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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 2 — Jan. 17, 2011
  • pp: 606–615
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Dynamic phase measurement in shearography by clustering method and Fourier filtering

Yuanhao Huang, Farrokh Janabi-Sharifi, Yusheng Liu, and Y. Y. Hung  »View Author Affiliations


Optics Express, Vol. 19, Issue 2, pp. 606-615 (2011)
http://dx.doi.org/10.1364/OE.19.000606


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Abstract

Quantitative phase extraction is a key step in optical measurement. While phase shifting technique is widely employed for static or semi-static phase measurement, it requires several images with known phase shifts at each deformed stage, thus is not suitable for dynamic phase measurement. Fourier transform offer a solution to extract phase information from a single fringe pattern. However, a high frequency spatial carrier which is sometimes not easy to generate is required to solve the phase ambiguity problem. In this paper, we aim to propose an ideal solution for dynamic phase measurement. Four images with known phase shift are captured at the reference stage to analyze the initial phase information. After the object starts continuous deformation, only one image is captured at each deformed stage. A clustering phase extraction method is then applied for deformation phase extraction utilizing the phase clustering effect within a small region. This method works well for speckle image with low and medium fringe density. When the fringe density is high, especially in the case of shearographic fringe, information insufficiency inherent with merely one deformed speckle image often results in poor quality wrapped phase map with plenty of phase residues, which make phase unwrapping a difficult task. In the light of this limitation, a Fourier transform based phase filtering method is proposed for fringe frequency analysis and adaptive filtering, and effectively removes most of the phase residues to reconstruct a high quality wrapped phase map. Several real experiments based on shearography are presented. Comparison between the proposed solution and standard phase evaluation methods is also given. The results demonstrate the effectiveness of the proposed integrated dynamic phase extraction method.

© 2011 OSA

1. Introduction

Optical measurement [1

1. P. K. Rastogi, ed., Photomechanics (Springer, Berlin, 2000).

] can be generally divided into interferometric methods and triangulation methods. Holographic interferometry, shearography, speckle method, moiré interferometry and photoelasticity belong to the former while techniques such as shadow moiré and fringe projection belong to the latter. In general, interferometric methods provide a much higher accuracy than triangulation methods and are suitable for micro- and nanoscale measurement. Triangulation methods, on the other hand, have a larger measurement range and can be flexibly applied for measurement ranging from microns to kilometers.

Among the interferometric techniques, shearography [2

2. Y. Y. Hung, Y. S. Chen, S. P. Ng, L. Liu, Y. H. Huang, B. L. Luk, R. W. L. Ip, C. M. L. Wu, and P. S. Chung, “Review and comparison of shearography and active thermography for nondestructive evaluation,”, ” Mater. Sci. Eng. Rep. 64(5-6), 73–112 (2009). [CrossRef]

,3

3. L. X. Yang, W. Steinchen, G. Kupfer, P. Mackel, and F. Vossing, “Vibration analysis by means of digital shearography,” Opt. Lasers Eng. 30(2), 199–212 (1998). [CrossRef]

] owns the valuable property of not requiring a reference laser beam. This leads to a simpler optical setup and relaxes the stringent environmental stability requirement. Another feature of shearography is that it enables direct measurement of surface strain instead of displacement. These advantages render shearography a practical technique for nondestructive testing and evaluation. With recent developments, shearography has gained wide acceptance in rubber industry for routine tire testing and aerospace industry for structural integrity evaluation. Other applications of shearography include vibration characterization, residual stress evaluation, 3-D shape measurement and leakage detection.

A key step in applying shearography for nondestructive evaluation is phase retrieval. Phase, which represents the desired physical quantity such as displacement, strain and stress, is usually modulated into sinusoidal intensity variation which is called fringes. Phase retrieval from fringes pattern with abundant speckle noise is a lasting topic in optical measurement. Phase shifting and Fourier transform are methods commonly employed for quantitative phase measurement. Accurate as it is, the phase shifting method [4

4. S. Zhang, D. Van Der Weide, and J. Oliver, “Superfast phase-shifting method for 3-D shape measurement,” Opt. Express 18(9), 9684–9689 (2010). [CrossRef] [PubMed]

,5

5. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22(16), 1268–1270 (1997). [CrossRef] [PubMed]

] requires the object under measurement remains static while capturing the phase-shifted fringe patterns. This renders a huge barrier for dynamic applications. Fourier Transform technique [6

6. C. Quan, C. J. Tay, and Y. H. Huang, “3-D deformation measurement using fringe projection and digital image correlation,” Optik (Stuttg.) 115(4), 164–168 (2004). [CrossRef]

], on the other hand, requires additional equipments to generate a high frequency carrier fringe, which either makes the experimental setup complicated or sometimes is not achievable. Temporal phase analysis [7

7. C. J. Tay, C. Quan, Y. Fu, and Y. H. Huang, “Instantaneous velocity displacement and contour measurement by use of shadow moiré and temporal wavelet analysis,” Appl. Opt. 43(21), 4164–4171 (2004). [CrossRef] [PubMed]

] using Wavelet transform has also been proposed for phase evaluation. However, a series of images need to be captured and a temporal carrier is also required to solve the phase ambiguity problem. Recently, there has been tremendous research effort on phase retrieval based on a single fringe pattern [8

8. K. M. Servin, J. L. Marroquin, and F. J. Cuevas, “Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms,” J. Opt. Soc. Am. A 18(3), 689–695 (2001). [CrossRef]

14

14. H. X. Wang and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern,” Opt. Express 17(17), 15118–15127 (2009). [CrossRef] [PubMed]

]. These methods try to solve the inherent ambiguity problem using smoothness and minimum-variation assumptions. Artful routing has to be implemented to avoid erroneous phase propagation, which makes the algorithm complex and less efficient. In the light of the aforementioned limitations, a convenient phase retrieval method without phase ambiguity problem would be desirable for dynamic shearographic phase measurement.

2. The clustering phase extraction method

2.1. Theoretical background

In holography and shearography, a general speckle pattern resulted from random interference of laser can be expressed as:

I(x,y)=a(x,y)+b(x,y)cos[φ(x,y)]
(1)

where I(x,y)is the intensity of the speckle, a(x,y) and b(x,y) are the background and fringe modulation terms respectively, and φ(x,y) is initial random speckle phase. For most applications in shearography, the deformation is in the range of microns and it is reasonable to assume that background and modulation are constant during the deformation. Thus the speckle intensity I(x,y) after a small deformation can be expressed as:

I(x,y)=a(x,y)+b(x,y)cos[φ(x,y)+Δ(x,y)]
(2)

Δ=±arccos[(Ia)/b]φ+2nπ
(3)

where a,and φ can be determined by four-step phase shifting technique before the object starts to deform, and n is an integer.

As we know, shearographic phase represents continuous deformation without abrupt jumps, thus it would be reasonable to assume that the phase values within a small area are close to each other. In Eq. (3), there are two candidate phase values for each pixel. To pick up the correct phase value, an area of 3x3 pixels surrounding the specified pixel is chosen and the corresponding 18 candidate phase values determined from Eq. (3) are queued as x0,x2x17in ascending sequence within the range of π to π (if some phase values are out of this range, adjust n in Eq. (3) to make them in range) as shown in Fig. 1
Fig. 1 Demonstration of clustering effect of wrapped phase values within an area of 3x3 pixels.
. Within these 18 phase values (represented by hollow dots), it can be observed that 9 correct phase data cluster together near the arrow position, while the other 9 wrong phase values distribute randomly. This observation inspires an algorithm to determine the clustering centre to represent the correct wrapped phase value for the specific pixel.

For this purpose, a wrapped phase distance function d(xm,xn) is first defined as:

d(xm,xn)={|xmxn|if|xmxn|π2π|xmxn|if|xmxn|>π}xm,xn[π,π)
(4)

The physical meaning of this distance function is well illustrated in Fig. 1 where the circle fromπ to π indicates the periodic feature of sinusoidal functions and the phase distances are measured as the shortest arc length. In this specific definition, the distance between x1=0.85π and x16=0.75π (see Fig. (1)) is not 1.6π but 0.4π. This is reasonable since x1 is quite close to x16 if the 2π jump which has not yet been settled by a phase unwrapping algorithm, is ignored. By using this wrapped phase distance function, a parameter calledSumDist(xk), which represents the accumulated distance of a point xk with the surrounding 8 points (4 to the left and 4 to the right), is calculated for each point as:

SumDist(xk)=i=44d(xk,xk+i)k=0,1,17
(5)

In this calculation, if index k+i exceeds 17 or is less than 0, it is subtracted or added by 18 to round it within 0 to 17. It is evident that if point xk resides within the cluster, the accumulated distance SumDist(xk) will be much smaller than points outside the cluster. Thus by looking for a smallest accumulated distance, the clustering centre xc is found as:

xc={xk|SumDist(xk)SumDist(xi);ik}
(6)

The clustering centre xc determined by Eq. (6) is an approximation of the wrapped phase in the center of the 3x3 pixels. It should be noted that the wrapped phase thus determined presents no phase ambiguity problem since the clustering algorithm has choose the correct phase value from Eq. (3). However, it should also be noted that a predetermined initial random phase φ without ambiguity is a necessity for the success of the clustering method. If φ is subject to phase ambiguity problem (i.e. φ is determined from arccosine function instead of phase shifting method), then the candidate phase values determined from Eq. (3) will have two clustering centers, making the resultant wrapped phase subject to phase ambiguity.

2.2. Comparison with phase shifting method

Shearographic experiment data have been used to verify the effectiveness of the proposed clustering phase extraction method. Figure 2(a)
Fig. 2 Typical shearographic fringe pattern of a central loaded rectangular plate (a) and the corresponding wrapped phase obtained by the clustering approach (b)
shows a typical shearographic fringe pattern which depicts the out-of-plane displacement derivative of a fully clamped rectangular plate subjected to a central loading. Figure 2(b) shows the wrapped phase map determined using the proposed clustering approach. It can be seen that the wrapped phase map agree well with fringe pattern without any phase ambiguity.

Quantitative comparison between the proposed clustering method and standard four-step phase shifting method has also been conducted. Both wrapped phase maps from clustering method and phase shifting method are unwrapped and the horizontal middle sections are plotted together as shown in Fig. 3
Fig. 3 Comparison of unwrapped phase maps of the clustering method and four-step phase shifting at the horizontal mid-section.
. It can be seen that the two cross section profiles agree pretty well with each other. Quantitative comparison shows that the maximum discrepancy is less than 0.8%. Thus it is confirmed that the proposed clustering method can effectively retrieve the phase map with accuracy similar to phase shifting method. While the proposed clustering method obtains similar wrapped phase map as standard four-step phase-shifting, a distinct advantage is that only one specklegram is required at each deformed stage. This advantage renders the proposed clustering method an ideal solution for continuous deformation measurement.

3. The Fourier phase filtering method

The quality of wrapped phase maps depends on several factors such as environmental noise, signal to noise ratio and resolution of the imaging element (normally CCD cameras), redundancy of information available, and the fringe density. Normally the initial noisy wrapped phase maps determined from either clustering method or phase shifting method can be improved by a fringe averaging method, which first converts the wrapped phase to both sine and cosine fringe maps, then averages the fringe maps on a small area (i.e. 3x3 pixels), and finally reconstructs a filtered wrapped phase map using the averaged sine and cosine fringe maps. However, for areas with rapid phase change, the clustering effect is not distinct and the clustering method may result in incorrect phase values which lead to severe phase residues after successive fringe averaging. Figure 4(a)
Fig. 4 (a) Initial wrapped phase map determined from the clustering method; (b) Wrapped phase map after 50 times fringe averaging; (c) The central area of (b) showing plenty of phase residues; (d) unwrapped phase map of (b) using Raster unwrapping algorithm.
shows an initial wrapped phase map determined by the clustering method. This wrapped phase map is subject to large amount of noise and difficult to unwrap. Figure 4(b) shows a clean wrapped phase map after applying the fringe averaging method for noise removal. However, in the central area indicated by a square and highlighted in Fig. 4(c), it can be seen that there are plenty of phase residues remain. This poses much difficulty for the subsequent phase unwrapping process. Figure 4(d) shows the phase unwrapping result using Raster’s unwrapping algorithm [17

17. C. G. Dennis, and D. P. Mark, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software, (Wiley, New York, 1998).

]. It can be seen that erroneous phase values propagate and damage the whole unwrapped phase map.

The phase residues in Fig. 4(c) cannot be removed by either fringe averaging or fringe median filtering. In fact, successive averaging and median filtering will only make the situation worse by enlarging the erroneous area. Noted in Fig. 4(c) that the fringe frequency is much lower than the residue frequency, we propose a frequency based Fourier filtering method to remove the unwanted phase residues while conserving the desired fringe information in the wrapped phase.

The wrapped phase map in Fig. 4(a) is converted to fringe patterns using both cosine and sine operations. In this process, the high frequency 2π phase jumps in wrapped phase map are replaced by low frequency continuous fringes due to the periodic feature of cosine and sine functions. The resultant fringe patterns f(x,y) and g(x,y)can be expressed as:

f(x,y)=cos(Δ(x,y))+r(x,y)g(x,y)=sin(Δ(x,y))+r(x,y)
(7)

Where cos(Δ(x,y))and sin(Δ(x,y)) corresponding to a low frequency fringe containing the deformation information, and r(x,y),r(x,y) corresponding to the high frequency phase residues.

A typical rewrapped fringe pattern is shown in Fig. 5(a)
Fig. 5 (a) Cosine fringe map obtained from the wrapped phase map determined from clustering method; (b) Fourier spectrum of (a); (c) low-pass filter determined by thresholding; (d) reconstructed cosine fringe map after Fourier filtering; (e) reconstructed wrapped phase map after Fourier phase filtering; (f) unwrapped phase map by Raster unwrapping algorithm.
. When applying a two-dimensional Fourier transformation to the fringe patterns, we have:

F(u,v)=12δ(uΔx2π,vΔy2π)+12δ(u+Δx2π,v+Δy2π)+R(u,v)G(u,v)=i2δ(u+Δx2π,v+Δy2π)i2δ(uΔx2π,vΔy2π)+R(u,v)
(8)

whereδ(u,v)is a two-dimensional Dirac delta function, Δx,Δyare the fringe frequency components in x,ydirections respectively, andR(u,v),R(u,v)are the Fourier transformation of high frequency residues. For a typical rewrapped shearographic fringe pattern shown in Fig. 5(a), the fringe frequencies Δx,Δyare confined to a small range and can be separated from the high frequency residues. Figure 5(b) shows the Fourier spectrum (complex modulus ofF(u,v)) of Fig. 5(a). The bright central ellipse corresponds to the low frequency fringe pattern while the residue spectrum distribute mostly at higher frequency area. Since the fringe pattern dominates in Fig. 5(a), the Fourier energy for fringe pattern is much larger than that for the residues. Thus the central bright part can be easily separated using feature extraction methods or thresholding method. In this study, a simple thresholding operation is applied to Fig. 5(b) with a default threshold value of 10/255 of the maximum power. Then the two binary images resulted from cosine and sine spectrums are combined. After that a dilation operation is applied and the resultant frequency filter is shown in Fig. 5(c). It can be seen that this low-pass filter contains the central bright spectrum, but seems a little bit larger than necessary and can be further improved. However, the filtering effect using this filter is quite satisfying already. By applying this filter and subsequently an inverse Fourier transform to Eq. (8), clean cosine and sine fringe patterns are reconstructed as follows:

f^(x,y)=ifft{F(u,v)*flt(u,v)}g^(x,y)=ifft{G(u,v)*flt(u,v)}
(9)

where ifft{}represent the inverse Fourier transform, ∗is pixel-wise product operation, and flt(u,v) is the low-pass filter shown in Fig. 5(c). A typical reconstructed fringe pattern using Eq. (9) is shown in Fig. 5(d). It can be observed that most of the noise has been removed and the original noisy fringe pattern has been smoothed. Based on both the reconstructed cosine and sine fringe maps, a new wrapped phase map is reconstructed as follows:

Δ(x,y)=arctan2(g^(x,y),f^(x,y))
(10)

where the pixel-wise function arctan2(y,x)retrieves a phase angle in the range of [π,π)from complex number x+yi. Figure 5(e) shows the reconstructed wrapped phase map after the whole Fourier filtering process. For clearer viewing, Fig. 5(e) is re-plotted in Fig. 6
Fig. 6 Reconstructed wrapped phase map by the proposed clustering method and Fourier phase filtering method.
. It can be seen that a noise-free wrapped phase map has been obtained. Any simple phase unwrapping algorithms can now be applied to easily obtain the unwrapped phase map. The result using Raster unwrapping algorithm is shown in Fig. 5(f).

It should be noted that phase residues is not a specific problem for clustering method. For standard phase shifting method where abundant information are available, the resultant phase maps may also be subject to phase residues in the cases of high phase gradient or large amount of environmental noise. In such cases, the proposed Fourier phase filtering method is an ideal tool to remove phase residues in wrapped phase map, and making the subsequent phase unwrapping process simple. It should also be noted that this Fourier filtering method is quite tolerant to change in parameters. For the example in Fig. 5, thresholding values ranging from 8/255 to 15/255 of the maximum power all result in acceptable filtering effect. It is also possible to design a more precise adaptive filter using feature extraction method based on the sudden changing of spectrum power between the high-energy central ellipse and the low-energy surrounding area. This more precise adaptive filter would be desirable for wrapped phase map with worse quality.

In our experimental study using shearographic setup on tripod, the noise level would be larger than interferometric systems placed on vibration isolated optical table. Under such situation, the maximum fringe density workable for our proposed combined method is about 12 pixels per fringe. When the deformation becomes larger and larger, the fringe would be too dense to accurately extract the phase. It would be thus desirable to capture intermediate speckle patterns to reduce the fringe density, and then combine the incremental phase change to deliver the final phase map. We will explain more details and give more results to handle such situations in a forthcoming paper.

4. Dynamic phase measurement experiment

5. Conclusions

A clustering method has been proposed for ambiguity-free phase extraction from one single specklegram. A Fourier phase filtering method is further proposed for efficient removal of phase residues to facilitate phase unwrapping process. The integrated method opens a door for simple and reliable dynamic phase measurement. With further refinement, the proposed methods will have great potential for practical applications, especially in aircraft and automobile industry for vibration characterization.

Acknowledgments

This work was financially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) through the Discovery Grant #203060-07 and the Ryerson University Post Doctoral Fellowship Award.

References and links

1.

P. K. Rastogi, ed., Photomechanics (Springer, Berlin, 2000).

2.

Y. Y. Hung, Y. S. Chen, S. P. Ng, L. Liu, Y. H. Huang, B. L. Luk, R. W. L. Ip, C. M. L. Wu, and P. S. Chung, “Review and comparison of shearography and active thermography for nondestructive evaluation,”, ” Mater. Sci. Eng. Rep. 64(5-6), 73–112 (2009). [CrossRef]

3.

L. X. Yang, W. Steinchen, G. Kupfer, P. Mackel, and F. Vossing, “Vibration analysis by means of digital shearography,” Opt. Lasers Eng. 30(2), 199–212 (1998). [CrossRef]

4.

S. Zhang, D. Van Der Weide, and J. Oliver, “Superfast phase-shifting method for 3-D shape measurement,” Opt. Express 18(9), 9684–9689 (2010). [CrossRef] [PubMed]

5.

I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22(16), 1268–1270 (1997). [CrossRef] [PubMed]

6.

C. Quan, C. J. Tay, and Y. H. Huang, “3-D deformation measurement using fringe projection and digital image correlation,” Optik (Stuttg.) 115(4), 164–168 (2004). [CrossRef]

7.

C. J. Tay, C. Quan, Y. Fu, and Y. H. Huang, “Instantaneous velocity displacement and contour measurement by use of shadow moiré and temporal wavelet analysis,” Appl. Opt. 43(21), 4164–4171 (2004). [CrossRef] [PubMed]

8.

K. M. Servin, J. L. Marroquin, and F. J. Cuevas, “Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms,” J. Opt. Soc. Am. A 18(3), 689–695 (2001). [CrossRef]

9.

Q. F. Yu, S. H. Fu, X. Yang, X. Y. Sun, and X. L. Liu, “Extraction of phase field from a single contoured correlation fringe pattern of ESPI,” Opt. Express 12(1), 75–83 (2004). [CrossRef] [PubMed]

10.

Q. F. Yu, S. H. Fu, X. L. Liu, X. Yang, and X. Y. Sun, “Single-phase-step method with contoured correlation fringe patterns for ESPI,” Opt. Express 12(20), 4980–4985 (2004). [CrossRef] [PubMed]

11.

K. G. Larkin, “Uniform estimation of orientation using local and nonlocal 2-D energy operators,” Opt. Express 13(20), 8097–8121 (2005). [CrossRef] [PubMed]

12.

J. C. Estrada, M. Servin, and J. L. Marroquín, “Local adaptable quadrature filters to demodulate single fringe patterns with closed fringes,” Opt. Express 15(5), 2288–2298 (2007). [CrossRef] [PubMed]

13.

O. Dalmau-Cedeño, M. Rivera, and R. Legarda-Saenz, “Fast phase recovery from a single close-fring pattern,” J. Opt. Soc. Am. A 25(6), 1361–1370 (2008). [CrossRef]

14.

H. X. Wang and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern,” Opt. Express 17(17), 15118–15127 (2009). [CrossRef] [PubMed]

15.

Y. H. Huang, S. P. Ng, L. Liu, Y. S. Chen, and Y. Y. Hung, “Shearographic phase retrieval using one single specklegram: a clustering approach,” Opt. Eng. 47(5), 054301 (2008). [CrossRef]

16.

Y. H. Huang, S. P. Ng, L. Liu, C. L. Li, Y. S. Chen, and Y. Y. Hung, “NDT&E using shearography with impulsive thermal stressing and clustering phase extraction,” Opt. Lasers Eng. 47(7-8), 774–781 (2009). [CrossRef]

17.

C. G. Dennis, and D. P. Mark, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software, (Wiley, New York, 1998).

18.

L. J. Chen, C. G. Quan, C. J. Tay, and Y. H. Huang, “Fringe contrast-based 3D profilometry using fringe projection,” Optik (Stuttg.) 116(3), 123–128 (2005). [CrossRef]

19.

Q. Kemao, S. H. Soon, and A. Asundi, “Smoothing filters in phase-shifting interferometry,” Opt. Laser Technol. 35(8), 649–654 (2003). [CrossRef]

20.

K. Qian, H. S. Seah, and A. Asundi, “Filtering the complex field in phase shifting interferometry,” Opt. Eng. 42(10), 2792–2793 (2003). [CrossRef]

21.

A. Dávila, G. H. Kaufmann, and D. Kerr, “Scale-space filter for smoothing electronic speckle pattern interferometry fringes,” Opt. Eng. 35(12), 3549–3554 (1996). [CrossRef]

22.

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45(2), 304–317 (2007). [CrossRef]

23.

K. M. Qian and S. H. Soon, “Two-dimensional windowed Fourier frames for noise reduction in fringe pattern analysis,” Opt. Eng. 44(7), 075601 (2005). [CrossRef]

24.

W. J. Gao, N. T. T. Huyen, H. S. Loi, and Q. Kemao, “Real-time 2D parallel windowed Fourier transform for fringe pattern analysis using Graphics Processing Unit,” Opt. Express 17(25), 23147–23152 (2009). [CrossRef]

OCIS Codes
(090.2880) Holography : Holographic interferometry
(100.2650) Image processing : Fringe analysis
(100.3020) Image processing : Image reconstruction-restoration
(100.5070) Image processing : Phase retrieval
(120.5060) Instrumentation, measurement, and metrology : Phase modulation
(070.2615) Fourier optics and signal processing : Frequency filtering

ToC Category:
Image Processing

History
Original Manuscript: December 3, 2010
Revised Manuscript: December 3, 2010
Manuscript Accepted: December 20, 2010
Published: January 5, 2011

Citation
Yuanhao Huang, Farrokh Janabi-Sharifi, Yusheng Liu, and Y. Y. Hung, "Dynamic phase measurement in shearography by clustering method and Fourier filtering," Opt. Express 19, 606-615 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-2-606


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References

  1. P. K. Rastogi, ed., Photomechanics (Springer, Berlin, 2000).
  2. Y. Y. Hung, Y. S. Chen, S. P. Ng, L. Liu, Y. H. Huang, B. L. Luk, R. W. L. Ip, C. M. L. Wu, and P. S. Chung, “Review and comparison of shearography and active thermography for nondestructive evaluation,”, ” Mater. Sci. Eng. Rep. 64(5-6), 73–112 (2009). [CrossRef]
  3. L. X. Yang, W. Steinchen, G. Kupfer, P. Mackel, and F. Vossing, “Vibration analysis by means of digital shearography,” Opt. Lasers Eng. 30(2), 199–212 (1998). [CrossRef]
  4. S. Zhang, D. Van Der Weide, and J. Oliver, “Superfast phase-shifting method for 3-D shape measurement,” Opt. Express 18(9), 9684–9689 (2010). [CrossRef] [PubMed]
  5. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22(16), 1268–1270 (1997). [CrossRef] [PubMed]
  6. C. Quan, C. J. Tay, and Y. H. Huang, “3-D deformation measurement using fringe projection and digital image correlation,” Optik (Stuttg.) 115(4), 164–168 (2004). [CrossRef]
  7. C. J. Tay, C. Quan, Y. Fu, and Y. H. Huang, “Instantaneous velocity displacement and contour measurement by use of shadow moiré and temporal wavelet analysis,” Appl. Opt. 43(21), 4164–4171 (2004). [CrossRef] [PubMed]
  8. K. M. Servin, J. L. Marroquin, and F. J. Cuevas, “Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms,” J. Opt. Soc. Am. A 18(3), 689–695 (2001). [CrossRef]
  9. Q. F. Yu, S. H. Fu, X. Yang, X. Y. Sun, and X. L. Liu, “Extraction of phase field from a single contoured correlation fringe pattern of ESPI,” Opt. Express 12(1), 75–83 (2004). [CrossRef] [PubMed]
  10. Q. F. Yu, S. H. Fu, X. L. Liu, X. Yang, and X. Y. Sun, “Single-phase-step method with contoured correlation fringe patterns for ESPI,” Opt. Express 12(20), 4980–4985 (2004). [CrossRef] [PubMed]
  11. K. G. Larkin, “Uniform estimation of orientation using local and nonlocal 2-D energy operators,” Opt. Express 13(20), 8097–8121 (2005). [CrossRef] [PubMed]
  12. J. C. Estrada, M. Servin, and J. L. Marroquín, “Local adaptable quadrature filters to demodulate single fringe patterns with closed fringes,” Opt. Express 15(5), 2288–2298 (2007). [CrossRef] [PubMed]
  13. O. Dalmau-Cedeño, M. Rivera, and R. Legarda-Saenz, “Fast phase recovery from a single close-fring pattern,” J. Opt. Soc. Am. A 25(6), 1361–1370 (2008). [CrossRef]
  14. H. X. Wang and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern,” Opt. Express 17(17), 15118–15127 (2009). [CrossRef] [PubMed]
  15. Y. H. Huang, S. P. Ng, L. Liu, Y. S. Chen, and Y. Y. Hung, “Shearographic phase retrieval using one single specklegram: a clustering approach,” Opt. Eng. 47(5), 054301 (2008). [CrossRef]
  16. Y. H. Huang, S. P. Ng, L. Liu, C. L. Li, Y. S. Chen, and Y. Y. Hung, “NDT&E using shearography with impulsive thermal stressing and clustering phase extraction,” Opt. Lasers Eng. 47(7-8), 774–781 (2009). [CrossRef]
  17. C. G. Dennis, and D. P. Mark, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software, (Wiley, New York, 1998).
  18. L. J. Chen, C. G. Quan, C. J. Tay, and Y. H. Huang, “Fringe contrast-based 3D profilometry using fringe projection,” Optik (Stuttg.) 116(3), 123–128 (2005). [CrossRef]
  19. Q. Kemao, S. H. Soon, and A. Asundi, “Smoothing filters in phase-shifting interferometry,” Opt. Laser Technol. 35(8), 649–654 (2003). [CrossRef]
  20. K. Qian, H. S. Seah, and A. Asundi, “Filtering the complex field in phase shifting interferometry,” Opt. Eng. 42(10), 2792–2793 (2003). [CrossRef]
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