AN AUTOMATED PHASE UNWRAPPING ALGORITHM FOR
ISOCLINIC PARAMETER IN PHASE-SHIFTING PHOTOELASTICITY
P. Pinit1 and E. Umezaki2
Graduate Student, Nippon Institute of Technology
4-1 Gakuendai, Miyashiro, Saitama 345-8501, Japan
2
Department of Mechanical Engineering, Nippon Institute of Technology
4-1 Gakuendai, Miyashiro, Saitama 345-8501, Japan
1
ABSTRACT
This paper presents the evaluation of the whole-field method for calculating and unwrapping the isoclinic parameter in the true
phase interval ranging from −π/2 to +π/2. The method is based on the four-step color phase shifting technique and the phase
unwrapping. It is applied to the circular disk model under three radial compressive loads and the M-shaped like model under
distributed compression. Results show that the isoclinic-angle map obtained is almost free from the influence of the
isochromatic parameter but the values of the global and local thresholds, which were used to define the phase jumps and local
discontinuities when dealing with the isotropic point, affected the accuracy of the isoclinic values.
Introduction
In phase-shifting technique, two important problems arise when considering the isoclinic parameter φ: the isochromaticisoclinic interaction and the wrapped phase data. The first problem physically occurs because at and near the skeleton of
isochromatic fringe the isoclinic parameter is indeterminate. For the wrapped phase data, it arises because the phase interval
is explicitly known at most between ±π/4 (or modulo +π/2) instead of its true phase interval −π/2 < φ ≤ +π/2 (or modulo +π). If
one lefts this problem still unsolved, the ambiguity exists on whether the isoclinic-angle map represents σ1 or σ2 orientations
over the entire domain and this effect propagates to the determination of the isochromatic parameter, especially in case of the
method using the circular polariscope. The phase unwrapping (PU) technique can solve this problem.
A number of researchers have attempted to solve such problems [1-7]. Those proposed methods used either the plane or
circular polariscope, and provided the isoclinic parameter both in the phase interval 0 ≤ φ ≤ +π/2 or −π/4 < φ ≤ +π/4 [1-2,7] and
in the true phase interval after unwrapping [3-6]. The methods work well with the benchmark problem (the circular disk under
diametral compression); however, they cannot be applied to models having the isotropic point despite the fact that the isotropic
point has strongly effect on the PU algorithm [8].
The developed PU algorithm with capability to deal with the problems having the isotropic point has been proposed [9]. In this
work, the PU algorithm has been extended to other models in order to confirm its performance. The problems of the circular
disk subjected to three radial compressive loads and the M-shaped like model under distributed compression are studied. The
effect of global and local thresholds used to define the global and local phase jumps, respectively, in the fringe field is also
described.
Determination of Isoclinic Parameter
Consider the dark-field plane polariscope system with white light source shown in Fig. 1 where the x axis is chosen to be a
reference axis. After the specimen is properly kept in the system and, loaded by a force, the general equation of the intensity I
with generic orientations m of the transmission axes of the polarizer and analyzer in a crossed fashion coming out of a digital
camera is given by
⎛ 1
I m (λ ) = ⎜
⎜ ∆λ
⎝
∫
λupper
λlower
F (λ ) I p (λ ) sin 2
⎞
dλ ⎟ sin 2 2(φ − θ m ) + I b (λ )
⎟
2
⎠
δ (λ )
(λ = R , G, B)
(1)
White
light source
Model
Analyzer
Polarizer
Digital
camera
Personal
computer
Figure 1. Dark-field plane polariscope system with the circular disk subjected to three radial compressive loads.
where λ denotes the primary wavelengths R, G and B of the white light corresponding to camera filters, m is the step number
of the crossed Polaroid's, λlower and λupper are the lower and upper limits of the spectrum acquired by the associated filter of the
camera, ∆λ = λupper − λlower, F(λ) is the spectral response of the filter associated with λ, Ip(λ) is the intensity coming out of the
polarizer, δ(λ) (= 2πN where N is the fringe order) is the fractional retardation or the isochromatic parameter, φ is the isoclinic
parameter or the angle of σ1 with respect to the reference axis and is counterclockwise, θm is the induced phase shift angle at
step m and is also counterclockwise and Ib(λ) is the background intensity. Note that for the dark-field setup, the value of the
induced phase shift angle θm is typically chosen to be equal.
Equation (1) is simply rewritten as
I m (λ ) = I mod (λ ) sin 2
δ (λ )
2
sin 2 2(φ − θ m ) + I b (λ )
(2)
where
I mod (λ ) =
∆λ ∫
1
λupper
λlower
F (λ ) I p (λ ) sin 2
δ (λ )
2
dλ
(3)
2
Elaborately manipulating to Eq. (2) with the trigonometric identity sin (γ/2) = (1 − cos γ)/2 and applying the four-step phase
shifting method such that the induced phase shift angle, θ1 = 0, θ2 = +π/8, θ3 = +π/4 and θ4 = +3π/8, yields the equation for
determining the isoclinic parameter as
φw =
π
8
−
⎛ I s − I 3s
1
tan −1 ⎜ 1s
⎜I −Is
4
4
⎝ 2
⎞
s
⎟ for I mod
≠0
⎟
⎠
(4)
where w denotes the wrapped value, i.e., wrapped into the phase interval 0 ≤ φ ≤ +π/4 due to the use of the arctangent and
I ms = I m ( R ) + I m (G ) + I m ( B)
s
I mod
= I mod (R ) + I mod (G ) + I mod ( B)
(5)
(6)
Before using Eq. (4), the summed value from Eq. (5) should be normalized by a factor such that the value does not exceed the
maximum gray level used [9]. This is done in order for reducing the effect of the isochromatic parameter and the variation of
s
the light when the crossed Polaroid's is being rotated. Also, I mod
can be determined using the following equation.
s
I mod
= ( I1s − I 3s ) 2 + ( I 2s − I 4s ) 2
(7)
45
P
12
P
30
+
R10
25
0.707P
45 D 45 D
50
Symmetrical
line
30
+
0.707P
V-block
P
(a)
(b)
Figure 2. Applied load directions and dimensions of models: (a) circular disk and (b) M-shaped like model. The magnitude of
force P is of 274 N. The black and white arrows indicate the applied load directions and the reaction at the supports,
respectively. (Geometrical unit: mm and images not in scale)
(a)
(b)
(c)
(d)
Figure 3. Raw color photoelastic fringe images of circular disk under compression collected at four different configurations of
the polariscope system: (a) θ1 = 0, (b) θ2 = +π/8, (c) θ3 = +π/4 and (d) θ4 = +3π/8. The image size is of 512 pixel × 480 pixel.
(a)
(b)
(c)
(d)
Figure 4. Raw color photoelastic fringe images of M-shaped like model under distributed compression collected at four
different configurations of the polariscope system: (a) θ1 = 0, (b) θ2 = +π/8, (c) θ3 = +π/4 and (d) θ4 = +3π/8. The image size is
of 500 pixel × 1000 pixel.
For the PU algorithm, it already has been developed [9]. However, for the sake of clarity, its main processes are briefly
described here. There are two main stages, i.e.,
• Expansion Stage: the PU algorithm performs unwrapping starting from the initial region (largest region of the wrapped
phase map of isoclinic parameter) by which the low-quality regions (the isotropic points and regions around them) are left. In
this way, the errors in the low-quality regions are suppressed and the initial region is continuously grown up.
• Shrinkage Stage: once completing the expansion stage, the PU algorithm performs unwrapping starting from the
outermost boundary of those left low-quality regions. These regions are smaller and smaller every time of the process.
Experimental Setup and Results
The dark-field plane polariscope system used for capturing the photoelastic fringe images is shown in Fig. 1. The optical
system mainly consisted of a plane polariscope setup with a white light source (halogen lamp), a digital camera model SLR
D70 of Nikon and a personal computer. The presented method was experimentally applied to the problem of the circular disk
subjected to three radial compressive loads and the M-shaped like model under distributed compression. These models were
made of 6-mm epoxy resin plate (Fig. 2). When performing the experiment, the models were subjected to the compressive
load P = 274 N and then four photoelastic fringe images for each model were digitally collected according to the four different
configurations of the Polaroid's in the crossed combination. For calculating and unwrapping the whole PU algorithm was
implemented in VC++ as a window-based program. The raw color fringe images of the circular disk and the M-shaped like
model models are respectively shown in Figs. 3 and 4. Figure 5 shows the wrapped and unwrapped phase maps of the
circular disk whereas Fig. 6 shows the same maps but the M-shaped like model. The wrapped and unwrapped phase values
are separately converted into 256 gray levels where pitch black represents 0 and pure white represents 255.
Discussion: Circular Disk Subjected to three Radial Compressive Loads
By using images shown in Fig. 3 with Eq. (4) and the PU algorithm, the wrapped and unwrapped phase maps of isoclinic
parameter are shown in Fig. 5. The wrapped phase map in Figs. 5a and b are respectively of [0,+π/2] and (−π/4,+π/4]. Figure
5c shows the unwrapped phase map in the interval (−π/2,+π/2]. Note that when unwrapping the global threshold Tglobal used to
define the phase jumps in the fringe field was set to +0.8π/2 and the local threshold Tlocal used to define the local discontinuities
when dealing with the isotropic point was set to +0.4π/2 (for theory Tglobal = +π/2). The window size used for detecting the
isotropic point(s) and expanding region around the point was 21 pixel × 21 pixel [9]. The unwrapped phase map is smooth and
is judiciously comparable with that reported in Ref. [10].
+π/2
+π/4
+π/2
0
−π/4
−π/2
(a)
(b)
(c)
isoclinics
isotropic point
(d)
s
I mod
first order isochromatics
(e)
Figure 5. Wrapped, unwrapped and
maps of circular disk subjected to three compressive radial loads: (a) wrapped phase
map in the interval [0,+π/2], (b) wrapped phase map in the interval (−π/4,+π/4], (c) unwrapped phase map in the interval
s
(−π/2,+π/2], (d) I mod
map (Eq. (7)), (e) green channel of color image separated from Fig. 3c.
One can see the effect of the isochromatic parameter, especially the first order isochromatics, and the edge stress in the
isoclinic-angle map. The edge stress occurred when the model was prepared and it is hard to avoid. The effect of the
isochromatic parameter (at and near the load application points and the supports) in the isoclinic-angle map occurred because
the distinction between the isoclinic fringe (black) and the first-order fringe (almost black) is very difficult (Figs. 5d and e). The
effect of the edge stress and the zero order isochromatics causes the ambiguity regions appearing near the edge of the model
(Fig. 5b). In case of the zero order isochromatics effect, one can clearly see the lines that are parallel to the edge of the model.
s
These lines occurred because I mod
= 0 . Then, the consequent result is that the obtained value of φw from Eq. 4 was unreliable.
Moreover, fortunately, with those values of Tglobal and Tlocal, the unwrapped phase map obtained was free from the erroneous
region. However, for the M-shaped like model, the erroneous region occurred (see the next subsection).
Close considering at the isotropic point (Fig. 5d) reveals that all isoclinics pass through this point (see the faded lines around
the point). As show in Fig. 5c, the isoclinic parameter gradually varies from −π/2 to +π/2 around the isotropic point in the
clockwise direction; therefore, this isotropic point is of the negative type [10]. If the techniques proposed, for example, by
Barone [5] and Kihara [6] (the PU algorithm starting from arbitrary points and using row-wise seeding and column-wise
scanning (or vice versa)) were used to unwrap the wrapped phase map of this problem (Fig. 5b), they would fail if the scanned
line passes through the isotropic point. Also, the obtained isoclinic-angle map would never be the same as that shown in Fig.
5c, especially the part below the isotropic point for row-wise scanning or the left or right part for column-wise scanning.
Discussion: M-shaped Like Model under Distributed Compression
With the PU algorithm applying to the images shown in Fig. 4, the wrapped and unwrapped phase maps of isoclinic parameter
are shown in Fig. 6. The wrapped phase map in Figs. 6a and b are, respectively, of [0,+π/2] and (−π/4,+π/4]. Figures 6c and d
show the unwrapped phase map in the interval (−π/2,+π/2]. Note that for Fig. 6c, Tglobal = +0.8π/2 and Tlocal = +0.4π/2 whereas
Tglobal = +0.6π/2 and Tlocal = +0.4π/2 were for Fig. 6d. The window sizes used for finding the isotropic and singular points and
expanding regions around them were 21 pixel × 21 pixel [9].
It can be seen that there are two isotropic points (Figs. 6c and d). Like the circular disk model, the isoclinic parameter gradually
varies from −π/2 to +π/2 in the clockwise direction; therefore, it is of the negative type. For Fig. 6c, it can be seen that the
erroneous region appeared at the left bottom part of the model whereas there was no such erroneous region in Fig. 6d.
However, as seen in Fig. 6d, there still has the small white region in the map (compare that region of Fig. 6a and d). The
residual stress appearing when the model was being prepared might cause this small region not the PU algorithm.
As explained in the previous subsection that the PU algorithm based on only the row-wise seeding and column-wise scanning
(or vice versa) would fail when the scanned line passes through the isotropic point. This situation would be worse if the
isotropic point had occupied a region rather than a point. This can be clearly seen at the lower isotropic point in Fig. 6b
(compare with that of Fig. 5b). Therefore, this would make such PU algorithm more unreliable. Further, such PU algorithm was
(a)
+π/2
+π/4
+π/2
+π/2
0
−π/4
−π/2
−π/2
(b)
(c)
(d)
Figure 6. Wrapped and unwrapped maps of M-shaped like model under distributed compression: (a) wrapped phase map in
the interval [0,+π/2], (b) wrapped phase map in the interval (−π/4,+π/4], (c) unwrapped phase map in the interval (−π/2,+π/2]
obtained with Tglobal = +0.8π/2 and Tlocal = +0.4π/2, (d) unwrapped phase map in the interval (−π/2,+π/2] obtained with Tglobal =
+0.6π/2 and Tlocal = +0.4π/2.
performed by starting from arbitrary points and that worked well with the benchmark problem but for the complicated model
(complicated geometrical shape) like this model, there might be the interaction between the user and the program in order that
the use can provide the correct starting points. This also means that, for the other models, the user must have seen the
wrapped isoclinic phase map of each model to properly select such starting points before the unwrapping process taking place.
This may be tedious for the user. However, for the PU algorithm used here [9], there was no need for such operation.
Another factor, apart from Tglobal, causes the presence of such erroneous region was thought to be the process of
determination of the secondary seed pixel locating at and near the boundary of the unwrapped region. This was because the
boundary of the unwrapped region was coincident with the boundary of the model. It should be noted here that the boundary of
the model is different from that of the unwrapped region [9]. Since, at the boundary of the model, the isoclinic value is generally
unreliable due to the difficulty of the identification the region(s) occupied by the model and the background, the error would
propagate from this secondary seed pixel. Therefore, rather than only changing the value of Tglobal, this erroneous region may
also be solved by well managing the seed pixel ordering process.
Conclusion
Already developed PU algorithm has been extended to evaluate the plane problems possessing the isotropic point(s). The
effect of some control parameters, the windows used to detect the isotropic point(s) and expand regions around it, has already
been discussed in detail in reference [9]. In this present work, Tglobal shows significant effect on the unwrapped isoclinic phase
map of the M-shaped like model whereas Tlocal provides no effect. Although Tlocal is unaffected on the unwrapped phase map
for the model presented here, it may affect that of other models.
In this work, with the presence of the influence of those parameters, the PU algorithm still renders the isoclinic-angle map in its
true interval. This confirms the performance of the PU algorithm. Preventing the pixels locating at and near the boundary of the
model to be the seed pixel may be also useful technique to help solving the erroneous region. Further, to obtain more accurate
results of isoclinic parameter, smoothing technique can be used to reduce the influence of the isochromatic parameter.
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