## Abstract

A non-iterative spatial phase-shifting algorithm based on principal component analysis (PCA) is proposed to directly extract the phase from only a single spatial carrier interferogram. Firstly, we compose a set of phase-shifted fringe patterns from the original spatial carrier interferogram shifting by one pixel their starting position. Secondly, two uncorrelated quadrature signals that correspond to the first and second principal components are extracted from the phase-shifted interferograms by the PCA algorithm. Then, the modulating phase is calculated from the arctangent function of the two quadrature signals. Meanwhile, the main factors that may influence the performance of the proposed method are analyzed and discussed, such as the level of random noise, the carrier-frequency values and the angle of carrier-frequency of fringe pattern. Numerical simulations and experiments are given to demonstrate the performance of the proposed method and the results show that the proposed method is fast, effectively and accurate. The proposed method can be used to on-line detection fields of dynamic or moving objects.

© 2012 OSA

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### Equations (13)

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(1)
$$I\left(x,y\right)\text{=A}\left(x,y\right)+B\left(x,y\right)\mathrm{cos}\left[2\pi \left({\kappa}_{x}x+{\kappa}_{y}y\right)+\varphi \left(x,y\right)\right]$$
(2)
$${I}_{1}\left(x,y\right)\text{=}I\left(x,y\right)\text{=A}\left(x,y\right)+B\left(x,y\right)\mathrm{cos}\left[2\pi \left({\kappa}_{x}x+{\kappa}_{y}y\right)\text{+}\varphi \left(x,y\right)\right]$$
(3)
$${I}_{2}\left(x,y\right)\text{=}I\left(x\text{+}1,y\right)\text{=A}\left(x,y\right)+B\left(x,y\right)\mathrm{cos}\left[2\pi \left({\kappa}_{x}x+{\kappa}_{y}y\right)\text{+}\varphi \left(x,y\right)\text{+}{\delta}_{1}\right]$$
(4)
$${I}_{3}\left(x,y\right)\text{=}I\left(x,y\text{+}1\right)\text{=A}\left(x,y\right)+B\left(x,y\right)\mathrm{cos}\left[2\pi \left({\kappa}_{x}x+{\kappa}_{y}y\right)\text{+}\varphi \left(x,y\right)\text{+}{\delta}_{2}\right]$$
(5)
$${I}_{4}\left(x,y\right)\text{=}I\left(x\text{+}1,y\text{+}1\right)\text{=A}\left(x,y\right)+B\left(x,y\right)\mathrm{cos}\left[2\pi \left({\kappa}_{x}x+{\kappa}_{y}y\right)\text{+}\varphi \left(x,y\right)\text{+}{\delta}_{3}\right]$$
(6)
$$\text{X=}{\left[{I}_{1},{I}_{2},\cdot \cdot \cdot ,{I}_{n},\cdot \cdot \cdot {I}_{N}\right]}^{T}$$
(7)
$${X}_{m}\text{=}\frac{1}{N}{\displaystyle \sum _{n=1}^{n=N}{I}_{n}}\approx A$$
(8)
$$C\text{=}\left[X-{X}_{m}\right]{\left[X-{X}_{m}\right]}^{T}$$
(9)
$$C{Q}_{i}={V}_{i}{Q}_{i},\text{\hspace{1em}}i=1,2,\mathrm{...},N$$
(10)
$$Q\text{=}{\left[{Q}_{1},{Q}_{2}\cdot \cdot \cdot ,{Q}_{n},\cdot \cdot \cdot {Q}_{N}\right]}^{T}$$
(11)
$$\text{D=}{Q}^{T}CQ$$
(12)
$$\Phi =\left\{\begin{array}{l}{\Phi}_{1}\\ {\Phi}_{2}\\ \cdot \cdot \cdot \\ {\Phi}_{N}\end{array}\right\}\text{=}Q\left(X-{X}_{m}\right)=\left\{\begin{array}{l}{Q}_{1}\\ {Q}_{2}\\ \cdot \cdot \cdot \\ {Q}_{N}\end{array}\right\}\left(X-{X}_{m}\right)$$
(13)
$$\varphi \left(x,y\right)=\mathrm{arc}\mathrm{tan}\left(\frac{{I}_{c}}{{I}_{s}}\right)-2\pi \left({\kappa}_{x}x+{\kappa}_{y}y\right)=\pm \mathrm{arc}\mathrm{tan}\left(\frac{{\Phi}_{1}}{{\Phi}_{2}}\right)-2\pi \left({\kappa}_{x}x+{\kappa}_{y}y\right)$$