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Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 20 — Oct. 4, 2004
  • pp: 4980–4985
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Single-phase-step method with contoured correlation fringe patterns for ESPI

Qifeng Yu, Sihua Fu, Xiaolin Liu, Xia Yang, and Xiangyi Sun  »View Author Affiliations


Optics Express, Vol. 12, Issue 20, pp. 4980-4985 (2004)
http://dx.doi.org/10.1364/OPEX.12.004980


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Abstract

Speckle fringe patterns of electronic speckle pattern interferometry (ESPI) are full of high-level speckle noise and are mainly processed by phase-shifting methods that normally require three speckle fringe patterns or more. The author proposes a novel method, the contoured correlation-fringe-pattern (CCFP) method, by which speckle-noise-free fringe patterns can be generated for ESPI. The application of this novel method is extended to the phase-shifting or phase-stepping for ESPI after its improvement. It generates speckle-noise-free phase fringes and remains valid for single phase-step condition, thus eliminates the two main disadvantages of the phase-shifting (stepping) methods of ESPI.

© 2004 Optical Society of America

1. Introduction

ESPI is a widely used metrology technique due to its simplicity and adaptability in field-site. However, the speckle fringe patterns of ESPI are inherently full of speckle noise [1

1. K. Creath, “Speckle: Signal or Noise?” in Proc. FRINGE 93, W. Jueptner and W. Osten, ed. (Akademie Verlag GmgH, Berlin, 1993), pp. 97–102.

], which causes difficulties in the post-processing of the phase extraction difficult for the ESPI fringe patterns. Generally, the most successful and most widely used processing method for ESPI data processing is the phase-shifting or phase-stepping algorithms [2

2. K. Creath, “Phase-measurement interferometry techniques,” Progress in Optics, XXVI, 349–393(1988). [CrossRef]

]. The resultant phase images of phase-shifting methods are, however, still full of speckle noise that makes phase unwrapping difficult or un-robust and thus sophisticated algorithms to suppress speckle noise are required on either speckle fringe patterns or speckle phase patterns. Most of speckle filtering algorithms suppress speckle noise at the expense of blurring phase or lowering the phase resolution [3

3. G. Kaufmann, A. Davila, and D. Kerr, “Interview-Smoothing of speckle interferometry fringe-patterns,” Optical Testing Digest, 2, 1997, http://www.spie.org/app/otd/index.cfm?fuseaction=displaynewsletter&newsletterid=62.

]. In addition, common phase-shifting methods for ESPI normally require multi-phase-steps with at least four speckle patterns (one reference speckle pattern and 3 deformed speckle patterns with phase-shifting and vice versa for two-phase-step method), which is inconvenient in application.

The single-phase-step method can use two fringe patterns with π/2 phase-shift under the condition of nearly uniformed background [4

4. D. Kerr, F. Mendoza Santoyo, and J. R. Tyrer, “Extraction of phase data from electronic speckle pattern interferometric fringes using a single-phase-step method: a novel approach,” J. Opt. Soc. Am. A, 7, 820–826(1990). [CrossRef]

]. In other cases a double-image phase-shifting method can be applied after the two fringe patterns are transformed as noise-free normalized fringe patterns [5

5. Qifeng Yu, Xiaolin Liu, Xiangyi Sun, and Zhihui Lei, “Double-image and single-image phase-shifting methods for phase measurement,” Optik, 109, 89–95(1998).

,6

6. Qifeng Yu, K. Andresen, W. Osten, and W. Jueptner, “Noise-free normalized fringe patterns and local pixel transforms for strain extraction,” Appl. Opt. 35, 3783–90(1996). [CrossRef] [PubMed]

] that is difficult to be applied to ESPI speckle fringe images. Single-image phase-shifting method needs additional phase-sign map for digitally π/2-phase-shifting that is difficult to achieve in practice [5

5. Qifeng Yu, Xiaolin Liu, Xiangyi Sun, and Zhihui Lei, “Double-image and single-image phase-shifting methods for phase measurement,” Optik, 109, 89–95(1998).

]. Some effective adaptive quadrature filters are developed to recover the local phase from a single fringe pattern [7

7. J.L. Marroquin, J.E. Figueroa, and M. Servin, “Robust quadrature filters,” J. Opt. Soc. Am. A, 14, 779–791(1997). [CrossRef]

,8

8. J.L. Marroquin, M. Servin, and R. Rodriguez-Vera, “Adaptive quadrature filters and the recovery of phase from fringe pattern images,” J. Opt. Soc. Am. A, 14, 1742–1753(1997). [CrossRef]

] and show good results. These algorithms are rather complicated and the computation with iteration is time-consuming especially when the fringes are closed.

Speckle fringe patterns of ESPI are generated traditionally by a subtraction between two original speckle patterns with a phase difference, and the resultant speckle fringe patterns are full of high-spatial-frequency and high-contrast speckle noise. The direct correlation (DC) method can generate a fringe pattern without these high-level speckle noises by performing a direct correlation between two original speckle patterns within rectangular windows and the correlation coefficients generate the fringe patterns [9

9. Douglas R. Schmitt and R. W. Hunt, “Optimization of fringe pattern calculation with direct correlations in speckle interferometry,” Appl. Opt. 36, 8848–8857(1997). [CrossRef]

,10

10. Yuming He, “Digital shearing speckle corelation fringes pattern formed by using linear correlation calculating method,” Acta Photonica Sinica, 24, 13–17(1995).

]. The DC method is applied to (5,1) and (1,5) phase-shifting algorithms that require six original speckle patterns to suppress the speckle noise [11

11. C. Kao, G. Yeh, S. Lee, C. Yang, and K. Wu, “Phase-shifting algorithms for electronic speckle pattern interferometry”, Appl. Opt. 41, 46–54 (2002). [CrossRef] [PubMed]

]. The top left part of Fig. 1 is the resultant speckle fringe pattern by common subtraction method and the top right part is that by the DC method.

Though DC method suppresses the high-spatial-frequency and high-contrast speckle noise, the resultant fringe patterns have some new large size speckles with obvious blurring effect that spoils the fine phase distribution, as shown in Fig. 1. The main source of the large size speckle noise and the blurring effect come from that the basic assumption of DC method is not true that the real phase is assumed to be constant inside its rectangular windows. The rectangular windows give approximate and one kind of average phase results.

The CCFP method is proposed by the authors to derive the fringe pattern with speckle-free, smooth, normalized and consistent fringes from two original speckle patterns [12

12. Qifeng Yu, Sihua Fu, Xia Yang, Xiangyi Sun, and Xiaolin Liu, “Extraction of phase field from a single contoured correlation fringe pattern of ESPI,” Opt. Express 12, 75–83(2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-75. [CrossRef] [PubMed]

], as shown in the bottom of Fig. 1. The main idea of the new method is that Yu’s contoured windows coinciding with fringe contours of equal-phase are first established and then the correlation between the two original speckle patterns is performed only on these contoured windows. Since CCFP method overcomes the main error source of DC method the resultant fringe patterns of CCFP have smooth and normalized fringes with the same phase field as that of subtraction method. The CCFP method, therefore, removes the speckle noise completely without phase blurring and the gray levels of the fringe patterns correspond directly to the phase field, which is a breakthrough for ESPI processing.

In this paper we extend the application of the CCFP method to the phase-shifting techniques for ESPI including the single-phase-step method [4

4. D. Kerr, F. Mendoza Santoyo, and J. R. Tyrer, “Extraction of phase data from electronic speckle pattern interferometric fringes using a single-phase-step method: a novel approach,” J. Opt. Soc. Am. A, 7, 820–826(1990). [CrossRef]

] or double-image phase-shifting method [5

5. Qifeng Yu, Xiaolin Liu, Xiangyi Sun, and Zhihui Lei, “Double-image and single-image phase-shifting methods for phase measurement,” Optik, 109, 89–95(1998).

]. The new method uses one reference speckle pattern and two speckle patterns with π/2 phase-shift and vice versa to derive the full phase field directly. With the new method all speckle noise suppression processing and the unwrapping approaches dealing with speckle noise become unnecessary. In the same principle, the CCFP method is applied to common multi-phase-step method of ESPI. The comparison between the results respectively with single-phase-step and multi-phase-step shows the results obtained by the single-phase-step method are good enough and almost the same as those with multi-phase-step methods. Therefore, the single-phase-step method with CCFP is recommended.

Fig. 1. The resultant fringe patterns of a ESPI derived by subtraction (top left), direct correlation with window size of 17×17(top right) and contoured correlation with window size of 67×5 pixels (bottom), respectively.

2. The Main principle of the method

The CCFP method [12

12. Qifeng Yu, Sihua Fu, Xia Yang, Xiangyi Sun, and Xiaolin Liu, “Extraction of phase field from a single contoured correlation fringe pattern of ESPI,” Opt. Express 12, 75–83(2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-75. [CrossRef] [PubMed]

] is briefly introduced as follows. In ESPI, two original speckle patterns with a phase difference a Δ θ(x, y) are first acquired and their intensities can be represented as follows:

f1(x,y)=I1(x,y)+I2(x,y)+2I1I2cos[Ψ1(x,y)Ψ2(x,y)]
(1)
f2(x,y)=I1(x,y)+I2(x,y)+2I1I2cos[Ψ1(x,y)Ψ2(x,y)+Δθ(x,y)]
(2)

where I1(x, y) and I2(x, y) are the intensity variables depending on the experimental setup, Ψ1(x, y) and Ψ2 (x, y) are the phases of random speckles, and Δθ(x, y) is the phase difference between the two speckle patterns resulted from some deformation. According to the speckle statistic theory [13

13. J.C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975), Chap.2.

], under the condition that the phase difference term of Δθ(x, y) is a constant over the whole window of (m× n) and the window size (m × n) is large enough, the following assumptions can be true:

<f1>=2<I>m×n,<f2>=2<I>m×n
(3)
<cosβ>m×n=<cos(β+Δθ)>m×n=0
(4)

where I is I1(x, y) or I2(x, y), <f1>, <f2> are the average values of f1, f2 within the windows, respectively, and β = Ψ1 - Ψ2. And let <I 1>m×n = <I 2>m×n = <I>m×n.

Under the above-mentioned conditions, the standard correlation between f1, f2 is performed only on the fringe contour windows (m× n) on which the phase difference Δθ(x, y) is constant and thus the CCFP is derived as follows:

C1(x,y)=<(f1<f1>m×n)(f2<f2>m×n)>m×n[<(f1<f1>m×n)2>m×n]12[<(f2<f2>m×n)2>m×n]12
=2<I>m×n2+2<I>m×n2cosΔθ4<I>m×n2=(cosΔθ+1)2
(5)

Eq. (5) indicates that the CCFP is a pure cosine fringe pattern without random speckle factors. The formulae of Eq. (5) is deducted under the condition that the phase of Δθ(x, y) is constant within the correlation windows, which is the essential difference from DC method and which makes the CCFP method accurate. The contoured windows that coincide with the fringe contours of the speckle fringe patterns can be determined by Yu’s contoured window method with the help of the speckle fringe orientation map (SFOM) that determined from the subtraction speckle fringe pattern [14

14. Qifeng Yu, X. Sun, and X. Liu, “Removing speckle noise and skeleton extraction from a single speckle fringe pattern,” Opt. Eng. 42, 68–74(2003). [CrossRef]

,15

15. Qifeng Yu, X. Sun, and X. Liu, “Spin filtering with curve windows for interferometric fringes,” Appl. Opt. 41, 2650–2654 (2002). [CrossRef] [PubMed]

]. The correlation coefficients generate the speckle-free and normalized fringe pattern of Eq. (5) without blurring effect.

With phase-shifting or phase-step method, a new speckle pattern with a π/2 phase shift relative to the pattern of Eq.(2) can be acquired as follows:

f3(x,y)=I1(x,y)+I2(x,y)+2I1I2cos[Ψ1(x,y)Ψ2(x,y)+Δθ(x,y)+π2]
(6)

In the same principle of Eq. (5) for f1, f3 a sinusoidal fringe pattern is derived:

C2(x,y)=<(f1<f1>m×n)(f3<f3>m×n)>m×n[<(f1<f1>m×n)2>m×n]12[<(f3<f3>m×n)2>m×n]12
=(cos(Δθ+π2)+1)2=(1sinΔθ)2
(7)

Thus the phase difference term, Δθ(x, y), is determined as follows:

Δθ(x,y)=arctan(12C2(x,y)2C1(x,y)1)=arctan(sin[Δθ(x,y)]cos[Δθ(x,y)])
(8)

This saw-tooth phase field is the same as that produced by common phase-shifting methods requiring at least three fringe patterns [2

2. K. Creath, “Phase-measurement interferometry techniques,” Progress in Optics, XXVI, 349–393(1988). [CrossRef]

] but without speckle noise for ESPI. Therefore, the wrapped phase pattern of Eq. (8) can be unwrapped by common unwrapping algorithms without extra speckle-noise suppression.

The key step of the method is that the correlation is performed only within the fringe contour windows for each point and then there is no blurring effect on the phase term, Δθ(x, y) for the whole procedure.

In the similar way the CCFP method can be applied to the common multi-phase-step phase-shifting methods and derive the same phase fringe patterns.

3. Experimental results and Discussions

A computed simulated example is first used to illustrate the correctness and the accuracy of the new single-phase-step method. The top of Fig. 2 is a simulated speckle fringe pattern and the bottom of Fig. 2 is the saw-tooth phase fringe pattern for the same pattern achieved by our new method. The phase distributions of the simulated ideal phase and the result derived by our new method are shown in Fig. 3 for the same cross-section in Fig. 2. Fig. 3 shows that both results coincide well with each other, which proofs that the new method has a high accuracy.

A practical speckle fringe pattern with common subtraction without post-processing is shown in the top of Fig. 4. The bottom of Fig. 4 is its speckle fringe orientation map (SFOM) that is the base for determination of the contoured windows. The top of Fig. 5 is the saw-tooth phase fringe pattern derived by the common four image (three-phase-step) phase-shifting method of ESPI in which the sine pattern and the cosine pattern of the phase fringe pattern are filtered and strongly blurred with an average filtering with a window of 7 × 7 pixels. The vestige of the speckle noise can be seen clearly on the image though the filtering blurs the phase seriously. The bottom of Fig. 5 is the phase fringe pattern derived by our new single-phase-step method with CCFP with a window sized of 51 × 3 pixels. The results show that the new method provides better and smoother phase pattern than common phase-shifting methods though the new method uses only three original speckle patterns instead of six speckle patterns by common methods.

Fig. 2. The top is a simulated speckle fringe pattern and the bottom is the saw-tooth phase fringe pattern for the same pattern achieved by our new method.
Fig. 3. The phase distributions of the simulated ideal phase and the result derived by our new method for the same cross-section in Fig. 2.
Fig. 4. The top is a practical speckle fringe pattern with common subtraction without post-processing, the bottom is its SFOM.
Fig. 5. The phase fringe patterns derived by (top) the common four image phase-shifting method, (bottom) the proposed single-phase-step method with a window size of 51×3 pixels.
Fig. 6. The phase fringe patterns derived by the proposed methods with contoured correlation fringe patterns with a window size of 51×3 pixels, (top) by the single-phase-step method, (bottom) by the three-phase-step method.
Fig. 7. The phase distributions of the same cross-section in the phase pattern derived by (a) subtraction method in the top of Fig. 5, (b) the single-phase-step method with contoured correlation fringe patterns in the bottom of Fig. 5 and (c) the three-phase-step method with contoured correlation fringe patterns in the bottom of Fig. 6.

The top and the bottom of Fig. 6 are the resultant phase fringe patterns derived by our new single-phase-step and three-phase-step methods with CCFP with a window sized of 51×3 pixels, respectively. Fig. 6 shows that with the CCFP method the single-phase-step method and three-phase-step method provide similar phase results without significant difference. Though the result of three-phase-step method is a little smoother, it requires two more fringe patterns and more computation cost than the single-phase-step method. On this consideration we recommend the single-phase-step method with CCFP.

4. Conclusions

The phase-shifting (phase-stepping) methods that are proved successful and widely used to extract the phase field for ESPI have two main disadvantages. One is that the high-level speckle noise on the speckle patterns requires sophisticated algorithms for suppressing in the state of either speckle fringe patterns or phase patterns and the speckle suppression blurs and decreases the phase resolution [3

3. G. Kaufmann, A. Davila, and D. Kerr, “Interview-Smoothing of speckle interferometry fringe-patterns,” Optical Testing Digest, 2, 1997, http://www.spie.org/app/otd/index.cfm?fuseaction=displaynewsletter&newsletterid=62.

]. The other is that phase-shifting methods normally require multi-phase-steps, i.e., three speckle fringe patterns or more, which is inconvenient for practical applications.

We extend our CCFP method to the phase-shifting methods and propose a single-phase-step method with CCFP for ESPI in this paper. By the method, the speckle-noise-free wrapped phase patterns can be generated directly and thus speckle suppression processing that normally blurs the phase field is unnecessary. The CCFP method performs correlation between two original speckle patterns only within fringe-contoured windows so that the new method has no phase blurring effect and provides better accuracy than that of common phase-shifting methods.

Furthermore, the proposed new method can take only single-phase-step, i.e., two fringe patterns with a π/2 phase shift and without additional assumptions, such as constant image background. Thus the requirement of the number of phase-steps is reduced to the minimum in comparison with that of common phase-shifting methods. The proposed single-phase-step method with CCFP, therefore, overcomes the two main disadvantages of the widely-used phase-shifting methods of ESPI. The new method provides the best accuracy and requires the least phase-shifting-step, which is a significant progress in phase-shifting techniques for ESPI.

Acknowledgments

We thank Dr. Jinglong Chen in Tianjin University for his kind help of providing some image.

References and links

1.

K. Creath, “Speckle: Signal or Noise?” in Proc. FRINGE 93, W. Jueptner and W. Osten, ed. (Akademie Verlag GmgH, Berlin, 1993), pp. 97–102.

2.

K. Creath, “Phase-measurement interferometry techniques,” Progress in Optics, XXVI, 349–393(1988). [CrossRef]

3.

G. Kaufmann, A. Davila, and D. Kerr, “Interview-Smoothing of speckle interferometry fringe-patterns,” Optical Testing Digest, 2, 1997, http://www.spie.org/app/otd/index.cfm?fuseaction=displaynewsletter&newsletterid=62.

4.

D. Kerr, F. Mendoza Santoyo, and J. R. Tyrer, “Extraction of phase data from electronic speckle pattern interferometric fringes using a single-phase-step method: a novel approach,” J. Opt. Soc. Am. A, 7, 820–826(1990). [CrossRef]

5.

Qifeng Yu, Xiaolin Liu, Xiangyi Sun, and Zhihui Lei, “Double-image and single-image phase-shifting methods for phase measurement,” Optik, 109, 89–95(1998).

6.

Qifeng Yu, K. Andresen, W. Osten, and W. Jueptner, “Noise-free normalized fringe patterns and local pixel transforms for strain extraction,” Appl. Opt. 35, 3783–90(1996). [CrossRef] [PubMed]

7.

J.L. Marroquin, J.E. Figueroa, and M. Servin, “Robust quadrature filters,” J. Opt. Soc. Am. A, 14, 779–791(1997). [CrossRef]

8.

J.L. Marroquin, M. Servin, and R. Rodriguez-Vera, “Adaptive quadrature filters and the recovery of phase from fringe pattern images,” J. Opt. Soc. Am. A, 14, 1742–1753(1997). [CrossRef]

9.

Douglas R. Schmitt and R. W. Hunt, “Optimization of fringe pattern calculation with direct correlations in speckle interferometry,” Appl. Opt. 36, 8848–8857(1997). [CrossRef]

10.

Yuming He, “Digital shearing speckle corelation fringes pattern formed by using linear correlation calculating method,” Acta Photonica Sinica, 24, 13–17(1995).

11.

C. Kao, G. Yeh, S. Lee, C. Yang, and K. Wu, “Phase-shifting algorithms for electronic speckle pattern interferometry”, Appl. Opt. 41, 46–54 (2002). [CrossRef] [PubMed]

12.

Qifeng Yu, Sihua Fu, Xia Yang, Xiangyi Sun, and Xiaolin Liu, “Extraction of phase field from a single contoured correlation fringe pattern of ESPI,” Opt. Express 12, 75–83(2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-75. [CrossRef] [PubMed]

13.

J.C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975), Chap.2.

14.

Qifeng Yu, X. Sun, and X. Liu, “Removing speckle noise and skeleton extraction from a single speckle fringe pattern,” Opt. Eng. 42, 68–74(2003). [CrossRef]

15.

Qifeng Yu, X. Sun, and X. Liu, “Spin filtering with curve windows for interferometric fringes,” Appl. Opt. 41, 2650–2654 (2002). [CrossRef] [PubMed]

OCIS Codes
(100.2650) Image processing : Fringe analysis
(120.6160) Instrumentation, measurement, and metrology : Speckle interferometry

ToC Category:
Research Papers

History
Original Manuscript: August 23, 2004
Revised Manuscript: September 27, 2004
Published: October 4, 2004

Citation
Qifeng Yu, Sihua Fu, Xiaolin Liu, Xia Yang, and Xiangyi Sun, "Single-phase-step method with contoured correlation fringe patterns for ESPI," Opt. Express 12, 4980-4985 (2004)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-20-4980


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References

  1. K. Creath, �??Speckle: Signal or Noise?�?? in Proc. FRINGE 93, W. Jueptner and W. Osten, ed. (Akademie Verlag GmgH, Berlin, 1993), pp. 97-102.
  2. K. Creath, �??Phase-measurement interferometry techniques,�?? Progress in Optics, XXVI, 349-393(1988). [CrossRef]
  3. G. Kaufmann, A. Davila and D. Kerr, �??Interview-Smoothing of speckle interferometry fringe-patterns,�?? Optical Testing Digest, 2, 1997.
  4. D. Kerr, F. Mendoza Santoyo and J. R. Tyrer, �??Extraction of phase data from electronic speckle pattern interferometric fringes using a single-phase-step method: a novel approach,�?? J. Opt. Soc. Am. A 7, 820-826 (1990). [CrossRef]
  5. Qifeng Yu, Xiaolin Liu, Xiangyi Sun and Zhihui Lei, �??Double-image and single-image phase-shifting methods for phase measurement,�?? Optik 109, 89-95 (1998).
  6. Qifeng Yu, K. Andresen, W. Osten and W. Jueptner, �??Noise-free normalized fringe patterns and local pixel transforms for strain extraction,�?? Appl. Opt. 35, 3783-90 (1996). [CrossRef] [PubMed]
  7. J.L.Marroquin, J.E.Figueroa and M.Servin, �??Robust quadrature filters,�?? J. Opt. Soc. Am. A 14, 779-791 (1997). [CrossRef]
  8. J.L.Marroquin, M.Servin and R.Rodriguez-Vera, �??Adaptive quadrature filters and the recovery of phase from fringe pattern images,�?? J. Opt. Soc. Am. A 14, 1742-1753 (1997). [CrossRef]
  9. Douglas R. Schmitt and R. W. Hunt, �??Optimization of fringe pattern calculation with direct correlations in speckle interferometry,�?? Appl. Opt. 36, 8848-8857 (1997). [CrossRef]
  10. Yuming He, �??Digital shearing speckle corelation fringes pattern formed by using linear correlation calculating method,�?? Acta Photonica Sinica 24, 13-17 (1995).
  11. C. Kao, G.Yeh, S. Lee, C. Yang and K. Wu, "Phase-shifting algorithms for electronic speckle pattern interferometry,�?? Appl. Opt. 41, 46-54 (2002). [CrossRef] [PubMed]
  12. Qifeng Yu, Sihua Fu, Xia Yang, Xiangyi Sun and Xiaolin Liu, �??Extraction of phase field from a single contoured correlation fringe pattern of ESPI,�?? Opt. Express 12, 75-83 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-75">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-75</a>. [CrossRef] [PubMed]
  13. J.C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, Berlin, 1975), Chap.2.
  14. Qifeng Yu, X. Sun and X. Liu, �??Removing speckle noise and skeleton extraction from a single speckle fringe pattern,�?? Opt. Eng. 42, 68-74 (2003). [CrossRef]
  15. Qifeng Yu, X. Sun and X. Liu, �??Spin filtering with curve windows for interferometric fringes,�?? Appl. Opt. 41, 2650-2654 (2002). [CrossRef] [PubMed]

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