## The Shape Transformation

During the past 10 years, a substantial amount of research has been dedicated to computational models for determining shape transformations x , based on tomographic images. We now give a brief review of this work. This review is by no means exhaustive, but is meant to give the reader representative examples of models that have been investigated, as well as their merits and limitations.

Many of the models that have been pursued have been based on physical concepts. In particular, the first attempt to develop algorithms for morphing one brain image to another dates back to the early 1980s and used a two-dimensional elastic transformation [9]. The goal there was to maximize some measure of similarity between the image being transformed and the target image. The image cross-correlation was used as the similarity measure. The optimal transformation, i.e., the transformation that resulted in maximal cross-correlation, was found via an iterative numerical optimization procedure. That method was later developed in three dimensions [10,11]. Several models that succeeded this method were similar, in that they attempted to maximize the similarity between two brain images, but they used different transformation models [2,1215].

The most important characteristic of the methods just described is that they take advantage of the full resolution of image data, i.e., the image similarity is examined on each individual point of the brain, and accordingly each point is freely transformed to maximize the similarity. Consequently, these methods have many degrees of freedom and therefore they are flexible to morph any brain to another brain. A main limitation, however, of these methods is that they are based on image similarity criteria. Images of similar brains can differ. For example, they can be MR images acquired with different protocols, or they can be digitized atlases [16]. Moreover, aging or diseases can change the signal characteristics of a tomo-graphic image of a brain. Such signal changes then can potentially adversely affect the spatial transformation. A second limitation of image matching methods is that they do not take into account geometric information, such as curvature or other shape indices, which often have a biological substrate.

The second major family of methods for shape transformations in brain imaging has been based on anatomical features. The main idea is to match distinct features, which are first extracted from the tomographic images, as opposed to using image similarity criteria. The features can be individual landmark points [1], curves [17-20], or surfaces [20,21]. In the remainder of this section we will briefly describe a feature-based method that was developed in our laboratory [19,22] and that is based on a surface-driven elastic transformation.

Consider a number of anatomical surfaces extracted from a tomographic brain image. Figure 3 shows some representative features: the outer cortical surface, the ventricular boundary, and various sulci. All of these surfaces are determined using

FIGURE 3 Examples of several anatomical surfaces extracted from magnetic resonance images via the methods described in [21, 23]. The top shows the boundary of the lateral ventricles, the middle shows the outer cortical surface, and the bottom image shows several sulci of a single brain. A mathematical representation of each of these surfaces is obtained simultaneously with these visual representations. Consequently, measurements of geometric properties of these surfaces, such as curvature, shape index, geodesics, or depth, can be measured quantitatively.

FIGURE 3 Examples of several anatomical surfaces extracted from magnetic resonance images via the methods described in [21, 23]. The top shows the boundary of the lateral ventricles, the middle shows the outer cortical surface, and the bottom image shows several sulci of a single brain. A mathematical representation of each of these surfaces is obtained simultaneously with these visual representations. Consequently, measurements of geometric properties of these surfaces, such as curvature, shape index, geodesics, or depth, can be measured quantitatively.

deformable parametric models, such as the ones described in [21,23,24]. A parametric model is always defined in a parametric domain, which in our models is either the unit square or the unit sphere. Consider, now, a particular surface, such as the outer cortical surface, whose parametric representation on the unit sphere has been extracted from a set of tomographic images for two different brains. The resulting surfaces are both defined in the same domain, in this particular case the unit sphere. However, homologous anatomical features do not necessarily have the same parametric coordi-

FIGURE 4 (a, b) A network of curves (sulci) overlaid on 3D renderings of the outer brain boundaries of two individuals. (c) The locations of these curves on the unit sphere for the two individuals (green corresponds to (a) and red to (b)). (d) An elastic reparameterization, i.e., a map of the unit sphere to itself, of the surface in (a), so that the two networks of curves have the same parametric coordinates (longitude and latitude) on the unit sphere. The warping of a "latitude grid" is shown for visual appreciation of the effect of the elastic reparameterization. See also Plate 21.

FIGURE 4 (a, b) A network of curves (sulci) overlaid on 3D renderings of the outer brain boundaries of two individuals. (c) The locations of these curves on the unit sphere for the two individuals (green corresponds to (a) and red to (b)). (d) An elastic reparameterization, i.e., a map of the unit sphere to itself, of the surface in (a), so that the two networks of curves have the same parametric coordinates (longitude and latitude) on the unit sphere. The warping of a "latitude grid" is shown for visual appreciation of the effect of the elastic reparameterization. See also Plate 21.

nates on the sphere. For example, the same cortical fold might have different longitude and latitude in the two brains. For illustration, Fig. 4 shows two spherical maps obtained from two different subjects, on which we have drawn the outlines of several sulci. In order to force corresponding features, such as the curves shown in Fig. 4, to have the same parametric coordinates, we reparameterize one of the two surfaces [19,22,25]. Effectively, this applies a local stretching or shrinking of the parametric grid of one of the two surfaces, so that certain anatomical features, such as prominent cortical folds, have the same parametric coordinates for both brains. In the context of the shape transformation we discussed earlier, we reparameterize the surfaces derived from the template (the atlas) so that their parametric grids match the surfaces derived from a brain under analysis.

Once a surface-to-surface map has been determined; the transformation T is then determined in the remainder of the brain by elastic interpolation. In particular, let x be a point on a surface defined as before from the template, and let g x be its counterpart in an individual brain. Define a force field that is applied to the template brain and is equal to

on the points lying on the anatomical surfaces used as features, and 0 elsewhere. Equation (1) implies that points that lie on a surface in the template should be transformed via the spatial transformation T to their counterparts, g x , in the individual brain. Otherwise, a force that is proportional to the distance between g x and T x is applied that tends to modify T so that these point correspondences are eventually satisfied. More specifically, the transformation T is the one satisfying the following differential equations, which describe the elastic deformation of the template under the influence of the external force field defined in (1):

The first term in this equation is a force field that attempts to match the features described earlier. The remaining terms describe the deformation of a linear elastic object [26]. These equations are solved numerically after discretization. In particular, the continuous transformation T is typically sampled on every other point in the image, which results in a number of unknown parameters of the order of 1 million. The discrete equivalent of the differential equation in (2) is a large, sparse linear system of equations, which is solved via well-known iterative techniques, such as successive overrelaxation [27].

Figure 5 shows a template of the corpus callosum, and three elastic adaptations of this template to match the anatomy of three different subjects. The corresponding MR images of these subjects are also shown in the same figure.

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