## Abstract

The two-dimensional regularized phase-tracking (RPT) technique is one of the most powerful approaches to demodulate a single interferogram with either open or closed fringes. However, it often fails in the cases of complex interferograms and needs well-defined scanning strategies. An improved algorithm based on the RPT is presented in this paper. We use a paraboloid phase model to approximate the phase function and modify the cost functional to search the smoothest phase solutions in the function space ${C}^{2}$. With these modifications, the phase tracker preserves the robustness of the RPT while at the same time it is no more sensitive to stationary points and is capable of demodulating complex interferograms with arbitrary scanning schemes. Moreover, the phase reconstructed by the proposed algorithm is normally more accurate than that of the RPT both for noiseless and noisy interferograms under the same conditions. Computer simulations and experimental results are both presented.

© 2010 Optical Society of America

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### Equations (16)

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(1)
$$I(x,y)=a(x,y)+b(x,y)\mathrm{cos}[\varphi (x,y)]+n(x,y),$$
(2)
$${U}_{T}=\sum _{(x,y)\in L}{U}_{x,y}^{\prime}({\widehat{\varphi}}_{0},{\widehat{\omega}}_{x},{\widehat{\omega}}_{y}),$$
(3)
$${U}_{x,y}^{\prime}({\widehat{\varphi}}_{0},{\widehat{\omega}}_{x},{\widehat{\omega}}_{y})=\sum _{(\epsilon ,\eta )\in ({N}_{x,y}\cap L)}(\{[{\widehat{I}}_{n}(\epsilon ,\eta )-\mathrm{cos}[\widehat{\varphi}(\epsilon ,\eta ,x,y)]{\}}^{2}+{\beta}_{1}[{\widehat{\varphi}}_{0}(\epsilon ,\eta )-\widehat{\varphi}(\epsilon ,\eta ,x,y){]}^{2}m(\epsilon ,\eta )),$$
(4)
$${\widehat{I}}_{n}(\epsilon ,\eta )\approx \mathrm{cos}[\varphi (\epsilon ,\eta )],$$
(5)
$$\widehat{\varphi}(\epsilon ,\eta ,x,y)={\widehat{\varphi}}_{0}(x,y)+[{\widehat{\omega}}_{\epsilon}(x,y)(\epsilon -x)\phantom{\rule{0ex}{0ex}}+{\widehat{\omega}}_{\eta}(x,y)(\eta -y)],$$
(6)
$$\widehat{\varphi}(\epsilon ,\eta ,x,y)={\widehat{\varphi}}_{0}(x,y)+{\widehat{\varphi}}_{1}(\epsilon ,\eta ,x,y)+{\widehat{\varphi}}_{2}(\epsilon ,\eta ,x,y),$$
(7)
$${\widehat{\varphi}}_{1}(\epsilon ,\eta ,x,y)=[{\widehat{{\varphi}^{\prime}}}_{\epsilon}(x,y)(\epsilon -x)+{\widehat{{\varphi}^{\prime}}}_{\eta}(x,y)(\eta -y)],\phantom{\rule[-0.0ex]{0em}{0.0ex}}\phantom{\rule{0ex}{0ex}}{\widehat{\varphi}}_{2}(\epsilon ,\eta ,x,y)=[{\widehat{{\varphi}^{\prime \prime}}}_{\epsilon \epsilon}(x,y)(\epsilon -x{)}^{2}\phantom{\rule{0ex}{0ex}}+2{\widehat{{\varphi}^{\prime \prime}}}_{\epsilon \eta}(x,y)(\epsilon -x)(\eta -y)\phantom{\rule{0ex}{0ex}}+{\widehat{{\varphi}^{\prime \prime}}}_{\eta \eta}(x,y)(\eta -y{)}^{2}]/2,$$
(8)
$$\widehat{\chi}(x,y)=[{\widehat{\varphi}}_{0}(x,y),{\widehat{{\varphi}^{\prime}}}_{\epsilon}(x,y),{\widehat{{\varphi}^{\prime}}}_{\eta}(x,y),{\widehat{{\varphi}^{\prime \prime}}}_{\epsilon \epsilon}(x,y),\phantom{\rule{0ex}{0ex}}{\widehat{{\varphi}^{\prime \prime}}}_{\epsilon \eta}(x,y),{\widehat{{\varphi}^{\prime \prime}}}_{\eta \eta}(x,y){]}^{T},$$
(9)
$${U}_{x,y}(\widehat{\chi}(x,y))=\sum _{(\epsilon ,\eta )\in ({N}_{x,y}\cap L)}(\{{\widehat{I}}_{n}(\epsilon ,\eta )-\mathrm{cos}[\widehat{\varphi}(\epsilon ,\eta ,x,y)]{\}}^{2}\phantom{\rule{0ex}{0ex}}+{\beta}_{1}[{\widehat{\varphi}}_{0}(\epsilon ,\eta )-\widehat{\varphi}(\epsilon ,\eta ,x,y){]}^{2}m(\epsilon ,\eta )\phantom{\rule{0ex}{0ex}}+R(\epsilon ,\eta ,x,y)),$$
(10)
$$R(\epsilon ,\eta ,x,y)={\beta}_{2}\{[{\widehat{{\varphi}^{\prime \prime}}}_{\epsilon \epsilon}(\epsilon ,\eta )-{\widehat{{\varphi}^{\prime \prime}}}_{\epsilon \epsilon}(x,y){]}^{2}+[{\widehat{{\varphi}^{\prime \prime}}}_{\epsilon \eta}(\epsilon ,\eta )-{\widehat{{\varphi}^{\prime \prime}}}_{\epsilon \eta}(x,y){]}^{2}+[{\widehat{{\varphi}^{\prime \prime}}}_{\eta \eta}(\epsilon ,\eta )-{\widehat{{\varphi}^{\prime \prime}}}_{\eta \eta}(x,y){]}^{2}\}m(\epsilon ,\eta ),$$
(11)
$${{\widehat{\varphi}}_{0}}^{k+1}(x,y)={{\widehat{\varphi}}_{0}}^{k}(x,y)-\tau \frac{\partial {U}_{x,y}(\widehat{\chi}(x,y))}{\partial {\widehat{\varphi}}_{0}(x,y)},\phantom{\rule[-0.0ex]{0em}{0.0ex}}\phantom{\rule{0ex}{0ex}}{{\widehat{{\varphi}^{\prime}}}_{\epsilon}}^{k+1}(x,y)={{\widehat{{\varphi}^{\prime}}}_{\epsilon}}^{k}(x,y)-\tau \frac{\partial {U}_{x,y}(\widehat{\chi}(x,y))}{\partial {\widehat{{\varphi}^{\prime}}}_{\epsilon}(x,y)},\phantom{\rule[-0.0ex]{0em}{0.0ex}}\phantom{\rule{0ex}{0ex}}{{\widehat{{\varphi}^{\prime}}}_{\eta}}^{k+1}(x,y)={{\widehat{{\varphi}^{\prime}}}_{\eta}}^{k}(x,y)-\tau \frac{\partial {U}_{x,y}(\widehat{\chi}(x,y))}{\partial {\widehat{{\varphi}^{\prime}}}_{\eta}(x,y)},\phantom{\rule[-0.0ex]{0em}{0.0ex}}\phantom{\rule{0ex}{0ex}}{{\widehat{{\varphi}^{\prime \prime}}}_{\epsilon \epsilon}}^{k+1}(x,y)={{\widehat{{\varphi}^{\prime \prime}}}_{\epsilon \epsilon}}^{k}(x,y)-\tau \frac{\partial {U}_{x,y}(\widehat{\chi}(x,y))}{\partial {\widehat{{\varphi}^{\prime \prime}}}_{\epsilon \epsilon}(x,y)},\phantom{\rule[-0.0ex]{0em}{0.0ex}}\phantom{\rule{0ex}{0ex}}{{\widehat{{\varphi}^{\prime \prime}}}_{\epsilon \eta}}^{k+1}(x,y)={{\widehat{{\varphi}^{\prime \prime}}}_{\epsilon \eta}}^{k}(x,y)-\tau \frac{\partial {U}_{x,y}(\widehat{\chi}(x,y))}{\partial {\widehat{{\varphi}^{\prime \prime}}}_{\epsilon \eta}(x,y)},\phantom{\rule[-0.0ex]{0em}{0.0ex}}\phantom{\rule{0ex}{0ex}}{{\widehat{{\varphi}^{\prime \prime}}}_{\eta \eta}}^{k+1}(x,y)={{\widehat{{\varphi}^{\prime \prime}}}_{\eta \eta}}^{k}(x,y)-\tau \frac{\partial {U}_{x,y}(\widehat{\chi}(x,y))}{\partial {\widehat{{\varphi}^{\prime \prime}}}_{\eta \eta}(x,y)},$$
(12)
$${U}_{x,y}(\widehat{\chi}(x,y))=\sum _{(\epsilon ,\eta )\in ({N}_{x,y}\cap L)}(\{{\widehat{I}}_{n}(\epsilon ,\eta )-\mathrm{cos}[\widehat{\varphi}(\epsilon ,\eta ,x,y)]{\}}^{2}\phantom{\rule{0ex}{0ex}}+{\beta}_{1}[{\widehat{\varphi}}_{0}(\epsilon ,\eta )-\widehat{\varphi}(\epsilon ,\eta ,x,y){]}^{2}m(\epsilon ,\eta )\phantom{\rule{0ex}{0ex}}+P(\widehat{\chi}(x,y))+R(\epsilon ,\eta ,x,y)),$$
(13)
$$P(\widehat{\chi}(x,y))=\{{\widehat{I}}_{n}(\epsilon ,\eta )-\mathrm{cos}[\widehat{\varphi}(\epsilon ,\eta ,x,y)+\alpha ]{\}}^{2},$$
(14)
$$I(x,y)=\mathrm{cos}[2\pi \times (21.6xy/{N}^{2}\{125.3[({x}^{2}+{y}^{2})/{N}^{2}{]}^{2}\phantom{\rule{0ex}{0ex}}-57.8({x}^{2}+{y}^{2})/{N}^{2}+6\}+0.6)]\times \text{Mask},\phantom{\rule{0.265em}{0ex}}$$
(15)
$$I(x,y)=[0.2+2\mathrm{cos}(2\pi \{0.81({x}^{2}+{y}^{2})/{N}^{2}\phantom{\rule{0ex}{0ex}}-2.5[\mathrm{cos}(3.6\pi x/N)+\mathrm{cos}(3.6\pi y/N)]+0.3\phantom{\rule{0ex}{0ex}}\times (\text{Rand}-0.5)\})]\times \text{Mask},$$
(16)
$${\epsilon}_{\mathrm{PV}}=\mathrm{max}(|\varphi (m,n)-\widehat{\varphi}(m,n)|),\phantom{\rule[-0.0ex]{0em}{0.0ex}}\phantom{\rule{0ex}{0ex}}{\epsilon}_{\mathrm{MSE}}=\frac{1}{M}\sum _{m,n}(|\varphi (m,n)-\widehat{\varphi}(m,n)|{)}^{2},$$