out-of-plane displacement measurement by electronic speckle pattern interferometry in presence of...

Out-of-plane displacement measurement by electronicspeckle pattern interferometry in presence of large

in-plane displacement

R.A. Mart�ıınez-Celorioa,*, B. Barrientosa, Francisco J. Sanchez-Mar�ıına,Luis Mart�ıı L�oopezb, J.A. Rayasa

a Centro de Investigaciones en �OOptica, AC, Apdo 1-948, CP 37150, Le�oon, Gto, Mexicob Centro de Neurociencias de Cuba, Apdo 6412, CP 10600, La Habana, Cuba

Received 23 January 2002; received in revised form 29 April 2002; accepted 29 April 2002

Abstract

We measured out-of-plane displacement in presence of large in-plane displacements and deformations by electronic

speckle pattern interferometry (ESPI). By means of digital speckle photography (DSP) the large in-plane displacement

is measured and then compensated by software from interferometric images, before calculating the phase distribution

related to the out-of-plane deformation. This means that decorrelation effects are not present and that the use of in-

termediate images is not necessary. The optical phase was extracted by spatial phase shifting, which enables the study of

rapid transient events. The proposed method was applied to different combinations of out-of-plane deformation and

large in-plane displacement. The compensation of large in-plane displacement allows us to maintain the contrast of

ESPI fringes. � 2002 Elsevier Science B.V. All rights reserved.

PACS: 42.40.Kw; 42.30.Ms; 42.30.Rx

Keywords: Spatial phase stepping; Digital speckle photography; ESPI

1. Introduction

ESPI is a well-established technique to mea-sure small displacements of diffusely reflectingobjects [1]. This technique produces fringe pat-terns by subtracting two different speckle patterns

and presents several advantages as, for example,non-contact nature, easy automation of 3-Dmeasurements, high sensitivity, full-field analysis,etc [2]. Its measurement range, however, is rela-tively small, around 30 lm [3]. This limitationarises from decorrelation effects between corre-sponding speckles of two object states. Large in-plane displacement worsens this limitation sinceit lowers fringe contrast reducing this way themeasurement range [4]. Furthermore, it may leadto erroneous results, especially in the magnitude

1 July 2002

Optics Communications 208 (2002) 17–24

www.elsevier.com/locate/optcom

*Corresponding author. Fax: +52-477-717-5000.

E-mail address: [email protected] (R.A. Mart�ıınez-

Celorio).

0030-4018/02/$ - see front matter � 2002 Elsevier Science B.V. All rights reserved.

PII: S0030-4018 (02 )01556-0

mail to: [email protected]

of the displacement, which ultimately may inhibitits use. To overcome this problem, some tech-niques have been implemented, as, for example,the construction of a phase map out of phaseincrements [5] and the use of variants of thetemporal phase unwrapping technique, TPU [6].However, the former inherently presents lowspatial resolution due to high-density fringe mapsand in the latter a large number of intermediateimages are needed and the error stemming fromrigid body displacement (RBD) is not eliminated[7–9].An effective technique for measuring large in-

plane displacements is double-exposure specklephotography (SP). This technique is based on thecorrelation of two speckle patterns representingdifferent states of the object under test, e.g., be-fore and after displacing the object [10–12]. Thus,SP permits the study of displacements greaterthan the average speckle diameter, which impliesan interval in which ESPI cannot be applied.However, with this technique the sensitivity isabout 1/10 of that of ESPI. Digital speckle pho-tography (DSP) is a technique that combines thefeatures of the recording scheme of classicaldouble-exposure speckle photography with therecording on a CCD sensor [10]. Recently,Andersson et al. [13] have compensated largein-plane displacements in ESPI. These authorscorrelate two speckle fields, and the result givesthe RBD component, which can then be removed,enabling the use of ESPI.In this paper we present a method that com-

bines ESPI and DSP in order to study out-of planedisplacements in presence of large in-plane dis-placements (LIPD, e.g., RBD, rotation, and de-formation) greater than the mean specklediameter. To demonstrate the method, we usedspatial phase stepping techniques (SPS) [14]. SPStechniques allow us to analyze rapid transientphenomena. We show that the fringe contrast andhigh sensitivity of ESPI are maintained. In thesecond section a description of the method ispresented. In Section 3, the experimental detailsare explained, and in Section 4 we show and dis-cuss the results. Finally, in Section 5 we point outthe advantages and disadvantages of the proposedmethod.

2. Description of the method

Fig. 1 shows a block diagram describing theproposed method for measuring out-of-plane dis-placements in presence of large in-plane displace-ments. To apply the method two pairs of specklepattern are recorded; the first one corresponds tothe initial state of the object, whereas the secondone to its final state. One speckle pattern of eachpair is a recording of the interference between thereference beam and the object image, whereas theother one is a recording of the object image withno reference, that is a DSP. In other words, thechanges that the object undergoes are character-ized simultaneously by ESPI and by double ex-posure DSP. Afterwards, the recorded specklephotographs are used to measure the existingLIPD. Once LIPD is known, it may be compen-sated in the speckle-displaced state. After com-pensating the LIPD, a spatial phase steppingmethod is applied, which in turn produces awrapped phase map. Finally, the displacementinformation is obtained out of this phase map viaa path-independent unwrapping algorithm. In thefollowing subsections, these steps are explainedmore widely.

2.1. Correlation by subtraction with spatial phasestepping

Fig. 2 depicts a typical ESPI configuration formeasuring out-of-plane displacements using spa-tial phase stepping, SPS. A laser beam is directedto the object under test and the scattered light iscollected by an optical system OS (a lens and anaperture). A coherent reference beam is made toimpinge on a CCD with an inclination h with re-spect to the optical axis of the system, and intro-duces a variable optical path difference betweenthe scattered object beam and the reference beam.This path difference is reflected as a phase differ-ence, a, between consecutive pixels in the x-direc-tion [14,15]. By digitally displacing one of thespeckle patterns �Dx; 0;Dx, (Dx is the pixel sepa-ration), and calculating the absolute value of thesubtraction from a second speckle pattern, threefringe patterns, I�1; I0 and Iþ1, with a phasestepping of a, are calculated, i.e.,

18 R.A. Mart�ıınez-Celorio et al. / Optics Communications 208 (2002) 17–24

I�1ði; jÞ ¼ jSP1ði; jÞ � SP2ði; j� 1Þj

¼ I01½1þ b1 cosð/ðx; yÞ � aÞ�;

I0ði; jÞ ¼ jSP1ði; jÞ � SP2ði; jÞj

¼ I02½1þ b2 cosð/ðx; yÞÞ�;

Iþ1ði; jÞ ¼ jSP1ði; jÞ � SP2ði; jþ 1Þj¼ I03½1þ b3 cosð/ðx; yÞ þ aÞ�;

ð1Þ

where SP1 and SP2 are the consecutives specklepatterns; I0i are the average irradiances; bi are

Fig. 1. Block diagram describing the procedure of the proposed method to measure out-of-plane.

Fig. 2. ESPI setup with out-of-plane sensitivity. SPS is implemented by the use of a rotary stage.

R.A. Mart�ıınez-Celorio et al. / Optics Communications 208 (2002) 17–24 19

modulation factors; /ðx; yÞ is the unknown phasethat must be obtained from the interfering specklepatterns and a is the constant phase shift. In orderto apply SPS, the average transversal correlationlength of the speckle must be at least three timesthe separation between pixels of the CCD array.When this condition is met, we can assume thatI01 ffi I02 ffi I03 and b1 ffi b2 ffi b3 in Eq. (1). Thisassumption holds if the in-plane speckle displace-ment is less than the transversal correlation lengthof the speckle field.

2.2. Compensation of large in-plane displacement bydigital speckle photography

Compensation of LIPD is achieved by calcu-lating an intensity correspondence function be-tween two different speckle patterns. This may bedone by using a correlation algorithm [16]. To dothis, first, the speckle images are divided intoN N subregions; then the corresponding subre-gions are correlated, and the result is the localLIPD with subpixel accuracy. The resulting cor-relation peak defines the relative displacementbetween the speckle fields in the subimages,(Dx;Dy), according to

Aðx; yÞ � Bðx; yÞ ¼ I�1fI½Aðx; yÞ�I ½Bðx; yÞ�g; ð2Þ

where A and B ¼ Aðxþ Dx; y þ DyÞ are the po-sition functions, and �; I; I�1; represent theoperators for correlation, Fourier transforma-tion, inverse Fourier transformation, and com-plex conjugation, respectively. In DSP thespeckle size must be greater than two pixels tosatisfy the sampling criteria [13], which is fullysatisfied from the spatial phase shifting require-ments.Once the correlation is calculated, the LIPD is

compensated by software in the displaced speckleimage, thus enabling the use of Eq. (1). The opticalphase obtained from Eq. (1) is proportional un-iquely to the out-of-plane displacement.

2.3. Phase assessment

The optical phase can be calculated by anystandard phase stepping algorithm, such as [17]

/ðx; yÞ ¼ arctanI�1ðx; yÞ � Iþ1ðx; yÞ

I�1ðx; yÞ þ Iþ1ðx; yÞ � 2I0ðx; yÞ

� �;

ð3Þ

where I i are given by Eq. (1), which implicitlyimplies LIPD compensation. The values of theoptical phase /ðx; yÞ are within the range [�p; p].Finally, a path-independent unwrapping phasealgorithm is used [18] and the out-of-plane defor-mation calculated.

3. Experiment

A 50 mW He–Ne laser beam (k ¼ 633 nm) wasused as the illumination source. It is split into twocomponents, the reference and the object beams, ina ratio of 1:10, respectively. The reference beam isspatially filtered and adjusted to fall onto the CCDsensor at a small angle (h � 0 rad) with respect tothe normal of the CCD sensor by means of a ro-tary stage (RS). This angle gives rise to a phasestep between consecutive pixels in the local CCDin the x-direction.To show the feasibility of the method to measure

out-of-plane deformation in presence of LIPDthree experiments were performed. Firstly, a RBDwas analyzed. In this case the object beam illumi-nates a rectangular aluminum plate of 20 cm15 cm (thickness of 1 mm). The plate can be rigidlydisplaced perpendicularly to the optical axis. At theback of the plate a cord–pulley–weight mechanismis loaded with different weights allowing us to de-form the plate in a controlled fashion. In a secondexperiment, the plate was in-plane rotated aboutone of the plate corners. Simultaneously, an out-of-plane deformation was applied at the center, simi-larly as in the previous case. Finally, a fracturedaluminum rectangular plate 27 cm 8 cm (thick-ness of 1 cm) was used. In this case, a point force atthe center of the back of the plate was applied. Thedirection of the force made a small angle with re-spect to the normal of the plate.We used a Kodak CCD and a zoom lens

(working at an f# ¼ 30) to record the speckleimages. A mean speckle size of 40.9 lm at theCCD plane was obtained. The separation between


the CCD pixels is 9 lm in both the x- and y-di-rections. An 8-bit frame grabber (Matrox MeteorII) was used to digitize the speckle images with aresolution of 640 480 pixels.

4. Results and discussion

A set of four speckle images was recorded ineach experiment. Two of them are recorded beforedisplacing the object and the rest afterwards.These images correspond to the object beam and aspeckle image (containing the interference of thereference and the object beams) in each state, re-spectively. In the first experiment, the object issubjected to an RBD of Dx ¼ 80 lm in the x-di-rection and Dy ¼ 60 lm in the y-direction. In ad-dition to this, a weight of 28 g was applied at thecenter of the plate, yielding an out-of-plane de-formation of 949.5 nm. In presence of a similarout-of-plane deformation, in the second experi-ment, the plate is rotated about one of its corners.In Figs. 3(a)–(c) we show the resulting in-plane

displacements for each of the three experiments. Asubregi�oon of 16 16 pixels was used. Figs. 3(a)and (b) hold identical loading conditions under thesame specimen, but the undergoing in-plane dis-placement was modified: a LIPD (RBD) and an in-plane rotation, respectively. As it is shown in Fig.3(a), the resulting in-plane displacement is practi-cally uniform throughout the object while in thesecond case this varies according to the appliedrotation. Fig. 3(c) shows in turn the resultingLIPD when a fractured aluminum plate was loa-ded in a non-perpendicular way with respect to thenormal of the plate. In this case, the abrupt changein the length of the arrows indicates the presenceof a vertical fracture located at approximatelyx ¼ 250 pixels.Once the in-plane displacement is calculated,

compensation in a pixel-by-pixel way in the dis-placed speckle patterns is performed allowing us tocorrelate the aforementioned images by means ofSPS.Before recording the second ESPI image, the

reference beam was inclined by an angle h � 13; 57mrad with respect to the axis of the system. Thisangle corresponds to a phase stepping of p=2 rad

between adjacent pixels in the x-direction. Aftercompensation, in each case, Eq. (1) is applied andthree fringe patterns are obtained. Figs. 4(a) and (c)show the resulting fringe patterns after compensa-tion. Fig. 4(a) shows the resulting fringe patternfrom the subtraction of a digitally displaced (onepixel to the left) displaced speckle image and thereference speckle image. Likewise, Figs. 4(b) and(c) show the corresponding images of the subtrac-tions of the displaced speckle image with digitaldisplacements of 0 and 1 pixels to the right and thereference speckle image, respectively. A p=2 phasestepping is clearly observable in these images.Furthermore, it is seen that the fringe contrastfound in ESPI techniques is retained. The sensi-tivity of the system is that of out-of-plane dis-placement measurements carried out by ESPItechniques as well. A low-pass filter was applied toeach of these fringe images in order to remove theintrinsic random phase of the speckle. Next, athree-bucket algorithm is applied and the wrappedoptical phase is obtained within the interval [�p; p].This result relates uniquely to the out-of-planedisplacement. Finally, Figs. 5(a) and (b) shows theresult of applying an unwrapping algorithm to thewrapped phase map [18], which corresponds to theundergoing out-of-plane deformation; one unitrepresents a displacement of 949.5 nm. In Fig. 5(a)the out-of-plane displacement for the first two ex-periments (LIPD) is shown, while in Fig. 5(b) thatcorresponding to the fractured plate is shown.In these experiments, it is not possible to use

SPS when LIPD compensation is not taken intoaccount. Several measurements with differentLIPD were carried out in order to estimate themagnitude of acceptable LIPD. Results show thatfor LIPD greater than 22.5 lm, the contrast of thesubtraction correlation fringes was reduced tohalf. This value corresponds to approximately halfthe transversal speckle size in this experiment.Considering these last results we calculated therelative error e, involved in the phase calculation.This error was obtained by the following expres-sion e ¼ jð/c � /sÞ=/cj, where /c and /s are theoptical phases in the presence and absence ofLIPD. In this case, the optical phase measurementin absence of LIPD was directly obtained by ESPI.A relative error of 8% was computed. This error


Fig. 3. Resulting RBD vector maps. In presence of LIPD: (a) RBD, (b) in-plane rotation; (c) deformation in an arbitrary direction was

applied on a fractured aluminum plate.


component arises mainly from the uncertainty ofthe geometric correspondence function betweenconsecutive object states. Another source error (inthe third experiment) has to do with the presenceof out-of-plane RBD, which is not detected by ourmethod. The precision of the correlation algo-rithm, despite presenting subpixel accuracy, isdictated by the subtraction correlation step, which

involves rounding errors. Other sources of errorare those arising from the filtering process andinaccuracy of setting the angle of inclination of thereference beam. The mean limitation of themethod is the low level of irradiances obtainedsince relatively large speckles are needed. One wayto overcome this problem is to use CCD sensorswith smaller pixels.

Fig. 4. Phase-shifted RBD-compensated speckle patterns obtained by subtraction correlation of a pair of speckle patterns (reference

and displaced state). In the displaced object state, a weight of 28 g was applied at the center of the plate. The phase shifts correspond to

(a) �p=2, (b) 0, (c) p=2 rad.


5. Conclusions

The proposed method shows that by combiningESPI and DSP it is possible to measure out-of-plane displacements in presence of large in-planedisplacement. This combination avoids the use ofintermediate speckle patterns when the decorrela-tion arises from LIPD. On computing the opticalphase, phase stepping techniques which enable thestudy of rapid transient events were implemented.It should be noted that the proposed methodpreserves the fringe contrast and the high sensi-tivity of ESPI techniques.

Acknowledgements

The authors wish to thank the Centro de In-vestigaciones en �OOptica, A. C. (Mexico) and TheCentro de Neurociencias de Cuba. We also thankMr. R. Mendoza for his interesting ideas.

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Fig. 5. 3-D plot of the unwrapped phase map representing uniquely out-of-plane displacement when (a) a RBD is applied (corre-

sponding to Figs. 3(a) and (b)) (one unit represents 949.5 nm), and (b) with large deformation in an arbitrary direction (one unit

represents 1550.1 nm).


out-of-plane displacement measurement by electronic speckle pattern interferometry in presence of...

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