## High-speed optical measurement for the drumhead vibration

Optics Express, Vol. 13, Issue 8, pp. 3110-3116 (2005)

http://dx.doi.org/10.1364/OPEX.13.003110

Acrobat PDF (897 KB)

### Abstract

A high-speed optical measurement for the vibrating drumhead is presented and verified by experiment. A projected sinusoidal fringe pattern on the measured drumhead is dynamically deformed with the membrane vibration and grabbed by a high-speed camera. The shape deformation of the drumhead at each sampling instant can be recovered from this sequence of obtained fringe patterns. The membrane vibration of Chinese drum has been measured with a high speed sampling rate of 1,000 frames/sec. and a standard deviation of 0.075 mm. The restored vibration of the drumhead is also presented in an animation.

© 2005 Optical Society of America

## 1. Introduction

1. M. Takeda and K. Motoh, “Fourier transform profilometry for the automatic measurement of 3D object shapes,” App. Opt. **22**, 3977–3982 (1983). [CrossRef]

3. Xianyu Su and Wenjing Chen, “Fourier transform profilomitry: a review,” Opt. Laser in Eng. **35**, 263–284 (2001). [CrossRef]

4. Xianyu Su, Wenjing Chen, Qichan Zhang, and Yiping Chao, “Dynamic 3-D shape measurement method based on FTP”, Opt. Laser Eng. , **36**, 49–64 (2001). [CrossRef]

6. Qi-Can Zhang and Xian-Yu Su, “An optical measurement of vortex shape at a free surface,” Opt. & Laser Tech. **34**, 107–113 (2002). [CrossRef]

8. Dan Russell, “Vibrational Modes of a Circular Membrane,” http://www.kettering.edu/~drussell/Demos/MembraneCircle/Circle.html.

9. ISVR - Institute of Sound and Vibration Research, University of Southampton, UK, “Animations of Acoustic Waves,” http://www.isvr.soton.ac.uk/SPCG/Tutorial/Tutorial/Tutorial_files/Web-standing-membrane.htm.

## 2. Fundamental concepts

### 2.1 FTP used in shape measurement for dynamic objects

*E*

_{p}-

*E*

_{c}-

*O*on a reference plane. The reference plane is fictitious and serves as a base plane for measuring the object height

*h*(

*x*,

*y*) of the measured object.

*d*is the distance between the projector system and the camera system,

*l*

_{0}is the distance between the camera system and the reference plane.

*p*

_{0}) on the reference plane observed through a CCD can be represented by

*r*

_{0}(

*x*,

*y*) is a non-uniform distribution of the reflectivity on the reference plane,

*A*

_{n}is the weighting factors of Fourier series,

*f*

_{0}(

*f*

_{0}=1/

*p*

_{0}) is the fundamental frequency of the observed grating image, and

*ϕ*

_{0}(

*x*,

*y*) is the original phase on the reference plane (i.e.

*h*(

*x*,

*y*)=0). The coordinate axes are chosen as shown in Fig. 1.

*g*(

*x*,

*y*). When a dynamic 3D object, whose height distributions are changing with time, is placed into the optical field, the intensity of these fringe patterns is obviously a function of time and can be marked as

*g*(

*x*,

*y*,

*z*(

*t*)), and the phase distribution which implicates the height variation of the measured dynamic object is also a function of time and can be noted as

*ϕ*(

*x*,

*y*,

*t*). The intensity distributions of these fringe patterns in difference time can be expressed as

*r*(

*x*,

*y*,

*t*) and

*ϕ*(

*x*,

*y*,

*t*) represent a non-uniform distribution of the reflectivity on the object surface and the phase modulation resulted from the object height variation at different times, respectively,

*m*is the number of all fringe images grabbed by the CCD camera. The interval between two neighboring frame is dependent on the sampling rate of the camera.

*n*=1) of Fourier spectra, and inverse Fourier transform are carried out to deal with each fringe pattern grabbed by CCD at different time. Complex signals at different time can be calculated.

*E*

_{p}

*HE*

_{c}and Δ

*CHD*, in Fig. 1, we can write

## 2.2 3D phase calculation and unwrapping in 3D phase space

*ϕ*(

*x*,

*y*,

*t*) and

*ϕ*

_{0}(

*x*,

*y*) are usually calculated by employing the function of arc tangent, so the result is consequentially wrapped in the range of (-

*π*,

*π*), which must be recovered to a natural and continuous distribution in order to restore the correct shapes of the measured object. In dynamic measurement, this wrapped phase is also a function of time, so we call it 3D wrapped phase. 3D phase unwrapping is conducted not only along the

*x*and

*y*directions, but also along the

*t*direction necessarily. Some points of discontinuity, which are resulted from noise, shadow and under-sampling, fail to be unwrapped along the

*x*or

*y*direction in its own frame, can be unwrapped successfully along the

*t*direction. So compared with 2D unwrapping [10

10. T. R. Judge and P.J. Bryyanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. & Leaser Eng. **21**, 199–239 (1994). [CrossRef]

14. Asundi and Wen-Sen Zhou, “Fast phase-unwrapping algorithm based on a gray-scale mask and flood fill,” App. Opt. **38**, 3556–3561 (1999). [CrossRef]

*g*ĝ*(

*x*,

*y*,

*t*-1) with

*g*ĝ (

*x*,

*y*,

*t*) in two neighboring sampling instants, we can obtain their 3D phase difference distribution like this

*t*axis is high enough, the time interval between two grabbed frame fringe patterns will be small, as a result, the phase difference between two neighboring pixels with the same (

*x*,

*y*) coordinates will be less than π in

*t*direction, i.e. the phase difference

*ϕ*(

*x*,

*y*,

*t*)-

*ϕ*(

*x*,

*y*,

*t*-1) is always smaller than

*ϕ*(

*x*,

*y*,

*t*)-

*ϕ*(

*x*,

*y*,0). It provides a new approach for 3D phase unwrapping.

*x*,

*y*), is selected to be unwrapped by 2D phase unwrapping procedure and to calculate its natural phase Φ

_{uw1}(

*x*,

*y*,1). The unwrapped phase Φ

_{uwk}at any time (

*t*=

*k*) can be obtained by calculating the sum of the

*k*-1 phase differences from

*t*=1 to

*t*=

*k*along the

*t*direction. It can be described as

*ϕ*

_{i}(

*x*,

*y*) represents the phase difference between two frames

*t*=

*i*and

*t*=

*i*-1.

*t*direction. So the whole unwrapping procedure is very simple and quick.

## 3. Experiment and results

*l*

_{0}) from the exit pupil of the projection system. The deformed grating image is observed by a high frame rate CCD camera (SpeedCam Visario, made in Switzerland, the sampling rate speed is up to 10,000 frames/sec.) with a zoom lens (Sigma Zoom, Φ82 mm, 1:28 DG, 24~70 mm, made in Japan). The distance between the projection system and the high-speed camera is 960 mm (

*d*). A sheet of dark and featureless cloth is placed farther behind the drum to minimize the glare and decrease the intensity of the reflected light, which will distract the attention from the drum, so that it is easy to segregate the drum from background.

## 4. Conclusion and discussion

## Acknowledgments

## References and links

1. | M. Takeda and K. Motoh, “Fourier transform profilometry for the automatic measurement of 3D object shapes,” App. Opt. |

2. | Jian Li, Xian-Yu Su, and Lu-Rong Guo, “An improved Fourier transform profilometry for automatic measurement of 3D object shapes,” Opt. Eng. |

3. | Xianyu Su and Wenjing Chen, “Fourier transform profilomitry: a review,” Opt. Laser in Eng. |

4. | Xianyu Su, Wenjing Chen, Qichan Zhang, and Yiping Chao, “Dynamic 3-D shape measurement method based on FTP”, Opt. Laser Eng. , |

5. | Qican Zhang, Xianyu Su, Wenjing Chen, Yiping Cao, and Liqun Xiang, “An Optical Real-time 3-D Measurement for Facial Shape and Movement,” in |

6. | Qi-Can Zhang and Xian-Yu Su, “An optical measurement of vortex shape at a free surface,” Opt. & Laser Tech. |

7. | M. Bertsch, “vibration patterns and sound analysis of the viennese timpani,” in |

8. | Dan Russell, “Vibrational Modes of a Circular Membrane,” http://www.kettering.edu/~drussell/Demos/MembraneCircle/Circle.html. |

9. | ISVR - Institute of Sound and Vibration Research, University of Southampton, UK, “Animations of Acoustic Waves,” http://www.isvr.soton.ac.uk/SPCG/Tutorial/Tutorial/Tutorial_files/Web-standing-membrane.htm. |

10. | T. R. Judge and P.J. Bryyanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. & Leaser Eng. |

11. | Xianyu Su and Wenjing Chen, “Reliability guided phase unwrapping algorithm-a review,” Opt. & Leaser Eng. |

12. | Xian-Yu Su, “Phase unwrapping techniques for 3D shape measurement,” in |

13. | Jie-Lin Li, Xian-Yu Su, and Ji-Tao Li, “Phase Unwrapping algorithm-based on reliability and edge-detection,” Opt. Eng. |

14. | Asundi and Wen-Sen Zhou, “Fast phase-unwrapping algorithm based on a gray-scale mask and flood fill,” App. Opt. |

**OCIS Codes**

(070.2590) Fourier optics and signal processing : ABCD transforms

(110.6880) Imaging systems : Three-dimensional image acquisition

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

(120.6650) Instrumentation, measurement, and metrology : Surface measurements, figure

(120.7280) Instrumentation, measurement, and metrology : Vibration analysis

**ToC Category:**

Research Papers

**History**

Original Manuscript: January 31, 2005

Revised Manuscript: April 5, 2005

Published: April 18, 2005

**Citation**

Qican Zhang and Xianyu Su, "High-speed optical measurement for the drumhead vibration," Opt. Express **13**, 3110-3116 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-8-3110

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### References

- M. Takeda, K. Motoh, �??Fourier transform profilometry for the automatic measurement of 3D object shapes,�?? App. Opt. 22, 3977-3982 (1983). [CrossRef]
- Jian Li, Xian-Yu Su, Lu-Rong Guo, �??An improved Fourier transform profilometry for automatic measurement of 3D object shapes,�?? Opt. Eng. 29, 1439-1444 (1990). [CrossRef]
- Xianyu Su, Wenjing Chen, �??Fourier transform profilomitry: a review,�?? Opt. Laser in Eng. 35, 263-284 (2001). [CrossRef]
- Xianyu Su, Wenjing Chen, Qichan Zhang, Yiping Chao, �??Dynamic 3-D shape measurement method based on FTP�??, Opt. Laser Eng., 36, 49-64 (2001). [CrossRef]
- Qican Zhang, Xianyu Su, Wenjing Chen, Yiping Cao, Liqun Xiang, �??An Optical Real-time 3-D Measurement for Facial Shape and Movement,�?? in The 3rd International Conference on Photonics and Imaging in Biology and Medicine 2003, Qingming Luo, Britton Chance, eds., Proc. SPIE 5254, 214-221(2003).
- Qi-Can Zhang, Xian-Yu Su, �??An optical measurement of vortex shape at a free surface,�?? Opt. & Laser Tech. 34, 107-113 (2002). [CrossRef]
- M. Bertsch, �??Vibration patterns and sound analysis of the viennese timpani,�?? in Proceedigs of ISMA '2001 (International Symposium on Musical Acoustics), Nr. Perugia: Musical and Architectural Acoustics Lab. FSSG-CNR Venezia, eds., Proc. ISMA 2001 2, 281-284 (2001).
- Dan Russell, �??Vibrational Modes of a Circular Membrane,�?? <a href= "http://www.kettering.edu/~drussell/Demos/MembraneCircle/Circle.html">http://www.kettering.edu/~drussell/Demos/MembraneCircle/Circle.html</a>.
- ISVR - Institute of Sound and Vibration Research, University of Southampton, UK, �??Animations of Acoustic Waves,�?? <a href= "http://www.isvr.soton.ac.uk/SPCG/Tutorial/Tutorial/Tutorial_files/Web-standing-membrane.htm">http://www.isvr.soton.ac.uk/SPCG/Tutorial/Tutorial/Tutorial_files/Web-standing-membrane.htm</a>.
- T. R. Judge, P.J. Bryyanston-Cross, �??A review of phase unwrapping techniques in fringe analysis,�?? Opt. & Laser Eng. 21, 199-239 (1994). [CrossRef]
- Xianyu Su, Wenjing Chen, �??Reliability guided phase unwrapping algorithm-a review,�?? Opt. & Laser Eng. 42, 245-261 (2004). [CrossRef]
- Xian-Yu Su, �??Phase unwrapping techniques for 3D shape measurement,�?? in International Conference on Holography and Optical Information Processing (ICHOIP '96), Guoguang Mu, Guofan Jin and Glenn T. Sincerbox, eds., Proc. SPIE 2866, 460-465 (1996).
- Jie-Lin Li, Xian-Yu Su, Ji-Tao Li, �??Phase Unwrapping algorithm-based on reliability and edge-detection,�?? Opt. Eng. 36, 1685-1690 (1997). [CrossRef]
- Asundi, Wen-Sen Zhou, �??Fast phase-unwrapping algorithm based on a gray-scale mask and flood fill,�?? App. Opt. 38, 3556-3561 (1999). [CrossRef]

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