A General Mathematical Model to Retrieve Displacement Information from Fringe Patterns

The extraction of the displacement field and its derivatives from fringe patterns entails the following steps: (1) information inscription; (2) data recovery; (3) data processing; (4) data analysis. Phase information is a powerful representation of the information contained in a signal. In a previous work, the above mentioned steps were formulated and discussed for a 1D signal, indicating that the extension to 2-D was a non trivial process. Proceeding along the same line of thought when one moves from the one dimension to two dimensions it is necessary to consider a 3D abstract space to generate the additional dimension that can handle the analysis of 2D signals and simultaneously extend the Hilbert transform to 2D. In this study the basic theory developed in the preceding reference is further elaborated to produce a version of the monogenic function yielding the necessary answers to the previously described processes. The monogenic signal, a 3D vector in a Cartesian complex space, is graphically represented by a Poincare sphere which provides a generalization of the Hilbert transform to a 2D version of what is called the generalized Hilbert transform or Riesz transform. These theoretical derivations are supported by the actual application of the theory and corresponding algorithms to 2D fringe patterns and by comparing the obtained results with known results.