## Abstract

A generalized regularized phase tracker (GRPT) for demodulation of a single fringe pattern was recently proposed. It is very successful for many fringe patterns. However, the GRPT has poor performance in the area where the fringe pattern is sparse. An improved GRPT (iGRPT) with two novel improvements is proposed to overcome the problem. First, the fixed window used in the GRPT is replaced by a spatially adaptive window. Second, a background regularization term and a modulation regularization term are incorporated in the cost function. With these two improvements, the proposed iGRPT can successfully demodulate sparse fringes and thus improves the demodulation capability of the GRPT. Simulation and experimental results are presented to verify the performance of the iGRPT.

© 2013 Optical Society of America

Full Article |

PDF Article
### Equations (15)

Equations on this page are rendered with MathJax. Learn more.

(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)
$$U\left(x,y\right)={\displaystyle \sum _{\left(\epsilon ,\eta \right)\in {N}_{x,y}}\left(w\left(x,y;\epsilon ,\eta \right)\left\{{\left[f\left(\epsilon ,\eta \right)-{f}_{e}\left(x,y;\epsilon ,\eta \right)\right]}^{2}+\lambda {\left[{\phi}_{0}\left(\epsilon ,\eta \right)-{\phi}_{e}\left(x,y;\epsilon ,\eta \right)\right]}^{2}m\left(\epsilon ,\eta \right)\right\}\right)},$$
(3)
$$w\left(x,y;\epsilon ,\eta \right)=\mathrm{exp}\left\{-\left[{\left(x-\epsilon \right)}^{2}+{\left(y-\eta \right)}^{2}\right]/\left(2{\sigma}^{2}\right)\right\},$$
(4)
$${f}_{e}\left(x,y;\epsilon ,\eta \right)={a}_{e}\left(x,y;\epsilon ,\eta \right)+{b}_{e}\left(x,y;\epsilon ,\eta \right)\mathrm{cos}\left[{\phi}_{e}\left(x,y;\epsilon ,\eta \right)\right],$$
(5)
$${a}_{e}\left(x,y;\epsilon ,\eta \right)={a}_{0}\left(x,y\right)+{a}_{x}\left(x,y\right)\left(\epsilon -x\right)+{a}_{y}\left(x,y\right)\left(\eta -y\right),$$
(6)
$${b}_{e}\left(x,y;\epsilon ,\eta \right)={b}_{0}\left(x,y\right)+{b}_{x}\left(x,y\right)\left(\epsilon -x\right)+{b}_{y}\left(x,y\right)\left(\eta -y\right),$$
(7)
$$\begin{array}{c}{\phi}_{e}\left(x,y;\epsilon ,\eta \right)={\phi}_{0}\left(x,y\right)+{\omega}_{x}\left(x,y\right)\left(\epsilon -x\right)+{\omega}_{y}\left(x,y\right)\left(\eta -y\right)+\frac{1}{2}{c}_{xx}\left(x,y\right){\left(\epsilon -x\right)}^{2}\\ +\frac{1}{2}{c}_{yy}\left(x,y\right){\left(\eta -y\right)}^{2}+{c}_{xy}\left(x,y\right)\left(\epsilon -x\right)\left(\eta -y\right),\end{array}$$
(8)
$${\omega}_{\text{TLF}}=\sqrt{{\omega}_{x}^{2}+{\omega}_{y}^{2}},$$
(9)
$$\tilde{\sigma}=2/{\omega}_{\text{TLF}},$$
(10)
$$\sigma =\{\begin{array}{ccc}{\sigma}_{\mathrm{min}}& if& \tilde{\sigma}\le {\sigma}_{\mathrm{min}}\\ \tilde{\sigma}& if& {\sigma}_{\mathrm{min}}<\tilde{\sigma}<{\sigma}_{\mathrm{max}}\\ {\sigma}_{\mathrm{max}}& if& \tilde{\sigma}\ge {\sigma}_{\mathrm{max}}\end{array},$$
(11)
$$\sigma =\{\begin{array}{ccc}{\sigma}_{\mathrm{min}}& if& \tilde{\sigma}\le \left({\sigma}_{\mathrm{min}}+{\sigma}_{\mathrm{max}}\right)/2\\ {\sigma}_{\mathrm{max}}& if& \tilde{\sigma}>\left({\sigma}_{\mathrm{min}}+{\sigma}_{\mathrm{max}}\right)/2\end{array}.$$
(12)
$$U\left(x,y\right)={\displaystyle \sum _{\left(\epsilon ,\eta \right)\in {N}_{x,y}}(w\left(x,y;\epsilon ,\eta \right)\{{\left[f\left(\epsilon ,\eta \right)-{f}_{e}\left(x,y;\epsilon ,\eta \right)\right]}^{2}}+\left({\lambda}_{a}{R}_{a}+{\lambda}_{b}{R}_{b}+{\lambda}_{\phi}{R}_{\phi}\right)m\left(\epsilon ,\eta \right)\}),$$
(13)
$${R}_{a}={\left[{a}_{0}\left(\epsilon ,\eta \right)-{a}_{e}\left(x,y;\epsilon ,\eta \right)\right]}^{2},$$
(14)
$${R}_{b}={\left[{b}_{0}\left(\epsilon ,\eta \right)-{b}_{e}\left(x,y;\epsilon ,\eta \right)\right]}^{2},$$
(15)
$${R}_{\phi}={\left[{\phi}_{0}\left(\epsilon ,\eta \right)-{\phi}_{e}\left(x,y;\epsilon ,\eta \right)\right]}^{2},$$