Finite element formulation for a digital image correlation method

Finite element formulation for a digital imagecorrelation method

Yaofeng Sun, John H. L. Pang, Chee Khuen Wong, and Fei Su

A finite element formulation for a digital image correlation method is presented that will determinedirectly the complete, two-dimensional displacement field during the image correlation process on digitalimages. The entire interested image area is discretized into finite elements that are involved in thecommon image correlation process by use of our algorithms. This image correlation method with finiteelement formulation has an advantage over subset-based image correlation methods because it satisfiesthe requirements of displacement continuity and derivative continuity among elements on images.Numerical studies and a real experiment are used to verify the proposed formulation. Results have shownthat the image correlation with the finite element formulation is computationally efficient, accurate, androbust. © 2005 Optical Society of America

OCIS codes: 120.0120, 100.2960.

1. Introduction

A digital image correlation (DIC) method1–4 is ap-plied to obtain the displacement field on the surface ofa planar specimen by comparing a pair of digitalimages taken before and after the imposed deforma-tion. In recent years the DIC method has been com-monly used for deformation measurements.5–10

Since the subset-based image correlation is used toobtain deformation parameters of a point one by one bymapping a subset with the point in the center over apair of digital images (see Fig. 1), and there is norequirement of inter-subset continuity during the cor-relation process, obviously, wrong data points9 (seeFig. 2) can be observed occasionally. In addition, thediscrete displacement field data that were obtaineddirectly from existing image correlation algorithmsusually need to be improved through postprocessing byuse of smoothing techniques11,12 such as finite elementsmoothing and least-squares polynomial fitting.

We believe that the two issues can be resolved byintroducing a finite element framework into the im-

Y. Sun ([email protected]) and J. H. L. Pang are with theSchool of Mechanical and Aerospace Engineering, Nanyang Tech-nological University, Singapore 639798. C. K. Wong and F. Su, aswell as Y. Sun, are with the Singapore Institute of ManufacturingTechnology, 71 Nanyang Drive, Singapore 638075.

Received 27 January 2005; revised manuscript received 12 April2005; accepted 14 April 2005.

0003-6935/05/347357-07$15.00/0© 2005 Optical Society of America

Fig. 1. Schematic of subset-based image correlation: (a) unde-formed and (b) deformed images.

Fig. 2. Wrong deformation data in the displacement vector field.

1 December 2005 � Vol. 44, No. 34 � APPLIED OPTICS 7357

age correlation process. We discretize the interestedimage area into finite elements, which are all in-volved in the common image correlation process withour developed algorithms. In this case, the interele-ment continuity requirement can be satisfied natu-rally and the underlying deformation field can beimproved by implicitly adopting finite elementsmoothing. The details of our finite element formula-tion for the DIC method are presented in the follow-ing sections. Results from a series of numericalstudies are also presented to verify our code and toprovide basic data with regard to the accuracyachieved by the method.

2. Background for Subset-Based Image Correlation

The subset-based image correlation method is used tomeasure full-field displacements with subpixel accu-racy from digital images by finding the matching sub-sets in images of undeformed and deformed states.One method that has proved to be efficient to solve thematching problem is the iterative least-squares corre-lation algorithm.13,14 The algorithm minimizes theleast-squares correlation coefficient between subsetprofiles in reference image f and deformed image g:

C ��

Sp�S�f�x, y� � g�x � u�x, y, a�, y � v�x, y, a��2

�Sp�S

f 2�x, y�,

(1)

where S represents a subset of points around thecenter �x0, y0�; Sp represents any single point in thesubset of the reference image at position �x, y�;u�x, y, a� and v�x, y, a� are displacement functionswith a parameter vector a that relates the position ofa subset in the reference image to its position in thedeformed image. Commonly used displacement func-tions are first-order displacement functions

u�x, y, a� � a0 � a2�x � x0� � a4�y � y0�,

v�x, y, a� � a1 � a3�x � x0� � a5�y � y0�. (2)

To efficiently solve for parameter vector a�a0, a1, a2,a3, a4, a5� or explicitly

au�x0, y0�, v�x0, y0�,�u�xx0, y0

,�v�xx0, y0

,

�u�yx0, y0

,�v�yx0, y0

�[where u�x0, y0� and v�x0, y0� are the displacements forthe subset center in the x and y directions, respec-tively] that minimizes the correlation coefficient[Eq. (1)], the Newton–Raphson method is used toiteratively find the solution with a given approxima-tion a0 to the solution. The next improved approxi-mation a can be obtained by use of the followingNewton–Raphson iteration formula:

��C�a0��a � a0� � ��C�a0�. (3)

The strain components of the subset center weretherefore determined in the imaging plane from

�xx ��u�x �

12��u

�x 2

� ��v�x 2�,

�yy ��v�y �

12��u

�y 2

� ��v�y 2�,

�xy �12��u

�y ��v�x �

12��u

�x�u�y �

�v�x

�v�y . (4)

As for the small rigid body rotations of less than10 deg, the angle of rotation � can be approximated bythe following equation3:

� �12��v

�x ��u�y . (5)

In addition, it is noted that the interpolation ofgray-scale values and their derivatives at nonintegerpixel locations are needed, taking into considerationthat the displacement can be any value but no gray-scale information is available between pixels. Typi-cally, bicubic spline interpolation is used for theinterpolation scheme:

g�x, y� � �i�0

3

�j�0

3

�ij x iy j. (6)

The coefficients �ij for the bicubic interpolation poly-nomial are determined by the gray-scale values of agrid of given points and continuity requirements.

3. Finite Element Formulation for Digital ImageCorrelation

The finite element formulation for DIC is introduced.We call our method the element-based image correla-tion method, or element-based method for short. Theelement-based method discretizes the interested im-age area into segments or elements linked by nodes(see Fig. 3). We obtained nodal displacements by min-imizing the least-squares correlation coefficient be-

Fig. 3. Schematic of element-based DIC: (a) undeformed and (b)deformed images.

7358 APPLIED OPTICS � Vol. 44, No. 34 � 1 December 2005

tween all the elements in the reference image and thecorresponding elements in the deformed image.

A. Correlation Coefficient and Displacement Functionsfor the Element-Based Method

Because element-based DIC involves all the elementsto be correlated at one time, Eq. (1) could be modifiedas follows:

C

��e�S

�Sp�e

�f�x, y�� g�x � u�x, y, pe�, y � v�x, y, pe��2

�Sp�S

f 2�x, y�,

(7)

where e is an element, S is a set of all the elements,and pe is the element displacement vector. As shownin Eq. (7), the relationship between a point in thereference image at (x, y) and its mapping in the de-formed image at �x, y� could be described as follows:

x � x � u�x, y, pe�, y � y � v�x, y, pe�. (8)

In the finite element method,15,16 the displace-ments of a point in an element are functions (knownas displacement functions of the element) of the dis-placements of the element nodes:

u�x, y, pe� � �i�1

m

Ni�x, y�ui,

v�x, y, pe� � �i�1

m

Ni�x, y�vi, (9)

where m is the number of nodes in the element wherethe point is located; Ni�x, y� are shape functions of theelement; ui and vi are the components of the displace-ment at the ith node. Thus, the element displacementvector pe � �u1 v1 . . . ui v . . . um vm�T.

B. Methodology for Determination of the MinimalCorrelation Coefficient

Equation (7) is used to find optimum values for the dis-placement parameters. For tidiness, let Sp represent apoint at �x, y�, and let F represent 1��Sp�S f 2�x, y�because it can be considered as a constant. Then Ccan be written as

C � F �e�S

�Sp�e

�f�Sp� � g�Sp, pe��2. (10)

The minimum of C can be obtained by solving the

following nonlinear equations:

�C � ��C�pi

i�1,2n

� �2F�e�S

GT� �Sp�e

�f�Sp�

� g�Sp, pe��g�Sp, pe�

�pje �

j�1,2m

� 0. (11)

where the global displacement vector p � �u1 v1 u2

v2 . . . ui vi . . . un vn�T; n is the total number of nodes;and G is known as the nodal assembly matrix. Therelationship between the element displacement vec-tor pe and the global displacement vector p is

pe � Gp. (12)

The size of matrix G is 2m 2n. We illustrate thedetermination of G by considering a finite elementmesh consisting of two quadrilateral elements (seeFig. 4). The numbers inside the mesh are the localindices of element nodes; those outside are the globalnode indices. For example, matrix G for element 1 inFig. 4 is

G �

1 0 0 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 1

. (13)

We used the Newton–Raphson iteration to find theroots of Eq. (11). This method uses the derivative of afunction to improve an approximation to the actualroot. The Newton–Raphson iteration formula is givenby

��C�p0��p � p0� � ��C�p0�, (14)

where p0 denotes an initial guess of global displace-ment vector p. The Hessian matrix ��C�p0� can bedetermined by

��C � � �2C�pi � pj

i�1,2n

j�1,2n

� �e�S

GT��2F �Sp�e

�f�Sp� � g�Sp, pe��2g�Sp, pe�

�phe � pk

e

�2F �Sp�e

�g�Sp, pe��ph

e

�g�Sp, pe��pk

e�

h�1,2m

k�1,2m

G. (15)


Following Vendroux and Knauss,13 when p is close tothe solution, consider f�x, y� � g�x, y, Gp�, whichleads to an approximation of the Hessian matrix:

�2C�pi � pj

� 2F �e�S

GT �Sp�e

�g�Sp, pe��ph

e


e G.

(16)

Given the displacement functions and the bicubic in-tensity interpolation from the deformed image, thefirst-order partial derivatives of the intensity withrespect to the displacement parameters are readilyderived:


e ��g�x, y, pe�

�x�x

�pke �

�g�x, y, pe��y

�y

�pke .

(17)

The ��g�x, y, pe��x and ��g�x, y, pe��y terms canbe derived from the bicubic interpolation polyno-mial in Eq. (6) of gray-scale values in the deformedimage and the ��x��pk

e� and ��y��pke�terms from

Eqs. (8). When the solution to p is obtained, that is,the displacements of all the element nodes are de-termined, the strain components and rotation anglecan be calculated from the known displacementfield.15,16

As mentioned above, initial guesses for the nodaldisplacements of set S are required for the correlationprocess. Vendroux and Knauss13 have shown that theNewton–Raphson iteration formula with the approx-imate Hessian matrix has a convergence radius of7 pixels, which can be confirmed by our studies. It issafer that the initial guess for the nodal displace-ments of any node is within 5 pixels of the actualdisplacements to ensure convergence. Hence, p0 canbe initially zero or a constant for the small deforma-tion case without rigid body translation or with largerigid body translation. In the case of large rotation orthe deformation between correlated images largerthan the convergence range, p0 can be estimated byperforming interpolation with user-defined displace-ment guesses for four or more arbitrary nodes dis-tributed in set S.

4. Validation and Results

To verify the correctness and accuracy of the element-based method and compare it with the existing

subset-based method, both DIC codes were writtenaround the above-mentioned scheme. Both codes im-plement the two-dimensional correlation methodwith bicubic spline interpolation and with the ap-proximate Hessian matrix formulation for the least-squares correlation coefficient. The element-basedmethod code uses rectangular elements with fournodes to approach the displacement functions andthe shape of subsets used in the subset-based imagecorrelation method. Although the element-basedmethod has the potential to adopt an arbitrary shapeand higher-order elements, we explore the advantageof the element-based method over the subset-basedmethod by meeting the requirement of displacementcontinuity among elements.

All the images that we used have 640 � 480 pixelsand 8-bit intensity data. The 32 � 32 pixels were usedas the subset size and the step size during subset-based image correlation to output results over fixed16 � 11 array points. Correspondingly, the fixed 16 �11 array points were selected as element nodes witha rectangular element size of 32 � 32 pixels duringelement-based image correlations. Therefore, thecomparison between the subset-based method andthe element-based method is surely based on thesame data points.

With regard to the element-based correlation pro-cess, we note that the correlation errors of computednodal displacements at the boundary are somewhatlarger than those in the interior because of the lack ofadjoining elements to impose continuity constraintsat the boundary. There are two approaches to elimi-nate this effect: one is to discard the results at theboundary and the other is to perform the second cor-relation by use of boundary elements with nodal dis-placements at the boundary as the unknowns andthose inside the boundary as the knowns. We usedthe former. Thus, the above-mentioned 16 � 11 arraypoints are interior nodes in the element-based corre-lation process.

Both codes had a CPU time of less than 2 s,completing the computation for each case of our

Fig. 4. Determination of nodal assembly matrices by use of thecorrespondence between global and element nodes.

Fig. 5. Reference image.


study on a PC with Intel Pentium 4 and Windows2000. According to the above algorithms, we knowthat the computation time for one Newton–Raphsoniteration consists of the time required for the sum-mation to assemble two matrices of the Newton–Raphson equation and the time required for thematrix solution. Most of the computation time isspent on the summation process. The summation is

performed one pixel by one pixel so its time is pro-portional to the number of pixels involved in thecorrelation process, or the total correlation area inthe element-based correlation, or the product of thesubset size and the total point number in thesubset-based correlation.

A. Numerical Simulations

Simulated deformed images were employed to vali-date and evaluate two DIC codes since simulatedimages have the advantage to eliminate assessinginstrumentation errors. Based on a reference image(Fig. 5), deformed images (e.g., Fig. 6) with imposedtranslation, stretch, skewness, or rotation were nu-merically generated by assigning intensity data fromthe reference image to the deformed image in corre-sponding positions, that is, g�x, y� � f�x, y�. Taking thedeformed image with imposed translation 0.5 pixelonly in the �x direction, for example, we wrote acomputer program to numerically generate a digitalimage with its intensity data g�x, y� � f�x � 0.5, y�.

Tables 1–3 show the results of two DIC codes ob-tained with numerically translated images, stretchedand skewed images, and a rotated image, respec-tively. A comparison of the element-based methodand the subset-based method results in Tables 1–3indicates that the element-based method has smaller

Fig. 6. Horizontal skew image.

Table 1. Results of Simulated Rigid Body Translations

Cases Method u� � u0 �pixels� v� � v0 �pixels� �u �pixels� �v �pixels� ��xx��yy

��xy

1a Element 8.39 � 10�4 9.68 � 10�3 10.73 � 10�3 12.02 � 10�3 2.23 � 10�4 2.22 � 10�4 2.97 � 10�4

Subset 1.13 � 10�4 8.37 � 10�3 7.75 � 10�3 7.99 � 10�3 7.35 � 10�4 7.36 � 10�4 9.57 � 10�4

2b Element 8.94 � 10�4 9.10 � 10�4 8.88 � 10�3 10.37 � 10�3 1.90 � 10�4 1.89 � 10�4 3.22 � 10�4

Subset 5.93 � 10�4 �1.10 � 10�4 7.09 � 10�3 6.83 � 10�3 7.80 � 10�4 7.02 � 10�4 9.82 � 10�4

au0 � 0.5 pixel, v0 � 0 pixel.bu0 � 0.5 pixel, v0 � 0.5 pixel.

Table 2. Results of Simulated Uniform Deformations

Cases Method ��xx � �xx0 ��yy � �yy

0 ��xy � �xy0 ��xx

��yy��xy

1a Element �2.80 � 10�5 �1.04 � 10�6 �2.71 � 10�6 3.57 � 10�4 1.39 � 10�4 2.72 � 10�4

Subset �1.3 � 10�4 �8.37 � 10�5 5.49 � 10�6 1.96 � 10�3 7.21 � 10�4 8.22 � 10�4

2b Element �2.43 � 10�5 �2.05 � 10�5 �8.67 � 10�6 3.70 � 10�4 3.88 � 10�4 3.43 � 10�4

Subset �1.10 � 10�4 �5.84 � 10�5 �5.15 � 10�5 1.93 � 10�3 2.23 � 10�3 1.08 � 10�3

3c Element 5.00 � 10�7 3.98 � 10�6 2.60 � 10�7 2.38 � 10�4 1.99 � 10�4 2.89 � 10�4

Subset 1.96 � 10�4 3.37 � 10�4 �2.1 � 10�4 7.29 � 10�4 7.55 � 10�4 1.09 � 10�3

a�xx0 � 0.0158730, �yy

0 � 0, �xy0 � 0.

b�xx0 � 0.0158730, �yy

0 � 0.0169492. �xy0 � 0.

c�xx0 � 0, �yy

0 � 0, �xy0 � 0.0174533.

Table 3. Results of Simulated Rigid Body Rotation

Method �� 0a (rad) �� (rad) ��xx

��yy��xy

Element 2.14 � 10�4 2.50 � 10�4 3.42 � 10�4 3.19 � 10�4 4.56 � 10�4

Subset 4.57 � 10�4 6.80 � 10�4 8.96 � 10�4 9.13 � 10�4 1.33 � 10�3

a�0 � 0.0174533.


standard deviations of strain components and rota-tion angle whereas the subset-based method hassmaller variations in displacement data. Figures 7and 8 graphically demonstrate the uniform strainvariations of the element-based method and thesubset-based method, respectively, from case 1 in Ta-ble 2. The large variations in strain shown in Fig. 8account for the additional smoothing process17 of full-field displacements that is often needed to obtaindisplacement gradients when using the subset- basedcorrelation method. In addition, corresponding stan-dard deviations of cases 1 and 2 in Table 1 are at thesame level even though case 1 and case 2 have dif-

ferent imposed translation. It is also true for a com-parison of cases 1, 2, and 3 in Table 2.

B. Rotation Experiment

A rotation experiment under controlled conditionswas also performed to evaluate the two DIC codes.We selected the rotation experiment because it is notnecessary to do CCD camera calibration, to evaluatethe calibration error, or to consider the rigid bodytranslation that results from vibration. The specimenwas mounted on a rotation stage (see Fig. 9) with anangular resolution of 0.01 deg (1.745 � 10�4 rad).Two images were acquired before and after the rota-tion angle change of 1 deg. Table 4 presents theelement-based and subset-based correlation results.The results show that the rotation error is 0.011 degfor the element-based correlation and 0.027 deg forthe subset-based correlation. Comparison of the re-sults in Tables 3 and 4 shows that intensity noiseslightly increases standard deviations in the rotationangle and strain when the subset-based correlation isused but not when the element-based correlation inused.

Fig. 7. Variation in strain �xx from the element-based method.

Fig. 8. Variation in strain �xx from the subset-based method.

Fig. 9. DIC specimen mounted on a high-resolution rotationstage.

Table 4. Results of Rigid Body Rotation Experiment

Method �� 0a (rad) �0 � rad� ��x ��y ��xy

Element 1.95 � 10�4 2.27 � 10�4 3.04 � 10�4 3.20 � 10�4 4.54 � 10�4

Subset 4.72 � 10�4 7.07 � 10�4 1.04 � 10�3 1.05 � 10�3 1.62 � 10�3

a�0 � 0.0174533.


5. Conclusion

A finite element formulation for digital image correla-tion has been developed by way of introducing a finiteelement framework into the image correlation process.Numerically generated deformed images were used toshow that (a) the proposed method is accurate and (b)in our research, the existing subset-based image cor-relation method has a 3–4 times variation in thestrains and rotation angle as the element-basedmethod although the latter has a slightly larger vari-ation of displacement data in translation measure-ments. A rotation experiment was also performed, andexperimental results agree well with those from sim-ulation studies. Such are outcome results from the factthat the element image correlation method, has theadvantage of meeting compatibility requirementsamong element deformation.

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Finite element formulation for a digital image correlation method

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