## Abstract

We present a novel ridge extraction algorithm for use with the two-dimensional continuous wavelet transform to extract the phase information from a fringe pattern. A cost function is employed for the detection of the ridge. The results of the proposed algorithm on simulated and real fringe patterns are illustrated.
Moreover, the proposed algorithm outperforms the maximum ridge extraction algorithm and it is found to be robust and reliable.

© 2007 Optical Society of America

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### Equations (6)

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(1)
$${\psi}_{F}\left(x,y\right)={\displaystyle \underset{j=0}{\overset{{N}_{\theta}-1}{\mathrm{\Sigma}}}\text{exp}\left(i{k}_{o}\left(x\text{\hspace{0.17em}cos \hspace{0.17em}}{\theta}_{j}+y\text{\hspace{0.17em} sin \hspace{0.17em}}{\theta}_{j}\right)\right)}\text{\hspace{0.17em}}\times \text{exp}\left(-\frac{1}{2}\sqrt{{x}^{2}+{y}^{2}}\right)\text{,}$$
(2)
$$S\left(a,b,s,\theta \right)={s}^{-1}{\displaystyle \int {\displaystyle \int {\psi}_{F}}}\left(\frac{x-a}{s},\frac{y-b}{s},{r}_{\theta}\right)\times f\left(x,y\right)\mathrm{d}x\mathrm{d}y\text{.}$$
(3)
$$\varphi \left(x,y\right)=3*{\left(1-x\right)}^{2}*\text{\hspace{0.17em}exp}\left(-{x}^{2}-{\left(y+1\right)}^{2}\right)-10*\left(\frac{x}{5}-{x}^{3}-{y}^{5}\right)*\text{exp}\left(-{x}^{2}-{y}^{2}\right)-\frac{1}{3}*\text{\hspace{0.17em}exp}\left(-{\left(x+1\right)}^{2}-{y}^{2}\right)\text{,}$$
(4)
$$I\left(x,y\right)=0.3\varphi \left(x,y\right)+\text{cos}\left(2\pi {f}_{o}x+\varphi \left(x,y\right)\right)+NOISE\text{,}$$
(5)
$$Cost\left[\varphi \left(b\right),b\right]=-{C}_{o}{{\displaystyle {\int}_{b}\left|S\left[\varphi \left(b\right),b\right]\right|}}^{2}\mathrm{d}b+{C}_{l}{{\displaystyle {\int}_{b}\left|\frac{\partial \varphi \left(b\right)}{\partial b}\right|}}^{2}\mathrm{d}b,$$
(6)
$$Cost={\displaystyle \underset{b=2}{\overset{W}{\mathrm{\Sigma}}}\{-{C}_{o}{\left|\left[S(\varphi \left(b\right),b)\right]\right|}^{2}+{C}_{l}\mid \varphi \left(b\right)-\varphi (b-1){\mid}^{2}\}\text{,}}$$