Vr • zc ¼F xc; yc ð Þ ð1:28Þ where F(xc, yc) indicates a 2D function. From Eq. (1.28), it is possible to get all the derivatives that appear in the preceding developments of the 3D complex field proceeding in a similar way to that utilized in the one dimensional case. Since we are dealing with Continuum Mechanics problems operating on tensorial entities, a family of orthogonal cosinusoidal signals must be defined in Cartesian coordinates as shown in Fig. 1.2. The orthogonal modulated fringe patterns project the displacement vectors in two orthogonal directions. These projections are not independent from each other since they are tied together by the compatibility conditions of the continuum and involve also the Eqs. (1.25)–(1.27) that both systems of fringes must satisfy at the same points of the image. Conclusions similar to the one dimensional case can be obtained: the successive derivatives of the gray levels must satisfy the above conditions to define a scalar potential. Hence, the passage from the actual signals to the continuum signals requires operations that must enforce the above conditions as close as it may be feasible. This conclusion is very important because the change of the orientation of the fringes is related to the curvature of the fringes, the larger is the local change of orientation the more important is the effect of the orientation on the derivatives of the gray levels function. 1.3 The Monogenic 2D Signal The extension of the one dimension approach of signal analysis to multiple dimensions has been the object of a large number of papers (see, for example, [5–8] and the references cited therein). This study will apply the complex Riesz transform presented in the preceding publications. In these four publications are introduced the required arguments to create a transform equivalent of the Hilbert transform in a multidimensional space. To achieve this purpose, the concept of monogenic function is introduced. The original derivation of the monogenic signal concept has its foundations on the algebra of quaternions that is connected to Lie algebra isomorphisms. In this paper, a variation of the original arguments is introduced. The derivations fit the mappings originally developed by Poincare and that for the particular field considered in this study, a 2D flat field, are graphically represented by a Poincare sphere [8] that it is utilized in the field of birefringent optics and in photoelasticity to define the different forms of polarization. The relationship of Poincare sphere and the concept of phase of the components of polarized light has been the object of several publications (see, for example, [9–11]). The connection between the preceding applications of the Poincare sphere and the phase concept and the current version introduced in this paper is a subject of a great deal of interest but is beyond the purpose of this paper. The motivation in the current version follows from the isomorphism pointed out in [12]. It has been shown that in order to introduce the concept of phase in one dimensional signals, it is necessary to resort to a 2D vectorial field, similarly for 2D signals it is necessary to introduce a vector field in the 3D complex space. The 3D phasor representing the gray levels in 2D has an amplitude that corresponds to the intensity of the signal, a phase that corresponds to the optical path information, and introduces a new variable that corresponds to the orientation of the 2D sinusoid in the physical plane defined by the normal n Eq. (1.18), a function of the angle θ defined in Fig. 1.1. Figure 1.3 illustrates the Poincare sphere notation. The vector amplitude is defined by the following components: (a) the components of the gray levels Ix, Iy associated with the versors i )and j), respectively; (b) to these two components it is added a third component Iq, corresponding to the versor k ). The complex amplitude vector in the complex space is given by, Isp ¼Ix i ) þIy j ) þIq k ) ð1:29Þ and corresponds to the radius of a Poincare sphere shown in Fig. 1.3a. This sphere represents the local phase and amplitude at a point in the 2D continuum of gray levels. In the coordinate plane i), j), sphere equator, Eq. (1.29) becomes: Ip ¼Ix i ) þIy j ) ð1:30Þ From Fig. 1.3, it follows: Ix ¼ Ip cos θ ð1:31Þ 1 A General Mathematical Model to Retrieve Displacement Information from Fringe Patterns 7
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