## Superfast multifrequency phase-shifting technique with optimal pulse width modulation |

Optics Express, Vol. 19, Issue 6, pp. 5149-5155 (2011)

http://dx.doi.org/10.1364/OE.19.005149

Acrobat PDF (920 KB)

### Abstract

The technique of generating sinusoidal fringe patterns by defocusing squared binary structured ones has numerous merits for high-speed three-dimensional (3D) shape measurement. However, it is challenging for this method to realize a multifrequency phase-shifting (MFPS) algorithm because it is difficult to simultaneously generate high-quality sinusoidal fringe patterns with different periods. This paper proposes to realize an MFPS algorithm utilizing an optimal pulse width modulation (OPWM) technique that can selectively eliminate high-order harmonics of squared binary patterns. We successfully develop a 556 Hz system utilizing a three-frequency algorithm for simultaneously measuring multiple objects.

© 2011 Optical Society of America

## 1. Introduction

*é*, holography, and fringe projection [1

1. S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?” Opt. Lasers Eng. **48**, 133–140 (2010). [CrossRef]

2. X.-Y. Su, W.-S. Zhou, G. V. Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. **94**(13), 561–573 (1992). [CrossRef]

3. S. Lei and S. Zhang, “Flexible 3-D shape measurement using projector defocusing,” Opt. Lett. **34**(20), 3080–3082 (2009). [CrossRef] [PubMed]

4. S. Lei and S. Zhang, “Digital sinusoidal fringe generation: defocusing binary patterns VS focusing sinusoidal patterns,” Opt. Lasers Eng. **48**(5), 561–569 (2010). [CrossRef]

4. S. Lei and S. Zhang, “Digital sinusoidal fringe generation: defocusing binary patterns VS focusing sinusoidal patterns,” Opt. Lasers Eng. **48**(5), 561–569 (2010). [CrossRef]

6. Y. Wang and S. Zhang, “Optimal pulse width modulation for sinusoidal fringe generation with projector defocusing,” Opt. Lett. **35**(24), 4121–4123 (2010). [CrossRef] [PubMed]

## 2. Principle

### 2.1. Three-step phase-shifting algorithm

7. D. Malacara, ed., *Optical Shop Testing*, 3rd ed. (John Wiley and Sons, 2007). [CrossRef]

*π*/3. Three fringe images can be described as, Where

*I*′(

*x*,

*y*) is the average intensity,

*I*″(

*x*,

*y*) the intensity modulation, and

*ϕ*(

*x*,

*y*) the phase to be solved for. The phase,

*ϕ*(

*x*,

*y*), and the texture,

*I*′(

*x*,

*y*), can be solved for from these equations Equation (4) provides the phase ranging [−

*π*,

*π*) with 2

*π*discontinuities. This 2

*π*phase jumps can be removed to obtain the continuous phase map by adopting a phase unwrapping algorithm [8]. However, such a spatial phase unwrapping algorithm has limitations: it could not used when large step height exists which may cause the phase change more than

*π*, or multiple objects need to be measured simultaneously.

### 2.2. Multifrequency phase-shifting (MFPS) algorithm

*π*). When a fringe pattern contains more than one stripes, the phase need to be unwrapped to obtain the continuous phase map. This means that if another set of wider fringe patterns (a single fringe stripe can cover the whole image) is used to obtain a phase map without 2

*π*discontinuities. The second phase map can be used unwrap the other one point by point without spatial phase unwrapping. To obtain the phase of the wider fringe patterns, there are two approaches: (1) use very long wavelength directly; and (2) use two short wavelengths to generate an equivalent long one. The former is not very commonly used because it is difficult to generate high-quality wide fringe patterns due to noises or hardware limitations. Thus the latter is more frequently adopted. This subsection will briefly explain the principle of this technique.

*λ*, and the height

*h*(

*x*,

*y*) can be written as Here

*C*is a system constant. Therefore, for

*λ*

_{1}<

*λ*

_{2}with absolute phase being Φ

_{1}and Φ

_{2}, respectively, their difference is Here,

*λ*

_{1}and

*λ*

_{2}. If

*λ*

_{2}∈ (

*λ*

_{1}, 2

*λ*

_{1}), we have

*ϕ*

_{1}and

*ϕ*

_{2}. We know that the relationship between the absolute phase is Φ and the wrapped phase

*ϕ*= Φ (mod 2

*π*) with 2

*π*discontinuities. Here the modulus operator is to convert the phase to a range of [0, 2

*π*). Taking the modulus operation on Eq.(7) will lead to Δ

*ϕ*

_{12}= ΔΦ

_{12}(mod 2

*π*). If the wavelengths are properly chosen, so that the resultant equivalent wavelength

9. K. Creath, “Step height measurement using two-wavelength phase-shifting interferometry,” Appl. Opt. **26**(14), 2810–2816 (1987). [CrossRef] [PubMed]

10. C. E. Towers, D. P. Towers, and J. D. Jones, “Optimum frequency selection in multifrequency interferometry,” Opt. Lett. **28**(11), 887–889 (2003). [CrossRef] [PubMed]

*λ*

_{3}) are used, the equivalent wavelength between

*λ*

_{1}and

*λ*

_{3}will be

### 2.3. Optimal pulse width modulation (OPWM) technique

4. S. Lei and S. Zhang, “Digital sinusoidal fringe generation: defocusing binary patterns VS focusing sinusoidal patterns,” Opt. Lasers Eng. **48**(5), 561–569 (2010). [CrossRef]

6. Y. Wang and S. Zhang, “Optimal pulse width modulation for sinusoidal fringe generation with projector defocusing,” Opt. Lett. **35**(24), 4121–4123 (2010). [CrossRef] [PubMed]

*n*times per-half cycle. For a periodic waveform with a period of 2

*π*, because it is an odd function, only the sine terms are left with the coefficients being described as:

*n*chops in the waveform provide

*n*degrees of freedom to eliminate

*n*− 1 selected harmonics whilst keeping the fundamental frequency component within a certain magnitude [11

11. V. G. Agelidis, A. Balouktsis, and I. Balouktsis, “On applying a minimization technique to the harmonic elimilation PWM control: the bipolar waveform,” IEEE Power Electron. Lett. **2**, 41–44 (2004). [CrossRef]

## 3. Experimental results

*λ*

_{1}= 60,

*λ*

_{2}= 90, and

*λ*

_{3}= 102 pixels. It can be found that the resultant equivalent fringe wavelength is 765 pixels. In other words, if we use the projector to generate 765 pixel wide fringe patterns, no spatial phase unwrapping is needed to recover absolute phase.

*λ*

_{1}and

*λ*

_{2}is shown in Fig. 2(h), and that for

*λ*

_{1}and

*λ*

_{3}is shown in Fig. 2(i). Finally, the phase map of the longest equivalent wavelength can be obtained from these two equivalent phase maps, and the result is shown in Fig. 2(j). It can be seen from this figure that this phase map has no 2

*π*discontinuities, thus no spatial phase unwrapping is needed.

*λ*

_{1}= 60 pixels can be unwrapped point by point that can then be used to recover 3D information. In this research, we use the calibration technique introduced in Ref. [12

12. S. Zhang, D. van der Weide, and J. Olvier, “Superfast phase-shifting method for 3-D shape measurement,” Opt. Express **18**(9), 9684–9689 (2010). [CrossRef] [PubMed]

13. M. Schaffer, M. Grosse, and R. Kowarschik, “High-speed pattern projection for three-dimensional shape measurement using laser speckles,” Appl. Opt. **49**(18), 3622–3629 (2010). [CrossRef] [PubMed]

*I*′(

*x*,

*y*) in Eq. (5), are generated by averaging three shortest wavelength phase-shifted fringe patterns. Media 2 shows the 3D reconstructed results at 25 fps. This experiment clearly demonstrated that by combining the OPWM technique with the defocusing technique, a superfast MFPS algorithm can be realized. This proposed technique can be used to measure multiple rapidly moving objects simultaneously.

## 4. Conclusion

## References and links

1. | S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?” Opt. Lasers Eng. |

2. | X.-Y. Su, W.-S. Zhou, G. V. Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. |

3. | S. Lei and S. Zhang, “Flexible 3-D shape measurement using projector defocusing,” Opt. Lett. |

4. | S. Lei and S. Zhang, “Digital sinusoidal fringe generation: defocusing binary patterns VS focusing sinusoidal patterns,” Opt. Lasers Eng. |

5. | S. Zhang, “Flexible 3-D shape measurement using projector defocusing: extended measurement range,” Opt. Lett. |

6. | Y. Wang and S. Zhang, “Optimal pulse width modulation for sinusoidal fringe generation with projector defocusing,” Opt. Lett. |

7. | D. Malacara, ed., |

8. | D. C. Ghiglia and M. D. Pritt, |

9. | K. Creath, “Step height measurement using two-wavelength phase-shifting interferometry,” Appl. Opt. |

10. | C. E. Towers, D. P. Towers, and J. D. Jones, “Optimum frequency selection in multifrequency interferometry,” Opt. Lett. |

11. | V. G. Agelidis, A. Balouktsis, and I. Balouktsis, “On applying a minimization technique to the harmonic elimilation PWM control: the bipolar waveform,” IEEE Power Electron. Lett. |

12. | S. Zhang, D. van der Weide, and J. Olvier, “Superfast phase-shifting method for 3-D shape measurement,” Opt. Express |

13. | M. Schaffer, M. Grosse, and R. Kowarschik, “High-speed pattern projection for three-dimensional shape measurement using laser speckles,” Appl. Opt. |

**OCIS Codes**

(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology

(120.2650) Instrumentation, measurement, and metrology : Fringe analysis

(320.7100) Ultrafast optics : Ultrafast measurements

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: January 10, 2011

Revised Manuscript: February 20, 2011

Manuscript Accepted: February 23, 2011

Published: March 3, 2011

**Citation**

Yajun Wang and Song Zhang, "Superfast multifrequency phase-shifting technique with optimal pulse width modulation," Opt. Express **19**, 5149-5155 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-6-5149

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### References

- S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?” Opt. Lasers Eng. 48, 133–140 (2010). [CrossRef]
- X.-Y. Su, W.-S. Zhou, G. V. Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94(13), 561–573 (1992). [CrossRef]
- S. Lei and S. Zhang, “Flexible 3-D shape measurement using projector defocusing,” Opt. Lett. 34(20), 3080–3082 (2009). [CrossRef] [PubMed]
- S. Lei and S. Zhang, “Digital sinusoidal fringe generation: defocusing binary patterns VS focusing sinusoidal patterns,” Opt. Lasers Eng. 48(5), 561–569 (2010). [CrossRef]
- S. Zhang, “Flexible 3-D shape measurement using projector defocusing: extended measurement range,” Opt. Lett. 35(7), 931–933 (2010).
- Y. Wang and S. Zhang, “Optimal pulse width modulation for sinusoidal fringe generation with projector defocusing,” Opt. Lett. 35(24), 4121–4123 (2010). [CrossRef] [PubMed]
- D. Malacara, ed., Optical Shop Testing, 3rd ed. (John Wiley and Sons, 2007). [CrossRef]
- D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (John Wiley and Sons, 1998).
- K. Creath, “Step height measurement using two-wavelength phase-shifting interferometry,” Appl. Opt. 26(14), 2810–2816 (1987). [CrossRef] [PubMed]
- C. E. Towers, D. P. Towers, and J. D. Jones, “Optimum frequency selection in multifrequency interferometry,” Opt. Lett. 28(11), 887–889 (2003). [CrossRef] [PubMed]
- V. G. Agelidis, A. Balouktsis, and I. Balouktsis, “On applying a minimization technique to the harmonic elimilation PWM control: the bipolar waveform,” IEEE Power Electron. Lett. 2, 41–44 (2004). [CrossRef]
- S. Zhang, D. van der Weide, and J. Olvier, “Superfast phase-shifting method for 3-D shape measurement,” Opt. Express 18(9), 9684–9689 (2010). [CrossRef] [PubMed]
- M. Schaffer, M. Grosse, and R. Kowarschik, “High-speed pattern projection for three-dimensional shape measurement using laser speckles,” Appl. Opt. 49(18), 3622–3629 (2010). [CrossRef] [PubMed]

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