74 C.A. Sciammarella and L. Lamberti Fig. 10.13 Eulerian components of the strain tensor along coordinate axes 10.9 Discussion and Conclusions Recently the authors have published papers [1, 2] about fringe pattern displacement information recovery. In these papers were addressed several aspects of fringe pattern processing that required additional analysis and developments to fully achieving the objective of a fast and accurate way of processing signals that contain displacement and deformation information in 2D. The current paper deals with the generalization of the outlined procedures in 2D to 3D. In [1], several fundamental aspects of fringe pattern analysis were dealt in the context of 1D signals. It is stressed that a fundamental concept of fringe pattern analysis is the concept of local phase. It is also pointed out that the definition of local phase in 1D requires the introduction of a 2D complex space that is geometrically represented by the unit circle. In [2], it is shown that the concept of local phase in 2D requires a 3D complex space. Furthermore, it is indicated that the unit circle is replaced by a well-known geometrical surface utilized in Photoelasticity to represent the different states of polarization of light, the Poincare sphere. Thus, the use of the Poincare is extended to patterns associated with displacements and metrological signals, a geometrical illustration of the concept of monogenic phasor. The current paper extends the concept of monogenic function to fringe patterns that correspond to 3D displacements. The addition of one dimension leads to a hyper-4D complex space and to a 4D hyper-sphere. The components of the radius of the 4D sphere are derived. The process of computing the components is illustrated with examples. An application to images produced by the MRI process applied to a human heart left ventricle provides the actual application of the theoretical developments yielding 3D displacements and displacement derivatives. The additional steps of the generalization of the monogenic phasor to the different dimensions are not trivial. Each one of the process of generalization required additional procedures that provided different aspects of the more general problem. For example, the 3D displacements case was handled by introducing a multidimensional generalization of the Fourier Transform that deals with vectorial quantities and has very interesting properties that are utilized practically in the process of data retrieval. The fact that the FT, the Hilbert transform and the Radon transform are connected to each other, opens new avenues for further future developments in the actual study of 3D deformations. References 1. Sciammarella, C.A., Lamberti, L.: Mathematical models utilized in the retrieval of displacement information encoded in fringe patterns. Opt. Lasers Eng. 77, 100–111 (2016) 2. Sciammarella, C.A., Lamberti, L.: Generalization of the Poincare sphere to process 2D displacement signals. Opt. Lasers Eng. 93, 114–127 (2017) 3. Fourier analysis for vectors. http: www.uio.no/studier/emner/matnat/math/MAT-INF2360/v12/fouriervectors
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