## Abstract

What we believe to be a novel technique of branch-cut placement in the phase unwrapping is proposed. This approach is based on what we named residue vector, which is generated by a residue in a wrapped phase map and has an orientation that points out toward the balancing residue of opposite polarity. The residue vector can be used to guide the manner in which branch cuts are placed in phase unwrapping. Also, residue vector can be used for the determination of the weighting values used in different existing phase unwrapping methods such as minimum cost flow and least squares. The theoretical foundations of the residue-vector method are presented, and a branch-cut method using its information is developed and implemented. A general comparison is made between the residue-vector map and other existing quality maps.

© 2007 Optical Society of America

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### Equations (35)

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(6)
$$\sum _{i}^{M}\hat{\nabla}\Phi \left({p}_{i}\right)}=0,$$
(7)
\Phi \left({p}_{i}\right)
(8)
{p}_{i}\text{\hspace{0.17em}}\in \text{\hspace{0.17em}}\left\{P\right\}
(9)
\widehat{\nabla}\Phi \left({p}_{i}\right)
(14)
$$\sum _{i}^{M}\widehat{\nabla}\Phi \left({p}_{i}\right)}=2\pi n.$$
(15)
\left|\widehat{\nabla}\Phi \right|>\pi
(20)
$$\sum _{b}^{{N}_{bc}}{\displaystyle \sum _{p}^{{N}_{bp}}\widehat{\nabla}\Phi \left(b,p\right)}}=0,$$
(21)
\widehat{\nabla}\Phi \left(b,p\right)
(25)
$$E={\displaystyle \sum _{i}^{N}{\displaystyle \sum _{j}^{M}\left|{k}_{x}\left(i,j\right)\right|}}+{\displaystyle \sum _{i}^{N}{\displaystyle \sum _{j}^{M}\left|{k}_{y}\left(i,j\right)\right|}},$$
(26)
$${k}_{x}\left(i,j\right)=\text{Int}\left(\frac{\Psi \left(i,j\right)-\Psi \left(i-1,j\right)-\widehat{\nabla}\Phi \left(i,j\right)}{2\pi}\right)$$
(27)
$${k}_{y}\left(i,j\right)=\text{Int}\left(\frac{\Psi \left(i,j\right)-\Psi \left(i,j-1\right)-\widehat{\nabla}\Phi \left(i,j\right)}{2\pi}\right)$$
(28)
\Psi \left(i,j\right)
(29)
\mathrm{I}\mathrm{n}\mathrm{t}(\cdot )
(30)
$$\text{min \hspace{0.17em}}E\text{\hspace{0.17em}}\stackrel{\text{If \hspace{0.17em} and \hspace{0.17em} only \hspace{0.17em} if}}{\leftrightarrow}\{\begin{array}{l}\text{max \hspace{0.17em}}{N}_{rf}\hfill \\ \text{min \hspace{0.17em}}{N}_{br}\hfill \end{array}.$$