## Abstract

An advanced spatial carrier phase-shifting (SCPS) algorithm based on least-squares iteration is proposed to extract the phase distribution from a single spatial carrier interferogram. The proposed algorithm divides the spatial carrier interferogram into four phase-shifted interferograms. By compensating for the effects of the variations of phase shifts between pixels and the variations of background and contrast, the proposed algorithm determines the local phase shifts and phase distribution simultaneously and accurately. Numerical simulations show that the accuracy of the proposed algorithm is obviously improved by compensating for the effects of background and contrast variations. The peak to valley of the residual phase error remains less than $0.002\text{\hspace{0.17em}}\mathrm{rad}$ when the magnitude of spatial carrier is in the range from $\pi /5$ to $\pi /2$ and the direction of the spatial carrier is in the range from $25\xb0$ to $65\xb0$. Numerical simulations and experiments demonstrate that the proposed algorithm exhibits higher precision than the existing SCPS algorithms. The proposed algorithm is sensitive to random noise, but the error can be reduced by *N* times if *N* measurements are taken and averaged.

© 2008 Optical Society of America

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### Equations (19)

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(1)
$$I(x,y)=A(x,y)+B(x,y)\mathrm{cos}[\phi (x,y)+{u}_{c}x+{v}_{c}y],$$
(2)
$$u(x,y)=\partial [\mathrm{\Phi}(x,y)]/\partial x=\partial \phi (x,y)/\partial x+{u}_{c},$$
(3)
$$v(x,y)=\partial [\mathrm{\Phi}(x,y)]/\partial y=\partial \phi (x,y)/\partial y+{v}_{c},$$
(4)
$${I}_{1}(x,y)=I(x,y)={A}_{1}+{B}_{1}\mathrm{cos}[\mathrm{\Phi}(x,y)],$$
(5)
$${I}_{2}(x,y)=I(x+1,y)={A}_{2}+{B}_{2}\mathrm{cos}[\mathrm{\Phi}(x,y)+u(x,y)],$$
(6)
$${I}_{3}(x,y)=I(x,y+1)={A}_{3}+{B}_{3}\mathrm{cos}[\mathrm{\Phi}(x,y)+v(x,y)],$$
(7)
$${I}_{4}(x,y)=I(x+1,y+1)={A}_{4}+{B}_{4}\mathrm{cos}[\mathrm{\Phi}(x,y)+u(x,y)+v(x,y)],$$
(8)
$${I}_{n}(x,y)-{A}_{n}(x,y)+{A}_{1}(x,y)={a}_{1}(x,y)+{b}_{1}(x,y)\frac{{B}_{n}(x,y)}{{B}_{1}(x,y)}\mathrm{cos}[{\theta}_{n}(x,y)]+{c}_{1}(x,y)\frac{{B}_{n}(x,y)}{{B}_{1}(x,y)}\mathrm{sin}[{\theta}_{n}(x,y)]\mathrm{.}$$
(9)
$$S(x,y)=\sum _{n=1}^{N}[{I}_{n}^{e}(x,y)-{I}_{n}(x,y){]}^{2},$$
(10)
$$\frac{\partial S(x,y)}{\partial {a}_{1}(x,y)}=0,\frac{\partial S(x,y)}{\partial {b}_{1}(x,y)}=0,\frac{\partial S(x,y)}{\partial {c}_{1}(x,y)}=0.$$
(11)
$$\left[\begin{array}{l}{a}_{1}\\ {b}_{1}\\ {c}_{1}\end{array}\right]=\left[\begin{array}{lll}N& \sum _{n=1}^{N}\frac{{B}_{n}}{{B}_{1}}\mathrm{cos}{\delta}_{n}& \sum _{n=1}^{N}\frac{{B}_{n}}{{B}_{1}}\mathrm{sin}{\delta}_{n}\\ \sum _{n=1}^{N}\frac{{B}_{n}}{{B}_{1}}\mathrm{cos}{\delta}_{n}& \sum _{n=1}^{N}\frac{{B}_{n}^{2}}{{B}_{1}^{2}}{\mathrm{cos}}^{2}{\delta}_{n}& \sum _{n=1}^{N}\frac{{B}_{n}^{2}}{{B}_{1}^{2}}\mathrm{cos}{\delta}_{n}\mathrm{sin}{\delta}_{n}\\ \sum _{n=1}^{N}\frac{{B}_{n}}{{B}_{1}}\mathrm{sin}{\delta}_{n}& \sum _{n=1}^{N}\frac{{B}_{n}^{2}}{{B}_{1}^{2}}\mathrm{sin}{\delta}_{n}\mathrm{cos}{\delta}_{n}& \sum _{n=1}^{N}\frac{{B}_{n}^{2}}{{B}_{1}^{2}}{\mathrm{sin}}^{2}{\delta}_{n}\end{array}{]}^{-1}\right[\begin{array}{l}\sum _{n=1}^{N}{I}_{n}^{t}-{A}_{n}+{A}_{1}\\ \sum _{n=1}^{N}({I}_{n}^{t}-{A}_{n}+{A}_{1})\frac{{B}_{n}}{{B}_{1}}\mathrm{cos}{\delta}_{n}\\ \sum _{n=1}^{N}({I}_{n}^{t}-{A}_{n}+{A}_{1})\frac{{B}_{n}}{{B}_{1}}\mathrm{sin}{\delta}_{n}\end{array}]\mathrm{.}$$
(12)
$$A(x,y)={a}_{1}(x,y),$$
(13)
$$B(x,y)=\sqrt{({c}_{1}(x,y){)}^{2}+({b}_{1}(x,y){)}^{2}},$$
(14)
$$\mathrm{\Phi}(x,y)=\phi (x,y)+{u}_{c}x+{v}_{c}y={\mathrm{tan}}^{-1}(-{c}_{1}(x,y)/{b}_{1}(x,y)),$$
(15)
$${I}_{n}^{t}(x,y)-{A}_{n}(x,y)={b}_{n}^{\prime}{B}_{n}(x,y)\mathrm{cos}[{\mathrm{\Phi}}_{n}(x,y)]+{c}_{n}^{\prime}{B}_{n}(x,y)\mathrm{sin}{\mathrm{\Phi}}_{n}(x,y),$$
(16)
$$\left[\begin{array}{l}{b}_{n}^{\prime}\\ {c}_{n}^{\prime}\end{array}\right]=\left[\begin{array}{l}\sum _{x=1}^{X}\sum _{y=1}^{Y}{B}_{n}^{2}{\mathrm{cos}}^{2}{\mathrm{\Phi}}_{n}\hspace{1em}\sum _{x=1}^{X}\sum _{y=1}^{Y}{B}_{n}^{2}\mathrm{sin}{\mathrm{\Phi}}_{n}\mathrm{cos}{\mathrm{\Phi}}_{n}\\ \sum _{x=1}^{X}\sum _{y=1}^{Y}{B}_{n}^{2}\mathrm{sin}\mathrm{\Phi}\mathrm{cos}{\mathrm{\Phi}}_{n}\hspace{1em}\sum _{x=1}^{X}\sum _{y=1}^{Y}{B}_{n}^{2}{\mathrm{sin}}^{2}{\mathrm{\Phi}}_{n}\end{array}{]}^{-1}\right[\begin{array}{l}\sum _{x=1}^{X}\sum _{y=1}^{Y}({I}_{n}-{A}_{n}){B}_{n}\mathrm{cos}{\mathrm{\Phi}}_{n}\\ \sum _{x=1}^{X}\sum _{y=1}^{Y}({I}_{n}-{A}_{n}){B}_{n}\mathrm{sin}{\mathrm{\Phi}}_{n}\end{array}]\mathrm{.}$$
(17)
$${\stackrel{-}{\theta}}_{n}={\mathrm{tan}}^{-1}(-{c}_{n}^{\prime}/{b}_{n}^{\prime})\mathrm{.}$$
(18)
$$|({\stackrel{-}{\theta}}_{n}^{i}-{\stackrel{-}{\theta}}_{1}^{i})-({\stackrel{-}{\theta}}_{n}^{i-1}-{\stackrel{-}{\theta}}_{1}^{i-1})|<\epsilon ,$$
(19)
$$I(x,y)=130\mathrm{exp}[-{r}_{a}({x}^{2}+{y}^{2})]+120\mathrm{exp}[-{r}_{b}({x}^{2}+{y}^{2})]\mathrm{cos}[2\pi ({x}^{2}+{y}^{2})+{u}_{c}x+{v}_{c}y]+n(x,y),$$