## Abstract

In the process of phase unwrapping for an image obtained by an interferometer or in-line holography, noisy image data may pose difficulties. Traditional phase unwrapping algorithms used to estimate a two-dimensional phase distribution include much estimation error, due to the effect of singular points. This paper introduces an accurate phase-unwrapping algorithm based on three techniques: a rotational compensator, unconstrained singular point positioning, and virtual singular points. The new algorithm can confine the effect of singularities to the local region around each singular point. The phase- unwrapped result demonstrates that accuracy is improved, compared with past methods based on the least-squares approach.

© 2010 Optical Society of America

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### Equations (23)

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(1)
$$\mathrm{\Phi}=\mathcal{W}\{\varphi \}\triangleq \varphi -\mathrm{Int}\left[\frac{\varphi}{2\pi}\right]2\pi ,$$
(2)
$$\mathit{g}(\mathit{r},{\mathit{r}}^{\prime})\triangleq \mathcal{W}\{\mathrm{\Phi}({\mathit{r}}^{\prime})-\mathrm{\Phi}(\mathit{r})\}\widehat{\mathit{s}}({\mathit{r}}^{\prime}-\mathit{r}),$$
(3)
$${\oint}_{c}\mathit{g}\xb7\widehat{\mathit{s}}\mathrm{d}l=0.$$
(4)
$${\oint}_{c}\mathit{g}\xb7\widehat{\mathit{s}}\mathrm{d}l=2\pi \sum _{k}{m}_{k}=2\pi M,\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}{m}_{k}\in \{-1,0,1\},$$
(5)
$$\mathit{g}=\nabla \varphi +\nabla \times \mathit{A}.$$
(6)
$${\oint}_{c}\mathit{g}\xb7\widehat{\mathit{s}}\mathrm{d}l={\oint}_{c}\nabla \times \mathit{A}\xb7\widehat{\mathit{s}}\mathrm{d}l.$$
(7)
$${\oint}_{c}\nabla \times \mathit{A}\xb7\widehat{\mathit{s}}\mathrm{d}l=2\pi \sum _{k}{m}_{k}.$$
(8)
$$\mathit{A}=\sum _{k}{\mathit{A}}_{k}.$$
(9)
$$\varphi ({\mathit{r}}^{\prime})-\varphi (\mathit{r})={\int}_{\mathit{r}}^{{\mathit{r}}^{\prime}}\nabla \varphi \xb7\widehat{\mathit{s}}\mathrm{d}l={\int}_{\mathit{r}}^{{\mathit{r}}^{\prime}}(\mathit{g}-\sum _{k}\nabla \times {\mathit{A}}_{k})\xb7\widehat{\mathit{s}}\mathrm{d}l.$$
(10)
$${\oint}_{c}\nabla \times {\mathit{A}}_{k}\xb7\widehat{\mathit{s}}\mathrm{d}l=2\pi {m}_{k}.$$
(11)
$$\nabla \times {\mathit{A}}_{k}\xb7\widehat{\mathit{s}}=\frac{\partial {a}_{z}}{\partial R}{\mathit{e}}_{R}\times {\mathit{e}}_{z}\xb7\widehat{\mathit{s}}=-\frac{\partial {a}_{z}}{\partial R}{\mathit{e}}_{R}\xb7\widehat{\mathit{n}},$$
(12)
$$\frac{\partial {a}_{z}}{\partial R}=-\frac{{m}_{k}}{R}.$$
(13)
$${\oint}_{c}\nabla \times {\mathit{A}}_{k}\xb7\widehat{\mathit{s}}\mathrm{d}l={\oint}_{c}\frac{{m}_{k}}{R}{\mathit{e}}_{R}\xb7\widehat{\mathit{n}}\mathrm{d}l.$$
(14)
$${C}_{k}({\mathit{r}}_{1},{\mathit{r}}_{2})\triangleq -{\int}_{{\mathit{r}}_{1}}^{{\mathit{r}}_{2}}\nabla \times {\mathit{A}}_{k}\xb7\widehat{\mathit{s}}\mathrm{d}l=-{\int}_{{\mathit{r}}_{1}}^{{\mathit{r}}_{2}}\frac{{m}_{k}}{R}{\mathit{e}}_{R}\xb7\widehat{\mathit{n}}\mathrm{d}l\phantom{\rule{0ex}{0ex}}={\int}_{{\mathit{r}}_{\u03f51}}^{{\mathit{r}}_{\u03f52}}\frac{{m}_{k}}{R}{\mathit{e}}_{R}\xb7\widehat{\mathit{n}}\mathrm{d}l={\int}_{{\theta}_{1,{s}_{k}}}^{{\theta}_{2,{s}_{k}}}\frac{{m}_{k}}{\u03f5}{\mathit{e}}_{R}\xb7(-{\mathit{e}}_{R})\u03f5\mathrm{d}\theta \phantom{\rule{0ex}{0ex}}=-{m}_{k}({\theta}_{2,{s}_{k}}-{\theta}_{1,{s}_{k}}),$$
(15)
$$\varphi ({\mathit{r}}^{\prime})=\varphi (\mathit{r})+\mathit{g}(\mathit{r},{\mathit{r}}^{\prime})\xb7\widehat{\mathit{s}}({\mathit{r}}^{\prime}-\mathit{r})+\sum _{k}{C}_{k}(\mathit{r},{\mathit{r}}^{\prime}).$$
(16)
$${\mathit{E}}_{\text{monopole}}=-\frac{1}{R}{\mathit{e}}_{R}.$$
(17)
$${\mathit{E}}_{\text{dipole}}=-\frac{1}{{R}^{2}}(2(\mathit{d}\xb7{\mathit{e}}_{R}){\mathit{e}}_{R}-\mathit{d}),$$
(18)
$$\mathrm{\Phi}(\mathit{r})=\mathcal{W}\{m\theta (\mathit{r},{\mathit{r}}_{s})+\overline{\varphi}+\delta \varphi (\mathit{r})\},\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\phantom{\rule{0ex}{0ex}}(m\in \{-1,+1\}),$$
(19)
$$\mathrm{\Delta}\triangleq \mathcal{W}\{\mathcal{W}\{{\mathrm{\Phi}}^{\prime}-\mathrm{\Phi}\}-m({\theta}^{\prime}-\theta )\},$$
(20)
$$\mathrm{\Delta}=\mathcal{W}\{\delta {\varphi}^{\prime}-\delta \varphi \}.$$
(21)
$$\text{minimize}\sum _{l=0}^{3}{\mathrm{\Delta}}^{2}=\sum _{l=0}^{3}(\mathcal{W}\{\mathcal{W}\{\mathrm{\Delta}{\mathrm{\Phi}}_{l}\}-m\mathrm{\Delta}{\theta}_{l}({\mathit{r}}_{s})\}{)}^{2},$$
(22)
$$\mathrm{\Delta}{\mathrm{\Phi}}_{l}\triangleq {\mathrm{\Phi}}_{l+1}-{\mathrm{\Phi}}_{l},\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}{\mathrm{\Phi}}_{4}={\mathrm{\Phi}}_{0},$$
(23)
$$\mathrm{\Delta}{\theta}_{l}\triangleq {\theta}_{l+1}({\mathit{r}}_{s})-{\theta}_{l}({\mathit{r}}_{s})>0,\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}{\theta}_{4}={\theta}_{0}+2\pi ,$$