## Abstract

This paper proposes a novel method to substantially reduce motion-introduced phase error in phase-shifting profilometry. We first estimate the motion of an object from the difference between two subsequent 3D frames. After that, by leveraging the projector’s pinhole model, we can determine the motion-induced phase shift error from the estimated motion. A generic phase-shifting algorithm considering phase shift error is then utilized to compute the phase. Experiments demonstrated that proposed algorithm effectively improved the measurement quality by compensating for the phase shift error introduced by rigid and nonrigid motion for a standard single-projector and single-camera digital fringe projection system.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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### Equations (11)

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(1)
$${I}_{k}\left({u}^{c},{v}^{c}\right)={I}^{\prime}\left({u}^{c},{v}^{c}\right)+{I}^{\u2033}\left({u}^{c},{v}^{c}\right)\mathrm{cos}\left[\mathrm{\Phi}\left({u}^{c},{v}^{c}\right)-{\delta}_{k}\right],$$
(2)
$$\left[\begin{array}{c}{a}_{0}\left({u}^{c},{v}^{c}\right)\\ {a}_{1}\left({u}^{c},{v}^{c}\right)\\ {a}_{2}\left({u}^{c},{v}^{c}\right)\end{array}\right]={\left[\begin{array}{ccc}M& {\displaystyle \sum \mathrm{cos}{\delta}_{k}}& {\displaystyle \sum \mathrm{sin}{\delta}_{k}}\\ {\displaystyle \sum \mathrm{cos}{\delta}_{k}}& {\displaystyle \sum {\mathrm{cos}}^{2}{\delta}_{k}}& {\displaystyle \sum \mathrm{cos}{\delta}_{k}\mathrm{sin}{\delta}_{k}}\\ {\displaystyle \sum \mathrm{sin}{\delta}_{k}}& {\displaystyle \sum \mathrm{cos}{\delta}_{k}\mathrm{sin}{\delta}_{k}}& {\displaystyle \sum {\mathrm{sin}}^{2}{\delta}_{k}}\end{array}\right]}^{-1}\left[\begin{array}{c}{\displaystyle \sum {I}_{k}}\\ {\displaystyle \sum {I}_{k}\mathrm{cos}{\delta}_{k}}\\ {\displaystyle \sum {I}_{k}\mathrm{sin}{\delta}_{k}}\end{array}\right],$$
(3)
$$\varphi \left({u}^{c},{v}^{c}\right)={\mathrm{tan}}^{-1}\left[\frac{{a}_{2}\left({u}^{c},{v}^{c}\right)}{{a}_{1}\left({u}^{c},{v}^{c}\right)}\right].$$
(4)
$$\mathrm{\Phi}\left({u}^{c},{v}^{c}\right)=\varphi \left({u}^{c},{v}^{c}\right)+k\left({u}^{c},{v}^{c}\right)\times 2\pi ,$$
(5)
$$\overline{{\delta}_{k}}\left({u}^{c},{v}^{c}\right)={\delta}_{k}\left({u}^{c},{v}^{c}\right)+{\u03f5}_{k}\left({u}^{c},{v}^{c}\right),$$
(6)
$${\u03f5}_{1}=\overline{{\mathrm{\Phi}}_{1}}-{\mathrm{\Phi}}_{1}.$$
(7)
$${\u03f5}_{2}=\overline{{\mathrm{\Phi}}_{2}}-{\mathrm{\Phi}}_{2}.$$
(8)
$${s}^{p}\left\{\begin{array}{c}{u}^{p}\\ {v}^{p}\\ 1\end{array}\right\}={\mathbf{A}}^{p}[{\mathbf{R}}^{p},{\mathbf{t}}^{p}]\left\{\begin{array}{c}{x}^{w}\\ {y}^{w}\\ {z}^{w}\\ 1\end{array}\right\}=\left[\begin{array}{cccc}{P}_{11}& {P}_{12}& {P}_{13}& {P}_{14}\\ {P}_{21}& {P}_{22}& {P}_{23}& {P}_{24}\\ {P}_{31}& {P}_{32}& {P}_{33}& {P}_{34}\end{array}\right]\left\{\begin{array}{c}{x}^{w}\\ {y}^{w}\\ {z}^{w}\\ 1\end{array}\right\}$$
(9)
$$\frac{\partial {u}^{p}}{\partial {x}^{w}}=\frac{{P}_{11}\left({P}_{32}{y}^{w}+{P}_{33}{z}^{w}+{P}_{34}\right)-{P}_{\begin{array}{l}1\\ 1\end{array}}\left({P}_{12}{y}^{w}+{P}_{13}{z}^{w}+{P}_{14}\right)}{{\left({P}_{31}{x}^{w}+{P}_{32}{y}^{w}+{P}_{33}{z}^{w}+{P}_{34}\right)}^{2}}.$$
(10)
$$\u03f5\left({u}^{c},{v}^{c}\right)=\frac{2\pi}{\lambda}\left(\frac{\partial {u}^{p}}{\partial {x}^{w}}\mathrm{\Delta}{x}^{w}+\frac{\partial {u}^{p}}{\partial {y}^{w}}\mathrm{\Delta}{y}^{w}+\frac{\partial {u}^{p}}{\partial {z}^{w}}\mathrm{\Delta}{z}^{w}\right).$$
(11)
$$\mathrm{\Delta}{x}^{w}=\frac{\overline{{x}^{w}}-{x}^{w}}{N};\mathrm{\Delta}{y}^{w}=\frac{\overline{{y}^{w}}-{y}^{w}}{N};\mathrm{\Delta}{z}^{w}=\frac{\overline{{z}^{w}}-{z}^{w}}{N},$$