Extended Depth-range Dual-wavelength Interferometry Based on Iterative Two-step Temporal Phase-unwrapping
Minmin Wang, Canlin Zhou, Shuchun Si, XiaoLei Li, Zhenkun Lei, YanJie Li
EExtended Depth-range Dual-wavelength Interferometry Based onIterative Two-step Temporal Phase-unwrapping
Minmin Wang a , Canlin Zhou a *, Shuchun Si a , XiaoLei Li b , ZhenkunLei c **, YanJie Li d a School of Physics, Shandong University, Jinan, China; b School of MechanicalEngineering, Hebei University of Technology, Tianjin, China; c Department ofengineering mechanics, Dalian University of Technology, Dalian, China; d School ofcivil engineering and architecture, Jinan University, Jinan, China * Corresponding author: Tel: +8613256153609; E-mail address:[email protected] ** Corresponding author: Tel: +8615841175236; E-mail address: [email protected] xtended Depth-range Dual-wavelength Interferometry Based onIterative Two-step Temporal Phase-unwrapping
Phase retrieval is one of the most challenging processes in many interferometrytechniques. To promote the phase retrieval, Xu et. al [X. Xu, Y. Wang, Y. Xu, W.Jin. 2016] proposed a method based on dual-wavelength interferometry.However, the phase-difference brings large noise due to its low sensitivity andsignal-to-noise ratio (SNR). Beside, special phase shifts are required in Xu’smethod. In the light of these problems, an extended depth-rangedual-wavelength phase-shifting interferometry is proposed. Firstly, the leastsquares algorithm is utilized to retrieve the single-wavelength phase from asequence of N-frame simultaneous phase-shifting dual-wavelengthinterferograms (SPSDWI) with random phase shifts. Then the phase-differenceand phase-sum are calculated from the wrapped phases of single wavelength,and the iterative two-step temporal phase-unwrapping is introduced to unwrapthe phase-sum, which can extend the depth-range and improve the sensitivity.Finally, the height of objects is achieved. Simulated experiments are conductedto demonstrate the superb precision and overall performance of the proposedmethod.Keywords: two-wavelength phase-shifting algorithm; temporal phaseunwrapping; interferometry; least squares algorithm; phase error
1. Introduction
Interferometry has been widely employed in optical phase and surface topographymeasurenent due to its advantages of high resolution, non-contact, and noiseresistance (1-3). Over the years, many phase demodulation methods have beenproposed to retrieve the phase from interferograms. Among these methods, thedual-wavelength interferometry (DWI) not only keeps the merits of traditionalingle-wavelength interferometry, but also can produce a non-wrappedphase-difference by the simple subtraction between two wrapped single-wavelengthphases (4). Hence the DWI have been naturally investigated by researchers and thereare several methods of DWI developed for phase retrieval, most of which can beclassified into two categories: spatial-Fourier-transform (SFT) based phase retrieval(5-8) and temporal-phase-shifting (TPS) based phase retrieval (9-11).The SFT method is a better choice for scenes that require real-time and highspeed as the wrapped phase can be retrieved from only a single-framedual-wavelength interferogram through Fourier transform. Onodera et. al (7)investigated a two-wavelength interferometry based on a Fourier-transform methodand analyzed the phase error caused by the difference between modulation intensitiesat two wavelengths. Min et. al (8) proposed a dual-wavelength digital holographicmicroscopy with a slightly off-axis configuration, where the high wavelengthselectivity of the Bayer mosaic filtered color CCD camera is used. Kühn et. al (12)developed a technique to perform two-wavelengths digital holographic microscopymeasurements with a single hologram acquisition. The real-time dual-wavelengthimaging is realized by using two reference waves with different wavelengths andpropagation directions for the hologram recording. Ichioka et. al (13) proposed adigital holographic configuration utilized for single-shot, dual-wavelength, off-axisgeometry and imaging polarimetry. The common problem of these SFT methods isthat their measuring accuracy is greatly influenced by the level of random noise, thefiltering window, and the Gibbs boundary effects.he TPS methods reveal higher accuracy to retrieve the phase compared withthe SFT method. Double-wavelength interferometry was improved by Polhemus [14]and later on by Cheng [15] using digital phase-shifting phase-demodulationAfterwards Kakue et. Al (16) applied the image-reconstruction algorithm of paralleltwo-step phase-shifting digital holography to the hologram so as to propose analgorithm that can improve the quality of the reconstructed image from the singlehologram. Abdelsalam and Kim (10) described a configuration used for two-wavelength phase-shifting in-line interferometry based on polarizing separation. Thismethod needs a sequence of phase-shifting interferograms for each single-wavelength,which is rather complicated and may consume a significant amount of time. Huang et.al (17) developed a dual-wavelength interferometry based on the spatialcarrier-frequency phase-shifting method, where the least squares method is iterativelyutilized. Thus it also has a heavy computational time and memory requirements.Zhang et. al (18) proposed a phase-shifting dual-wavelength interferometry based onthe two-step demodulation algorithms. The phase is retrieved through five frames ofsimultaneous phase-shifting dual-wavelength interferograms with the special phaseshifts by using subtraction and the Gram-Schmidt two-step demodulation algorithm.Xu et. al (19) improved Zhang’s approach, and proposed an approach of quantitativephase extraction based on two intensities with dual wavelength after filtering thecorresponding dc terms for each wavelength. However, a special phase shift isrequired in Xu’s method. Besides, the previous DWI including Xu’s method rely onlyon the phase-difference. Since the phase-difference has low sensitivity andignal-to-noise ratio (SNR), the retrieved phase may be quite noisy.In this paper, we present an extended depth-range dual-wavelengthinterferometry for phase retrieval based on combined the iterative two-step temporalphase unwrapping and the least squares method. It is an improvement of Xu’s method(19). The limitation of being exclusively applicable to interferograms with specialphase shifts is solved by using the least squares method. By introducing in thephase-sum and the iterative two-step temporal phase unwrapping, the sensitivity andSNR is greatly improved. The suitability and superiority of the proposed method isillustrated on simulated experiments.
2. Theory
Assume that two lasers with different wavelengths travel the same inlinephase-shifting interference system at the same time. The phase shifts of two referencewaves come into being simultaneously by a piezoelectric transducer (PZT). A seriesof interferograms are then captured by a monochrome CCD, which aremathematically described as ( , ) ( , ) ( , ) cos[ ( , ) ] n i i i i ni
I x y a x y b x y x x y (1)where a ( x , y ) is the background intensity; the subscript i =
1, 2 denotes two singlewavelengths; ( , ) i b x y is the intensity modulation amplitudes at a wavelength of λ i ; , i n is the phase step at λ i . ( , ) 2 ( , ) / i i x y h x y is the phase to be retrieved at λ i ; h ( x , y ) is optical path difference. n = 1, 2, 3... N , n is the image index, and N is the totalnumber of images. The relationship between the phase shifts and the wavelengths isgiven by , 2, 2 1 / / n n (2) The wrapping-free phase extraction method proposed by Xu et. al has beenthoroughly described and discussed (19). This section only introduces its main ideabriefly. In Xu’s method, firstly, the corresponding dc terms are eliminated by solvingthe intensity difference between I and other intensity values. Then, the phase shifts of δ , n ( δ , n ) are set as 2π and 4π, thus two intensities of λ ( λ ) can be separated, and thewrapped phase of ( ) can be retrieved. After determining the single wavelengthphase, the wrapped phase-difference is obtained by doing subtraction between twosingle wavelength phase, which is then unwrapped to further calculate the height ofthe object. Clearly, Xu’s method is only applicable for interferograms with specialphase shifts, which means it will fail once the phase-shifting inducer (i.e. PZT) is notcapable of producing a specific phase shift. Besides, the sensitivity and SNR of thephase-difference is rather low, so Xu’s method is easily destroyed by the phase errors(harmonics and noise). To solve these problems, we present a robust dual-wavelength simultaneousphase-shifting interferometry that is based on combined the least squares algorithmand the iterative two-step temporal phase-unwrapping method.
To overcome the limitation of phase shifts without sacrificing the accuracy,here, we review the generalized phase shifting algorithm (20-23). Eq. (1) can beewritten as n 1 1 1, 1 1 1,2 2 2, 2 2 2, ( , ) ( , ) ( , ) cos[ ( , )]cos( ) ( , ) sin[ ( , )]sin( )( , ) cos[ ( , )]cos( ) ( , ) sin[ ( , )]sin( ) n nn n
I x y a x y b x y x y b x y x yb x y x y b x y x y (3)Instead of making the subtraction operation between the intensity distributionof two interferograms, we define a new set of variables as ( , ) ( , )cos[ ( , )] b x y b x y x y , ( , ) ( , )sin[ ( , )] c x y b x y x y , ( , ) ( , )cos[ ( , )] d x y b x y x y , and ( , ) ( , )sin[ ( , )] e x y b x y x y ,Eq. (3) can be described as n 1, 1,2, 2, ( , ) ( , ) ( , ) cos( ) ( , ) sin( )( , ) cos( ) ( , ) sin( ) n nn n I x y a x y b x y c x yd x y e x y (4)The deviation square sum for all N fringe patterns can be expressed as
21, 1, '1 2, 2, n ( , ) ( , ) cos( ) ( , ) sin( )( , ) ( , ) cos( ) ( , ) sin( ) ( , )
N n nn n n a x y b x y c x yE x y d x y e x y I x y (5)where 'n I is the intensity of interferograms measured in the experiments. Accordingto the principle of the least squares method, the following extreme value conditionshould be satisfied ( , ) 0( , ) E x ya x y , ( , ) 0( , ) E x yb x y , ( , ) 0( , ) E x yc x y , ( , ) 0( , ) E x yd x y , ( , ) 0( , ) E x ye x y (6)In this way, five unknowns of a ( x, y ), b ( x, y ), c ( x, y ), d ( x, y ), e ( x, y ) can beresolved simultaneously from these equations. Solve Eq. (6) to get the followingmatrices X U Q (7)where , 1, 2, 2,1 1 1 121, 1, 1, 1, 1, 2, 1, 2,1 1 1 1 121, 1, 1, 1, 2, 1, 1, 2,1 1 1 1 122, 1, 2, 2, 1, 2, 2, 2,1 1 N N N Nn n n nn n n nN N N N Nn n n n n n n nn n n n nN N N N Nn n n n n n n nn n n n nN Nn n n n n n n nn n n
N c s c sc c c s c c c sU s c s s c s s sc c c c s c c s N N Nn nN N N N Nn n n n n n n nn n n n n s c s s s c s s [ ( , ) ( , ) ( , ) ( , ) ( , )] T X a x y b x y c x y d x y e x y N N N N N1, 1, 2, 2,n 1 n 1 n 1 n 1 n 1 [ ]
Tn n n n n n n n n
Q I I c I s I c I s (8)where , , cos i n i n c and , , sin i n i n s (i = 1, 2).By taking N = 5 and solving Eq. (7) through five interferograms with arbitraryphase shifts, the wrapped phases can be obtained by ( , )( , ) arctan( )( , ) w c x yx y b x y , ( , )( , ) arctan( )( , ) w e x yx y d x y (9) The phase-difference and highly wrapped phase-sum can be calculated by ( , ) ( , ) ( , ) d w w w x y x y x y (10a) ( , ) ( , )+ ( , ) w w ws x y x y x y (10b)If the wavelength of and are chosen close, their phase-difference d iscontinuous after using simple addition (given in Eq. (11)) since the syntheticwavelength is much larger (20). ( , ) ( , ) 0( , ) ( , ) 2 w wd dd wd x y while x yx y x y else (11)Plenty of dual-wavelength interferometry including Xu’s method rely only onhe phase-difference as the non-wrapped phase-difference contains the searchedphase-profile and is already unwrapped. However, we think that the phase-sum shouldalso be included to improve the sensitivity and precision of the retrieved phase (25).The advantages of the phase-sum are illustrated as follows.First, the sensitivity of the phase-sum and phase-difference is compared. Theequivalent fringe periods of the phase-difference and phase-sum are expressed as ; s for (12a) ; s for (12b)If the wavelength of and are chosen close enough, namely , d is much larger than or , while s is much shorter than either or . Forthat reason, the phase-difference has low sensitivity while the phase-sum has highsensitivity. Then the sensitivity gain G between the phase-difference and phase-sum is sd G (13)Next, we illustrate the superiority of the phase-sum by analyzing andcontrasting the SNR of the phase-sum and phase-difference. In practice whiteGaussian noise may pollute the demodulated phases and . Thus a measuringphase-noise should be added to the demodulated phases (26). This means thedemodulated phases can be expressed as ( , ) 2 ( , ) / ( , ) x y h x y e x y (14a) ( , ) 2 ( , ) / ( , ) x y h x y e x y (14b)where e ( x , y ) and e ( x , y ) are two noise samples. The phase-difference and phase-sumre re-expressed as ( , ) 2 ( , )( ) / ( , ) ( , ) d x y h x y e x y e x y ( , ) 2 ( , )( ) / ( , ) ( , ) s x y h x y e x y e x y (15)Then the SNR for d and s are
22 2 2 22 1 1 2 ( , ) 22 1( , ) x yd x y h x y de x y e x y d (16a)
22 2 2 22 1 1 2 ( , ) 22 1( , ) x ys x y h x y de x y e x y d (16b)where is the well-defined fringe-data. As both noise samples are generated by thesame Gaussian zero-mean stationary random process, in the average, the energy of ( , ) ( , ) e x y e x y and ( , ) ( , ) e x y e x y are equal (26). Therefore, the SNR gainbetween s and d is, sd G (17)The phase-sum has G higher SNR than the phase-difference. Given the theoreticalanalysis mentioned above, the phase-sum should be introduced to retrieve the phasewith high sensitivity and high precision. The two-step temporal phase unwrapping algorithm is used to retrieve thecontinuous phase in this paper. The non-wrapped phase-difference d is utilized asthe first estimation to temporarily unwrap the phase-sum s (27-28), then we canobtain the unwrapped phase as ( , ) ( , ) ( , ) ( , ) ws d s d x y G x y W x y G x y (18)here W is the wrapping phase operator, and ( , ) [ ( , )] ws s x y W x y . Eq. (18) iseffective, only when the following condition is fulfilled, [ ( , ) ( , )] ( , ) s d x y G x y (19) We know from Eq. (17) that the greater the sensitivity gain G is, the greaterthe SNR gain of the demodulation phase. Therefore, in order to improve themeasurement accuracy, the G value should be chosen as large as possible. However,as can be seen from Eq. (18), the phase d is scaled up by G to further unwrap thephase s . Thus the noise in the phase d is also magnified. Besides, when and are chosen close to each other, G is very large, so it may be hard to satisfy thecondition shown in Eq. (19), so noise may destroy the phase retrieval process. Toincrease the sensitivity gain G as well as reducing the noise, here, we utilized theiterative two-step temporal phase unwrapping algorithm. First, the phase w isunwrapped by the phase-difference d using Eq. (20a). ( , ) ( , ) ( , ) ( , ) wd d x y G x y W x y G x y (20a) ( , ) ( , ) ( , ) ( , ) ws s x y G x y W x y G x y (20b)The non-wrap phase is then utilized to unwrap the phase-sum ws using Eq.(20b). The sensitivity gain for this case is d G and s G . In this way,the continuous phase-sum is obtained with high sensitivity gain G = G × G and highSNR. Finally, the thickness of the phase object can be obtained by Eq. (21). ( , ) ( , ) / 2 s s h x y x y (21)bviously, using the proposed method, the special phase shifts of the fringe patternsare no longer needed, and the sensitivity and SNR are improved.The main steps of the proposed algorithm can be summarized as follows:(1) capture a sequence of N-frame ( N ) SPSDWI with random phase shifts;(2) calculate the wrapped phase-maps w and w of the single wavelength fromfringe patterns using the generalized phase-shifting algorithm;(3) compute the phase-difference d and the highly-wrapped phase-sum ws fromthe wrapped phase-maps w and w according to Eq. (10) and Eq. (11);(4) unwrap w with d by Eq. (20a), and obtain the continuous phase ;(5) unwrap ws with by Eq. (20b), and obtain the continuous phase data s with a high sensitivity gain G and high SNR;(6) obtain the thickness of the phase object from the continuous phase s by Eq. (21).These are the main procedures of the proposed algorithm with larger sensitivity gain G by cascading with 2 level different sensitivities ( G = G × G ). Some followingsimulated experiments are used to verify the proposed algorithm.
3. Simulated experiments and result analysis
To demonstrate the feasibility of the proposed method, the simulatedexperiments are carried out in MATLAB. Five-frame interferograms with size of 256× 256 pixels are generated, in which the background is ( , ) 120 x y a x y e andthe modulation amplitudes are ( , ) 50 x y b x y e and ( , ) 60 x y b x y e at 532 nm and 632.8 nm, respectively. The synthetic beatwavelength of the phase-difference is equal to 3339.78 nm, while the equivalentavelength of the phase-sum is equal to 289.02 nm. A spherical cap with the height of
480 10 ( ) set h x y , in which -1.27 mm ≤ x , y ≤ 1.28 mm is employed as themeasured object. White noise with a variance of 0.6 and a mean of 0 was added to allthe five fringe patterns to test the robustness of the proposed method. The phase shiftsfor the fringe patterns are chosen as =0, =2π, =4π, =2π, =4πat n , and
1, 1 2 n at n , so they are available for both Xu’s method and theproposed method, as shown in Fig. 1.First, we utilized Xu’s method to process the simulated patterns. The wrappedsingle wavelength phases can be calculated from these phase-shifting interferogramsusing Eq. (9), as shown in Figures 2(a) and 2(b). Figure 3(a) shows thephase-difference between those two wrapped single wavelength phases. As there is noexplanation about how to retrieve the continuous phase-difference from the wrappedphase-difference in Xu’s method (19), we utilized the Unwrap function in MATLAB,and the non-wrapped phase-difference is shown in Figure 3(b). Finally, the thicknessof the phase object was determined, as shown in Figure 3(c). It is easy to see that Xu’smethod accommodates noise with the RMSE of 0.0053 um. Since the sensitivity ofthe phase-difference is far too small , it is almost impossible for this case to accuratelycalculate the thickness distribution of the phase object.Next, we employed the proposed method. The least squares method is utilizedto estimate the wrapped single wavelength phases. Then except for thephase-difference, the highly-wrapped phase-sum is obtained, as shown in Figure 4(a).Using the continuous phase-difference to unwrap the phase-sum by the 2-stepemporal phase unwrapping algorithm (Eq. 18), the retrieved phase and its thicknessare shown in Figure 4(b) and 4(c), respectively. As the sensitivity gain
G = G = G = G × G . Thus, the noise is reducedas G is relatively small. By applying the proposed method to the wrapped phase maps,it acts to significantly reduce the error with the RMSE of 9.21 × -6 um.To better illustrate the difference, Figures 6 show the horizontal cross sectionsof the retrieved surfaces along the middle row. Through comparing Figures 6, we cansee that the proposed method is far more robust to noises when utilizing the iterativetwo-step temporal phase unwrapping as we have 11.56-times ( G = G × G = 5.28 ×2.19 = 11.56) more sensitivity in the phase-sum compared to the phase-difference.
4. Conclusion
In this paper, we presented an extended depth-range dual-wavelengthinterferometry for phase retrieval based on combined the iterative two-stepemporal phase unwrapping and the least squares method. It is an extension ofXu’s method. Comparing with Xu’s method, the proposed method solves thelimitation of being exclusively applicable to interferograms with special phaseshifts by utilizing the least squares method and generalized phase shiftingalgorithm. Besides, by introducing in the phase-sum and the iterative two-steptemporal phase-unwrapping, the proposed method greatly extends thedepth-range and improves the sensitivity, meanwhile reducing the noise impact.Based on simulated experiments presented in Section 3, the high precision of theproposed method have been successfully demonstrated and confirmed.
Acknowledgment
This work was supported by the [National Natural Science Foundation of China] under Grant[number 11672162,11302082 and 11472070]. The support is gratefully acknowledged.
References (1) Ichioka, Y.; Inuiya, M. Appl. Opt. , 11, 1507-1514.(2) Cheng, Y.; Wyant, J.C. Appl. Opt. , 23, 4539-4543.(3) Serrano-García; DI/ Martinez-García; A/ Toto-Arellano; NI/ Otani; Yukitoshi.AOT. , 3, 401-406.(4) Barada, D.; Kiire, T.; Sugisaka, J.-i.; Kawata, S.; Yatagai, T. Appl. Opt. , 50,237.(5) Massig, J. H.; Heppner, J. Appl. Opt. , 40, 2081-2088.(6) Kühn, J.; Colomb, T.; Montfort, F.; Charrière, F.; Emery, Y.; Cuche, E.; Marquet,P.; Depeursinge, C. Opt. Express. , 15, 7231-7242.7) Onodera, R.; Ishii, Y. Appl. Opt. 1998, 37, 7988-94.(8) Min, J.; Yao, B.; Gao, P.; Guo, R.; Ma, B.; Zheng, J.; Lei, M.; Yan,S.; Dan, D.;Duan, T. Appl. Opt. , 51, 191-196.(9) Ishii, Y.; Onodera, R. Opt. Lett. , 16, 1523-1525.(10) Abdelsalam, D. G.; Kim, D. Appl. Opt. , 50, 6153 .(11) Zhang, W.; Lu, X.; Fei, L.; Zhao, H.; Wang, H.; Zhong, L. Opt. Lett. , 39,5375- 5378.(12) Kühn, J.; Colomb, T.; Montfort, F.; Charrière, F.; Emery, Y.; Cuche, E.; Marquet,P.; Depeursinge, C. Opt. Express. , 15, 7231.(13) Abdelsalam, D.; Magnusson, R.; Kim, D. Appl. Opt. , 50, 3360.(14) Polhemus, C. Appl. Opt. , 12, 2071-2074.(15) Cheng, Y. Y.; Wyant, J.C. Appl. Opt. , 23, 4539-4543.(16) Kakue, T.; Moritani, Y.; Ito, K.; Shimozato, Y.; Awatsuji, Y.; Nishio, K. Opt.Express. , 18, 9555(17) Huang, L.; Lu, X.; Zhou, Y.; Tian, J.; Zhong, L. Appl optics. , 55, 2363.(18) Zhang, W.; Lu, X.; Fei, L.; Zhao, H.; Wang, H.; Zhong, L. Opt. Lett. , 39,5375-8.(19) Xu, X.; Wang, Y.;Xu, Y.; Jin, W. Opt. Lett. , 41, 2430.(20) Wang, Z.; Han, B. Opt. Lett. , 29, 1671.(21) Xia, J.; Chen, Z.; Sun, H.; Liang, P.Y.; Ding, J.P. Appl. Optics. , 55,2843-2847.(22) Fan, J.; Xu, X.; Lv, X.; Liu, S.; Zhong, L. Chinese J. Lasers. , 43, 0308007.23) Wang, M.; Du, G.; Zhou, C.; Zhang, C.; Si, S.; Li, H. Opt Commun. , 385,43-53.(24) Fan, J.; Xu, X.; Zhang, W.; Lv, X.; Zhao, H.; Liu, S.; Zhong, L.Chinese J. Lasers. , 42, 1008004-10.(25) Servin, M.; Padilla, J. M.; Garnica, G. Extended depth-range profilometry usingthe phase-difference and phase-sum of two close-sensitivity projected fringes.arXiv.org. arXiv:1704.04662.(26) A. Papoulis, Probability, Random Variables and Stochastic Processes, 4th ed.(McGraw Hill, 2000).(27) Du, G.; Zhang, C.; Zhou, C.; Si, S.; Li, H.; Li, Y. Opt. Lasers Eng. , 79,22-28.(28) Servin, M.; Padilla, J. M.; Gonzalez, A.; Garnica, G. Opt. Express. , 23,15806-15.