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# Registration Required Before Measurement or Analysis

Target figures such as curves or surfaces should be registered or normalized appropriately before measuring the distances between them or analysis of their the statistics. If you represent figures in different images which are represented using functions of voxels, f (x), the values must be compared at the corresponding voxels;

i. e., each voxel in one of the images should be mapped to a voxel in each of the other images before the comparison. Registration of given images is hence required, and use of anatomical landmarks is vital. Medical images can be registered by detecting anatomical landmarks and by deforming the images so that the detected corresponding landmarks have (approximately) identical coordinates. Landmark detection will be described in Sect. 2.3.3.

For statistical analysis of figures, if the figures are represented using parametric functions, x(s, t; (в)), the locations of points on the figures that have identical values of (s, t) that indicate the locations on the figures must be compared. For example, when using the NURBS functions to represent surfaces, the distances among the figures are calculated as shown in (2.140), in which the distance between two points on different figures that have identical values of the parameters, (s, t), is integrated. The computed distance, therefore, is plausible from the point of view of medical imaging only when the parameters, (s, t), that indicate the locations are appropriately set for each of the given figures; i.e., the points that are on different figures and that have identical values of the parameters, (s, t), should appropriately correspond. The parameters, (в, ), should correspond when employing spherical harmonics for the surface representation, and the index numbers,

j, of the points should correspond when PDMs are used for the representations.

When parametric functions are employed for the representation, correspondences among given figures must be made in order to compare them and to analyze their statistics. Several approaches for making these correspondences can be utilized . Diffeomorphism-based frameworks figure prominently in this methodology and will be described in Sect. 2.3.4.

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