## Abstract

It is well known that having 3 temporal phase shifting (PS) interferograms we do not have many possibilities of using an algorithm with a desired frequency spectrum, detuning, and harmonic robustness. This imposes severe restrictions on the possibilities to demodulate such set of temporal interferograms. It would be nice to apply for example a 7 step PS algorithm to these 3 images in order to have more possibilities to phase demodulate them; even further, it would be even better to apply a quadrature filter having a spatial spread given by a real number to these 3 interferograms. In this paper we propose to do just that; namely we show how to demodulate a set of *M*-steps phase shifting images with a quadrature filter having a real-number as spatial spread. The interesting thing in this paper is to use a higher than *M* spread quadrature filter to demodulate our interferograms; in traditional PS interferometry one is stuck to the use of *M* step phase shifting formula to obtain the searched phase. Using a less than *M* PS formula is not interesting at all given that we would not use all the available information. The main idea behind the “squeezing” phase shifting method is to re-arrange the information of the *M* phase shifted fringe patterns in such a way to obtain a single carrier frequency interferogram (a spatio-temporal fringe image) and use any two dimensional quadrature filter to demodulate it. In particular we propose the use of Gabor quadrature filters with a spread given by real-numbers along the spatial coordinates. The Gabor filter may be designed in such way that we may squeeze the frequency response of the filter along any desired spatio-temporal dimension, and obtain better signal to noise demodulation ratio, and better harmonic rejection on the estimated phase.

© 2008 Optical Society of America

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### Equations (12)

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(1)
$$I(x,y,-\alpha )=a(x,y)+b(x,y)\mathrm{cos}\left[\varphi (x,y)-\alpha \right]$$
(2)
$$I(x,y,0)\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}=a(x,y)+b(x,y)\mathrm{cos}\left[\varphi (x,y)\right]$$
(3)
$$I(x,y,+\alpha )=a(x,y)+b(x,y)\mathrm{cos}\left[\varphi (x,y)+\alpha \right]$$
(4)
$${\varphi}_{w}(x,y)={\mathrm{tan}}^{-1}\left[\frac{\left(1-\mathrm{cos}\left(\alpha \right)\right)\left[I(x,y,-\alpha )-I(x,y,+\alpha )\right]}{\mathrm{sin}\left(\alpha \right)\left[2I(x,y,0)-I(x,y,-\alpha )-I(x,y,+\alpha )\right]}\right]$$
(5)
$$\begin{array}{c}I\prime (3x,y)=I(x,y,-\alpha )\\ \phantom{\rule{.4em}{0ex}}I\prime (3x+1,y)=I(x,y,0)\\ \phantom{\rule{.9em}{0ex}}I\prime (3x+2,y)=I(x,y,+\alpha )\end{array}\}\phantom{\rule{.9em}{0ex}}(0,0)\le (x,y)\le (L,L)$$
(6)
$$\begin{array}{c}I\prime (x,3y)=I(x,y,-\alpha )\\ \phantom{\rule{.4em}{0ex}}I\prime \left(x,3y+1\right)=I(x,y,0)\\ \phantom{\rule{.9em}{0ex}}I\prime (x,3y+2)=I(x,y,+\alpha )\end{array}\}\phantom{\rule{.9em}{0ex}}(0,0)\le (x,y)\le (L,L).$$
(7)
$$I\prime (Mx+m,y)=I(x,y,m\alpha ),\phantom{\rule{.9em}{0ex}}\alpha =\frac{2\pi}{M},\phantom{\rule{.9em}{0ex}}(0,0)<(x,y)<(L,L),\phantom{\rule{.9em}{0ex}}m=0,...,M-1$$
(8)
$$I\prime (x\prime ,y)=a(x\prime ,y)+b(x\prime ,y)\mathrm{cos}\left[\alpha x\prime +\varphi (x\prime ,y)\right],\phantom{\rule{.2em}{0ex}}(0,0)<(x\prime ,y)<(ML,L),\alpha =\frac{2\pi}{M}$$
(9)
$${G}_{\alpha}(x,y)=\left[\mathrm{cos}\left(\alpha x\right)+i\mathrm{sin}\left(\alpha x\right)\right]{e}^{-\left[\frac{{x}^{2}}{{\left(M\sigma \right)}^{2}}+\frac{{y}^{2}}{{\sigma}^{2}}\right]},\alpha =\frac{2\pi}{M}.$$
(10)
$$\varphi (x,y)=\mathrm{arctan}\left\{\frac{\mathrm{Im}\left[I\prime (x,y)**{G}_{\alpha}(x,y)\right]}{\mathrm{Re}\left[I\prime (x,y)**{G}_{\alpha}(x,y)\right]}\right\},(0,0)<(x,y)<(L,L),$$
(11)
$${G}_{\alpha}(x,y)=\left[\mathrm{cos}\left({\alpha}_{x}x\right)+i\mathrm{sin}\left({\alpha}_{x}x\right)\right]{e}^{-\left[\frac{{x}^{2}}{{\left(M{\sigma}_{x}\right)}^{2}}+\frac{{y}^{2}}{{{\sigma}_{y}}^{2}}\right]},\alpha =\frac{2\pi}{M},$$
(12)
$$I\prime \left(x,My+m\right)=I(x,y,m\alpha ),\phantom{\rule{.9em}{0ex}}\alpha =\frac{2\pi}{M},\phantom{\rule{.9em}{0ex}}(0,0)<(x,y)<(L,L),\phantom{\rule{.9em}{0ex}}m=0,...,M-1$$