Displacement field analysis based on the combination digital speckle correlation method with radial basis function interpolation

Displacement field analysis based on the combinationdigital speckle correlation method with radial basis

function interpolation

Chen Tang,1,* Linlin Wang,1,2 Si Yan,3 Jian Wu,1 Liyan Cheng,1 and Cancan Li1

1Department of Applied Physics, University of Tianjin, Tianjin 300072, China2Engineering Training Center, Shenyang Institute of Aeronautical Engineering, Shenyang 110136, China

3School of Electronic and Information Engineering, University of Tianjin, Tianjin 300072, China

*Corresponding author: [email protected]

Received 9 April 2010; revised 1 August 2010; accepted 4 August 2010;posted 4 August 2010 (Doc. ID 126797); published 13 August 2010

The digital speckle correlation method (DSCM) has been widely used to resolve displacement and de-formation gradient fields. The computational time and the computational accuracy are still two challen-ging problems faced in this area. In this paper, we introduce the radial basis function (RBF) interpolationmethod to DSCM and propose a method for displacement field analysis based on the combination ofDSCM with RBF interpolation. We test the proposed method on two computer-simulated and twoexperimentally obtained deformation measurements and compare it with the widely used Newton–Raphson iteration (NR method). The experimental results demonstrate that our method performs betterthan the NR method in terms of both quantitative evaluation and visual quality. In addition, the totalcomputational time of our method is considerably shorter than that of the NR method. Our method isparticularly suitable for displacement field analysis of large regions. © 2010 Optical Society of AmericaOCIS codes: 120.6150, 120.6650.

1. Introduction

The digital speckle correlation method (DSCM) in-troduced by Peters and Ranson [1] is an effective non-contact optical metrology technique. It has beenextensively investigated and widely used for defor-mation measurements in numerous fields [2–4]. Itconsists of recording with a camera some digitalimages of a specimen undergoing a mechanicaltransformation and applying an image correlation al-gorithm. The image correlation algorithm is of funda-mental importance for the successful application ofDSCM, and it is supposed to provide an apparenttwo-dimensional (2D) displacement field accordingto some principle of optical flow conservation [5,6].The optical flow of an image sequence is a set of vec-tor fields relating each image to the next one. The

basic principle underlying DSCM is the fact thatthe image is only deformed by the in-plane displace-ment field, without any further alteration of the graylevels. Therefore, if g denotes the deformed and f thereference images, these functions are related by [6]

gðxÞ ¼ f ðxþ uðxÞÞ þ ηðxÞ; ð1Þ

where ηðxÞ is noise induced by image acquisition. Thegeneral purpose of DSCM algorithms is to determinethe displacement field u from the gray-level distribu-tions f and g. As such, this so-called “optical flowdetermination” problem is an ill-posed inverse pro-blem that is only solved approximately with addi-tional assumptions [5].

In the past two decades, there have been differentsubpixel correlation algorithms developed, includingthe correlation coefficient curve-fitting method [7],the Newton–Raphson iteration (NR method) [8],the gradient-based method [9,10], the double Fourier

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transform [11], the genetic algorithm (GA) [12], andthe artificial neural network [13]. Among thesealgorithms, the correlation coefficient curve-fitting(CCCF) method , the NR method, and the gradient-based method are the three most commonly used al-gorithms due to their simplicity, effectiveness, andaccuracy [14]. Pan et al. examined the performancesof the three methods [14]. The conclusion is that theNR method is more accurate, but much slower thanother two algorithms. The double Fourier transformmethod, without the multiplication in the spectralspace and the inverse Fourier transform, are per-formed by a two-step Fourier transform. However,to obtain subpixel resolution, the method must besupplemented by the interpolation-based oversam-pling technique or the recursive iteration methodbased on the discrete spectrum [15]. The GA is aglobal optimization method and does not involve rea-sonable initial guesses of displacement or deforma-tion gradient, or the calculation of second-orderspatial derivatives of the digital images. However, ac-cording to our own experience in this method, theperformance of a GA depends largely on the selectionof the genetic operators. In addition, GAs may re-quire a large number of iterations for a near opti-mum solution to evolve. In the artificial neuralnetwork method, subpixel resolution is achieved bynonintegral pixel shifting and by training the artifi-cial neural network to estimate the fractional part ofthe displacement. The method can facilitate very fastanalysis without knowledge of the analytical form ofthe image correlation function.

Although there have been various algorithms, thecomputational efficiency and accuracy are still two ofthe challenging problems faced in this area. Accord-ing to our own experience in this area, the CCCFmethod is one of the fastest algorithms. Its MATLABcode is available from http://www.mathworks.com. Ittakes only about 0:034 s to obtain the displacement ofa point with this code. But if the MATLAB code isused to calculate the displacement field of a regionof size 150 × 150 pixels, it still takes about 12 min.Moreover, the accuracy of the CCCF method is low.The NR method is more accurate than the CCCFmethod. If the NRmethod is used to calculate the dis-placement field of a region of size 150 × 150 pixels, itwill take about 5h, as is shown in Section 3. Thisshortcoming hinders seriously the successful appli-cation of the DSCM, especially for large regionanalysis. In addition, no matter which type of theabove-mentioned techniques is used for the DSCM,the discrete displacement field data obtained usuallyneed to be improved through postprocessing by use ofsmoothing techniques, such as least-squares polyno-mial fitting [16] and wavelet transform [17], whichfurther increases the computational time and com-putational complexity.

In this paper, we put the emphasis on improvingthe computational efficiency and accuracy of imagecorrelation. We use a different and powerful mathe-matical tool for this purpose. We introduce the radial

basis function (RBF) interpolation method to theDSCM, and propose a method of displacement fieldanalysis based on the combination of the DSCM withRBF interpolation. RBF methods are modern meth-ods and can provide a very general and flexible wayof interpolation in multidimensional spaces, even forunstructured data, where it is often impossible to ap-ply polynomial or spline interpolation. They havebeen known, tested, and analyzed for several yearsnow, and many positive properties have been identi-fied [18]. The major features are that the methodsare able to handle arbitrarily scattered data and givevery good accuracy of approximation. In our method,the displacements of some points (called interpola-tion points) are calculated using the NR method,and then the whole displacement of the region tobe calculated is obtained by the RBF interpolation.Our method takes far less computational time anddoes not require any postprocessing. Moreover, ourmethod is flexible. The main advantages of our meth-od are demonstrated via application to the twocomputer-simulated and two practical deformationmeasurements and comparisons with the NR meth-od. In the following sections, we will first review theNR method briefly as it relates to our work, and thendescribe in detail the method of displacement fieldanalysis based on the combination of the DSCMand RBF interpolation. Section 3 contains experi-ments and discussion. Section 4 provides the sum-mary for this paper.

2. Displacement Field Analysis Based on theCombination of the DSCM with RBF Interpolation

A. Brief Review of the NR Method

The underlying principle of the DSCM is to matchtwo speckle patterns before and after deformationwith the predefined correlation criterion, which mea-sures how well subsets match.

In the NR method, the correlation function is

S

�x; y;u; v;

∂u∂x

;∂u∂y

;∂v∂x

;∂v∂y

�

¼P

mi¼1

Pmj¼1½f ðx; yÞgðx�; y�Þ�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP

mi¼1

Pmj¼1 f

2ðx; yÞPmi¼1

Pmj¼1 g

2ðx�; y�Þq ; ð2Þ

where, to the given subset of m ×m, f ðx; yÞ andgðx�; y�Þ are the gray values of the subset centeredat the source and target point located in the referenceand deformed images, respectively. The NR iterationassumes the local deformation could be representedby two displacements and four displacementgradients:

�x� ¼ xþ uþ ∂u

∂xΔxþ ∂u∂yΔy

y� ¼ yþ vþ ∂v∂xΔxþ ∂v

∂yΔy; ð3Þ

where u and v are the displacements for the subsetcenters in the x and y directions, respectively. The

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terms Δx and Δy are the distances from the subsetcenter to the point ðx; yÞ. The NR method is based onthe calculation of correction terms that improve initi-al guesses. The correction for guess i is given by

ΔPi ¼ −H−1ðPÞ ×∇ðPiÞ; ð4Þwhere Pi ¼ ðu; v; ∂u

∂x ;∂u∂y ;

∂v∂x ;

∂v∂yÞT and ∇ðPiÞ is the Jaco-

bian matrix. Each term in the Jacobian is the deri-vative of the correlation function evaluated atguess i. HðPiÞ is the Hessian matrix, which are thesecond derivatives of the correlation function.

B. Radial Basis Function Interpolation

The RBF method is now one of the primary tools forinterpolating multidimensional scattered data. Be-cause of its good approximation properties and easeof implementation, the method is a popular choice inmany different areas, ranging from statistics to theapproximation of partial differential equations.The details of the method can be found in the pub-lished literature [19]. The basic theory that will beimportant for the forthcoming sections is here brieflysummarized. There are two separate versions of theRBF method, including the “basic” RBF method and“augmented” RBF method. Throughout this paper,we let X , Xi ∈ Rd, where d is some positive integer,and f i be a scalar value.

For a given set containingN distinct points fXigNi¼1and the corresponding data values ff igNi¼1, the basicRBF interpolant is given by

sðXÞ ¼XNi¼1

wiϕð‖X − Xi‖Þ; ð5Þ

where wi are the unknown RBF weighting coeffi-cients and ‖X − Xi‖ denotes the Euclidean distancebetween every two points of X and Xi. fϕð‖X −

Xi‖Þji ¼ 1; 2;…:Ng is a radially symmetric realvalued function called the basis function. The com-monly used types of radial basis function include

the thin-plate spline, ϕðrÞ ¼ r2 log r;the Gaussian, ϕðrÞ ¼ expð−r2=σ2Þ; andthe multiquadric, ϕðrÞ ¼ ðr2 þ β2Þ−1=2.

Among the three basis functions, the thin-platesplines function, introduced by Duchon [20], is suita-ble for fitting functions of two variables. Therefore,we chose the thin-plate spline as the basis functionto fit the displacement field of the interested regions.

The expansion coefficients wi in Eq. (5) are deter-mined from the interpolation conditions sðXiÞ ¼ f i;i ¼ 1; 2;…N:This leads to the following symmetriclinear system:

½A�½W � ¼ ½F �; ð7Þ

where the entries of A are given by ϕi;j ¼ϕð‖Xi − Xj‖Þ, i; j ¼ 1; 2;…;N. W ¼ fw1;w2; � � � ;wNgTand F ¼ ff 1; f 2; � � � ; f NgT .

As noted, the theoretical results on solvability ofthe system in Eq. (7) are based on the concept of aconditionally positive definite function [19]. Thiswas accomplished by placing some mild restrictionson the center locations and by adding polynomialterms to the basic RBF interpolation problem. Withthe addition of polynomial terms, the augmentedRBF interpolant is expressed as

sðXÞ ¼XNi¼1

wiϕð‖X − Xi‖Þ þXMk¼1

ckpkðXÞ; X ∈ Rd;

ð8Þ

where fpkðXÞgMk¼1 is a basis for the M-dimensionalspace, ΠmðRdÞ. To account for the conditions fromthe additional polynomial terms, the following addi-tional constraints are imposed:

XNj¼1

cjpkðXjÞ ¼ 0; k ¼ 1; 2;…;M: ð9Þ

The expansion coefficients wi and cj are then deter-mined from the interpolation conditions of Eq. (8),along with the constraints of Eq. (9), which leadsto the following linear system:

24 A P

PT 0

3524W

C

35 ¼

24F

0

35; ð10Þ

where A is the matrix in Eq. (7) and P is the n ×mmatrix with entries pkðXjÞ for j ¼ 1; 2;…;Nand k ¼ 1; 2;…;M.

If the polynomial part of the RBF in Eq. (8) has theform pðXÞ ¼ c1 þ c2xþ c3y, it is concluded that theaugmented RBF interpolant is uniquely solvable,and the conditions on data points fXigNi¼1 are satis-fied based on Micchelli’s theorem [19]. Thus, the lin-ear system for coefficients wi and cj can be expressedas

266666666664

ϕ11 ϕ12 … ϕ1N 1 x1 y1ϕ21 ϕ22 … ϕ2N 1 x2 y2... ..

.… ..

. ... ..

. ...

ϕN1 ϕN2 … ϕNN 1 xN yN1 1 … 1 0 0 0x1 x2 … xN 0 0 0y1 y2 … yN 0 0 0

377777777775

266666666664

w1

w2

..

.

wN

c1c2c3

377777777775¼

266666666664

f 1f 2...

f N000

377777777775: ð11Þ

Once coefficients w1;…;wN and c1, c2, and c3 arefound, Eq. (8) can be used to estimate value of thefunction at any point.

Now, the accuracy of RBF interpolation is demon-strated via application to a 2D example and compar-ison with MATLAB cubic interpolation. Let F be afunction of the variables x and y and be of the form


Fðx; yÞ ¼ sinðyÞ × cosðxÞ; 0 ≤ x ≤ 2π; 0 ≤ y ≤ 2π:ð12Þ

We chose randomly 200 interpolation points. Boththe RBF interpolation and the MATLAB cubic inter-polation are applied to approximate the function F ata set of points ðxi; yiÞ, xi ¼ yi ¼ i × 3π

127, ði ¼ 0;…; 84Þwith the 200 interpolation points. Figure 1(a) showsthe exact values of the function at the points ðxi; yiÞ

and the 200 interpolation points, marked by �. Theevaluated values of the function at points ðxi; yiÞ, ob-tained by the MATLAB cubic interpolation and theRBF interpolation, are given in Figs. 1(b) and 1(c),respectively. Figures 1(d) and 1(e) show the absoluteerror of the MATLAB cubic interpolation and RBFinterpolation, respectively. It is obvious that the ab-solute error of RBF interpolations is smaller thanthat of cubic interpolation. The RBF interpolationcan provide much better approximation of the origi-nal function as compared with the widely used cubicinterpolation.

C. Displacement Field Analysis Based on theCombination of DSCM with RBF Interpolation

The main steps of the method are introduced asfollows.

First, we chose a finite set of distinct N interpola-tion points Xi ¼ fxi; yig, i ¼ 1; 2;…;N, where ðxi; yiÞdenote the position of the ith interpolation point.These points can be either regularly or irregularlydistributed in space. The corresponding displace-ments fui; vig; i ¼ 1; 2;…;N are calculated by theDSCM. Here, the NR method is used to calculatethe displacements of the interpolation points becauseof its high accuracy.

Second, for calculating the whole field displace-ments u of the region of interest, we substitute thedisplacements ui of the interpolation points for f iði ¼1; 2;…;NÞ in Eq. (11). We chose the thin-platespline as the basis function. In this case, the ϕij inEq. (11) has the form ϕij ¼ ððxi − xjÞ2 þ ðyi − yjÞ2Þ logffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðxi − xjÞ2 þ ðyi − yjÞ2q

, ði; j ¼ 1; 2;…;NÞ. Using thelinear system of Eq. (11), the coefficients wi and cjin Eq. (8) are evaluated. Thus, the approximatingfunction sðx; yÞ for the displacements u is obtained.

Third, by inserting the x and y coordinates of theregion of interest into this approximating functionsðx; yÞ, the displacements u in the region of interestcan be easily calculated. The component v in the re-gion of interest can be calculated in a similar way.

3. Experiments and Discussion

For evaluating the real performance of our method,in this section, we test the proposed method on thecomputer-simulated uniaxial tensile and rigid bodyrotation, and the experimentally obtained rigid bodytranslation and three-point bending deformationmeasurements and compare with the widely used,accurate NR method.

The computer-simulated speckle images are gener-ated by means of the method reported in Ref. [10]based on the following equations:

I1½i; j� ¼14πa2I0

XSk¼1

�erf

�i − xka

�− erf

�iþ 1 − xk

a

��

×�erf

�j − yka

�− erf

�jþ 1 − yk

a

��; ð13Þ

Fig. 1. Two-dimensional interpolation example. (a) The exact va-lues of function and the interpolation points marked by “�.” (b) Theevaluated values of the function by MATLAB cubic interpolation.(c) The evaluated values of function by RBF interpolation. (d) Theabsolute error of theMATLAB cubic interpolation. (e) The absoluteerror of the RBF interpolation.


I2½i; j� ¼14πa2I0J

XSk¼1

�erf

�ξ1 − ξ0;ka

�− erf

�ξ2 − ξ0;ka

��

×�erf

�η1 − η0;ka

�− erf

�η2 − η0;ka

��; ð14Þ

where

ξ1 ¼ ð1 − uxÞi − uyj; ð15Þ

ξ2 ¼ ð1 − uxÞðiþ 1Þ − uyðjþ 1Þ; ð16Þ

ξ0;k ¼ xk þ u0; ð17Þ

η1 ¼ −vxiþ ð1 − vyÞj; ð18Þ

η2 ¼ −vxðiþ 1Þ þ ð1 − vyÞðjþ 1Þ; ð19Þ

η0;k ¼ yk þ v0; ð20Þ

J ¼ det�� 1 − ux −uy

−vx 1 − vy

��; ð21Þ

s is the total number of speckle granules,R is the sizeof the speckle granules, ðxk; ykÞ are the positions ofeach speckle granule with a random distribution,and I0 is the random peak intensity of each specklegranule. Random noises with a given signal-to-noiseratio (SNR) are then added. Here, the chosen para-meters are a ¼ 4, s ¼ 2500, SNR ¼ 28 dB, andI0 ¼ 250.

For comparing the computational accuracy of thetwo methods, the mean square error (MSE) for thetwo computer-simulated deformations is calculatedby

MSE ¼ 1M ×N

XMk¼1

XNl¼1

ðuðk; lÞ − u0ðk; lÞÞ2; ð22Þ

where uðk; lÞ and u0ðk; lÞ are, respectively, the calcu-lated and theoretical values.M andN are the sizes ofthe region to be calculated.

Further, for comparing the computational effi-ciency of the two methods, the time ratio Tr iscalculated by

Tr ¼TNR

TRBF; ð23Þ

where TNR and TRBF denote the computational timesof the NR method and our method, respectively. Allthe computations are completed under the same con-ditions of a personal Pentium Dual E6300 computerwith 2:8GHz and 2GB memory by use of MATLAB.In all tests, we let the markers “þ” denote the choseninterpolation points, and the square ABCD or the

rectangle ABCD denote the region to be calculated,which are shown in the reference speckle images.The unit of displacement obtained by the NR methodand our method is in pixels. The subset sizes in theNR method are chosen as 41 × 41 pixels.

Figures 2(a) and 2(b) are the computer-simulatedspeckle image pairs generated with a preassigneduniaxial tensile with u ¼ ð0:025x; 0ÞT of 300 × 300pixels. We chose irregularly 20 interpolation points[see the markers “þ” in Fig. 2(a)]. We calculate thedisplacements of these interpolation points usingthe NR method. The calculated region is a square

Fig. 2. Computer-simulated uniaxial tensile and its displace-ment field: (a) the reference speckle image, (b) the deformedspeckle image, (c) the contour image of the displacement u bythe NR method, and (d) the contour image of the displacementu by our method.


ABCD of 150 × 150 pixels, from the point Að51; 51Þ tothe point Dð200; 200Þ, shown in Fig. 2(a). Using theRBF interpolation method, we obtain the displace-ments of each pixel in the square ABCD. In addition,the NR method is used to calculate the displace-ments of each pixel in the same area. Figures 2(c)and 2(d) show the contour of the displacement com-ponent u in the horizontal direction obtained by theNR method and our method, respectively. Here thedeformation gradient εx is calculated by εx ¼ ∂u=∂x.The MSEs of the displacements u and the deforma-tion gradient εx, obtained by the NR method are, re-spectively, 2:63e − 003 and 6:12e − 006, while thedisplacements u and the deformation gradient εx, ob-tained by our method are, respectively, 2:22e − 003and 5:03e − 007. In our method, it takes 15:4 s to ob-

tain the displacements of the 20 interpolation pointsusing the NR method, and the RBF interpolation forthis calculated region takes 1:0 s. Therefore, totalcomputation time of our method is 16:4 s, whilethe NR method takes 18213:8 s to obtain the displa-cements of this calculated region. The time ratio Tris 1110.6.

Other computer-simulated speckle image pairs of300 × 300 pixels are generated with a preassigned ri-gid body rotation with u ¼ ð0:0105y;−0:0105xÞT ,which is shown in Figs. 3(a) and 3(b). Here we alsotake 20 interpolation points, and the calculated re-gion is a square ABCD of 150 × 150 pixels, fromthe point Að51; 51Þ to the point Dð200; 200Þ, shownin Fig. 3(a). Figures 3(c-1) and 3(c-2) show the con-tours of the displacements u and v, obtained by

Fig. 3. Computer-simulated rigid body rotation and its displacement field: (a) the reference speckle image, (b) the deformed speckleimage, (c-1) the contour image of the displacement u by the NR method, (c-2) the contour image of the displacement v by the NR method,(d-1) the contour image of the displacement u by our method, and (d-2) the contour image of the displacement v by our method.


the NR method, respectively. The contours of the dis-placements u and v, obtained by our method are,respectively, given in Figs. 3(d-1) and 3(d-2). The

evaluated rotation angle θ at pixel ðx; yÞ can be cal-culated by θðx; yÞ ¼ jtan−1ðxþ u=yþ vÞ − tan−1ðx=yÞj.The MSEs of the displacements u, v, and the rotation

Fig. 4. Experimentally obtained rigid body translation and its displacement field: (a) the reference speckle image, (b) the deformedspeckle image, (c) the displacement in the x direction by the NR method, and (d) the displacement in the x direction by our method.

Fig. 5. Experimentally obtained three-point bending and its displacement field: (a) the reference speckle image, (b) the deformed speckleimage, (c-1) the contour image of the displacement u by the NR method, (c-2) the contour image of the displacement v by the NR method,(d-1) the contour image of the displacement u by our method, and (d-2) the contour image of the displacement v by our method.


angle θ, obtained by the NRmethod are, respectively,1:31e − 004, 4:41e − 004, and 3:51e − 009, while theMSEs of the displacements u, v, and the rotation an-gle θ, obtained by our method are, respectively,1:21e − 004, 3:88e − 044, and 1:77e − 009. The totalcomputation times of our method and the NRmethodare, respectively, 16.7 and 18191:9 s. The time ratioTr is 1089.3.

Subsequently, two experimentally obtained defor-mation configurations are used to generate speckleimage pairs: (i) rigid body translation and (ii) three-point bending. Figures 4(a) and 4(b) are experimen-tally obtained speckle image pairs with rigid bodytranslation of 440 × 364 pixels. In this experiment,an aluminum block is bolted to the base of an X–Ytranslation table and is made to move 1.63 pixels inthe x direction. The interpolation points and the cal-culated region ABCD are shown in Fig. 4(a). Here thenumber of the interpolation points is 42, and the cal-culated region is a square of 200 × 200 pixels,from point Að130; 80Þ to point Bð329; 279Þ.Figures 4(c) and 4(d) show the displacement compo-nent in the x direction obtained by the NR methodand our method, respectively. The total computationtimes of our method and the NR method are, respec-tively, 15.9 and 29987:9 s. The time ratio Tr is1886.0.

Figures 5(a) and 5(b) are the experimentally ob-tained speckle image pairs generated by three-pointbending of 701 × 576 pixels. In this experiment, thespecimen is firmly attached to a loading instrument.When the loading is 10N, its surface is imaged; thenthe loading is increased gradually to 100N, and itssurface is imaged for the second time. We take 28interpolation points, and the calculated region ABCDis a rectangle of 400 × 180 pixels from pointAð151; 226Þ to point Bð550; 405Þ. Similar results forFigs. 5(a) and 5(b) as for Figs. 3(a) and 3(b) are givenin Figs. 5(c-1) and 5(c-2). The material of this speci-men, shown in Fig. 6, is plexiglas. The total compu-tation times of our method and the NR method are,respectively, 11.7 and 20899:1 s. The time ratio Tris 1786.2.

As can be seen, our method performs better thanthe widely used NR method in terms of both quanti-tative evaluation and visual quality. One can findthat the processing time by our method is consider-ably shorter than that of the NR method in all testcases. The NR method is about 1089–1886 timesslower than our method.

4. Conclusion

In this paper, we introduced the RBF method to theDSCM and proposed a method of displacement fieldanalysis based on the combination of the DSCM withRBF interpolation. We demonstrated the perfor-mance of our method via application to the computer-simulated and experimentally obtained deformationmeasurements and comparison with the widely usedNR method. One advantage of the proposed methodis that the processing time of our method is consid-erably shorter. Our method is especially suitable fordisplacement field analysis of large areas. A secondadvantage is that our method does not require anysmoothing algorithm to process the discrete displace-ment field data obtained. A third advantage is thatour method is flexible. Once coefficients w1; � � � ;wNand c1, c2, and c3 in Eq. (8) are calculated by Eq.(11), Eq. (8) can model the x and y coordinates, thedisplacement field relationship, and can subse-quently be used to estimate the value of the displace-ments at any point. In addition, from the main stepsof our method as described in Subsection 2.C, one canfind that our implementation is very easy. The ex-perimental results do demonstrate the performanceof our method.

This work is supported by the National NaturalScience Foundation of China (NSFC) (grant60877001). We thank the topical editor and the anon-ymous reviewer for the constructive and helpfulcomments.

Reference

1. W. H. Peters and W. F. Ranson, “Digital imaging techniques inexperimental stress analysis,” Opt. Eng. 21, 427–431 (1982).

2. T. F. Begemann, “Three-dimensional deformation field mea-surement with digital speckle correlation,” Appl. Opt. 42,6783–6796 (2003).

3. E. B. Li, A. K. Tieu, and W. Y. D. Yuen, “Application of digitalimage correlation technique to dynamic measurement of thevelocity field in the deformation zone in cold rolling,” Opt.Lasers Eng. 39, 479–488 (2003).

4. M. A. Sutton, J. J. Orteu, and H. Schreier, Image Correlationfor Shape, Motion and Deformation Measurements (Springer,2009).

5. M. Bornert, F. Brémand, P. Doumalin, J. C. Dupré, M. Fazzini,M. Grédiac, F. Hild, S. Mistou, J. Molimard, J. J. Orteu, L.Robert, Y. Surrel, P. Vacher, and B. Wattrisse, “Assessmentof digital image correlation measurement errors: methodologyand results,” Exp. Mech. 49, 353–370 (2009).

6. S. Roux, J. Réthoré, and F. Hild, “Digital image correlation andfracture: an advanced technique for estimating stress inten-sity factors of 2D and 3D cracks,” J. Phys. D 42, 214004 (2009).

7. B. Wattrisse, A. Chrysochoos, J. M. Muracciole, and M.Némoz-Gaillard, “Analysis of strain localization during tensiletests by digital image correlation,” Exp. Mech. 41, 29–39(2001).

8. H. A. Bruck, S. R. McNeil, M. A. Sutton, and W. H. Peters,“Digital image correlation using Newton–Raphson method ofpartial differential correction,”Exp.Mech. 29, 261–267 (1989).

9. C. Q. Davis and D. M. Freeman, “Statistics of subpixel regis-tration algorithms based on spatiotemporal gradients or blockmatching,” Opt. Eng. 37, 1290–1298 (1998).

Fig. 6. Specimen of three-point bending.


10. P. Zhou and K. E. Goodson, “Subpixel displacement and defor-mation gradient measurement using digital image/specklecorrelation,” Opt. Eng. 40, 1613–1620 (2001).

11. D. J. Chen, F. P. Chiang, Y. S. Tan, and H. S. Don, “Digitalspeckle-displacement measurement using a complex spec-trum method,” Appl. Opt. 32, 1839–1849 (1993).

12. H. Jin and H. Bruck, “Pointwise digital image correlationusing genetic algorithms,” Exp. Tech. 29, 36–39 (2005).

13. M. C. Pitter, C. W. See, and M. G. Somekh, “Subpixel micro-scopic deformation analysis using correlation and artificialneural networks,” Opt. Express 8, 322–327 (2001).

14. B. Pan, H. M. Xie, B. Q. Xu, and F. L. Dai, “Performance of sub-pixel registration algorithms in digital image correlation,”Meas. Sci. Technol. 17, 1615–1621 (2006).

15. J. Zhang, G. Jin, S. Ma, and L. Meng, “Application of animproved subpixel registration algorithm on digital speckle

correlation measurement,” Opt. Laser Technol. 35, 533–542(2003).

16. Z. Feng and R. E. Rowlands, “Continuous full-field represen-tation and differentiation of three-dimensional experimentalvector data,” Comput. Struct. 26, 979–990 (1987).

17. X. Dai, Y. C. Chan, and A. C. K. So, “Digital speckle correlationmethod based on wavelet-packet noise-reduction processing,”Appl. Opt. 38, 3474–3482 (1999).

18. M. D. Buhmann, Radial Basis Functions: Theory andImplementations (Cambridge U. Press, 2003).

19. G. B. Wright, “Radial basis function interpolation: numericaland analytical developments,” Ph.D. dissertation (Universityof Colorado, 2003).

20. J. Duchon, “Splines minimizing rotation-invariant semi-norms in Sobolev spaces,” Laboratoire de MathematiquesAppliquees. (Springer-Verlag, 1977), pp. 85–100.


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Displacement field analysis based on the combination digital speckle correlation method with radial basis function interpolation