derive a spherical map for each respective surface model [Fig. 9(b)/(c)] and then perform the matching process in the spherical parametric space, O. The parameter shift function u(r): O—O, is given by the solution Fpq: r—r + u(r) to a curve-driven warp in the biperiodic parametric space O =[0, 2re) x [0, re) of the external cortex (cf. [26, 35, 41, 123]). This warp can be set up in a variety of ways. Spherical harmonic functions are an orthonormal basis on the sphere (they are eigenfunctions of the Laplacian operator) and can be used extend this curve-based deformation to the whole surrounding spherical map . An alternative, mathematically more challenging approach is to use the regularization approach introduced earlier, with several modifications.
At first glance it seems that the previous approach, using ordinary differential operators to constrain the mapping, can be applied directly. For points r = (r, s) in the parameter space, a system of simultaneous partial differential equations can be written for u(r) using
3D Deformation fc^ Field
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