ANALYSIS OF DSPI FRINGE PATTERNS ISSUED FROM TRANSITORY
MECHANICAL LOADINGS
V. Valle, E. Robin and F. Brémand
Laboratoire de Mécanique des Solides, UMR 6610
Université de Poitiers
Poitiers, FRANCE
ABSTRACT
The aim of this paper concerns the phase demodulation of a fringes pattern in dynamic regime. During a dynamic loading, we
can not use a phase shifting technique because the mechanical parameters involve according to the time. This problem can be
avoided by the development of algorithms which can extract the mechanical information from only one fringes pattern. In this
way, we present two algorithms, the MPC [1] (Modulated Phase Correlation) and the pMPC [2] (polynomial Modulated Phase
Correlation). These algorithms allow us to extract the phase from a single fringes pattern obtained, for example, by shadow
moiré, photoelasticity or interferometry. These processes are based on the use of the virtual fringes pattern which locally
approaches the real fringes morphology. The similarity degree between real and virtual fringes pattern is estimated by digital
correlation technique. When the best similarity is obtained, we suppose that the virtual phase function is very near of the real
phase function. One shows with examples, the low sensivity to the noise of these techniques. So, we propose to use the MPC
or the pMPC algorithms in order to extract the relief in dynamic from a single interferogram obtained by digital speckle pattern
interferometry. In this paper we present experimental tests of impact loading on plates. The frame rate is adapted to the
dynamic conditions and varies from 6000 to 15000 frames per second. The figure 1 shows one of the fringes patterns recorded
in dynamic and the result of wrapped phase demodulated with one of our algorithms.
Introduction
The optical methods based on the fringes analysis are employed in experimental mechanics in purpose to obtain mechanical
information such as out-of plane displacements [1][2], principal stresses [3][4]…These techniques are usually used in statics.
Indeed, the demodulation process is performed by phase shifting methods [5][6] using several fringes patterns. In dynamics,
the kinematical parameters evolve during the test; this is why these demodulation techniques are not easily usable. An
alternative way can be found in the development of a demodulation algorithm performing the analysis of a single fringe pattern.
Several methods exist such as the Phase Locked Loop [7], or the Regularized Phase Tracking [8]. In this paper, we present
two new demodulation methods: the Modulated Phase Correlation (MPC) [10] and the polynomial Modulated Phase
Correlation (pMPC) [11]. An example of application is shown and concerns the analysis of DSPI fringes pattern obtained in
dynamics.
General principle
The basic principle of our methods consists in the construction of a virtual function which has locally the best degree of
similitude with the analyzed fringe pattern. The degree of similitude is evaluated with the help of a correlation technique [9].
The Digital Image Correlation method is widely employed in order to measure the in-plan displacements fields. It consists to
estimate the best degree of similitude between two images before and after loading. The degree of similitude is the highest
when the correlation function is minimized. Let us consider a periodic signal to be analyzed and a virtual periodic signal. If we
suppose that the virtual signal is only defined and initially centred in a zone of interest, the displacement obtained by
correlation can be considered as a phase shifting. The fringe pattern demodulation can be performed by re-iterating the
process along X coordinate and Y coordinate. However, locally on the zone of interest, the real and the virtual fringe patterns
must have the same morphology. This is why an evolving virtual fringe pattern should be used.
Demodulation processes: MPC and pMPC
The fringe pattern is usually simulated from equation 1:
I ( x, y ) = A( x, y ) cos[φ ( x, y )] + B ( x, y ) ,
(1)
where I(x,y), A(x, y), φ(x,y) and B(x, y) represent respectively at the point of coordinates (x, y), the grey level, the
modulation amplitude, the phase function and the illumination background. Taking into account of its general character, the
relationship (1) can not be directly used. This is why we suppose that it is possible to locally approach the fringe pattern
morphology (fig. 1(a)) with the help of virtual parallel fringe patterns (fig. 1(b)).
On the zone of interest, the determination of the adequate virtual fringe pattern is performed by a correlation
function minimization. This point constitutes the basic principle of both methods: MPC and pMPC. Indeed, when the best
degree of similitude between the virtual and the real fringes pattern is found, we suppose that the virtual phase function is
close to the wanted phase function. The use of parallels virtual fringes involves an ambiguity concerning the fringes
orientations determination. As figure 1(c) shows it, two virtual fringe patterns (mod π) can approach the morphology of the
studied fringe pattern.
Figure 1. Approach of the studied fringe pattern: analyzed fringe pattern (a), approach by a virtual parallel fringe pattern (b),
(c) orientation ambiguity with a virtual parallel fringe pattern.
Consequently the virtual fringe pattern takes an orientation modulus (π) instead of modulus (2π). In this condition, we prefer
employed the term inclination instead of orientation. In this way the demodulation process by MPC and pMPC is divided in two
steps (fig. 2). The first concerns the determination of the unwrapped demodulated phase, which is then not oriented (mod π).
From the inclination field, the second step deals with the correctly oriented unwrapped demodulated phase determination
oriented (mod 2π). It is performed with the help of a classical unwrapping algorithm which seeks the π discontinuities [12].
0
(a)
2π
-π/2
(b)
π/2
0
2π 0
(c)
2π
Figure 2. Example of phase demodulation by pMPC : (a) fringes pattern analyzed, (b) first step: obtaining of unwrapped phase
without orientation and its inclination field, (c) second step: obtaining of unwrapped phase with orientation from the unwrapping
of the inclination field.
The difference between the MPC and pMPC methods concerns the first step: the correlation function is not the same. The
direct consequence concerns the minimization processes which can be associated to an optimisation technique with (pMPC)
and not with (MPC). In these conditions, the pMPC algorithm is faster than the MPC algorithm, but in other hand, the MPC
algorithm is more accurate than the pMPC method. The next sections present the used correlation function and explain the
minimization processes.
Modulated Phase Correlation (MPC)
The virtual fringe pattern Fi used by the MPC method can be mathematically written in the zone of interest by:
 2π

2π
Fi (ξ i , γ i ) = Ai . cos
cos(α i )(xi + ξ i ) +
sin(α i )( yi + γ i ) + ϕ i  + Bi ,
pi
 pi

(2)
where Ai and Bi represent the modulation amplitude and the background illumination; pi represents the pitch and αi the
inclination of the fringes; φi represents the local phase at the point of coordinates (xi, yi). Fi generates parallel and directional
fringe pattern as shown on figure 1(b).
The best degree of similitude between the virtual and the real fringes pattern is found when the variables Ai, Bi, pi,
αi and φi minimize the following correlation function:
ψ i ( Ai , Bi , α i , pi , ϕ i ) = ∫∫ (Fi (ξ i , γ i ) − I (xi + ξ i , yi + γ i )) dξ i dγ i ,
2
(3)
Li
where I(xi+ξI, yi+γi) represents the grey level value of the pixel at the coordinate (ξi, γi) in the zone of interest centred at the
coordinate (xi, yi); Li is the zone of interest (33 by 33 pixels). As the correlation function is not unimodal, one can not employ
an optimisation technique. In these conditions we propose to minimize the relationship (3) by scanning all possibilities of
quintuple (Ai, Bi, pi, αI, φi) and we keep the solution which minimizes ψi. Then we use an incremental law for each variable, in
practice 10 increments are used. The incremental laws can not allow us to scan all possibilities. So, in order to improve the
results, an interpolation of ψi around of the solution for the variables αI and φi is performed.
polynomial Modulated Phase Correlation (pMPC)
In order to accelerate the minimization process, the pMPC method employs an optimisation technique. In these conditions, the
relationship (2) can not be used because of the cosine function. Then in the pPMC correlation function, we locally approach
the cosine morphology with a specific polynomial form.
The pMPC virtual fringes pattern, on the zone of interest, is given by:
Pn ( X ) =
N
∑C
in
Xn
X = ( xi + ξ i ) cos(α i ) + ( y i + γ i ) sin(α i ) ,
with
(6)
n=0
where Cin with n∈{0…N} represent the polynomial form coefficients and α i is the inclination.
So the pMPC correlation function can be written by:
ψ i ( Ai , Bi , α i , pi , ϕ i ) = ∫∫ (Pn ( X ) − I (xi + ξ i , y i + γ i )) dξ i dγ i ,
2
(7)
Li
where Pn and I are respectively the virtual fringe pattern and the grey level value of the pixel at the coordinate (ξi, γi) in the
zone of interest centred at the coordinate (xi, yi). In practice we use a polynomial form, which has a degree equals to 4.
When ψi (Ai, Bi, pi, αI, φi) is minimized, one has only access to the inclination values since the demodulated phase
(φi) does not appear in equation (6). We determine the demodulated phase values from the following remarks.
Although the computation is performed on the zone of interest, only the analyzed pixel and its neighbourhood are
important. Indeed, one must approach, as well as possible, the real fringes morphology at the centre of the zone of interest.
This is why we propose to use a Mac Laurin expansion of the equation (2) as:
π

n
Ai (2πX ) cos ϕ i + n 
 2π

2 A sin (ϕ i )π
2

Ai cos
X + ϕ i  + Bi = Bi + Ai cos(ϕ i ) − i
X + ... +
+ O ( X n +1 )
n
p
p
p
n
!
i
 i

i
(8)
In these conditions, we have two polynomial forms: the virtual fringes pattern numerically obtained by the optimization and the
Mac Laurin expansion obtained analytically. So, in this area we propose to perform identification between the specific
polynomial form and the Mac Laurin expansion.
The analytic resolution leads to the demodulated phase φi at the pixel (xi, yi), by:
2C i21 (C i22 − C i1 )
3C C − C i 2 C i1
, Bi = i 0 i 3
,
6C i 3
3C i 3
 πC i1 
2π 2 C i1

,
pi = −
ϕ i = arctan
3C i 3
 pi Ci 2 
Ai = −
(9)
Application in the case of DSPI fringes
Usually in DSPI method, we can extract the kinematical quantities directly from speckle pattern, using phase shifting
technique. This principle implies to capture a minimum of three images of the same loading condition. This is achieved by a
synchronisation between the image capture and the loading cycle. With this procedure, it’s very difficult to study transitory
dynamic phenomena. Here, we propose to store the speckle pattern at each time of the transient loading, and then to analyse
the fringes issued from the subtraction between the first speckle image and the image at each time of the loading (Figure 3-a)
3-b)). We can note on Figure 3-C) that the fringes patterns created by this principle are very noisy. We propose to test the
efficiency of the DSPI fringe demodulation using the two presented algorithms, and to compare the results (Figure 3-d) with
those obtained by adding a low pass filter pre-processing.
a)
b)
c)
d)
Figure 3. a) DSPI Image at t=0µs, b) at t=66µs, c) Fringe pattern calculated, d) demodulated by MPC method
Experimental results: Fringes DSPI filtering
We can see on the figure 4-a), the image of an experimental DSPI fringe pattern (Zf=1x1). The other images are respectively
obtained by filtering the first image with a low-pass filter. These low-pass filters, noted by Zf=nxn, correspond to average filters
that we can expressed by:
Zf = nxn
:
n −1
2
n−1
i=−
2
I f ( x, y ) = ∑
∑
n −1
2
n −1
j =−
2
I ( x + i, y + j )
(Zf=1x1 corresponds to the case of no filtering, Zf=3x3 relates to the case of each point of the filtered image are calculated by
the average of 9 neighbourhood points, and so on…)
On figures 4b) to 4e), we can see the images corresponding to different values of filtering. From theses images, one has
applied our algorithm and extracted the phase for each case. We have calculated the RMS value between each calculated
phase and the phase issued from the largest (Zf=11x11) low-pass filter. This procedure can give an estimation of the preprocessing filter influence on the demodulation accuracy. On Figure 4-g), we can see the result of these comparisons.
0.4
0.35
0.3
RM S (rad)
0.25
0.2
0.15
0.1
0.05
0
Zf=1x1
Zf=3x3
Zf=5x5
Zf=7x7
Zf=9x9
g)
Figure 4. a) DSPI Fringe pattern without filtering, b) c), d), e) and f) with different widths of low-pass filtering
g) Accuracies estimation of demodulation process by MPC method.
We can note that there is a sensitive improvement of the accuracy by using a low-pass filter, but it is not necessary to employ
a larger one. The accuracy of the methods can be estimate to 0.15 rad (1/40 wave length). However, it is important to note that
the two algorithms work properly even if there is no pre-processing filtering.
Experimental results: Fringes DSPI demodulation
In order to present an example of demodulation by MPC and pMPC, we propose to perform a relief measuring during an
impact loading.
The first test concerns the dynamic behaviour of a textile. The optical method used is the digital speckle pattern interferometry.
We dispose of a high speed camera (PHOTRON Ultima APX) having a frame rate equal to 6000 frames per second and a
-1
1,5 W laser (Argon) with a wave lenth λ = 514.5 nm. The sensitivity is equal to 257 nm.rad between two consecutive fringes.
The recorded fringe patterns are filtered with the help a low pass filter with a width of 3x3 (i.e:Zf=3x3). The fringe patterns and
the corresponding phase images are presented on figures 5a). The demodulation method employed is the pMPC.
Figure 5: a) Results of relief extraction by pMPC method of a textile plate under impact, b) Experimental setup
We have plotted the wave propagation in function of time. Figure 6 represents the out-of plane displacement evolution versus
time and a radial position, centred at the loading point. On this figure, we show the transient effect of surface propagation.
1.5µm
220
210
Radial Position (pixel)
200
190
180
170
160
150
140
130
120
0µm
0
1
2
3
4
5
6
7
8
9
10
11
12
Time (Each moment at 1.66e-4 s)
Figure 6 : Example of temporal out-of plane displacement evolution
The second test concerns the dynamic behaviour of an orthotropic material: the corrugated paperboard. The frame rate of the
camera is equal to 15000 frames per second. The recorded fringe patterns are filtered with the help a low pass filter with a
width of 3x3 (i.e:Zf=3x3). The fringe patterns and the corresponding phase images are presented on figures 7. The
demodulation method employed is the MPC.
Figure 7: Result of relief extraction by MPC method of a corrugated board under impact
On this figure, we noted the orientation of orthotropic axis corresponding exactly to the ovalisation of the fringe patterns.
The two methods are particularly few sensitive to noise (Figure 5-a) and Figure 7) and the results obtained by MPC and pMPC
are very similar on a visual point of view.
Conclusions and prospects
We presented two demodulation processes from a single fringe pattern based on correlation technique. The basic principle of
these algorithms (MPC and pMPC) consists in to search locally the best degree of similitude between the studied fringe
pattern and the virtual fringe pattern. When the best degree is obtained, we suppose that the virtual phase is locally very close
to the real phase. As the examples show it, these algorithms allow us the study of fringe pattern obtained experimentally and in
particular the fringes issued from DSPI with only one image. The direct consequence is the new possibility that the methods
based on the fringes analysis can be used in dynamics. Moreover, the MPC and the pMPC algorithms can be employed in
static when the phase shifting method can not be performed.
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