Algorithm for phase extraction from a set of interferograms with arbitrary phase shifts

Abstract

The generalized analytical quadrature filter from a set of interferograms with arbitrary phase shifts is obtained. Both symmetrical and non symmetrical algorithms for any order are reported. The analytic expression is obtained through the convolution of a set of two-frame algorithms and expressed in terms of the combinatorial theory. Finally, the solution is applied to obtain several generalized tunable quadrature filters.

Practical eight-frame algorithms for fringe projection profilometry

“…is the column vector of frames, and N and D are the desired numerator and denominator row vectors. In a previous work [1,16,18], it was proved that the Fourier transform ( ) H ω of this filter is,…”

“…where each k α is the cut off frequency or zero of the Fourier impulse response of the filter. In other words, the design of a filter becomes a geometrical problem, and it is reduced to choose a set of M-1 frequencies that are the necessary conditions to be a specific filter [16,18]. Therefore, from Eq.…”

“…Then, an option to obtain the required ratio N/D with symmetric coefficients is obtained from the expression [16,18] ( )…”

“…Assuming that the phase step is π/4, it is well known that a filter that eliminates harmonics corresponds to the cut-off frequencies k α = 0, π/4, π/2, 3π/4, π, 5π/4 and 3π/2 [1,11,16,18].…”

“…Additionally, in a previous work [16,18], it is demonstrated that, from an individual algorithm an infinite number of equivalent phase shifted algorithms can be obtained. However, for the particular case, where each individual two-frame filter is phase shifted by an angle of / 2 k k θ α = , the corresponding phase shifted filter r r N D becomes [16,18],…”