## Abstract

The predictions of success rate and depth uncertainty for the
negative exponential sequence used for temporal phase unwrapping of
shape data are generalized to include the effect of a reduced sequence
and speckle noise in single-channel and multichannel systems,
respectively. To cope with the reduction of the sequence, a scaling
factor is introduced. A thorough investigation is made of the
performance of this algorithm, called the reduced temporal
phase-unwrapping algorithm. Two different approaches are
considered: a single-channel approach in which all the necessary
images are acquired sequentially in time and a multichannel approach in
which the three channels of a color CCD camera are used to carry the
phase-stepped images for each fringe density in parallel. The
performance of these two approaches are investigated by numerical
simulations. The simulations are based on a physical model in which
the speckle contrast, the fringe modulation, and random noise are
considered the sources of phase errors. Expressions are found that
relate the physical quantities to phase errors for the single-channel
and the multichannel approaches. In these simulations the
single-channel approach was found to be the most
robust. Expressions that relate the measurement accuracy and the
unwrapping reliability, respectively, with the reduction of the fringe
sequence were also found. As expected, the measurement accuracy is
not affected by a shorter fringe sequence, whereas a significant
reduction in the unwrapping reliability is found as compared with the
complete negative exponential sequence.

© 2001 Optical Society of America

Full Article |

PDF Article
### Equations (16)

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

(1)
$$\mathbf{t}=\left[{2}^{k-1},s\right],$$
(2)
$$\mathrm{\Delta}{\mathrm{\Phi}}_{u}\left[s-\mathbf{t}\left(i\right),s-\mathbf{t}\left(i+1\right)\right]=U\left\{\mathrm{\Delta}{\mathrm{\Phi}}_{w}\left[s-\mathbf{t}\left(i\right),s-\mathbf{t}\left(i+1\right)\right],\mathrm{\Delta}{\mathrm{\Phi}}_{u}\left[s,s-\mathbf{t}\left(i\right)\right]\right\},$$
(3)
$$\mathrm{\Delta}{\mathrm{\Phi}}_{u}\left[s,s-\mathbf{t}\left(i+1\right)\right]=\mathrm{\Delta}{\mathrm{\Phi}}_{u}\left[s,s-\mathbf{t}\left(i\right)\right]+\mathrm{\Delta}{\mathrm{\Phi}}_{u}\left[s-\mathbf{t}\left(i\right),s-\mathbf{t}\left(i+1\right)\right],$$
(4)
$$U\left\{{\mathrm{\Phi}}_{1},{\mathrm{\Phi}}_{2}\right\}={\mathrm{\Phi}}_{1}-2\mathrm{\pi}\mathrm{NINT}\left[\frac{{\mathrm{\Phi}}_{1}-T{\mathrm{\Phi}}_{2}}{2\mathrm{\pi}}\right]$$
(5)
$$T=\frac{\mathbf{t}\left(i+1\right)-\mathbf{t}\left(i\right)}{\mathbf{t}\left(i\right)}$$
(6)
$$\stackrel{\u02c6}{\mathrm{\omega}}=\frac{s{\mathrm{\Phi}}_{u}\left(s\right)+{\displaystyle \sum _{i=1}^{m-1}}\left[s-\mathbf{t}\left(i\right)\right]{\mathrm{\Phi}}_{u}\left[s-\mathbf{t}\left(i\right)\right]}{{s}^{2}+{\displaystyle \sum _{i=1}^{m-1}}{\left[s-\mathbf{t}\left(i\right)\right]}^{2}},$$
(7)
$${I}_{\mathrm{reg}}\left(j\right)={I}_{\mathrm{mod}}\left(j\right){I}_{s}\left(j\right)+2R\left(j\right)\mathrm{\u220a}\left(j\right),$$
(8)
$${I}_{\mathrm{mod}}\left(j\right)=1+Mcos\left[{\mathrm{\omega}}_{\mathbf{t}}x+\mathrm{\alpha}\left(j\right)\right]$$
(9)
$${I}_{s}\left(j\right)=\frac{1}{N}\sum _{i=1}^{\mathrm{ceil}\left(N\right)}k|a|_{i}{}^{2},$$
(10)
$$k=\left\{\begin{array}{ll}N-\mathrm{floor}\left(N\right)& \mathrm{if}i=\mathrm{ceil}\left(N\right)\\ 1& \mathrm{otherwise}\end{array}\right.,$$
(11)
$${s}_{\mathrm{\Phi}}=R\left({a}_{1}C+\frac{{a}_{2}}{M}\right),$$
(12)
$${s}_{\mathrm{\Phi}}=C\left(0.15+\frac{0.76}{M}\right)+R\left(6.91R+\frac{0.24}{M}-3.31C\right).$$
(13)
$${s}_{w}=\frac{{s}_{\mathrm{\Phi}}}{s}$$
(14)
$${s}_{f}=\frac{{s}_{\mathrm{\Phi}}}{{\left\{{s}^{2}+{\displaystyle \sum _{i=1}^{m-1}}{\left[s-\mathbf{t}\left(i\right)\right]}^{2}\right\}}^{1/2}}$$
(15)
$$\mathrm{S}.\mathrm{R}.=exp\left(-\frac{s_{\mathrm{\Phi}}{}^{3}{T}^{2.1}}{4}\right),$$
(16)
$${S}_{\mathrm{\Phi}g}\left(\mathrm{S}.\mathrm{R}.,T\right){=}^{3}\sqrt{-\frac{4ln\mathrm{S}.\mathrm{R}}{{T}^{2.1}}},$$