Simultaneous measurement of out-of-plane displacementand slope using a multiaperture DSPI system and fast
Fourier transform
Basanta Bhaduri, Nandigana Krishna Mohan,* and Mahendra Prasad KothiyalDepartment of Physics Applied Optics Laboratory, Indian Institute of Technology Madras, Chennai TN 600 036, India
Digital speckle pattern interferometry (DSPI) anddigital shearography (DS) are two powerful noncon-tact, whole-field, highly sensitive optical techniquesfor the measurement of displacement and its deriva-tives in experimental mechanics [1,2]. These tech-niques extract accurate quantitative displacementdata by using the phase-shifting method [3]. Tempo-ral phase shifting (TPS) is a widespread method inDSPI and DS, in which the phase-shifted data areacquired in a temporal sequence of camera frames.But this method is susceptible to external distur-bances like vibration, temperature fluctuation, or therapid motion of the test object itself. The spatialphase-shifting (SPS) technique [4] and the spatialcarrier phase measurement technique [5] are simpleways to eliminate the external disturbances. In thismethod the spatial carrier is introduced into the in-terference pattern and a Fourier transform (FT) is
performed on the digitally recorded pattern. If thebandwidth of the amplitude variations of the inter-fering waves is less than the spatial frequency offset,then the Fourier spectrum will exhibit three distinctbands. One of the outer bands can be windowed offfrom the rest, shifted to zero frequency, and inversetransformed to yield a complex function whose phaseangle is the interference phase function of the origi-nal interferogram. The FT method for phase evalua-tion has been employed in DSPI and DS as well[6–8].
A few optical configurations have been reportedfor the real-time evaluation of out-of-plane dis-placement and its slope change from a single setup[9–11]. The primary advantage of such a DSPI sys-tem is the possibility of obtaining information aboutthe object displacement and its gradient from a sin-gle arrangement using a low-cost device suitable forindustrial applications. However these arrange-ments need to switch over from out-of-plane dis-placement to shear mode of operation or vice versafor extracting the desired information. For the si-multaneous measurement of out-of-plane displace-
ment and its slope, a TV holo-shearography setupwas first demonstrated by Mohan et al. [12]. Theexisting methods use the temporal phase-shiftingtechnique for quantitative data analysis, hence re-quiring at least three images before and after theobject is loaded.
We describe a DSPI system that employs a three-aperture mask in front of the imaging lens to obtainquantitative out-of-plane displacement and slope in-formation simultaneously by storing two speckle im-ages, once before and once after the object is loaded.Methods based on using the multiaperture maskwith shearing elements in front of the imaging sys-tem have been reported by recording and filteringthe specklegram on a holographic plate for thesimultaneous extraction of the displacement and itsderivatives from a single setup [13,14]. Further, amultiaperture mask placed in front of the imaginglens is useful to generate a spatial carrier fringe in-side the speckle [15,16]. In the proposed method adiffusely reflected object wave is allowed to fall on onehalf of the mask, where two apertures are situatedalong the vertical direction. A wedge plate (WP) witha small wedge angle is attached to one of these aper-tures for shearing the object wave. A smooth spheri-cal beam is allowed through the third aperture, whichis situated on the other half of the mask, to act as areference beam. These three waves—that is, the twosheared beams from the object and a smooth refer-ence beam—combine coherently at the image planewhere a CCD is placed. The FFT of the image willproduce seven distinct halos including a central haloin the spectrum. By selecting the appropriate halos,one can directly obtain two out-of-plane displace-ment phase maps and a slope phase map from thetwo speckle images, one before and the second afterloading the object. The obtained out-of-plane dis-placement phase maps are shifted to each other alongthe x direction. Further it is also shown that by sub-tracting these out-of-plane displacement phase mapsone can independently generate the slope phase map.Experimental results on a circular diaphragm clampedalong the edges and loaded at the center are presented.
2. Theory
Figure 1 is a schematic of the three-aperture maskthat contains three circular apertures of the diameterda. The aperture A1 is separated from the aperturesA0 and A2 by a distance D in the x and y directions,respectively. The three-aperture mask made of a thinhylam sheet is attached to the mount of the imaginglens that images the object onto the CCD plane. Theaperture A0 is used to introduce the smooth referencebeam while apertures A1 and A2 allow the objectbeam to be imaged. A WP with a small wedge angleis attached to aperture A2 such that a shear in theobject beam is introduced in the x direction. Theshear �x in the object plane can be written as [14],�x � U�n � 1��, where U is the object distance fromthe imaging lens, n is the refractive index of thewedge material, and � is the wedge angle.
Thus three beams, namely, the speckled objectwave, the sheared speckled object wave, and thesmooth reference wave, enter through the aper-tures independently and interfere at the imageplane. The interference pattern generates multiplecarrier fringes (crossed-grid structure) within thespeckle image recorded with a CCD camera. Con-sidering that the images formed on the CCD detec-tor are due to the superposition of the wavefrontscoming from A0, A1, and A2, one can express thecomplex wave amplitudes of these individualwavefronts as
where |u|s are the modulus of the amplitudes, ��x, y�and ��x � �x, y� are sheared object wave phases, f0x
and f0y are the horizontal and vertical carrier fringefrequencies within the speckle, respectively.
For the present case we have assumed that f0x
� f0y � f0. This carrier fringe frequency f0 can be givenby f0 � 1��, where � is the carrier fringe width. Thecarrier fringe width is given by � � V�D and theaverage size of the speckle �ds� is given by ds
� 1.22V�da, where � is the wavelength of the lightused, V is the image distance, D is the aperture sep-aration, and da is the aperture diameter [15]. For asmooth reference beam no spatial phase variation isassumed. Thus the total intensity I(x, y) on the CCDdetector is
It should be noted that the fourth and fifth terms inthis equation contain the information concerning thephase ��x, y�, the sixth and seventh terms representthe information concerning the phase ��x � �x, y�and the eighth and ninth terms contain the informa-tion concerning the phase difference ���x, y� � ��x ��x, y��. In order to get these phases, a FT is per-formed on the recorded intensity. Since a multiplica-tion in the space domain corresponds to a convolutionin the spatial frequency domain, the FT of Eq. (2) canbe expressed as
i � FT�I� � U0 � U0* � U1 � U1* � U2 � U2*� U0 � U1* � U1 � U0* � U0 � U2* � U2 � U0*� U1 � U2* � U2 � U1*, (4)
where Uj � FT�uj�; j � 0, 1, 2 and R denotes theconvolution operation. The spatial frequencies fx andfy have been dropped for simplicity.
The first term in Eq. (4) is a delta function at theorigin that corresponds to the dc bias of the image.The second and third terms are the speckle haloscentered at the origin, corresponding to the spectrumof the image speckle patterns due to two apertures, A1and A2. The fourth and fifth terms are the specklehalos due to the FTs of the interference of the fieldspassing through aperture pairs A0 and A1. They ap-pear shifted in the horizontal direction due to theangular offset between the reference and the objectbeam, u1�x, y�. The sixth and seventh terms are thespeckle halos due to the FTs of the interference of thefields passing through aperture pairs A0 and A2. Theyappear shifted both in the horizontal and verticaldirections due to the angular offset between the ref-erence and object beam, u2�x, y�. The eighth and ninthterms are the speckle halos due to the FTs of theinterference of the object beams passing through ap-erture pairs A1 and A2. They appear shifted in thevertical direction due to the angular offset betweenthe object beams.
Figure 2 is a schematic of the halos in the spatialfrequency domain obtained with a three-aperturemask. The size and the shape of the sidebands de-pend on the size and the shape of the apertures andthe nature of the interfering waves [8]. The sidebandsformed due to the interference between the two
speckled object waves are twice as large as thoseformed due to the interference between the speckledobject wave and the smooth reference wave [17].Likewise, the speckle halos around the origin arelarger. If the sidebands do not overlap each other, itis possible to separate them by using bandpass filter-ing. The sampling criterion states that the bandwidthin an image sampled with a CCD of pixel pitch dp is1�dp equally distributed over positive and negativefrequencies giving a maximum frequency �fmax� of1��2dp� in the image [6]. Thus the width of the spec-trum for each speckle image limited to 2fs � fmax�3� 1��6dp�, where fs is the maximum spatial frequencycontent of the speckle image, in order to avoid alias-ing [6]. So the minimum size of the speckle is sixtimes greater than the pixel pitch of the CCD for thepresent case.
Now if we select the fifth term U1 � U0* from Eq.(4) and perform the inverse FT, we will obtain u0u1*.The phase term ���x, y� � 2�f0x� is then calculated byusing the relation
where Im and Re represent the imaginary and realparts, respectively, N and D are the numerator andthe denominator of the arctangent function, respec-tively, and subscript B represents the state of theobject before deformation.
Similarly, if we select the eighth term U1 � U2*from Eq. (4) and perform the inverse FT, we will
Fig. 2. Schematic of the halos in the spatial frequency domainobtained with a three-aperture mask.
where N� and D� are the numerator and the denom-inator of the arctangent function, respectively, andsubscript B represents the state of the object beforedeformation. The object wave fields after object de-formation can be given by
Now using the same procedure we can calculatethe phase terms ����x, y� � 2�f0x� and ����x, y�� ���x � �x, y� � 2�f0y�. Thus the phase difference,���x, y� due to out-of-plane displacement can be ob-tained by [18]
where subscripts B and A represent the state of theobject before and after deformation, respectively.
The phase difference ����x, y� is related to theslope �w�x, y���x as [1, 2]
����x, y� �4�
�w�x, y��x �x, (11)
where �x is the shear magnitude in the object plane.It is interesting to note that we can get the phase
difference ���x � �x, y� due to out-of-plane displace-ment using a similar procedure as the one describedhere by selecting the seventh term in Eq. (4). Thisphase is the same as the one obtained with Eq. (8)with the difference that it is shifted in the x directionby �x. This contribution is mainly due to the inter-ference of the sheared object wave with the smoothreference via the apertures A0 and A2, respectively.Subtraction between the two sheared out-of-planedisplacement phase maps also yields the slope phasemap as it can be represented as
Fig. 3. (Color online) Schematic of a three-aperture digitalspeckle pattern interferometric arrangement: O, object; RM, ref-erence mirror; BS, beam splitter; M, mirrors; P, front-surfaces-coated right angle prism; A, three-aperture mask; NDF, neutraldensity filter; L1, lens; L2, imaging lens.
Fig. 4. Magnified portion of the speckle pattern obtained using amulti-aperture arrangement revealing the multiple spatial carrierfringes within the speckle.
The resulting phase maps obtained in this methodare noisy, so filtering is necessary to improve them.For the present case, we have filtered using a 5 � 5window phase filter after sine-cosine division [2]. Thefiltered phase distributions are the wrapped or
modulo-2� phase map, which require unwrapping toget desired a continuous phase function [19].
3. Experiment and Results
The experimental arrangement is shown in Fig. 3.The arrangement contains a three-aperture mask (A)for spatial carrier frequency generation inside thespeckle. As mentioned in Section 2, the speckle sizeshould be equal to six CCD pixels. In our experimen-tal setup, this requires an aperture size of 2.65 mm.Further for a carrier fringe width of 3 pixels pitch ofCCD (both in the x and the y directions), spacing Dbetween the apertures both in the x and the y direc-tions will be 4.35 mm. The aperture mask used in ourexperiment has an aperture diameter of 2.8 mm andaperture separation of 4.2 mm in both the x and they directions. It does not meet the requirement exactlybut was nevertheless used to prove the principle ofthe method.
The experiments were conducted on a white mattepaint-coated 60 mm diameter circular diaphragmwith its edge rigidly clamped and loaded at the cen-ter. The object is illuminated by a 20 mW He–Nelaser beam after collimation. A reference mirror (RM)is placed very near to the object so that the laserbeam is also incident on the RM. The scattered ob-ject wave enters through the two apertures with thehelp of a mirror (M1) and the front-surfaces-coated
Fig. 5. Moiré correlation fringes obtained as the real-time sub-traction of the deformed frame from the initial frame.
Fig. 6. Spectrum of the recorded interferogram obtained with fastFourier transform. Halos 1 and 2 are used for inverse Fouriertransformation.
Fig. 7. (Color online) Out-of-plane displacement evaluation: (a)filtered phase map, (b) unwrapped 2D and (c) 3D plots.
right angle prism (P). A WP with a small wedge angle��20 arc min) is attached to one of these two aper-tures to introduce shear along the x direction. Theshear in the object plane is approximately 3 mm. Theintensity and the size of the specularly reflectedsmooth wave from the RM are controlled with thesupport of a neutral density filter (NDF) and a smallconvex lens �L1�. The reference beam is directed viathe mirror �M2� and the front-surfaces-coated rightangle prism (P) to the third aperture. The three-aperture mask is placed close to an imaging lens, L2.The three beams enter the apertures independentlyand combine coherently at the CCD plane (imageplane). We used a Jai CV-A (Pulnix) CCD camera,with a 1376 1035 matrix and 4.65 4.65 �m2 pixelsize. Precise alignment of the mirrors, the lens L1, theWP, and the front-surfaces-coated right angle prismare essential for the proper superposition of thebeams onto the CCD plane. Figure 4 shows a magni-fied portion of the speckle pattern obtained with amulti-aperture arrangement revealing the multiplespatial carrier fringes within the speckle. Softwarebased on LabVIEW was developed to record the im-ages before and after object deformation. The soft-ware also displays the real-time subtraction of theimages, thus the correlation speckle fringe can beviewed during loading. There will be a superpositionof the three correlation fringe patterns, two out-of-plane displacement fringe patterns, and one slope
fringe pattern. Figure 5 shows such a Moiré correla-tion speckle fringe pattern as observed during theload.
Figure 6 shows the spectrum of the recorded inter-ferogram obtained with a FFT, which reveals thediffraction halos as obtained theoretically (Fig. 2). Ifwe select halo 1 in Fig. 6 and perform inverse FFT forboth frames before and after object deformation, weget the phase map due to out-of-plane displacementaccording to Eq. (8). Figure 7(a) shows the filteredphase map whereas Figs. 7(b) and 7(c) show the cor-responding unwrapped 2D and 3D plots, respectively.Similarly, if we select halo 2 in Fig. 6 and performinverse a FFT we get the phase map due to slopeaccording to Eq. (10). As halo 2 is formed due to thesuperposition of two speckle sheared waves, its size isalmost double compared with halo 1, which is due tothe superposition of a speckle object wave and asmooth reference wave [17]. There is a superpositionof the halos along the vertical direction because thecarrier frequency is less than the required value asour aperture separation is less than the designedvalue. As a result, the obtained phase map for theslope will be noisier as compared with the out-of-plane displacement phase map as shown in Fig. 8.Figure 8(a) shows the filtered phase map whereasFigs. 8(b) and 8(c) show the corresponding un-wrapped 2D and 3D plots, respectively.
Fig. 8. (Color online) Slope evaluation using halo 2 in Fig. 6: (a)filtered phase map, (b) unwrapped 2D and (c) 3D plots.
Fig. 9. (Color online) Slope evaluation by subtracting the out-of-plane displacement phase maps obtained from halos 1 and 3 inFig. 6: (a) filtered phase map, (b) unwrapped 2D and (c) 3D plots.
There is yet another possibility to get the slopeinformation from this experiment. If we select halo 3in Fig. 6 and perform an inverse FFT for both framesbefore and after object deformation, we get the phasemap due to out-of-plane displacement that is similarto Fig. 6(a) but with a shift of �x in the x direction.Now subtracting the phase map corresponding to theout-of-plane displacement obtained from halo 1 fromthe phase map obtained from halo 3, we can indepen-dently generate the slope phase map according to Eq.(12). Figure 9(a) shows the filtered slope phase mapwhereas Figs. 9(b) and 9(c) show the correspondingunwrapped 2D and 3D plots, respectively. It can beseen from Fig. 9 that the results are much bettercompared with that of Fig. 8.
4. Conclusion and Discussion
We have presented the simultaneous quantitativemeasurement of out-of-plane displacement and slopeusing the FT method with a single three-apertureDSPI arrangement. The advantage of this technique isthat speckle phase can be measured from a singleframe. This has an advantage when the deformationvaries continuously. The method can be used for thequantitative analysis of quasi-dynamic behavior of theobject. Further, the proposed technique is simple andcost effective as it requires only a three-aperture maskin front of the imaging lens instead of a conventionalPZT-driven phase-shifting unit.
The resolution of the camera plays an importantrole as it not only determines the image resolutionbut also the aperture sizes to be used for the gener-ation of large speckles. With a lower resolution cam-era, the aperture sizes need to be smaller indimension. Thus the optical system requires a highpower laser to increase the reflected light from object.
The slope phase map obtained independently bysubtracting the two sheared out-of-plane displacementphase maps generated via halos 1 and 3, respectively,as shown in Fig. 6, is much better compared withthat obtained by selecting halo 2 alone. This givesthe advantage that the reference wave could bemade brighter, so as to drive the no-longer-relevantspeckle interference (and speckle noise) into thebackground and improve the signal-to-noise ratio(SNR) [20]. Further the speckles can be madesmaller, since the central halo and the sidebandsdue to the interference of the object waves becomecomparatively weak against the halos formed withthe smooth reference wave, and the overlap of thesehalos with the signal bands would not greatly dis-turb the phase reconstruction. This would improvethe spatial resolution and make the instrumentmore light efficient.
The limitation of the method is that the aperturemask needs to be redesigned every time the geometryis changed to accommodate the different sized ob-jects. Also for testing large objects we need a higherpower laser.
We are thankful to the reviewers for their usefulsuggestions.
References1. P. K. Rastogi, ed., Digital Speckle Pattern Interferometry and
Related Techniques (Wiley, 2001).2. W. Steinchen and L. Yang, Digital Shearography: Theory and
Application of Digital Speckle Pattern Shearing Interferometry(SPIE Press, 2003).
3. K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt.24, 3053–3058 (1985).
4. M. Kujawinska, “Spatial phase measurement methods,” in In-terferogram Analysis-Digital Fringe Pattern MeasurementTechniques, D. W. Robinson and G. T. Reid, eds. (Institute ofPhysics, 1993), Chap. 5.
5. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transformmethod of fringe-pattern analysis for computer-based topogra-phy and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
6. G. Pedrini, Y. L. Zou, and H. Tiziani, “Quantitative evaluationof digital shearing interferogram using the spatial carriermethod,” Pure Appl. Opt. 5, 313–321 (1996).
7. H. O. Saldner, N.-E. Molin, and K. A. Stetson, “Fourier-transform evaluation of phase data in spatially phase-biasedTV holograms,” Appl. Opt. 35, 332–336 (1996).
8. T. F. Begemann and J. Burke, “Speckle interferometry: three-dimensional deformation field measurement with a single in-terferogram,” Appl. Opt. 40, 5011–5022 (2001).
9. T. W. Ng, “Digital speckle pattern interferometer for combinedmeasurements of out-of-plane displacement and slope,” Opt.Commun. 116, 31–35 (1995).
10. P. A. Fomitchov and S. Krishnaswamy, “A compact dual-purpose camera for shearography and electronic speckle-patterninterferometry,” Meas. Sci. Technol. 8, 581–583 (1997).
11. B. Bhaduri, N. Krishna Mohan, and M. P. Kothiyal, “A dual-function ESPI system for the measurement of out-of-planedisplacement and slope,” Opt. Lasers Eng. 44, 637–644 (2006).
12. N. Krishna Mohan, H. O. Saldner, and N.-E. Molin, “Electronicspeckle pattern interferometry for simultaneous measurementof out-of-plane displacement and slope,” Opt. Lett. 18, 1861–1863 (1993).
13. R. S. Sirohi, ed., Speckle Metrology (Marcel Dekker, 1993).14. R. K. Mohanty, C. Joenathan, and R. S. Sirohi, “Speckle and
speckle-shearing interferometers combined for the simulta-neous determination of out-of-plane displacement and slope,”Appl. Opt. 24, 3106–3109 (1985).
15. R. S. Sirohi, J. Burke, H. Helmers, and K. D. Hinsch, “Spatialphase shifting for pure in-plane displacement and displacement-derivative measurements in electronic speckle pattern inter-ferometry (ESPI),” Appl. Opt. 36, 5787–5791 (1997).
16. B. Bhaduri, N. Krishna Mohan, M. P. Kothiyal, and R. S.Sirohi, “Use of spatial phase shifting technique in digitalspeckle pattern interferometry (DSPI) and digital shearogra-phy (DS),” Opt. Express 14, 11598–11607 (2006).
17. J. Burke, “Application and optimization of the spatial phaseshifting technique in digital speckle interferometry,” Ph.D.dissertation (Carl von Ossietzky University, Oldenburg, Ger-many, 2000).
18. J. M. Huntley, “Random phase measurement errors in digitalspeckle interferometry,” Opt. Lasers Eng. 26, 131–150 (1997).
19. D. C. Ghiglia and L. A. Romero, “Robust two-dimensionalweighted and unweighted phase unwrapping that uses fasttransforms and iterative methods,” J. Opt. Soc. Am. A 11,107–117 (1994).
20. J. Burke and H. Helmers, “Spatial versus temporal phaseshifting in ESPI: noise comparison in phase maps,” Appl. Opt.39, 4598–4606 (2000).
