## Study on subset size selection in digital image correlation for speckle patterns

Optics Express, Vol. 16, Issue 10, pp. 7037-7048 (2008)

http://dx.doi.org/10.1364/OE.16.007037

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### Abstract

Digital Image Correlation (DIC) has been established as a flexible and effective technique to measure the displacements on specimen surface by matching the reference subsets in the undeformed image with the target subsets in the deformed image. With the existing DIC techniques, the user must rely on experience and intuition to manually define the size of the reference subset, which is found to be critical to the accuracy of measured displacements. In this paper, the problem of subset size selection in the DIC technique is investigated. Based on the Sum of Squared Differences (SSD) correlation criterion as well as the assumption that the gray intensity gradients of image noise are much lower than that of speckle image, a theoretical model of the displacement measurement accuracy of DIC is derived. The theoretical model indicates that the displacement measurement accuracy of DIC can be accurately predicted based on the variance of image noise and Sum of Square of Subset Intensity Gradients (SSSIG). The model further leads to a simple criterion for choosing a proper subset size for the DIC analysis. Numerical experiments have been performed to validate the proposed concepts, and the calculated results show good agreements with the theoretical predictions.

© 2008 Optical Society of America

## 1. Introduction

2. W. Tong, “An evaluation of digital image correlation criteria for strain mapping applications,” Strain. **41**, 167–175 (2005). [CrossRef]

*et al.*, studied the systematic errors in DIC caused by undermatched shape functions [3

3. H. W. Schreier and M. A. Sutton, “Systematic errors in digital image correlation due to undermatched subset shape functions,” Exp. Mech. **42**, 303–310 (2002). [CrossRef]

4. H. W. Schreier, J. R. Braasch, and M. A. Sutton, “Systematic errors in digital image correlation caused by intensity interpolation,” Opt. Eng. **39**, 2915–2921 (2000). [CrossRef]

*et al.*, studied the performance of sub-pixel registration algorithms on displacement estimations [5

5. B. Pan, H. M. Xie, B. Q. Xu, and F. L. Dai, “Performance of sub-pixel registration algorithms in digital image correlation,” Meas. Sci. Technol. **17**, 1615–1621 (2006). [CrossRef]

*et al.*, [6

6. D. S. Zhang, M. Luo, and D. D. Arola, “Displacement/strain measurements using an optical microscope and digital image correlation,” Opt. Eng. **45**, 033605 (2006). [CrossRef]

*et al.*, [7

7. S. Yoneyama, H. Kikuta, A. Kitagawa, and K. Kitamura, “Lens distortion correction for digital image correlation by measuring rigid body displacement,” Opt. Eng. **45**, 023602 (2006). [CrossRef]

*et al.*, pointed out that the subset size should have a lower limit in order to suppress the influence of random noise [10

10. Y. F. Sun and H. J. Pang, “Study of optimal subset size in digital image correlation of speckle pattern images,” Opt. Lasers Eng. **45**, 967–974 (2007). [CrossRef]

*et al.*, [9

9. Z. Y. Wang, H. Q. Li, J. W. Tong, and J. T. Ruan, “Statistical analysis of the effect of intensity pattern noise on the displacement measurement precision of digital image correlation using self-correlated images,” Exp. Mech. **47**, 701–707 (2007). [CrossRef]

## 2. Fundamental principle of DIC

*x*

_{0},

*y*

_{0}) from the reference image is chosen and used to determine its corresponding location in the deformed image. In routine practice of DIC, a CC criterion or SSD correlation criterion [2

2. W. Tong, “An evaluation of digital image correlation criteria for strain mapping applications,” Strain. **41**, 167–175 (2005). [CrossRef]

*u*and

*v*at point (

*x*

_{0},

*y*

_{0}). More details of the DIC technique can be found in Ref. [11

11. B. Pan, H. M. Xie, Z. Q. Guo, and T. Hua, “Full-field strain measurement using a two-dimensional Savitzky-Golay digital differentiator in digital image correlation,” Opt. Eng. **46**, 033601 (2007). [CrossRef]

## 3. Displacement measurement accuracy of DIC

*u*

_{0}and

*v*

_{0}are the integral pixel displacements in

*x*and

*y*directions, which can be accurately determined with ease and are assumed as known values;

^{Δ}u and

^{Δ}v are the corresponding sub-pixel displacements in

*x*and

*y*directions, respectively.

*g*(

*x*′,

*y*′) yields

*g*and

_{x}*g*are the first-order derivatives of grayscale intensities, and in this study they are calculated by central difference of neighboring points in

_{y}*x*and

*y*directions, respectively.

^{Δ}u and

^{Δ}v. Thus we have

*η*

_{1}and

*η*

_{2}, are considered as random and are additively added to the images. Thus, the grayscale intensity distribution of the noisy images can be written as:

*f*and

*g*in Eqs. (6) and (7) with

*f*′ and

*g*′, respectively, the following equations can be obtained,

*η*(

*x*,

*y*)=

*η*

_{2}(

*x*,

*y*)-

*η*

_{1}(

*x*,

*y*). Similar to the Ref. [9

9. Z. Y. Wang, H. Q. Li, J. W. Tong, and J. T. Ruan, “Statistical analysis of the effect of intensity pattern noise on the displacement measurement precision of digital image correlation using self-correlated images,” Exp. Mech. **47**, 701–707 (2007). [CrossRef]

*η*(

*x*,

*y*): First, the mean value of the noise is a constant, that is,

*E*(

*η*)=

*const*; Second, the noise values at different pixels are independent.

*u*′ and Δ

*v*′ will be identical to those of Δ

*u*and Δ

*v*. Hence, we get

*u*′ and Δ

*v*′ are not affected by the image noise, and this conclusion has been verified in the work of Wang,

*et al.*, [9

9. Z. Y. Wang, H. Q. Li, J. W. Tong, and J. T. Ruan, “Statistical analysis of the effect of intensity pattern noise on the displacement measurement precision of digital image correlation using self-correlated images,” Exp. Mech. **47**, 701–707 (2007). [CrossRef]

*g*)

_{x}^{2}∑∑(

*g*)

_{y}^{2})

^{2}, the variances of Δ

*u*′ and Δ

*v*′ can be derived and expressed as

*D*(

*η*) is the variance of image noise, ∑∑(

*g*)

_{x}^{2}and ∑∑(

*g*)

_{y}^{2}are the Sum of Square of Subset Intensity Gradients (SSSIG) in

*x*and

*y*direction, respectively.

*H*is of form:

*g*and

_{x}*g*, defined as

_{y}*H*approximates to the value of 1, so it can be neglected in Eqs. (14) and (15).

*g*(

_{x}*x*+

*u*

_{0}),

*y*+

*v*

_{0})≈

*f*(

_{x}*x*,

*y*),

*g*(

_{y}*x*+

*u*

_{0},

*y*+

*v*

_{0})≈

*f*(

_{y}*x*,

*y*) in Eqs. (14) and (15), the Standard deviation (SD) error of the displacement measurement of DIC can be further expressed as:

*et al.*, [9

**47**, 701–707 (2007). [CrossRef]

**47**, 701–707 (2007). [CrossRef]

**47**, 701–707 (2007). [CrossRef]

## 4. Selection of subset size

## 5. Experimental validations

### 5.1 Numerical experiments

*D*(

*η*)=4, was added to the speckle images to form the deformed images.

*d*is the displacement component

_{i}*u*or

*v*of point

*i*(i=1, 2, …, 2400). The SD error reflects the deviation of the measured displacements from their mean value.

*H*from Eqs. (14) and (15), since H approximates to the value of 1.

### 5.2 Effect of subset size on displacement measurement

*u*-displacement and the

*v*-displacement for the three test image pairs with subset sizes ranging from 17×17 pixels to 71×71 pixels. It is obvious that, for all the three test image pairs, the SD errors decrease as the subset size increases. According to Eqs. (18) and (19), the improvement of displacement measurement accuracy can be attributed to the increase of SSSIG, when a larger subset size is used for the calculation.

### 5.3 Effect of SSSIG on displacement measurement

*u*-displacement and the

*v*-displacement for the three test speckle image pairs with subset sizes ranging from 11×11 pixels to 71×71 pixels at an increment of 6 pixels. It is clear that the SD errors decrease as SSSIG increases in all the three cases. In contrast with Fig. 5, it is noted that the relationship between the SD errors and SSSIG is independent of the image contrast. Moreover, the experimental results are in good agreement with the theoretical results. Therefore, the SSSIG, instead of the subset size itself, is the determinant parameter that directly affects the displacement measurement accuracy.

### 5.4 Selection of subset size

^{5}. Based on the image noise variance of 4 and Eqs. (18) or (19), the theoretical SD error is determined as 0.007 pixels. The results obtained by the proposed algorithm are listed in Table 2. It can be seen from the table that the calculated displacements are in good agreement with pre-assigned theoretical ones, and the SD errors of the

*u*-displacement and

*v*-displacement also match well with the theoretical value.

^{5}, the average subset size is 12 pixels, 14 pixels, and 36 pixels respectively for test image pairs A, B, and C. This reveals that a small subset can lead to a high accuracy of displacement measurement for images with larger contrast. It is also noteworthy that because the values of SSSIG in

*x*direction and

*y*direction can not be precisely controlled through increasing the subset size, the average SSSIG of the optimal subset size for each of the six cases is different, as can be seen from Table 2. This explains why the STD errors of the displacements are not exactly equal to each other.

## 6. Discussion and Conclusion

- Increasing the subset SSSIG. In addition to using larger subsets when applicable, the subset SSSIG can also be increased by using a high bit-depth (e.g., 12-bit or 16-bit) CCD camera or increasing the contrast of the speckle patterns.
- Decreasing the image noise. Image noise can be reduced or suppressed through the use of high-performance hardware, such as a cooled CCD and a high-quality camera lens. In addition, frame averaging during image acquisition is another commonly used effective scheme to alleviate the influence of image noise.

## Acknowledgment

## References and links

1. | M. A. Sutton, S. R. McNeill, J. D. Helm, and Y. J. Chao, “Advances in two-dimensional and three-dimensional computer vision,” in |

2. | W. Tong, “An evaluation of digital image correlation criteria for strain mapping applications,” Strain. |

3. | H. W. Schreier and M. A. Sutton, “Systematic errors in digital image correlation due to undermatched subset shape functions,” Exp. Mech. |

4. | H. W. Schreier, J. R. Braasch, and M. A. Sutton, “Systematic errors in digital image correlation caused by intensity interpolation,” Opt. Eng. |

5. | B. Pan, H. M. Xie, B. Q. Xu, and F. L. Dai, “Performance of sub-pixel registration algorithms in digital image correlation,” Meas. Sci. Technol. |

6. | D. S. Zhang, M. Luo, and D. D. Arola, “Displacement/strain measurements using an optical microscope and digital image correlation,” Opt. Eng. |

7. | S. Yoneyama, H. Kikuta, A. Kitagawa, and K. Kitamura, “Lens distortion correction for digital image correlation by measuring rigid body displacement,” Opt. Eng. |

8. | |

9. | Z. Y. Wang, H. Q. Li, J. W. Tong, and J. T. Ruan, “Statistical analysis of the effect of intensity pattern noise on the displacement measurement precision of digital image correlation using self-correlated images,” Exp. Mech. |

10. | Y. F. Sun and H. J. Pang, “Study of optimal subset size in digital image correlation of speckle pattern images,” Opt. Lasers Eng. |

11. | B. Pan, H. M. Xie, Z. Q. Guo, and T. Hua, “Full-field strain measurement using a two-dimensional Savitzky-Golay digital differentiator in digital image correlation,” Opt. Eng. |

**OCIS Codes**

(100.2000) Image processing : Digital image processing

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

**ToC Category:**

Image Processing

**History**

Original Manuscript: March 4, 2008

Revised Manuscript: April 20, 2008

Manuscript Accepted: April 21, 2008

Published: May 1, 2008

**Citation**

Bing Pan, Huimin Xie, Zhaoyang Wang, Kemao Qian, and Zhiyong Wang, "Study on subset size selection in digital image correlation for speckle patterns," Opt. Express **16**, 7037-7048 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-10-7037

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### References

- M. A. Sutton, S. R. McNeill, J. D. Helm, and Y. J. Chao, "Advances in two-dimensional and three-dimensional computer vision," in Topics in Applied Physics, 1st ed., P. K. Rastogi, ed., (Springer, Berlin: Springer, 2000) 323-372.
- W. Tong, "An evaluation of digital image correlation criteria for strain mapping applications," Strain. 41, 167-175 (2005). [CrossRef]
- H. W. Schreier and M. A. Sutton, "Systematic errors in digital image correlation due to undermatched subset shape functions," Exp. Mech. 42, 303-310 (2002). [CrossRef]
- H. W. Schreier, J. R. Braasch, and M. A. Sutton, "Systematic errors in digital image correlation caused by intensity interpolation," Opt. Eng. 39, 2915-2921 (2000). [CrossRef]
- B. Pan, H. M. Xie, B. Q. Xu, and F. L. Dai, "Performance of sub-pixel registration algorithms in digital image correlation," Meas. Sci. Technol. 17, 1615-1621 (2006). [CrossRef]
- D. S. Zhang, M. Luo, and D. D. Arola, "Displacement/strain measurements using an optical microscope and digital image correlation," Opt. Eng. 45, 033605 (2006). [CrossRef]
- S. Yoneyama, H. Kikuta, A. Kitagawa, and K. Kitamura, "Lens distortion correction for digital image correlation by measuring rigid body displacement," Opt. Eng. 45, 023602 (2006). [CrossRef]
- http://www.correlatedsolutions.com.
- Z. Y. Wang, H. Q. Li, J. W. Tong, and J. T. Ruan, "Statistical analysis of the effect of intensity pattern noise on the displacement measurement precision of digital image correlation using self-correlated images," Exp. Mech. 47, 701-707 (2007). [CrossRef]
- Y. F. Sun and H. J. Pang, "Study of optimal subset size in digital image correlation of speckle pattern images," Opt. Lasers Eng. 45, 967-974 (2007). [CrossRef]
- B. Pan, H. M. Xie, Z. Q. Guo, and T. Hua, "Full-field strain measurement using a two-dimensional Savitzky-Golay digital differentiator in digital image correlation," Opt. Eng. 46, 033601 (2007). [CrossRef]

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