The current paper was devoted to put into the general theory of monogenic function, 2D fringe pattern analysis. This effort has an interesting sequel, the Poincare sphere well known to photoelasticians appears again on a different topic, the retrieval of displacement fields and their derivatives. This finding opens a new and interesting field of research in view of the fact that the phase concept has been attached in the past to the Poincare sphere [9–11]. As part of these developments the relationship between the FFT transform and the generalized Hilbert transform is analyzed. An important conclusion arrived to in [1], the equivalence of the multi phase method and the in-quadrature method, is extended to two dimensional signals. A more detailed analysis of these questions is presented in the discussion of the local phase concept closely related to the narrow band condition that makes more likely that the generalized Hilbert-transform to be a valid representation of the captured signals. Finally, the role of other transforms (Gabor transform and wavelet transform) in fringe pattern analysis is considered. The analysis of an actual fringe pattern, the fringes corresponding to a disk under diametrical compression by all the above described methods provides results that are numerically very close and also close to the finite element solution in spite of the quite different algorithms utilized in each case. This is also a very valuable outcome that provides a comparison of different approaches from the point of view of reliability and accuracy of the final results. Fig. 1.18 Strains εx and εy of the disk under diametrical compression obtained from the FFT by differentiation in the frequency space [25] 0 20 40 60 80 100 120 140 160 180 200 220 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 Strain (microeps) Position along horizontal diameter (mm) Gabor/Morlet FFT Hilbert 2D FEM Fig. 1.19 Comparison of strains εx obtained by Gabor transform—Morlet wavelet and windowed FFT with finite element computation (control path corresponds to the horizontal diameter of the disk) 1 A General Mathematical Model to Retrieve Displacement Information from Fringe Patterns 23
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