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etal. [76] extract a 3D skeletonized representation of deep sulci and parse it into an attributed relational graph of connected surface elements. They then define a syntactic energy on the space of associations between the surface elements and anatomic labels, from which estimates of correct labelings (and therefore correct matches across subjects) can be derived.

2.13 Surface-Based Approaches

Ultimately, accurate warping of brain data requires the following.

(1) Matching entire systems of anatomic surface boundaries, both external and internal

(2) Matching relevant curved and point landmarks, including ones within the surfaces being matched (e.g., primary sulci at the cortex, tissue type boundaries at the ventricular surface)

In our own model-driven warping algorithm [111,113,115], systems of model surfaces are first extracted from each dataset and used to guide the volumetric mapping. The model surfaces include many critical functional interfaces, as well as numerous cytoarchitectonic and lobar boundaries in three dimensions. Both the surfaces and the landmark curves within them are reconfigured and forced to match their counterparts in the target datasets exactly. We will discuss this approach in some detail.

Since much of the functional territory of the human cortex is buried in the cortical folds or sulci, a generic structure is built to model them (Figs 4, 5; [111]), incorporating a priori topological and shape information about the deep sulcal pattern. The underlying data structure consists of a connected system of surface meshes, in which the individual meshes are parametric and have the form of complex 3D sheets that divide and join at curved junctions to form a network of connected surfaces. Separate surfaces are used to model the deep internal trajectories of features such as the parieto-occipital sulcus, the anterior and posterior calcarine sulcus, the Sylvian fissure, and the cingulate, marginal, and supracallosal sulci in both hemispheres. Additional major gyral and sulcal boundaries are represented by parameterized curves lying in the cortical surface. The ventricular system is modeled as a closed system of 14 connected surface elements whose junctions reflect cytoarchitectonic boundaries of the adjacent tissue (for details, see Thompson and Toga [115]). Information on the meshes' spatial relations, including their surface topology (closed or open), anatomical names, mutual connections, directions of parameterization, and common 3D junctions and boundaries, is stored in a hierarchical graph structure. This ensures the

FIGURE 4 Connected surface systems used to drive the warp. Models of deep structures are used to guide the mapping of one brain to another (data from [111]). Deep sulcal surfaces include the anterior and posterior calcarine (CALCa/p), cingulate (CING), parieto-occipital (PAOC), and callosal (CALL) sulci and the Sylvian fissure (SYLV). Also shown are the superior and inferior surfaces of the rostral horn (VTSs/i) and inferior horn (VTIs/i) of the right lateral ventricle. Ventricles and deep sulci are represented by connected systems of rectangularly parameterized surface meshes, whereas the external surface has a spherical parameterization that satisfies the discretized system of Euler-Lagrange equations used to extract it. Connections are introduced between elementary mesh surfaces at known tissue-type and cytoarchitectural field boundaries, and at complex anatomical junctions (such as the PAOC/CALCa/CALCp junction shown here). Color-coded profiles show the magnitude of the 3D deformation maps warping these surface components (in the right hemisphere of a 3D T1-weighted SPGR MRI scan of an Alzheimer's patient) onto their counterparts in an identically acquired scan from an age-matched normal subject. See also Plate 83.

continuity of displacement vector fields defined at mesh junctions.

Surface parameterization, or imposition of an identical regular structure on anatomic surfaces from different subjects (Fig. 5), provides an explicit geometry that can be exploited to drive and constrain the correspondence maps which associate anatomic points in different subjects. Structures that can be extracted automatically in parametric form include the external cortical surface (discussed in Section 3), ventricular surfaces, and several deep sulcal surfaces. Recent success of sulcal extraction approaches based on deformable surfaces [122] led us to combine a 3D skeletonization algorithm with deformable curve and surface governing equuationations to automatically produce parameterized models of cingulate, parieto-occipital, and calcarine sulci, without manual initialization [136]. Additional, manually segmented surfaces can also be given a uniform rectilinear parameterization using algorithms described in

FIGURE 5 Mesh construction and matching. The derivation of a standard surface representation for each structure makes it easier to compare and analyze anatomical models from multiple subjects. An algorithm converts points (dots, top right panel) on an anatomical structure boundary into a parametric grid of uniformly spaced points in a regular rectangular mesh stretched over the surface [110]. Computation of anatomic differences between subjects requires transformation tools that deform connected systems of mesh-based surface models representing structures in one subject's anatomy, into correspondence with their counterparts in the anatomy of another subject. This mapping is computed as a surface-based displacement map (right panel), which deforms each surface locally into the shape of its counterpart. Maintenance of information on surface connectivity guarantees accurate mapping of curved junctions among surfaces, under both the surface-based and subsequent volumetric transformations. Note. Matching of surfaces with a spherical parameterization requires separate methods, which deal with the matching of curved internal landmarks (Section 3).

FIGURE 5 Mesh construction and matching. The derivation of a standard surface representation for each structure makes it easier to compare and analyze anatomical models from multiple subjects. An algorithm converts points (dots, top right panel) on an anatomical structure boundary into a parametric grid of uniformly spaced points in a regular rectangular mesh stretched over the surface [110]. Computation of anatomic differences between subjects requires transformation tools that deform connected systems of mesh-based surface models representing structures in one subject's anatomy, into correspondence with their counterparts in the anatomy of another subject. This mapping is computed as a surface-based displacement map (right panel), which deforms each surface locally into the shape of its counterpart. Maintenance of information on surface connectivity guarantees accurate mapping of curved junctions among surfaces, under both the surface-based and subsequent volumetric transformations. Note. Matching of surfaces with a spherical parameterization requires separate methods, which deal with the matching of curved internal landmarks (Section 3).

Thompson et al. [109,110] and used to drive the warping algorithm. Each resultant surface mesh is analogous in form to a uniform rectangular grid, drawn on a rubber sheet, which is subsequently stretched to match all data points. Association of points on each surface with the same mesh coordinate produces a dense correspondence vector field between surface points in different subjects. This procedure is carried out under very stringent conditions,2 which ensure that landmark curves and points known to the anatomist appear in corresponding locations in each parametric grid.

2.16 Displacement Maps

For each surface mesh M;p in a pair of scans Ap and Aq we define a 3D displacement field

carrying each surface point r;p(u, v) in Ap into structural correspondence with rrq(u, v), the point in the target mesh parameterized by rectangular coordinates (u, v). This family of high-resolution transformations, applied to individual meshes in a connected system deep inside the brain, elastically transforms elements of the surface system in one 3D image to their counterparts in the target scan.

As in approaches based on matching points and curves, the surface-based transformation can be extended to the full volume in a variety of ways. In one approach [111], weighted linear combinations of radial functions, describing the influence of deforming surfaces on points in their vicinity, extend the surface-based deformation to the whole brain volume (see Fig. 6). For a general voxel x in the scan Ap to be transformed, we let Sjp(x) be the distance from x to its nearest point(s) on each surface mesh M;p, and let the scalars yjp(x) e [0,1] denote the weights {1AP(x)}/ Ei=1toi{1AP(x)}. Then Wpq(x), the displacement vector which takes a general point x in scan Ap onto its counterpart in scan Aq, is given by the linear combination of functions:

Wpq(x) = Ei=1toL ^(x) • Dp(np,p(x)), for all xe Ap. (18)

2 For example, the calcarine sulcus (see Fig. 4) is partitioned into two meshes (CALCa and CALCp). This ensures that the complex 3D curve forming their junction with the parieto-occipital sulcus is accurately mapped under both the surface displacement and 3D volumetric maps reconfiguring one anatomy into the shape of another. Figure 5b illustrates this procedure, in a case where three surface meshes in one brain are matched with their counterparts in a target brain. A separate approach (discussed later, Section 3) is used to match systems of curves lying within a surface with their counterparts in a target brain.

FIGURE 6 Volume warp calculation. The volumetric transformation Wpq (x), of an arbitrary point x in a scan, p, to its counterpart in another scan, q, is expressed as a weighted linear combination of distortion functions associated with each surface. Within a surface Sf, the relative contribution of each point in the projected patch {np{[B(x; rc)]} to the elastic transformation at x is given a relative weight wi. The distortion at x due to surface Si is given by D{pq(x) = {fr e B wpWipq dr }/ {IreB Wp dr }, where the W¡pq are the displacement maps defined on each surface (Fig. 5(b)). The volume warp Wpq(x) is a weighted average (over i) of the Dp (x), depending on the relative distance yi (x) of x from its near-points on each surface S{. (Adapted from [111].)

Here the Dtpq are distortion functions (Fig. 6) due to the deformation of surfaces close to x, given by

Wrpq[np,p(r)] is the (average) displacement vector assigned by the surface displacement maps to the nearest point(s) nplp(r) to r on Mrp. Rc is a constant, and B(x; rc) is a sphere of radius rc = min{Rc, min{^;p(r)}}. The wf are additional weight functions defined as w,p(x, S,p(r)) = exp(-{d(npp(r), x)/S,p(x)}2), (20)

where d(a, b) represents the 3D distance between two points a and b. The Jacobian of the transformation field at each point x is tracked during the computation, as recommended by Christensen et al. [16]. In rare cases where the transformation is locally singular, the vector field computation is discretized in time, and the deformation field is reparameterized at successive time steps, as suggested in Christensen et al. [17]. Intermediate surface blends (1 — t)rlp(u, v) + trlq(u, v), t e [0,1], are generated for every surface, and these surfaces are uniformly l'.i i'

FIGURE 6 Volume warp calculation. The volumetric transformation Wpq (x), of an arbitrary point x in a scan, p, to its counterpart in another scan, q, is expressed as a weighted linear combination of distortion functions associated with each surface. Within a surface Sf, the relative contribution of each point in the projected patch {np{[B(x; rc)]} to the elastic transformation at x is given a relative weight wi. The distortion at x due to surface Si is given by D{pq(x) = {fr e B wpWipq dr }/ {IreB Wp dr }, where the W¡pq are the displacement maps defined on each surface (Fig. 5(b)). The volume warp Wpq(x) is a weighted average (over i) of the Dp (x), depending on the relative distance yi (x) of x from its near-points on each surface S{. (Adapted from [111].)

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