REVIEW OF SCIENTIFIC INSTRUMENTS 80, 093107 共2009兲

Strain, curvature, and twist measurements in digital holographic interferometry using pseudo-Wigner–Ville distribution based method G. Rajshekhar, Sai Siva Gorthi, and Pramod Rastogia兲 Applied Computing and Mechanics Laboratory, IMAC IS ENAC EPFL, Swiss Federal Institute of Technology, 1015 Lausanne, Switzerland

共Received 29 July 2009; accepted 30 August 2009; published online 24 September 2009兲 Measurement of strain, curvature, and twist of a deformed object play an important role in deformation analysis. Strain depends on the first order displacement derivative, whereas curvature and twist are determined by second order displacement derivatives. This paper proposes a pseudo-Wigner–Ville distribution based method for measurement of strain, curvature, and twist in digital holographic interferometry where the object deformation or displacement is encoded as interference phase. In the proposed method, the phase derivative is estimated by peak detection of pseudo-Wigner–Ville distribution evaluated along each row/column of the reconstructed interference field. A complex exponential signal with unit amplitude and the phase derivative estimate as the argument is then generated and the pseudo-Wigner–Ville distribution along each row/column of this signal is evaluated. The curvature is estimated by using peak tracking strategy for the new distribution. For estimation of twist, the pseudo-Wigner–Ville distribution is evaluated along each column/row 共i.e., in alternate direction with respect to the previous one兲 for the generated complex exponential signal and the corresponding peak detection gives the twist estimate. © 2009 American Institute of Physics. 关doi:10.1063/1.3234260兴

I. INTRODUCTION

Strain evaluation and determination of flexural and torsional moments are important aspects of deformation measurements with significant applications in material characterization, reliability analysis, and quality control. The flexural and torsional moments correspond to curvature and twist of a deformed object. In the past few decades, optical interferometric techniques have become popular tools for deformation analysis mainly because of their noncontact nature and whole field measurement capability. Digital shearography is an important optical technique, which directly provides the strain information but its sensitivity depends on the amount of shearing introduced in the measurement process.1 Another important optical technique is digital holographic interferometry 共DHI兲, whose popularity was ensured by the advent of high resolution charge coupled device 共CCD兲 camera, which greatly facilitated the digital recording of hologram and numerical reconstruction of object wave fields, thereby directly providing the complex signal corresponding to the object wave.2 Digital holography has found many applications in the field of nondestructive testing.2–4 In recent years, various methods using DHI have been proposed for slope and curvature measurement.5–8 These methods rely on approximating the phase differentiation operation for a complex signal by multiplying the complex signal with its pixel shifted complex conjugate. Wrapped phase derivative maps are obtained using these methods. However, these wrapped first or higher order phase derivatives are observed to be highly susceptible to noise and hence several filtering schemes such as average a兲

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0034-6748/2009/80共9兲/093107/5/$25.00

filtering, sine cosine filtering, etc., have been proposed.9,10 In the above methods, filtering is done for several iterative cycles until satisfactory results are obtained. As large iterative filtering cycles tend to smear out dense fringe patterns due to excessive filtering and additionally could be time consuming, the number of iterative cycles has to be carefully chosen, which could be difficult in many practical situations. Also the first and higher order phase derivatives are obtained in a wrapped form, which necessitates application of an efficient unwrapping operation. The paper presents a pseudo-Wigner–Ville distribution 共PSWVD兲 based method to estimate first and higher order phase derivatives in DHI. PSWVD is a space frequency distribution which localizes the spectral content of a signal in space and provides a joint space-frequency representation of the signal energy. It has better immunity against interference terms compared with the conventional WVD due to the presence of a smoothing window11 and is a popular tool for instantaneous frequency estimation in many areas of signal processing. It can be noted that the phase derivative is usually referred to as instantaneous frequency in signal processing.12 Our proposed method relies on the implementation of PSWVD as a phase derivative estimation technique for the reconstructed interference field obtained in DHI. The theory of the proposed method is described in Sec. II. Simulation and experimental results are shown in Sec. III followed by conclusions in Sec. IV and acknowledgments. II. THEORY

For deformation analysis in DHI, holograms are recorded prior to and after deformation of object, i.e., for two object states. The complex signal corresponding to object

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© 2009 American Institute of Physics

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Rev. Sci. Instrum. 80, 093107 共2009兲

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wave is generated using numerical reconstruction.2 The product of complex signal corresponding to predeformed object state and conjugate of complex signal corresponding to postdeformed object state gives the reconstructed interference field, which can be represented as I共x,y兲 = a共x,y兲exp关j␾共x,y兲兴 + ␩共x,y兲,

共1兲

where a共x , y兲 is the slow varying amplitude term, ␾共x , y兲 is the interference phase, and ␩共x , y兲 represents the noise assumed to be additive white Gaussian noise 共AWGN兲. Here x and y refer to the pixel values along the N ⫻ N fringe pattern, i.e., x , y 苸 关1 , N兴. So y and x denote the rows and columns of an N ⫻ N matrix with elements I共x , y兲. The first and second order phase derivatives are given as

␻x共x,y兲 =

⳵ ␾共x,y兲 , ⳵x

共2兲

␻xx共x,y兲 =

⳵2␾共x,y兲 , ⳵ x2

共3兲

␻xy共x,y兲 =

⳵2␾共x,y兲 . ⳵x ⳵ y

共4兲

the curvature estimate ␻xx共x , y兲 and twist estimate ␻xy共x , y兲 can be given as

␻xx共x,y兲 =

⳵␻x共x,y兲 , ⳵x

共9兲

␻xy共x,y兲 =

⳵␻x共x,y兲 . ⳵y

共10兲

For curvature and twist estimation, direct numerical differentiation of strain estimate using Eqs. 共9兲 and 共10兲 could give erroneous estimates since small errors in strain estimation would propagate forward in second order derivative calculation. Hence to extract curvature and twist estimates from the strain estimate, a PSWVD based method is reused for which we construct a new complex signal with unit amplitude and the argument given by estimated phase derivative ␻x共x , y兲. The new complex signal can be given as

Since first and second order phase derivatives correspond to strain, curvature, and twist, their estimates would be referred to as strain estimate ␻x共x , y兲, curvature estimate ␻xx共x , y兲, and twist estimate ␻xy共x , y兲 in the rest of the paper. For implementation of PSWVD, consider an arbitrary row y for which Eq. 共1兲 can be written as I共x兲 = a共x兲exp关j␾共x兲兴 + ␩共x兲.

共5兲 11

The PSWVD of I共x兲 can be given as ⬁

W共x,⍀兲 =

w共␶兲I共x + ␶兲Iⴱ共x − ␶兲exp共− j2⍀␶兲, 兺 ␶=−⬁

共6兲

where ⴱ denotes complex conjugate, ␶ represents the spatial lag, ⍀ represents the angular frequency, and w is a real window of fixed length. For our analysis, we used a Gaussian window of the form w共x兲 =

冉 冊

1 − x2 2 1/2 exp 共2␲␴ 兲 2␴2

∀ x 苸 关− ␴/2, ␴/2兴,

共7兲

where length of the window is ␴ + 1. Now PSWVD concentrates the energy of the signal around the phase derivative12 and hence at any point x, the phase derivative or strain estimate ␻x共x兲 corresponds to the frequency at which W共x , ⍀兲 becomes maximum. In other words

␻x共x兲 = arg maxW共x,⍀兲. ⍀

共8兲

Equation 共8兲 gives the phase derivative or strain estimate along x for a given row y. The above procedure can be applied for all rows y 苸 关1 , N兴 to estimate the overall strain ␻x共x , y兲 along x for the entire fringe pattern. Noting that the second order phase derivatives can be obtained by differentiating the first order phase derivative, the curvature corresponds to derivative of ␻x共x , y兲 along x, whereas twist corresponds to derivative of ␻x共x , y兲 along y. Hence effectively,

FIG. 1. 共Color online兲 共a兲 Simulated fringe pattern 共256⫻ 256兲 at SNR of 20 dB, 共b兲 strain estimate ␻x共x , y兲 in radians/pixel, 共c兲 wrapped strain estimate, 共d兲 curvature estimate ␻xx共x , y兲 in radians/ pixel2, 共e兲 wrapped curvature estimate, 共f兲 twist estimate ␻xy共x , y兲 in radians/ pixel2, and 共g兲 wrapped twist estimate.

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Rev. Sci. Instrum. 80, 093107 共2009兲

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FIG. 2. 共Color online兲 共a兲 Experimentally recorded digital hologram in an off-axis configuration. 共b兲 Intensity of the numerically reconstructed hologram using discrete Fresnel transform. 共c兲 Phase in the marked inspection area of the reconstructed object wave field before deformation. 共d兲 Real part of the reconstructed interference field, which constitutes the experimental fringe pattern.

I⬘共x,y兲 = exp关j␻x共x,y兲兴.

共11兲

From Eq. 共11兲, it is clear that I⬘共x , y兲 effectively has ␻x共x , y兲 as its phase. Since the PSWVD acts as a phase derivative estimation technique as shown in Eqs. 共6兲 and 共8兲, it can be applied on I⬘共x , y兲 to estimate derivatives of ␻x共x , y兲. For curvature estimation, consider an arbitrary row y for which PSWVD of I⬘共x兲 and curvature estimate ␻xx共x兲 can be written as ⬁

W⬘共x,⍀兲 =

w共␶兲I⬘共x + ␶兲I⬘ⴱ共x − ␶兲exp共− j2⍀␶兲, 兺 ␶=−⬁ 共12兲

␻xx共x兲 = arg maxW⬘共x,⍀兲. ⍀



共15兲

Equations 共14兲 and 共15兲 can be repeated for all columns x to estimate the overall twist estimate ␻xy共x , y兲 for the entire fringe pattern. A major benefit of the proposed method is that the strain estimate ␻x共x , y兲, curvature estimate ␻xx共x , y兲 and twist estimate ␻xy共x , y兲 are directly obtained in unwrapped form thereby eliminating the need of an unwrapping algorithm. Note that strain and curvature are evaluated along x direction in our analysis; this is just for the sake of illustration of the proposed method. The method can be applied for strain and curvature evaluation along y direction by starting the method for an arbitrary column in Eq. 共5兲 and subsequently interchanging the role of rows and columns in the method’s procedure.

共13兲 III. SIMULATION AND EXPERIMENTAL RESULTS

Equations 共12兲 and 共13兲 can be repeated for all rows y to estimate the overall curvature estimate ␻xx共x , y兲 for the entire fringe pattern. For twist estimation, consider an arbitrary column x for which PSWVD of I⬘共y兲 and twist estimate ␻xy共y兲 can be written as ⬁

W⬙共y,⍀兲 =

␻xy共y兲 = arg maxW⬙共y,⍀兲.

w共␶兲I⬘共y + ␶兲I⬘ⴱ共y − ␶兲exp共− j2⍀␶兲, 兺 ␶=−⬁ 共14兲

Figure 1共a兲 shows a simulated fringe pattern 共corresponding to the simulated reconstructed interference field兲 with AWGN at a signal to noise ratio 共SNR兲 of 20 dB. Figure 1共b兲 shows the strain estimate ␻x共x , y兲 evaluated by the proposed method. The wrapped form of ␻x共x , y兲 is shown in Fig. 1共c兲. The curvature estimate ␻xx共x , y兲 evaluated from the proposed method is shown in Fig. 1共d兲. Its wrapped form is shown in Fig. 1共e兲. The twist estimate ␻xy共x , y兲 evaluated using the proposed method and its wrapped form are shown in Figs. 1共f兲 and 1共g兲. It needs to be emphasized that the

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Rev. Sci. Instrum. 80, 093107 共2009兲

FIG. 3. 共Color online兲 共a兲 Strain estimate ␻x共x , y兲 in radians/pixel for experimental fringe pattern, 共b兲 wrapped strain estimate, 共c兲 curvature estimate ␻xx共x , y兲 in radians/ pixel2, 共d兲 wrapped curvature estimate, 共e兲 twist estimate ␻xy共x , y兲 in radians/ pixel2, and 共f兲 wrapped twist estimate.

proposed method directly provides unwrapped estimates and their wrapped forms are shown for illustration purpose only. We used a Gaussian window with ␴ = 32 in Eq. 共7兲 for our analysis. The root mean square errors for strain, curvature, and twist estimation were found to be 1.90 ⫻ 10−3 rad/ pixel, 8.14⫻ 10−5 rad/ pixel2, and 3.81 −5 2 ⫻ 10 rad/ pixel . Note that two-dimensional median filtering was applied to smooth the estimates obtained by the proposed method and the first and the last 15 pixels along the borders were neglected to ignore errors near the boundaries. The computational time taken for the MATLAB implementation of the proposed method was approximately 30 s on a 2.66 GHz Intel Core 2 Quad Processor machine with 3.23 Gbyte random access memory.

The practical applicability of the proposed method is validated with experimental results. Two digital holograms are recorded by illuminating the test object with a Coherent Verdi laser 共532 nm兲 for different object states, i.e., before loading and after loading. The image of the digital hologram recorded before loading the object is shown in Fig. 2共a兲. Numerical reconstruction of the hologram is performed using discrete Fresnel transform.2 The intensity of the numerical reconstruction of the hologram in Fig. 2共a兲 is shown in Fig. 2共b兲. Since these holograms are recorded in an off-axis configuration, the real and virtual reconstructions of the object and the undiffracted pattern are separated, as shown in Fig. 2共b兲. The inspection area shown in Fig. 2共b兲 constitutes the region of interest. The phase of the reconstructed object

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093107-5

Rev. Sci. Instrum. 80, 093107 共2009兲

Rajshekhar, Gorthi, and Rastogi

wave field corresponding to the inspection area before deformation is shown in Fig. 2共c兲. Similarly, the reconstructed object wave field after deformation is obtained. From the two reconstructed object wave fields, the reconstructed interference field of the form given in Eq. 共1兲 is formed. Figure 2共d兲 shows the real part 共which constitutes a fringe pattern兲 of the reconstructed interference field corresponding to the loading of a circularly clamped object 共6 cm in diameter兲, located at a distance of 110 cm from the CCD camera 共SONY XCLU1000, 1628⫻ 1236兲. The proposed method based on PSWVD was applied to estimate the strain, curvature and twist from the experimental fringe pattern shown in Fig. 2共d兲. The strain estimate and its wrapped form are shown in Figs. 3共a兲 and 3共b兲. The curvature estimate and its wrapped form are shown in Figs. 3共c兲 and 3共d兲. The twist estimate and corresponding wrapped form are shown in Figs. 3共e兲 and 3共f兲. IV. CONCLUSIONS

The paper presents a robust and efficient method to estimate strain, curvature and twist in digital holographic interferometry. The method based on PSWVD does not require prior filtering of fringe pattern and directly provides un-

wrapped estimates in a computationally efficient manner. Simulation and experimental results validate the applicability of the proposed method for strain, curvature, and twist measurement and establish it as a potential technique in non destructive testing and evaluation. ACKNOWLEDGMENTS

This work is funded by Swiss National Science Foundation under Grant No. 200020- 121555. 1

W. Steinchen and L. Yang, Digital Shearography: Theory and Application of Digital Speckle Pattern Shearing Interferometry 共SPIE, Bellingham, WA, 2003兲. 2 U. Schnars and W. P. O. Juptner, Meas. Sci. Technol. 13, R85 共2002兲. 3 U. Schnars, H. J. Hartmann, and W. P. O. Juptner, Proc. SPIE 2545, 250 共1995兲. 4 S. Seebacher, W. Osten, and W. P. O. Juptner, Proc. SPIE 3479, 104 共1998兲. 5 Y. Zou, G. Pedrini, and H. Tiziani, Opt. Commun. 111, 427 共1994兲. 6 M. Y. Y. Hung, L. Lin, and H. M. Shang, Appl. Opt. 40, 4514 共2001兲. 7 C. Liu, Opt. Eng. 42, 3443 共2003兲. 8 U. Schnars and W. P. O. Juptner, Appl. Opt. 33, 4373 共1994兲. 9 W. Chen, C. Quan, and C. J. Tay, Appl. Opt. 47, 2874 共2008兲. 10 C. Quan, C. J. Tay, and W. Chen, Opt. Commun. 282, 809 共2009兲. 11 L. Cohen, Time Frequency Analysis 共Prentice-Hall, Englewood Cliffs, NJ,1995兲. 12 B. Boashash, Proc. IEEE 80, 519 共1992兲.

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Strain, curvature, and twist measurements in digital ...

Sep 24, 2009 - of shearing introduced in the measurement process.1 Another important optical technique is digital holographic interferom- etry (DHI), whose popularity was ensured by the advent of high resolution charge coupled device (CCD) camera, which greatly facilitated the digital recording of hologram and nu-.

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