68 C.A. Sciammarella and L. Lamberti Fig. 10.6 Resultant projected vector in the 3-D space x1 x2 x3 Ip I2 I1 I3 Ip12 1 2 0 Fig. 10.7 Vector displacement at a voxel obtained as a sum of the vectors of the family of three tagging planes From Fig. 10.6 one obtains the relationships between the vector Ip and its projections, I1 DIp cos ™1 cos ™2 (10.16) I2 DIp cos ™1 sin ™2 (10.17) I3 DIp sin ™1 (10.18) Equations (10.16) to (10.18) give the relationship between the vector Ip and its projections in the Cartesian coordinates of versors ei, (iD1,23). Calling vector Ip1 one of the three projection vectors (Ip1, Ip2 and Ip3), that correspond to the three families of tagging planes, the resultant vector is given by the vectorial equation, IpT .x/ DIp1 .x/ CIp2 .x/ CIp3 .x/ (10.19) For each tagging plane with normal ni, there is a vector Ipi. The vectorial equation (10.19) provides the vector sum. Then, one has three projection equations for each one of the components of the light intensity vectors Ipi (x), IpT .x/ D Ip11 CIp12 CIp13 e1 C Ip11 CIp12 CIp13 e2 C Ip11 CIp12 CIp13 e3 (10.20) These resultant intensity vectors are converted into projected displacements vectors through the application of the theory of complex signals. Equation (10.20) is graphically represented in Fig. 10.7.
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