Digital speckle-pattern interferometry: fringe retrievalfor large in-plane deformations with digitalspeckle photography

Angelica Andersson, Anna Runnemalm, and Mikael Sjodahl

The compensation of large in-plane motions in digital speckle-pattern interferometry ~DSPI! with the useof digital speckle photography ~DSP! is demonstrated. Ordinary recordings of DSPI are recombined andanalyzed with DSP. The DSP result is used to compensate for the bulk speckle motion prior to calcu-lation of the phase map. This results in a high fringe contrast even for deformations of several specklediameters. In addition, for the case of an in-plane deformation, it is shown that the absolute phasechange in each pixel may be unwrapped by use of the DSP result as an initial guess. The principles ofthis method and experiments showing the in-plane rotation of a plate and the encounter of two roundedplates are presented. © 1999 Optical Society of America

OCIS codes: 120.3940, 120.6160, 120.4290, 120.3180.

1. Introduction

Digital speckle-pattern interferometry ~DSPI! @some-times referred to as electronic speckle-pattern inter-ferometry ~ESPI!, TV holography, or electro-opticholography# is a well-established technique for themeasurement of small object deformations. If thein-plane deformation of the object exceeds 1 specklediameter, assuming the speckle size to be larger thanthe pixel pitch, uncorrelated speckles are comparedand the fringes are lost. The measurement range inone measurement step is therefore limited to only afew micrometers. In particular, measurements ofin-plane deformations suffer from this problem, sinceloading the object in plane easily introduces rigidbody motions. However, if the temporal samplingrate is high enough, the reference image may be up-dated so that the total deformation becomes the sumof the incremental unwrapped deformations. Con-sequently, deformations beyond 1 speckle diametermay be measured. One further step is to increasethe temporal sampling rate even more so that at leasttwo incremental phase maps are obtained for every

The authors are with the Division of Experimental Mechanics,Luleå University of Technology, SE-971 87 Luleå, Sweden. A.Andersson’s e-mail address is angelica.andersson@mt.luth.se.

Received 17 February 1999; revised manuscript received 14 May1999.

0003-6935y99y255408-05$15.00y0© 1999 Optical Society of America

5408 APPLIED OPTICS y Vol. 38, No. 25 y 1 September 1999

2p revolution, a technique known as temporal phaseunwrapping1 or temporal speckle-pattern inter-ferometry.2 The total phase change then becomesthe sum of the incremental phase changes. There-fore the absolute phase in each pixel is unwrappedalong the time axis independently from all other pix-els without the need to bother about losses in fringecontrast. However, if the required sampling rate isnot available or if the loading apparatus does notallow for incremental loading, the above techniquescannot be used. One other feature with the tempo-ral techniques above is that they measure the changein phase in a specific pixel on the detector as theobject deforms rather than measuring the change inphase in a specific speckle ~which relates to a specificpoint on the object!. Therefore the total phasechange in one pixel may have gained contributionsfrom many different points on the object as the objectdeforms.

In contrast to DSPI, digital speckle photography~DSP! is a technique that measures the absolute de-formation of a speckle pattern over several tens ofspeckle diameters.3 In DSP the bulk motion of aspeckle pattern is determined from the peak positionof the digital cross correlation between subimages,typically of 32 3 32 pixels, from the reference and thedeformed images, respectively. Our purpose in thispaper is to show that DSPI combined with DSP offersa possibility to measure large deformations with in-terferometric accuracy without having to require in-termediate images. It was previously shown that

Mm

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the phase coding of the images used in DSPI is can-celed by a recombination of the phase-stepped imagesused, thus leaving only the speckle modulation.4These images are analyzed by DSP to find the bulkmotion of the speckle pattern between the exposures.Compensating for the speckle motion in the phasecalculation results in a high-quality phase map, al-though the speckle motion is several speckle diame-ters. In addition, it is shown that, in the case ofin-plane deformations, it is possible to use the resultfrom the DSP calculation to unwrap the absolutephase in each pixel independently from all other pix-els. First, in Section 2, we describe the principlebehind the method, and then, in Section 3, the exper-imental results are presented.

2. Theory

A condition for DSP is that 2 max@Px, Py# # l~1 1!F# ~the Nyquist sampling criterion!, whereax@Px, Py# means the largest of the two parameters,

px and py are the pixel pitches in the x and the ydirections, respectively. l is the wavelength of theaser, M is the magnification, and F# multiplied byhe focal length of the lens gives the diameter of theperture.5 This shows that the speckle size ~s! muste greater than 2 pixels to satisfy the sampling cri-eria. The optical configuration is shown in Fig. 1.t is the classical Leendertz configuration for in-planeeformations6 and has the sensitivity vector along

the x axis. A beam splitter divides the laser lightinto two coherent and collimated beams that illumi-nate the object. One of these beams can be phasestepped and is used to obtain four phase-stepped im-ages at each state of object deformation. The irra-diance of the images of the undeformed object isexpressed as7

In~x1, y1! 5 I0~x1, y1! 1 Im~x1, y1!cos@f~x1, y1! 1 npy2#,

n 5 0, 1, 2, 3, (1)

where I0 is the background irradiance, Im is the mod-ulation irradiance, and f is the random specklephase.

Fig. 1. Optical configuration of the system for measurements oflarge in-plane deformations. The system is sensitive to deforma-tions in the x direction. M, mirror; BS, beam splitter; PS, phase-stepped mirror; BE, beam expander; CCD, video camera.

1

After deformation four new phase-stepped imagesof the object are taken:

In~x2, y2! 5 I0~x2, y2! 1 Im~x2, y2!cos@f~x2, y2! 1 npy2

1 V~x2, y2!#, n 5 0, 1, 2, 3, (2)

where V is the phase change along the sensitivityector caused by the deformation, i.e., the quantity ofnterest.

To cancel the background irradiances,4,8 the follow-ing new images are created from Eqs. ~1! and ~2!:

C1 5 I0~x1, y1! 2 I2~x1, y1! 5 2Im~x1, y1!cos@f~x1, y1!#,

(3)

S1 5 I3~x1, y1! 2 I1~x1, y1! 5 2Im~x1, y1!sin@f~x1, y1!#,

(4)

C2 5 I0~x2, y2! 2 I2~x2, y2!

5 2Im~x2, y2!cos@f~x2, y2! 1 V~x2, y2!#, (5)

S2 5 I3~x2, y2! 2 I1~x2, y2!

5 2Im~x2, y2!sin@f~x2, y2! 1 V~x2, y2!#. (6)

These images contain information about both thespeckle modulation and the phase in each pixel beforeand after deformation, respectively. The phase in-formation is not of interest when DSP is used.Therefore a reference image and an image of thedeformed object are created as

2Im~x1, y1! 5 ~C12 1 S1

2!1y2, (7)

2Im~x2, y2! 5 ~C22 1 S2

2!1y2. (8)

In these images, there is no phase information, and aspeckle pattern that will follow the motion of thesurface of the object is obtained. Therefore the rel-ative motion between the two speckle patterns inEqs. ~7! and ~8! can effectively be analyzed by DSP.

two-dimensional deformation field is obtained thatives the speckle motion ~Dx, Dy! in the plane of the

detector. DSP is reliable as long as the speckle de-correlation is smaller than 50%. If the decorrelationincreases, the reliability is lost, and the phase cannotbe retrieved.

In Eqs. ~3!–~6! the phase term is described for bothf~x, y! and V~x, y!. f~x, y! is the random specklephase and is of no interest in this case. V~x, y!,however, is the phase term that is wanted. For aLeendertz arrangement that has an in-plane sensi-tivity direction, the relation between the phase termV~x, y! and the deformation u~x, y! along the sensi-tivity vector is described by

V~x, y! 5 ~4pyl!u~x, y!sin u, (9)

where u is the angle to the optical axis for the twoillumination beams. In DSPI the speckles are gen-erally assumed to be stationary during an experi-ment; i.e., x2 5 x1 5 x and y2 5 y1 5 y. The wrapped

September 1999 y Vol. 38, No. 25 y APPLIED OPTICS 5409

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phase change, Vw~x, y! ~i.e., all values lie in the in-terval @2p, p#!, is obtained from Eqs. ~3!–~6! as9

Vw~x, y! 5 arctanSC1 S2 2 C2 S1

C1 C2 1 S1 S2D , (10)

ithout being disturbed by the random speckle phase. If the deformation exceeds 1 speckle diameter,

he speckle correlation drops to zero, and the infor-ation is lost.7 Therefore DSPI is limited to defor-

mations smaller than 1 speckle diameter. However,if S1~x, y!, C1~x, y!, S2~x 1 Dx, y 1 Dy!, and C2~x 1 Dx,

1 Dy! ~where Dx and Dy are the in-plane compo-nents rounded to the closest integral pixel value ofthe speckle motion obtained from the DSP calcula-tion! are used in Eq. ~10!, the result is visible fringeseven in the presence of large motions. This meansthat Vw can be obtained for deformations consider-ably larger than 1 speckle diameter. One factor thatcomplicates use of DSP to estimate the speckle bulkmotion is that the spatial resolution of DSP is typi-

Fig. 2. Rotation of a plate. ~a! Wrapped phase map obtained wh~c! Wrapped phase map obtained with DSPI when the informatiodirection.

410 APPLIED OPTICS y Vol. 38, No. 25 y 1 September 1999

cally 1y16 that of DSPI because of the finite size of thesubimage ~typically 32 3 32 pixels! used in DSP.

his means that the variation in the speckle motionust be less than 1 speckle diameter within the size

f the subimage used in the DSP calculation. Inractical applications, however, the deformation issually smooth, so interpolating in the DSP resultives a sufficient estimate of Dx and Dy in each pixel.From Eq. ~10! a wrapped phase map is obtainedhere no phase information is lost even if the defor-ation exceeds 1 speckle diameter. Once therapped phase map is obtained, it can be unwrapped.owever, the result obtained from DSP may also besed to unwrap the absolute phase ~as opposed to theelative phase! in each pixel, provided it is accuratenough. If the deformation field obtained by DSP,.e., uDSP, is used, an estimate of the absolute phase in

each point is given by @compare with Eq. ~9!#

V~x, y! 5 ~4pyl!uDSP~x, y!sin u, (11)

rdinary DSPI is used. ~b! Deformation field obtained with DSP.~b! is used. ~d! Unwrapped deformation in each pixel in the x

en on from

ˆ

pV

r

2

fisibia2tsbstdsio

tcpwwtmetef

fFtttbpaarcnodspmwpiw

where V~x, y! is an estimate of the absolute value ofthe phase in each pixel. Two values of the phase areobtained: first, a wrapped value Vw of the real

hase and, second, an estimate of the absolute phaseˆ . V is then given by

V~x, y! 5 2pm 1 Vw~x, y!, (12)

where m, which is an integer, is given by

m 5 INTFV~x1, y1! 2 Vw~x1, y1!

2pG , (13)

provided that the estimate V is within 2p of the cor-ect phase V. If Eqs. ~12! and ~13! are used on every

point of the object, a map of the deformation is ob-tained that has the same advantages as temporalphase unwrapping. This means that phase errorsdo not spread in the measurements.

3. Experiments

In these experiments, a Nd:YAG laser with a wave-length of 532 nm was used to illuminate the objectand a NEC Model TI-234A camera with a pixelpitch of 8.3 mm horizontally and 9.7 mm verticallywas used to detect the deformations. The CCDcamera is connected to an image processing system,which was developed at Recognition Technology,Inc. A telecentric lens ~Melles Griot! with fixedmagnification, M 5 0.5, and with an F# of approx-imately 24 was used. This gave a speckle size s of

pixels in the vertical direction.As a first experiment an aluminum plate was

astened onto a rotation table to perform a puren-plane rotation. The plate was rotated manuallyo much that the fringes were lost in parts of themage. This means that ordinary DSPI could note used to achieve useful information in the wholemage. In this experiment only the deformationlong the x axis was measured. In Figs. 2~a! and~c! the difference between the phase maps whenhe compensation from DSP was used or not can beeen. Figure 2~b! shows the deformation obtainedy DSP. Note that the center of rotation was out-ide the field of view. In Fig. 2~d! a cross section ofhe unwrapped absolute, as opposed to the relative,eformation is shown. Here it can be seen that thetraight line does not cross the origin, which alsomplies that the center of the rotation was outsideur image.In the second experiment two plates were moved

oward each other in the x direction so that theontact point was in the center of the image. Thelates were made of poly~methyl methacrylate! andere machined to have rounded tips. The endsere rounded to achieve a point contact. At first,

he two plates were separated by approximately 0.1m ~approximately 3 speckle diameters or 6 pix-

ls!. Therefore only a pure translation occurred inhe beginning when the plates were moved towardach other. When the plates came in contact, aorce in the x direction was introduced, and a de-

1

ormation took place. The DSP result shown inig. 3~a! shows that the relative displacement of the

wo plates is approximately 6 pixels. Thereforehe compensation that has to be made to retrievehe fringes in DSPI is approximately 63 pixels foroth halves. In Fig. 3~b! the compensated wrappedhase map is shown. The fringe contrast is high,lthough the relative displacement is 3 speckle di-meters. In this experiment excessively large er-ors were introduced into the DSP result in theontact point between the plates. Therefore it wasot possible to unwrap the phase map with the helpf Eqs. ~12! and ~13!. As a result only the relativeisplacement and the wrapped phase map are pre-ented in this paper. However, it is theoreticallyossible to achieve an unwrapped absolute phaseap of the plates by use of the values in two pointsell away from the contact point ~one point on eachlate!. The image could then be divided into twomages @one plate in each#, which could be un-rapped by use of a spatial algorithm.

Fig. 3. Two plates that were moved toward each other. ~a! Meshof the measured displacement in the x direction, by use of DSP.The 6-pixel displacement is approximately 3 speckle diameters.~b! Retrieved phase map.

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5

4. Conclusions

The eight images used in this combined technique arethe same images that are usually used in DSPI. Thenew feature is that DSP can be used to obtain atwo-dimensional deformation field of the absolute de-formation. This can be used to determine whichspeckles are to be compared in the DSPI technique sothat a large deformation can be detected at the sametime the phase information is retained. Thereforefringes are obtained, although the deformation ex-ceeds 1 speckle diameter. In addition, the absolutephase in each pixel can be obtained independentlyfrom all other pixels ~and discontinuous phase jumpscan be determined!, provided that the accuracy in theDSP result is within 6p of the correct phase.

We conclude with some remarks concerning thespeckle size relative to the pixel size. When smallspeckles are used ~s # p!, the fringe contrast is lim-ited by p ~the pixel pitch! rather than s ~the specklesize!.10 Therefore it is always sufficient to locate thespeckle displacement with a resolution of 1 pixel.This means that, if only the fringe contrast is of in-terest, it is not necessary for the speckles to be re-solved. Aliasing can be accepted, since the DSPcalculations will attract the nearest integral pixelvalue.5 However, if the DSP result should be usedfor absolute phase unwrapping, the Nyquist sam-pling criterion must not be violated.

The J. C. Kempe foundation financed the DSPIsystem.

412 APPLIED OPTICS y Vol. 38, No. 25 y 1 September 1999

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unwrapping algorithm for automated interferogram analysis,”Appl. Opt. 32, 3047–3052 ~1993!.

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8. K. A. Stetson, “Theory and applications of electronic hologra-phy,” in Proceedings of the International Conference on Holo-gram Interferometry and Speckle Metrology, K. A. Stetson andR. J. Pryputniewicz, eds. ~Society for Experimental Mechanics,Bethel, Conn., 1990!, pp. 294–300.

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10. A. Papoulis, Probability, Random Variables, and StochasticProcesses ~McGraw-Hill, New York, 1965!.