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# Geometrical Representation

In this section, some representations of geometrical figures such as points, curves, surfaces, and regions (volumes) in a three-dimensional space, are described. As described in the previous subsection, representation methods of geometric figures in given images can be roughly classified into two categories. In the first category, targets are represented using functions defined on an image space. In the second category, they are represented using functions defined on the parameter space, Sect. 2.3.2.1, with the resulting representation depicting the organ. Figure 2.11 shows some examples of different representations of an organ. An organ can be represented by labeling the voxels inside the organ, as will be described in Sect. 2.3.2.1, and the resultant representation represents the region with a set of the labeled voxels as shown in the top left panel in Fig. 2.11. Another method to represent an organ is by assigning negative values to the voxels inside the organ volume and positive values to the outside, using a level-set function, which implicitly represents the boundary of the region with its zero-crossing (see the bottom left panel in Fig. 2.11); this level-set representation will also be described

Fig. 2.11 Different representations of a target organ

in Sect. 2.3.2.1. The boundary of an organ can also be explicitly represented by parametric functions such as spline surfaces. These parametric representations will be described in Sect.2.3.2.2. A set of points distributed on a target surface (see the bottom right panel of Fig. 2.11) is also widely employed for the representation of an organâ€™s surface. The distribution of the points is described by the coordinates of each of the points, and those coordinates are the parameters of the representation. Such a representation with the set of the points will also be described in Sect. 2.3.2.2.

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