Determination of displacements and their derivatives from 3D fringe patterns via extended monogenic phasor method

For 1D signals, it is necessary to resort to a 2D abstract space because the concept of phase utilized in the retrieval of fringe pattern analysis information relies on the use of a vectorial function. Fourier and Hilbert transforms provide in-quadrature signals that lead to the very important basic concept of local phase. A 3D abstract space must hence be generated in order to analyze 2D signals. A 3D vector space in a Cartesian complex space is graphically represented by a Poincare sphere. In this study, the extension of the associated spaces is extended to 3D. A 4D hypersphere is defined for that purpose. The proposed approach is illustrated by determining the deformations of the heart left ventricle.