## Statistical behavior analysis and precision optimization for the laser stripe center detector based on Steger's algorithm |

Optics Express, Vol. 21, Issue 11, pp. 13442-13449 (2013)

http://dx.doi.org/10.1364/OE.21.013442

Acrobat PDF (880 KB)

### Abstract

Triangulation laser range scanning, which has been wildly used in various applications, can reconstruct the 3D geometric of the object with high precision by processing the image of laser stripe. The unbiased line extractor proposed by Steger is one of the most commonly used algorithms in laser stripe center extraction for its precision and robustness. Therefore, it is of great significance to assess the statistical performance of the Steger method when it is applied on laser stripe with Gaussian intensity profile. In this paper, a statistical behavior analysis for the laser stripe center extractor based on Steger method has been carried out. Relationships between center extraction precision, image quality and stripe characteristics have been examined analytically. Optimal scale of Gaussian smoothing kernel can be determined for each laser stripe image to achieve the highest precision according to the derived formula. Flexible three-step noise estimation procedure has been proposed to evaluate the center extraction precision of a typical triangulation laser scanning system by simply referring to the acquired images. The validity of our analysis has been verified by experiments on both artificial and natural images.

© 2013 OSA

## 1. Introduction

1. S. Cui and X. Zhu, “A generalized reference-plane-based calibration method in optical triangular profilometry,” Opt. Express **17**(23), 20735–20746 (2009). [CrossRef] [PubMed]

2. Z. Zhang and L. Yuan, “Building a 3D scanner system based on monocular vision,” Appl. Opt. **51**(11), 1638–1644 (2012). [CrossRef] [PubMed]

4. L. Marc, K. Pulli, B. Curless, S. Rusinkiewicz, D. Koller, L. Pereira, M. Ginzton, S. Anderson, J. Davis, J. Ginsberg, J. Shade, and D. Fulk, “The digital Michelangelo project: 3D scanning of large statues,” in *Proceedings of the 27th annual Conference on Computer Graphics and Interactive Techniques*, ACM Press/Addison-Wesley Publishing Co., 131–144, (2000).

5. C. Steger, “An unbiased detector of curvilinear structures,” IEEE Trans. Pattern Anal. Mach. Intell. **20**(2), 113–125 (1998). [CrossRef]

7. F. Zhou, G. Zhang, and J. Jiang, “Constructing feature points for calibrating a structured light vision sensor by viewing a plane from unknown orientations,” Opt. Lasers Eng. **43**(10), 1056–1070 (2005). [CrossRef]

11. F. Bouchara and S. Ramdani, “Statistical behavior of edge detectors,” Signal Image Video Process. **1**(3), 273–285 (2007). [CrossRef]

12. K. Astrom and A. Heyden, “Stochastic modeling and analysis of sub-pixel edge detection,” in Proceedings of the 13th International Conference on Pattern Recognition, (1996), 86–90. [CrossRef]

## 2. Laser stripe profile model and center position distribution

*y*-axis of the image. The intensity profile

*f*of a laser stripe can be described by a Gaussian function with background [10]:where the term

_{w}(x)*A*stands for the peak intensity of the laser stripe while the term

*σ*is equal to the standard deviation width (STDW) of laser stripe, and

_{w}*h(x)*stands for the image background which can be easily eliminated by background subtraction. For the laser stripe center extractor based on Steger method, the intensity profile

*f*is convolved with Gaussian kernel

_{w}(x)*g*(

_{σ}*x*) and differential operators to calculate the derivatives of

*f*. The width of

_{w}(x)*g*(

_{σ}*x*) is determined by scale parameter

*σ*. The center of laser stripe is given by the first-order zero-crossing-point (FOZCP) of the convolution result which also reaches local extreme points in the zero-order and second-order derivatives as shown in Fig. 1. The reason to involve Gaussian kernel instead of using differential operators solely is to smooth the influence from noise.

15. J. Forest, J. Salvi, E. Cabruja, and C. Pous, “Laser stripe peak detector for 3D scanners. A FIR filter approach,” in Proceedings of the 17th International Conference on Pattern Recognition, (2004), 646–649. [CrossRef]

*f(x, y)*can be defined as:

*i(x, y)*stands for the noise-free laser stripe image. Random noise

*n*(

*x, y*) can be described as one-dimensional stochastic process in multi-dimensional random field and is commonly treated as Gaussian white noise of zero expectation. According to

*Central Limit Theorem*, when affected by random noise, each center extraction process can be considered as an independent identically distributed random variable. Its distribution satisfies the normal distribution:Where

*4σ*.

_{l}## 3. Statistical behavior analysis for laser stripe center extraction

*f*:

_{w}(x)*SNR*), the variance of center position extraction will be:

*SNR*is of linear inverse proportion. Therefore the statistical behavior of the Steger extractor applied to laser stripe with Gaussian intensity profile can be evaluate once the

*SNR*of laser stripe image is identified. If the laser peak value

*A*is known then the image

*SNR*can be easily obtained by estimating the variance of the random noise

*SNR*and STDW are not posterior knowledge. We can only estimate them from the acquired image. Therefore we proposed a flexible three steps estimation method:

*w*is performed. Given a prior estimated center

_{c}*X*where peak value

_{c}*A*of intensity occurs, two points

*X*and

_{a}*X*that have the intensity of a certain fraction of

_{b}*A*(i.e. 5%-20%) are chosen as the boundary points by searching through the neighborhood of

*X*. As shown in Fig. 2,

_{c}*X*and

_{a}*X*are chosen with

_{b}*0.2A*. The line-width is derived as

*w*(

_{c}= X_{b}-X_{a}*X*] is defined as the stripe region

_{a}-w_{c}, X_{b}-w_{c}*l*.

_{a}*l*to obtain the approximated noiseless Gaussian profile

_{a}*l*. Using this fitted Gaussian profile, the true stripe width variance

_{b}*σ*is also derived.

*n = l*-

_{b}*l*and then the estimated noise variance

_{a}*σ*is obtained.

_{n}^{2}## 4. Experiments

*A*= 100. White noise with variance

*SNR*to further investigate the consistency between the actual distribution and estimated distribution according to the proposed method. The contrast result between predicted and extracted center position variance is shown in Fig. 3(c). As can be seen, the shapes of predicted and extracted variance are resembled to each other. However, the absolute extraction error is larger in low image

*SNR*regions, where there may be a threshold effect that leads to undistinguished stripe peak signals from background noise. In general, a processed image is acceptable if its

*SNR*is greater than 30 dB [16

16. T.-S. Chen, C.-C. Chang, and M.-S. Hwang, “A virtual image cryptosystem based upon vector quantization,” IEEE Trans. Image Process. **7**(10), 1485–1488 (1998). [CrossRef] [PubMed]

*SNR*of around 60 dB [17

17. J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, “Image denoising using scale mixtures of Gaussians in the wavelet domain,” IEEE Trans. Image Process. **12**(11), 1338–1351 (2003). [CrossRef] [PubMed]

*SNR*estimation method. More importantly, the actual precisions are coincided with the estimated precisions, the difference between them are within one tenth of a pixel. Overall, Eq. (9) gives an excellent estimation of the line position variance both in synthetic and real images.

## 5. Discussion

*SNR*are established. With these relationships, once the two of them are determined, the left one can be easily obtained. Three practical applications of this triple relation are introduced:

*SNR*can be determined based on user proposed minimum precision requirement. Means of reducing lens’ F-number/refining illuminations/replacing image sensor that could possibly improve image quality should be implemented if the acquired images could not meet this

_{min}*SNR*. Most of the times, a stable laser can also offer great improvement.

_{min}## 6. Conclusion

## Acknowledgments

## References and links

1. | S. Cui and X. Zhu, “A generalized reference-plane-based calibration method in optical triangular profilometry,” Opt. Express |

2. | Z. Zhang and L. Yuan, “Building a 3D scanner system based on monocular vision,” Appl. Opt. |

3. | Y. Zhang, S. Wang, X. Zhang, F. Xie, and J. Wang, “Freight train gauge-exceeding detection based on three-dimensional stereo vision measurement,” Mach. Vis. Appl. |

4. | L. Marc, K. Pulli, B. Curless, S. Rusinkiewicz, D. Koller, L. Pereira, M. Ginzton, S. Anderson, J. Davis, J. Ginsberg, J. Shade, and D. Fulk, “The digital Michelangelo project: 3D scanning of large statues,” in |

5. | C. Steger, “An unbiased detector of curvilinear structures,” IEEE Trans. Pattern Anal. Mach. Intell. |

6. | C. Steger, “Unbiased Extraction of Curvilinear Structures from 2D and 3D Images,” Dissertation, Fakultät für Informatik, Technische Universität München, 1998. |

7. | F. Zhou, G. Zhang, and J. Jiang, “Constructing feature points for calibrating a structured light vision sensor by viewing a plane from unknown orientations,” Opt. Lasers Eng. |

8. | R. Yang, S. Cheng, W. Yang, and Y. Chen, “Robust and accurate surface measurement using structured light,” IEEE Trans. Instrum. Meas. |

9. | R. D. Wedowski, G. A. Atkinson, M. L. Smith, and L. N. Smith, “A system for the dynamic industrial inspection of specular freeform surfaces,” Opt. Lasers Eng. |

10. | C. Steger, “Unbiased extraction of lines with parabolic and Gaussian profiles,” Comput. Vis. Image Underst. |

11. | F. Bouchara and S. Ramdani, “Statistical behavior of edge detectors,” Signal Image Video Process. |

12. | K. Astrom and A. Heyden, “Stochastic modeling and analysis of sub-pixel edge detection,” in Proceedings of the 13th International Conference on Pattern Recognition, (1996), 86–90. [CrossRef] |

13. | C. Steger, “Analytical and empirical performance evaluation of sub-pixel line and edge detection,” in |

14. | R. B. Fisher and D. K. Naidu, “A comparison of algorithms for sub-pixel peak detection,” in |

15. | J. Forest, J. Salvi, E. Cabruja, and C. Pous, “Laser stripe peak detector for 3D scanners. A FIR filter approach,” in Proceedings of the 17th International Conference on Pattern Recognition, (2004), 646–649. [CrossRef] |

16. | T.-S. Chen, C.-C. Chang, and M.-S. Hwang, “A virtual image cryptosystem based upon vector quantization,” IEEE Trans. Image Process. |

17. | J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, “Image denoising using scale mixtures of Gaussians in the wavelet domain,” IEEE Trans. Image Process. |

**OCIS Codes**

(100.0100) Image processing : Image processing

(140.0140) Lasers and laser optics : Lasers and laser optics

(150.6910) Machine vision : Three-dimensional sensing

**ToC Category:**

Image Processing

**History**

Original Manuscript: March 18, 2013

Revised Manuscript: May 9, 2013

Manuscript Accepted: May 17, 2013

Published: May 28, 2013

**Citation**

Li Qi, Yixin Zhang, Xuping Zhang, Shun Wang, and Fei Xie, "Statistical behavior analysis and precision optimization for the laser stripe center detector based on Steger's algorithm," Opt. Express **21**, 13442-13449 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-11-13442

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### References

- S. Cui and X. Zhu, “A generalized reference-plane-based calibration method in optical triangular profilometry,” Opt. Express17(23), 20735–20746 (2009). [CrossRef] [PubMed]
- Z. Zhang and L. Yuan, “Building a 3D scanner system based on monocular vision,” Appl. Opt.51(11), 1638–1644 (2012). [CrossRef] [PubMed]
- Y. Zhang, S. Wang, X. Zhang, F. Xie, and J. Wang, “Freight train gauge-exceeding detection based on three-dimensional stereo vision measurement,” Mach. Vis. Appl.24(3), 461–475 (2012).
- L. Marc, K. Pulli, B. Curless, S. Rusinkiewicz, D. Koller, L. Pereira, M. Ginzton, S. Anderson, J. Davis, J. Ginsberg, J. Shade, and D. Fulk, “The digital Michelangelo project: 3D scanning of large statues,” in Proceedings of the 27th annual Conference on Computer Graphics and Interactive Techniques, ACM Press/Addison-Wesley Publishing Co., 131–144, (2000).
- C. Steger, “An unbiased detector of curvilinear structures,” IEEE Trans. Pattern Anal. Mach. Intell.20(2), 113–125 (1998). [CrossRef]
- C. Steger, “Unbiased Extraction of Curvilinear Structures from 2D and 3D Images,” Dissertation, Fakultät für Informatik, Technische Universität München, 1998.
- F. Zhou, G. Zhang, and J. Jiang, “Constructing feature points for calibrating a structured light vision sensor by viewing a plane from unknown orientations,” Opt. Lasers Eng.43(10), 1056–1070 (2005). [CrossRef]
- R. Yang, S. Cheng, W. Yang, and Y. Chen, “Robust and accurate surface measurement using structured light,” IEEE Trans. Instrum. Meas.57(6), 1275–1280 (2008). [CrossRef]
- R. D. Wedowski, G. A. Atkinson, M. L. Smith, and L. N. Smith, “A system for the dynamic industrial inspection of specular freeform surfaces,” Opt. Lasers Eng.50(5), 632–644 (2012). [CrossRef]
- C. Steger, “Unbiased extraction of lines with parabolic and Gaussian profiles,” Comput. Vis. Image Underst.117(2), 97–112 (2013).
- F. Bouchara and S. Ramdani, “Statistical behavior of edge detectors,” Signal Image Video Process.1(3), 273–285 (2007). [CrossRef]
- K. Astrom and A. Heyden, “Stochastic modeling and analysis of sub-pixel edge detection,” in Proceedings of the 13th International Conference on Pattern Recognition, (1996), 86–90. [CrossRef]
- C. Steger, “Analytical and empirical performance evaluation of sub-pixel line and edge detection,” in Empirical Evaluation Methods in Computer Vision, K.W. Bowyer and P. J. Phillips, ed. (IEEE Computer Society Press, 1998).
- R. B. Fisher and D. K. Naidu, “A comparison of algorithms for sub-pixel peak detection,” in Advances in Image Processing, Multimedia and Machine Vision, J. Sanz, ed. (Springer-Verlag, Heidelberg, 1996).
- J. Forest, J. Salvi, E. Cabruja, and C. Pous, “Laser stripe peak detector for 3D scanners. A FIR filter approach,” in Proceedings of the 17th International Conference on Pattern Recognition, (2004), 646–649. [CrossRef]
- T.-S. Chen, C.-C. Chang, and M.-S. Hwang, “A virtual image cryptosystem based upon vector quantization,” IEEE Trans. Image Process.7(10), 1485–1488 (1998). [CrossRef] [PubMed]
- J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, “Image denoising using scale mixtures of Gaussians in the wavelet domain,” IEEE Trans. Image Process.12(11), 1338–1351 (2003). [CrossRef] [PubMed]

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