## Measurements on Surfaces

Up to now, we have been concerned with shape transformations that map a 3D anatomical template to a 3D brain image. It is often more intuitive, however, to consider certain anatomical structures as surfaces embedded in three dimensions. Examples are the cortical mantle, a sulcus, or the boundary of a subcortical structure. For example, one might be

FIGURE 7 Correlation analysis between and any other measurement can be performed. For example, this figure shows the regions in which a significant correlation between measured for men (left) and women (right).

FIGURE 7 Correlation analysis between and any other measurement can be performed. For example, this figure shows the regions in which a significant correlation between measured for men (left) and women (right).

interested in measuring the geodesic, i.e., the minimum distance, between two points on the cortical mantle [32], or the depth [33] or the curvature [21] of a sulcal ribbon. In the remainder of this section, we briefly describe a few examples of measurements performed on the cortical sulci, which we model as thin convoluted ribbons that are embedded in 3D. Figure 3 (bottom) shows examples of four different sulcal ribbons, which were extracted from the MRI of a normal brain.

In order to perform measurements on a surface embedded in 3D, we first need to obtain a mathematical representation of the surface. We have previously developed two methods for determining a parametric representation of sulcal ribbons [21,23]. The parameterization of a surface is represented by a map from a planar domain, such as the unit square, to the three-dimensional space. Effectively, a parameterization places a grid of the surface, in our case on the sulcal ribbon. This allows for various measurements to be made. In particular, we can obtain estimates of the local curvature of the sulcus or of the depth of the sulcus. Depth measurements are of particular interest, since they can potentially provide useful information to algorithms for the automated identification of the cortical sulci. In order to measure the depth of a sulcus, we have developed a dynamic programming algorithm, which is described in detail in [33]. This algorithm simulates the placement of a flexible probe along the depth of the sulcus, and perpendicularly to the outer (exposed) edge of the sulcus.

Figure 8 shows an example of a sulcus together with a number of depth probes determined via this dynamic programming algorithm. The resulting depth measurements are also shown.

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