## Abstract

The regularized phase tracker (RPT) is one of the most powerful approaches for demodulation of a single fringe pattern. However, two disadvantages limit the applications of the RPT in practice. One is the necessity of a normalized fringe pattern as input and the other is the sensitivity to critical points. To overcome these two disadvantages, a generalized regularized phase tracker (GRPT) is presented. The GRPT is characterized by two novel improvements. First, a general local fringe model that includes a linear background, a linear modulation and a quadratic phase is adopted in the proposed enhanced cost function. Second, the number of iterations in the optimization process is proposed as a comprehensive measure of fringe quality and used to guide the demodulation path. With these two improvements, the GRPT can directly demodulate a single fringe pattern without any pre-processing and post-processing and successfully get rid of the problem of the sensitivity to critical points. Simulation and experimental results are presented to demonstrate the effectiveness and robustness of the GRPT.

© 2012 OSA

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### Equations (7)

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(1)
$$f\left(x,y\right)=a\left(x,y\right)+b\left(x,y\right)\mathrm{cos}\left[\phi \left(x,y\right)\right]+n\left(x,y\right),$$
(2)
$${f}_{n}\left(\epsilon ,\eta \right)\approx \mathrm{cos}\left[\phi \left(\epsilon ,\eta \right)\right],$$
(3)
$${\phi}_{e}\left(x,y;\epsilon ,\eta \right)={\phi}_{0}\left(x,y\right)+{\omega}_{x}\left(x,y\right)\left(\epsilon -x\right)+{\omega}_{y}\left(x,y\right)\left(\eta -y\right),$$
(4)
$${a}_{e}\left(x,y;\epsilon ,\eta \right)={a}_{0}\left(x,y\right)+{a}_{x}\left(x,y\right)\left(\epsilon -x\right)+{a}_{y}\left(x,y\right)\left(\eta -y\right),$$
(5)
$${b}_{e}\left(x,y;\epsilon ,\eta \right)={b}_{0}\left(x,y\right)+{b}_{x}\left(x,y\right)\left(\epsilon -x\right)+{b}_{y}\left(x,y\right)\left(\eta -y\right),$$
(6)
$$\begin{array}{c}{\phi}_{e}\left(x,y;\epsilon ,\eta \right)={\phi}_{0}\left(x,y\right)+{\omega}_{x}\left(x,y\right)\left(\epsilon -x\right)+{\omega}_{y}\left(x,y\right)\left(\eta -y\right)+\frac{{c}_{xx}\left(x,y\right)}{2}{\left(\epsilon -x\right)}^{2}\\ +\frac{{c}_{yy}\left(x,y\right)}{2}{\left(\eta -y\right)}^{2}+{c}_{xy}\left(x,y\right)\left(\epsilon -x\right)\left(\eta -y\right),\end{array}$$
(7)
$$\begin{array}{c}{a}_{0}\left(x,y\right)={a}_{0}\left({x}_{s},{y}_{s}\right)+{a}_{x}\left({x}_{s},{y}_{s}\right)\left(x-{x}_{s}\right)+{a}_{y}\left({x}_{s},{y}_{s}\right)\left(y-{y}_{s}\right)\\ {a}_{x}\left(x,y\right)={a}_{x}\left({x}_{s},{y}_{s}\right)\\ {a}_{y}\left(x,y\right)={a}_{y}\left({x}_{s},{y}_{s}\right)\\ {b}_{0}\left(x,y\right)={b}_{0}\left({x}_{s},{y}_{s}\right)+{b}_{x}\left({x}_{s},{y}_{s}\right)\left(x-{x}_{s}\right)+{b}_{y}\left({x}_{s},{y}_{s}\right)\left(y-{y}_{s}\right)\\ {b}_{x}\left(x,y\right)={b}_{x}\left({x}_{s},{y}_{s}\right)\\ {b}_{y}\left(x,y\right)={b}_{y}\left({x}_{s},{y}_{s}\right)\\ {\phi}_{0}\left(x,y\right)={\phi}_{0}\left({x}_{s},{y}_{s}\right)+{\omega}_{x}\left({x}_{s},{y}_{s}\right)\left(x-{x}_{s}\right)+{\omega}_{y}\left({x}_{s},{y}_{s}\right)\left(y-{y}_{s}\right)\\ +{c}_{xx}\left({x}_{s},{y}_{s}\right){\left(x-{x}_{s}\right)}^{2}/2+{c}_{yy}\left({x}_{s},{y}_{s}\right){\left(y-{y}_{s}\right)}^{2}/2\\ +{c}_{xy}\left({x}_{s},{y}_{s}\right)\left(x-{x}_{s}\right)\left(y-{y}_{s}\right)\\ {\omega}_{x}\left(x,y\right)={\omega}_{x}\left({x}_{s},{y}_{s}\right)+{c}_{xx}\left({x}_{s},{y}_{s}\right)\left(x-{x}_{s}\right)+{c}_{xy}\left({x}_{s},{y}_{s}\right)\left(y-{y}_{s}\right)\\ {\omega}_{y}\left(x,y\right)={\omega}_{y}\left({x}_{s},{y}_{s}\right)+{c}_{yy}\left({x}_{s},{y}_{s}\right)\left(y-{y}_{s}\right)+{c}_{xy}\left({x}_{s},{y}_{s}\right)\left(x-{x}_{s}\right)\\ {c}_{xx}\left(x,y\right)={c}_{xx}\left({x}_{s},{y}_{s}\right)\\ {c}_{yy}\left(x,y\right)={c}_{yy}\left({x}_{s},{y}_{s}\right)\\ {c}_{xy}\left(x,y\right)={c}_{xy}\left({x}_{s},{y}_{s}\right).\end{array}$$