1.8 Process of Phase Recovery The preceding section covered the analysis of fringe patterns as 2D signals. One of the important concepts in this analysis is the definition of local phase. While the local phase concept is clear in the continuum theory, in the passage from the continuum to the actual signals one should note that the concept of local phase depends of the selected widows to smooth real signals. In [1], it was concluded that in order to satisfy the quasi-harmonic condition that allows in 1D signals the utilization of in-quadrature signals for the computation of local phase, it is required to encode the displacement signal into a high frequency carrier. A similar argument must be introduced for 2D signals. In order to achieve this objective, we must consider again the concept of local phase, that in the general theory of 1D signal analysis is known as the definition of the instantaneous frequency, subject that has been the object of extensive studies [16, 17]. The phase computation is a pointwise operation that provides the local phase through the computation of the arc tan function and can be derived by the introduction of the concept of analytic functions [4]. Within the analytic function theory one has signals that are amplitude modulated or frequency modulated. Fringe patterns are both amplitude and frequency modulated signals. This creates a very difficult problem because amplitude and simultaneously phase modulated signals do not have a uniquely defined analytic signal representation [16]. The amplitude and the phase have their own frequency spectra and these spectra can overlap. In [1], it is observed that harmonics providing different optical information besides amplitude and phase can also overlap in the frequency space with amplitude and phase information. As a consequence of these facts, phase and amplitude recovery information is not possible unless steps are taken to minimize the effect of the mixing of different harmonics. Two important tools are utilized to get solution to these two problems of real signals. The Bedrosian-Nuttal’s theorems [18–20] provide solutions to the overlapping problem. If the Bedrosian-Nuttal’s theorems are satisfied, the amplitude of the general phasor A(x) and the phase ϕ(x) are separated in the frequency space. The analytical function theory can provide the in-quadrature components that lead to a real signal with a well defined local frequency, f xð Þ¼ 1 2π d dx argz xð Þ ð1:66Þ The possibility of satisfying the Bedrosian-Nuttal’s theorems is related to the bandwidth of the involved signals. To get an intuitive picture of the problem of defining local phase at a given point of an image, let us return to the one dimensional signal. The simple harmonic function has been traditionally utilized to define instantaneous phase. A periodic motion is represented by a body that moves with constant speed along a circular path (see Fig. 1.10), S xð Þ¼A0 cos ϕ xð Þ ð1:67Þ where Ao is the radius of the circle, ϕ(x) ¼ωx, ωis the constant angular frequency connected with the spatial frequency fx ¼ω/2π. In [1], it is concluded that for patterns where the amplitude modulation frequencies are much smaller than the frequencies of the phase modulation fa fϕ, the in-quadrature signals provide accurate values of the phase. This conclusion is proven by the analysis of 1D signals extracted from moire´ patterns both computer generated with known frequencies and actual optically produced moire´ patterns. It is necessary to extend this conclusion to 2D signals, Fig. 1.10 Simple harmonic motion as a model to define instantaneous frequency 16 C.A. Sciammarella and L. Lamberti
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