## Abstract

The windowed Fourier transform (WFT) has been recognized as an effective tool for extracting phase map from a single fringe pattern. This paper presents a new WFT-based algorithm, which is guided by the principal component analysis (PCA) of the fringe pattern. With it, the principal direction of frequency at each pixel is determined first by using the PCA of the gradients of the fringe pattern, and then the windowed Fourier ridge for each pixel is detected by searching a one-dimensional space of frequency, so that the computational burden for phase extraction is alleviated significantly. In addition, this technique enables automatically identifying and excluding the regions of singular points from the fringe pattern by using the eigenvalues resulting from the PCA just mentioned. Numerical simulation and experiment are carried out to demonstrate the validity of this method.

© 2013 Optical Society of America

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### Equations (24)

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(1)
$$sf(u,v,\xi ,\eta )={\int}_{-\infty}^{+\infty}{\int}_{-\infty}^{+\infty}f(x,y)g(x-u,y,-v)\mathrm{exp}[-j(\xi x+\eta y)]\mathrm{d}x\mathrm{d}y,$$
(2)
$$f(x,y)=\frac{1}{4{\pi}^{2}}{\int}_{-\infty}^{+\infty}{\int}_{-\infty}^{+\infty}{\int}_{-\infty}^{+\infty}{\int}_{-\infty}^{+\infty}[Sf(u,v;\xi ,\eta )\phantom{\rule{0ex}{0ex}}\times g(x-u,y-v)\mathrm{exp}(j\xi x+j\eta y)\mathrm{d}\xi \mathrm{d}\eta \mathrm{d}u\mathrm{d}v].$$
(3)
$$f(x,y)=a(x,y)+b(x,y)\mathrm{cos}[\phi (x,y)],$$
(4)
$$\frac{\partial f(x,y)}{\partial x}\approx \nabla {f}_{x}(x,y)=\frac{f(x+1,y)-f(x-1,y)}{2}$$
(5)
$$\frac{\partial f(x,y)}{\partial y}\approx \nabla {f}_{y}(x,y)=\frac{f(x,y+1)-f(x,y-1)}{2}.$$
(6)
$${\sigma}_{x}^{2}=\frac{1}{{(2m+1)}^{2}}\sum _{u=-m}^{m}\sum _{v=-m}^{m}{[\nabla {f}_{x}(x-u,y-v)]}^{2}$$
(7)
$${\sigma}_{y}^{2}=\frac{1}{{(2m+1)}^{2}}\sum _{u=-m}^{m}\sum _{v=-m}^{m}{[\nabla {f}_{y}(x-u,y-v)]}^{2}.$$
(8)
$${\sigma}_{xy}=\frac{1}{{(2m+1)}^{2}}\sum _{u=-m}^{m}\sum _{\text{\hspace{0.17em}}v=-m}^{m}\nabla {f}_{x}(x-u,y-v)\nabla {f}_{y}(x-u,y-v).$$
(9)
$$\mathbf{C}=\left[\begin{array}{cc}{\sigma}_{x}^{2}& {\sigma}_{xy}\\ {\sigma}_{xy}& {\sigma}_{y}^{2}\end{array}\right].$$
(10)
$$|\lambda \mathbf{I}-\mathbf{C}|=0$$
(11)
$${\lambda}_{1}=[({\sigma}_{x}^{2}+{\sigma}_{y}^{2})+\sqrt{{({\sigma}_{x}^{2}-{\sigma}_{y}^{2})}^{2}+4{\sigma}_{xy}^{2}}]/2,$$
(12)
$${\lambda}_{2}=[({\sigma}_{x}^{2}+{\sigma}_{y}^{2})-\sqrt{{({\sigma}_{x}^{2}-{\sigma}_{y}^{2})}^{2}+4{\sigma}_{xy}^{2}}]/2.$$
(13)
$$(\mathbf{C}-{\lambda}_{1}\mathbf{I})\mathbf{W}=0,$$
(14)
$$\{\begin{array}{l}{w}_{1}=-{\sigma}_{xy}/\sqrt{{\sigma}_{xy}^{2}+{({\lambda}_{1}-{\sigma}_{x}^{2})}^{2}}\\ {w}_{2}=({\lambda}_{1}-{\sigma}_{x}^{2})/\sqrt{{\sigma}_{xy}^{2}+{({\lambda}_{1}-{\sigma}_{x}^{2})}^{2}},\end{array}$$
(15)
$$\{\begin{array}{l}{w}_{1}=({\lambda}_{1}-{\sigma}_{y}^{2})/\sqrt{{\sigma}_{xy}^{2}+{({\lambda}_{1}-{\sigma}_{y}^{2})}^{2}}\\ {w}_{2}=-{\sigma}_{xy}/\sqrt{{\sigma}_{xy}^{2}+{({\lambda}_{1}-{\sigma}_{y}^{2})}^{2}}\end{array}.$$
(16)
$$P=\frac{{\lambda}_{1}}{{\lambda}_{1}+{\lambda}_{2}}.$$
(17)
$$\widehat{\mathbf{W}}={({\widehat{w}}_{1},{\widehat{w}}_{2})}^{T}={(\mathrm{cos}\text{\hspace{0.17em}}\theta ,\mathrm{sin}\text{\hspace{0.17em}}\theta )}^{T}.$$
(18)
$$\theta =\mathrm{arctan}\text{\hspace{0.17em}}\frac{{\widehat{w}}_{2}}{{\widehat{w}}_{1}}.$$
(19)
$$[\frac{\partial}{\partial u},\frac{\partial}{\partial v}]\phi (u,v)=\mathrm{arg}\text{\hspace{0.17em}}\underset{\xi ,\eta}{\mathrm{max}}|Sf(u,v;\xi ,\eta )|.$$
(20)
$$\phi (u,v)=\mathrm{Im}\left[\mathrm{ln}\left(Sf{(u,v;\xi ,n)|}_{[\xi ,\eta ]=[\frac{\partial}{\partial u},\frac{\partial}{\partial v}]\phi (u,v)}\right)\right],$$
(21)
$$\xi =\zeta \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta =\zeta {\widehat{w}}_{1}\eta =\zeta \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta =\zeta {\widehat{w}}_{2},$$
(22)
$$\left|\frac{\partial \phi}{\partial \widehat{\mathbf{W}}}\right|=\mathrm{arg}\text{\hspace{0.17em}}\underset{\zeta}{\mathrm{max}}|Sf(u,v;\zeta )|\mathrm{.}$$
(23)
$$\phi (u,v)=\mathrm{Im}\left[\mathrm{ln}\left(sf{(u,v;\zeta )|}_{\zeta =\left|\frac{\partial \varphi}{\partial \widehat{\mathbf{W}}}\right|}\right)\right]\mathrm{.}$$
(24)
$$\left|\frac{\partial \varphi}{\partial \widehat{\mathbf{W}}}\right|=\mathrm{arg}\underset{\zeta}{\mathrm{max}}|\sum _{x=u-n}^{u+n}\sum _{y=v-n}^{v+n}[f(x,y)-\overline{f}]\phantom{\rule{0ex}{0ex}}\times h(x-u,y-v)\mathrm{exp}[j\zeta (x{\widehat{w}}_{1}+y{\widehat{w}}_{2})]|,$$