## Phase-stepped fringe projection by rotation about the camera’s perspective center |

Optics Express, Vol. 19, Issue 19, pp. 18458-18469 (2011)

http://dx.doi.org/10.1364/OE.19.018458

Acrobat PDF (1230 KB)

### Abstract

A technique to produce phase steps in a fringe projection system for shape measurement is presented. Phase steps are produced by introducing relative rotation between the object and the fringe projection probe (comprising a projector and camera) about the camera’s perspective center. Relative motion of the object in the camera image can be compensated, because it is independent of the distance of the object from the camera, whilst the phase of the projected fringes is stepped due to the motion of the projector with respect to the object. The technique was validated with a static fringe projection system by moving an object on a coordinate measuring machine (CMM). The alternative approach, of rotating a lightweight and robust CMM-mounted fringe projection probe, is discussed. An experimental accuracy of approximately 1.5% of the projected fringe pitch was achieved, limited by the standard phase-stepping algorithms used rather than by the accuracy of the phase steps produced by the new technique.

© 2011 OSA

## Introduction

^{−2}) experienced in a CMM.

2. G. Sansoni, M. Carocci, and R. Rodella, “Three-dimensional vision based on a combination of gray-code and phase-shift light projection: analysis and compensation of the systematic errors,” Appl. Opt. **38**(31), 6565–6573 (1999). [CrossRef] [PubMed]

3. 3M, “3M MPro 110 Micro projector,” http://www.3mselect.co.uk/p-1783-3m-mpro-110-micro-projector-uk-model.aspx (accessed 5 February 2010).

*et al*. presented an approach that avoided internal moving projector parts by translating the object with respect to a fixed camera and fringe projector [4]. The object was moved in a direction parallel to the camera image plane and perpendicular to the projected fringes by a distance equivalent to an integral number of pixels at the camera image plane. The image translation was compensated so that each pixel imaged the same point on the object before and after the motion, but the phase of the projected fringes at that point was stepped. This system relied on telecentric imaging optics, or a long standoff compared to the depth of field, so that the magnification remained constant throughout the measurement volume.

## Theory

*O*at the perspective center of the camera. The perspective center of the projector is at

*R*is the rotation matrix between the two coordinate systems. Although Fig. 1 shows the camera and projector coordinate Y-axes parallel to each other (i.e. coplanar optic axes) it should be emphasized that it is

*not*a requirement: the mathematical development below does not require it and experimentally the axes are determined by the system calibration. The projector and camera optic axes intersect with included angle

*α,*at standoff distance

*S*from the camera’s perspective center.

*x*in the projector image plane is projected to

_{p}*c*is the principle distance of the projector (the distance between the perspective center and the image plane along the optic axis). Without loss of generality, the projector is assumed to project fringes extending in the

_{p}*p*in its ‘image’ plane. The phase at any point

*X*and

_{P}*Z*gives an expression for phase at a point

_{P}*X*in the camera coordinate system.

*x*in the coordinate system of the rotated camera and

*c*is the principal distance of the camera. That is, the image point after rotation is the projection of the original image point on to the image plane of the rotated camera. Importantly, the relationship between new and old image coordinates for a given point is not dependent on the position of the point in object space. Therefore the image point from the rotated camera can be translated by

*X*following the rotation may be expressed in the rotated camera coordinate system relative to the camera perspective center aswhere

*ω*is the small rotation angle and

*X*due to the motion of the projector is

*Z*from the projector;

_{P}*magnitude*of the phase change for a given angle of rotation varies with distance from the projector and is therefore not constant with position in the image. Standard algorithms which are valid for a range of phase step sizes, e.g. Carré’s algorithm [8

8. P. Carré, “Installation et utilisation du comparateur photoelectrique et interferential du Bureau International des Poids et Mesures,” Metrologia **2**(1), 13–23 (1966). [CrossRef]

9. J. Novak, “Five-step phase-shifting algorithms with unknown values of phase shift,” Optik (Stuttg.) **114**(2), 63–68 (2003). [CrossRef]

*Z*from the projector and

_{P}*S*from the camera and

*Z*from the projector to be calculated.

_{P}## Experimental demonstration

^{3}measurement volume. The period of the projected fringes at the intersection of the camera and projector axes was approximately 4 mm (approximately 30 pixels in the camera image). For the system geometry used, a nominal phase step of 90° in the projected fringe pattern required a rotation of about the camera’s perspective centre of 0.25°. The motion required by the CMM to effect this rotation about the camera’s perspective centre comprised a rotation about the Revo head's centre of rotation of 0.25° and a translation in an arc of 300μm. The head and CMM motors are capable of positioning correct to 0.5 arc sec (or 1.4×10

^{−4}degrees) and 5μm respectively. From Eq. (6) the phase step variation throughout the measurement volume varied between 0.8 and 1.4 times the nominal value for a given rotation about the camera perspective center.

10. J.-Y. Bouguet, “Camera calibration toolbox for Matlab”, http://www.vision.caltech.edu/bouguetj/calib_doc/index.html (accessed 5 February 2010).

### Rotation about the camera perspective center

### Phase to height calibration

11. M. Reeves, A. J. Moore, D. P. Hand, and J. D. C. Jones, “Dynamic shape measurement system for laser materials processing,” Opt. Eng. **42**(10), 2923–2929 (2003). [CrossRef]

8. P. Carré, “Installation et utilisation du comparateur photoelectrique et interferential du Bureau International des Poids et Mesures,” Metrologia **2**(1), 13–23 (1966). [CrossRef]

9. J. Novak, “Five-step phase-shifting algorithms with unknown values of phase shift,” Optik (Stuttg.) **114**(2), 63–68 (2003). [CrossRef]

### Object measurement

^{2}was 60μm, equivalent to approximately 1.5% of a fringe period

## Discussion

### Speckle noise

17. A. J. Moore, R. McBride, J. S. Barton, and J. D. C. Jones, “Closed-loop phase stepping in a calibrated fiber-optic fringe projector for shape measurement,” Appl. Opt. **41**(16), 3348–3354 (2002). [CrossRef] [PubMed]

### Inter-frame intensity variation

19. K. E. Torrance and E. M. Sparrow, “Theory of off-specular reflection from roughened surfaces,” J. Opt. Soc. Am. **57**(9), 1105–1114 (1967). [CrossRef]

### Non-linear phase step error

*X*is proportional to the fringe width at that point. The factor of proportionality is at most of order 1/|

*X*| for a practical system and varies with position within the measurement volume. Therefore the non-linear phase step error is small provided that the fringe pitch is small compared to the standoff distance. This condition is desirable for fringe projection systems because height resolution increases with decreasing fringe period. The response of Carré’s and Novak’s algorithms to non-linear phase steps was simulated. The mean error over one fringe period is shown in Fig. 5(c), and again dominated the rms error.

## Conclusions

## Acknowledgments

## References and links

1. | K. Creath, “Comparison of phase-measurement algorithms,” Proc. SPIE |

2. | G. Sansoni, M. Carocci, and R. Rodella, “Three-dimensional vision based on a combination of gray-code and phase-shift light projection: analysis and compensation of the systematic errors,” Appl. Opt. |

3. | 3M, “3M MPro 110 Micro projector,” http://www.3mselect.co.uk/p-1783-3m-mpro-110-micro-projector-uk-model.aspx (accessed 5 February 2010). |

4. | D. M. Kranz, E. P. Rudd, D. Fishbaine, and C. E. Haugan, “Phase profilometry system with telecentric projector,” International Patent, Publication Number WO01/51887 (2001). |

5. | M. A. R. Cooper with S. Robson, “Theory of close-range photogrammetry,” in |

6. | J. G. Fryer, “Camera Calibration,” in |

7. | J. Heikkila and O. Silven, “A four-step camera calibration procedure with implicit image correction,” in |

8. | P. Carré, “Installation et utilisation du comparateur photoelectrique et interferential du Bureau International des Poids et Mesures,” Metrologia |

9. | J. Novak, “Five-step phase-shifting algorithms with unknown values of phase shift,” Optik (Stuttg.) |

10. | J.-Y. Bouguet, “Camera calibration toolbox for Matlab”, http://www.vision.caltech.edu/bouguetj/calib_doc/index.html (accessed 5 February 2010). |

11. | M. Reeves, A. J. Moore, D. P. Hand, and J. D. C. Jones, “Dynamic shape measurement system for laser materials processing,” Opt. Eng. |

12. | J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. |

13. | G. Sansoni, S. Corini, S. Lazzari, R. Rodella, and F. Docchio, “Three-dimensional imaging based on Gray-code light projection: characterization of the measuring algorithm and development of a measuring system for industrial applications,” Appl. Opt. |

14. | S. Zhang and S.-T. Yau, “Generic nonsinusoidal phase error correction for three-dimensional shape measurement using a digital video projector,” Appl. Opt. |

15. | N. J. Weston, Y. R. Huddart, A. J. Moore, and T. C. Featherstone, “Phase analysis measurement apparatus and method”, International patent pending WO2009 / 024757(A1) (2008). |

16. | H. Lui, G. Lu, S. Wu, S. Yin, and F.T. S.U. Yu, “Speckle-induced phase error in laser-based phase-shifting projected fringe profilometry,” J. Opt. Soc. Am. A |

17. | A. J. Moore, R. McBride, J. S. Barton, and J. D. C. Jones, “Closed-loop phase stepping in a calibrated fiber-optic fringe projector for shape measurement,” Appl. Opt. |

18. | H. Ragheb and E. R. Hancock, “Surface radiance: empirical data against model predictions,” in |

19. | K. E. Torrance and E. M. Sparrow, “Theory of off-specular reflection from roughened surfaces,” J. Opt. Soc. Am. |

20. | K. Creath, “Temporal phase measurement methods,” in |

**OCIS Codes**

(120.3940) Instrumentation, measurement, and metrology : Metrology

(120.4630) Instrumentation, measurement, and metrology : Optical inspection

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

(110.2650) Imaging systems : Fringe analysis

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: July 13, 2011

Revised Manuscript: August 13, 2011

Manuscript Accepted: August 15, 2011

Published: September 6, 2011

**Citation**

Y. R. Huddart, J. D. Valera, N. J. Weston, T. C. Featherstone, and A. J. Moore, "Phase-stepped fringe projection by rotation about the camera’s perspective center," Opt. Express **19**, 18458-18469 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-19-18458

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### References

- K. Creath, “Comparison of phase-measurement algorithms,” Proc. SPIE680, 19–28 (1986).
- G. Sansoni, M. Carocci, and R. Rodella, “Three-dimensional vision based on a combination of gray-code and phase-shift light projection: analysis and compensation of the systematic errors,” Appl. Opt.38(31), 6565–6573 (1999). [CrossRef] [PubMed]
- 3M, “3M MPro 110 Micro projector,” http://www.3mselect.co.uk/p-1783-3m-mpro-110-micro-projector-uk-model.aspx (accessed 5 February 2010).
- D. M. Kranz, E. P. Rudd, D. Fishbaine, and C. E. Haugan, “Phase profilometry system with telecentric projector,” International Patent, Publication Number WO01/51887 (2001).
- M. A. R. Cooper with S. Robson, “Theory of close-range photogrammetry,” in Close range photogrammetry and machine vision, K. B. Atkinson, ed., (Whittles Publishing, Caithness, UK, 2001).
- J. G. Fryer, “Camera Calibration,” in Close range photogrammetry and machine vision, K. B. Atkinson, ed., (Whittles Publishing, Caithness, UK, 2001).
- J. Heikkila and O. Silven, “A four-step camera calibration procedure with implicit image correction,” in Proceedings of the 1997 Conference in Computer Vision and Pattern Recognition (CVPR ’97) (IEEE Computer Society, Washington, DC, 1997), pp. 1106–1112.
- P. Carré, “Installation et utilisation du comparateur photoelectrique et interferential du Bureau International des Poids et Mesures,” Metrologia2(1), 13–23 (1966). [CrossRef]
- J. Novak, “Five-step phase-shifting algorithms with unknown values of phase shift,” Optik (Stuttg.)114(2), 63–68 (2003). [CrossRef]
- J.-Y. Bouguet, “Camera calibration toolbox for Matlab”, http://www.vision.caltech.edu/bouguetj/calib_doc/index.html (accessed 5 February 2010).
- M. Reeves, A. J. Moore, D. P. Hand, and J. D. C. Jones, “Dynamic shape measurement system for laser materials processing,” Opt. Eng.42(10), 2923–2929 (2003). [CrossRef]
- J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt.13(11), 2693–2703 (1974). [CrossRef] [PubMed]
- G. Sansoni, S. Corini, S. Lazzari, R. Rodella, and F. Docchio, “Three-dimensional imaging based on Gray-code light projection: characterization of the measuring algorithm and development of a measuring system for industrial applications,” Appl. Opt.36(19), 4463–4472 (1997). [CrossRef] [PubMed]
- S. Zhang and S.-T. Yau, “Generic nonsinusoidal phase error correction for three-dimensional shape measurement using a digital video projector,” Appl. Opt.46(1), 36–43 (2007). [CrossRef] [PubMed]
- N. J. Weston, Y. R. Huddart, A. J. Moore, and T. C. Featherstone, “Phase analysis measurement apparatus and method”, International patent pending WO2009 / 024757(A1) (2008).
- H. Lui, G. Lu, S. Wu, S. Yin, and F.T. S.U. Yu, “Speckle-induced phase error in laser-based phase-shifting projected fringe profilometry,” J. Opt. Soc. Am. A16(6), 1484–1495 (1999). [CrossRef]
- A. J. Moore, R. McBride, J. S. Barton, and J. D. C. Jones, “Closed-loop phase stepping in a calibrated fiber-optic fringe projector for shape measurement,” Appl. Opt.41(16), 3348–3354 (2002). [CrossRef] [PubMed]
- H. Ragheb and E. R. Hancock, “Surface radiance: empirical data against model predictions,” in Proceedings of the 2004 International Conference on Image Processing (ICIP), (Institute of Electrical and Electronics Engineers, 2005), pp. 2689–2692.
- K. E. Torrance and E. M. Sparrow, “Theory of off-specular reflection from roughened surfaces,” J. Opt. Soc. Am.57(9), 1105–1114 (1967). [CrossRef]
- K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis, D.W. Robinson and G.T. Reid, eds., 94–140 (Institute of Physics 1993).

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