Maturational, neuroplastic, and degenerative processes in the central nervous system (CNS) are regionally specific, causing localized variability in brain morphology, within and across groups of healthy individuals and persons with various diseases, [

1], [

2] that can be measured in terms of differences in regional volumes, shapes, and asymmetries of brain regions [

3]. Understanding normal and pathological morphology of the brain, therefore, requires detailed statistical analyses of the shapes of cortical and subcortical regions [

4]-[

13]. Statistical analyses of complex shapes depend critically, however, on the choice of method for formal geometric representation of the morphology of these surfaces.

Geometric representations provide a quantitative, summary descriptor for a morphological surface that can then be used for formal statistical analyses across surfaces in individuals or groups of individuals [

14]-[

17]. A number of geometric representations have been proposed to describe and analyze a shape. For example, after selecting points along the boundary of a region as a descriptor of the surface, principal components analysis (PCA) can quantify its global variability [

18]. Alternatively, a linear sum of the Fourier basis functions can be used to describe the surface, thereby allowing a statistical analysis of an overall shape that is based on the weights of the Fourier basis functions [

19], [

20]. Statistical analyses using discrete, multiscale medial descriptors (

*m*-reps) [

21]-[

23] of the surface can capture both coarse- and fine-scale variability in shape [

14], [

24], [

25]. A representation of shape based on biological landmarks and their warps along thin-plate splines can be used to analyze the global characteristics of shape [

26]. A parametric description of a surface [

27] using spherical harmonics as basis functions can model the surface of a simply connected structure [

28]. Finally, the statistical learning of shapes using distance maps has also been proposed [

29]. All of these methods aim to reduce the complexity of a surface by retaining only a quantitative, summary descriptor that best describes the shape of that surface; hence, each local feature of the surface influences to some extent the value of this global descriptor. Whereas multiscale descriptors can be used to capture fine-scale variability in surface morphology [

30], [

24], [

14], the statistical dependences of these various features are either poorly described or unknown, thus limiting the utility or validity of these descriptors for analyzing surface morphologies.

To overcome the limitations of multiscale descriptors, we use the entire set of points on a discretized surface as its descriptor, thereby allowing the detection of small local differences in surface morphology across individuals or groups of individuals. The statistical variability in morphology at a particular point on the surface can be studied by statistically comparing sets of corresponding points across subjects; using these descriptors for statistical comparisons, therefore, requires the accurate determination of points of correspondence for the surface across individuals. These correspondences across surfaces can be identified by first using affine transformations to coarsely register the surfaces within a common coordinate space [

15]. Point correspondences across the surfaces can then be determined using a number of methods including geodesics [

31], nearest neighbors [

8], warpings based on fluid dynamics or elastic deformations [

3], [

32], measures of curvature [

33], [

34], robust point matching [

35], [

36], or hierarchical attribute matching [

37].

These approaches, however, require the selection of a reference brain with points already identified on the surface of interest to serve as markers for labeling points of correspondence on the surfaces of analogous regions in brains that have been registered to this reference. After determining these points of correspondence, Euclidean distance is calculated between each point on the surface of the reference region and the corresponding point on the surface of each coregistered surface; for a group of subjects, this process generates for the group a set of Euclidean distances calculated for each point on the surface of the reference region. The reference region combined with the sets of distances at every point on its surface yields a complete statistical description of the surface’s morphology for the subjects who belong to that group. To compare surface morphologies across groups of subjects, this same statistical descriptor is calculated for subjects in the second group, which can then be used to compare the sets of distances at each point on the reference surface.

Students’

*t*-statistics [

38] can be used for statistical comparisons of surface morphology across individuals and groups of subjects. An image depicting the

*t*-statistics computed at each point on the surface of a reference region is called a

*t*-map. Usually the null hypothesis to be tested is that no between-group differences will be detected in the signed Euclidean distances (distances for the points on the undeformed surface of each subject’s region that are positioned inside the boundary of the reference region are labeled as negative, whereas distances for points positioned outside of it are labeled as positive) at each point on the reference surface. However, because the number of points on the surface of a reference region will be large (ranging typically from several thousand to several hundred thousand), the false positive (or Type I) error rate quickly becomes unacceptably high [

39]. Although Bonferroni correction [

40], [

41] can be employed to correct for the number of statistical comparisons performed, Bonferroni correction is not the ideal statistical adjustment [

42] because the sets of Euclidean distances are not statistically independent of one another. Bonferroni correction therefore tends to be overly conservative, leading to an unacceptable false negative (or Type II) error rate. On the other hand, nonparametric statistical methods [

43] do not account for the intrinsic curvature of the two-dimensional (2-D) surfaces embedded in

*R*^{3} Euclidean space.

Our statistical model for morphological analyses describes a correlated, smooth random field defined on the surface of the reference region. Comparison of these random fields is desired for valid statistical analyses of morphological surfaces across groups of subjects. Under the null hypothesis of no shape difference across the entire surface, and for a given significance level (

*α*) [

38], if a correct threshold can be computed for the random field, then the

*t*-map can be thresholded to yield statistically significant differences in surface morphology. The smallest significance level at which a voxel is statistically significant is called the probability value, or

*p*-value, for that point. A

*p*-value for a zero-mean and homogeneous Gaussian random field (GRF),

*f*(

*t*), that is defined on

*N*-dimensional Euclidean space,

*R*^{N}, has been developed previously [

44]; this work has been used extensively to detect statistically significant voxels in Functional Magnetic Resonance Imaging (FMRI) datasets [

45], [

46] and has been extended to non-Gaussian random fields [

46], [

47]. These results have also been used to detect statistically significant differences in the shapes of brain regions between groups. This prior work, however, does not account for the intrinsic curvature of surfaces. Additionally, nonparametric statistical methods [

43] have been used for the analyses of surface morphometry and although they make minimal assumptions about the distribution of the samples, how nonparametric methods account for correlations across the surface in Euclidean distances is unclear. Additionally, current nonparametric methods also ignore intrinsic curvature of the surfaces, and therefore, would not correctly address the issue of multiple comparisons. The degree of improvement gained through analyses that account for the curvature of surfaces, as compared to the analyses that ignore it, will depend upon the surface morphology of regions under investigation. For example, because amygdala is a small region, and hence has considerable curvature across its surface, analyses that account for curvature will be more accurate because the random fields are correctly modeled. Because our statistical model for the comparison of morphological surfaces defines a random field on the surface of the reference region, i.e., on a 2-D manifold in three-dimensional (3-D) space, we use previously described, detailed methods [

48] to derive expressions for the statistical analysis of differences in surface morphology across groups of individuals.