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Measurement of in-plane displacements using the phase
singularities generated by directional wavelet transforms of a
speckle pattern image
Ana Laura Vadnjal,1,∗Pablo Etchepareborda,1Alejandro Federico,1and Guillermo H. Kaufmann2
1Electr´onica e Inform´atica, Instituto Nacional de Tecnolog´ıa Industrial,
P.O. Box B1650WAB, B1650KNA San Mart´ın, Argentina.
2Instituto de F´ısica Rosario (CONICET-UNR) and Centro Internacional Franco
Argentino de Ciencias de la Informaci´on y de Sistemas (CONICET-UNR-AMU),
Ocampo y Esmeralda, S2000EZP Rosario, Argentina.
compiled: May 31, 2017
We present a method to determine micro and nano in-plane displacements based on the phase singu-
larities generated by application of directional wavelet transforms to the speckle pattern image. The
spatial distribution of the obtained phase singularities of the transformed image configures a network
that is characterized by two quasi-orthogonal directions. The displacement value is determined by
identifying the intersection points of the network before and after the displacement of the tested ob-
ject. The performance of the system is evaluated using simulated speckle patterns and experimental
data. A comparison of the presented approach with the Optical Vortex Metrology and Digital Image
Correlation methods is carried out, and the performances and noise robustness are also determined.
The advantages and limitations produced by each method is revised and discussed.
OCIS codes: 030.6140, 100.2000, 120.6150, 350.5030.
1. Introduction
Digital Image Correlation (DIC) is a well known
method that has been used as an effective non-
contact optical metrology technique for full-field
displacement measurements and presents a low per-
formance when spatial resolution is increased [1–3].
During the last two decades, the measurement of
in-plane displacements using digital speckle pattern
images (abbreviated as Speckle Images, SIs) was
mainly based on the cross-correlation function of
the intensity of the speckle distributions and with-
out considering the use of the phase information.
Recently, the introduction of a complex signal asso-
ciated with SI has established a new point of view.
The complex signal is obtained by vortex filtering
the speckle pattern intensity that generates phase
singularities invariant to in-plane translations and
rotations [4]. Therefore, this pseudo-phase infor-
mation was used as a turning point with respect
∗Corresponding author: avadnjal@inti.gov.ar
to the usual practice of speckle digital correlation
and Optical Vortex Metrology (OVM) technique
was established. Remarkable applications that use
the pseudo-phase information are fluid analysis [5],
object tracking and image matching [4], measure-
ments of in-plane micro displacements [6, 7] and
nano displacements [8].
In the DIC and OVM techniques, two SIs corre-
sponding to the first and second deformation states
of the object are acquired. In the DIC method,
the cross-correlation function between both images
is carried out to determine the displacement of the
local maximum. To obtain sub-pixel registration
accuracy, an additional algorithm is usually intro-
duced [9–11]. In the OVM technique, the displace-
ment is measured by quantifying the displacement
of the phase singularities that are detected by vor-
tex filtering of the SIs. Each singularity is identified
from the initial image with its counterpart in the
post-displacement image. This identification can
be done by searching for the nearest singularity be-
tween images. However, the distance between sin-
2
gularities within the same image is not predictable
and subsequently the matching might be wrong.
For this reason, phase singularities are character-
ized using the parameters of the core structure of
optical vortices such as vorticity, zero crossing an-
gle, eccentricity and topological charge [12]. In
most practical cases, it is important to know that
the identification of phase singularities using infor-
mation from both states of the object is not al-
ways possible. The set of singularities obtained
form clusters with parameter values that are fre-
quently modified by the displacement of the tested
object. Even, new phase singularities may be cre-
ated or some of them may disappear. Consequently,
to obtain an effective identification it is necessary
the introduction of a merit function and also the in-
tervention of an external operator who defines the
threshold values of the core structure parameters
of this function [7]. This way of matching uses one
threshold value associated with each core structure
parameter. Moreover, it needs the introduction of
an additional parameter that restricts the search-
ing region of the corresponding pair and also the
scale factor. Therefore, a total of 6 parameters are
simultaneously required. Typical variations in the
experimental conditions are compensated by means
of the resetting of the threshold values, which is a
time consuming task. It is important to note that
in Ref. [5] a function of the threshold values was
introduced by using the Poincar´e sphere, which is a
3D vector representation of the core structure pa-
rameters. As consequence, this procedure alleviates
the task of the operator.
The presented approach uses the phase singu-
larities obtained by application of two directional
continuous wavelet transforms (DCW T s) to each
recorded SI, which correspond to the initial and
post-displacement states of the object under test.
This method is based on the property of covariance
of the continuous wavelet transform, which implies
that the wavelet coefficients are covariant under
translations, dilations and rotations [13]. The dis-
tance between phase singularities obtained by the
introduced wavelet transform method is adjustable,
so that the matching technique by means of the
election of the nearest singularity between images
is acceptable. Moreover, regarding to the variations
of the experimental conditions, it will be shown that
the phase singularities generated by application of
two directional wavelet transforms are less sensitive
to variations of the experimental conditions than
those produced by vortex filtering. This feature
generates a more robust framework and a less labo-
rious intervention of the operator when micro and
nano in-plane displacement need to be determined.
The wavelet function is chosen such that the gen-
erated phase singularities are spatially arranged by
shaping a mesh composed by singularity lines with
quasi-orthogonal directions. The intersections of
the phase singularity lines of the wavelet transform,
named Phase Singularities Crossing Points (PSCP),
can be easily tracked in both speckle images. The
presented method only needs two parameters to
be set up by the operator: the scale factor and
wave vector parameter. The scale factor is the well
known dilation parameter of the DC W T , which de-
termines the spatial resolution of the method. The
wave vector parameter is the Morlet frequency of
the DC W T and determines the network pitch size
with its corresponding number of PSCP.
In micro and nano in-plane displacement mea-
surements using SIs, advantages and limitations in-
troduced by the use of the pseudo-phase informa-
tion are not clear when this approach is compared
with the traditional cross-correlation method. In
our knowledge, the DIC and OVM methods have
not been compared. An additional contribution of
this work is to provide a comparison between the
pseudo-phase information methods (OVM, PSCP)
and the traditional DIC approach. Thus, the read-
ers can get a suitable point of view about the use of
the PSCP system in micro and nano in-plane dis-
placement measurements using Sis, respect to the
application of the others. We also revise and discuss
the advantages and limitations produced by these
methods.
2. Theoretical description
In order to avoid confusions between the different
meanings of the involved phase singularities associ-
ated to each of the methods to be evaluated, i.e. the
phase singularities that correspond to the real opti-
cal vortices exhibited in the SIs, the phase singular-
ities obtained by Laguerre-Gauss filtering process
in the application of the OVM technique and the
wavelet phase singularities produced by using di-
rectional wavelet transform filtering, is convenient
to begin this Section with a preliminar discussion.
The spatial position of the phase singularity of an
optical vortex corresponding to a given SI can be
determined by noting that the real and imaginary
parts of the associated electric field are simultane-
ously null [12]. By using this feature, a simulated SI
is shown in Fig. 1(a) with its corresponding optical
vortex positions highlighted by white circles. The
same SI is shown in Fig. 1(b) with the phase sin-
gularities generated by applying a Laguerre-Gauss
3
filter and its corresponding spatial positions are il-
lustrated with white squares. Figure 1(c) shows the
spatial positions of the phase singularities obtained
in Figs. 1(a) and 1(b) simultaneously. It is seen
that the spatial positions of the phase singularities
detected by the OVM technique do not correspond
with those of the real optical vortices. In fact, the
amount of phase singularities obtained by Laguerre-
Gauss filtering is a function of the filter scale factor,
while the amount of the phase singularities corre-
sponding to the real optical vortices for a particular
SI is fixed and in addition are essentially character-
ized by null values of the intensity.
(c)
Fig. 1. (Color online) (a) Simulated intensity image of
a speckle pattern with its corresponding optical vortices
marked in white solid circles for real (light cyan) and
imaginary (dark red) contours of level zero of the asso-
ciated electric field; (b) the same speckle image as in
(a) with phase singularities (white squares) detected by
the OVM method with real (light cyan) and imaginary
(dark red) contours of level zero corresponding to the
Laguerre-Gauss filtering values; (c) spatial positions of
the optical vortices (black solid circles) and OVM phase
singularities (red squares) simultaneously shown.
The PSCP method proposes the use of phase sin-
gularities generated by applying directional wavelet
transform to the SI. These wavelet transform phase
singularities are sensitive to the displacement of the
object. Figures 2(a) and 2(d) illustrate two SIs
corresponding to the initial and post-displacement
states of the object test respectively, with a rigid-
body in-plane displacement of 10 pixels to the right.
Figures 2(b) and 2(e) show the phase of the wavelet
coefficients in the range of (−π, π] obtained using
the directional wavelet transform of the speckle pat-
terns shown in Figs. 2(a) and 2(d), respectively.
The spatial positions of the phase singularities cor-
responding to Fig. 2(b) are shown in Fig. 2(c).
Figure 2(f) depicts simultaneously the spatial po-
sitions of the phase singularities corresponding to
the initial and post-displacement states represented
by Figs. 2(b) and 2(e). Figure 2 clearly illus-
trates the mechanism involved in an in-plane trans-
lation by direct inspection of the positions of the
wavelet transform phase singularities before (light
cyan) and after (dark red) the SI translation. Below
we present a theoretical description of the PSCP
method. For that purpose, we begin by introduc-
ing the directional continuous wavelet transform.
The DC W T is characterized by a rotation pa-
rameter θ∈(0,2π], in addition to the usual trans-
lation parameter b= (bx, by)∈R2and the dilation
parameter or scale factor a∈R>0. The DC W T of
a speckle image I(r), r= (x, y)∈R2, with respect
to a given mother wavelet ψ(r), is expressed in the
plane R2or the Fourier space by
DCWT [I(r)] (a, b, θ)
=1
a2ZR2
d2rψ∗a−1R−θ(r−b)I(r)
=ZR2
d2keik·bˆ
ψ∗[a R−θ(k)] ˆ
I(k),(1)
where R−θdenotes the usual rotation matrix in
the plane, (ˆ) and (∗) stand for the Fourier trans-
form and complex conjugation, respectively. The
DC W T is covariant under translations (1), dila-
tions (2) and rotations (3). This means that the
correspondence DC W Tψ:I(r)7→ DC W T (b, a, θ)
implies the following ones [13]:
1. I(r−b0)7→ DC W T (b−b0, a, θ),
2. a−1
0I(a−1
0r)7→ DC W T (a−1
0b, a−1
0a, θ),
3. I[rθ0(r)] 7→ DC W T [r−θ0(b), a, θ −θ0].
In this work, the proposed mother wavelet ψ(r) is
defined by
ψ(r) = 2− |r/σ −iσk0|2
σ2!exp (ik0·r)···
···exp −|r|2/2σ2,(2)
ˆ
ψ(k) = σ2|k|2exp −σ2
2|k−k0|2,(3)
4
(a)
(b)
(d)
(e)
Fig. 2. (Color online) Illustration of the mechanism involved in the movement of the positions of the DC W T phase
singularities for a given in-plane translation. (a) Simulated SI; (b) phase of the DC W T of (a); (c) spatial positions
of the wavelet phase singularities from (b); (d) SI shown in (a) displaced 10 pixels to the right; (e) phase of the
DC W T of (d); (f ) spatial positions of the wavelet phase singularities obtained from (b) (light cyan) and (e) (dark
red).
where k0=k0(cos θ0,sin θ0) is the wave vector
and modulates the mother wavelet. σis the stan-
dard deviation of the Gaussian function and de-
termines the local wavelet width. As a starting
point, we define θ0= 0 and σ= 1. The reader
should note that the selected wavelet is the nega-
tive Laplacian of the 2D Morlet wavelet, which ful-
fills the admissibility condition ˆ
ψ(0) = 0 [13]. The
DC W T can be seen as a band-pass filter with circu-
lar shape, which is characterized by the frequency
center k0and radius r=√2/(2σ) (see Fig. 3).
The filter is adjusted by varying the scale factor a
and the rotation angle θfollowing the relationship
ψ∗a−1R−θ(r−b)for a given position bof the
SI. When the scale factor varies, the cutoff frequen-
cies of the filter will change and the passband will
shrink or expand. Equations (2) and (3) show that
the performance of the DC W T is a function of both
k0and σ. Typically, a larger k0obtains better per-
formance for high frequency details in the SI. σde-
termines the anti-noise performance. For example,
a small σimproves large local variations, while a
large value tends to smooth out the local variations
and makes the DC W T more robust to noise and
defects. Note that k0and σare coupled together,
see Eq. (3). The selection of either k0or σmust
be based on fixing another of them. Thus, a prac-
tical criterion will be discussed later. For the inter-
ested reader, the DC W T can be computed in Mat-
lab using the wavelet toolbox YAWTB offered in
http://sites.uclouvain.be/ispgroup/yawtb/.
r
k0
00
kx
(a)
ky
(b)
(c) (d)
Fig. 3. (Color online) (a) Amplitude of ψ(r); (b) Phase
of ψ(r); (c) ˆ
ψ(k); (d) Scheme of the parameters in ˆ
ψ(k).
The PSCP method applies two successive wavelet
transforms to each SI produced by the tested ob-
ject using the same fixed scale factor, and with ro-
tation parameters of θ= 0 and θ=π/2. There-
fore, for each rotation parameter, two complex im-
ages are obtained corresponding to the initial and
post-displacement states. Figures 4(a) and 4(b) il-
lustrate the phase of the wavelet coefficients ob-
5
tained in the range (−π, π], when the two succes-
sive DC W T were applied to the same experimental
SI with θ= 0 and θ=π/2, respectively. The dis-
placement is measured by tracking the PSCP de-
fined from the initial state and post-displacement
state of the object. The tracking procedure of the
PSCP approach can be summarized in the following
steps:
•A Canny edge detector is applied to the phase
images obtained in Figs. 4(a) and 4(b) to find
the spatial configuration of the phase singu-
larities. The two resulting images describe the
curves with quasi-parallel directions that are
shown in Figs. 4(c) and 4(d), respectively.
•A network with quasi-orthogonal directions
(see Fig. 4(e)) is obtained by overlapping Figs.
4(c) and 4(d).
•The matching of the PSCP pairs is done by
looking for the nearest crossing point between
images obtained from each state of the object.
•The displacement of the object is estimated
by measuring the distance between the PSCP
matched pairs.
In practice the election of the network size, which
varies as a function of k0, should take into account
the limitations of the matching procedure of the
nearest PSCP and the involved in-plane displace-
ment value. Quite good results are obtained by
selecting the network size larger than the double of
the measured displacement. It is noticed that for
rigid-body displacements, the scale factor acan be
fixed within a large range of values without signif-
icant changes in the performance of the method.
Therefore, k0is the main parameter of the PSCP
approach to be selected by the operator. The se-
lected rotation parameters, 0 and π/2, makes the
implementation of the algorithm easier. It must be
also noted that a sub-pixel interpolation of the SI
should be used to obtain sub-pixel accuracy. To
compare the obtained structures of the phase sin-
gularities, Fig. 4(f) shows the pseudo-phase singu-
larities obtained by applying the OVM technique
to the same analyzed SI. The displayed curves are
contour lines at level 0 of the real (light cyan) and
imaginary (dark red) parts of the vortex filtering of
the speckle image. It is important to highlight that
the phase singularities followed a clustered pattern
concentrated at left side of the image. Therefore,
the right side of the image does not seem to be well
represented. In addition, it is seen that the PSCP
method gives a more representative spatial config-
uration within the measurement region (compare
Fig. 4(e) with Fig. 4(f)).
3. Numerical results
In this Section, a comparison between the DIC,
OVM and PSCP methods is presented using simu-
lated SIs. For this numerical analysis, simulated SIs
of 512×512 pixels were generated using the method
reported in Ref. [14], with mean values of speckle
sizes of s= 1, s= 3 and s= 6 pixels. In OVM
and PSCP methods, the obtained SIs were interpo-
lated with a factor of int = 5. A translated SI was
obtained by shifting the original image 3 pixels to
the right direction and 2 pixels up. A rotational
movement was also simulated, with an angle of ro-
tation of 0.6◦and the center of rotation located in
the center of the image (x= 256, y = 256).
The DIC method was applied using the cross-
correlation function with the sub-pixel algorithm,
named iterative spatial domain cross-correlation al-
gorithm with spline interpolation (see Ref. [3]).
The sizes of the sliding windows were 32 ×32 and
64×64 pixels extracting the central part of the cor-
relation. The reference images were taken with a 5
pixel step between them.
The OVM method was applied by using the pro-
cedure presented in Ref. [7], with the searching
region restricted to a window of 10 ×10 pixels.
Threshold values ǫ={ǫ1, ǫ2, ǫ3}for eccentricity
(ǫ1), vorticity (ǫ2) and zero crossing angle (ǫ3) were
defined for each case, and there was no threshold
value for the topological charge.
In the OVM and PSCP methods, the in-plane
translation estimate (∆x, ∆y) was calculated as the
mode of the distances measured by all the pairs of
matched singularities. The measure of dispersion
was defined as the mean absolute deviation (MAD)
in each direction
MAD (X, Y ) =
1
nPn
i=1 |∆xi−mode(X)|,|∆yi−mode(Y)|,(4)
where nis the number of pairs and {X;Y}=
{∆x1,...,∆xn; ∆y1,...,∆yn}is the data set with
∆xiand ∆yibeing the horizontal and vertical dis-
tances measured for each detected pair i, respec-
tively. For the DIC method, the mean of the dis-
placements measured by each cross-correlation was
used as the displacement estimate, and the stan-
dard deviation as the measure of dispersion.
In the simulated rotational deformation, the cen-
ter of rotation (xc, yc) was estimated by applying
6
(a)
(b)
(c)
(d)
(e)
Fig. 4. (Color online) PSCP and OVM analysis of an experimental SI. (a) Phase of the coefficients of the directional
wavelet transform of the speckle pattern with θ= 0; (b) idem to (a) with θ=π/2; (c) result of the Canny edge
detector applied to image (a); (d) result of the Canny edge detector applied to image (b); (e) overlapping of (c) and
(d), which defines the PSCP; (f) phase singularities obtained by the OVM technique (solid circles).
the least-square fitting method
xc=Pib2
i(−Piaici)−(Piaibi) (−Pibici)
Pia2
iPib2
i−(Piaibi)2,
(5)
yc=Pia2
i(−Pibici)−(Piaibi) (−Piaici)
Pia2
iPib2
i−(Piaibi)2,
(6)
where ai= 2 (xi−x′
i), bi= 2 (yi−y′
i) and ci=
x′2
i+y′2
i−x2
i−y2
i, being xiand yithe coordi-
nates of a detected point in the initial image and x′
i
and y′
ithe same point coordinates in the final im-
age. The rotation angle was estimated by the mean
of the angle calculated by the law of cosines using
the triangles formed by the matched pairs and the
estimated center of rotation. For the rotational de-
formation, the comparison was made between OVM
and PSCP methods.
To introduce noise in the SIs and to simulate the
decorrelation effects in the processes, the field U
corresponding to the speckle beam in the final state
of the object was affected by an independent speckle
field U′according to the following expression
I=U(1 −αd[0,1] ) + αd[0,1]U′
2,(7)
where Iis the intensity of the SI, αis a real constant
in [0,1] and d[0,1] is a matrix composed by random
numbers uniformly distributed in the range [0,1].
Only the displaced SI was degraded before the DIC,
OVM and PSCP methods were applied.
Table 1 shows the performances of all tested
methods in the in-plane translation displacement
as function of the αparameter and the mean of the
speckle size s. The performances of the OVM and
PSCP methods were obtained using 1100 phase sin-
gularities approximately. The threshold parameters
for the OVM method were optimized to obtain the
best performance. In the PSCP method, a= 75
and k0= 6 (1/pixel) were used for all the tested
cases.
It is seen from Table 1 that the OVM method
estimates very well and with low dispersion when
it is applied to the cases without noise and decor-
relation (α= 0). However, the method is quite
sensitive when the mean of speckle size increases
and α6= 0. In the OVM and PSCP methods,
the dispersion increases with noise and decorrela-
tion. There is high agreement in displacement esti-
mation and stability for larger speckle sizes in the
application of the PSCP method. No readjusting
of the PSCP parameters was necessary during the
test. The DIC method works very well and does
not vary its performance with noise and decorrela-
tion, although the error in the estimation increases
with the mean of speckle size, specially if a smaller
sliding window is used.
7
Table 2 shows the performance of the methods in
the rotational movement. About 1300 phase singu-
larities were used to evaluate the performance. As
well as in the translation displacement, the param-
eters in OVM were optimized to obtain the best
result, and the PSCP parameters were a= 50 and
k0= 4 remaining unchanged for all the cases. As
it can be seen, the methods had very similar per-
formance through all the studied cases of the rota-
tional movement, showing a slightly decrease with
noise and decorrelation.
(a)
(b)
Fig. 5. 100x100 pixels region of detected pairs before
(solid circles) and after (empty circles) the introduction
of an in-plane translation of 3 pixels to the right direction
and 2 pixels up using: (a) the PSCP approach; (b) the
OVM method.
Figures 5(a) and 5(b) display the phase singular-
ities detected by the PSCP and the OVM methods,
respectively, in a given region corresponding to the
analyzed SIs of 100 ×100 pixels and mean of the
speckle size of s= 6 without noise and decorrela-
tion (α= 0). Figures 4(f) and 5(b) show that the
application of the OVM technique produces clus-
ters of phase singularities, which can be quite dis-
advantageous when sequences of localized in-plane
deformations of the tested object are measured. To
confirm this assumption, a locally noisy and decor-
related SI of 512 ×512 pixels was generated and a
localized deformation over a 40 ×40 pixels region
of the image was simulated. The spatial structure
of the phase singularities detected by the PSCP
method is shown in Fig. 6(a), before (dark red)
and after (light cyan) the object in-plane deforma-
tion. The pairs of the pseudo-phase singularities de-
tected by means of the OVM technique are shown
in Fig. 6(b). This last image displays the con-
tour lines at level 0 of the real and the imaginary
parts of the filtered SI, before (dark red) and after
(light cyan) the object deformation. Few singulari-
ties were obtained in the distorted region when the
OVM technique was used. Therefore, it is easier to
interpret the distortion field in Fig. 6(a) than Fig.
6(b). In the PSCP method, the size of the grid can
be selected by modifying the value of k0. There-
fore, a suitable spatial structure of the phase sin-
gularities within the distorted region can be easily
determined. This result presents an important ad-
vantage with respect to the application of the OVM
method. In practice, the scale factor ashould be
smaller than the size of the distorted region for lo-
calized deformations. Note that these localized de-
formation measurements are not possible by using
the DIC method.
Fig. 6. (Color online) Phase singularities determined be-
fore (solid circles) and after (empty circles) the localized
deformation over 40 ×40 pixels by using: PSCP (a) and
OVM (b) approaches.
4. Experimental results
Below, three experimental cases are presented and
discussed. In the first case (A), two speckle im-
ages corresponding to a given in-plane translation
of an object were acquired with an optical setup
(Fig. 7) based on a linearly polarized laser beam
of λ= 682.5nm, which illuminated the sample af-
ter passing through a Leica microscope objective
of 100×magnification and 0.85 numerical aper-
ture. The backscattered beam, that was reflected
backwards by the microscope objective, was redi-
rected by a polarization beam splitter (BS) towards
a Foculus FO442B CCD camera with a pixel size of
6.45µm. Taking into account the pixel separation
of the CCD sensor, the unit pixel for the in-plane
displacement corresponded to an object displace-
ment of 97nm. The second test (B) was done with
the same optical setup as (A). The third test (C)
was also performed with the same optical setup but
using a Leica microscope objective of 2.5×magnifi-
cation and 0.7 numerical aperture. In this last case,
the unit pixel of the in-plane displacement corre-
sponded to an object displacement of 3.9µm.
Table 3 shows the results obtained for three rigid-
body in-plane displacements of an object using the
DIC, OVM and PSCP methods. The mode was
chosen as the estimate of the displacement and the
8
Table 1. Performance test in translation movement. The OVM parameters used were (a)ǫ={0.01,0.08,0.08}and
a= 2.8; bǫ={0.2,0.3,0.3}and a= 2.6; (c)ǫ={0.2,0.3,0.3}and a= 2.5; dǫ={0.01,0.08,0.08}and a= 2.7;
(e)ǫ={0.2,0.3,0.3}and a= 2.4; fǫ={0.3,0.4,0.4}and a= 2.2. The PSCP parameters used were a= 75 and
k0= 6 (1/pixel).
DIC(64 ×64) DI C (32 ×32) O V M P SC P
s [pixels] α∆x[pixels] ∆y[pixels] ∆x[pixels] ∆y[pixels] ∆x[pixels] ∆y[pixels] ∆x[pixels] ∆y[pixels]
0 3.00 ±0.00 2.00 ±0.00 3.00 ±0.00 2.00 ±0.00 a3.00 ±0.02 2.00 ±0.02 3.00 ±0.25 2.00 ±0.26
1 0.01 3.00 ±0.00 2.00 ±0.00 3.00 ±0.00 2.00 ±0.00 b3.02 ±0.19 2.00 ±0.18 3.00 ±0.38 2.00 ±0.32
0.02 3.00 ±0.00 2.00 ±0.00 3.00 ±0.00 2.00 ±0.00 c2.99 ±0.24 2.03 ±0.22 3.00 ±0.42 2.00 ±0.37
0 3.00 ±0.02 2.00 ±0.02 3.00 ±0.05 2.00 ±0.05 a3.00 ±0.03 2.00 ±0.03 3.00 ±0.13 2.00 ±0.12
3 0.05 3.00 ±0.02 2.00 ±0.02 3.00 ±0.05 2.00 ±0.05 b2.99 ±0.24 2.00 ±0.22 3.00 ±0.20 2.00 ±0.20
0.1 3.00 ±0.02 2.00 ±0.02 3.00 ±0.05 2.00 ±0.05 c2.99 ±0.33 1.96 ±0.30 3.00 ±0.27 2.00 ±0.28
0 3.00 ±0.11 2.00 ±0.10 2.99 ±0.38 2.00 ±0.36 d3.00 ±0.03 2.00 ±0.03 3.00 ±0.08 2.00 ±0.08
6 0.1 3.00 ±0.11 2.00 ±0.10 2.99 ±0.40 1.99 ±0.38 e3.02 ±0.31 1.95 ±0.29 3.00 ±0.25 2.00 ±0.24
0.2 3.00 ±0.13 2.00 ±0.12 2.97 ±0.45 1.98 ±0.39 f3.05 ±0.50 2.02 ±0.48 3.00 ±0.37 2.00 ±0.42
Table 2. Performance test in rotation movement. The
OVM parameters used were (a)ǫ={0.01,0.08,0.08}and
a= 2.2; bǫ={0.2,0.3,0.3}and a= 2.4; (c)ǫ=
{0.2,0.3,0.3}and a= 2.3; dǫ={0.2,0.3,0.3}and
a= 2.2; (e)ǫ={0.01,0.08,0.08}and a= 2.1; fǫ=
{0.2,0.3,0.3}and a= 2.1; (g)ǫ={0.2,0.3,0.3}and
a= 1.8. The PSCP parameters used were a= 50 and
k0= 4 (1/pixel).
OV M P SC P
s [pixels] α φ [degrees] φ[degrees]
0a0.56 ±0.10 0.58 ±0.13
1 0.01 b0.59 ±0.13 0.60 ±0.14
0.02 c0.58 ±0.12 0.59 ±0.17
0a0.56 ±0.10 0.58 ±0.12
3 0.05 c0.58 ±0.11 0.59 ±0.13
0.1d0.59 ±0.14 0.60 ±0.14
0e0.58 ±0.09 0.57 ±0.12
6 0.1f0.59 ±0.13 0.59 ±0.13
0.2g0.60 ±0.16 0.62 ±0.18
MAD as a measure of the dispersion for the OVM
and PSCP techniques, and the mean and standard
deviation for the DIC method. The OVM and the
PSCP parameters were adjusted to obtain about
2700, 4165 and 1500 phase singularities for the cases
(A), (B) and (C), respectively. The images were
interpolated by a factor of int = 5, the searching
area was a window of 10×10 pixels and no threshold
value for the topological charges was used.
In the OVM technique, the parameters that were
used were ǫ={0.2,0.3,0.3}for (A) and (C), ǫ=
{0.1,0.1,0.1}for (B), with a= 1.4, a= 1.1 and a=
Table 3. Measurement of experimental in-plane displace-
ments.
Case Method ∆x[nm] ∆y[nm]
A DIC(64) 186 ±5 6 ±3
DIC(32) 173 ±12 4 ±6
OVM 194 ±28 4 ±26
PSCP 197 ±27 0 ±20
B DIC(64) 291 ±13 89 ±12
DIC(32) 296 ±56 88 ±42
OVM 291 ±26 92 ±23
PSCP 291 ±24 97 ±16
C DIC(64) 13223 ±60 47 ±25
DIC(32) 13229 ±136 45 ±67
OVM 13206 ±984 186 ±873
PSCP 13260 ±1279 0 ±714
2.15 for (A), (B) and (C), respectively. In the PSCP
method, a= 40 and k0= 16 (1/pixel) for (A) and
(C), and a= 75 and k0= 12 (1/pixel) for (B). The
mode of the distances between the matched pairs of
the phase singularities was evaluated in the xand
ydirections. The sizes corresponding to the sliding
windows used in the DIC method were 32 ×32 and
64 ×64 pixels.
It is observed from Table 3 that in (A) and (B)
cases the OVM, PSCP and DIC (64×64) had come
to equivalent estimations with similar performance,
while in case (C) there is less dispersion in the DIC
method than in the others.
5. Conclusions
We have evaluated a new method to measure in-
plane displacements based on the phase singulari-
9
Fig. 7. (Color online) Optical setup consisting of a lin-
early polarized laser beam, a Leica microscope objective,
a polarization beam splitter (BS) and a CCD Camera.
ties that are obtained by the application of a direc-
tional wavelet transform to a SI. These singularities
are not associated with core structure parameters
as it is the case in the use of the OVM technique.
In the PSCP approach, the detection conditions are
relaxed, so that a more robust and less laborious ap-
proach is obtained compared with the OVM tech-
nique. In addition, the spatial quasi-uniform dis-
tribution of the PSCP with variable network size
presents an advantage for detecting localized defor-
mations, including also the possibility of qualita-
tive characterizations by simple inspection. Only
two parameters, aand k0, must be selected by the
operator. The scale factor acan be fixed within
a large range of values for rigid-body translations.
The amplitude of the wave vector k0modifies the
network size and gives a criteria of selection to make
easier the work of the operator. Most important,
there is no need to modify the working parameters
with the increasing mean of the speckle size and
it can also be used in noisy environments, favoring
its application in industrial measurements. The use
of the pseudo-phase information is better adapted
than the traditional cross-correlation approach for
characterizing localized deformations. The PSCP
method has shown the best performance to char-
acterize localized deformations. We have also com-
pared the performance of the OVM, DIC and PSCP
methods at different measurement contexts and an-
alyzed its advantages and limitations. The DIC
method has the best performance for small mean
values of the speckle size with less sensitivity to
noise and decorrelation. However, when localized
deformations need to be characterized this method
is not suitable because the use of a smaller sliding
window decreases the performance, specially when
the mean of the speckle size increases. The OVM
method presents a very good performance in noise-
less and no decorrelated measurements.
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