Research ArticleScheimpflug Camera-Based Stereo-Digital Image Correlation forFull-Field 3D Deformation Measurement

Cong Sun ,1,2 Haibo Liu ,1,2 Yang Shang ,1,2 Shengyi Chen,3 and Qifeng Yu1,2

1College of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, China2Hunan Provincial Key Laboratory of Image Measurement and Vision Navigation, Changsha 410073, China3China Satellite Maritime Tracking and Control Department, Jiangyin 214431, China

Correspondence should be addressed to Haibo Liu; liuhaibo@nudt.edu.cn

Received 10 October 2019; Accepted 28 November 2019; Published 17 December 2019

Academic Editor: Manuel Aleixandre

Copyright © 2019 Cong Sun et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

To further extend the scope of stereo-digital image correlation (stereo-DIC) to more challenging environments, a novelScheimpflug camera-based stereo-DIC is developed for full-field 3D deformation measurement, wherein the Scheimpflugcondition, consisting of tilting the sensor plane with respect to the lens plane for the sake of larger depth of field (DOF) of thecamera, is employed. The geometric imaging model of the Scheimpflug camera is described, on the basis of which a robust andeffective stepwise calibration strategy is performed to calculate the intrinsic and extrinsic parameters of the stereo Scheimpflugrig. With the aid of a specially tailored stereo triangulation method and well-developed subset-based DIC algorithms, the three-dimensional shape and displacement of the specimen can be retrieved. Finally, practical experiments, including rigid motiontests and three-point bending tests, demonstrate the effectiveness and accuracy of the proposed approach.

1. Introduction

Since Kahn-Jetter and Chu [1] and Luo et al. [2] laid thefoundation and established the main framework based onimage correlation and stereovision in the early 1990s,stereo-DIC has been significantly improved in the aspectsof accuracy, computational efficiency, and robustness [3, 4].Actually, stereo-DIC has evolved into a quite maturenoncontact image-based optical technique for full-field 3Dshape, displacement, and deformation measurements. Dueto its outstanding superiorities, stereo-DIC has been exten-sively adopted in various fields, ranging from regular metalto composite materials [5] or even biological tissues [6], fromnano- [7] to macroscopic scale [8], from static or quasistaticloading to high-speed dynamic loading [9], and from labora-tory conditions to industrial extreme environments [10].

Nevertheless, the application of stereo-DIC might beconstrained in view of the limited field of view (FOV) inthe required working distance. In order to deal with thechallenge, multicamera stereo-DIC, in which three or morecameras are employed to capture the surface images of the

specimen from multiviews, has been developed [11–14].Generally, the multi-DIC system can be regarded as acombination of multiple regular stereo-DIC systems, firstlyeach stereo-DIC measures the local 3D shape at differentregions of the specimen surface, and then the obtainedresults via individual stereo-DIC are stitched or mappedto the universal reference system with precalibrationparameters. In spite of the fact that the multi-DIC systemproduces a sufficiently larger effective FOV, the uncer-tainty of the system might increase and the robustnessdeteriorates with the growth of the camera number. More-over, multicamera calibration with high precision andflexibility is still a challenging problem.

In this work, an alternative approach of the Scheimpflugcamera-based stereo-DIC method is provided. The Scheimp-flug camera, adopting the Scheimpflug condition by tiltingthe lens with respect to the image plane, enables extendingthe measurement range of the stereo-DIC system withoutzooming the aperture, which has been widely employed inthe field of particle image velocimetry (PIV) [15], linestructured light [16], and portable 3D laser scanner [17].

HindawiJournal of SensorsVolume 2019, Article ID 5391827, 11 pageshttps://doi.org/10.1155/2019/5391827

However, to the best of our knowledge, Scheimpflug camerashave been rarely involved in stereo-DIC literatures so far.

The outline of the paper is as follows. Section 2 elaboratesthe geometric imaging model including lens distortion of theScheimpflug camera, on the basis of which a robust andeffective stepwise calibration strategy is presented to calculatethe intrinsic and extrinsic parameters of the stereo Scheimp-flug rig. In Section 3, the tailored Scheimpflug stereo triangu-lation method and well-developed subset-based DICalgorithms utilized in this work are introduced in detail. Forvalidation, practical experiments, including rigidmotion testsand three-point bending tests, are performed in Section 4 toevaluate the performance of the proposed method. Finally,brief conclusions are drawn in Section 5.

2. Scheimpflug Camera Model andStereo Calibration

2.1. Scheimpflug Camera Model. As depicted in Figure 1(a),the Scheimpflug principle, traditionally credited to TheodorScheimpflug in 1902, states that the object plane (the planethat is in focus), the thin lens’s plane, and the image planemust all meet in a single line, the Scheimpflug line [18, 19].The principle is applicable to both thin prism and thick prismmodels, with minor corresponding modifications. To ensureits further high-quality measurement in stereo-DIC systems,accurate calibration of the Scheimpflug stereo rig is theprerequisite and most essential step. Yet the conventionalcalibration methods are not valid in this case because theassumptions used by classical calibration methodologies arenot satisfied anymore for cameras undergoing the Scheimp-flug condition [20]. Therefore, more and more researchersdevote themselves to the related research, and a variety ofmethods have been proposed to accommodate various appli-cations [21–25]. Inspired by the conclusion drawn in litera-ture [26], the Scheimpflug imaging model based on RMM(rotation matrix model) is adopted in this section.

RMM models the lens-image sensor configuration by anexplicit rotation matrix about the optic axis and includes itas a part of the intrinsic calibration parameter set. As illus-trated in Figure 1(b), O‐XCYCZC is the camera coordinatesystem with the origin at the optical center, while the inter-section OðCx, CyÞ of the optical axis ZC with the ideal imageplane IO is referred to as the principal point. Denoted as O‐xtyt , the image coordinate system of the tilted Scheimpflugimage plane IS shares the same origin with the O‐xy in theimage plane IO. And the symbols α and β indicate the rota-tional angles about axes XC and YC , respectively. The 3Dworld point PW ðXW, YW, ZWÞ projects on IO at point piðx, yÞ, and the light ray intersects IS at point pt ðxt , ytÞ.

The conventional perspective projection can be brieflyexpressed as

λpi

1

" #=K

R T

0 1

" #PW

1

" #, ð1Þ

where R and T indicate the rotation matrix and translationvector of the camera frame relative to the world frame,

respectively. Usually, extrinsic parameters fγ, κ, θ, tx, ty , tzgrefer to the three angle components defining the successiverotation about axes R = RXðγÞRYðκÞRZðθÞ and the threetranslation components. Besides, λ is the projective depth,and K3×3 is the camera intrinsic matrix with respect to theideal image plane, which is defined as

K =f x 0 Cx

0 f y Cy

0 0 1

2664

3775, ð2Þ

where f x and f y denote the horizontal and vertical focallength in pixel units, respectively.

Taking the tilt effect of the Scheimpflug condition intoaccount, the rotation which maps the plane IO onto the planeIS can be split into successive rotation about axes XC and YCwith angles α and β.

RS =cos β 0 sin β

0 1 0−sin β 0 cos β

2664

3775

1 0 00 cos α −sin α

0 sin α cos α

2664

3775: ð3Þ

Therefore, the projection pt in the tilted Scheimpflugimage plane of the world point PW can be modeled asfollows, and the detailed derivation process can be foundin Reference [25].

λ1pt

1

" #=KΛRT

SR T

0 1

" #PW

1

" #, ð4Þ

where λ1 is the arbitrary scale factor and Λ is theScheimpflug array:

Λ =cos α cos β 0 sin β

0 cos α cos β −cos β sin α

0 0 1

2664

3775: ð5Þ

Substituting Equations (1), (2), (3), and (4) and let-ting the normalized coordinates in IO be denoted as pnðxn, ynÞT , thus, the transformation that relates pn to theimage point pt can be achieved.

λ2pt

1

" #=KΛRT

Spn

1

" #: ð6Þ

Likewise, λ2 is the arbitrary scale factor. In view ofthe inevitable lens distortion, the radial and tangentiallens distortions are taken into account on the ideal imageplane IO [38].

xd = k1r2 + k2r

4� �xn + 2p1xnyn + p2 r2 + 2x2n

� �,

yd = k1r2 + k2r

4� �yn + 2p2xnyn + p1 r2 + 2y2n

� �,

ð7Þ

2 Journal of Sensors

where r2 = xn2 + yn

2 and ðk1, k2Þ, ðp1, p2Þ represent theradial and tangential distortion coefficients, respectively.Thus, relying on Equations (1), (2), (3), (4), (5), (6),and (7), the Scheimpflug imaging model including lensdistortion is established.

2.2. Stepwise Calibration of the Stereo Scheimpflug Camera. Inconsideration of the sensitivity and dependency of theScheimpflug camera’s parameters, the stepwise calibrationstrategy analogous to the literature [23] is employed to allowan adequate scope for parameter values. Besides, on accountof the unique advantages of the planar calibration object, aplanar checkerboard is adopted in this section. Sufficientimages of the calibration object ought to be acquired by twocameras synchronously from multiple views.

A stereo rig composed of two Scheimpflug cameras isconsidered. As depicted in Figure 2, O1 and O2 are theoptical centers of the left and right cameras, respectively.A 3D world point P projects on ILO at point pLi , the lightray intersects ILS at point pLt in the left images, and thecamera baseline, respectively, intersects image planes ILOand ILS at eL and eLt ; likewise, in the right images. Givenall the calibrated intrinsic and extrinsic parameters of thestereo rig (αg, βg, Kg, Rg, Tg, g = L, R), we have

λLpLt

1

" #=KLΛLRT

LSRL TL

0 1

" #PW

1

" #, ð8Þ

λRpRt

1

" #=KRΛRRT

RSRR TR

0 1

" #PW

1

" #: ð9Þ

Supposing that a set of image correspondences pLt ⟷ pRtis given, then the corresponding world point set PW can becalculated combining Equations (8) and (9).

In general, the stepwise calibration scheme of stereoScheimpflug cameras can be divided into five consecutivesteps.

Step 1. Initialization of the intrinsic and extrinsic parametersof the two individual cameras. In terms of intrinsic parame-ters, the Scheimpflug angles (α, β) can be roughly estimatedaccording to literature [25] or just obtained from the data-sheet. Meanwhile, identical with the frequently used method,the principal point ðCx, CyÞ is initialized at the center of theimage plane. Furthermore, the distortion coefficients areassumed to be zero. For the extrinsic parameters Φðγk, κk, θk, txk, tyk, tzkÞ, they can be calculated from thehomographies of all captured images using Zhang’s method[20], without taking the Scheimpflug condition and lens dis-tortion into account.

Step 2. Estimation of the Scheimpflug angles of two indi-vidual cameras. The projection from PW to the Scheimp-flug image plane, depending on Equations (1), (2), (3),(4), (5), (6), and (7), can be denoted by ~pt . To estimatethe Scheimpflug angles, the following objective functionρ is minimized:

ρi =min 〠N

k=1〠M

j=1pj,kt − ~pj,kt ξið Þ

��� ���2, ð10Þ

with the corresponding featured image point numberj = 1, 2,⋯,M, the captured image number k = 1, 2,⋯,N ,and the optimization parameter vector ξ1 = f f x, f y, α, β,Φg.pj,kt is the observation of marker j for position k. It should benoted that the parameters of the principal point and distortionare not involved in this optimization. Moreover, the minimi-zations in this step and hereinafter are performed bymeans ofa suitable version of the Levenberg-Marquardt algorithms.

Step 3. Estimation of the principal points of two individualcameras. A similar objective function as in Step 2 isperformed, while the optimization parameters are supposedto be changed into ξ2 = f f x, f y, Cx, Cy,Φg. Likewise, theScheimpflug angles and distortion coefficient parametersare excluded from optimization this time.

Scheimpflug line

Imag

ing

sens

or

Image plane

LensLens axis

Lens plane

Object plane

(a)

Yc

Xc

x

Scheimpflugimage plane IS

y

PWpt

oc

pi

yt

xt

Ideal imageplane IO

Zc

O

𝛽

(b)

Figure 1: Schematic of the imaging system applied with the Scheimpflug principle: (a) Scheimpflug principle in a 2D view; (b) opticalgeometry of the Scheimpflug camera in a 3D view.

3Journal of Sensors

Step 4.Determination of the distortion coefficients. Given theresults obtained from the precious steps, the distortionparameters are taken into consideration in this step. Andthe optimization parameters are supposed to be ξ3 = f f x,f y, k1, k2, p1, p2,Φg.

Consequently, the intrinsic parameter vector η, includingf f x, f y, α, β, Cx, Cy, k1, k2, p1, p2g, of the two cameras can beacquired through the separated stepwise calibrationprocedure.

Step 5. Calculation of the relative orientation R̂ðγ, κ, θÞ andposition T̂ðtx, ty, tzÞ between the two cameras. Suppose thatthe optimized intrinsic and extrinsic parameter sets of twoindividual cameras are defined as ηl, ηr and Φl,Φr , respec-tively. Without loss of generality, the left camera is assumedto be the reference camera. Hence, initialization of the rightcamera pose with respect to the left can be expressed as

R̂ = Rkl Rk

r

� �−1,

T̂ = Tkl − Rk

l Rkr

� �−1Tkr :

ð11Þ

Thus, the previous results can be merged in the referencecoordinate system, on the basis of which an additional opti-mization is implemented.

ρi′=min 〠2

l=1〠N

k=1〠M

j=1pj,k,lt − ~pj,k,lt ξið Þ

��� ���2, ð12Þ

where l refers to the number of cameras. And the optimizationparameter vector converts into ξ4 = fγ, κ, θ, tx, ty , tz ,Φl,Φrg.Note that the optimized intrinsic parameters set ηl, ηr isemployed for the optimization. Hence, all the intrinsic andextrinsic parameters of the stereo Scheimpflug cameras havebeen obtained.

3. Stereo-DIC with Scheimpflug Cameras

To accurately determine the full-field surface displacementand strain of the specimen, the well-developed subset-based DIC algorithms are utilized to register the imagesof the specimen coated with a random speckle pattern inthe reference and deformed states and to match the sameregion between the stereo images. Typically, a squarereference subset centered at the predefined calculationpoint is chosen from the left reference image andemployed to search its corresponding location in the rightimage. To this end, a robust zero-mean normalized sum ofsquared difference (ZNSSD) criterion [27], integrated witha second-order shape function [28], is adopted to quanti-tatively assess the similarity between the reference andtarget subsets. To enhance subpixel registration accuracy,the state-of-the-art inverse compositional Gauss-Newton(IC-GN) algorithm [29] and a biquintic B-spline interpola-tion scheme are utilized. Giving top priority to the accu-racy rather than efficiency, the matching strategy adoptedhere is to correlate all the deformed images of the stereorig to the left reference image [4].

Therefore, the desired disparity data of the region ofinterest (ROI) can be obtained. Together with the precali-brated parameters of the stereo Scheimpflug cameras, the3D world coordinates of the measurement points on thespecimen surface can be recovered via the tailored bundleadjustment-based stereo triangulation. It should be notedthat the lens distortion must be taken into considerationduring the triangulation process. Furthermore, the threedisplacement components, resulting from external loading,can be calculated by subtracting the 3D coordinates of thesame interest points at the initial state from those of thedeformed. In common with literature [30], a variation ofthe triangular Cosserat point element method proposedin [31] is utilized to calculate the full-field strain distribu-tion. It is worth noting that the average rigid-body motionof the specimen should be subtracted to give a betterinsight into the relative deformation of the specimensurface.

P

ILo IRo

IRs

pRi

eRt

pRt

eR

pLi

eLt

eL

eL

pLt

ILs

OL OR

(R,T)

Figure 2: Schematic of epipolar geometry of stereo Scheimpflug cameras.

4 Journal of Sensors

4. Experiment Verification

4.1. Experimental Apparatus. To verify the performance ofthe developed Scheimpflug camera-based stereo-DIC system,a series of experiments, including stereo calibration ofScheimpflug cameras, rigid motion tests, and three-pointbending tests of the specimen, were performed under con-trolled laboratory conditions.

As shown in Figure 3(a), experiments in this study havebeen conducted with a wide range of system configurations,which mainly include (1) two 8-bit IMPERX CCD Cameras(IGV-B2520M-SC000) employed to simultaneously recordthe images of the specimen surface from two directions,which have the spatial resolution of 2456 ∗ 2058 pixels, andequipped with Kowa lenses (focal length 50mm) andcustom-made Scheimpflug adapters, (2) a static loadingframe (CTLD-2) of 2 kN capacity, (3) a high-precision verti-cal linear stage (Newport GTS30V) with the precision of0.1μm, and (4) a customized aluminum alloy three-pointbending specimen, which is sketched in Figure 3(b), andthe surface indicated by a yellow rectangle in the middle asdepicted in Figure 3(c) is chosen as the ROI.

Note that all equipment aforementioned is mounted on aNewport optical bench, and specimen preparation consists ofspraying black/white colour paint over the specimen surfaceto yield a suitable random speckle pattern. In order to obtainentirely focused images with small distortions, the camerasand lenses are adjusted with the aid of the Scheimpflugarrangement. Besides, uniform illumination of the specimen

surface is guaranteed via holding two LED lamps on each sideof the camera. Moreover, the specially tailored experimentaldata processing software is developed on the basis of thewell-developed open-source toolbox Ncorr [32] and Multi-DIC [30].

An example is presented in Figure 4 to compare theimaging performance of the Scheimpflug camera with thatof the conventional camera under the same experimentalconfiguration. The conventional camera here refers to thesame cameras indicated in Figure 3(a) but without tiltingthe lens. Figures 4(a) and 4(b), respectively, show the speci-men surface image obtained from the left camera based onthe Scheimpflug imaging model and that based on the con-ventional pinhole imaging model; meanwhile, the enlargedlocal images are depicted on their right.

As illustrated in Figure 4(b), it is apparent that a largerrange of sharp images of the ROI on the specimen surfacecan be acquired using the Scheimpflug camera. Besides, aneffective global parameter (Mean Intensity Gradient (MIG))[33] is adopted to further quantitatively assess the overallquality of the obtained speckle patterns. Twenty image pairsobtained via the Scheimpflug camera and the conventionalcamera are utilized for comparisons, and the MIG of cap-tured speckle patterns from the Scheimpflug camera(16.8765) is obviously higher than that from the conven-tional camera (8.7836), which indicates that the Scheimp-flug camera accordingly produces the smaller mean biaserror and standard deviation error in the further DICmeasurements. Thus, it can be deduced that the imaging

Static loadingframe (CTLD-2)

LED lamp

30

19

60 60

134

(a)

(b)

(c)

2045

15

114

119

12Unit (mm)

Z

X

Y

ROI

�ree-pointbending specimen

Two IMPERX CCD Camerasequipped with Kowalenses (f=50mm) and Scheimpflug adapters

Figure 3: (a) Experimental setup for the three-point bending test; (b) specimen geometry; (c) ROI on the specimen surface (yellow rectangle).

5Journal of Sensors

performance of the Scheimpflug camera, under the exper-imental configuration in this paper, is superior to that ofthe conventional camera.

4.2. Stereo Scheimpflug Camera Calibration. Prior to speci-men testing, the Scheimpflug camera-based stereo-DICsystem should be well calibrated. As sketched in Figure 3(a),two Scheimpflug cameras are symmetrically arranged, withthe normal to the specimen surface such that it approximatelybisects the stereo angle. In the calibration experiment, aprinted checkerboard pattern attached to a cardboard ratherthan high-precision calibration targets is adopted. The cali-bration pattern consists of a plane of 12 ∗ 9 grids with 5mmcheckerboard spacing. Totally, 30 images of the checker-board under different orientations and positions arecaptured. Employing the proposed stepwise stereo cameracalibration technique, the intrinsic and extrinsic parameters,as well as the reprojection errors of the stereo rig, can beretrieved in Table 1.

To further verify the calibration results of the Scheimp-flug rig, the 3D coordinates and structure of 20 arbitrarilyplaced checkerboards are reconstructed via the rig with thecalibrated parameters and simultaneously obtained imagepairs. And we fit the reconstructed coordinates with an idealflat plane and calculated the distance between the recon-structed points and the ideal plane. Moreover, a typicalreconstruction result is given in Figure 5.

Figure 5(a) depicts the reconstructed 3D checkerboardpoints and the fitted plane while the error distribution ofthe reconstructed checkerboard points is presented inFigure 5(b). As indicated in Figure 5(a), the reconstructed3D points agree well with the fitted plane in spite of minordeviations. Furthermore, it can be found out in Figure 5(b)that the deviations of reconstructed points are approximatelysymmetrical about the center of the fitted plane, while thedeviations at the corners are more significant. Nonetheless,the maximum deviation of the reconstructed points is

approximately 0.030mm, which is still quite small comparedwith the depth of the checkerboard. Besides, the RMSE (RootMean Squared Error) of the distance between all the recon-structed 3D points and the corresponding fitted planes under20 different poses is 0.0133mm. Consequently, the experi-mental results verify the effectiveness and accuracy of thecalibration method.

4.3. Rigid-Body Motion Test. To quantify the metrologicalperformance of the developed Scheimpflug camera-basedstereo-DIC system, a null strain test was performed on thespecimen subjected to a rigid-body motion [34]. As thesefields are theoretically equal to zero, we can thus obtain theoverall error of the displacement measurements. The three-point bending specimen is positioned on the high-precision

(a)

(b)

Figure 4: (a) The specimen surface image obtained by the conventional camera; (b) the specimen surface image obtained by the Scheimpflugcamera.

Table 1: The calibration results of the Scheimpflug stereo cameras.

Intrinsic parameters Left camera Right camera

f x 10912.3651 10025.7085

f y 10896.6062 10977.6092

α (°) 0.6156 0.3511

β (°) -4.5583 4.4574

Cx 868.9786 1503.2014

Cy 989.9657 1000.1329

k1 -0.3621 -0.3208

k2 0.8389 1.3490

p1 -0.0013 -0.0022

p2 -0.0108 -0.0104

Reprojection error (pixel) 0.3479 0.3528

R (in Rodriguez form) (0.0037, 0.3948, -0.0115)

T (mm) (-144.8040, 1.3722, 27.9974)

6 Journal of Sensors

vertical linear stage, and the stage is moved along the verticaldirection from 0 to 10mm and back to 0mm by steps of1mm with regard to the reference position. Stereo imagesare simultaneously captured for each position of the speci-men. According to the criterion developed in [35], a subsetsize of 64 ∗ 64 pixels coupled with a step size of 5 pixels ischosen during the stereo-DIC processing; therefore, thefull-field displacements and strains of the specimen surfacecan be calculated.

Given that the reference frame of the vertical linear stage isnot identical to the reference frame of the proposed stereo-DICsystem, the scale of the displacement D =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX2 + Y2 + Z2

p,

rather than the 3D ðX, Y , ZÞ displacement components, is com-pared between the proposed stereo-DIC measurements and thereadout of the vertical linear stage.

As shown in Figure 6(a), the average displacement mea-surements of the specimen are compared with referencevalues obtained from the vertical linear stage, and the corre-sponding standard deviation of displacements is illustrated inFigure 6(b).

It can be observed that the displacement measurementsfor the Scheimpflug-based stereo-DIC system is in well con-formity with the reference imposed vertical displacements. Itis worth noting that as the imposed vertical displacementsincrease from 1 to 10mm, the average errors of the measuredspecimen displacements are enlarged from 5.8μm (unit ofmicron) to 50.9μm. In the meanwhile, the standard devia-tions of the specimen climb from 1.9μm to 10.3μm. Despiteall these, the average errors and standard deviations ofdisplacement measurements are rather low, even in the case

−0.04 2020

−0.03−0.02

10

−0.01

10

Dist

ance

s of r

econ

struc

ted

poin

ts to

the fi

tted

plan

e(m

m)

0

Y (mm) X (mm)

0.01

0

0.02

0

0.03

−10−20 −10

(a)

2520 20

1510

10

Y (mm) X (mm)

0 5

0−10

340

−5−20 −10

Z (m

m) 360

−0.03

−0.02

−0.01

0

0.01

0.02

25

2015

1010

Y (mm) X (mm)

0 5

0−10

−5

(b)

Figure 5: (a) The reconstructed 3D checkerboard points and the fitted 3D plane; (b) the error distribution of the reconstructed points withrespect to the fitted plane.

7Journal of Sensors

of large vertical translations. As the translation is increased,the percent of average errors and standard deviations indisplacement measurements has remained approximatelyconstant. Consequently, these results indicate that the dis-placement measurement for the Scheimpflug-based stereo-DIC system is fairly accurate.

4.4. Three-Point Bending Experiment. To further verify thevalidity of the proposed method, a three-point bendingexperiment is performed on the specially customized speci-men, which is widely adopted to identify the flexural andtensile strength of quasibrittle materials. The experimentalsetup is exhibited in Figure 3, the specimen is positioned onthe center of the loading frame, and a pair of images capturedat a small load (about 20N) is taken as the reference config-uration. It needs to be emphasized that the small load appliedhere is to overcome excessive rigid-body motion due to anyslack in loading fixtures [36]. As the applied load increasesfrom 20 up to 2000N with the increment of 200N, a seriesof image pairs are recorded as the deformed configuration.The same settings as in the rigid-body motion test are appliedto the subset-based matching operation. Therefore, full-fielddistributions of the specimen surface displacement anddeformation can be obtained and utilized to explain theunderlying deformation mechanisms in the three-pointbending test. Moreover, Figures 7(a)–7(c), respectively,depict the typical evolution of full-field distributions ofdisplacement components in three directions along with theincrease in the applied load (left 600N, middle 1200N, andright 1800N).

It is worthwhile to notice that all the experimental resultsin this paper are obtained in the left camera coordinate sys-

tem. As shown in Figure 3(a), the Z-axis points to the speci-men surface, while the Y-axis is vertically downward and theX-axis forms a right-hand coordinate system. Even thoughthe small angle between the Z-axis and the normal of thespecimen surface exists, for convenience of analysis, it isignored and assumed that the Z-axis is perpendicular to thespecimen surface, while the X-axis and Y-axis are, respec-tively, parallel to the longitudinal and transverse directionsof the ROI. Namely, the static loading direction is approxi-mately parallel to the Y-axis.

Globally, displacement maps of the components in threedirections in Figures 7(a)–7(c) exhibit typical patterns thatare characteristics of three-point bending tests. As illustratedin Figure 7(a), the displacement fields of the X-componentsare distributed symmetrically in the center and the displace-ments in the middle part of the specimen surface are rela-tively low and gradually increase towards both ends of thespecimen surface. With the increase in the applied loads,the upper surface of the specimen tends to move towardsthe middle, while the lower surface moves towards the oppo-site direction. As for Y-components in Figure 7(b), they areaxially symmetrically distributed, and the middle parts ofthe specimen surface move downward along the Y-axis,while the sides move upward. In the case of the displace-ments of Z-components, they are much smaller than thosein the other two directions. In general, the displacements atthe edge of the specimen surface are larger than those at thecentral part. Actually, for the arrangement in these experi-ments, the displacements of Z-components can be approxi-mated as the out-of-plane displacements on the specimensurface. Moreover, Figure 7(d) shows the full-field distribu-tion of the specimen surface displacement magnitude. It

0 2 4 6 8 10 12 14 16 18 20Frames

0

2

4

6

8

10

12

�e s

cale

of d

ispla

cem

ent (

mm

)

�e measured displacementsReadouts of vertical linear stage

(a)

0 2 4 6 8 10 12 14 16 18 20Frames

1

2

3

4

5

6

7

8

9

10

11

Erro

rs (m

m)

×10−3

(b)

Figure 6: (a) The scale of the specimen displacements measured by the proposed method against reference values; (b) the standard deviationof the specimen displacements related to the vertical translations.

8 Journal of Sensors

600 N 1200 N

(a)

(b)

(c)

(d)

1800 N

3000

2500

2000

1500

1000

3000 3500 400 4500

−0.01

−0.005

0.005

0.01

0

25002000150010003000

2500

2000

1500

1000

3000 3500 400 4500

−0.01

−0.005

0.005

0.01

0

−0.01

−0.005

0.005

0.01

0

25002000150010003000

2500

2000

1500

1000

3000 3500 400 45002500200015001000

3000

2500

2000

1500

1000

3000 3500 400 4500

−0.015

−0.01

−0.005

0.005

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0

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Figure 7: The full-field distribution of the specimen surface displacement components and magnitude at selected loads (unit (mm)): (a) X-component displacements; (b) Y-component displacements; (c) Z-component displacements; (d) displacement magnitude.

2000

600 N 1200 N

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Figure 8: (a) The full-field distribution of Lagrangian strain magnitude maps at given loads; (b) the relative area change of the specimensurface at given loads.

9Journal of Sensors

can be observed that two fixed minimum points appearbetween the three loading heads and do not change as theapplied loads increase.

Besides, the full-field distribution of Lagrangian strainmagnitude maps corresponding to the typical applied loadsis shown in Figure 8(a); meanwhile, Figure 8(b) illustratesthe relative surface area change of the specimen. It can befound that the strain mainly occurs at the upper and lowerparts of the specimen surface, while the strain on the neutralplane is rather close to zero. Further, as revealed inFigure 8(b), the upper surface area of the specimen is com-pressed while the lower surface area is stretched, and thistrend is more pronounced as the applied load increases. Con-sidering the geometric size of the aluminum alloy specimenand the limited capacity of the static loading frame, the defor-mation and strain levels of the specimen are relatively small.Despite all that, the experimental results of three-point bend-ing further validate the effectiveness of the proposed method.Further analysis of the specimen’s mechanical properties isbeyond the scope of this work but can be found in [37].

5. Conclusions

In this article, a novel Scheimpflug camera-based stereo-DICmethod is developed for the full-field 3D shape and deforma-tion measurement. In contrast with conventional cameras,the Scheimpflug camera enables extending the measurementrange of the stereo-DIC system without zooming the aper-ture, which exhibits significant advantages over conventionalcameras in stereo-DIC measurements. Also, a robust andeffective stepwise calibration approach is presented to calcu-late the intrinsic and extrinsic parameters of the stereoScheimpflug rig. Consequently, with the aid of the tailoredstereo triangulation method and well-developed subset-based DIC algorithms, the three-dimensional shape anddeformation of the specimen can be retrieved. Finally, exper-imental results of rigid-body motion tests and three-pointbending tests successfully demonstrate the accuracy andeffectiveness of the proposed approach. Due to the attractiveadvantages of employing Scheimpflug cameras, the proposedtechnique offers new avenues for performing 3D deforma-tion measurements and indicates great potentials in furtherfields of stereo-DIC measurements.

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request.

Conflicts of Interest

The authors declare that there is no conflict of interestregarding the publication of this paper.

Acknowledgments

This research was funded by the National Natural ScienceFoundation of China (NSFC) under Grant Nos. 11872070,11727804, and 11902349.

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