66 C.A. Sciammarella and L. Lamberti Equation (10.2) gives the displacement vector of a point P of the continuum shown in Fig. 10.3, in a Cartesian system of axes, dp Du1e1 Cu2e2 Cu3e3 (10.2) where e1,e2, e3 are the versors of the coordinate system. The projected vectors in each reference plane are given by, dp12 Du1 e1 Cu2 e2 (10.3) dp13 Du1 e1 Cu3 e3 (10.4) dp23 Du2 e2 Cu3 e3 (10.5) It is necessary to connect the displacement vector to the signal carrier of displacement information a scalar that represents light intensities in terms of gray levels. This is done through the equation, dpij.x/ D ¥i.x/ 2 p (10.6) where ¥i(x) is the local phase, the ratio ®i.x/ 2 D n is the fringe order. Since we are speaking of local fringe orders, n will be a real number n 0 , 1. p is the signal pitch. Equation (10.6) is the transformation scale between the intensities and the displacements. Hence, one can write 8 ˆ ˆ< ˆ ˆ: dp12 D k¥12k 2 p dp13 D k¥13k 2 p dp23 D k¥23k 2 p (10.7) For a generic point P the resultant vector is the sum of three components vectors, dp .x; t/ D dp12 .x1; x2; t/ Cdp23 .x2; x3; t/ Cdp13 .x1; x3; t/ (10.8) It can be observed in Fig. 10.3 that the third component dp23 is determined if the other two components are known. The angle that the projected vector dp12 makes with axis x1 is, ™12 Darctg u2 u1 (10.9) Similar equations can be written for ™13 and ™23. To proceed with the determination of the components of the displacement vector it is necessary to review some of the properties of the Fourier transform, FT, that also apply to the Hilbert transform. 10.3 Properties of the Fourier Transform The recovery of the displacement information is done working with recorded patterns of gray levels (scalar quantities) that result from the modulation of sinusoidal carriers, hence in the following analysis the utilized variables are light intensities I(x). In the derivation presented in this section, the multidimensional FT is applied, a generalization of the 1D to multiple spaces that deals with vectorial quantities [3, 4]. While gray levels are scalars, after the FT is applied to gray levels, one gets vector quantities, intensity vectors corresponding to the projection of the displacement vector along a selected coordinate axis. In [2], it was shown that the Fourier transform and the Hilbert transform can be utilized as two possible alternatives to decode displacements from recorded fringe patterns. In this paper, to generalize the procedures derived in [1, 2], we will utilize the FT; similar arguments could be derived applying the generalized Hilbert transform.
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