64 C.A. Sciammarella and L. Lamberti p p p space Reciprocal f f f Fig. 10.1 Physical space tagged with planes separated by the pitch p and reciprocal frequency space with plane distances f Fig. 10.2 A prismatic body tagged with a 3D system of planes; (a) Descriptive geometry: 3 quadrants represent sections of the body; (b) cross sections of the coordinate planes in space and frequency information. Let us outline the basic process of extracting 3-D information utilizing deterministic signals. The relationship between p and f is, f D 1 p (10.1) Figure 10.1 shows the physical space tagged with planes separated by the pitch p, spatial distance between planes that define the voxels containing the volume information. Corresponding to the voxels in the physical space there are voxels in the reciprocal space or frequency space with plane distances f. Figure 10.2 provides a 3D scheme for spatial and frequency analysis of displacements and the corresponding derivatives. At this point, it is not necessary to specify how the reference system is introduced to the medium under analysis. There are several alternative procedures that can be utilized. The reference system is associated with a Cartesian system of coordinates. Figure 10.2 illustrates a tagged 3D body utilizing Descriptive Geometry representation. The planes cover the whole volume, Fig. 10.2a. Figure 10.2b shows an enlarged view of the cross sections set of planes corresponding to the voxels that will carry the displacement information and the dots that define in the frequency space the family of orthogonal lines corresponding to these planes. The coordinate system X1, X2 andX3 is a Lagrangian or material system of reference. The Eulerian system of reference x1, x2 and x3 is selected with planes parallel to the Lagrangian system. To keep the developments very general, large deformations are included, one of the problems that arise is to compare the very different geometries generated in the process of deformation. Figure 10.3 shows the displacement vector of a voxel and its projections in the coordinate system. Figure 10.4 illustrates the process of large deformation of a vessel recorded in a time sequence; time t D0 corresponds to the reference configuration. A parameter must be introduced to connect different stages of the process of deformation,
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