CORRELI Q4 : A Software for “Finite-element” Displacement Field Measurements by Digital Image Correlation Franc ¸ois HILD and St ´ ephane ROUX April 2008 Internal report no. 269 LMT-Cachan ENS de Cachan/CNRS-UMR 8535/Universit´ e Paris 6/PRES UniverSud Paris 61 avenue du Pr´ esident Wilson, F-94235 Cachan Cedex, France Email: {francois.hild,stephane.roux}@lmt.ens-cachan.fr 1
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CORRELIQ4:
A Software for “Finite-element”
Displacement Field Measurements
by Digital Image Correlation
Francois HILD and Stephane ROUX
April 2008
Internal report no. 269
LMT-Cachan
ENS de Cachan/CNRS-UMR 8535/Universite Paris 6/PRES UniverSud Paris
61 avenue du President Wilson, F-94235 Cachan Cedex, France
This internal report completes the first ones (i.e., no. 230 and 254) on digital image correlation.
The technique developed herein is based upon a multi-scale approach to determine “finite ele-
ment” displacement fields by digital image correlation. The displacement field is first estimated
on a coarse resolution image and progressively finer details are introduced in the analysis as the
displacement is more and more securely and accurately determined. Such a scheme has been
developed to increase the robustness, accuracy and reliability of the image matching algorithm.
The details of the program are then presented. The procedure, CORRELIQ4, is implemented
in MatlabTM. The different steps are presented. The procedure is used on one example deal-
ing with Portevin-Le Chatelier bands in an aluminum alloy, and with an artificially deformed
picture of stone wool. Other published applications using the present code are listed.
The software has been protected under the IDDN.FR.001.110004.000.S.P.2008.000.21000.
Resume : rapport interne n 269
Ce rapport interne complete les precedents (i.e., n 230 et 254) sur la correlation d’images
numeriques. La technique presentee ici est basee sur une determination multi-echelles par
correlation d’images d’un champ de deplacement de type “element fini”. Le champ de deplace-
ment est d’abord determine sur une image de resolution grossiere et des details plus fins sont
rajoutes au fur et a mesure que les evaluations sont obtenues de maniere plus robuste et sure.
L’algorithme developpe ici a pour but d’augmenter la robustesse, la precision et la fiabilite de la
technique de correlation. Les details du programme, CORRELIQ4, implante dans MatlabTM,
sont ensuite presentes. Ce programme est enfin utilise dans un exemple correspondant a une
bande de Portevin-Le Chatelier dans un alliage d’aluminium, et un autre correspondant a
l’analyse d’une image artificiellement deformee d’une laine de roche. D’autres applications
publiees utilisant le code de correlation presente ici sont listees.
Le logiciel est protege sous l’IDDN.FR.001.110004.000.S.P.2008.000.21000.
2
1 Introduction
The analysis of displacement fields from mechanical tests is a key ingredient to bridge the gap
between experiments and simulations. Different optical techniques are used to achieve this
goal [1]. Among them, digital image correlation (DIC) is appealing thanks to its versatility
in terms of scale of observation ranging from nanoscopic to macroscopic observations with
essentially the same type of analyses. Most developments based on correlation exploit mainly
locally constant or linearly varying displacements [2].
In Solid Mechanics, the measurement stage is only the first part of the analysis. The
most important application is the subsequent extraction of mechanical properties, or quantita-
tive evaluations of constitutive law parameters [3]. By having an identical description for the
displacement field during the measurement stage and for the numerical simulation is the key
for reducing the noise or uncertainty propagation in the identification chain. During the latter,
there is usually a difference between the kinematic hypotheses made during the measurement
and simulation stages. To avoid this source of noise, it is proposed to develop a DIC approach
in which the measured displacement field is consistent with a finite element simulation. Con-
sequently, the measurement mesh has also a mechanical meaning. Let us emphasize that in
the present study, only the displacement field measurement is considered (i.e., it is a DIC tech-
nique), and no (finite-element) mechanical computation is performed. No constitutive law has
been chosen, nor any identification performed. However the displacement evaluation is directly
matched to a format ready to use for any further finite element modeling work.
In the following, it is proposed to develop a Q4-DIC technique in which the displacements
are assumed to be described by Q4P1-shape functions relevant to finite element simulations [4].
The pattern-matching algorithm is based upon the conservation of the optical flow. Variational
formulations are derived to solve this ill-posed problem. A spatial regularization was introduced
by Horn and Schunck [5] and consists in a looking for smooth displacement solutions. The
quadratic penalization is replaced by “smoother” ones based upon robust statistics [6, 7, 8]. In
the present approach, the sought displacement field directly satisfies continuity. In its direct
application, the conservation of the optical flow is a non-linear problem that is expressed in
terms of the maximization of a correlation product when the sought displacement is piece-wise
constant [9]. Other kinematic hypotheses are possible and a perturbation technique of the
minimization of a quadratic error leads to a linear system as in finite element problems. To
increase the measurable displacement range, a multi-scale setting is used as was proposed for
a standard DIC algorithm [10].
The report is organized as follows. Section 2 presents the general principles of a DIC
approach. It is particularized to Q4P1-shape functions and it is thus referred to as Q4-DIC. In
3
Section 3, all the details are given to run correli_q4.m as a MatlabTM file and discussed for an
artificially deformed picture of stone wool. A picture of an aluminium alloy sample constitutes
another test case discussed in Section 4 for the quantitative analysis of a Q4P1 kinematics as
it offers a good illustration of a heterogeneous strain field (i.e., a localized band is observed
in a tensile test). Section 5 briefly summarizes other applications with the same correlation
technique, and Section 6 with extensions of Q4-DIC. Details can be found in the listed papers.
2 Q4-Digital Image Correlation (Q4-DIC)
In this section the principle of the perturbation approach is introduced. Let us underline that
this approach applies to a wide class of functions, and is not confined to finite element shape
functions. Other examples have been explored [11, 12, 13], using mechanically based functions,
or using spectral decompositions of the displacement field [15, 14]. However, the discussion will
be specialized to Q4P1-shape functions, which provide a versatile tool for the analysis of very
different mechanical problems, ideally suited to finite element modeling.
2.1 Principle of DIC with an arbitrary displacement basis
Let us deal with two images, which characterize the original and deformed surface of a material
subjected to a known loading. An image is a scalar function of the spatial coordinate that gives
the gray level at each discrete point (or pixel) of coordinate x. The images of the reference
and deformed states are respectively called f(x) and g(x). Let us introduce the displacement
field u(x). This field allows one to relate the two images by requiring the conservation of the
optical flow
g(x) = f [x + u(x)] (1)
Assuming that the reference image are differentiable, a Taylor expansion to the first order yields
g(x) = f(x) + u(x).∇f(x) (2)
Let us underline here that the differentiability of the original image is not simply an academic
question, but we will come back to this point later on. The measurement of the displacement is
an ill-posed problem. The displacement is only measurable along the direction of the intensity
gradient. Consequently, additional hypotheses have to be proposed to solve the problem. For
example, if one assumes a locally constant displacement (or velocity), a block matching pro-
cedure is found. It consists in maximizing the cross-correlation function [16, 9]. To estimate
u, the quadratic difference between right and left members of Eq. (2) is integrated over the
4
studied domain Ω and subsequently minimized
η2 =
∫∫
Ω
[u(x).∇f(x) + f(x)− g(x)]2 dx (3)
The displacement field is decomposed over a set of functions Ψn(x). Each component of the
displacement field is treated in a similar manner, and thus only scalar functions ψn(x) are
introduced
u(x) =∑α,n
aαnψn(x)eα (4)
The objective function is thus expressed as
η2 =
∫∫
Ω
[∑α,n
aαnψn(x)∇f(x).eα + f(x)− g(x)
]2
dx (5)
and hence its minimization leads to a linear system
∑
β,m
aβm
∫∫
Ω
[ψm(x)ψn(x)∂αf(x)∂βf(x)]dx =
∫∫
Ω
[g(x)− f(x)] ψn(x)∂αf(x)dx (6)
that is written in a compact form as
Ma = b (7)
where ∂αf = ∇f.eα denotes the directional derivative. The matrix M and the vector b are
directly read from Eq. (6)
Mαnβm =
∫∫
Ω
[ψm(x)ψn(x)∂αf(x)∂βf(x)]dx (8)
and
bαn =
∫∫
Ω
[g(x)− f(x)] ψn(x)∂αf(x)dx (9)
Let us note that the role played by f and g is symmetric, and up to second order terms,
exchanging those two functions will lead to simply exchanging the sign of the displacement.
Thus in order to compensate for variations of the texture and to cancel the induced first order
error in u, one substitutes f in the expression of the matrix M by the arithmetic average
(f + g)/2. This symmetrization turns out to make the estimate of a more stable and accurate,
although it requires more computation time associated with the computation and assembly of
all elementary matrices and vectors.
Last, the present development is similar to a Rayleigh-Ritz procedure frequently used in
elastic analyses [4]. The only difference corresponds to the fact that the variational formulation
is associated to the (linearized) conservation of the optical flow and not the principal of virtual
work.
5
2.2 Particular case: Q4P1-shape functions
A large variety of functions Ψ may be considered. Among them, finite element shape functions
are particularly attractive because of the interface they provide between the measurement of the
displacement field and a numerical modeling of it based on a constitutive equation. Whatever
the strategy chosen for the identification of the constitutive parameters, choosing an identical
kinematic description suppresses spurious numerical noise at the comparison step. Moreover,
since the image is naturally partitioned into pixels, it is appropriate to choose a square or
rectangular shape for each element. This leads us to the choice of Q4-finite elements as the
simplest basis. Each element is mapped onto the square [0, 1]2, where the four basic functions are
(1−x)(1−y), x(1−y), (1−x)y and xy in a local (x, y) frame. The displacement decomposition
(4) is therefore particularized to account for the previous shape functions of a finite element
discretization. Each component of the displacement field is treated in a similar manner, and
thus only scalar shape functions Nn(x) are introduced to interpolate the displacement ue(x)
in an element Ωe
ue(x) =ne∑
n=1
∑α
aeαnNn(x)eα (10)
where ne is the number of nodes (here ne = 4), and aeαn the unknown nodal displacements. The
objective function is recast as
η2 =∑
e
∫∫
Ωe
[∑α,n
aeαnNn(x)∇f(x).eα + f(x)− g(x)
]2
dx (11)
and hence its minimization leads to a linear system (6) in which the matrix M is obtained
from the assembly of the elementary matrices M e whose components read
M eαnβm =
∫∫
Ωe
[Nm(x)Nn(x)∂αf(x)∂βf(x)]dx (12)
and the vector b corresponds to the assembly of the elementary vectors be such that
beαn =
∫∫[g(x)− f(x)] Nn(x)∂αf(x)dx (13)
Thus it is straightforward to compute for each element e the elementary contributions to M
and b. The latter is assembled to form the global “mass” matrix M and “force” vector b,
as in standard finite element problems [4]. The only difference is that the “mass” matrix and
the “force” vector contain picture gradients in addition to the shape functions, and the “force”
vector includes also picture differences. The matrix M is symmetric, positive (when the system
is invertible) and sparse. These properties are exploited to solve the linear system efficiently.
Last, the domain integrals involved in the expression of M e and be require imperatively a pixel
6
summation. The classical quadrature formulas (e.g., Gauss point) cannot be used because of
the very irregular nature of the image texture. This latter property is crucial to obtain an
accurate displacement evaluation.
2.3 Sub-pixel interpolation
In the previous subsection, the gradient ∇f(x) is used freely in the Taylor expansion leading
to Eq. (2). However, f represents the texture of the initial image, discretized at the pixel
level. Therefore, the definition of a gradient requires a slight digression. Previous works have
underlined the importance of sub-pixel interpolation. In Ref. [17] a cubic spline was argued to
be very convenient and precise. Here a different route is proposed, namely, a Fourier decom-
position. The latter provides a C∞ function that passes by all known values of f at integer
coordinates. From such a mapping one easily defines an interpolated value of the gray level
at any intermediate point. Moreover, one also exploits the same mapping for computing a
gradient at any point. Finally, powerful Fast Fourier Transform (FFT) algorithms allow for a
very rapid computation.
There is however a weakness in this procedure related to the treatment of edges. Fourier
transforms over a finite interval implicitly assume periodicity. Thus left-right or up-down
differences induce spurious oscillations close to edges. To reduce edge effects, each element
is enlarged to an integer power of two size, including a frame around each element. This
enlarged element is only used for FFT purposes, and once gradients are estimated, the original
element is cut out the enlarged zone, and thus the region where most of spurious oscillations are
concentrated is omitted. Moreover, at present, an “edge-blurring” procedure is implemented,
i.e., each border is replaced by the average of the pixel values of the original and opposite
border ones. This again reduces the discontinuity across boundaries [10]. There exist a few
alternative routes to limit or circumvent part of this artefact, namely, ad hoc windowing [10],
neutral padding [18], symmetrization, or linear trend removal. Such options have not been
tested.
2.4 Multi-scale approach
Even though a way of interpolating between gray level values at a sub-pixel scale was intro-
duced above, the very use of a Taylor expansion requires that the displacement be small when
compared with the correlation length of the texture. For a fine texture and a large initial
displacement, this requirement appears as inappropriate to converge to a meaningful solution.
Thus one may devise a generalization to arbitrarily expand the correlation length of the tex-
7
ture. This is achieved through a coarse-graining step. Again many ways may be considered,
such as a low pass filtering in Fourier or Wavelet spaces. A rather crude, but efficient way, is
to resort to a simple coarse-graining in real space [10] obtained by forming super-pixels of size
2n × 2n pixels, by averaging the gray levels of the pixels contained in each super-pixel.
First one generates a set of coarse-grained pictures of f and g for super-pixels of size
2× 2 pixels, 4× 4 pixels, 8× 8 pixels and 16× 16 pixels. Starting from the coarser scale, the
displacement is evaluated using the above described procedure. This determination is iterated
using a corrected image g where the previously determined displacement is used to correct for
the image. These iterations are stopped when the total displacement no longer varies. At this
point, one may estimate that a gross determination of the displacement has been obtained,
and that only small displacement amplitudes remain unresolved. This lack of resolution is
due to the fact that the small scale texture was filtered out. Thus finer scale images are used
taking into account the previously estimated displacement to correct for the g image. Again
the displacement evaluation is iterated up to convergence. This process is stopped once the
displacement is stabilized at the finer scale resolution, i.e., dealing with the original images.
Along the iterations, the “correction” of the deformed image by the previously determined
displacement field are possible with different degrees of sophistication. For reasons of compu-
tation efficiency, only the most crude correction is performed in the present implementation,
namely, each element is simply translated by the average displacement in the element. Inte-
ger rounded displacements are taken into account by a mere shift of coordinates, and sub-pixel
translation is performed by a phase shift in Fourier space [19]. This is a very low cost correction
since Fourier transforms are already required to compute gradients.
At the present stage, the implementation is such that the same number of super-pixel is
contained in each element. Thus as a finer resolution image is considered, the displacement is to
be determined on a physically finer grid. The transfer of the displacement from one scale to the
next one is performed using a linear interpolation, consistent with the Q4P1-shape functions
that are used. This multi-resolution scheme is thus also a mesh refinement procedure which is
performed uniformly (up to now) over the entire map.
This multi-resolution scheme was previously implemented using an FFT-correlation ap-
proach to estimate the displacement field [10]. In this context, it leads to much more robust
results. Large displacements and strains are measured using this algorithm, whereas a single
scale procedure revealed to be severely limited. Similarly, using the present Q4-decomposition,
this multi-resolution analysis revealed very precious to significantly increase the robustness and
accuracy of the measurement.
8
3 CORRELIQ4: User’s guide
The following section describes all the steps that can be followed when using the Q4-DIC code.
First start MatlabTM: at least the version 5.3 is needed (or any newer one; versions 7.xx have
been tested). Choose the directory in which the CORRELI files are put as the current directory.
Type the command correli_q4 at the MATLAB prompt. The first step is to choose in a first
menu (Fig. 1) to run a priori (performance) analyses, a Q4-DIC computation, to visualize a
result, or to create a movie:
• Texture: texture analysis;
• Uncertainty: uncertainty analysis;
• Resolution: resolution analysis;
• Computation click: choice of the Region Of Interest (ROI) by mouse click;
• Computation restart: get the ROI coordinates of a previous computation;
• Computation data: give the coordinates (in pixels) of the ROI to analyze;
• Visualization: visualize results of any previous computation.
• Movies: generate movies (only active for versions greater than or equal to 7.0).
!"#%$&'$(
)* +,# +%'$(
Figure 1: Menu to choose the type of analysis.
9
3.1 A priori analyses
The aim of the present section is to evaluate the a priori performances of the Q4-DIC technique
applied to the picture that corresponds to the reference configuration of the experiment to be
analyzed in Section 4. Figure 2 shows the texture used to measure displacement fields. It is
obtained by spraying a white and black paint prior to the experiment. Another picture (of
stone wool) will also be analyzed.
1 2 3 4 5
x 104
0
0.005
0.01
0.015
0.02
0.025
0.03
Gray level
Frequency
Figure 2: View of a reference picture and corresponding gray level histogram. The tension axis
is vertical. The width of the sample is 30 mm.
3.1.1 Texture characteristics
The quality of the displacement measurement is primarily based on the quality of the image
texture. Hence before discussing the result of the analysis, the characteristics of the texture
are presented. The gray level was encoded on a 16-bit depth (even though the original depth
was equal to 12 bits) in the image acquisition, and the true gray level dynamic range takes
advantage of this encoding, as judged from the gray level histogram shown in Fig. 2. Such a
histogram is a good indication of the global image quality to check for saturation problems.
However, such a global characterization of the image is only of limited interest. It is mostly
useful at the stage of image acquisition to set, say, the exposure time and / or the aperture.
Many acquisition softwares offer such functions. However, since the actual dynamic range of
gray levels is an important element to appreciate the quality of a picture, it is included here as
a possible diagnostic tool of poor performance.
What is more significant is the average of texture properties as estimated from sampling of
sub-images in elements. This is more a characteristic of the patterns than of image acquisition.
The point is to evaluate whether the sub-images carry enough information to allow for a proper
10
analysis. Each element is characterized by its own gray level dynamic range, or its standard
deviation of gray level. The latter quantity, averaged over all elements of a given size, and
normalized by the maximum gray level used in the image, is shown in Fig. 3-a. Even for the
smallest element sizes, this ratio is already as large as 0.06, and increases to about 0.13 for large
element sizes. The higher the ratio, the smaller the detection threshold as shown in Eq. (2);
the standard deviation being an indirect way of characterizing the sensitivity of the technique.
One thus concludes that the gray level amplitude is large enough to allow for a good quality of
the analysis even for element sizes as small as 4 pixels.
2 4 8 16 32 641.0
1.5
2.0
2.5
3.0
3.5
Element size (pixel)
-b-
x (
pix
el)
2 4 8 16 32 640.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
Mea
n gray l
evel
fluctuat
ion
Element size (pixel)
-a-
Figure 3: Fluctuation of gray level values averaged over elements of different sizes normalized
by the gray level dynamic range of the image (a). Average of largest (+) and smallest ()correlation radii determined on elements of varying sizes (b).
Another significant criterion is the correlation radius of the image texture. The latter is
computed from a parabolic interpolation of the auto-correlation function at the origin. The
inverse of the two eigenvalues of the curvature give an estimate of the two correlation radii, ξ1
and ξ2, shown in Fig. 3-b when averaged over all elements of a given size. The texture is rather
isotropic (i.e., similar eigenvalues), and remains small (varying from 1-2 to about 3 pixels) for
all element sizes. This indicates a very good texture quality that reveals small scale details
even for small element sizes. If one wants at least one disk and its complementary surrounding
to get a good estimate, the correlation radii should be less than one fourth of the element size.
This is achieved for element sizes greater than 6 pixels in the present case.
11
How to get these results?
To show the difference with a natural texture, another picture will be considered. It corresponds
to that of a stone wool sample [10]. Choose the Texture option in the first menu. Another
menu appears in which the format of the pictures has to be given (Fig. 4):
• Unknown format: it should one of the following formats .bmp, .CR2, .hbf, .hmf, .jpg,
.png, .tif.
• Image .bmp: this is a classical 8-bit coded format. Make sure the pictures are stored as
B/W .bmp files.
• Image .hbf or Image .hmf: these are HOLO3 (www.holo3.com) formats.
• Image .jpg: this is a classical 8-bit coded format. Make sure the pictures are stored as
B/W .jpg files.
• Image .png: this is a classical 8-bit coded format. Make sure the pictures are stored as
B/W .png files.
• Canon EOS 350: this is a .CR2 raw file.
• Image .tif: this is a classical 8-bit or 16-bit coded format. Make sure the pictures are
stored as B/W .tif files.
Figure 4: Menu to choose the format of the picture and then the reference picture.
The reference image, which is not necessarily located in the same directory as CORRELIQ4,
has to be chosen (Fig. 4). The next step consists in selecting the region of interest (ROI). Three
options are possible:
12
• Computation Click. Follow the instructions and click to choose the two end points of
the ROI (Fig. 5).
Figure 5: Choice of the Region Of Interest (ROI) by mouse click. Menu to choose the result
file of a previous computation for which the same ROI will be considered.
• Computation Restart. Indicate the results (.mat) file in which the ROI size is given
(Fig. 5).
• Computation Data. In the MATLAB Command Window, the user has to answer to four
questions to choose the size of the ROI. The minimum and maximum values are given
and they correspond to the image size:
minimum horizontal coordinate 1 <= datum <=1016?
minimum vertical coordinate 1 <= datum <= 1008?
maximum horizontal coordinate 1 <= datum <= 1016?
maximum vertical coordinate 1 <= datum <= 1008?
The maximum values are automatically given as those corresponding to the reference
image.
Once the picture is chosen, the software is running. Seven different results are shown. Let
us discuss them as they appear on the screen. The gray level histogram is first given for the
considered ROI (Fig. 6). In the present case, only a very small part (i.e., ca. 40 gray levels) of
the dynamic range of the CCD camera (i.e., 256 gray levels) is used. If possible, this should be
avoided and the situation shown in Fig. 2 is more desirable. If the test has not yet started, it
is time to reconsider the parameters of the camera. Otherwise, it is too late, and one will have
to do with what one has.
The fluctuation properties are given next (Fig. 7). The normalized RMS value with respect
to the dynamic range of the picture is plotted as a function of the element size. A practical
limit is chosen to be at least 1% of the dynamic range of the camera. Below this value, it is
13
120 130 140 150 160 170 1800
0.5
1
1.5
2
2.5x 10
4 Gray level histogram
Gray level
Num
ber
of s
ampl
esRegion of interest
50 100 150 200 250
50
100
150
200
250
Figure 6: Histogram of the chosen ROI.
believed that the measurement is not possible (i.e., there are not enough gradients to capture
displacements). With this limit, it is concluded that about 30% of all 4-pixel elements do not
meet this criterion. Such small size should therefore not be used. With 16-pixel and larger
elements, this first criterion is always satisfied.
4 8 16 32 64 1280
10
20
30
40
50
60
70
80
90
100Fluctuation criterion
ZOI size (pixels)
ZO
I pe
rcen
tage
Unvalidated ZOIValidated ZOI
4 8 16 32 64 1280
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
ZOI size (pixels)
Rel
ativ
e fl
uctu
atio
n
Mean fluctationMin. fluctationLimit
Figure 7: Percentage of validated elements. Minimum and mean relative RMS fluctuations as
functions of the element size in the analyzed picture of Fig. 6.
Last, the principal correlation radii are shown in values expressed in pixels, and normalized
by the element size (Fig. 8). A practical limit is chosen to be at most 25% of the element size.
Above this value, it is believed that the measurement is not secure. With this limit, it is
concluded that about 50% of all 8-pixel elements do not meet this criterion. Such small size
should therefore not be used. With 16-pixel and larger elements, this second criterion is always
satisfied. Last, let us note that the two correlation radii are significantly different. This result
14
indicates that the texture is anisotropic (Fig. 6). The interested reader will find additional
details in Ref. [18] concerning the determination of dominant orientations of a texture.
4 8 16 32 64 1280
10
20
30
40
50
60
70
80
90
100Correlation radius criterion
ZOI size (pixels)
ZO
I pe
rcen
tage
Unvalidated ZOIValidated ZOI
4 8 16 32 64 1280
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
ZOI size (pixels)
Dim
ensi
onle
ss c
orre
latio
n ra
dii
R1meanR1max R2meanR2max Limit
4 8 16 32 64 1280
2
4
6
8
10
12
14
16
18
20
ZOI size (pixels)
Cor
rela
tion
radi
i (pi
xels
)
R1meanR1max R2meanR2max
Figure 8: Percentage of validated elements. Maximum and mean correlation radii as functions
of the element size.
With these two simple criteria, it is concluded that at least 16-pixel elements are to
be chosen (i.e., both are satisfied simultaneously). It is worth noting that this first a priori
analysis concerns only the texture itself. It is therefore a qualitative analysis. In the following,
two quantitative analyses are proposed.
3.1.2 Displacement uncertainty
Prior to any computation, it is important to estimate the a priori performance of the approach
on the actual texture of the image. If one changes the picture, one may not get exactly the
same performance since it is related to the local details of the gray level distribution as shown
in Section 2. This is performed by using the original image f only, and generating a translated
image g by a prescribed amount upre. Such an image is generated in Fourier space using a simple
phase shift for each amplitude. This procedure implies a specific interpolation procedure for
inter-pixel gray levels, to which one resorts systematically (see Section 2.3). The algorithm is
then run on the pair of images (f, g), and the estimated displacement field uest(x) is measured.
One is mainly interested in sub-pixel displacements, where the main origin of errors comes
from inter-pixel interpolation. Therefore the prescribed displacement is chosen along the (1, 1)
direction so as to maximize this interpolation sensitivity. To highlight this reference to the
pixel scale, one refers to the x- (or y-) component of the displacement upre ≡ upre.ex varying
from 0 to 1 pixel, rather than the Euclidian norm (varying from 0 to√
2 pixel).
The quality of the estimate is characterized by two indicators, namely, the systematic
error, δu = ‖〈uest〉 − upre‖, and the standard uncertainty σu = 〈‖uest − 〈uest〉‖2〉1/2. The
change of these two indicators is shown in Fig. 9 as functions of the prescribed displacement
amplitude for different element sizes ` ranging from 4 to 128 pixels. Both quantities reach a
15
maximum for one half pixel displacement, upre = 0.5 pixel, and are approximately symmetric
about this maximum. Integer valued displacements (in pixels) imply no interpolation and are
exactly captured through the multi-scale procedure discussed above. This confirms that these
errors are due to interpolation procedures. The results are shown in a semi-log scale to reveal
the strong sensitivity to the element size, however a linear scale would show that both δu and σu
follow approximately a linear increase with upre from 0 to 0.5 pixel (and a symmetric decrease
from 0.5 to 1 pixel).
10-6
10-5
10-4
10-3
10-2
10-1
0 0.2 0.4 0.6 0.8 1
4
8
16
32
64
128
Prescribed displacement (pixel)
-a-
Mea
n d
ispla
cem
ent
erro
r (p
ixel
)
10-5
10-4
10-3
10-2
10-1
100
0 0.2 0.4 0.6 0.8 1
4
8
16
32
64
128
Sta
nd
ard u
nce
rtai
nty
(pix
el)
Prescribed displacement (pixel)
-b-
Figure 9: Mean error δu and standard deviation σu as a function of the prescribed displacement
upre for different element sizes ` ranging from 4 to 128 pixels.
To quantify the effect of the element size, the error and standard uncertainty, are averaged
over upre within the range [0, 1] as functions of the element size `. These data are shown in
Fig. 10. A power-law decrease
〈σu〉 = A1+ζ`−ζ
〈δu〉 = B1+υ`−υ(14)
for 8 ≤ ` ≤ 128 pixels is usually observed as shown by a regression line on the graph. Both
amplitudes are typically close to 1 pixel (more precisely A = 1.15 pixel and B = 1.07 pixel).
The exponents are measured to be ζ = 1.96 and υ = 2.34. The data for ` = 128 pixels seem
to depart from the power-law trend with a tendency to saturate. These results quantify the
trade-off the experimentalist has to face in the analysis of a displacement field, namely, either
the measurement is accurate but estimated over a large zone, or it is spatially resolved but at
the cost of a less accurate determination. This is a significant difference with classical finite
element techniques for which convergence is achieved when the element size decreases. This
is not the case when measurements are concerned. Let us however underline the following
conclusions:
16
• Elements as small as ` = 4 pixels may be used with an average error and standard
uncertainty of the order of 0.1 pixel,
• Systematic errors of the order of 10−2 and 10−3 pixel is reached for element sizes respec-
tively equal to 8 and 16 pixels.
• Standard uncertainties of the order of 2× 10−2 and 6× 10−3 pixel is reached for element
sizes respectively equal to 8 and 16 pixels.
• The systematic error in the determination of a displacement is such that evaluations will
be “attracted” toward integer values. Correspondingly, transitions at half-integer pixel
values for the displacement will appear as more abrupt. This phenomenon will be referred
to as “integer locking” in the sequel, and will be discussed in detail. Let us underline
that this spurious bias is revealed in this technique because the latter is used down to
extremely small element sizes.
"!$# % &')( *+% ,- .
/ 0'/
1'1"2-3 465 789 3 465 7
: ; <
"!$# % &'=( *+% ,- .
/ >$/
Figure 10: Average error 〈δu〉 and standard uncertainty 〈σu〉 as functions of the element size
`. For the displacement uncertainty, the results obtained by Q4-DIC are compared with those
obtained by FFT-DIC. The dashed lines correspond to power-law fits.
For comparison purposes, the displacement uncertainties obtained with the present tech-
nique are compared with those of a standard FFT-DIC technique [10]. In that case, a weaker
power-law decrease is observed with A = 1.00 pixel and ζ = 1.23 (Fig. 10b). This result shows
that by using a continuous description of the displacement field, it enables for a decrease of the
displacement uncertainty when the same element size is used. Conversely, for a given displace-
ment uncertainty, the Q4-DIC algorithm allows one to reduce significantly the element size,
17
thereby increasing the number of measurement points when compared to a classical FFT-DIC
technique.
How to get these results?
The same instructions as presented for the texture analysis (Section 3.1.1) hold. We reproduce
them for a more linear reading. Choose the Uncertainty option in the first menu. Another
menu appears in which the format of the pictures is chosen (Fig. 4):
• Unknown format: it should one of the following formats .bmp, .CR2, .hbf, .hmf, .jpg,
.png, .tif.
• Image .bmp: this is a classical 8-bit coded format. Make sure the pictures are stored as
B/W .bmp files.
• Image .hbf or Image .hmf: these are HOLO3 (www.holo3.com) formats.
• Image .jpg: this is a classical 8-bit coded format. Make sure the pictures are stored as
B/W .jpg files.
• Image .png: this is a classical 8-bit coded format. Make sure the pictures are stored as
B/W .png files.
• Canon EOS 350: this is a .CR2 raw file.
• Image .tif: this is a classical 8-bit or 16-bit coded format. Make sure the pictures are
stored as B/W .tif files.
The reference image, which is not necessarily located in the same directory as CORRELIQ4,
has to be chosen (Fig. 4). The next step consists in selecting the region of interest (ROI). Three
options are possible:
• Computation Click. Follow the instructions and click to choose the two end points of
the ROI (Fig. 5).
• Computation Restart. Indicate the results (.mat) file in which the ROI size is given
(Fig. 5).
• Computation Data. In the MATLAB Command Window, the user has to answer to four
questions to choose the size of the ROI. The minimum and maximum values are given
and they correspond to the image size:
18
minimum horizontal coordinate 1 <= datum <=1016?
minimum vertical coordinate 1 <= datum <= 1008?
maximum horizontal coordinate 1 <= datum <= 1016?
maximum vertical coordinate 1 <= datum <= 1008?
The maximum values are automatically given as those corresponding to the reference
image.
Once the picture is chosen, the software is running. It consists in artificially moving a
picture with a constant rigid body motion. In practice, increments of 0.5 pixel are sufficient
since the mean error and corresponding standard deviation are piece-wise linear functions of the
prescribed displacement (Fig. 9). Four different results are shown. The mean displacement error
and corresponding standard deviation are plotted as functions of the prescribed displacement
for different element sizes (Fig. 11). In the present case, there is a significant decrease as the
element size becomes greater than or equal to 16 pixels.
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Prescribed displacement (pixel)
Stan
dard
unc
erta
inty
(pi
xel)
4 8 163264
0 0.2 0.4 0.6 0.8 10
0.005
0.01
0.015
0.02
0.025
0.03
0.035
Prescribed displacement (pixel)
Mea
n er
ror
(pix
el)
4 8 163264
Figure 11: Mean displacement error and standard deviation as functions of the prescribed
displacement for different element sizes.
The average displacement error and standard deviation are also given and power law
fitted. Figure 12 shows first that the mean error is an order of magnitude less than the standard
deviation. This result shows that the measurements are basically unbiased but mildly scattered
around the true value. Second, the standard deviation decreases as the element size increases
as already discussed above (namely, either the measurement is accurate but generally estimated
over a large zone, or it is spatially resolved but at the cost of a less accurate determination).
With this second analysis, it is concluded that 16-pixel elements can be chosen and that
the displacement uncertainty is of the order of 2× 10−2 pixel. When compared with the results
of Fig. 10, it is concluded that the artificial texture (Fig. 2) yields lower levels of displacement
uncertainties than the stone wool texture.
19
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Figure 12: Mean displacement error and standard deviation as functions of the element size.
3.1.3 Noise sensitivity
Last, the effect of noise associated to the image acquisition (e.g., digitization, read-out noise,
black current noise, photon noise [20]) on the displacement measurement is assessed. This anal-
ysis allows one to estimate the displacement resolution [21]. The reference image is corrupted
by a Gaussian noise of zero mean and standard variation σg ranging from 1 to 8 gray levels
at each pixel with no spatial correlation. No displacement field is superimposed on the image,
and the displacement field is then estimated. The standard deviation of the displacement field,
σu, is shown in Fig. 13 as a function of the noise amplitude σg and for different element sizes
ranging from 4 to 128 pixels. The quantity σu is linear in the noise amplitude and inversely pro-
portional to the element size. The latter properties are derived from the central limit theorem.
A theoretical analysis of this problem is discussed in Refs. [11, 22], and leads to the
following estimate of the standard deviation of the displacement field induced by a Gaussian
white noise
σu =12√
2σgp
7〈|∇f |2〉1/2`(15)
where p is the physical pixel size. For the present application, one computes 〈|∇f |2〉1/2 ≈5340 pixel−1, hence σu ≈ 4.5× 10−4σgp/`. This theoretical expectation (neglecting the spatial
correlation in the image texture) is consistent with the direct estimates shown in Fig. 13 (e.g.,
for ` = 4 pixels and σg = 8 gray levels, the direct estimate is 1.2×10−3 pixel to be compared with
9× 10−4 pixel given by the above formula). In practice, with the used CCD camera, the noise
level is given with a maximum range less than 3 gray levels. Consequently, the contribution of
20
0 2 4 6 80
0.5
1
1.5x 10
Noise level (gray level)
Stan
dard
unc
erta
inty
(pix
el)
4
8
16
3264128
-3
Figure 13: Standard deviation of the displacement error versus noise amplitude for different
element sizes (4, 8, 16, 32, 64 and 128 pixels) from top to bottom.
image noise is negligibly small when compared to that induced by the sub-pixel interpolation.
How to get these results?
The same instructions as presented for the texture analysis (Section 3.1.1) hold. We reproduce
them for a more linear reading. Choose the Resolution option in the first menu. Another
menu appears in which the format of the pictures has to be chosen (Fig. 4):
• Unknown format: it should one of the following formats .bmp, .CR2, .hbf, .hmf, .jpg,
.png, .tif.
• Image .bmp: this is a classical 8-bit coded format. Make sure the pictures are stored as
B/W .bmp files.
• Image .hbf or Image .hmf: these are HOLO3 (www.holo3.com) formats.
• Image .jpg: this is a classical 8-bit coded format. Make sure the pictures are stored as
B/W .jpg files.
• Image .png: this is a classical 8-bit coded format. Make sure the pictures are stored as
B/W .png files.
• Canon EOS 350: this is a .CR2 raw file.
• Image .tif: this is a classical 8-bit or 16-bit coded format. Make sure the pictures are
stored as B/W .tif files.
21
The reference image, which is not necessarily located in the same directory as CORRELIQ4,
has to be chosen (Fig. 4). The next step consists in selecting the region of interest (ROI). Three
options are possible:
• Computation Click. Follow the instructions and click to choose the two end points of
the ROI (Fig. 5).
• Computation Restart. Indicate the results (.mat) file in which the ROI size is given
(Fig. 5).
• Computation Data. In the MATLAB Command Window, the user has to answer to four
questions to choose the size of the ROI. The minimum and maximum values are given
and they correspond to the image size:
minimum horizontal coordinate 1 <= datum <=1016?
minimum vertical coordinate 1 <= datum <= 1008?
maximum horizontal coordinate 1 <= datum <= 1016?
maximum vertical coordinate 1 <= datum <= 1008?
The maximum values are automatically given as those corresponding to the reference
image.
Once the picture is chosen, the software is running. It consists in adding varying levels of
random Gaussian noise to the reference picture and then performing the correlation calculation
with respect to the reference picture. Two different results are shown. The mean displacement
error and corresponding standard deviation are plotted as functions of the noise level (standard
deviation) for different element sizes (Fig. 14). The mean error has usually a very erratic trend
(this is only a check) and very low values. Conversely, the standard deviation has a clearer
tendency already discussed above.
In the present case however, the results have to be analyzed very carefully. The levels
are very high. This is to be related to the low dynamic range of the analyzed picture (Fig. 6)
that makes the texture very sensitive to acquisition noise. Figure 15 shows a zoom around
the 0-2 gray level standard deviation range. It is concluded that in the present case, the main
limitation in terms of performance is induced by the texture itself, and not the correlation
algorithm. This is an extreme case. In many situations, the result is just the opposite (namely,
the correlation code bears most of the “responsibility,” in particular when artificial textures
are considered as shown above).
With this third analysis, it is concluded that 16-pixel elements can be chosen and that
the displacement uncertainty is of the order of 2× 10−2 pixel, provided the camera sensor has
22
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) *+,-*./ +00. ,12 3-+4+567 1. 8+29
!#"
$ % &'( &)( *'+,)% &- '% ./ 0- 1,23 4
56 78:97 4; 6 95
Figure 14: Mean displacement error and standard deviation as functions of the noise level for
different element sizes.
!#"
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4 5687
9 :
7 46:;5
Figure 15: Mean displacement error and standard deviation as functions of the noise level for
different element sizes.
low noise features. Otherwise, it might be desirable to increase the element size, or work harder
to get pictures with a better quality (this is, unfortunately not always possible...).
3.2 Computation
Any of the following options has to be selected to run directly a computation:
• Computation Click. Follow the instructions and click to choose the two end points of
the ROI (Fig. 5).
• Computation Restart. Indicate the results (.mat) file in which the ROI size is given
(Fig. 5).
• Computation Data. In the MATLAB Command Window, the user has to answer to four
23
questions to choose the size of the ROI. The minimum and maximum values are given
and they correspond to the image size:
minimum horizontal coordinate 1 <= datum <=1016?
minimum vertical coordinate 1 <= datum <= 1008?
maximum horizontal coordinate 1 <= datum <= 1016?
maximum vertical coordinate 1 <= datum <= 1008?
The maximum values are automatically given as those corresponding to the reference
image.
Another menu appears in which the format of the pictures has to be selected (Fig. 4):
• Unknown format: it should one of the following formats .bmp, .CR2, .hbf, .hmf, .jpg,
.png, .tif.
• Image .bmp: this is a classical 8-bit coded format. Make sure the pictures are stored as
B/W .bmp files.
• Image .hbf or Image .hmf: these are HOLO3 (www.holo3.com) formats.
• Image .jpg: this is a classical 8-bit coded format. Make sure the pictures are stored as
B/W .jpg files.
• Image .png: this is a classical 8-bit coded format. Make sure the pictures are stored as
B/W .png files.
• Canon EOS 350: this is a .CR2 raw file.
• Image .tif: this is a classical 8-bit or 16-bit coded format. Make sure the pictures are
stored as B/W .tif files.
The reference image, which is not necessarily located in the same directory as CORRELIQ4, has
to be selected (Fig. 4). In the next menu (Fig. 16), the parameters of the correlation method
have to be chosen:
• The size of the elements (typically 16 × 16 pixels). This is one of the most critical one.
The a priori analysis should therefore help in choosing it. In the present version, only
even numbers can be selected.
• The shift δ between to neighboring zones. This option is not active with a Q4 algorithm.
24
• The number of scales is the second most important parameter. As discussed above, the
present algorithm is based upon a coarse-graining approach. When large displacements
and / or strains are suspected to occur, it is desirable to increase the number of scales.
The maximum number of scales depends on the size of the ROI in comparison to that
of the elements (e.g., for a 512 × 512pixel-ROI, and 16-pixels elements, the maximum
number of scales is 5, i.e., at least 2× 2 elements are needed for the coarsest scale).
• The number of iterations (the higher the number, the higher the accuracy and the com-
putation time). This value is given to avoid endless iterations. This criterion should not
be reached. In case it is reached a message warns the user:
• The number of images to analyze (the reference image is not included in that count);
• When a sequence of more than one image is considered, the correlation can be performed
either by considering always the same reference image for strains (in absolute value) less
than or equal to 10% or by changing (or updating) the reference image. In the last case,
the Image update Y/N has to be activated.
• Last, it is possible to store the correlation results for any couple of analyzed pictures.
This is made possible by activating the Independent calculation Y/N button.
Validate the choice at the end (press the Validation button).
The sequence of deformed pictures has to be selected. The user selects the whole sequence
by using the same procedure as for the reference picture (Fig. 17).
It is then possible to mask part of the ROI. A menu appears and seven different operations
are possible:
• Define exclusion polygon allows the user by mouse click to enter different exclusion
regions. Follow the instructions on top of the picture that appears.
• Remove polygon allows the user by mouse click to remove any of the existing exclusion
polygon.
25
• Define exclusion circle allows the user, for instance to mask a hole. Follow the
instructions on top of the picture that appears.
• Remove exclusion circle allows the user by mouse click to remove any of the existing
exclusion circle.
• Define inclusion circle allows the user to choose a circular ROI from a rectangular
shape chosen before. This would be the case for the analysis of a Brazilian test [12].
Follow the instructions on top of the picture that appears.
• Remove inclusion circle allows the user by mouse click to remove any of the existing
inclusion circle.
• Redraw when the user is not happy with the (random) colors.
• Exit to end the mask procedure.
When the computation is completed, the result file has to be saved (Fig. 18). The extension
is ‘.mat’ to be readable for a later visualization. The computation is now ended and the
visualization stage starts. One can choose to perform another computation or to visualize any
Figure 16: dialog box for choosing the correlation parameters.
26
Figure 17: dialog box to choose the deformed picture(s).
Figure 18: dialog box to choose the type of mask and to save the results
results. Type the command correli_q4 at the MATLAB prompt. It can be noted that during
the computations, different messages may appear. Some messages are only given to indicate
that the computation is running normally.
3.3 Visualization
If Visualization is chosen, the result file (‘.mat’ extension) to be displayed has to be selected
(Fig. 19).
Figure 20 appears, in which different options can be chosen. In the present case, the
vertical displacement field (expressed in pixels) is plotted for the stone wool texture for which
an artificially deformed picture was created. A uniform nominal strain level of 0.25 was applied.
This is an extreme case for the present correlation software that needs all the scales to capture
properly the displacement field. The element size was equal to 16 pixels. It is worth noting
that the maximum displacements are greater than 3 times the element size. Had the multiscale
algorithm not been implemented, it would have been impossible to capture these levels (i.e., 5
27
Figure 19: dialog box to read the results of a previous computation.
scales were used in the present case). When a sequence is analyzed, any image can be selected
Figure 20: Visualization of the measurement results. In the present case, the displacement field
along the vertical direction.
by using the two buttons bellow the Image No. message (Fig. 20).
28
At the top left corner are options related to the strain measures:
• infinitesimal: infinitesimal strains (i.e., symmetric part of displacement gradient, see
Eq. (24));
• nominal: nominal (Cauchy-Biot) strains (Eq. (58), when m = 1/2);
• Green Lag.: Green-Lagrange strains (Eq. (58), when m = 1);
• logarithmic: logarithmic (Hencky) strains (Eq. (58), when m → 0+);
• RdB (internal development);
• Eigen value?: the eigen values of the selected strain measure are shown.
It is worth noting that in the case of large strains (see details in the Appendix), the out-of-
plane displacement is assumed to be small and is therefore neglected. Other assumptions can
be made. They will have to be implemented on demand.
Just below are options related to the type of component to visualize. The type of com-
ponent to display has to be chosen, namely, in-plane strains, in-plane displacements, or out-of-
plane rotation. The frame is always the same, namely, horizontal direction: 2, vertical direction:
1. By choosing error, an error indicator (i.e., correlation residual |u(x).∇f(x)+ f(x)− g(x)|normalized by the dynamic range of the picture) of the result is plotted. The closer to 0, the
better the result (when no lighting variations occur, levels below few percentages are usually
achieved).
Two pictures are plotted. The left picture is always the very first reference picture. For
the reference image, the chosen field is plotted. In the case of displacements, the average value
is always in the middle of the scale. To change the scale, the dialog box in between the two
images is to be used (Fig. 21).
The number of contours can be changed. If 0 is set, a grayscale is used. If 11 is chosen,
a fancy color scale appears, otherwise a conventional (i.e., hot) one is used. The amplitude of
the displacement field is chosen with respect to the average value. For strains, two routes can
be followed:
• the maximum and minimum values are given by the user in the middle part of the dialog
box;
• the w or w/o mean button is activated and the average value corresponds to the middle
of the scale, the range of which is chosen as the maximum strain.
29
Figure 21: Contour and scale options. Mesh options. Amplification of the displacements.
When the fill option is activated, the contours are filled on the reference image. The rigid
body motion can be subtracted to the overall displacement by pushing the corresponding button
(Rigid body motion Y/N). When the error indicator is plotted, there is no need to change any
scale parameters, it is performed automatically. The w or w/o mean button can also be used.
When selecting mesh, the undeformed and deformed meshes appear on the relevant images
(Fig. 22). When selecting vector, the displacement vectors are shown (Fig. 21). It can be noted
that both options can be used simultaneously. A slider enables the displacements to be amplified
by a factor that can be chosen by the user (Fig. 21). When the amplification is greater than 1,
the underlying image disappears.
To further comment on the results on the artificially deformed picture, the corresponding
(nominal) strain field is shown in Fig. 22. Since the w or w/o mean button is activated, the
average strain can be read directly as the median value (i.e., 0.2396 ≈ 0.24 for a prescribed
value of 0.25).
3.4 Virtual Gauges
Type the command gauge at the MATLAB prompt. This procedure allows for the computation
of the average strain in a user-selected ROI. one needs to indicate the result file in which all
the data needed are stored. Then the Region Of Interest (ROI) is selected by mouse. Click and
maintain to select the ROI. If the ROI is larger than the mesh, then the size is automatically
reset to the maximum size.
In the MATLAB Command Window, average values of different strain tensors are given.
30
Figure 22: Visualization of post-processed results. In the present case the normal (nominal)
strains along the vertical direction.
The corresponding eigen strains are also computed. The results can also be saved in an ASCII
file. The computation is now ended. Type any command at the MATLAB prompt.
4 Application to a tensile test
In this section, an application of the previously proposed algorithm is carried out to analyze
a tensile test performed on an aluminum alloy sample. In the plastic regime, the formation
of localization strain bands is observed. The fact that for a given displacement uncertainty,
smaller element sizes can be chosen in the present case (Q4-DIC) when compared with those
of a standard FFT-DIC technique (Fig. 10b), enables one to better capture kinematic details
in the localization band.
31
4.1 Material and method
The studied aluminum alloy is of type 5005 (i.e., more than 99 wt% of Al content and a small
amount of Mg; these values were determined by electron probe micro-analysis). As shown in
Fig. 2, the sample is a coated with sprayed black and white paints to create the random texture
for the displacement field measurement. The sample size is 140× 30× 2 mm3. It is positioned
within hydraulic grips of a 100 kN servo-hydraulic testing machine. Its alignment is checked
with DIC measurements (i.e., no significant rotation of the sample is observed in the elastic
domain). To have a first strain evaluation, an extensometer was used. Its pins are observed on
the right edge of the sample (Fig. 2).
An artificial light source is used to minimize gray level variations so that the conservation
of the optical flow is considered as practically achieved. A CCD camera (12-bit digitization,
noise less than 3 gray levels, resolution: 1024 × 1280 pixels) with a conventional zoom is
positioned in front of the sample. In the present case, the physical size of one pixel is 25 µm.
Two loading sequences are carried out. First, in the elastic domain, a controlled displacement
rate of 5 µm/s is applied and pictures are taken for 12 µm-increments. Elastic properties may
be identified [23]. This will not be discussed herein. Second, a controlled displacement rate of
10 µm/s is applied to study strain localization and pictures are taken for 60 µm-increments.
The following analysis of the displacement field is an increment between two image acquisitions
in the “plastic” regime.
Figure 23 shows the change of the average longitudinal strain with the number of pictures
(or equivalently with time). This result was obtained by using the Q4-DIC analysis. Until the
extensometer pins slipped (at about a 5% strain), the average strain measured by DIC and
that given by the extensometer were close, even though the same surface was not analyzed.
This response is typical of a Portevin-Le Chatelier (PLC) phenomenon or jerky flow [24].
From a microscopic point of view, PLC effects are related to dynamic interactions between
mobile dislocations and diffusing solute atoms [25, 26]. From a macroscopic perspective, it is
related to a negative strain rate sensitivity that leads to localized bands that are simulated [27].
Many experimental studies [28] however are based upon average strain measurements. There
are also full-field displacement measurements performed by using, for instance, laser speckle
interferometry [29]. Yet the spatial resolution did not allow for an analysis of the displacement
field within the band. Additional insight is gained by using IR thermography [30].
32
! "$#%'&)(#*,+&-%
Figure 23: Mean strain for a region of interest of 1000× 700 pixels as a function of the number
of picture. The box shows the two pictures that are analyzed.
4.2 Kinematic fields
Let us now analyze the displacement field in between two states (0.3% mean strain apart,
see Fig. 23). The same region of interest of size 1000 × 700 pixels is studied using the above
method, with different element sizes ranging from 16 down to 4 pixels. Figures 24 and 25 show
the resulting displacement fields (component Ux and Uy, respectively). Although the test is
pure tension, the analysis reveals without ambiguity the presence of a localization band whose
width is about 150 pixels, and across which the displacement discontinuity is about 2 pixels
along the tension axis, and about 1 pixel perpendicular to it. Let us concentrate here on this
single pair of images to validate the algorithm on a real experimental test and evaluate its
performances.
One notes that all element sizes may be used. As expected, the smallest element sizes are
noisier, yet the agreement between all these determinations is excellent. Let us underline that
FFT-DIC usually deals with element sizes equal or larger than 32 pixels, exceptionally 16 pixels
for very favorable cases are used when locally constant displacement fields are sought. Using
the Q4-DIC technique allows one to reduce the element size by a factor of 4, which means that
the number of pixels in the element has been cut down by a factor of 16.
Let us however note that one should be cautious about the fact that displacements have a
tendency to be attracted toward integer values, especially for small element sizes. Therefore the
direct evaluation of strains along the tension axis εyy as obtained by the Q4P1-shape functions
or equivalently as a simple finite difference is expected to be artificially increased at half-integer
displacement components. Figure 26 shows such strain fields for 4 different element sizes from
33
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Figure 24: Map of Uy displacement for different element sizes: (a) ` = 16, (b) ` = 12, (c)
` = 10, (d) ` = 8, (e) ` = 6 and (f) ` = 4 pixels. The physical size of one pixel is equal to
25 µm.
16 down to 8 pixels. For a size of 16 pixels, the localization band appears as a genuine zone
of increased strains as compared to a “silent” (or elastic) background. For smaller element
sizes, the edges of the shear band appear to concentrate still a higher strain. The same effect
is apparent for element sizes 12, 10 and 8 pixels. The strain maps obtained for smaller element
sizes are not shown, since the noise level becomes much higher and thus the measurement
cannot be trusted. The same artefact of strain enhancement at the edges of the shear band is
however observed.
34
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-f-
800 1000
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Figure 25: Map of Ux displacement for different element sizes: (a) ` = 16, (b) ` = 12, (c)
` = 10, (d) ` = 8, (e) ` = 6 and (f) ` = 4 pixels. The physical size of one pixel is equal to
25 µm.
4.3 Integer locking
One notes on the previous figure that the Uy-displacement is half-integer valued at the edge of
the shear band. The larger strains at the edge of the band could therefore be interpreted as an
artefact due to integer locking. Integer valued displacements being favored, an artificial gap is
created for half-integer values displacement, and thus any gradient (finite difference operator)
will underline this effect very markedly.
To test this interpretation, the following test is proposed. An artificially translated image
by 0.5 pixel is computed from the original one, using a fast Fourier transform, as the latter
35
200 400 600
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800200 400 600
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0.015
0.01
0.005
0
0.005
0.01
0.015
0.02
0.025
εxx
y (pixel)
x (
pix
el)
200 400 600
-d-
800 1000
100
200
300
400
500
600
700
800
Figure 26: Map of the strain component εxx for different element sizes: (a) ` = 16, (b) ` = 12,
(c) ` = 10 and (d) ` = 8 pixels.
provides a simple and numerically efficient way of interpolating the image at any arbitrary sub-
pixel value. A genuine strain enhancement is thus expected to be identified at a fixed position
in the reference image frame of coordinates, whereas a numerical artefact would be moved to
a different location. Figures 27a and b show the Uy displacement component starting from
the original image or from the translated one (and where the 1/2 pixel translated has been
corrected for). A good agreement is observed for the displacement field thus revealing a rather
poor sensitivity to such a rigid translation. Figures 27c and d show the corresponding εyy strain
maps. On the latter set of figures, although high strain values tend to concentrate along two
lines in both figures, the precise location of these bands is not stable. This is a signature of the
integer locking phenomenon. Therefore the strain enhancement at the edge of the shear band
is to be considered as an artefact.
Let us underline that such a phenomenon results from the fact that the elements are
reduced to a very small size, and still provide a very accurate determination of the displacement
field, without much noise. Such a success encourages the user to decrease the size of the elements
to very small values. By doing so, the determination of the displacement is much more prone
36
200 400 600
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100
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2
1.5
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Ux (pixel)
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8000.04
0.03
0.02
0.01
0
0.01
0.02
0.03
0.04
εxx
y (pixel)
x (
pix
el)
200 400 600
-d-
800 1000
100
200
300
400
500
600
700
800
Figure 27: Map of Ux displacement, estimated for a element size ` = 12 pixels: (a) for the
original image, (b) for an artificially translated image of 0.5 pixel. Maps of the corresponding
normal strain component εxx: (c) for the original image, (d) for an artificially translated image
of 0.5 pixel. The physical size of one pixel is equal to 25 µm.
to slight sub-pixel shifts (Fig. 10), here characterized as an attraction toward integer values,
which appear as very significant upon differentiation (in the computation of the strain). This
phenomenon should be identified before any further interpretation of the strain map to ensure
its validity.
4.4 Error maps
As mentioned earlier, a very important output of the displacement measurement obtained from
a minimization procedure is that the optimization functional provides not only a global quality
factor of the determined field, but more importantly a spatial map of residuals, so that one
may appreciate a specific problem that may be spatially localized.
Figure 28 shows the different error maps obtained for different element sizes. This error
is the remanent difference in gray levels that is still unexplained by the estimated displacement
field. The first observation is that the error level does not vary very much with the element
37
size. This is consistent with the fact that the displacement field is quite comparable for different
element sizes. However, there is a slight increase in the error as the element size decreases. This
is explained by the fact that the performance of the correlation algorithm degrades as the spatial
resolution improves. This observation is in good agreement with the results to be expected from
the analysis of Fig. 10.
Figure 28: Map of the residual error η for different element sizes: (a) ` = 16, (b) ` = 12, (c)
` = 10 (d) ` = 8, (e) ` = 6 and (f) ` = 4 pixels.
5 Applications of CORRELIQ4
In the following, different references are given in which the software was used. The abstract is
given for the reader to select the relevant ones. The name of the .pdf file is also given.
38
5.1 Reference paper
G. Besnard, F. Hild and S. Roux, “Finite-element” displacement fields analysis from digital
Let us consider two neighboring points P and Q of a body Ω in its reference state, and a
corresponding frame R0. If a rigid body motion is applied, it corresponds to an isometry,
namely the distance between the points P ′ and Q′ after the transformation characterized by a
tensor A
P 0Q0 = A PQ (47)
is identical to the initial one
‖P 0Q0‖2 = PQ At A PQ = ‖PQ‖2, (48)
so that
AtA = I2 (49)
53
where I2 is the second order unit tensor. The first order derivative of the vector P 0Q0 is
expressed asdP 0Q0
dt= V (Q′/R0)− V (P ′/R0) (50)
or equivalentlydP 0Q0
dt= A At P 0Q0 (51)
where the tensor AAt is antisymmetric. As shown above, an antisymmetric tensor acting on
a vector can also be written as the cross product of a vector, here the rotation vector ω of Ω
with respect to R0, with P 0Q0
dP 0Q0
dt= ω(Ω/R0)× P 0Q0. (52)
Consequently, the vector field associated to a rigid body kinematics is such that
V (Q′/R0) = V (P ′/R0) + ω(Ω/R0)× P 0Q0 (53)
The motion of a body Ω is described by the function x0 = x0(x, t) giving the position
x0 at a time t of a particle that occupied the position x prior to a deformation. At a fixed
time, this function defines the deformation of the body between its reference configuration Cand the current one C ′. In Solid Mechanics, one chooses as reference configuration, the initial
configuration C and the current configuration in the same frame. It is however preferable,
to avoid any confusion, to distinguish these two frames, and in particular the Lagrangian
coordinates x for C and the Eulerian coordinates x′ for C ′. For the sake of simplicity, the two
systems are chosen orthogonal and normalized. As performed above, it is common to introduce
the displacement vector u(x, t)
x′(x, t) = x + u(x, t) (54)
Contrary to infinitesimal motions, no particular hypotheses are made concerning u. The func-
tion x′(x, t) defines the global motion of the body. Locally, i.e., for two points P and Q
separated by dx, one uses the deformation gradient tensor F relating an infinitesimal vector
dx in the reference configuration (C) to dx′ in the deformed configuration (C ′)
dx′ = F dx (55)
so that the tensor F can be related to the displacement gradient ∇u by
F = I2 + ∇u. (56)
The deformation gradient tensor defines the local motion. To measure a strain, i.e., a shape
change, it is necessary to eliminate the rotation. To characterize shape changes, one needs to
54
determine length variations as well as angle variations. In both cases, the information is given
by scalar products. Let us form the scalar product of dx′ with itself
dx′.dx′ = dx.F t F .dx (57)
where C = F t F denotes the right Cauchy-Green strain tensor. The Lagrangian strain measures
are formed by using C. It is possible to define various strain measures. The use of one rather
than another is a matter of choice or convenience. The only requirements are:
• they vanish for any rigid body motion (i.e., C = I2 for any rigid body motion),
• usually, they are positive for an expansion and negative for a contraction.
For Lagrangian measures, they can be expressed by using the strain tensors Em [32]
Em =
12m
(Cm − 1) when m 6= 0
12ln (C) when m → 0+
(58)
When m = 1, the Green-Lagrange strain tensor is obtained (see Section 7.2), m = 1/2 is the
Cauchy-Biot (or nominal) strain tensor and yields ∆L/L0 for a uniaxial elongation, where L0
is a reference (or gauge) length and ∆L the length variation. The case m → 0+ corresponds to
the logarithmic (or Hencky) strain tensor. The latter is the only additive strain measure. By
definition, all these strains are equal to zero for a rigid translation (i.e., F = I2 and C = I2)
or a rigid rotation (i.e., F = R and C = I2), where R is an orthogonal tensor. When the
amplitude of the body motion is small as well as the strain gradient, all measures converge
towards the infinitesimal strain tensor ε defined by