10 Extension of the Monogenic Phasor Method to Extract Displacements and Their Derivatives from 3-D Fringe Patterns 67 Fig. 10.5 Planes in the 3D and projected displacement vectors x3 x2 x1 Ip1 I p0 0 n=0 n=1 Let us consider the 3D case, the following equations for the FT applies, ^f . 1; 2; 3/ DFŒf .x1; x2; x3/ D Z 1 1 Z 1 1 Z 1 1 e 2 i .x1 1Cx2 2Cx3 3/f .x1; x2; x3/dx1dx2dx3 (10.10) If we consider the exponent of the exponential function in (10.10), this exponent equals 1 whenever x Ÿ is an integer, that is, when x1Ÿ1 Cx2Ÿ2 Cx3Ÿ3 Dn (10.11) Equation (10.11) corresponds to the tangent planes to a family of curved surfaces that correspond to the tagging planes shown in Fig. 10.3 and that after deformations have become curved surfaces. These curved surfaces are the equivalent in 3D to the isothetic lines in 2D that locally can be represented by the tangent planes shown in Fig. 10.5. Equation (10.11) represents the orthogonal normal vectors to the 3D isothetic surfaces. It can be shown that the modulus of the intensity light vector is given by, Ip D 1 qŸ2 1 CŸ 2 2 CŸ 2 3 (10.12) where, k kDqŸ2 1 CŸ 2 2 CŸ 2 3 (10.13) Ÿ is the corresponding frequency vector in the frequency space illustrated in Fig. 10.1. The exponent of Eq. (10.10) can be written, e 2 i . x1 1Cx2 2Cx3 3/ De 2 ix1 1 e 2 ix2 2 e 2 ix3 3 (10.14) Each of the three exponential terms is of the form cos¥Cisin¥;, and the corresponding frequencies are Ÿi, (i D1,2 3). The first exponential term varies along x1 with a frequency Ÿ1, likewise the other two terms respectively vary along x2, x3 with frequencies Ÿ2 and Ÿ3. This implies three separate transforms along the three axes xi. The above expression gives the relationship between the vector representing the signal in 3D and its projections in the Cartesian coordinates versors ei, (i D1,2 3). Figure 10.6 represents the 3D vector Ip(x). The modulus of this vector is, Ip D qI2 1 CI 2 2 CI 2 3 D 1 qŸ2 1 CŸ 2 2 CŸ 2 3 (10.15) Ip is the 3D normal vector of the curved surfaces that provide the 3D space equivalent of the 2D isothetic lines. In the 2D case, the projected displacement vector remains in the 2D plane and has components along the x1 and the x2 axes. In the 3D case, the projected displacement vector has components along x1, x2 and x3, Fig. 10.6.
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