∂V • yc ∂xc ∂V • xc ∂yc ¼ 0 ð1:13Þ These equations mean that the potential function in the complex plane must satisfy the Cauchy-Riemann equations, ∂V • xc ∂xc ¼ ∂V • yc ∂yc ð 1:14Þ ∂V • xc ∂yc ¼ ∂V • yc ∂xc ð 1:15Þ Equation (1.12) implies that the gray level potential V • to define a local phase must be a solution of Laplace’s equation in the complex plane. The solutions of the Laplace’s equation are part of the theory of potentials; these solutions are known to be harmonic functions. The field is conservative and the vectorial field is the gradient of a potential scalar field. The meaning of Eqs. (1.14) and (1.15) is that, in order to define a local phase, successive derivatives of gray levels must satisfy the above conditions. Furthermore, considering the full complex field, these equations are the conditions for V • ρc ð Þ dρc, where ρc ¼xc i ) þyc j ) is a given direction in the complex plane, to be an exact differential or, in other words, that is a potential such that the integral of the field is independent of the pathway followed. This conclusion leads to the complex function, z xð Þ¼Ve • xð Þþ j )Vo • xð Þ ð1:16Þ where Vo • xð Þ represents the odd component of the signal. Through Eq. (1.16) one gets the connection between the Hilbert transform, holomorphic functions and the levels of gray as a potential function leading to the definition of a local phase. For example, if Ve • is of the form given by Eq. (1.8), through the Hilbert transform we will obtain, Vo • xð Þ¼Iq sin 2π p x þϕ0 ð1:17Þ Each one of these derivatives can be computed from the information recorded in the image sensor. For each point of the ℒ1 domain, one can plot the gray levels as V • xð Þ. From Eq. (1.8) ∂V • ∂xc ∂V xð Þ • ∂x and ∂V xc ð Þ • ∂yc ∂V xð Þ • ∂y can be obtained and, finally, complementary derivatives are obtained from Eqs. (1.13) and (1.14). In summary, to represent the deformation of a continuous field the derivatives must satisfy the above relationships for a one dimensional signal. However, recorded signals will be contaminated by different signals that we designate as noise. Whatever processes that are applied to the signal to remove noise they must get successive derivatives satisfying the above conditions. All previous developments correspond to gray levels in one dimension. To introduce the definition of local phase for the 2D sinusoidal signal shown in Fig. 1.1 it is necessary to resort to a 3D complex space. Figure 1.1 showed a 2D cosinusoidal function which has the same parameters as a 1D signal but also additional parameter, the orientation θ. The normal nto the fringe trajectory shown in Fig. 1.1 provides the orientation of the signal at a given point of the physical space and the angleθ defines the orientation of the segment of curve with respect to a selected reference system. Some notations that will be useful in the developments that follow are now introduced. The unit normal to fringes in a point of a sinusoidal signal (Fig. 1.1b) is, n ¼x cos θi þy sin θj ð1:18Þ The above relationship is converted into cycles per unit length by multiplying Eq. (1.18) by 2π/p. Introducing the concept of wave vector for the sinusoidal signal, it can be written: k ¼ 2π p cos θi þ 2π p sin θj ¼kxi þkyj ð1:19Þ The wave vector is an alternative way to define the orientation of a segment of a sinusoidal signal in 2-D and relates it to the projections of the trajectory into the reference axis x–y. From Eq. (1.19), the local value of θ is given by 1 A General Mathematical Model to Retrieve Displacement Information from Fringe Patterns 5
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