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0recent years, the development of efficient tools forld velocimetry, aiming at fluid flow visualization,1s been quite important. Digital particle imagelocimetry,25 digital image correlation DIC,6,7 orre generally correlation imaging velocimetry8,9ve proven to be an efficient and robust tool whoseecision can be extended much below the pixel ac-racy. The spirit of the method is to look for theximum correlation between small zones extractedm the deformed and reference images. The trans-ion, which corresponds to the maximum correla-n, can be obtained for different positions of the

of a displacement field that is piecewise constant.The correlations are evaluated either in the referencespace6,10 or in the Fourier space,2,3,1113 and the ex-tensive use of fast Fourier transforms FFT is veryeffective in reducing the computation cost.

Some applications concern soft solids where straingauges can be difficult to position or may disturb theresponse of the material because of stiffness varia-tions e.g., mineral wool,9 paper and wood,14 and poly-mers.15,16 In most experiments in solid mechanics,the random texture used to determine the displace-ment field is not changed or deliberately altered dur-ing the test, e.g., cracks may appear, but thealterations are localized and produce few if any arti-facts. It can be noted that the texture is either nat-ural e.g., many materials at high magnification orartificial e.g., by spraying paint. In some applica-tions, however, this is no longer the case, e.g., duringa compression test of mineral-wool samples wherepictures are obtained in transmission. As the sam-ple is compressed, the average texture intensity de-creases when the aperture is not altered. Inaddition, the change in intensity due to the compres-sion alters the absorption and scattering of light inthe material, and hence increases the distortion be-tween the images in a complex and heterogeneousmanner. As a consequence, classical methods fail to

. Hild and B. Raka are with LMT-Cachan, ENS de CachanRS-UMR 8535Universite Paris 61, avenue du President Wil-, F-94235 Cachan Cedex, France. M. Baudequin and S. Rouxwith the Laboratoire Surface du Verre et Interfaces, UMR

RSSaint-Gobain 89 quai L. Lefranc, F-93303 Aubervilliers Ce-, France. F. Cantelaube is with CRIR, 19 rue Emile Zola,0290 Rantigny, France. The authors e-mail addresses arencois.Hild,[email protected];ud.Baudequin,[email protected],[email protected] 3 January 2002; revised manuscript received 28 May2.003-693502326815-14$15.0002002 Optical Society of America

10 November 2002 Vol. 41, No. 32 APPLIED OPTICS 6815ultiscale displacement fieldompressed mineral-wool sam

age correlation

ancois Hild, Bumedijen Raka, Maud Baudequin,rence Cantelaube

We propose a multiscale approach todisplacement field is first estimateintroduced in the analysis as the dSuch a scheme has been developedmatching algorithm. The procedura representative image that is deforIn particular, the maximum measupictures taken during compressionprocedure are analyzed and their reare tested. 2002 Optical Society

OCIS codes: 070.2590, 100.2000easurements ofles by digital

phane Roux, and

rmine the displacement field by digital image correlation. Thea coarse resolution image and progressively finer details are

cement is more and more securely and accurately determined.crease the robustness, accuracy, and reliability of the image-

sed on two different types of examples. The first one deals withprecisely and purposefully to assess the intrinsic performances.strain is determined. The second case deals with a series of

riments on mineral-wool samples. The different steps of thetive role is assessed. Both reflection and transmission imagesmerica.5010, 120.3940, 120.6150, 040.1520.Introduction ne of interest. This allows for the determinationmp

Ste

deted onisplato in

e is umedrableexpespecof A, 100

zo

measure accurately the strain fluctuations in suchsituations.

One purpose of the present study is to propose amultiscale approach to increase the maximum detect-able displacement compared to conventional DICtechniques. Because texture variations are ex-pected to occur, a special procedure is developed todeal with this effect, which can be probed in partic-ular, for the above-mentioned mineral-wool samples.Section 2 introduces two basic tools needed to developa DIC technique, namely: displacements measuredby a correlation algorithm and the choice of a straindescriptor in the finite deformation framework. InSection 3 the basic DIC algorithm is recalled. Asubpixel procedure is used to increase the sensitivityfor accurate strain measurements. The maximumdetectable displacement is evaluated on a deformedpicture, which is artificially generated. A multiscaleapproach is then developed to increase the maximumdetectable displacement, without affecting the mini-mum measurable strain. Section 4 deals with arti-ficially deformed images to analyze the effect ofdifferent steps introduced in the multiscale correla-tion algorithm. Section 5 is devoted to the analysisof two sets of compression experiments performed onmineral-wool samples. This material allows forlarge strains and various exposure modes. More-over, being heterogeneous, it displays fluctuations inlocal strains. It thus constitutes a difficult test casefor the analysis. Mineral wool is a material forwhich better mechanical performances can bereached by mastering the microstructure and homo-geneity. In particular, for medium to high-densityproducts, the wool is crimped to get a better mechan-ical performance. Crimping is a uniaxial compres-sion along the line of movement performed on thewool by forcing it through progressively slower ad-

vancing compression belts. This industrial process,suited to continuous production, induces buckling ofthe denser planes and hence changes the orientationof initially horizontal fibers to vertical or at leastmore isotropic distributions. Hence, it is of interestto be able to resolve the local strain, or irreversiblepart of the strain, to relate it to the local microstruc-ture, such as material density or fiber orientation.This is especially important for crimped products,where quality factors for the texture can be derivedfrom the local analysis of the irreversible strain, andits relationship to morphological features.

2. Preliminaries

A. Displacement Measurement by Image Correlation

To determine the displacement field of one imagewith respect to a reference image, one considers asubimage e.g., a square region which will be re-ferred to as a zone of interest ZOI or interrogationwindow. The aim of the correlation method is tomatch the ZOI in the two images Fig. 1. The ref-erence image corresponds to the texture of a stone-wool sample. The displacement of one ZOI withrespect to the other one is a two-dimensional shift ofan intensity signal digitized by a CCD camera. Toestimate a shift between two signals, one of the stan-dard approaches is based on a correlation function.One considers signals gx which are merely pertur-bations of a shifted copy f x u of some referencesignal f x

gx f x u bx, (1)

where u is an unknown in-plane displacement vectorassumed to be constant locally i.e., independent ofthe position x and bx is a random noise e.g., pho-

Fig. 1. a Initial and b artificially deformed Eyy 0.05 images of a stone-wool sample. The region of interest ROIref and two zonesof interest ZOIref and ZOIdef are depicted. Four centers of ZOIs are shown on the reference picture : separation between neighboringZOIs.

6816 APPLIED OPTICS Vol. 41, No. 32 10 November 2002

ton noise, readout noise, dark current noise for CCDcameras.17 To evaluate the displacement u, onemay minimize the norm of the difference betweenf x v and gx with respect to a trial displacementv

minv

g f v2 (2)

where denotes a dummy variable. The functionalg f v2 can be used in other instances whenthe displacement field, instead of being constant, iseither affine or developed in terms of a truncatedFourier series.9 These variational formulations aremainly based on the optical flow equation. A spatialregularization was introduced by Horn andSchunck18 and it consists in looking for smooth dis-placement solutions. However, this method is notappropriate for problems dealing with discontinuitiesin the apparent displacement.19 In the latter case,the quadratic penalization is replaced by smootherpenalizations based on robust statistics.2022 Fur-thermore, when dealing with deformable solids, otherregularization techniques are to be introduced, suchas the one based on the strain energy.23

In fact, any class of displacements may be intro-duced and a Taylor expansion yields a linear systemin the unknown displacement components.24 Ifone chooses the usual quadratic norm f 2

f x2dx, it corresponds to the so-called maxi-mum quadratic difference MQD method used in ex-perimental fluid mechanics25 when implementeddirectly with a constant trial displacement. Withthis norm, the previous minimization problem isequivalent to maximizing the quantity hv

hv g f v

gx f x vdx, (3)

where denotes the cross-correlation operator. Fur-thermore, when b is a white noise, the previous esti-mate is optimal. The displacement maximizing thecross-correlation product will be denoted by w andcorresponds to an evaluation of the unknown dis-placement vector u. The computation of a cross-correlation product can be performed either in theoriginal space or in Fourier space, by using an FFT

g f FFT1FFTgFFTf, (4)

where the complex conjugate is overlined. In thefollowing sections, extensive use of the cross-correlation product will be made. It will always beevaluated via Fourier transforms to speed up thecomputations. Numerous works have addressed theissue of the periodicity of the image implicitly as-sumed when using an FFT. In particular, there liesthe main difference between the MQD method andstandard correlation-based evaluations, as discussedin detail in Ref. 25. In the present work, we dealwith this problem through windowing filters e.g.,Hanning window or edge blurring as discussed be-low.

B. Strain Measures

Finite strain measures are needed because the strainlevel will be greater than a few percents. Thesemeasures are built by using the deformation gradienttensor F relating an infinitesimal vector dx0 in thereference configuration 0 to dx in the deformedconfiguration

dx Fdx0 (5)

so that the tensor F can be related to the displace-ment gradient u by

F 1 u, (6)

where 1 is the second-order unit tensor. For La-grangian measures, they can be expressed by usingthe strain tensors Em

Em 1

2mCm 1 when m 0

12

lnC when m 3 0, (7)

where C Ft. F denotes the right CauchyGreendeformation tensor, and t the transposition operator.When m 1, the GreenLagrange strain tensor isobtained, m 12 is the CauchyBiot or nominalstrain tensor and yields LL0 for a uniaxial elonga-tion, where L0 is a reference or gauge length and Lis the length variation. The case m 3 0 corre-sponds to the logarithmic or Hencky strain tensor.The latter is the only additive strain measure. Bydefinition, all these strains are equal to zero for arigid translation i.e., F 1 and C 1 or a rigidrotation i.e., F R and C 1, where R is anorthogonal tensor. When the amplitude of the bodymotion is small as well as the strain gradient, allmeasures converge toward the so-called infinitesimalstrain tensor defined as

12u ut. (8)

In the following, for the sake of simplicity, the nom-inal strain tensor will be used and will be denoted byE E12.

3. Multi-Scale Approach in Digital Image Correlation

A. Conventional Approach

In practice, two images are considered: The firstone, referred to as the reference image and the secondone, called the deformed image. The following algo-rithm is summarized in the flowchart of Fig. 2. Todetermine the average displacement, one extracts thelargest value of P of an inscribed region of interestROI of size 2P 2P centered in the reference image.The same ROI is considered in the deformed image.A first FFT correlation is performed to determine theaverage displacement w0 of the deformed image withrespect to the reference image. This displacementvector is expressed by an integer number of pixels

10 November 2002 Vol. 41, No. 32 APPLIED OPTICS 6817

and is obtained as the maximum of the cross-correlation function evaluated for each pixel of theROI. This first prediction enables one to determinethe maximum number of pixels that belong to bothimages. The ROI in the deformed image is now cen-tered at a point corresponding to the displaced centerof the ROI in the reference image by an amount w0.

The user usually chooses the size of the ZOIs bysetting the value of p P so that its size is 2p 2p

pixels. To map the whole image, the second param-eter to set is the shift x y between twoconsecutive ZOIs: 1 2p pixels. This param-eter defines the mesh formed by the centers of eachZOI used to analyze the displacement field Fig. 1.The analysis is then performed for each ZOI indepen-dently. A first FFT correlation is carried out and afirst value of the in-plane displacement correction wis obtained. The components of w are again inte-ger numbers so that the ZOI in the deformed imagecan be displaced by an additional amount w. Thedisplacement residues are now less than 12 pixel ineach direction. A subpixel iterative scheme can beused.

This first procedure is well adapted for small strainlevels. If no additional positioning is performed, tobe successful, the norm of the displacement correc-tion w has to be limited. Otherwise, the correla-tion fails because there is not enough information todetermine the correction. To illustrate this effect, aconventional analysis is performed between the ref-

erence image 512 512 pixels, 8-bit digitizationshown in Fig. 1 and an artificially deformed image.The reference image corresponds to a stone-woolsample to be artificially deformed along the verticaldirection i.e., j. The deformed image is computedby using Eq. 1, and a linear interpolation of the graylevels to evaluate the pixel value in the deformedimage. No noise is added i.e., b 0 but thesignal is digitized by using the same 8-bit coding.The same 512 512 pixel ROI is considered anddifferent values for p are investigated. In thepresent case, the initial correction w0 is found to bevanishingly small. Consequently, the middle heightof the image is a line of symmetry for the displace-ment field. Figure 3 shows the standard deviation ofthe vertical displacement wy for each vertical positiony of the ZOIs when 16 pixels. The smaller thestandard deviation, the more accurate the displace-ment evaluation. For symmetry reasons, the stan-dard deviation wy is interpolated by a parabolicpolynomial defined by

wy y p p255.5 y2, (9)

Fig. 2. Flowchart of the principal steps in a conventional corre-lation algorithm.

Fig. 3. Standard deviation of the vertical displacement vs. verti-cal position y of the ZOI when a strain Eyy 0.05 is prescribed Fig.1. A classical digital image correlation procedure is used fordifferent sizes 2p 2p pixels of the zones of interest 16pixels.

Table 1. Average Nominal Strain E yy and Corresponding StandardDeviation Eyy

a

Pixels E yy Eyy

p 4 0.0014 0.003p 5 0.0419 0.0006p 6 0.0456 0.0004p 7 0.0470 0.0001

aFor different sizes of ZOIs 16 pixels predicted by a con-ventional DIC technique. A pixel procedure is used. The pre-scribed strain Eyy is equal to 0.05.


where and are constants dependent upon the sizeof the ZOIs i.e., 2p 2p pixels. An arbitrary limitwmax is defined such that the evaluation is consideredto be accurate if it is below the limit and is question-able if it is above the limit. A value of one pixel ischosen. Consequently, displacement corrections wthat vary within the range7 pixels can be measuredaccurately when p 4 and 13 pixels when p 5.For higher values of p, no intersection is found whenthe average strain is equal to 0.05 Fig. 3. As a first

approximation, a displacement correction greaterthan3 2p3 pixels leads to a loss of accuracy in itsevaluation. It can be noted that when a value of 0.6pixel is chosen, the displacement estimate is moreaccurate than in the previous case and the quarterrule is obtained i.e., the maximum measurable dis-placement is equal to one quarter of the size of theZOI27. These results show that the first pixel eval-uation w0 is crucial in enabling the admissible rangeof measurable displacements to increase. Table 1shows that the higher p, the more accurate the strainevaluation. However, the gain between p 6 andp 7 is minimal. If one increases p too much, thenone averages over a larger surface the displacement.The measurement of fluctuations with respect to anaverage field is limited e.g., in tensioncompressiontests. Consequently, a good compromise in this ex-ample is given by p 6. Last, for images such thatthe maximum distance of horizontal or vertical cen-ters of ZOIs is of the order of 1024, a maximum mea-surable strain is equal to 3 2p12, which is of theorder of 0.05 when p 6.

It can be noted that small strain levels can be ac-curately measured with a subpixel algorithm. Anadditional cross correlation is performed. A sub-pixel correction of the displacement w is obtained bydetermining the maximum of a parabolic interpola-tion of the correlation function. The interpolation isperformed by considering the maximum pixel and itseight nearest neighbors. Therefore, one obtains asubpixel value. By use of the shift modulationproperty of Fourier transform, which is nothing but aphase change, the deformed ZOI can be moved by anamount w. Because an interpolation is used, onemay induce some small errors that require reitera-tion by considering the new deformed ZOI until con-vergence. The procedure tests whether themaximum of the interpolated correlation function in-

creases as the number of iterations increases. Oth-erwise, the iteration scheme is stopped.

To limit the influence of edge effects, a windowingof the ZOI can be performed16

ZOI ZOI (10)

where ZOI denotes the windowed ZOI, V the dyadicproduct, and the one-dimensional modified Han-ning window

The value 2p2 is optimal to minimize the error dueto edge effects and to have a sufficiently large numberof data unaltered by the window. In the following, asecond window is introduced. It consists in con-structing a periodic ZOI, ZOI, by averaging the pixelson the edges of the initial ZOI i.e., blurring the edg-es

ZOIW ZOI W, (12)

where the matrix W is equal to the identity matrixapart from the four corners, which are equal to 12.The effect of the two windows on the accuracy of thestrain measurement will be tested below. The pro-cedure, CORRELILMT, is implemented in MATLAB.28The subpixel procedure allows for strain levels lessthan 104 to be measured accurately with an 8-bitcamera.16,29 For many materials e.g., metals andalloys, composites, a strain range that varies be-tween 104 and 5 102 is sufficient to analyzetensioncompression tests. However, mineral woolor polymers can experience strain levels greater thanthe previous levels. The aim of the following devel-opments is to increase the maximum detectable dis-placement without altering the intrinsic sensitivity ofthe subpixel correlation procedure. To achieve thisgoal, a multiscale approach is now developed.

B. Definition of Image and Displacement Scales

To increase the maximum measurable displacementbetween two pictures, two major steps are added.Contrary to adaptive local window-shifting tech-niques,30,31 in the present approach, different scalesare introduced: the finer scale is that of the originalpicture i.e., typically 0.5 to 1 Mpixels with classicalCCD cameras. It will be referred to as scale no. 0Fig. 4. In this image, the ROI is a smaller part ofit: it defines the region over which the multiscaleprocedure will be applied. This corresponds to scale

I 12 1 cos 4I2p 1 when 0 I 2p2,1 when 2p2 I 3 2p2,

12 1 cos 4I2p 1 when 3 2p2 I 2p 1.

(11)


no. 1 Fig. 4. Let d01 denote the vector joining theorigin of the image frame to that of the ROI. If norigid body rotations are assumed, the displacementfield ux0 in a homogeneous deformation test is ex-pressed as

ux0 Ax0 a, (13)

with

A Axx 00 Ayy , (14)where A is an in-plane tensor that characterizes theaverage deformation in the present case, F A,and a a constant displacement vector. The rightCauchyGreen deformation tensor can be written as

C A2, (15)

and hence, in this particular case, the tensor A isnothing but the nominal strain tensor, E A.

In the frame of the ROI, where a current point islocated by the vector x1 x0 d01, the previousdisplacement is written as

ux1 Ax1 a Ad01. (16)

One determines the largest values for P and Q ofthe sub-ROI of the size 2P 2Q centered and in-scribed in the ROI. This sub-ROI is adapted to FFTalgorithms and defines scale no. 2 Fig. 4. Let d12denote the vector joining the origin of the ROI to that

of the sub-ROI. In the frame of the sub-ROI x2 x1 d12, one has

ux2 Ax2 a Ad01 d12. (17)

From scale no. 3 on, each scale transition is charac-terized by the definition of superpixels Fig. 4. Thesuperpixels are defined recursively from one scale tothe next by averaging the gray levels of 2 2 neigh-boring pixels. Alternatively, one could use the Fou-rier transform images to filter neighboring pixels.Both procedures should give comparable results, andin the following the real-space coarse-graining super-pixels are used. On scale no. n 3, the displace-ment field is now expressed as

uxn Axn 22na Ad01 d12. (18)

Equations 13, 16, 17, and 18 show that anyscale transition leaves the magnitude of strain un-changed whereas the average displacement de-creases starting from scale no. 3. In Fig. 4, thedifferent notations used are summarized. A refer-ence sub-ROI is shown as well as two superimages oncoarser scales. This procedure is carried out untilthe minimum size of the subimage is equal to 128pixels. The size of the ZOI is such that p 5 onscales no. 2 to n. On scale no. 1, the user chooses thesize of the ZOIs. An iterative correlation algorithmis now applied on each scale. The aim is to firstevaluate interpolated displacements i.e., A and a onthe higher scales down to scale no. 1 for which the

Fig. 4. Different notations used in the multiscale approach. Sub-ROI scale no. 2 of a reference image and corresponding superimageson scales nos. 3, 4.


displacement amplitude may be greater than the sizeof the ZOI on scale no. 1.

The maximum detectable displacement amplitudeis denoted by wmax 3 2

p2 pixels see Section3.A. This value is independent of the scale consid-ered. It is an intrinsic property of the correlationprocedure. It follows that the maximum measur-able strain E1 maxAxx, Ayy on scale no. 1, when nomultiscale procedure is used, is expressed as

E1wmax

N 1, (19)

where N is the maximum number of horizontal orvertical centers of ZOIs. If N 1 512 pixels,then four scales can be defined so that the size of theROI on scale no. 4 is: L4 128 pixels. Conse-quently, the maximum detectable strain E4 on scaleno. 4 is

E4wmax

L4, (20)

which can be of the order of four times the value onscale no. 1. Because the strain tensor A remainsunaltered by any of the considered scale transitions,this result shows the benefit of using a multiscaleapproach. The strain resolution is unaltered when asubpixel procedure is used and the maximum valuedoubles for each scale transition n 3. Further-more, an iterative scheme will improve the assess-ment for large displacements as discussed in the nextsection.

C. Multiscale Correlation Algorithm

The multiscale correlation algorithm is summarizedin the flowchart of Fig. 5. Starting on the highesti.e., coarser scale, for which the subimage size is atleast equal to 128 superpixels, a first correlation isperformed. From this first computation, the dis-placement field is interpolated according to Eq. 18as

unxn Bnxn bn, (21)

where Bn and bn are first estimates of the straintensor of the form given in Eq. 14 and rigid bodydisplacement on scale no. n, respectively. To checkthis first estimate, the center of each ROI of the de-formed image is relocated according to these firstestimates. A new correlation is performed and cor-rections to the previous displacement field are eval-uated. The iterations are stopped as soon as there isno new correction between two iterations for any ZOIanalyzed. This iterative procedure was imple-mented to make the displacement evaluation morerobust. This robustness is a key to the success of theprocedure. If ever the displacement is not properlyevaluated on higher scales, there is no chance of get-ting a good final result: see Section 4. Conse-quently, the first evaluations have to be performedvery carefully.

When the iterative procedure stops on scale no. n,

it proceeds to scale no. n 1. Having identified afirst estimate of the displacement field, it is used toposition the ZOIs on scale no. n 1, according to thetransformation rules inverted from those developedin the previous section. The same iterative schemeis followed for scales no. n 1 to 2. In practice, theuser can choose between a first evaluation that isvery fast because it remains on the pixel or super-pixel level, and an accurate evaluation requiring sev-eral iterations. On scale no. 1, the user selectseither a pixel or a subpixel sensitivity of the displace-ment by following the above described subpixel pro-cedure, thereby allowing for the measurement ofsmall levels of strains as well as large levels that areevaluated in the following sections.

When a sequence of more than two images is ana-lyzed, two routes can be followed. The first one con-sists in considering the same reference image. Itfollows that the errors are not cumulated, but thereexists a maximum-strain level above which themethod fails see Section 4. The second one consid-ers that the reference image is the deformed image ofthe previous step i.e., updating procedure. Underthese hypotheses, there is no real limitation, apartfrom the fact that the errors are now cumulated.Strains of the order of 1 and more are routinely ob-

Fig. 5. Flowchart of the principal steps in the multiscale corre-lation algorithm.


served in a full-field assessment even with a conven-tional DIC technique.16

D. Control of the Displacement Estimate

The previous procedure has to be further improved toaccount for severe texture variations see Section5.B. In practice, small variations always occur butthey are usually limited in magnitude. However, acompressive test dealing with mineral-wool samplesrequires explicit treatment of this effect. Two dif-ferent tests are therefore added.

First, it can be noted that, if the gray level is con-stant over a given ZOI, then a displacement estima-tion based on a correlation technique is impossiblethe signal derivative is the key to minimizing thefunctional 2. Consequently, the standard devia-tion of any ZOI has to be greater than one gray level.It can be noted that on the images of Fig. 6, thiscondition is not satisfied in the lower part. Whenthe standard deviation is less than the chosen limit,the interpolated displacement obtained by the mul-tiscale procedure is used instead of any other esti-mate. This is true on any scale considered.

Second, to check that the correlation procedure wassuccessful, an error indicator, which is independentof the correlation product or its normalized counter-part, is introduced. The following indicator

MEANZOIrefMEANROIref MEANZOIdefMEANROIdef MEANZOIrefMEANZOIdef

12 (22)

turned out to be satisfactory, where the subscript refrefers to the reference image and def to the deformedimage when convergence was obtained. Hence asecond test associated with is added: When 4gray levels for 8-bit pictures, the correlation is con-sidered as unsuccessful and the correlated displace-ment is again preferred. This indicator allows forthe correction due to the average intensity changeduring a test and is expressed in terms of gray levelsbecause of the square-root part.

4. Test on Artificially Deformed Images

In this section, the reference image of Fig. 1 is used toevaluate the intrinsic performances of the algorithm.The images are again deformed artificially with thesame procedure as that used to obtain the results ofFig. 3. Figure 4 shows the sub-ROI of the referenceimage and the corresponding superimages for scalesnos. n 3, 4. When a strain of 0.05 is applied anda subpixel evaluation is chosen with no iteration, theeffect of windowing can be discussed. When no win-dow is used, an average strain of 0.0493 0.000132 isobtained. When the modified Hanning window Eq.11 is implemented, an average strain of 0.0494 0.0001 is measured. Finally, when the edge-blurring procedure Eq. 12 is preferred, the averagestrain is equal to 0.0496 7 105. The last mea-surement is the most accurate. The other ones arealso acceptable thanks to the multiscale procedure.

Figure 7 shows the results obtained by the multi-scale procedure for a nominal strain of 0.35 when p 4 and 16 pixels. The predicted value is 0.35010.0001 when the subpixel option is used and 0.34960.0001 when the pixel option is used. For the sub-pixel option, the standard deviation ranges between0.13 and 0.26 pixel average value: 0.2 pixel and isless than the prescribed limit. It follows that all themeasurements are considered to be accurate. Thiscase is out of reach for a conventional DIC techniquesee Section 3.A. Because the prescribed and inter-polated displacement fields are both bilinear, thepresent approach accommodates displacement am-plitudes greater than the size of the ZOI. In theexample, the displacement amplitude i.e., 120 pix-els is 7.5 times greater than the ZOI size i.e., 16pixels. The limitation is now given by the highestscale i.e., n as opposed to scale no. 1 in the conven-tional DIC technique in the present case, n 4.Figure 8 shows how the average strains evolve duringthe iterations on each scale. It can be seen that eventhough the first estimate is only approximately halfof the actual solution, it is sufficient to obtain a goodresult at the end of the computation. In this partic-ular example, the iterations are needed and conver-gence is not obtained otherwise. The benefit of the

Fig. 6. Initial a and last deformed b images of a glass-wool sample. The reference length L0 and the length variation L are drawn.


multiscale procedure is clearly shown as well as theiterative scheme for each scale.

A nominal strain of 0.5 is now considered. Thisamplitude is too large for the procedure describedabove: it fails in one step with no additional treat-ment. Two different strategies are used to measurethat strain:

First, only the reference and deformed picturesare considered. However, the user is allowed to give

a reference length L0 and deformed lengths L0 Lfor the deformed pictures. This information is usedto evaluate a first displacement field on scale no. n 4. The average strain is equal to 0.503 0.001 andthe maximum error is equal to 2.1 gray levels theaverage is equal to 0.6 gray level,

Second, an intermediate picture is added itsdeformation corresponds to 0.25 and the referencepicture is updated. This alternative is also success-ful average strain: 0.502 0.001 because strainsof the order of 0.3 are measured without any problemwith a good accuracy the maximum error is hereequal to 0.8 gray level and the average equals 0.2gray level.

This simulation shows that even though errors arecumulated when using more than one deformed im-age in an updating procedure, the second strategy ismore robust than the first one. This is due to thefact that a 0.5 deformation is too high locally: Thebasic hypothesis of a constant local displacement isno longer valid. These two strategies are investi-gated in the following section, which is devoted to theanalysis of actual experiments.

5. Experiments on Mineral-Wool Samples

The above tests are quite satisfactory. However, byconstruction, there is no noise in the signal, and moreimportantly, the basic hypothesis 1 is met with b 0. However, in real cases, this assumption is onlyroughly satisfied, in particular when the images areobtained in a transmission mode. There is always aslight shift in contrast and in light intensity that mayaffect the overall efficiency. Furthermore, by con-struction, CCD cameras produce noise. Finally, theinterpolated and prescribed displacements had thesame shape. Therefore it is important to test themethod by using realistic images for which strainheterogeneities occur. Compression experiments onstone- and glass-wool samples are analyzed below.Reflection and transmission illuminations are consid-ered.

A. Images in Reflection

The first part deals with a series of pictures of threedeformed stages of a 60 200 200 mm3 stone-woolsample. The last deformed image is shown in Fig. 9as well as the reference image. Because the productis crimped, the fibers are no longer oriented along apreferential direction. The density of the materialstudied herein is equal to 80 kgm3. The images areobtained in reflection. It follows that the correctionsdue to intensity variations are not needed in this firstcase. However, they are still used to check whetherany bias is introduced. The pictures are taken byusing a standard CCD camera 336 751 pixels, 8-bitcoding. Deformed picture no. 1 corresponds to anapparent strain of 0.083, which is evaluated bymeasuring the distance variation between upper andlower platens. Apparent strains of the order of0.113 and 0.2 are obtained for pictures nos. 2 and3, respectively. It can be noted that the upper

Fig. 7. Vertical displacement open circles and correspondingstandard deviation filled circles vs. vertical position y of the ZOIwhen a strain Eyy 0.35 is prescribed. The multiscale correlationprocedure is used p 4, 16 pixels.

Fig. 8. Average strain vs. iteration number for different sizes2p 2p pixels of the zones of interest and shifts . The dashedcurve corresponds to a simulation in which no iteration is used fora given scale. The arrows show the first time a scale, the level ofwhich is indicated by the numbers close to the arrows, is invoked.A strain Eyy 0.35 is applied.


platen is not necessarily in contact with the samplefor the reference picture, thereby inducing an uncer-tainty in the evaluation of the reference length L0.Consequently, the chosen ROI does not include thetwo platens Fig. 9. Even though these apparentstrains are only orders of magnitude of the actualstrain levels experienced by the stone-wool sample,the maximum measurable strain for a gauge lengthof 7 32 224 pixels is of the order of 0.1 when p 5. Consequently, a direct computation with no up-dating or additional information will fail a priori; aposteriori it did for the last picture for which theactual average strain is of the order of 0.13. Thetwo strategies introduced in Section 4 are also used inthe present case.

Figure 10 shows the deformed mesh as well as thecontours of the error defined in Eq. 22 for the thirddeformed image. The first strategy leads to a max-imum error equal to 10.7 gray levels average: 1.2that is greater than that given by the second i.e.,updating strategy average: 0.7 and maximum:7.2 gray levels. The same tendency as that ob-served in Section 4 is found. The average strainlevel is equal to 0.034 0.0008 for the first pictureto compare with an apparent strain of0.083. Forthe second image, the average is equal to 0.056 0.001 to relate with an apparent strain of 0.113and for the third image, 0.128 0.005 to comparewith an apparent strain of 0.2. These resultsshow that a simple measurement based on apparentstrains is not able to evaluate average strains in thesample. This difference is probably due to the eval-

uation of the reference gauge length L0: The refer-ence picture does not correspond to the point wherethe upper platen is in full contact with the sampledark zones close to the platens in Fig. 9a that dis-appear later, see Fig. 9b. In practice, this momentis very difficult to determine. With the present re-sults, it is possible to evaluate the gauge length byusing the measured average strains and by keepingthe measured separations of the platens for the de-formed images. It follows that, for the two first im-ages, a value of 283 1 pixels is predicted. Thisreference length is drawn in Fig. 9a and seems rea-sonable. For the third image, a value of 271 pixels isfound. This presumably indicates that the test is nolonger homogeneous as shown in Fig. 10. This hy-pothesis is confirmed by estimating the coefficient ofvariation i.e., the ratio of the standard deviation tothe average strain for the three stages: 0.025,0.027, and 0.035, respectively.

In the present example, the multiscale procedureaccommodates values of p as small as 4. However,with this ZOI size, six analyzed points had a standarddeviation of texture less than the prescribed limit.A conventional correlation technique would have re-quired at least p 7, thereby limiting the informa-tion concerning the strain heterogeneity in this kindof experiment. This corresponds to a second advan-tage of the multiscale approach when compared toclassical DIC techniques. The results presentedhere are obtained in less than one min for 3 184ZOIs p 5, 32 pixels on a Pentium II PC.

Fig. 9. Reference and deformed images of a stone-wool sample. The estimated reference length L0 is drawn.


B. Images in Transmission

The second case used to test the method is a 140 200 50 mm3 sample of light glass wool i.e., 6 kgm3in the present case, which in its deformed state isuniaxially compressed. Images are obtained intransmission with no special preparation Fig. 6.Moreover, owing to the structure of the medium, thedisplacement field is essentially along the main con-traction axis, and hence it is wellsuited to the multi-scale analysis proposed in this study. Because of itslow density, this material is used for its thermal in-sulation properties. For the same reason, it is se-verely compressed i.e., more than 90% and storedunder high strains. The material must be able torecover most of its initial thickness after decompres-sion to provide the required thermal insulation per-formances. The experimental procedure consists inapplying a maximum displacement to the upperplaten and to measure the residual strain when the

displacement is removed. A sequence of 11 de-formed states is considered.

At the end of the sequence, the apparent residualstrain i.e., LL0, see Fig. 6 is of the order of 0.25.This value should make it possible to perform a directmeasurement between the reference and deformedimages shown in Fig. 6. However, the texture vari-ation during the experiment prevents this approachto be successful. Again, the two strategies of Section4 are used. First, when the user is allowed to give areference and deformed gauge length, the computa-tion can be carried out in one step. However, a lot ofpoints have their error greater than 4 gray levels,thereby limiting the analysis of the fluctuations ofstrains. An average strain of 0.2417 0.0002 isobtained and a corresponding standard deviation of0.0377. The average strain is consistent with a di-rect observation of the apparent strain. The aver-age error is equal to 9.7 gray levels maximum value:71 gray levels. It follows that the displacements ofnumerous ZOIs are set to the interpolated value be-cause the estimate is not accurate enough. Conse-quently, the estimate of the average strain is likely tobe correct, but the fluctuations are not completelycaptured.

Because a sequence of 11 pictures is taken, an up-dating procedure can be followed. The ROI contains840 ZOIs so that 9240 analyses on scale no. 1 arecarried out in less than two min when no iterationsare used on each scale, and less than 5 min wheniterations are allowed. An average strain of0.2501 0.0004 is obtained and a correspondingstandard deviation of 0.0470. The average error isequal to 2.5 gray levels maximum value: 22 graylevels. This second set of results is more accuratethan the first one. This is important when consid-ering the fluctuations. Even though errors are cu-mulated when using the updating procedure, it ismore robust because of texture variations. Thechange of the average strain and corresponding stan-dard deviation for the whole sequence is shown inFig. 11. When the maximum applied strain de-creases, the average residual strain decreases andthe corresponding standard deviation increases.Apart from the initial stages of compression, the stan-dard deviation can be shown to increase in proportionto its mean value. More importantly, the volumeelements that display a large respectively small re-sidual strain as compared with the average, are thesame in the sequence of images, thus revealing thatweak or strong spots in the material remain weakor strong for all applied strains, and hence theirmechanical performance is presumably due to a givenmicrostructure.

When a subpixel procedure is used on scale no. 1,an average strain of 0.2505 0.0004 and a corre-sponding standard deviation of 0.0529 are measured.The average error is equal to 2.5 gray levels. Thereference and deformed meshes are shown in Fig. 11.Figure 12 compares the deformed meshes for the finalimage of the sequence. First, the result obtainedwith an analysis of one image: The fluctuations are

Fig. 10. Reference and deformed meshes of a stone-wool samplepredicted by using two strategies of the multiscale approach andcorresponding error contours: a no image update, b with twoimage updates. The black square depicts one ZOI p 5, 32pixels.


Fig. 11. a Average residual strain vs. standard deviation of a compressed glass-wool sample. b Reference and deformed meshes atthe end of the sequence p 5, 32 pixels.

Fig. 12. Comparison of reference and deformed meshes at the end of the sequence of a compressed glass-wool sample p 5, 32pixels. Four different options are used in the multiscale approach: a no image update, bwith image update and no iteration, cwithimage update and no limitation on , and d with image update and no test on the local texture fluctuations.


minimal as explained above Fig. 12a. Second,Fig. 12b shows the results when no iterations areallowed in an updating procedure. The averagestrain is equal to 0.2573 0.0004 and the averageerror is equal to 2.5 gray levels. In this particularexample, the iterations are not necessarily needed toget a reasonable estimate for the strain field. This isdue to the fact that the strain variation between twopictures does not exceed 0.05. The effect of discard-ing the test on the error to keep a displacement esti-mate is shown in Fig. 12c. The fluctuationsincrease because the standard deviation now equals0.0601 for an average strain of0.2572 0.0004 andan average error of 2.5 gray levels. In this case thefluctuations are higher, but still on the same order asa subpixel procedure. Last, Fig. 12d shows the ef-fect of considering analyses for which the standarddeviation of the ZOI can be very small. Few addi-tional artifacts can be observed and lead to an in-crease of the standard deviation i.e., 0.0607 for anaverage strain equal to 0.2557 0.0004 and anaverage error of 2.5 gray levels. Altogether, thesefour results are consistent with each other. Yet, ifthe strain fluctuations are sought, the two additionaltests introduced in Section 3.D are very useful.

6. Conclusions

An improved procedure is introduced to determinedisplacement fields from the comparison of two im-ages. It is based on a multiscale approach of a ho-mogeneous strain field in a tension or compressiontest. When a subpixel procedure is used, strains ofthe order of 0.0001 can be measured. The multiscaleapproach enables the user to measure strain levelsthat are more important than with a conventionaltechnique; the more the number of scales, the higherthe maximum measurable strain. Furthermore, thetexture variation is considered by checking the stan-dard deviation of the zones of interest and introduc-ing an error indicator, which is independent of thecorrelation procedure.

The method has been tested on two cases: A firstartificial example where an actual texture has beendeformed numerically, and a second based on realpictures of compressed mineral-wool samples. Thefirst case gives accurate estimates of the displace-ment field when compared with a known displace-ment. Strain levels as high as 0.35 or as low as0.26 can be measured when only two images areconsidered and no strain fluctuations occur. Forhigher strain levels, the updating procedure is veryrobust, even though the errors are cumulated. Inthis example, it is also shown that the iteration pro-cedure used on each scale is beneficial to the overallperformance of the algorithm.

The second example shows a first situation dealingwith stone-wool samples for which the images areobtained in reflection. In this case, fluctuations oc-cur and are measured with the multiscale approach.In particular, it is shown that the higher the appliedstrain, the higher the corresponding standard devia-tion. The updating strategy turned out to be the

most reliable because the region of interest was nottoo large the number of scales involved did not allowfor a direct measurement. The correlation tech-nique yields local information regarding the strainsthat can be compared to the local texture. The sec-ond situation, which is more complicated, corre-sponds to a glass-wool sample compressed atdifferent levels. Here, the images were obtained intransmission. Consequently, severe texture varia-tions occur. Again, the updating procedure was themost robust strategy. It was possible to correlatethe average permanent strain with the correspondingstandard deviation.

Last, it can be noted that the measurement of dis-placement and strain fields in experiments onmineral-wool samples are only possible, up to now,thanks to digital image correlation. When largestrains andor texture variations occur, which is usu-ally the case, robust techniques are needed. Other-wise, the evaluation of the strain fluctuations cannotbe estimated. The iterative multiscale strategy wassuccessful in the examples studied herein. It wasalso used to analyze multiaxial experiments on elas-tomers e.g., polyethylene terephthalate for whichstrains as high as 0.3 were measured between twoimages.33

The authors acknowledge useful discussions withY. Berthaud.

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