Two-frame algorithm to design quadrature filters in phase shifting interferometry

Abstract

The main purpose of this paper is to present a method to design tunable quadrature filters in phase shifting interferometry. From a general tunable two-frame algorithm introduced, a set of individual filters corresponding to each quadrature conditions of the filter is obtained. Then, through a convolution algorithm of this set of filters the desired symmetric quadrature filter is recovered. Finally, the method is applied to obtain several tunable filters, like four and five-frame algorithms.

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,…”

“…Then, at least two of those parameters are necessary to eliminate the D.C. component and the fundamental frequency. Two additional conditions are used to compensate the linear bias variation and the linear phase shift detuning error [16]. The remaining three conditions are used to compensate other errors generated by other effects, like a non-linearity response and to obtain a better SNR.…”

“…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].…”