## Single-camera microscopic stereo digital image correlation using a diffraction grating |

Optics Express, Vol. 21, Issue 21, pp. 25056-25068 (2013)

http://dx.doi.org/10.1364/OE.21.025056

Acrobat PDF (2117 KB)

### Abstract

A simple, cost-effective but practical microscopic 3D-DIC method using a single camera and a transmission diffraction grating is proposed for surface profile and deformation measurement of small-scale objects. By illuminating a test sample with quasi-monochromatic source, the transmission diffraction grating placed in front of the camera can produce two laterally spaced first-order diffraction views of the sample surface into the two halves of the camera target. The single image comprising negative and positive first-order diffraction views can be used to reconstruct the profile of the test sample, while the two single images acquired before and after deformation can be employed to determine the 3D displacements and strains of the sample surface. The basic principles and implementation procedures of the proposed technique for microscopic 3D profile and deformation measurement are described in detail. The effectiveness and accuracy of the presented microscopic 3D-DIC method is verified by measuring the profile and 3D displacements of a regular cylinder surface.

© 2013 Optical Society of America

## 1. Introduction

2. B. Pan, K. M. Qian, H. M. Xie, and A. Asundi, “Two-dimensional Digital Image Correlation for In-plane Displacement and Strain Measurement: A Review,” Meas. Sci. Technol. **20**(6), 062001 (2009). [CrossRef]

2. B. Pan, K. M. Qian, H. M. Xie, and A. Asundi, “Two-dimensional Digital Image Correlation for In-plane Displacement and Strain Measurement: A Review,” Meas. Sci. Technol. **20**(6), 062001 (2009). [CrossRef]

3. J. J. Orteu, “3-D computer vision in experimental mechanics,” Opt. Lasers Eng. **47**(3-4), 282–291 (2009). [CrossRef]

4. M. A. Sutton, J. H. Yan, V. Tiwari, W. H. Schreier, and J. J. Orteu, “The effect of out-of-plane motion on 2D and 3D digital image correlation measurements,” Opt. Lasers Eng. **46**(10), 746–757 (2008). [CrossRef]

5. B. Pan, L. P. Yu, and D. F. Wu, “High-accuracy 2D digital image correlation measurements with bilateral telecentric lenses: error analysis and experimental verification,” Exp. Mech. , doi:. [CrossRef]

6. P. F. Luo, Y. J. Chao, M. A. Sutton, and W. H. Peters III, “Accurate measurement of three-dimensional displacement in deformable bodies using computer vision,” Exp. Mech. **33**(2), 123–132 (1993). [CrossRef]

7. B. Pan, D. F. Wu, and L. P. Yu, “Optimization of a three-dimensional digital image correlation system for deformation measurements in extreme environments,” Appl. Opt. **51**(19), 4409–4419 (2012). [CrossRef] [PubMed]

12. S. Xia, A. Gdoutou, and G. Ravichandran, “Diffraction assisted image correlation: a novel method for measuring three-dimensional deformation using two-dimension digital image correlation,” Exp. Mech. **53**(5), 755–765 (2013). [CrossRef]

12. S. Xia, A. Gdoutou, and G. Ravichandran, “Diffraction assisted image correlation: a novel method for measuring three-dimensional deformation using two-dimension digital image correlation,” Exp. Mech. **53**(5), 755–765 (2013). [CrossRef]

## 2. Principles of the single-camera microscopic 3D-DIC method

### 2.1 Measuring system

12. S. Xia, A. Gdoutou, and G. Ravichandran, “Diffraction assisted image correlation: a novel method for measuring three-dimensional deformation using two-dimension digital image correlation,” Exp. Mech. **53**(5), 755–765 (2013). [CrossRef]

13. M. Trivi and H. J. Rabal, “Stereoscopic uses of diffraction gratings,” Appl. Opt. **27**(6), 1007–1009 (1988). [CrossRef] [PubMed]

*O*(0,0,0) located at the intersecting point of the optical axis and the diffraction grating. Assume that

*XY*plane coincides with the grating plane with

*X*axis perpendicular to and

*Y*axis parallel to the grating rulings. The positive direction of

*Z*axis is defined to point from the grating to the test object. Using this coordinate system, the real coordinate of a point on the test object surface is denoted as

*P*(

*X*,

*Y*,

*Z*). Moreover, the coordinates of its corresponding two first-order diffracted points are represented by

*P*

_{-1}(

*X*

_{-1},

*Y*

_{-1},

*Z*

_{-1}) and

*P*

_{+1}(

*X*

_{+1},

*Y*

_{+1},

*Z*

_{+1}), respectively. By using a backward ray-tracing method, the distances between the grating and points

*P*

_{-1},

*P*

_{+1}are found to be [12

**53**(5), 755–765 (2013). [CrossRef]

*p*denoting the pitch of the grating.

*Y*axis, the

*Y*and

*X*coordinates of point

*P*,

*P*

_{-1},

*P*

_{+1}can be determined as

*P*

_{-1}and

*P*

_{+1}should be

*c*,

_{x}*c*); the image distance of the optical system is

_{y}*L*

_{img}; the object distance of the optical system is

*Z*

_{obj}. Also, it is quite important to mention here that, using a long working distance microscope, the object distance

*Z*

_{obj}is much larger than the height variations of the test micro-scale sample surface. For this reason, it is reasonable to assume that the object distance of each point on the sample surface equals to Z

_{obj}. Based on these assumptions and approximations, the image coordinates (in units of pixels) of

*P*,

*P*

_{-1},

*P*

_{+1}, i.e.,

*x*

_{+1},

*y*

_{+1}), (

*x*

_{-1},

*y*

_{-1}) are the image coordinates of the two virtual points located in the positive first-order ( + 1) and the negative first-order (−1) diffracted images of the test specimen, respectively;

*P*

_{sz}is the pixel size of the camera;

*M = L*

_{img}/(

*Z*

_{obj}×

*P*

_{sz}) (in unit of pixel/mm) is the magnification factor of the optical system, which can be calibrated in advance. Note that the coordinate correspondence of the two diffracted points can be obtained by matching the positive first-order and the negative first-order images of the test specimen using well-established subset-based 2D-DIC algorithm [2

2. B. Pan, K. M. Qian, H. M. Xie, and A. Asundi, “Two-dimensional Digital Image Correlation for In-plane Displacement and Strain Measurement: A Review,” Meas. Sci. Technol. **20**(6), 062001 (2009). [CrossRef]

14. B. Pan and K. Li, “A fast digital image correlation method for deformation measurement,” Opt. Lasers Eng. **49**(7), 841–847 (2011). [CrossRef]

### 2.2 3D profile measurement

*X*and

*Z*coordinates can be determined. Likewise, the

*Y*coordinates can be determined by adding the

*y*-coordinate of Eq. (6) to that of Eq. (5). In such a way, the real 3D coordinates of a point are written as

### 2.3 3D deformation measurement

*P*(

*X*,

*Y*,

*Z*) is denoted as

*P*moves to

*P’*(

*X + U*,

*Y + V*,

*Z + W*). Then, according to the grating equation and Eq. (1), the exact coordinates of

*P*

_{-1}and

*P*

_{+1}can be written as

*θ*of the grating is calculated as 3.99°, thus we have cos

*θ*= 0.9976≈1. In addition, the object distance

*Z*

_{obj}of the long working distance microscope being used is much larger than the out-of-plane displacement component

*W*(i.e.,

*W*<<

*Z*

_{obj}). Based on these approximations, Eqs. (11) and (12) can be simplified as

*x*-directional displacements of the negative first-order diffracted images (i.e.,

*u*

_{-1}) from those of the positive first-order diffracted images (i.e.,

*u*

_{+1}) gives

*x*- and

*y*-directional displacements of the negative first-order diffracted images (i.e.,

*u*

_{-1},

*v*

_{-1}) to those of the positive first-order diffracted images (i.e.,

*u*

_{+1},

*v*

_{+1}) and taking consideration of Eqs. (5), (6), (13) and (14) as well as the assumption

*W*<<

*Z*

_{obj}, we have

*P*can be obtained as where

*Z*is the distance of each measurement point to the grating, which can be obtained after shape reconstruction.

*Z*

_{obj}is the object distance of the imaging system, which can be determined by the calibration approach described in the following section.

**53**(5), 755–765 (2013). [CrossRef]

*M*,

*Z*

_{obj}) are involved in determining the three displacements of the measured object surface. As will be shown below, the formulas derived in this work can provide more accurate results.

### 2.4 Determine image displacements of the diffracted images using 2D-DIC

*x*

_{-1},

*y*

_{-1}) are searched in the right (positive first-order) diffracted image using subset-based 2D-DIC to determine its corresponding image coordinates (

*x*

_{+1},

*y*

_{+1}), as schematically shown in the top half part of Fig. 3. Afterwards, these mapped image coordinates are submitted to Eqs. (7)-(9) for reconstruction of the profile of the ROI.

*x’*

_{-1},

*y’*

_{-1}) and (

*x*

_{-1},

*y*

_{-1}) provides the in-plane displacements (

*u*

_{-1},

*v*

_{-1}) of the point in negative first-order images, while the difference between (

*x’*

_{+1},

*y’*

_{+1}) and (

*x*

_{+1},

*y*

_{+1}) gives the desired in-plane displacements (

*u*

_{+1},

*v*

_{+1}) of the point in positive first-order images.

### 2.5 Calibration of the imaging system

*M*and the object distance

*Z*

_{obj}, must be calibrated. In practice,

*M*can be easily determined by measuring the image displacements of a speckle pattern with prescribed in-plane translations. Specifically, a glass plate decorated with speckle pattern can be translated using a precision translation stage along the horizontal direction (in millimeters), the in-plane motions (in pixels) of the zero-order image of the sample can be detected by regular 2D-DIC method. Afterwards, the magnification

*M*of the imaging system can be computed by fitting the actual and measured displacements using linear least squares.

## 3. Experimental validation

### 3.1 Experimental details

*z*-directional) rigid body translations were also performed on the same cylinder. The motions were applied by a two-axis translation stage with a positioning accuracy of 5 μm, and the 3D displacements were then calculated using the presented technique. Subsequently, the measured displacements were compared with applied values to validate the effectiveness and accuracy of the proposed technique for 3D deformation measurements.

### 3.2 Shape measurement of a cylinder surface

*Z*-coordinates of the 3D surface are inverted. The final profile is indicated in Fig. 6(b), which is in good agreement with the practical situation, confirming the correctness of the proposed technique for profile measurement. The radius of the cylinder can be computed by fitting the reconstructed height data using an nonlinear iterative least square approach given in Ref [17

17. P. F. Luo and J. N. Chen, “Measurement of curved-surface Deformation in cylindrical coordinates,” Exp. Mech. **40**(4), 345–350 (2000). [CrossRef]

### 3.3 3D displacement measurement

*Z*directions were exerted using a two-axis translation stage. The prescribed translations in Z-directions range from −0.5 mm to 0.5 mm with a 0.1 mm increment between consecutive translations. Diffraction speckle image of the cylinder sample was recorded at each translation. Note that the image in its original position (

*W*= 0 mm) was used as the reference image. All the rest of the images were subsequently compared with the reference image using the same calculation settings to determine the full-field image displacements within the specified ROI. Afterwards, real displacements of each object point were obtained using Eqs. (18)-(20)).

*Z*-directional displacements and the prescribed displacements. For comparison, the average of the

*Z*-displacements calculated by Eq. (8) in Ref [12

**53**(5), 755–765 (2013). [CrossRef]

**53**(5), 755–765 (2013). [CrossRef]

*Z*-directional) displacements. Figure 7(b) illustrates the displacement vector of each calculated point in the specified ROI when the applied

*Z*-displacement is 0.5mm. We can clearly see that all of the measured displacement vectors are along the positive

*Z*-direction, and the value of each displacement vector approximately equals to 0.5 mm, which are consistent with the prescribed displacements.

*Z*-directional displacements measured by Eq. (18) of this work and by Eq. (8) of Ref [12

**53**(5), 755–765 (2013). [CrossRef]

*Z*-displacements measured by the proposed technique is only 1.56%, with a maximum relative error of 4%. By contrast, the mean relative error of the displacements measured by existing work [12

**53**(5), 755–765 (2013). [CrossRef]

**53**(5), 755–765 (2013). [CrossRef]

**53**(5), 755–765 (2013). [CrossRef]

*Z*/(

*Z*

_{obj}-

*Z*). When the corresponding values (i.e., Z

_{obj}= 214.46mm, Z≈32mm) are substituted in this equation, a factor of 17.58% is obtained. This value is in approximate agreement with the percentage errors given in the last column of Table 1.

## 4. Conclusion

## Acknowledgements

## References and links

1. | M. A. Sutton, J. J. Orteu, and H. W. Schreier, |

2. | B. Pan, K. M. Qian, H. M. Xie, and A. Asundi, “Two-dimensional Digital Image Correlation for In-plane Displacement and Strain Measurement: A Review,” Meas. Sci. Technol. |

3. | J. J. Orteu, “3-D computer vision in experimental mechanics,” Opt. Lasers Eng. |

4. | M. A. Sutton, J. H. Yan, V. Tiwari, W. H. Schreier, and J. J. Orteu, “The effect of out-of-plane motion on 2D and 3D digital image correlation measurements,” Opt. Lasers Eng. |

5. | B. Pan, L. P. Yu, and D. F. Wu, “High-accuracy 2D digital image correlation measurements with bilateral telecentric lenses: error analysis and experimental verification,” Exp. Mech. , doi:. [CrossRef] |

6. | P. F. Luo, Y. J. Chao, M. A. Sutton, and W. H. Peters III, “Accurate measurement of three-dimensional displacement in deformable bodies using computer vision,” Exp. Mech. |

7. | B. Pan, D. F. Wu, and L. P. Yu, “Optimization of a three-dimensional digital image correlation system for deformation measurements in extreme environments,” Appl. Opt. |

8. | H. W. Schreier, D. Garcia, and M. A. Sutton, “Advances in light microscope stereo vision,” Exp. Mech. |

9. | M. A. Sutton, X. Ke, S. M. Lessner, M. Goldbach, M. Yost, F. Zhao, and H. W. Schreier, “Strain field measurement on mouse carotid arteries using microscopic three-dimensional digital image correlation,” J. Biomed. Mater. Res. |

10. | Z. X. Hu, H. Y. Luo, Y. J. Du, and H. B. Lu, “Fluorescent stereo microscopy for 3D surface profilometry and deformation mapping,” Opt. Express |

11. | |

12. | S. Xia, A. Gdoutou, and G. Ravichandran, “Diffraction assisted image correlation: a novel method for measuring three-dimensional deformation using two-dimension digital image correlation,” Exp. Mech. |

13. | M. Trivi and H. J. Rabal, “Stereoscopic uses of diffraction gratings,” Appl. Opt. |

14. | B. Pan and K. Li, “A fast digital image correlation method for deformation measurement,” Opt. Lasers Eng. |

15. | B. Pan, H. M. Xie, and Z. Y. Wang, “Equivalence of digital image correlation criteria for pattern matching,” Appl. Opt. |

16. | H. Lu and P. D. Cary, “Deformation measurement by digital image correlation: implementation of a second-order displacement gradient,” Exp. Mech. |

17. | P. F. Luo and J. N. Chen, “Measurement of curved-surface Deformation in cylindrical coordinates,” Exp. Mech. |

**OCIS Codes**

(100.2000) Image processing : Digital image processing

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

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: August 6, 2013

Revised Manuscript: September 30, 2013

Manuscript Accepted: October 2, 2013

Published: October 14, 2013

**Citation**

Bing Pan and Qiong Wang, "Single-camera microscopic stereo digital image correlation using a diffraction grating," Opt. Express **21**, 25056-25068 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-21-25056

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### References

- M. A. Sutton, J. J. Orteu, and H. W. Schreier, Image correlation for shape, motion and deformation measurements (Springer, 2009).
- B. Pan, K. M. Qian, H. M. Xie, and A. Asundi, “Two-dimensional Digital Image Correlation for In-plane Displacement and Strain Measurement: A Review,” Meas. Sci. Technol.20(6), 062001 (2009). [CrossRef]
- J. J. Orteu, “3-D computer vision in experimental mechanics,” Opt. Lasers Eng.47(3-4), 282–291 (2009). [CrossRef]
- M. A. Sutton, J. H. Yan, V. Tiwari, W. H. Schreier, and J. J. Orteu, “The effect of out-of-plane motion on 2D and 3D digital image correlation measurements,” Opt. Lasers Eng.46(10), 746–757 (2008). [CrossRef]
- B. Pan, L. P. Yu, and D. F. Wu, “High-accuracy 2D digital image correlation measurements with bilateral telecentric lenses: error analysis and experimental verification,” Exp. Mech., doi:. [CrossRef]
- P. F. Luo, Y. J. Chao, M. A. Sutton, and W. H. Peters, “Accurate measurement of three-dimensional displacement in deformable bodies using computer vision,” Exp. Mech.33(2), 123–132 (1993). [CrossRef]
- B. Pan, D. F. Wu, and L. P. Yu, “Optimization of a three-dimensional digital image correlation system for deformation measurements in extreme environments,” Appl. Opt.51(19), 4409–4419 (2012). [CrossRef] [PubMed]
- H. W. Schreier, D. Garcia, and M. A. Sutton, “Advances in light microscope stereo vision,” Exp. Mech.44(3), 278–288 (2004). [CrossRef]
- M. A. Sutton, X. Ke, S. M. Lessner, M. Goldbach, M. Yost, F. Zhao, and H. W. Schreier, “Strain field measurement on mouse carotid arteries using microscopic three-dimensional digital image correlation,” J. Biomed. Mater. Res.84A(1), 178–190 (2008). [CrossRef]
- Z. X. Hu, H. Y. Luo, Y. J. Du, and H. B. Lu, “Fluorescent stereo microscopy for 3D surface profilometry and deformation mapping,” Opt. Express21(10), 11808–11818 (2013). [CrossRef] [PubMed]
- http://www.correlatedsolutions.com
- S. Xia, A. Gdoutou, and G. Ravichandran, “Diffraction assisted image correlation: a novel method for measuring three-dimensional deformation using two-dimension digital image correlation,” Exp. Mech.53(5), 755–765 (2013). [CrossRef]
- M. Trivi and H. J. Rabal, “Stereoscopic uses of diffraction gratings,” Appl. Opt.27(6), 1007–1009 (1988). [CrossRef] [PubMed]
- B. Pan and K. Li, “A fast digital image correlation method for deformation measurement,” Opt. Lasers Eng.49(7), 841–847 (2011). [CrossRef]
- B. Pan, H. M. Xie, and Z. Y. Wang, “Equivalence of digital image correlation criteria for pattern matching,” Appl. Opt.49(28), 5501–5509 (2010). [CrossRef] [PubMed]
- H. Lu and P. D. Cary, “Deformation measurement by digital image correlation: implementation of a second-order displacement gradient,” Exp. Mech.40(4), 393–400 (2000). [CrossRef]
- P. F. Luo and J. N. Chen, “Measurement of curved-surface Deformation in cylindrical coordinates,” Exp. Mech.40(4), 345–350 (2000). [CrossRef]

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