Having defined the framework for curve shape matching, we now construct a sulcal atlas of a large population of curves in the shape space. However, in order to do so, we need the notion of a shape average based on the sulcal and gyral curves. Owing to the nonlinearity of the shape space, the computation of an average shape is not straightforward. There are two well known approaches of computing statistical averages in such spaces. The extrinsic shape average is computed by projecting the elements of the shape space in the ambient linear space, where an Euclidean average is computed, and subsequently projected back to the shape space. On the other hand, the intrinsic average, also known as the Karcher mean ([1
]) is computed directly on the shape space, and makes use of distances and lengths that are defined strictly on the manifold. It uses the geodesics defined via the exponential map, and iteratively minimizes the average geodesic variance of the collection of shapes. The extrinsic average is simple and efficient to calculate, however it ignores the nonlinearity of the shape space, as well as depends upon the method used for embedding the manifold in the Euclidean space. As a simple example, a unit circle can be embedded inside 2
in several ways, and each embedding possibly leads to different values of extrinsicmeans on the circle. We will adopt the intrinsic approach by computing the Karcher mean for a given set of shapes. Algorithm 1 describes the procedure for computing the Karcher mean of a collection of shapes under the q
Using Algorithm 1, we now construct a sulcal shape atlas for a population of 176 subjects. These subjects underwent high-resolution T1-weighted MRI scans with spatial resolution 1×1×1 mm3
on a Siemens 1.5T scanner. After preprocessing the raw data, and registering it stereotaxically to a standard atlas space, the cortical surfaces for these subjects were extracted using an automated algorithm [6
]. For each of these subjects, a total of 28 landmark curves were manually traced. shows the original 28 landmark curves for each of the 176 subjects for the left hemisphere overlaid together. For the remainder of this paper, we will demonstrate the results on the left hemisphere of the brain to simplify things, although the same procedure can be repeated for the right hemisphere as well. also shows the intrinsic sulcal shape averages of the 28 landmark curves using Algorithm 1, as well as the respective extrinsic Euclidean averages for the same. While computing the extrinsic average, each curve for the same landmark type was mapped to its q
representation, thus making it scale and translation invariant. The Euclidean average of all the q
functions was then computed after a pairwise rotational alignment. Both the Karcher mean shapes as well as the Euclidean averages were then mapped back to the curve space in order to visualize them.
Lateral, frontal, and medial views of, Top row: 28 landmark sulci and gyri for 176 subjects, Middle row: Euclidean sulcal shape averages for each landmark, Bottom row: Karcher means for each landmark. (Best viewed in color).
It is observed that the intrinsic averages although smooth, have preserved important features along the landmarks, thus representing the average local shape geometry along the sulci and gyri. This also implies that the shape average has not only captured the salient geometric features, but has also reduced the shape variability in the population. In order to demonstrate this property, we plot the variance of the shape deformation for each landmark type as captured by the velocity vector along the geodesic path, both for Euclidean extrinsic, and Riemannian intrinsic averages. Thus for each of the 28 landmark average shapes for 176 subjects, i, i
= 1, … , 28, we plot the quantity
, where χ
) given in Algorithm 1, line 6. This quantity measures the invariant deformation between a pair of shapes, and only depends upon the intrinsic geometry of the shapes. shows a comparison of the plots of
) for each of the landmarks, taken along the length of the curve, for both Euclidean shape averages, as well as intrinsic shape averages. This can also be thought of as the geodesic variance. From the color-coded map, it is observed that the intrinsic average has reduced the variance in terms of shape geometry deformation, and thus is a better representative of the population. This has important implications for surface registration as is demonstrated in the following section.
Comparison of the geodesic variance for the entire sulcal population for each of the 28 landmarks, both for Euclidean shape averages, as well as elastic shape averages, along the length of the curves.