## Abstract

This paper presents a comparative study on three sinusoidal fringe pattern generation techniques with projector defocusing: the squared binary defocusing method (SBM), the sinusoidal pulse width modulation (SPWM) technique, and the optimal pulse width modulation (OPWM) technique. Because the phase error will directly affect the measurement accuracy, the comparisons are all performed in the phase domain. We found that the OPWM almost always performs the best, and SPWM outperforms SBM to a great extent, while these three methods generate similar results under certain conditions. We will briefly explain the principle of each technique, describe the optimization procedures for each technique, and finally compare their performances through simulations and experiments.

© 2012 Optical Society of America

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### Equations (30)

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(1)
$${I}_{1}(x,y)={I}^{\prime}(x,y)+{I}^{\prime}(x,y)\mathrm{cos}(\varphi -2\pi /3),$$
(2)
$${I}_{2}(x,y)={I}^{\prime}(x,y)+{I}^{\prime}(x,y)\mathrm{cos}(\varphi ),$$
(3)
$${I}_{3}(x,y)={I}^{\prime}(x,y)+{I}^{\prime}(x,y)\mathrm{cos}(\varphi +2\pi /3),$$
(4)
$$\varphi (x,y)={\mathrm{tan}}^{-1}[\sqrt{3}({I}_{1}-{I}_{3})/(2{I}_{2}-{I}_{1}-{I}_{3})].$$
(5)
$${S}^{m}({f}_{c})=\{\begin{array}{ll}2{f}_{c}x-2N& x\in [N/{f}_{c},(2N+1)/(2{f}_{c}))\\ -2{f}_{c}x+2N+2& x\in [(2N+1)/(2{f}_{c}),(N+1)/{f}_{c})\end{array},$$
(6)
$${S}^{i}({f}_{0})=0.5+0.5\text{\hspace{0.17em}}\mathrm{cos}(2\pi {f}_{0}x).$$
(7)
$${I}^{b}(x,y)=\{\begin{array}{ll}1& {S}^{m}({f}_{c})>{S}^{i}({f}_{0}),\\ 0& \text{otherwise}\end{array}\mathrm{.}$$
(8)
$${a}_{0}=\frac{1}{2\pi}{\int}_{\theta =0}^{2\pi}f(\theta )\mathrm{d}\theta =0.5,$$
(9)
$${a}_{k}=\frac{1}{\pi}{\int}_{\theta =0}^{2\pi}f(\theta )\mathrm{cos}(k\theta )\mathrm{d}\theta ,$$
(10)
$${b}_{k}=\frac{1}{\pi}{\int}_{\theta =0}^{2\pi}f(\theta )\mathrm{sin}(k\theta )\mathrm{d}\theta \mathrm{.}$$
(11)
$${b}_{k}=\frac{4}{\pi}{\int}_{\theta =0}^{2\pi}f(\theta )\mathrm{sin}(k\theta )\mathrm{d}\theta .$$
(12)
$${b}_{k}=\frac{4}{\pi}{\int}_{0}^{{\alpha}_{1}}\mathrm{sin}(k\theta )\mathrm{d}\theta +\frac{4}{\pi}{\int}_{{\alpha}_{2}}^{{\alpha}_{3}}\mathrm{sin}(k\theta )\mathrm{d}\theta +\dots +\frac{4}{\pi}{\int}_{{\alpha}_{n}}^{\pi /2}\mathrm{sin}(k\theta )\mathrm{d}\theta $$
(13)
$$=\frac{4}{k\pi}[1-\mathrm{cos}\text{\hspace{0.17em}}k{\alpha}_{1}+\mathrm{cos}\text{\hspace{0.17em}}k{\alpha}_{2}-\mathrm{cos}\text{\hspace{0.17em}}k{\alpha}_{3}+\dots +\mathrm{cos}\text{\hspace{0.17em}}k{\alpha}_{n}].$$
(14)
$${b}_{1}=1-\mathrm{cos}({\alpha}_{1})+\mathrm{cos}({\alpha}_{2})-\mathrm{cos}({\alpha}_{3})+\mathrm{cos}({\alpha}_{4})=\pi /4,$$
(15)
$${b}_{5}=1-\mathrm{cos}(5{\alpha}_{1})+\mathrm{cos}(5{\alpha}_{2})-\mathrm{cos}(5{\alpha}_{3})+\mathrm{cos}(5{\alpha}_{4})=0.0,$$
(16)
$${b}_{7}=1-\mathrm{cos}(7{\alpha}_{1})+\mathrm{cos}(7{\alpha}_{2})-\mathrm{cos}(7{\alpha}_{3})+\mathrm{cos}(7{\alpha}_{4})=0.0,$$
(17)
$${b}_{11}=1-\mathrm{cos}(11{\alpha}_{1})+\mathrm{cos}(11{\alpha}_{2})-\mathrm{cos}(11{\alpha}_{3})+\mathrm{cos}(11{\alpha}_{4})=0.0.$$
(18)
$$\begin{array}{cc}\mathrm{\Delta}{\varphi}^{b}(x,y)=[{\varphi}^{b}(x,y)-{\varphi}^{s}(x,y)]& \mathrm{mod}\text{\hspace{0.17em}}2\pi \end{array},$$
(19)
$$\begin{array}{cc}\mathrm{\Delta}{\varphi}^{p}(x,y)=[{\varphi}^{m}(x,y)-{\varphi}^{s}(x,y)]& \mathrm{mod}\text{\hspace{0.17em}}2\pi ,\end{array}$$
(20)
$$\begin{array}{cc}\mathrm{\Delta}{\varphi}^{o}(x,y)=[{\varphi}^{o}(x,y)-{\varphi}^{s}(x,y)]& \mathrm{mod}\text{\hspace{0.17em}}2\pi .\end{array}$$
(21)
$$y(x)=\{\begin{array}{ll}0& x\in [(2n-1)\pi ,2n\pi )\\ 1& x\in [2n\pi ,(2n+1)\pi )\\ \end{array}\mathrm{.}$$
(22)
$$y(x)=0.5+\sum _{k=0}^{\infty}\frac{2}{(2k+1)\pi}\text{\hspace{0.17em}}\mathrm{sin}[(2k+1)\pi ]\mathrm{.}$$
(23)
$${I}_{1}^{h}(x,y)={I}^{\prime}(x,y)+{I}^{\prime \prime}(x,y)\mathrm{cos}(\varphi -2\pi /3)+\dots {I}_{k}(x,y)\mathrm{cos}[(2k+1)(\varphi -2\pi /3)],$$
(24)
$${I}_{2}^{h}(x,y)={I}^{\prime}(x,y)+{I}^{\prime \prime}(x,y)\mathrm{cos}(\varphi )+\dots {I}_{k}(x,y)\mathrm{cos}[(2k+1)\varphi ],$$
(25)
$${I}_{3}^{h}(x,y)={I}^{\prime}(x,y)+{I}^{\prime \prime}(x,y)\mathrm{cos}(\varphi +2\pi /3)+\dots {I}_{k}(x,y)\mathrm{cos}[(2k+1)(\varphi +2\pi /3)],$$
(26)
$$\varphi (x,y)={\mathrm{tan}}^{-1}[\sqrt{3}({I}_{1}^{h}-{I}_{3}^{h})/(2{I}_{2}^{h}-{I}_{1}^{h}-{I}_{3}^{h})]$$
(27)
$$={\mathrm{tan}}^{-1}[\sqrt{3}({I}_{1}-{I}_{3})/(2{I}_{2}-{I}_{1}-{I}_{3})].$$
(28)
$${I}^{k}(x,y)=0.5+\frac{2}{3\pi}\text{\hspace{0.17em}}\mathrm{sin}({f}_{0}x)+\frac{2}{(2k+1)\pi}\text{\hspace{0.17em}}\mathrm{sin}[(2k+1){f}_{0}x].$$
(29)
$$G(x,y)=\frac{1}{2\pi {\sigma}^{2}}{e}^{-\frac{{(x-\overline{x})}^{2}+{(y-\overline{y})}^{2}}{2{\sigma}^{2}}}\mathrm{.}$$
(30)
$$G(x)=\frac{1}{\sqrt{2\pi}\sigma}{e}^{-\frac{{(x-\overline{x})}^{2}}{2{\sigma}^{2}}}\mathrm{.}$$