Optics and Lasers in Engineering 51 (2013) 716–734

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Optics and Lasers in Engineering

0143-81

http://d

n Corr

E-m

journal homepage: www.elsevier.com/locate/optlaseng

Using deconvolution to improve the metrological performanceof the grid method

Michel Grediac a,n, Frederic Sur b, Claudiu Badulescu c, Jean-Denis Mathias d

a Clermont Universite, Universite Blaise Pascal, Institut Pascal, UMR CNRS 6602, BP 10448, 63000 Clermont-Ferrand, Franceb Laboratoire Lorrain de Recherche en Informatique et ses Applications, UMR CNRS 7503, Universite de Lorraine, CNRS, INRIA projet Magrit, Campus Scientifique, BP 239, 54506

Vandoeuvre-l�es-Nancy Cedex, Francec Laboratoire Brestois de Mecanique et des Syst�emes, ENSTA Bretagne, 2 rue Franc-ois Verny, 29806 Brest Cedex 9, Franced IRSTEA, Laboratoire d’Ingenierie pour les Syst�emes Complexes, Campus universitaire des Cezeaux, 24 avenue des Landais, BP 50085, 63172 Aubi�ere Cedex, France

a r t i c l e i n f o

Article history:

Received 7 November 2012

Received in revised form

8 January 2013

Accepted 10 January 2013Available online 21 February 2013

Keywords:

Deconvolution

Grid method

Strain measurement

66/$ - see front matter & 2013 Elsevier Ltd. A

x.doi.org/10.1016/j.optlaseng.2013.01.009

esponding author. Tel.: þ33 4 73 28 80 77; fa

ail address: michel.grediac@univ-bpclermont

a b s t r a c t

The use of various deconvolution techniques to enhance strain maps obtained with the grid method is

addressed in this study. Since phase derivative maps obtained with the grid method can be

approximated by their actual counterparts convolved by the envelope of the kernel used to extract

phases and phase derivatives, non-blind restoration techniques can be used to perform deconvolution.

Six deconvolution techniques are presented and employed to restore a synthetic phase derivative map,

namely direct deconvolution, regularized deconvolution, the Richardson–Lucy algorithm and Wiener

filtering, the last two with two variants concerning their practical implementations. Obtained results

show that the noise that corrupts the grid images must be thoroughly taken into account to limit its

effect on the deconvolved strain maps. The difficulty here is that the noise on the grid image yields a

spatially correlated noise on the strain maps. In particular, numerical experiments on synthetic data

show that direct and regularized deconvolutions are unstable when noisy data are processed. The same

remark holds when Wiener filtering is employed without taking into account noise autocorrelation. On

the other hand, the Richardson–Lucy algorithm and Wiener filtering with noise autocorrelation provide

deconvolved maps where the impact of noise remains controlled within a certain limit. It is also

observed that the last technique outperforms the Richardson–Lucy algorithm. Two short examples of

actual strain fields restoration are finally shown. They deal with asphalt and shape memory alloy

specimens. The benefits and limitations of deconvolution are presented and discussed in these two

cases. The main conclusion is that strain maps are correctly deconvolved when the signal-to-noise ratio

is high and that actual noise in the actual strain maps must be more specifically characterized than in

the current study to address higher noise levels with Wiener filtering.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Full-field measurement techniques are now wide spread in theexperimental mechanics community. One of the reasons is theirability to measure heterogeneous states of strain, thus enabling todetect localized events that often occur in specimens under testbut whose location is not known a priori. Classic transducers suchas strain gauges only give local information, so they are not wellsuited to reach this goal despite a resolution generally recognizedto be better than that of full-field measurement techniques.Assessing the real metrological performances of full-field measure-ment techniques remains an open problem, as illustrated by theincreasing number of papers on this topic. They mainly deal with

ll rights reserved.

x: þ33 4 73 28 80 27.

.fr (M. Grediac).

digital image correlation for which both the surface marking [1–3]and the algorithm which are used [4,5] constitute an issue. One ofthe difficulties comes from the fact that several parametersinfluence these performances, some of them being not reallyintrinsic, but extrinsic such as lighting or image contrast forinstance [6]. Another reason is the fact that metrological perfor-mances depend, among others, on two conflicting concepts:resolution and spatial resolution. The first one is defined here bythe smallest strain or displacement that can be detected, thesecond one by the smallest distance between independent mea-surements. It is well known that the better the resolution, theworst the spatial resolution. In addition, displacement is generallythe physical quantity which is provided by most techniques butstrain is often the desired physical quantity for material character-ization purposes. The reason is that this quantity is directlyinvolved in constitutive equations of engineering materials. Sincedisplacement maps are generally noisy, they are smoothed using

M. Grediac et al. / Optics and Lasers in Engineering 51 (2013) 716–734 717

various strategies prior to the differentiation necessary to get thestrain components [7,8]. Hence the number of independent para-meters influencing the final result is in general significant andchanging one of the parameters has a direct impact on the obtainedstrain value. This does not help to give a clear idea on the actualmetrological performances of full-field measurement techniques ifstrain components are to be considered instead of displacements.Finally, it must be pointed out that finding a reference state ofstrain is also an issue. Specific testing devices or procedures havebeen proposed in the literature [9,10] but synthetic images areused in most cases as in [4] for digital image correlation or in [11]for techniques for which fringes must be processed.

This paper is devoted to a specific problem: the actual strainvalue determination in case of localized phenomena which occurin many situations, especially when heterogeneous materials aretested since the different phases may potentially exhibit variousmechanical responses. ‘‘Localized’’ means here that the dimen-sions of the phenomenon to be observed and for which actualstrain components are to be determined are similar to, or lowerthan the size of the region involved in the calculation of the straincomponents by local image processing, in other words that thestrain gradient is significant.

The measurement method used herein is the grid method, whichconsists first in depositing a regular bi-directional grid, in takingimages of this grid and finally processing these images to retrieve thein-plane strain components. The drawback of this technique is that aregular marking (more precisely a pseudo-regular since printing thegrids induces some defects) of the surface under investigation mustbe obtained by any means, by engraving, transferring or bonding forinstance. This surface preparation is in general more difficult thanconsidering a random marking. The advantage is however that aregular marking is much easier to duplicate or to control than arandom one. Moreover, a regular marking enables one to usetechniques based on the Fourier transform, for which a broadliterature is available, especially in image processing [12].

Processing grid images to retrieve strains consists first intaking pictures of the grid before and after the load is applied,and to quantify the changes in the images due to deformation.The ‘‘classic’’ procedure for processing these grids consists inextracting the phase using a windowed Fourier transform becausephase change is proportional to displacement [13]. The obtaineddisplacement field is then smoothed and differentiated using asuitable strategy to deduce the strain field [14,8,15] for instance.A consequence of this ‘‘classic’’ procedure is that several para-meters influence the result, and thus the final metrologicalperformance in terms of quality of the strain measurement. Onthe contrary, retrieving directly the phase and differentiating itdirectly without any smoothing, or retrieving the phase derivativedirectly from the grid images, as proposed in [16,17] for instance,reduces the number of parameters that govern the extractingprocedure to one since only the size of the window used in thewindowed Fourier transform used in this case drives the proce-dure. As a result, the link between the obtained strain field andthe grid images is straightforward. This link has been character-ized in [18] in the case of perfect grids. This enables one to tacklethe problem of actual strain measurement in case of strong straingradient with a suitable tool, namely deconvolution.

Deconvolution techniques are widely used in image processingbut it seems that they have only seldom been employed in thefield of mechanics [19,20]. Concerning displacement and strainmeasurement, one of the main reasons is certainly that the‘‘classic’’ procedure involves several non-linear steps and numer-ous parameters. Therefore, the retrieved mechanical quantities doprobably not appear as the convolution of any kernel with theactual quantities. Knowing the kernel is however crucial to per-form efficient deconvolution [21].

In this context, the aim of this paper is to show thatdeconvolution can be applied to enhance the ability of the gridmethod to detect and reliably quantify strain gradients. The paperis organized as follows. After a short recall on the grid method, thebasics of deconvolution are described along with the character-istics of various deconvolution algorithms. The performance ofthese algorithms is then briefly reviewed and a numericalexample is processed to study the limit and interest of deconvo-lution in the current context, especially when noise corrupts thegrid images that are processed. Two short examples of actualstrain map restoration picked in the recent literature are thenshown to illustrate the procedure.

2. Determining in-plane strain components from grid images

The objective here is to briefly recall the procedure used todetermine in-plane strain components from grid images. In suchimages, the light intensity sðx,yÞ at each point (x,y) is a quasi-periodic signal characterized by two phases Fx and Fy which aredefined along the x- and y-directions, respectively. The variationof these phases between two images, denoted as DFx and DFy,are related to the in-plane displacements ux and uy through thefollowing equations [13]:

ux ¼�p

2p� DFx

uy ¼�p

2p� DFy

8><>: ð2:1Þ

The in-plane strain components are therefore deduced fromthe phase derivatives variations:

Exx ¼�p

2p� D

@Fx

@x

Eyy ¼�p

2p�D

@Fy

@y

Exy ¼�p

2p� D@Fx

@yþD

@Fy

@x

� �

8>>>>>>><>>>>>>>:

ð2:2Þ

Considering a given grid picture, the phases can be determinedby calculating a windowed Fourier transform which writes asfollows:

Cðx,y,yÞ ¼Z þ1�1

Z þ1�1

sðx,ZÞgsðx�x,y�ZÞ

e�2ipf ðx cosðyÞþZ sinðyÞÞ dx dZ¼ Rðx,y,yÞþ iJðx,y,yÞ ð2:3Þ

where gs is a 2D window function of width s. It is symmetric,positive, and integrates to 1. In practice, this window can be atriangle, as suggested in [13], or a Gaussian, as in [17] to ensuredifferentiability. A Gaussian function is used in the current work.y can be either equal to 0 or p=2, depending on the directionunder consideration: x or y, respectively. The actual phase varia-tions DFxðx,yÞ and DFyðx,yÞ are approximated by calculating thefollowing quantities denoted D ~Fx ðx,yÞ and D ~Fy ðx,yÞ:

D ~Fx ðx,yÞ ¼D arctanJðx,y,0Þ

Rðx,y,0Þ

� �

D ~Fy ðx,yÞ ¼D arctanJ x,y,

p2

� �R x,y,

p2

� �0B@

1CA

8>>>>>><>>>>>>:

ð2:4Þ

2.1. The apparent phase and phase derivatives as the convolution of

their actual counterparts

An important result shown in [18] is the fact that each of theD ~Fi quantities, i¼ x,y, is nearly equal to the convolution of its

M. Grediac et al. / Optics and Lasers in Engineering 51 (2013) 716–734718

actual counterpart DFi, i¼ x,y, by the envelope gs used in thewindowed Fourier transform defined above. The same remarkholds for the phase derivatives. Basically, this is true becausedisplacements are small with respect to the pitch of the grid [18].Thus

D ~Fi CDFings, i¼ x,y ð2:5Þ

with y¼ 0 for i¼x and y¼ p=2 for i¼y. In the same way

D@ ~Fi

@xCD

@Fi

@xngs

D@ ~Fi

@yCD

@Fi

@yngs, i¼ x,y

8>>><>>>: ð2:6Þ

In terms of displacement and strain, plugging Eqs. (2.5) and(2.6) in Eqs. (2.1) and (2.2) leads to the displacement and straincomponents ~ui and ~Eij provided by the procedure, which can alsobe approximated by the convolution of their actual counterpartsui and Eij by gs. Thus

~ui Cuings ð2:7Þ

with i¼ x,y, and

~Eij CEijngs ð2:8Þ

with ij¼ xx,yy,xy.This property has two main consequences:

1.

Any sudden variation of the actual displacement (resp. strain)distribution is smoothed by the phase (resp. phase derivative)extraction procedure. Hence resulting maps are automaticallyblurred in zones featuring high gradients. Consequently, thequantities provided by the grid image processing are lowerthan the actual ones over these zones.

2.

Since such maps can be considered as maps of the actualquantity convolved by gs, one can restore them and find theactual quantity by deconvolution.

Remark 1 above mainly concerns strain components in prac-tice, displacement gradients being very small within the frame-work of small deformations. Consequently, Remark 2 also mainlyconcerns strain components in practice. It must however bepointed out that it is only valid if no filtering (with polynomialsfor instance) is performed prior to strain calculation to removenoise. Indeed, the classic route for calculating strain components([22] for instance) consists in:

1.

extracting the phase and deducing the displacement field; 2. smoothing the displacement field with a suitable procedure to

reduce noise; and

3. differentiating the smoothed displacement field to deduce the

strain field.

In this case, the number of parameters that govern theprocedure to go from the grid images to the strain map increases.Moreover these parameters generally spatially change, so it is notpossible to perform deconvolution. The idea here is to reduce thisnumber as much as possible to be able to perform deconvolutionand thus to retrieve the actual strain in case of strong straingradients. For instance, Steps 1 and 2 above can be merged merelyby adjusting the size of the Gaussian envelope used in thewindowed Fourier transform. This is done by changing the valueof s. Even the three steps above can be merged by using explicitderivation, as in [16,17] but the phase obtained at the end of Step2 can also be numerically differentiated. In conclusion, only onemain parameter may potentially govern the smoothness of the

strain field obtained at the end of the procedure: s. The larger thevalue of s, the higher the smoothness of the strain field but theworst the spatial resolution in strain (defined here by the shortestdistance between two points where independent strain measure-ments are obtained) and the stronger the weakening of the strainvalue in case of high strain gradients.

2.2. Synthetic image

A numerical example is discussed here to illustrate the factthat the difference between the phase derivative provided by thewindowed Fourier transform and the actual one increases whenthe phase derivative gradient increases. The same example is thenused throughout the paper to discuss the performance of variousdeconvolution techniques.

It is assumed that the phase derivative ð@FyÞ=ð@yÞ is a sinewave along the y-direction of the image, with a period slightlylinearly changing along the x-axis. The idea under this assump-tion is to provide an image containing a rich (within certainlimits) and regular frequency signature, but with a frequencynearly constant over the Gaussian kernel. The other phase Fx isassumed to be null. In practice, the period pwave of this sine waveis modelled by the following equation:

pwave ¼ pminiwaveþ

pmaxiwave�pmini

wave

Lx� x ð2:9Þ

where pminiwave is the minimum value of the sine period and pmaxi

wave itsmaximum value. Lx is the width of the image. Its height is denotedLy in the following. The phase amplitude is chosen in such a waythat the amplitude of its derivative denoted AF0 is constant. Thus

Fx ¼ 0

Fy ¼ AF0pwave

2psin

2ppwave

y�Ly

2

� �� �8><>: ð2:10Þ

and

@Fx

@x¼@Fx

@y¼ 0

@Fy

@x¼ 0,

@Fy

@y¼ AF0 cos

2ppwave

y�Ly

2

� �� �8>>><>>>: ð2:11Þ

The interest of this phase derivative distribution is to see at aglance the response of various phase derivative calculation andrestoration strategies. It is symmetric with respect to the hor-izontal midline. This feature will turn out to be useful in thefollowing since the cross-section of the phase derivative mapalong this symmetry axis will be plotted to assess the perfor-mance of various deconvolution techniques. pmini

wave and pmaxiwave are

chosen in such a way that the sine wave period gently decreasesfrom the left to the right, so the period variation over the kernelremains negligible.

The phase and the derivative maps are shown in Fig. 1, whereAF0 ¼ 1E�03 (which represents a typical value when strains aremeasured with this method), Lx¼4000 pixels, Ly¼1000 pixels,pmini

wave¼10 pixels and pmaxiwave¼80 pixels. The scale along the x- and

y-axes is different to avoid too high an aspect ratio of theresulting image.

This phase is then used to build up a synthetic grid image forwhich the light intensity can be written as follows:

sðx,yÞ ¼A

2ð2þsin3

ð2pfxÞþsin3ð2pfyþFyðx,yÞÞÞþnðx,yÞ ð2:12Þ

where nðx,yÞ is a Gaussian white noise of standard deviation N.Note that the signal is not encoded with a sine to illustrate that apure sine is not rigorously necessary to correctly retrieve thephase derivative since only the first harmonic contains the soughtinformation [13,18]. The simulated grid image (not shown here) is

x

y

500 1000 1500 2000 2500 3000 3500 4000

200

400

600

800

1000−0.015

−0.01

−0.005

0

0.005

0.01

0.015

x

y

500 1000 1500 2000 2500 3000 3500 4000

200

400

600

800

1000−1

−0.5

0

0.5

1x 10−3

Fig. 1. Reference phase derivative and phase maps. (a) Phase Fy . (b) Phase derivative @Fy=@y.

M. Grediac et al. / Optics and Lasers in Engineering 51 (2013) 716–734 719

built with Aðx,yÞ ¼ A¼ 212, 5 pixels/grid period and quantizationof the grey level to 12 bits. Such values are typical in realexperiments carried out with a CCD camera. Note that thequantization step makes it impossible to perfectly retrieve theactual phase or phase derivative from the grid image.

2.3. Convolution of the phase derivative

The objective here is to illustrate and to quantify the fact thatthe phase derivative which is retrieved with the windowedFourier transform is different from the actual one because ofconvolution. Parameter s, which drives the window size, is equalto s¼ 5 pixels in these simulations, which is a typical value usedin practice [17]. A top view of the envelope gs is superimposed onthe top left-hand side of the map shown in Fig. 1a and b in orderto more clearly compare the size of this envelope and the periodof the waves. Since the scale along the x- and y-axes is different,the circle turns into an ellipse in each map. The actual diameter ofthe circle is equal to 6� s¼ 30 pixels, 73s being the usual widthconsidered for a Gaussian envelope. It can be seen to the nakedeye that the envelope covers several wave periods on the left-hand side (up to 3), thus correctly assessing the phase derivativeis difficult.

The estimated phase derivative is shown in Fig. 2a. As may beeasily seen since the amplitude of the reference phase derivativeis constant, the estimated phase derivative is all the moreimpaired by the grid image processing that the frequency of thereference phase increases (see the left-hand side of the figure).The cross-section of the map along its symmetry axis (thehorizontal midline defined by y¼500) is plotted in Fig. 2b. Thephase derivative should be constant along this mid-line (as thereference value) but it is clear that the estimated value is not,

especially for high frequencies of the sine wave (left-hand side ofthe curve). The difference between estimated and reference phasederivative is finally shown in Fig. 2c to enable the reader to clearlyassess this phenomenon. This simulation clearly illustrates thefact that deconvolution is necessary to try to restore as well aspossible the phase derivatives to get reliable information fromstrain maps where significant strain gradients occur.

The phase derivative is all the more sensitive to this phenom-enon that the width of the kernel envelope used for the wind-owed Fourier transform is significant compared to the period ofthe phase. This feature is illustrated in Fig. 3a, where the cross-section of the phase derivative map along its symmetry axis(y¼500) is plotted for various values of s. The higher the value ofs, the wider the Gaussian envelope and the worst the spatialresolution. In addition, the estimated phase derivative becomesmore and more impaired by the procedure when s increases. Thetrade-off between resolution and spatial resolution is also illu-strated in Fig. 3b, where a noise has been added to the grid imagesused for plotting Fig. 3a. It is clear that the noise level observed onthe phase derivative decreases when s increases, but the resultbecomes more and more impaired by the procedure at thesame time.

Since the period of the phase derivative waves linearlydecreases from the left to the right, the same results as abovecan be plotted with respect to a normalized period defined by theratio x between the period of the phase derivative waves and s.The vertical lines correspond to boundary effects that can alreadybe visible along the very left- and right-hand sides of Fig. 3a andb. They are shifted toward the center of Fig. 3c because of thechange in variable. They enable us to more clearly distinguish thedomain over which they are defined. All the curves are perfectlysuperimposed but the number of normalized periods decreases

x

y

500 1000 1500 2000 2500 3000 3500 4000

100200300400500600700800900

1000 −1

−0.5

0

0.5

1x 10−3

2 4 6 8 10 12 14 160

0.2

0.4

0.6

0.8

1

x 10−3

reference phase derivativeestimated phase derivative

x

y

500 1000 1500 2000 2500 3000 3500 4000

100200300400500600700800900

1000 −1

−0.5

0

0.5

1x 10−3

Normalized period (period/σ)

Fig. 2. Estimated phase derivative. (a) Estimated phase derivative @Fy=@y. (b) Cross-section of the estimated phase derivative map along the x-axis. (c) Difference between

estimated and reference phase derivatives.

M. Grediac et al. / Optics and Lasers in Engineering 51 (2013) 716–734720

when s increases, and therefore when noise decreases, thusillustrating that the size of the observed field is a third parameterwhich influences the quality of the result in addition to the spatialresolution and the resolution. The spatial resolution being definedhere by the ‘‘width’’ of the Gaussian envelope 6s, it can be seenthat a loss of information slightly greater than 40% is obtainedwhen the period of the waves is equal to this spatial resolutionðx¼ 6Þ and nearly lower than 20% when this period is equal totwice this spatial resolution ðx¼ 12Þ. This highlights that aprocedure able to restore the phase derivative would be veryuseful to get the actual value instead of quantities which aresystematically impaired by the procedure employed toretrieve them.

The effect of convolution on the phase derivative is finallyillustrated by a portion of the cross-section of the phase

derivative map along the y-axis for x¼1500. The resultobtained for various values of s is shown in Fig. 4. As maybe seen, the difference between estimated and actual phasederivatives increases when s increases. Even for small valuesof s, this difference is significant (about 20% of the referencevalue).

The objective is now to take advantage of the fact that@ ~Fy=@y can be considered as a convolution of @Fy=@y by gs toretrieve @Fy=@y from @ ~Fy=@y by deconvolution. Various popu-lar deconvolution techniques are briefly described in thefollowing section. They are then applied to the examplepresented above and the obtained results are discussed. Someexamples of strain maps picked in the recent literature servefinally as examples to briefly illustrate the interest but also thelimits of deconvolution.

0 500 1000 1500 2000 2500 3000 3500 40000

0.2

0.4

0.6

0.8

1

x 10−3

X

phas

e de

rivat

ive

σ = 4σ = 5σ = 6σ = 7σ = 8σ = 9reference

0 500 1000 1500 2000 2500 3000 3500 40000

0.2

0.4

0.6

0.8

1

x 10−3

X

phas

e de

rivat

ive

2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

x 10−3

normalized period

phas

e de

rivat

ive

σ = 4σ = 5σ = 6σ = 7σ = 8σ = 9reference

σ = 4σ = 5σ = 6σ = 7σ = 8σ = 9reference

Fig. 3. Influence of s on the estimated phase derivative. (a) Without noise N¼0. (b) With noise N¼1. (c) Phase derivative vs. normalized period, without noise.

M. Grediac et al. / Optics and Lasers in Engineering 51 (2013) 716–734 721

3. Phase and phase derivative restoration by deconvolution

3.1. Introduction

The grid image is actually impaired by digital noise. Under themild assumption that the image noise is a Gaussian white noise, itis shown in [18] that it yields a structured noise on the phase orits derivatives. In this section, the restoration problem is definedas follows:

u¼ u0ngsþn ð3:1Þ

where u can be either the phase or one of its derivatives retrievedby the windowed Fourier transform, u0 is the ideal phase (resp.one of its phase derivatives) and n is the noise which corrupts thequantity to be deconvolved. n is a spatially correlated Gaussian

noise, which has been characterized for the phases and the phasederivatives in [18]. In this section, knowing u and with someinformation about noise n, the aim of restoration is to estimate u0.

A large amount of literature is dedicated to deconvolution,either blind or non-blind. Let us name just the survey article [23]in the specific, yet representative field of astronomy. In thecurrent case however, we are within the scope of non-blinddeconvolution since the point spread function (PSF) which blursthe maps is well known a priori: this is merely the windowenvelope defined by gs. Even in this case which is easier to tacklethan the case of blind deconvolution, it must be pointed out thatnon-blind deconvolution is an ill-posed inverse problem which isprone to noise amplification and artifacts such as the so-calledringing artifact [12]. Contrary to most of the studies in the field(although a recent survey paper [24] discusses the more limited

0 50 100 150 200 250 300 350 400−1

−0.5

0

0.5

1x 10−3

Y

phas

e de

rivat

ive

σ = 4σ = 5σ = 6σ = 7σ = 8σ = 9reference

Fig. 4. Cross-section of the phase derivative map at x¼1500.

M. Grediac et al. / Optics and Lasers in Engineering 51 (2013) 716–734722

denoising problem), the noise is not white in the problem ofinterest [18].

Four non-blind deconvolution techniques are briefly recalledhere: direct deconvolution, regularized (Tikhonov) deconvolution[25], Richardson–Lucy deconvolution [26,27] and Wiener filtering[28]. All these techniques have been used in turn to deconvolvethe synthetic image presented and discussed above, the objectivebeing to assess the efficiency of these techniques and theirrobustness even though the hypotheses under which some ofthem are developed are not fully satisfied.

3.2. Brief presentation of the deconvolution techniques used

In this section, bf denotes the Fourier transform of any function f.All the deconvolution methods described below have been tested inthis study using standard Matlab functions for deconvolution fromthe Image Processing Toolbox [29], namely deconvreg, deconvwnrand deconvlucy. The arguments used in each case are briefly givenfor each deconvolution method.

3.2.1. Direct deconvolution

Neglecting the noise, u¼ u0ngs yields with Fourier transform:

cu0 ðx,yÞ ¼buðx,yÞcgs ðx,yÞ

ð3:2Þ

Image u0 is then retrieved by inverse Fourier transform. Thismethod is called direct deconvolution. Since gs is here a smoothfunction, cgs quickly vanishes for high frequencies. Dividing by cgstherefore accentuates high frequencies and possibly produces theringing artifact along localized, ‘‘edge-like’’ features of the phaseor its derivatives.

In the presence of noise, direct deconvolution yieldsbu=cgsþbn=cgs . This still amplifies high frequency noise artifacts inthe retrieved image, which is even often overwhelmed by thenoise. When no noise occurs, the Wiener filtering described belowis equivalent to direct deconvolution. This latter technique wasconsequently implemented here using the Matlab functiondeconvwnr devoted to Wiener filtering, with the noise-to-signalratio set to 0.

3.2.2. Regularized deconvolution

A popular way to impose smoothness in deconvolution is tolimit noise amplification by using Tikhonov regularization [25]. Inthis framework, the deconvolved image u0 minimizes the follow-ing quantity:

F ðuÞ ¼ Ju�u0ngsJ22þlJf ðu0ÞJ

22 ð3:3Þ

where Jf ðu0ÞJ2 is a measure of ‘‘roughness’’ of u0 and l controlsthe degree of roughness. In this paper, the popular Laplacianfunction f ðu0Þ ¼Du0 is used for this purpose.

Note that if l¼ 0, since Ju�gsnu0J22 ¼ Jbu�cu0cgsJ2

2 (by Parsevaltheorem), minimizing F ðuÞ yields bu ¼cu0 �cgs . In this case, thesolution is therefore the same as in direct deconvolution. TheMatlab function deconvreg with standard arguments was usedhere. In particular l was set automatically.

3.2.3. Richardson–Lucy deconvolution

Richardson–Lucy deconvolution [26,27] is an iterative proce-dure defined by

uiþ1ðx,yÞ ¼ uiðx,yÞ �uðx,yÞ

uingsðx,yÞngs ð3:4Þ

where gs ðx,yÞ ¼ gsð�x,�yÞ. Under Poisson noise assumption, ui

converges to a maximum likelihood estimate of u0. It is importantthat uZ0 to keep consistency with Poisson noise hypothesis.

In the current study case however, u is not non-negative sincewe deal with phase and phase derivatives. Two possible work-around are tested in this study:

1.

The Richardson–Lucy deconvolution is used on expðuÞ and thelogarithm is then taken. This method is denoted as LR 1.

2.

The Richardson–Lucy deconvolution is used on u�minðuÞ andthe result is normalized so that its mean is equal to the meanof the corrupted image u. This method is denoted as LR 2.

Note however that in both methods, the noise on the non-negative image gray-values do not fit a Poisson distribution. This stillproves to give satisfactory empirical results. Both versions wereimplemented with the Matlab function deconvlucy using standardarguments. The most critical one is the number of iterations whoseinfluence is discussed in Section 3.5 below.

3.2.4. Wiener filtering

Wiener filtering [28] amounts to retrieve u0 via

cu0 ðx,yÞ ¼cgs nðx,yÞ

9cgs ðx,yÞ92þMðx,yÞ=Sðx,yÞ

buðx,yÞ ð3:5Þ

where zn is the conjugate of the complex number z. E denotes theexpectation. Mðx,yÞ ¼ Eð9bnðx,yÞ92

Þ is the mean power spectraldensity of the random noise n and Sðx,yÞ ¼ 9cu0 ðx,yÞ92

is the powerspectral density of the (deterministic) ideal image u0. Note that inthe noise-free case, Wiener filtering boils down to direct

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100200300400500600700800900

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Fig. 5. Difference between deconvolved and reference phase derivatives obtained with various deconvolution strategies. Noise amplitude¼0. (a) Direct deconvolution.

(b) Regularized deconvolution. (c) Richardson–Lucy, logarithmic (RL 1). (d) Richardson–Lucy, linear (RL 2).

initial direct regul. RL 1 RL 20

5

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NR

0 500 1000 1500 2000 2500 3000 3500 4000

0

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15x 10−4

x

phas

e de

rivat

ive

referenceestimateddirectregularizedRL 1RL 2

Fig. 6. Comparison of the results obtained with various deconvolution techniques. No noise. (a) PSNR obtained with various deconvolution techniques. (b) Cross-section of

the error maps along the horizontal symmetry axis.

M. Grediac et al. / Optics and Lasers in Engineering 51 (2013) 716–734 723

M. Grediac et al. / Optics and Lasers in Engineering 51 (2013) 716–734724

deconvolution. Implementing Wiener filter necessitates to esti-mate both Mðx,yÞ and Sðx,yÞ. Two approaches have been investi-gated here:

1.

Fig(b)

aut

Assuming that n is a white noise, then Mðx,yÞ ¼VarðnÞ. In [18],it is demonstrated that an accurate estimate of the variance ofthe noise on the phase maps fx and fy is

VarðnÞ ¼N2DxDy

8ps2K2ð3:6Þ

where N2 is the variance of the Gaussian white noise impair-ing the grid image. K ¼ 9d19A=2, where d1 is the first Fouriercoefficient of the sin3 function, and A is as in Eq. (2.12). DxDy

is the area of one pixel. It is equal to one here since all thedimensions are given in pixels, so it vanishes in this expres-sion.It is also shown that the variance of the noise on the phasederivative maps in the x- and y-directions is

VarðnÞ ¼N2DxDy

16ps4K2ð3:7Þ

Besides, Sðx,yÞ is approximated by 9buðx,yÞ92. This method is

denoted as hereafter WF 1.

x

y

500 1000 1500 2000 2500 3000 3500 4000

x500 1000 1500 2000 2500 3000 3500 4000

x500 1000 1500 2000 2500 3000 3500 4000

100200300400500600700800900

1000 −1

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y

100200300400500600700800900

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1000 −1

−0.5

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0.5

1x 10−3

. 7. Difference between deconvolved and reference phase derivatives obtained wit

Regularized deconvolution. (c) Richardson–Lucy, logarithmic (RL 1). (d) Richardso

ocorrelation (W 2).

2.

The autocorrelation function r of the noise on u is assumed tobe given a priori, using the results presented in [18]. The well-known Wiener–Khinchin theorem yields Mðx,yÞ ¼ brðx,yÞ. Con-cerning S, it is still approximated by 9bu92

. This method isdenoted as WF 2, hereafter.

Both Wiener filter variants were implemented with the Matlabfunction deconvwnr, each time with arguments suited to thecorresponding variant.

3.3. Deconvolution of the phase derivative of the synthetic example.

Comparison between various techniques

The objective here is to apply deconvolution to restore thephase derivative map discussed in Section 2.2 using themethods presented in Section 3.2. The efficiency of thesemethods is assessed by calculating the difference betweendeconvolved and reference phase derivatives, first assumingthat no noise corrupts the grid images, and then with anadditional noise to observe robustness. These so-called errormaps, obtained by subtracting the phase derivative maps andthe reference map, are used to assess the obtained result. s¼ 5pixels is chosen in all these simulations to keep the spatialresolution constant. The Richardson–Lucy technique being

x500 1000 1500 2000 2500 3000 3500 4000

x500 1000 1500 2000 2500 3000 3500 4000

x500 1000 1500 2000 2500 3000 3500 4000

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h various deconvolution strategies. Noise amplitude¼3. (a) Direct deconvolution.

n–Lucy, linear (RL 2). (e) Wiener without autocorrelation (W 1). (f) Wiener with

x

y

500 1000 1500 2000 2500 3000 3500 4000x

500 1000 1500 2000 2500 3000 3500 4000

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Fig. 8. Difference between deconvolved and reference phase derivatives obtained with various deconvolution strategies. Noise amplitude¼5. (a) Direct deconvolution.

(b) Regularized deconvolution. (c) Richardson–Lucy, logarithmic (RL 1). (d) Richardson–Lucy, linear (RL 2). (e) Wiener without autocorrelation (W 1). (f) Wiener with

autocorrelation (W 2).

initial direct regul. RL 1 RL 2 W 1 W 20

5

10

15

20

25

30

35

40

N = 0N = 3N = 5

Fig. 9. PSNR for various deconvolution techniques and noise levels.

M. Grediac et al. / Optics and Lasers in Engineering 51 (2013) 716–734 725

iterative, nit ¼ 10 iterations are taken in all cases for these firstattempts but the influence of this parameter is discussed inmore details in Section 3.5 below.

Results obtained without noise added to the grid image aregathered in Fig. 5. The Wiener filter boiling down to directdeconvolution when no noise is present, it is not employed here.

sig = 4 sig = 5 sig = 6 sig = 7 sig = 8 sig = 9 sig = 10 sig = 11 sig = 125

10

15

20

25

30

35

40

PS

NR

W 2, N = 0W 2, N = 1W 2, N = 2W 2, N = 3W 2, N = 4W 2, N = 5initial, N = 0

Fig. 10. PSNR for W 2, various values of s and noise levels.

init 1 5 10 15 20 30 40 50 75 100 150 200 300 500

sigma = 4

sigma = 5

sigma = 6

sigma = 7

sigma = 8

sigma = 9

sigma = 10

sigma = 11

sigma = 12

sigma = 1310

15

20

25

30

35

40

init 1 5 10 15 20 30 40 50 75 100 150 200 300 500

sigma = 4sigma = 5sigma = 6sigma = 7sigma = 8sigma = 9

sigma = 10sigma = 11sigma = 12sigma = 13

0

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10

15

20

25

30

35

40

init 1 5 10 15 20 30 40 50 75 100 150 200 300 500

sigma = 4

sigma = 5

sigma = 6

sigma = 7

sigma = 8

sigma = 9

sigma = 10

sigma = 11

sigma = 12

sigma = 130

5

10

15

20

25

30

35

40

Fig. 11. PSNR for RL 1, various values of s and nit. (a) N¼0. (b) N¼3. (c) N¼5.

M. Grediac et al. / Optics and Lasers in Engineering 51 (2013) 716–734726

x

y

200 400 600 800 1000 1200 1400 1600 1800 2000

100

200

300

400

500

600 −1

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y

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x

y

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100

200

300

400

500

600 −5

0

5x 10−4

Fig. 12. Error maps obtained with RL 1, noise amplitude N¼3, s¼ 8 pixels. (a) Difference between estimated and reference phase derivative maps. PSNR¼9.68 dB.

(b) Difference between deconvolved and reference phase derivative maps. RL 1, nit ¼ 10 iterations. PSNR¼27.19 dB. (c) Difference between deconvolved and reference

phase derivative maps. RL 1, nit ¼ 500 iterations. PSNR¼17.37 dB.

M. Grediac et al. / Optics and Lasers in Engineering 51 (2013) 716–734 727

Comparing the four error maps with that shown in Fig. 2c showsthat a wide portion of the phase derivative map is restored. Somesignificant edge effects are clearly visible in the direct case and, toa lesser extend, in the regularized deconvolution case. Boundaryeffects along the vertical border occur only in the first case.Remarkably, the regularized deconvolution provides a map ofnearly null error (to the naked eye) over a wider portion of theimage compared to the other techniques. Hence even smalldetails that take place on the left-hand side of the phasederivative map are restored in this case. For the three othertechniques, a zone roughly defined by xo500 is not correctlyrestored. Beyond this threshold value, the error is nearly null in all

cases. The fringe pattern is however slightly recognizable in theRL 2 error map.

The performance of the different methods can be compared bycalculating the peak signal to noise ratio (PSNR) over a certain zonedefined by the white rectangle shown in Fig. 5d. The PSNR value iscalculated over this rectangle throughout this paper. This quantity isequal to PSNR¼ 20 log10ðD=RMSEÞ, where RMSE represents the rootmean square error calculated over this zone and D is the dynamics ofthe grid image. The zone is defined away from the edge of the map toavoid boundary effects. The PSNR value reflects here the quality ofdeconvolution in a zone where the error is stabilized with respect tothe effect of the sine wave frequency on the deconvolved map. The

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−1

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0.5

1

Fig. 13. Phase derivatives: comparison between estimated and deconvolved phase derivatives. (a) Estimated phase derivatives, no noise, s¼ 5 pixels. (b) Estimated phase

derivatives, N¼5, s¼ 5 pixels. (c) Deconvolved phase derivatives, N¼5, s¼ 5 pixels, RL 1 no noise. (d) Deconvolved phase derivatives, W 2, N¼5, s¼ 5 pixels.

(e) Deconvolved phase derivatives, RL 1, N¼5, s¼ 8 pixels. (f) Deconvolved phase derivatives, W 2, N¼5, s¼ 8 pixels.

M. Grediac et al. / Optics and Lasers in Engineering 51 (2013) 716–734728

PSNR values obtained with the four deconvolution techniques arerepresented in Fig. 6a along with their counterpart calculated withthe image before deconvolution (see Fig. 2c) for comparison purposes(first bar on the left-hand side of the figure). As may be seen, alltechniques lead to an improvement of the PSNR value, but the bestresults are obtained with RL 1.

The influence of the frequency of the phase derivative on thequality of the results can be assessed by examining the cross-sectionof the previous figures along the horizontal symmetry axis, wherethe reference phase derivative is constant whatever the sine wavefrequency. The obtained result is shown in Fig. 6b, where the resultgiven by all the methods are superimposed. The boundary effectsalong the vertical border are clearly visible with direct deconvolu-tion. It is also worthy noting that this technique provides a noisydistribution away from the borders. This is consistent with theconclusion drawn from the PSNR maps. On the left-hand side, the RL1 and RL 2 curves sharply increase at a fairly linear rate which isnearly the same for both variants. In addition, RL 1 then significantlyoverestimates the phase derivative (between x¼500 and x¼1200).In all cases however it is worth mentioning that the actual value isreached before x¼500. At that point, it can be checked with Eq. (2.9)that the period of the sine wave pwave is slightly lower than 2s,which is equal to one-third of the spatial resolution since the latter

is assumed to be equal to 6s. This result is very interesting sinceconvolution induces a bias of about 20% in the phase derivative for asine wave period equal to twice the spatial resolution, as discussedin Section 2.3 above.

The robustness of these different deconvolution strategiesto the noise in the grid image is a key-issue. It has beenexamined by processing in turn the grid image corrupted byvarious noise levels. Figs. 7 and 8 show a set of maps obtainedwith the same procedures as in Fig. 5. The only difference isthe fact that a noise (zero mean, amplitude N¼3 and N¼5 forFigs. 7 and 8, respectively) has been added to the grid imagemodelled by Eq. (2.12). It clearly appears that the strategiesdescribed above exhibit a different response when the gridimage is corrupted by noise. Figs. 7a, b, e, and 8a, b and eclearly show that the direct, regularized and W 1 deconvolu-tion strategies are not robust whereas the other ones (RL 1, RL2 and W 2) still provide acceptable results. The last one (W 2)gives the less noisy phase derivative map, but details featuringa high frequency are not so well restored compared to the twotechniques based on the Richardson–Lucy algorithm.

The performance of the techniques in each case can also beassessed with the PSNR value. This quantity is represented inFig. 9. W 1 and W 2 appear only in case of noise, as justified above.

x

y

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0

100

200

300

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600

700 −0.015

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−0.005

0

0.005

0.01

0.015

Fig. 14. Asphalt specimen under compression [30]. (a) Front face of the specimen.

(b) exx strain map measured with s¼ 5 pixels.

M. Grediac et al. / Optics and Lasers in Engineering 51 (2013) 716–734 729

The obtained results confirm the trends observed to the naked eyein Figs. 7 and 8:

�

The PSNR value decreases when the noise amplitude increases.This is quite logical. � RL 1 provides the best results when no noise occurs. � W 2 outperforms the other techniques when noise corrupts

the grid images.

The latter result is obtained in a case for which noise has beencorrectly characterized to feed the filter. Indeed WL 1, which alsorelies on the Wiener filter, provides poor results because noise isnot correctly characterized in this case (autocorrelation is nottaken into account). This means that, in practice, the Wiener filtershould be used with caution if noise is not correctly assessed. RL1 or RL 2 should be used instead in this case since they still giveacceptable results.

3.4. Optimal choice of s for W 2

The previous results are obtained with a given value of s:s¼ 5 pixels, but one can easily guess that this parameter alsoinfluences the results. Indeed, increasing s reduces noise leveland therefore facilitates deconvolution. However this also reducesthe amplitude of the convolved phase derivative level and it is nowonder that retrieving the actual one becomes more difficult ifthe value of s becomes too large. In addition, this also contributessome blobs to appear in the phase derivative map because thelarger the s, the larger the range of noise correlation, as discussedin [18]. These ‘‘blobs’’ may then be mistaken for fictitious localevents that are then restored by deconvolution and this shouldobviously be avoided as far as possible. As a conclusion, choosingat best the value for s is a trade-off between various constraints.

This issue has been investigated here by examining the influenceof N and s the PSNR value.

Fig. 10 shows the PSNR value for various noise levels andvalues of s. These curves are plotted for W 2 and for the initialphase derivative maps. Six noise levels are considered in eachcase. As may be seen, the curves obtained with W 2 regularlydecrease when the noise level increases, as expected. Moreinterestingly, there is an optimal value lying between s¼ 7pixels and s¼ 9 pixels, depending on the noise level. Con-sidering that s¼ 8 pixels is optimal whatever the noise seemsto be a sound choice for the following calculations.

The curves obtained without deconvolution decrease whens increases. This is quite logical since it means that convolu-tion smoothes the resulting phase derivative map, thusincreasing the error made when the size of the kernelincreases because details are obtained with a lower quality.It is also worth mentioning that the effect of noise is hiddenbehind this phenomenon because the error due to smoothingis much greater than that due to noise. Finally, it is also clearthat the PSNR value is greater for W 2 than for the initial imagewhatever the noise level, so it means that deconvolution withW 2 remains stable in this case.

3.5. Optimal choice of the number of iterations for RL 1

The Richardson–Lucy algorithm being iterative, the numberof iterations nit must be chosen. For the same reason as aboves must also be chosen. Since this quantity directly influencesthe noise level in the phase derivative map, it must beadjusted to limit this effect whilst limiting blobs due tocorrelated noise, as explained above. This issue has beeninvestigated by calculating the PNSR value for various valuesof s and nit, the noise amplitude N being given. Three caseshave been investigated herein: N¼0, 3, 5. The obtained resultsare shown in Fig. 11. Comparing the PSNR values in the threecases confirms that noise significantly impairs the results, butit is worth mentioning that the maximum PSNR value for thehighest noise level (nearly 23.92 dB for s¼ 10 pixels, nit ¼ 20)remains much higher than that obtained with the raw phasederivative map (7.34 dB, see first column in Fig. 11c, s¼ 10pixels). This shows that deconvolution still dramaticallyimproves the phase derivative map despite noise. Anothershrinking fact is that the optimal choice for the s and nit

parameters corresponds to a series of points located along theridge of a 3D surface. This ridge slightly moves downwards leftas the noise level increases. If the noise level remainsunknown in practice, it is possible to choose a ‘‘reasonablevalue’’ for s from these figures: s¼ 8 pixels, and then launchthe deconvolution programme with nit ¼ 10.

The fact that a maximum value for PSNR exists for a givenvalue of s means that beyond this maximum, the procedurerestores some details in the phase derivative map which aresequels of the noise correlated by image processing and notzones of actual phase derivative concentration. This feature isillustrated in Fig. 12, where the error map over the whiterectangle shown in Fig. 5d is plotted. The benefit of deconvo-lution is clearly visible since the residual fringes (differencebetween estimated and reference phase derivative) have dis-appeared, details in the phase derivative map being nowrestored, but noise amplitude also increases when nit

increases. This is quantified by the PSNR value (reported inthe caption of each sub-figure) which is greater for nit ¼ 500iterations than that obtained before deconvolution: details arerestored but noise also increases for large values of nit.

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Fig. 15. Asphalt specimen under compression. Deconvolution of Image 1 for load level 1. (a) Enlargement of Zone A (initial, s¼ 8 pixels), Image 1. (b) Enlargement of Zone

A (W2, s¼ 8 pixels), Image 1. (c) Image 1 deconvolved with W 2, s¼ 8 pixels. (d) Image 1 deconvolved with RL 1, s¼ 8 pixels. (e) Image 1 deconvolved with W 2, s¼ 8

pixels. (f) Image 1 deconvolved with RL 1, s¼ 5 pixels.

M. Grediac et al. / Optics and Lasers in Engineering 51 (2013) 716–734730

3.6. Conclusion

Some examples of restored phase derivative maps are shownin Fig. 13 to conclude this section. The estimated phase derivativemap is shown in Fig. 13a. One can see that the informationprogressively vanishes when moving from the right to the left,which illustrates the negative impact of convolution when thefrequency of the details increases. The effect of noise on the gridimage can be assessed in Fig. 13b. The two figures above servethen as input images for the deconvolution techniques. ApplyingRL 1 in the noiseless case leads to the map in Fig. 13c. Comparedto the map located just above, it can be seen that a wide portionof the image is restored. A thin band located at the very leftremains unaccessible even after deconvolution. The influence ofnoise can be assessed in Fig. 13d when W 2 (the best technique inthis case after the results obtained above) has been employed, butwith s¼ 5 pixels which is not the optimized value. Again, detailsare restored but noise clearly corrupts the resulting map. Thisresult is improved by increasing s, as suggested by the conclusionof the simulations in Sections 3.4 and 3.5 above. RL 1 provides lessnoisy results than W 2, but it can be checked that the error is

greater for RL 1. In addition, the zone which remains unrestored isgreater for RL 1 than for W 2.

4. Examples

4.1. Introduction

The objective here is to briefly illustrate the effect of theprocedure described above on actual strain maps obtained withthe grid method. Two short examples have been chosen for thispurpose. They are extracted from recent studies in which the gridmethod was employed to obtain the strain field during mechan-ical tests. It must however be pointed out that deconvolution wasnot an objective as these studies were undertaken, so someinformation is not available here, especially concerning noise. Inaddition, parameter s used to obtain the maps was not alwaysoptimal for deconvolution purposes. Nevertheless the authorsthink these examples still give a first idea on the strengths andcurrent limitations of this approach.

0 10 20 30 40 50 60−2

0

2

4

6

8

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u

ε xx

initialW 2RL 1

0 10 20 30 40 50 60−2

02468

1012

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1

2

3x 10−3

u

initialW 2RL 1

initialW 2RL 1

initialW 2RL 1

X: 26Y: 0.008658

Fig. 16. Asphalt specimen under compression. Cross-section of the binder band in Zone A. (a) Cross-section of Zone A (s¼ 8 pixels), Image 1. (b) Cross-section of Zone A

(s¼ 5 pixels), Image 1. (c) Cross-section of Zone A (s¼ 8 pixels), Image 2. (d) Cross-section of Zone A (s¼ 5 pixels), Image 2.

M. Grediac et al. / Optics and Lasers in Engineering 51 (2013) 716–734 731

4.2. Example 1: strain field on asphalt specimens

In this example addressed in full details in [30], asphaltspecimens were subjected to various compressive tests and thestrain fields were measured on one of their faces. Asphalt ismainly made up of aggregates bound by a bitumen matrix. Theproblem here is the fact that aggregates are separated by narrowbands of binder in some zones (see Fig. 14a) whose width can belower than 1 mm. The pitch of the grid bonded on the surface isequal to 0.2 mm. Assuming that 5 pixels/period are used toencode the grid image (this choice is a trade-off between spatialresolution, resolution and size of the field under interest), 25pixels are used to capture the details of the strain field over adistance equal to 1 mm. The Gaussian envelope used to calculatethe phases and their derivatives generally exhibits a standarddeviation equal at least to s¼ 5 pixels to limit the effect of noise[17], so the ‘‘width’’ of the Gaussian envelope is equal to 30 pixelsð6� sÞ in this case. This width is greater than the size of thebinder band under investigation in these zones (this is worse ifs45 pixels), so the procedure for extracting the phase deriva-tives from the grid images is expected to directly influence thestrain level measured within the band.

Fig. 14b shows a typical exx strain map obtained with s¼ 5pixels during a compression test. This image (denoted Image 1 inthe following) is obtained at an advanced stage of the test. Hencethe strain level in the binder bands reaches up to more than 1%,which is quite significant. Noise influence due to the camerasensor is expected to be limited here, so this situation is a priori

favorable for deconvolution. The shape of the bigger aggregates isrecognizable in this figure. The observed pattern is consistentwith the picture shown in Fig. 14a because the strain level isnearly null in the aggregates compared to that in the binderbands, bitumen being much softer than aggregates.

We focus now as an example on the zone defined by the smallblack rectangle. An enlargement is shown in Fig. 15a. On closecomparison between the width of the apparent band in Fig. 14band the corresponding zone in the picture shown in Fig. 14a, it

clearly appears that the first one is greater, thus illustrating theeffect of blurring and meaning that the strain level displayed hereis lower than the actual one.

The W 2 and LR 1 methods described and discussed abovehave been applied in this case. For the first technique, the noiseamplitude of the CCD chip was not measured and characterizedhere. It is however one of the input data of the Wiener filter [18],it is roughly assessed from the data sheet of the camera supplier.For the second technique, the grid image has been processed withnit ¼ 10, as suggested in Section 3.5 above. In both cases, s is firstchosen to be s¼ 8 pixels instead of 5, as suggested above. Theobtained results are shown in Fig. 15c and d for W 2 and RL 1,respectively. It can be seen that the strain maps are slightly lessblurred than in Fig. 14b, but the strain field seems more noisyover the aggregates, which means that noise and other phenom-ena such as grid defects were not sufficiently erased beforedeconvolution. Fig. 15b shows the enlarged zone after deconvolu-tion. The same color scale is used for the maps obtained beforeand after deconvolution (Fig. 15a and b, respectively). As may beseen, the band seems to be thinner and the strain amplitude atthe center is greater, as expected.

To make things clearer, a cross-section along the line super-imposed in Fig. 15a is shown in Fig. 16a along with the curveobtained with RL 1 (the abscissa measured along this line is denotedas u). Interestingly, the profile is similar to that obtained with W 2.This shows that the results obtained with both methods areconsistent. A strain level enhancement clearly appears: the profileis thinner at the base of the distribution and the maximum valuereached at the center is now greater. The strain value at the center isgreater after deconvolution: 30% with RL 2 and 33% with W 2. Thisillustrates the fact that deconvolution is vital if a quantitativeinformation and not only a qualitative one is sought in this zone.

s¼ 8 pixels being expected to be optimal, s¼ 5 pixels hasbeen tested to visualize the effect of this parameter. The obtainedresults are shown in Fig. 15e and f for W 2 and RL 1, respectively.As may be seen, the corresponding maps are noisier, thusconfirming that an optimal value for s exists. The strain profile

M. Grediac et al. / Optics and Lasers in Engineering 51 (2013) 716–734732

is shown in Fig. 16b. The initial profile is now higher than withs¼ 8 pixels, as expected since blurring is lower, but the restoredprofiles are noisier merely because the initial profile is alsonoisier. It is however worth noting that the restored profilesfeature nearly the same maximum value as with s¼ 5 pixels.

A second image obtained during the same test has also beeninvestigated. It is referred to as Image 2 in the following. It hasbeen chosen at the very beginning of the test, for a much lowerload level to examine what happens when the strain level islower, and therefore the impact of noise greater. Obtained results

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Fig. 17. Deconvolution of Image 2. (a) Deconvolved image 2. RL 1, s¼ 8 pixels.

(b) Deconvolved image 2. W 2, s¼ 8 pixels.

Fig. 18. Shape memory alloy under tension. Eyy map [31]. (a) Before deconvolution

[31]. (b) After deconvolution. (For interpretation of the references to color in this

figure caption, the reader is referred to the web version of this article.)

are shown in Fig. 17a and b. As may be seen, some significantblobs appear. They are due to noise and some other phenomenasuch as grid defects which were not sufficiently erased in thestrain map before deconvolution and/or, concerning the Wienerfilter, too rough a characterization of noise. The band profileobtained after deconvolution with s¼ 8 pixels is shown inFig. 16c. The impact of noise is visible. Interestingly, the magni-tude of the restoration is comparable to that observed for Image 1above, but it is worth noting that we have here a strain levelwhich is approximately 10 times lower (compare the strain levelsin Fig. 16b and c). This suggests that the ‘‘degree of restoration’’mainly depends on the geometry of the specimen, in other wordsthe actual thickness of the band in the current example. Finally,Fig. 16d shows what happens when s¼ 5 pixels: the impact ofnoise becomes now too significant and the information is lost.

In conclusion, it can be said that the results obtained here arepromising. Keeping in mind that noise was not properly char-acterized here, there is room for further improvements. Inparticular, it is shown in [18] that the phase derivative variancecan be predicted if the noise variance of the CCD chip is known.This result has been used in the simulations discussed in Section3.3 above. This is however not enough when actual grids areconsidered: they feature indeed some pitch variations or somelocal lack of ink which should be characterized specifically. Thelens also induces a PSF which certainly influences the strain maps.Tackling these issues is however beyond the scope of thecurrent paper.

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Fig. 19. Shape memory alloy under tension. Zoom of the white rectangle.

(a) Before deconvolution. (b) After deconvolution. (For interpretation of the

references to color in this figure caption, the reader is referred to the web version

of this article.)

0 20 40 60 80 100 120 1400

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before deconvolutionafter deconvolution

Fig. 20. Shape memory alloy under tension. Cross-section of the strain distribution.

M. Grediac et al. / Optics and Lasers in Engineering 51 (2013) 716–734 733

4.3. Example 2: strain field on a shape memory alloy

This second example deals with a tensile test performed on aspecimen of monocrystalline CuAlBe shape memory alloy. It isaustenite at room temperature, so when a tensile test is per-formed, martensite bands or needles suddenly appear and pro-pagate throughout the specimen, as discussed in [31]. Thesebands are generally very thin when they appear, so only slightbands are visible in the strain field. It must be pointed out thatthis case is very favorable for deconvolution since the strainamplitude within these bands is immediately very high (somepercents) because they are due to a phase change, so the noiseamplitude is very limited compared to the measured strainthanks to the width of the kernel used in [31] to process the gridimages: s¼ 5 pixels.

A typical Green–Lagrange Eyy strain field (obtained (i) bycalculating the Hencky strain tensor by adding small strainincrements and then (ii) by deducing directly the Green–Lagrangetensor) is shown in Fig. 18a, where y represents the verticaldirection along which the tensile force is applied. The blue colorcorresponds to austenitic regions (only very slightly elasticallystretched during the test) whereas the red color corresponds tomartensitic regions since the phase change gives rise to a suddenand significant deformation.

In the white rectangle enlarged in Fig. 19a, it is clear that a hellblue band containing some slight streaks has appeared. The coloris not the same as that obtained in wider bands located slightlyhigher. It can be checked that later on, during the same test, thisband becomes progressively wider and reaches the same color asthat observed above in the specimen [31].

The width of the band in the white rectangle being lower thanthe width of the Gaussian envelope used for processing the gridimages (a circle featuring a 6� s diameter is superimposed to themap), one can reasonably guess that the actual strain level in thisband is greater than the apparent one. Applying here the RL2 method reveals that two needles and not only one were hiddenbehind the unique blurred band in Fig. 19a, which could not reallybe guessed. In addition, the strain level in these needles is verysimilar to that observed above in the figure (see Fig. 19b), whereaustenite has already transformed into martensite. This is quitelogical since it means that the martensite variant obtained inneedles is the same as that obtained above in wider bands, asexpected. Fig. 20 finally shows the strain distribution along thewhite portion of line superimposed to the preceding figure. Strainenhancement clearly appears. It can be seen that the thickness ofthe needles (estimated at half the maximum) is equal to about 10pixels, which is equal here to 0.4 mm, the size of the pixels beingequal to 40 mm in this case.

5. Conclusion

In this paper, various deconvolution techniques are presentedand used to restore strain fields obtained with the grid method. Asynthetic example serves as a common thread between variousnumerical examples which enable to assess the performance ofthese deconvolution techniques. The influence of noise on thequality of the deconvolved maps is a major issue since sometechniques are not robust whereas some other ones still give goodresults when noise corrupt grid images. Thanks to the regularmarking, it is possible to establish a link between noise variancein the camera sensor and strain variance. Simulations have shownthat taking this information in the Wiener filter led to the bestresults if noise is correctly characterized. Two examples of actualstrain fields were also studied. In these cases, it is clear that someadditional phenomena such as slight grid pitch variations, localgrid bonding defects or PSF of the optical device itself corrupt theactual strain fields when the signal to noise ratio decreases. Thesephenomena could not be taken into account in the current work.Characterizing them separately should be undertaken to provide amore complete and realistic information on the deterioration ofthe final strain maps. This should then improve actual imageenhancement procedures based on deconvolution. Another exten-sion could be to deconvolve simultaneously the three in-planestrain components under the constraints of the compatibilityequations which link the second-derivatives of the straincomponents.

Acknowledgments

Dr. D. Delpueyo, Prof. X. Balandraud and Prof. E. Toussaint aregratefully acknowledged for fruitful discussions about the experi-mental results shown in the two examples.

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