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Quantitative phase retrieval in dynamic laser speckle interferometry
Optics and Lasers in Engineering 50 (2012) 534–539
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Quantitative phase retrieval in dynamic laser speckle interferometry Y.H. Huang a,c, S.Y. Hung b, Farrokh Janabi-Sharifi c, W. Wang d, Y.S. Liu a,n a
State Key Lab. of CAD&CG, Zhejiang University, Hangzhou 310027, PR China Department of Mechanical Engineering, Oakland University, USA Department of Mechanical and Industrial Engineering, Ryerson University, Canada d Department of Mechanical Engineering, Zhejiang University, Hangzhou, PR China b c
a r t i c l e i n f o
a b s t r a c t
Available online 24 August 2011
The rapid progress of modern manufacturing and inspection technologies has posed stringent requirements on optical techniques for vibration characterization and dynamic testing. Due to its simplicity, accuracy and whole-field characters, laser speckle interferometry has served as one of the major techniques for dynamic measurement. In this paper, a two-step phase shifting method is developed for quantitative speckle phase measurement, which helps to eliminate the specklegrams needed for phase evaluation and facilitate dynamic measurement. Unlike previously reported two-step methods using fringe patterns with known phase shift of p/2, a small unknown phase shift is employed instead in the proposed method, which eliminates the need for phase shifting devices. Further investigation shows that small phase shifts are preferable over large phase shifts in this method. Shearographic experiments conducted have demonstrated the effectiveness of the proposed technique. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Shearography Phase shifting Dynamic measurement Speckle metrology
1. Introduction Speckle interferometry techniques such as electronic speckle pattern interferometry [1] and shearography [2] have been widely accepted in laboratories as well as industries for deformation, strain measurement and vibration characterization. Normally the desired physical quantities, which are related to the phase information in some forms dependent on the optical configuration, are encoded in fringe patterns in cosine form. One of the most popular methods for quantitative phase retrieval is phase shifting technique [3] where three or more interferograms with known phase shifts are captured and processed pixel-wise to determined the three unknowns for each pixel. In general, more interferograms will provide abundant information and make the resultant phase map more accurate and noise-free. However, when dealing with dynamic measurement, the testing object is subject to continuous movement, which makes it difficult to capture several interferograms with predetermined phase shifts. Even in semi-static measurement, hostile industrial circumstance will introduce environmental noise such as vibration and air refractive index fluctuation, which will introduce phase drifting over time and result in error in phase measurement. Thus phase retrieval techniques requiring less interferograms would be desirable in dynamic measurement or under uncontrollable environmental noise.
n
Corresponding author. E-mail address: [email protected] (Y.S. Liu).
0143-8166/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2011.06.025
In history, researchers have proposed various two-step phase shifting algorithms [4–11] to accommodate this need. However, most of algorithms deal with digital holography [4], clean fringe patterns from fringe projection [5,7], or speckle-noise-free interferometry fringe [7]; and only a few papers [8–11] attempt to extract phase information from speckle interferometric fringe patterns overwhelmed by speckle noise. In their works [8–11], the researchers try to remove speckle noise and the background intensity using various filters and reconstruct the desired phase using two clean fringe patterns with p/2 phase shift. In this paper, we propose an alternative two-step phase shifting method for phase retrieval in speckle interferometry. Different from previous techniques, the proposed method is based on speckle patterns analysis instead of fringe patterns normalization. Another distinct feature of this two-step method is that it does not need to know the exact amount of the phase shift. All it requires is a small unknown phase shift (i.e. r p/3) between two speckle patterns to solve for the phase ambiguity problem. The unknown phase shifts can be obtained by capturing specklegrams of a continuously moving object, thus no phase shifting devices such as PZT mirror or liquid crystal phase retarder are required, which makes this method extremely simple and practical. In Section 2, the theoretical background for the two-step method is given. In Section 3, two shearographic experiments are introduced. In Section 4, the phase maps obtained by the two-step method are illustrated; the method is further compared with previously reported phase extraction methods. The effect of the unknown phase shift amount and other parameters is also
Y.H. Huang et al. / Optics and Lasers in Engineering 50 (2012) 534–539
evaluated theoretically and experimentally. Section 5 concludes this paper.
2. Theoretical background In speckle interferometry, a speckle pattern can be generally represented as Iðx,yÞ ¼ aðx,yÞ þbðx,yÞcos½jðx,yÞ
ð1Þ
where a(x,y), b(x,y) and j(x,y) are pixel-wise background intensity, modulation and random phase, respectively. Conventional phase shifting techniques tend to introduce three or more known phase shifts to j(x,y) in Eq. (1) using various phase shifters and solve for the three knowns using optimization methods. The phase information j(x,y) obtained in this stage is an initial random phase resulting from random interference between the reference and object beam reflected off an optically rough surface. After the testing object undergoes a deformation, another speckle pattern is digitized as I0 ðx,yÞ ¼ aðx,yÞ þ bðx,yÞcos½jðx,yÞ þ Dðx,yÞ
ð2Þ
where D(x,y) is a phase change related to the deformation in some form depending on the optical configuration. The same phase shifting technique can be applied again to deliver another random phase j(x,y) þ D(x,y). The subtraction between this deformed phase and the initial random phase j(x,y) will result in a sawteeth wrapped phase map D(x,y), which contains the desired deformation information. According to our study and previous reports [12,13] on speckle statistics and properties, the random phase j(x,y), which has a range of ( p,p], is a zero-mean highly random value over any small area of the image plane. However, the background a(x,y)and modulation b(x,y) is found to be much more constant on the whole image, especially within an individual speckle. As an evidence, the mean value of j(x,y) is found to be 0.0625 rad and the standard deviation is 1.8144 rad; while that for a(x,y) are 140.3841 and 28.8485(gray value), and that for b(x,y) are 27.1455and 15.1677 in one of our typical experiment. Based on this observation, if we make a local average processing (i.e. on 3 3 pixels) on the speckle pattern shown in Eq. (1), since a(x,y) and b(x,y) are almost constant within such a small area (this area is dependent on speckle size. If the speckle size is less than 3 pixels, one can choose a smaller aperture to obtain bigger speckles) while j(x,y) change abruptly within ( p,p], the second term of Eq. (1) tends to be zero-mean while the first term is kept. Thus it would be reasonable to make an estimation of the pixel-wise values of a(x,y) as follows: i ¼X 1,j ¼ 1
_ aðx,yÞ ¼ 1=9
Iðx þ i,yþ jÞ
ð3Þ
i ¼ 1,j ¼ 1
On the other hand, if we subtract the newly obtained background _ aðx,yÞfrom Eq. (1), then we get an approximation for bðx,yÞcos½jðx,yÞ. Again, b(x,y) is almost constant within a small area while j(x,y) is highly random. Thus if we choose the maximum absolute value of bðx,yÞcos½jðx,yÞ over a small area (i.e. 5 5 pixels), we would get a reasonable approximation of b(x,y) since one of the highly random j value may hit the extreme positions for cosðjÞ. Thus b(x,y) can be approximated as _ bðx,yÞ ¼
i ¼ 2,j ¼ 2
Max
i ¼ 2,j ¼ 2
9Iðx þ i,y þ jÞ_ aðx þ i,y þ jÞ9
ð4Þ
where the function Max( ) is designed to extract the maximum _ value. Since we have obtained both estimations _ aðx,yÞ and bðx,yÞ for approximating the speckle background and modulation, we can now obtain the absolute value of initial random phase j(x,y)
535
from Eq. (1) as _ 9jðx,yÞ9 ¼ arccosf½Iðx,yÞ_ aðx,yÞ=bðx,yÞg
ð5Þ
where arccosðUÞ is the normal arccosine function. The sign information for j(x,y) is lost in the speckle pattern due to the even function property of cosine operation, thus additional information is required to solve this sign ambiguity problem and choose a correct initial phase from the absolute value. We solve this problem by introducing a second speckle pattern Id(x,y), which has a small phase change d(x,y) relative to Eq. (1): Id ðx,yÞ ¼ aðx,yÞ þ bðx,yÞcos½jðx,yÞ þ dðx,yÞ
ð6Þ
If d(x,y) is a positive small number (for negative d, similar derivation can be obtained), the term cos½jðx,yÞ þ dðx,yÞ in Eq. (6) can be approximated as cos½jðx,yÞdðx,yÞsin½jðx,yÞ by ignoring the higher order terms in the Taylor expansion. Subtracting Eq. (6) from Eq. (1) will result in an intensity of Isub ðx,yÞ ¼ Iðx,yÞId ðx,yÞ bðx,yÞdðx,yÞsin½jðx,yÞ
ð7Þ
Since b(x,y) and d(x,y) are positive values and sinðUÞis an odd function, the sign of j(x,y) in Eq. (5) is now determined to be the same as the sign of Isub(x,y). Thus, an unambiguous initial random phase j(x,y) is now fully determined as _ jðx,yÞ ¼ acosf½Iðx,yÞ_ aðx,yÞ=bðx,yÞg signðIsub ðx,yÞÞ ð8Þ where function sign( )extracts the positive or negative sign information of an input value. Till now an unambiguous random phase has been determined from two speckle patterns with a small phase shift. After the object undergoes a deformation, another two speckle patterns with small phase shift can be used again to determine the deformed random phase [j(x,y) þ D(x,y)]. The subtraction between the initial and deformed random phase will then give a wrapped phase map D(x,y)representing the deformation.
3. Experiment work To verify the proposed two-step phase shifting method for phase evaluation, shearographic experiments are conducted. Fig. 1 shows a typical shearographic setup for strain and displacement derivative determination. The object under testing is illuminated by an expanded coherent laser light. The rough surface of the object reflects laser light to produce an interference speckle image via a liquid crystal phase retarder (phase shifting device) and a Wollaston prism (image shearing device). By incorporating the image shearing device, laser light scattered from neighboring points M and B are brought to interfere on the CCD chip and the relevant optical path difference of points M and B are recorded in terms of phase. The relationship between the phase and deformation in this setup can be found in Ref. [14]. When the object undergoes a deformation, the deformed amount can be revealed by a fringe pattern, which is obtained by subtracting the deformed speckle image from the undeformed speckle image. Quantitative evaluation of the deformation field can be fulfilled by various phase retrieval methods. Our first experiment is a static deformation experiment, which allows speckle images with accurate phase shift amounts to be captured and evaluated. This static experiment is intended to compare the two-step method with previous reported phase retrieval methods. In the experiment, a plate fully clamped by all the boundaries was loaded in the center by a micrometer (refer to Fig. 1b in Ref. [15]). Before and after the static loading, various phase shifts were introduced by the phase shifter and speckle images were captured for phase evaluation using conventional four-step phase shifting, the proposed two-step phase shifting,
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Fig. 1. Shearographic setup employed in our experiment.
and the clustering method [3] where four phase shifted images are captured before the deformation and only a single deformed image is captured after the deformation. The results from these methods are then compared to evaluate the performance of various methods. In the second experiment, a steel tube with an internal crack as shown in Fig. 6(b) of Ref. [16] was insulated and continuously pressurized by a pump. Speckle images are continuously captured with a frame rate of 30 frames per second. It is estimated that the deformation induced phase shifts between successive frames are about p/3. These successive speckle pairs are then evaluated by the two-step method to deliver the random speckle phase at each instant. Subsequent subtraction of speckle phases at different stages will then result in deformation phase. On the other hand, the phase shifters are also employed to capture speckle images with additional phase shifts of p/2. The interval between the four-step shifted images is 0.05 s. Though this is a short time, the deformation induced phase shift during this interval is already comparable with p/2. This induces a very large phase shift error, and the resultant phase map obtained from four-step phase shifting is in poor quality and unreliable.
4. Results and discussions Fig. 2 shows the result of the static experiment for the central loaded plate. Fig. 2(a) shows a wrapped phase map from four-step phase shifting method depicting the out-of-plane displacement derivative of the central loaded plate. The wrapped phase map is subjected to large amount of noise due to air fluctuation, temperature change, environmental vibration or laser drifting. However, the phase information is still visible at this stage. Fig. 2(b) shows the same wrapped phase map, but determined from the proposed two-step method. It can be observed that the inherent information deficiency in this two-step phase shifting method makes the noise more severe than conventional four-step phase shifting methods. For further comparison, a wrapped phase map determined from the clustering method [3] is also shown in Fig. 2(c). All the three wrapped phase maps at their initial stage are too noisy for phase unwrapping, thus noise filtering techniques should be employed to get a clean wrapped phase map
before phase unwrapping can be successfully applied. One effective filtering method for noise removal is to convert the wrapped phase map into cosine and sine fringe maps, then apply low-pass filters (such as 3 3 averaging filter) to suppress noise, and finally reconstruct a clean wrapped phase map by applying an arctangent operation pixel-wisely on the filtered cosine and sine fringe maps. Fig. 2(d)–(f) shows the filtered results of Fig. 2(a)–(c), respectively. It can be seen that most of the noise have been removed. However, the wrapped phase map from clustering method (Fig. 2(f)) still has plenty of phase residues in the central area with dense fringes, which will pose severe problem for subsequent phase unwrapping, while the phase map from four-step phase shifting (Fig. 2(d)) is totally residue-free and can be unwrapped using any simple algorithms. The result from the proposed two-step method (Fig. 2(e)) is in-between and a few phase residues are presented in the central area, which may be easily unwrapped by Goldstein’s branch-cut algorithm [17]. By applying the Fourier filtering method [18] to Fig. 2(e), a high quality wrapped phase map has been obtained as shown in Fig. 2(g). Quantitative comparison between Fig. 2(g) and (d) reveals that the discrepancy is less than 1%. The proposed two-step phase shifting method has been proposed for phase retrieval with a double pulse laser [19] or dealing with sparsely sampled speckle sequence [20]. In the previous papers, only small unknown phase shifts are employed, and no thorough analysis and comparison between small and large phase shifts are given. In this paper, we develop more general formula to accommodate both small and large phase shift, and verify the potential advantages of small phase shift in this method. For the proposed two-step phase shifting method, Eqs. (1)–(8) use a small unknown phase shift between two speckle patterns. What if we use a large known phase shift instead (i.e. p/2) as in previously reported papers [8–11]? Will it reduce the noise level and improve the quality of resultant wrapped phase map due to the availability of the phase shift amount? We conducted both simulations and experiments to find out the answer. For a large known phase shift, the first-order Taylor expansion in Eq. (7) is invalid. Thus a new criterion is set as follows: _ Discrepancyð_ j Þ ¼ 9b½cos_ j cosð_ j þ dÞIsub 9
ð9Þ
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Fig. 2. Wrapped phase maps determined by various methods. (a), (b) and (c) wrapped phases determined by conventional four-step phase shifting, the proposed two-step phase shifting and the clustering method [3], respectively; (d)–(f) corresponding wrapped phase maps after noise filtering, noted that two-step method (e) results in less phase residues than clustering method (f) in central areas; (i) Reconstructed wrapped phase map of (e) by adaptive Fourier filtering [18], the discrepancy to (d) is less than 1%; (g) and (h) filtered wrapped phase map obtained by Eq. (10) with phase shifts of p/3and 2p/3, respectively, worse results than (e) has been observed.
where (x,y) is omitted for simplicity and d is a known phase shift. Function Discrepancyð_ j Þ indicates the discrepancy between a phase value _ j and the true initial random phase value j, and a
j Þ indicates a better estimation. Designate smaller Discrepancyð_ the positive and negative phase value obtained by Eq. (5) as jp and jn, respectively, we then substitute them into Eq. (9) and
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choose a potentially more correct phase value by (
j¼
jp if Discrepancyðjp Þ r Discrepancyðjn Þ jn otherwise
ð10Þ
Eq. (10) is theoretically valid for both large and small phase shift as long as the phase shift amount is known. We thus simulated a series of small known phase shifts of 0.05p, 0.1p, 0.15p and 0.2p, and use both Eqs. (8) and (10) for phase determination. The results indicate that both equations give similar wrapped phase map as shown in Fig. 2(e), while Eq. (8) does not require knowing the amount of phase shift but Eq. (10) needs to know the pixel-wise phase shift. When the phase shift amount becomes larger, Eq. (8) is no longer a good approximation, thus only Eq. (10) was used. Fig. 2(h) and (i) shows the wrapped phase map obtained using Eq. (10) with simulated p/3 and 2p/3 phase shift (for experimental p/3phase shift, the result is similar to Fig. 2(h)), it can be observed that the wrapped phase map quality is poorer than Fig. 2(e), which uses a small phase shift. The reason is that for a larger phase shift, the value of _ ½cos _ j cosð_ j þ dÞ in Eq. (9) is larger, thus the error contained in b (approximation of b) is amplified, which makes the judgment _ using Eq. (10) erroneous. In other words, if b is a very good approximation of b, then small known phase shift and large known phase shift should deliver similar results. However, the inherent information deficiency in two-step phase shifting cannot guarantee a good b approximation, thus a small phase shift would
be preferred. A more thorough quantitative investigation shows that phase shifts from 0.02p to 0.33p deliver similar results to Fig. 2(e) with the maximum discrepancy of 1.69%, and that from 0.33p to 0.66p result in maximum phase error of 5.31% with noisy wrapped phase maps difficult to unwrap. After 0.66p the phase maps are becoming extremely noisy and information is totally lost at the point of 0.82p. We also used experimental obtained interferograms with 0.5p phase shift to test the proposed algorithm, and the result is quite similar to the simulated specklegrams. As the proposed two-step method uses speckle statistics to approximate the background intensity a and fringe modulation b, the effect of window size selection on a and b approximation should also be investigated. Theoretically the window size should be smaller than the speckle size as speckle statistics varies dramatically within adjacent speckle. But the window size could not be too small to achieve a good statistical approximation. Thus the optimal window size should be the compromise between these two effects. In both simulation and experimental investigations, however, it is found that the selection of window sizes for a estimation in Eq.(3) and b estimation in Eq. (4) does not have much influence on the results. In a typical experiment, window size ranging from 3 3 pixels to 19 19 pixels deliver similar result with maximum discrepancy of 1.21% for a estimation. For b estimation, the maximum discrepancy is 1.24% with windows ranging from 5 5 pixels to 19 19 pixels, but the discrepancy reaches 2.42% with window of 3 3 pixels. Thus it is concluded that the proposed two-step method is quite tolerant for varying
Fig. 3. Experiment results of a steel tube with crack. (a) and (b) shearographic fringe patterns captured during continuous deformation of a defective tube subjecting to pressurization; (c) and (d) the corresponding wrapped phase map of (a) and (b) obtained by two-step phase shifting; (e) wrapped phase map obtained by four-step phase shifting, noted that the deformation induced phase shift errors have rendered the result unreliable.
Y.H. Huang et al. / Optics and Lasers in Engineering 50 (2012) 534–539
parameters. The reason for this may rely on the fact the approximation for a and b (refer to Eqs. (3) and (4)) is not accurate no matter what window sizes are chosen due to information deficiency from only a single interferogram. We have also tried more complex approximation methods where curve surfaces fitting or local statistical analysis [13] are used to determine a and b. However, the results are similar or even worse than the simple approximation in Eq. (3) and (4). Thus these simple approximations are adopted and window sizes of 3 3 and 5 5 are chosen routinely for a and b estimation in this paper for computation efficiency. The reason why less accurate intensity approximations can result in acceptable phase accuracy (as shown in Fig. 2(e)) is that the phase sign plays a dominant role over the random phase errors caused by a and b approximations. Once the phase sign is correctly determined, most of the random phase errors can be removed during the filtering process. In the proposed method, the determination of phase sign (refer to Eq. (7)) does not involve approximation of a and b and the result is reliable, which guarantees the reliability of the resultant phase. The proposed two-step phase shifting method has many merits, including not requiring an accurate phase shift element (i.e. PZT mirror or LCD phase retarder), good tolerance to phase shift amount and evenness over the spatial domain, simple pixelwise operation, and workable even when the deformation is small and no apparent fringe patterns appear (which is superior over methods based on fringe normalization). Most importantly, it offers a simplest way for dynamic deformation measurement. We demonstrate this using another dynamic shearographic experiment for nondestructive testing of a pressurized defective tube. Fig. 3(a) and (b) shows two fringe patterns of a defective tube subjecting to a continuous pressurization. We capture the speckle patterns with a frame rate of 30 frames per second without the use of any phase shifting device. The image sequence has a small uneven phase shift between any successive frames, thus can be used by our proposed two-step phase shifting method for deformation evaluation. Fig. 3(c) and (d) shows the corresponding wrapped phase map of Fig. 3(a) and (b) obtained by the proposed two-step phase shifting method. The experiment result using four-step phase shifting with the help of a LCD phase retarder (refer to Fig. 1) is also shown in Fig. 3(e). As the desired p/2 phase shift has been severely modified by the deformation induced phase change, the result from four-step phase shifting is quite different from that from the proposed method and being no longer reliable. However, wrapped phase map with poor quality are still obtainable, which may be quite misleading to practitioners. Thus special caution should be given to conventional four-step phase shifting when dealing with dynamic measurement. It should also be noted that speckle decorrelation problem plays an important role for dynamic measurement [21]. In the proposed two-step method, however, speckle phases are calculated using adjacent speckles with small phase change; thus the decorrelation problem has been effectively avoided.
5. Conclusion A two-step phase shifting method has been proposed for quantitative phase evaluation with minimal experimental equipments. It is revealed that small unknown phase shift is inherently advantageous
539
over large known phase shift in this method. The effectiveness of the method has been demonstrated using both static and dynamic shearographic experiments. Though only shearographic experiments are included in this paper, this method is potentially applicable to most techniques in experimental mechanics, including ESPI, fringe projection and shadow moire´.
Acknowledgment The first author is thankful for Prof. Michael Y.Y. Hung for his valuable guidance and inspiring discussion. The financial supports from 863 High-Technology Project of China (#Q2010AA1000692006) and Zhejiang Provincial Natural Science Foundation of China (#R1110377) are acknowledged. References [1] Wang WC, Hwang CH, Lin SY. Vibration measurement by the time-averaged electronic speckle pattern interferometry methods. Appl Opt 1996;35:4502–9. [2] Hung YY, Chen YS, Ng SP, Liu L, Huang YH, Luk BL, et al. Review and comparison of shearography and active thermography for nondestructive evaluation. Mater Sci Eng R-Rep 2009;64:73–112. [3] Huang YH, Ng SP, Liu L, Chen YS, Hung MYY. Shearographic phase retrieval using one single specklegram: a clustering approach. Opt Eng 2008;47:054301. [4] Tahara T, Ito K, Fujii M. Experimental demonstration of parallel two-step phase-shifting digital holography. Opt Express 2010;18:18975–80. [5] Meng XF, Peng X, Cai LZ. Wavefront reconstruction and three-dimensional shape measurement by two-step dc-term-suppressed phase-shifted intensities. Opt Lett 2009;34:1210–2. [6] Jia P, Kofman J, English C. Two-step triangular-pattern phase-shifting method for three-dimensional object-shape measurement. Opt Eng 2007;46:083201. [7] Shaked NT, Zhu YZ, Rinehart MT. Two-step-only phase-shifting interferometry with optimized detector bandwidth for microscopy of live cells. Opt Express 2009;17:15585–91. [8] Yu QF, Fu SH, Liu XL. Single-phase-step method with contoured correlation fringe patterns for ESPI. Opt Express 2004;12:4980–5. [9] Santoyo FM, Kerr D, Tyrer JR. Interferometric fringe analysis using a single phase step technique. Appl Opt 1988;27:4362–4. [10] Kerr D, Santoyo FM, Tyrer JR. Extraction of phase data from electronic speckle pattern interferometric fringes using a single-phase-step method - a novelapproach. J Opt Soc Am A 1990;7:820–6. [11] Almazan-Cuellar S, Malacara-Hernandez D. Two-step phase-shifting algorithm. Opt Eng 2003;42:3524–31. [12] Goodman JW. Some fundamental properties of speckle. J Opt Soc Am 1976;66:1145–50. [13] Tay CJ, Quan C, Chen L, Fu Y. Phase extraction from electronic speckle patterns by statistical analysis. Opt Commun 2004;236:259–69. [14] Hung YY, Huang YH, Liu L, Ng SP, Chen YS. Computerized tomography technique for reconstruction of obstructed phase data in shearography. Appl Opt 2008;47(17):3158–67. [15] Huang YH, Liu L, Chen YS, Li CL, Hung YY. Unified approach for rough phase measurement without phase unwrapping by changing sensitivity factor. J Mod Opt 2009;56(9):1070–7. [16] Huang YH, Ng SP, Liu L, Li CL, Chen YS, Hung YY. NDT&E using shearography with impulsive thermal stressing and clustering phase extraction. Opt Lasers Eng 2009;47(7–8):774–81. [17] Ghiglia Dennis C, Pritt Mark D. Two-dimensional phase unwrapping: theory, algorithms, and software. New York: Wiley; 1998. [18] Huang YH, Farrokh Janabi-Sharifi YS, Liu, Hung YY. Dynamic phase measurement in shearography by clustering method and Fourier filtering. Opt Express 2011;19(2):606–15. [19] Huang YH, Hung SY, Liu YS, Chen YS, Zhang CY, Lu J. Two-step phase shifting with an unknown phase shift for quantitative phase retrieval in speckle interferometry. Opt Express, submitted for publication. [20] Huang YH, Liu YS, Hung SY, Li CG, Janabi-Sharifi Farrokh. Dynamic phase evaluation in Sparse-sampled temporal speckle pattern sequence. Opt Lett 2011;36(4):526–8. [21] Joenathan C, Haible P, Tiziani HJ. Speckle interferometry with temporal phase evaluation: influence of decorrelation, speckle size, and nonlinearity of the camera. Appl Opt 1999;38(7):1169–78.
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Quantitative phase retrieval in dynamic laser speckle interferometry Y.H. Huang a,c, S.Y. Hung b, Farrokh Janabi-Sharifi c, W. Wang d, Y.S. Liu a,n a
State Key Lab. of CAD&CG, Zhejiang University, Hangzhou 310027, PR China Department of Mechanical Engineering, Oakland University, USA Department of Mechanical and Industrial Engineering, Ryerson University, Canada d Department of Mechanical Engineering, Zhejiang University, Hangzhou, PR China b c
a r t i c l e i n f o
a b s t r a c t
Available online 24 August 2011
The rapid progress of modern manufacturing and inspection technologies has posed stringent requirements on optical techniques for vibration characterization and dynamic testing. Due to its simplicity, accuracy and whole-field characters, laser speckle interferometry has served as one of the major techniques for dynamic measurement. In this paper, a two-step phase shifting method is developed for quantitative speckle phase measurement, which helps to eliminate the specklegrams needed for phase evaluation and facilitate dynamic measurement. Unlike previously reported two-step methods using fringe patterns with known phase shift of p/2, a small unknown phase shift is employed instead in the proposed method, which eliminates the need for phase shifting devices. Further investigation shows that small phase shifts are preferable over large phase shifts in this method. Shearographic experiments conducted have demonstrated the effectiveness of the proposed technique. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Shearography Phase shifting Dynamic measurement Speckle metrology
1. Introduction Speckle interferometry techniques such as electronic speckle pattern interferometry [1] and shearography [2] have been widely accepted in laboratories as well as industries for deformation, strain measurement and vibration characterization. Normally the desired physical quantities, which are related to the phase information in some forms dependent on the optical configuration, are encoded in fringe patterns in cosine form. One of the most popular methods for quantitative phase retrieval is phase shifting technique [3] where three or more interferograms with known phase shifts are captured and processed pixel-wise to determined the three unknowns for each pixel. In general, more interferograms will provide abundant information and make the resultant phase map more accurate and noise-free. However, when dealing with dynamic measurement, the testing object is subject to continuous movement, which makes it difficult to capture several interferograms with predetermined phase shifts. Even in semi-static measurement, hostile industrial circumstance will introduce environmental noise such as vibration and air refractive index fluctuation, which will introduce phase drifting over time and result in error in phase measurement. Thus phase retrieval techniques requiring less interferograms would be desirable in dynamic measurement or under uncontrollable environmental noise.
n
Corresponding author. E-mail address: [email protected] (Y.S. Liu).
0143-8166/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2011.06.025
In history, researchers have proposed various two-step phase shifting algorithms [4–11] to accommodate this need. However, most of algorithms deal with digital holography [4], clean fringe patterns from fringe projection [5,7], or speckle-noise-free interferometry fringe [7]; and only a few papers [8–11] attempt to extract phase information from speckle interferometric fringe patterns overwhelmed by speckle noise. In their works [8–11], the researchers try to remove speckle noise and the background intensity using various filters and reconstruct the desired phase using two clean fringe patterns with p/2 phase shift. In this paper, we propose an alternative two-step phase shifting method for phase retrieval in speckle interferometry. Different from previous techniques, the proposed method is based on speckle patterns analysis instead of fringe patterns normalization. Another distinct feature of this two-step method is that it does not need to know the exact amount of the phase shift. All it requires is a small unknown phase shift (i.e. r p/3) between two speckle patterns to solve for the phase ambiguity problem. The unknown phase shifts can be obtained by capturing specklegrams of a continuously moving object, thus no phase shifting devices such as PZT mirror or liquid crystal phase retarder are required, which makes this method extremely simple and practical. In Section 2, the theoretical background for the two-step method is given. In Section 3, two shearographic experiments are introduced. In Section 4, the phase maps obtained by the two-step method are illustrated; the method is further compared with previously reported phase extraction methods. The effect of the unknown phase shift amount and other parameters is also
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evaluated theoretically and experimentally. Section 5 concludes this paper.
2. Theoretical background In speckle interferometry, a speckle pattern can be generally represented as Iðx,yÞ ¼ aðx,yÞ þbðx,yÞcos½jðx,yÞ
ð1Þ
where a(x,y), b(x,y) and j(x,y) are pixel-wise background intensity, modulation and random phase, respectively. Conventional phase shifting techniques tend to introduce three or more known phase shifts to j(x,y) in Eq. (1) using various phase shifters and solve for the three knowns using optimization methods. The phase information j(x,y) obtained in this stage is an initial random phase resulting from random interference between the reference and object beam reflected off an optically rough surface. After the testing object undergoes a deformation, another speckle pattern is digitized as I0 ðx,yÞ ¼ aðx,yÞ þ bðx,yÞcos½jðx,yÞ þ Dðx,yÞ
ð2Þ
where D(x,y) is a phase change related to the deformation in some form depending on the optical configuration. The same phase shifting technique can be applied again to deliver another random phase j(x,y) þ D(x,y). The subtraction between this deformed phase and the initial random phase j(x,y) will result in a sawteeth wrapped phase map D(x,y), which contains the desired deformation information. According to our study and previous reports [12,13] on speckle statistics and properties, the random phase j(x,y), which has a range of ( p,p], is a zero-mean highly random value over any small area of the image plane. However, the background a(x,y)and modulation b(x,y) is found to be much more constant on the whole image, especially within an individual speckle. As an evidence, the mean value of j(x,y) is found to be 0.0625 rad and the standard deviation is 1.8144 rad; while that for a(x,y) are 140.3841 and 28.8485(gray value), and that for b(x,y) are 27.1455and 15.1677 in one of our typical experiment. Based on this observation, if we make a local average processing (i.e. on 3 3 pixels) on the speckle pattern shown in Eq. (1), since a(x,y) and b(x,y) are almost constant within such a small area (this area is dependent on speckle size. If the speckle size is less than 3 pixels, one can choose a smaller aperture to obtain bigger speckles) while j(x,y) change abruptly within ( p,p], the second term of Eq. (1) tends to be zero-mean while the first term is kept. Thus it would be reasonable to make an estimation of the pixel-wise values of a(x,y) as follows: i ¼X 1,j ¼ 1
_ aðx,yÞ ¼ 1=9
Iðx þ i,yþ jÞ
ð3Þ
i ¼ 1,j ¼ 1
On the other hand, if we subtract the newly obtained background _ aðx,yÞfrom Eq. (1), then we get an approximation for bðx,yÞcos½jðx,yÞ. Again, b(x,y) is almost constant within a small area while j(x,y) is highly random. Thus if we choose the maximum absolute value of bðx,yÞcos½jðx,yÞ over a small area (i.e. 5 5 pixels), we would get a reasonable approximation of b(x,y) since one of the highly random j value may hit the extreme positions for cosðjÞ. Thus b(x,y) can be approximated as _ bðx,yÞ ¼
i ¼ 2,j ¼ 2
Max
i ¼ 2,j ¼ 2
9Iðx þ i,y þ jÞ_ aðx þ i,y þ jÞ9
ð4Þ
where the function Max( ) is designed to extract the maximum _ value. Since we have obtained both estimations _ aðx,yÞ and bðx,yÞ for approximating the speckle background and modulation, we can now obtain the absolute value of initial random phase j(x,y)
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from Eq. (1) as _ 9jðx,yÞ9 ¼ arccosf½Iðx,yÞ_ aðx,yÞ=bðx,yÞg
ð5Þ
where arccosðUÞ is the normal arccosine function. The sign information for j(x,y) is lost in the speckle pattern due to the even function property of cosine operation, thus additional information is required to solve this sign ambiguity problem and choose a correct initial phase from the absolute value. We solve this problem by introducing a second speckle pattern Id(x,y), which has a small phase change d(x,y) relative to Eq. (1): Id ðx,yÞ ¼ aðx,yÞ þ bðx,yÞcos½jðx,yÞ þ dðx,yÞ
ð6Þ
If d(x,y) is a positive small number (for negative d, similar derivation can be obtained), the term cos½jðx,yÞ þ dðx,yÞ in Eq. (6) can be approximated as cos½jðx,yÞdðx,yÞsin½jðx,yÞ by ignoring the higher order terms in the Taylor expansion. Subtracting Eq. (6) from Eq. (1) will result in an intensity of Isub ðx,yÞ ¼ Iðx,yÞId ðx,yÞ bðx,yÞdðx,yÞsin½jðx,yÞ
ð7Þ
Since b(x,y) and d(x,y) are positive values and sinðUÞis an odd function, the sign of j(x,y) in Eq. (5) is now determined to be the same as the sign of Isub(x,y). Thus, an unambiguous initial random phase j(x,y) is now fully determined as _ jðx,yÞ ¼ acosf½Iðx,yÞ_ aðx,yÞ=bðx,yÞg signðIsub ðx,yÞÞ ð8Þ where function sign( )extracts the positive or negative sign information of an input value. Till now an unambiguous random phase has been determined from two speckle patterns with a small phase shift. After the object undergoes a deformation, another two speckle patterns with small phase shift can be used again to determine the deformed random phase [j(x,y) þ D(x,y)]. The subtraction between the initial and deformed random phase will then give a wrapped phase map D(x,y)representing the deformation.
3. Experiment work To verify the proposed two-step phase shifting method for phase evaluation, shearographic experiments are conducted. Fig. 1 shows a typical shearographic setup for strain and displacement derivative determination. The object under testing is illuminated by an expanded coherent laser light. The rough surface of the object reflects laser light to produce an interference speckle image via a liquid crystal phase retarder (phase shifting device) and a Wollaston prism (image shearing device). By incorporating the image shearing device, laser light scattered from neighboring points M and B are brought to interfere on the CCD chip and the relevant optical path difference of points M and B are recorded in terms of phase. The relationship between the phase and deformation in this setup can be found in Ref. [14]. When the object undergoes a deformation, the deformed amount can be revealed by a fringe pattern, which is obtained by subtracting the deformed speckle image from the undeformed speckle image. Quantitative evaluation of the deformation field can be fulfilled by various phase retrieval methods. Our first experiment is a static deformation experiment, which allows speckle images with accurate phase shift amounts to be captured and evaluated. This static experiment is intended to compare the two-step method with previous reported phase retrieval methods. In the experiment, a plate fully clamped by all the boundaries was loaded in the center by a micrometer (refer to Fig. 1b in Ref. [15]). Before and after the static loading, various phase shifts were introduced by the phase shifter and speckle images were captured for phase evaluation using conventional four-step phase shifting, the proposed two-step phase shifting,
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Fig. 1. Shearographic setup employed in our experiment.
and the clustering method [3] where four phase shifted images are captured before the deformation and only a single deformed image is captured after the deformation. The results from these methods are then compared to evaluate the performance of various methods. In the second experiment, a steel tube with an internal crack as shown in Fig. 6(b) of Ref. [16] was insulated and continuously pressurized by a pump. Speckle images are continuously captured with a frame rate of 30 frames per second. It is estimated that the deformation induced phase shifts between successive frames are about p/3. These successive speckle pairs are then evaluated by the two-step method to deliver the random speckle phase at each instant. Subsequent subtraction of speckle phases at different stages will then result in deformation phase. On the other hand, the phase shifters are also employed to capture speckle images with additional phase shifts of p/2. The interval between the four-step shifted images is 0.05 s. Though this is a short time, the deformation induced phase shift during this interval is already comparable with p/2. This induces a very large phase shift error, and the resultant phase map obtained from four-step phase shifting is in poor quality and unreliable.
4. Results and discussions Fig. 2 shows the result of the static experiment for the central loaded plate. Fig. 2(a) shows a wrapped phase map from four-step phase shifting method depicting the out-of-plane displacement derivative of the central loaded plate. The wrapped phase map is subjected to large amount of noise due to air fluctuation, temperature change, environmental vibration or laser drifting. However, the phase information is still visible at this stage. Fig. 2(b) shows the same wrapped phase map, but determined from the proposed two-step method. It can be observed that the inherent information deficiency in this two-step phase shifting method makes the noise more severe than conventional four-step phase shifting methods. For further comparison, a wrapped phase map determined from the clustering method [3] is also shown in Fig. 2(c). All the three wrapped phase maps at their initial stage are too noisy for phase unwrapping, thus noise filtering techniques should be employed to get a clean wrapped phase map
before phase unwrapping can be successfully applied. One effective filtering method for noise removal is to convert the wrapped phase map into cosine and sine fringe maps, then apply low-pass filters (such as 3 3 averaging filter) to suppress noise, and finally reconstruct a clean wrapped phase map by applying an arctangent operation pixel-wisely on the filtered cosine and sine fringe maps. Fig. 2(d)–(f) shows the filtered results of Fig. 2(a)–(c), respectively. It can be seen that most of the noise have been removed. However, the wrapped phase map from clustering method (Fig. 2(f)) still has plenty of phase residues in the central area with dense fringes, which will pose severe problem for subsequent phase unwrapping, while the phase map from four-step phase shifting (Fig. 2(d)) is totally residue-free and can be unwrapped using any simple algorithms. The result from the proposed two-step method (Fig. 2(e)) is in-between and a few phase residues are presented in the central area, which may be easily unwrapped by Goldstein’s branch-cut algorithm [17]. By applying the Fourier filtering method [18] to Fig. 2(e), a high quality wrapped phase map has been obtained as shown in Fig. 2(g). Quantitative comparison between Fig. 2(g) and (d) reveals that the discrepancy is less than 1%. The proposed two-step phase shifting method has been proposed for phase retrieval with a double pulse laser [19] or dealing with sparsely sampled speckle sequence [20]. In the previous papers, only small unknown phase shifts are employed, and no thorough analysis and comparison between small and large phase shifts are given. In this paper, we develop more general formula to accommodate both small and large phase shift, and verify the potential advantages of small phase shift in this method. For the proposed two-step phase shifting method, Eqs. (1)–(8) use a small unknown phase shift between two speckle patterns. What if we use a large known phase shift instead (i.e. p/2) as in previously reported papers [8–11]? Will it reduce the noise level and improve the quality of resultant wrapped phase map due to the availability of the phase shift amount? We conducted both simulations and experiments to find out the answer. For a large known phase shift, the first-order Taylor expansion in Eq. (7) is invalid. Thus a new criterion is set as follows: _ Discrepancyð_ j Þ ¼ 9b½cos_ j cosð_ j þ dÞIsub 9
ð9Þ
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Fig. 2. Wrapped phase maps determined by various methods. (a), (b) and (c) wrapped phases determined by conventional four-step phase shifting, the proposed two-step phase shifting and the clustering method [3], respectively; (d)–(f) corresponding wrapped phase maps after noise filtering, noted that two-step method (e) results in less phase residues than clustering method (f) in central areas; (i) Reconstructed wrapped phase map of (e) by adaptive Fourier filtering [18], the discrepancy to (d) is less than 1%; (g) and (h) filtered wrapped phase map obtained by Eq. (10) with phase shifts of p/3and 2p/3, respectively, worse results than (e) has been observed.
where (x,y) is omitted for simplicity and d is a known phase shift. Function Discrepancyð_ j Þ indicates the discrepancy between a phase value _ j and the true initial random phase value j, and a
j Þ indicates a better estimation. Designate smaller Discrepancyð_ the positive and negative phase value obtained by Eq. (5) as jp and jn, respectively, we then substitute them into Eq. (9) and
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choose a potentially more correct phase value by (
j¼
jp if Discrepancyðjp Þ r Discrepancyðjn Þ jn otherwise
ð10Þ
Eq. (10) is theoretically valid for both large and small phase shift as long as the phase shift amount is known. We thus simulated a series of small known phase shifts of 0.05p, 0.1p, 0.15p and 0.2p, and use both Eqs. (8) and (10) for phase determination. The results indicate that both equations give similar wrapped phase map as shown in Fig. 2(e), while Eq. (8) does not require knowing the amount of phase shift but Eq. (10) needs to know the pixel-wise phase shift. When the phase shift amount becomes larger, Eq. (8) is no longer a good approximation, thus only Eq. (10) was used. Fig. 2(h) and (i) shows the wrapped phase map obtained using Eq. (10) with simulated p/3 and 2p/3 phase shift (for experimental p/3phase shift, the result is similar to Fig. 2(h)), it can be observed that the wrapped phase map quality is poorer than Fig. 2(e), which uses a small phase shift. The reason is that for a larger phase shift, the value of _ ½cos _ j cosð_ j þ dÞ in Eq. (9) is larger, thus the error contained in b (approximation of b) is amplified, which makes the judgment _ using Eq. (10) erroneous. In other words, if b is a very good approximation of b, then small known phase shift and large known phase shift should deliver similar results. However, the inherent information deficiency in two-step phase shifting cannot guarantee a good b approximation, thus a small phase shift would
be preferred. A more thorough quantitative investigation shows that phase shifts from 0.02p to 0.33p deliver similar results to Fig. 2(e) with the maximum discrepancy of 1.69%, and that from 0.33p to 0.66p result in maximum phase error of 5.31% with noisy wrapped phase maps difficult to unwrap. After 0.66p the phase maps are becoming extremely noisy and information is totally lost at the point of 0.82p. We also used experimental obtained interferograms with 0.5p phase shift to test the proposed algorithm, and the result is quite similar to the simulated specklegrams. As the proposed two-step method uses speckle statistics to approximate the background intensity a and fringe modulation b, the effect of window size selection on a and b approximation should also be investigated. Theoretically the window size should be smaller than the speckle size as speckle statistics varies dramatically within adjacent speckle. But the window size could not be too small to achieve a good statistical approximation. Thus the optimal window size should be the compromise between these two effects. In both simulation and experimental investigations, however, it is found that the selection of window sizes for a estimation in Eq.(3) and b estimation in Eq. (4) does not have much influence on the results. In a typical experiment, window size ranging from 3 3 pixels to 19 19 pixels deliver similar result with maximum discrepancy of 1.21% for a estimation. For b estimation, the maximum discrepancy is 1.24% with windows ranging from 5 5 pixels to 19 19 pixels, but the discrepancy reaches 2.42% with window of 3 3 pixels. Thus it is concluded that the proposed two-step method is quite tolerant for varying
Fig. 3. Experiment results of a steel tube with crack. (a) and (b) shearographic fringe patterns captured during continuous deformation of a defective tube subjecting to pressurization; (c) and (d) the corresponding wrapped phase map of (a) and (b) obtained by two-step phase shifting; (e) wrapped phase map obtained by four-step phase shifting, noted that the deformation induced phase shift errors have rendered the result unreliable.
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parameters. The reason for this may rely on the fact the approximation for a and b (refer to Eqs. (3) and (4)) is not accurate no matter what window sizes are chosen due to information deficiency from only a single interferogram. We have also tried more complex approximation methods where curve surfaces fitting or local statistical analysis [13] are used to determine a and b. However, the results are similar or even worse than the simple approximation in Eq. (3) and (4). Thus these simple approximations are adopted and window sizes of 3 3 and 5 5 are chosen routinely for a and b estimation in this paper for computation efficiency. The reason why less accurate intensity approximations can result in acceptable phase accuracy (as shown in Fig. 2(e)) is that the phase sign plays a dominant role over the random phase errors caused by a and b approximations. Once the phase sign is correctly determined, most of the random phase errors can be removed during the filtering process. In the proposed method, the determination of phase sign (refer to Eq. (7)) does not involve approximation of a and b and the result is reliable, which guarantees the reliability of the resultant phase. The proposed two-step phase shifting method has many merits, including not requiring an accurate phase shift element (i.e. PZT mirror or LCD phase retarder), good tolerance to phase shift amount and evenness over the spatial domain, simple pixelwise operation, and workable even when the deformation is small and no apparent fringe patterns appear (which is superior over methods based on fringe normalization). Most importantly, it offers a simplest way for dynamic deformation measurement. We demonstrate this using another dynamic shearographic experiment for nondestructive testing of a pressurized defective tube. Fig. 3(a) and (b) shows two fringe patterns of a defective tube subjecting to a continuous pressurization. We capture the speckle patterns with a frame rate of 30 frames per second without the use of any phase shifting device. The image sequence has a small uneven phase shift between any successive frames, thus can be used by our proposed two-step phase shifting method for deformation evaluation. Fig. 3(c) and (d) shows the corresponding wrapped phase map of Fig. 3(a) and (b) obtained by the proposed two-step phase shifting method. The experiment result using four-step phase shifting with the help of a LCD phase retarder (refer to Fig. 1) is also shown in Fig. 3(e). As the desired p/2 phase shift has been severely modified by the deformation induced phase change, the result from four-step phase shifting is quite different from that from the proposed method and being no longer reliable. However, wrapped phase map with poor quality are still obtainable, which may be quite misleading to practitioners. Thus special caution should be given to conventional four-step phase shifting when dealing with dynamic measurement. It should also be noted that speckle decorrelation problem plays an important role for dynamic measurement [21]. In the proposed two-step method, however, speckle phases are calculated using adjacent speckles with small phase change; thus the decorrelation problem has been effectively avoided.
5. Conclusion A two-step phase shifting method has been proposed for quantitative phase evaluation with minimal experimental equipments. It is revealed that small unknown phase shift is inherently advantageous
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over large known phase shift in this method. The effectiveness of the method has been demonstrated using both static and dynamic shearographic experiments. Though only shearographic experiments are included in this paper, this method is potentially applicable to most techniques in experimental mechanics, including ESPI, fringe projection and shadow moire´.
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