Mathematical models utilized in the retrieval of displacement information encoded in fringe patterns

•Critical analysis of fringe patterns as signals containing displacement information.•The number of significant harmonics in a signal depends on the modulation index β.•A mathematical framework to handle AM-FM signals and mixed signals is developed.•In-quadrature signal processing needs carrier separating displacement and background.•Multi-phase and in-quadrature processing give same results if properly implemented.

All the techniques that measure displacements, whether in the range of visible optics or any other form of field methods, require the presence of a carrier signal. A carrier signal is a wave form modulated (modified) by an input, deformation of the medium. A carrier is tagged to the medium under analysis and deforms with the medium. The wave form must be known both in the unmodulated and the modulated conditions. There are two basic mathematical models that can be utilized to decode the information contained in the carrier, phase modulation or frequency modulation, both are closely connected. Basic problems connected to the detection and recovery of displacement information that are common to all optical techniques will be analyzed in this paper, focusing on the general theory common to all the methods independently of the type of signal utilized. The aspects discussed are those that have practical impact in the process of data gathering and data processing.