PHASE UNWRAPPING FOR ABSOLUTE FRINGE ORDER IN
PHOTOELASTICITY
P. Pinit1, Y. Nomura2 and E. Umezaki3
Graduate Student, Nippon Institute of Technology
4-1 Gakuendai, Miyashiro, Saitama 345-8501, Japan
3
Department of Mechanical Engineering, Nippon Institute of Technology
4-1 Gakuendai, Miyashiro, Saitama 345-8501, Japan
1,2
ABSTRACT
A new phase unwrapping (PU) algorithm for processing fractional fringes photoelastically obtained using the acos(⋅) operator is
presented. The PU algorithm works with plane-polarized RGB light. The three fractional fringe values at a point are used to
generate the three-dimensional plane. One point in the fringe fields is automatically chosen to be the seed point using a
vectorial matching. Simulation of the circular disk under compression demonstrates the performance of the proposed PU
algorithm and results show well agreement with theory.
Introduction
A number of methods based on phase-shifting technique have been proposed for determination of the isochromatic parameter
[1-9]. These methods used the circular [1-3], semicircular [4] or plane polariscope [5-9]. For the methods using the circular and
semicircular polariscopes, the fractional fringe is given through the atan(⋅) operator; hence, its profile is of a saw-tooth type and
this eases the phase unwrapping (PU) for the absolute fringe order. With these techniques, good results were given; however,
the results might be affected by the mismatch of the quarter-wave plates used. Further, the methods [1,4] need the externally
supplied absolute fringe orders at least one point to complete the PU process; hence, they lacked the autonomous feature. In
the methods using the plane polariscope [5-9], the profile of the fractional fringe obtained is of a triangular type due to the use
of acos(⋅) operator. In this case, the isochromatic parameter is determined with sign ambiguity and this makes the PU much
more difficult. Chen [2], Sarma et al. [5] and Plouzennec et al. [8-9] had attempted to directly unwrap the fractional fringe of
triangular type. Chen and Sarma et al. used the unload fringe pattern; hence, the method cannot be applied to the frozen
slices. The Plouzennec et al.'s methods provided good results but fails if there is no zero-fringe order(s) in the fringe field.
In this paper, a new PU algorithm working on the fractional fringe of triangular type is presented. The performance of the
proposed method is theoretically examined with the problem of the circular disk under compressive load.
Determination of Fractional Fringe Order: Dark-field Plane Polariscope
The intensity I with generic orientations m of the transmission axes of the polarizer and analyzer in a crossed fashion coming
out of the plane polariscope is given by [10]
I m ,λ = I p,λ sin 2
δλ
2
sin 2 2(φ − θ m ) + I b,λ
( λ = R , G , B)
(1)
where λ denotes the plane-polarized R, G and B lights, Ip,λ is the intensity coming out of the polarizer, φ is the isoclinic
parameter or the angle of σ 1 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. δλ is the relative retardation relating to the principal-stress
difference (σ 1 − σ 2 ) in plane-stress state by
δλ
C h
h
= N λ = λ (σ 1 − σ 2 ) =
(σ 1 − σ 2 )
2π
λ
f σ ,λ
(2)
where Nλ is the fringe order, Cλ is the stress-optic coefficient, fσ,λ is the well-known material stress fringe value obtained by
2
calibration and h is the model thickness. With sin (γ/2) = (1 − cos γ)/2, Eq. (1) can be rewritten as
I m ,λ = I eff,λ − 21 I mod,λ sin(ϕ + β m )
where
I mod,λ = I p,λ sin 2
δλ
(3)
(4a)
2
ϕ = 21 π − 4φ
(4b)
β m = 4φ m
(4c)
I eff ,λ = I mod,λ + I b,λ
(4d)
1
2
Applying four-step phase shift method such that βm = (m−1)π/2 for m = 1, 2, 3, 4, yields
I 1, λ = I eff , λ − 12 I mod, λ sin ϕ
(5a)
I 2 , λ = I eff , λ − I mod, λ cos ϕ
(5b)
I 3 , λ = I eff , λ + I mod, λ sin ϕ
(5c)
I 4 , λ = I eff , λ + 12 I mod, λ cos ϕ
(5d)
I mod, λ = ( I 1, λ − I 3 , λ ) 2 + ( I 2 , λ − I 4 , λ ) 2
(6a)
I eff , λ = 14 ( I 1, λ + I 2 , λ + I 3 , λ + I 4 , λ )
(6b)
1
2
1
2
Combining Eqs. 5a to 5d, yields
With Eqs. 4(a) and 6(a), δλ or Nλ can be determined if Ip,λ is known. To solve this, an additional fringe image is required.
Determination of Fractional Fringe Order: Bright-field Plane Polariscope
Consider the intensity equation of a bright-field arrangement as following.
I m′ ,λ = I p,λ − I mod,λ sin 2 2 (φ − θ m ) + I b,λ
(7)
Equation (7) can be recast in the same way as done with Eq. 3 as
′ λ + 21 I mod,λ sin(ϕ + β m )
I m′ ,λ = I eff,
(8)
where ϕ and βm are the same as shown in Eqs. (4b) and (4c), respectively, and
′ λ = I p,λ + I b,λ − 21 I mod,λ
I eff,
(9)
In case of β3 = π, the bright-field intensity equation can be easily obtained as
′ λ − 21 I mod,λ sin ϕ
I 3,′ λ = I eff,
(10)
Determination of Fractional Fringe Order: Fractional Fringe Order
Subtracting Eq. (5a) from Eq. (10) and manipulating, yield
I p,λ = I mod,λ + I 3,′ λ − I 1,λ
(11)
2
Then, using Eqs. (4a) with sin (γ) = (1 − cos 2γ)/2, (6a) and (11), the fractional retardation is given by
⎛
⎜
⎝
δ λf = cos −1 ⎜ 1 −
2 I mod,λ ⎞
⎟ for I mod,λ ≤ I p ,λ
I p ,λ ⎟⎠
(12)
z, N B
P⊥t
N Bu
d
Plane
O
N Gu
y, N G
N Ru
x, N R
Figure 1. Generated three-dimensional (3D) plane and vector normal to the plane P⊥t . The fringe order space is analogous to
the Cartesian coordinates.
or
N λf =
⎛ 2 I mod,λ
1
cos −1 ⎜ 1 −
⎜
2π
I p ,λ
⎝
⎞
⎟ for I mod,λ ≤ I p ,λ
⎟
⎠
(13)
Equations (12) or (13) are mathematically limited in the interval [0,+π] or [0,0.5], respectively, due to the range of the arccosine
function. Note that [0,+π] means 0 ≤ N λf ≤ +π. The superscript f denotes the fractional value. Due to the multiple-valued
function of the trigonometric function, the absolute fringe order N λu can be written as
N λu = N int ± N λf for N λu ≥ 0 and N int ≥ 0
(14)
The superscript u denotes the absolute fringe value or unwrapped value and Nint is an integral fringe value. The upper and
lower signs are used when Nint = 0, 1, 2, ... and Nint = 1, 2, 3, ..., respectively. Since the state of stress at the same point on the
model is the same for different wavelengths [2]; hence, from Eq. (2), the following relation is valid.
N λu f σ ,λ = h (σ 1 − σ 2 ) = K
(15)
where K is a constant. Then, for R, G, and B wavelengths,
N Ru f σ ,R = N Gu f σ ,G = N Bu f σ ,B = h (σ 1 − σ 2 ) = K
(16)
Phase Unwrapping for Absolute Fringe Order
In this section, PU algorithm is developed and it involves the following steps:
Step 1: Compute the fractional fringe values for each wavelength using Eq. (13).
Step 2: Generate a theoretical reference plane by means of the vector normal to the plane (Fig. 1). The vector normal
to the plane can be expressed as
P⊥ = N G N B i + N R N B j + N R N G k
(17)
where NR, NG and NB are, respectively, the fringe order of any values on the x-, y- and z-axis in Cartesian coordinates and i, j
and k are the base unit vectors in such coordinates (Fig. 1). For reference vector, if one value of the absolute fringe is known,
the other two absolute fringe orders can also be given (Eq. (16)). For instance, if N R = N Ru , then,
B
P⊥t = N Gu N Bu i + N Ru N Bu j + N Ru N Gu k
(18)
Step 3: Detect the valid points. These valid points are the point candidates to be the seed point or pixel that must
satisfy the following conditions: N Rf < N Gf < N Bf and N Rf ≥ 0.1 and N Bf ≤ 0.4 . It should be note that the second condition is
user-dependent but N Rf ≥ 0 .
Step 4: Extract the seed point using the vectorial matching. For all pixels detected from step 3, use each set of those
three fringe order values to generate the vector normal to the plane and then its unit vector (P⊥ and n⊥). Then, the pixel
possess a maximum of n t⊥ ⋅ n ⊥(vector dot product) is chosen to be the seed point for unwrapping. Note that only resultant of
the dot product being larger than or equal to 0.8 is considered in order to speed up the process. Generally, the generated
plane at this seed point is not parallel to the theoretical reference plane obtained from step 2 (Eq. 18); therefore, in order to
make them parallel, orthogonal projection can be applied. By this, the very accurate values of the absolute fringe orders at the
seed pixel can be obtained.
Step 5: Perform unwrapping starting from the seed pixel (also seed plane). The seed pixel is used as the central pixel
of the 8-neighbors mask window. Then, for wrapped pixels inside the mask window, minimize the error function as
E=
⎧
2
u 2⎫
⎨[ d − d c ] + ∑ [ I λ − cos( 2πN λ )] ⎬
d , N λ ∈[ 0 , +∞ ] ⎩
λ
⎭
min
u
(19)
where d is the shortest distance from the generated plane to the origin of the coordinates of the pixel being considered (Fig. 1)
and dc is also the shortest distance but it is of the central pixel of the mask window. Iλ (= cos δ λf ) can be obtained from Eq. (12).
It should be noted that, in term of d, the generated plane of the considered pixel is automatically parallel to the reference plane
and also n ⊥ = n t⊥ . Then, the absolute fringe order in term of d for each wavelength can be written as
N λu =
d
n ⊥ ,λ
=
d
(20)
n ⊥t ,λ
Therefore, the error function to be minimized can be recast as
⎧⎪
E = min ⎨[ d − d c ] 2 +
d∈[ 0 , +∞ ] ⎪
⎩
∑[ I
λ
λ
2⎫
⎪
− cos( 2πd / n ⊥t ,λ )] ⎬
⎪⎭
(21)
For the pixel being considered, let (dc − 1) ≤ d ≤ (dc +1) where (dc +1) is the upper limit and (dc − 1) is the lower limit. Then,
reduce the d value with the known step s = 1/10k where k (= 0, 1, 2,…) is the loop number until reaches the lower limit. Note
that while reducing d value the value of E is computed. E value and its associated d value are registered into a twodimensional array. For the next loop (k + 1), d values at the array index former and later the index pointing to minimal E at loop
k are reassign to be the upper and lower limits of the loop k + 1. Process is repeated until the difference between the upper
and lower limits satisfies a predefined stopping criterion. After stopping, the absolute fringe order for each wavelength is
obtained from Eq. (20). It should be noted that the unwrapping order for the wrapped pixels in the 8-neighbors mask window is
controlled by the value of Imod,avg,
I mod,avg =
I mod,R + I mod,G + I mod,B
(22)
3
Then, the pixel having a maximum of Imod,avg is unwrapped first and the pixel with a minimum of Imod,avg is lastly unwrapped.
This process is performed until all pixels in the domain unwrapped.
Numerically Simulated Results: Simulation Conditions
Numerical simulations were performed in order to confirm the ability of the proposed method. The benchmark problem, i.e., the
circular disk subjected to diametral compressive load was used. The diameter and thickness of circular disk model were of 30
mm and 6 mm, respectively. The material stress fringe values fσ,λ used to simulated the fringe patterns were of epoxy resin
and they were, respectively, 11.1995, 10.0100 and 7.9970 N/(mm⋅fringe) of fσ,R, f σ,G and f σ,B. These values were calibrated at
wavelengths R = 612 nm, G = 547 nm and B = 437 nm.
By these material stress fringe values, the reference vector, P⊥t , normal to the reference plane can be determined. With Eqs.
(16) and (18) by assuming N Ru = 1.0000, yields N Gu = 1.1190 and N Bu =1.4004. Hence,
P⊥t = 1.5670 i + 1.4004 j + 1.1188 k
(23)
n t⊥ = 0.6581i + 0.5882 j + 0.4670 k
(24)
and
When simulating those photoelastic fringe images, the circular disk was virtually loaded by a force P having the magnitude of
274 N.
For the end of the PU algorithm at one considered point, the stopping criteria (see step 5), i.e., the difference between the
-5
upper and lower limits, was set to be 1×10 . This parameter is user-dependent.
Numerically Simulated Results: Results
The theoretically simulated photoelastic fringes obtained the well-known equation in Ref. [11] are shown in Fig. 2. These
images were computationally simulated according to the four steps of the dark-field (Figs. 2(a-d)) and a single step of brightfield (Fig. 2e) plane polariscope arrangements for a circular disk with the conditions previously described. Figure 2f was
computationally obtained by using Eq. (13) with images in Figs 2(a-e).
(a)
(b)
(c)
(d)
(e)
(f)
Figure 2. Theoretically simulated color photoelastic fringe images of circular disk under compression for different configurations
of the plane polariscope: as dark-field setup (a) β1 = 0, (b) β2 = +π/2, (c) β3 = +π, (d) β4 = +3π/2 and as bright-field setup (e) β3 =
+π. (f) fractional fringe image computed using Eq. (13) with (a)-(e). The image size is of 512 pixel × 480 pixel.
14
12
6
3
0
(a)
(b)
Figure 3. Continuous unwrapped phase map of N Gu : (a) computationally simulated result obtained using the PU algorithm with
the fringe images in Fig. 2f and (b) theoretical unwrapped phase map generated from the well-known equation in Ref. [11]. The
map is shown in this color scale for the sake of clarity.
4
Absolute fringe order, fringe
3.5
240th row
3
2.5
2
1.5
1
0.5
0
0
64
128
192
256
320
384
448
512
Pixel
th
Figure 4. Profiles of simulated unwrapped phase map along the 240 row of Fig. 3a. Note that actual values of absolute fringe
orders were used to plot these profiles.
12
Absolute fringe order, order
Absolute fringe order, order
15
15
9
6
3
0
0
80
160
240
320
y−pixel
400
480
0
80
160
240
320
400
480
12
9
6
3
0
0
x−pixel
(a)
40
80 120 160 200 240 280 320 360 400 440 480
y−pixel
(b)
Figure 5. Topographical maps of the continuous unwrapped phase map of Fig. 3a: (a) three-dimensional view and (b) y-z view
Upon completing unwrapping process as mentioned above, i.e., detecting the valid pixels, identifying the seed pixel by
vectorial matching method, generating its 3D plane and making it to be parallel to the reference plane (Eq. (24)) and
performing minimization starting from such seed pixel (plane) all over the entire field, the simulated unwrapped phase map
was obtained.
u
The unwrapped phase map of the only absolute fringe order N G is shown in Fig. 3a whereas Fig. 3b shows the same absolute
fringe order map but of theory for the purpose of comparison. Note that the maps are shown in this color scale in order for the
sake of clarity. It should be noted further that due to the virtue of Eq. (16), the maximum of absolute fringe order values is of B
wavelength and its value is about 17.
One can see that the map of the simulated unwrapped phase map (Fig. 3a) contains the erroneous region near the upper load
application point. It was thought that there were two causes making this erroneous region exists. The first one was the way to
determine the seed pixel and the other one was unwrapping order controlled by the value of Imod,avg. At and near the load
application point, the fringe density is by nature very high; therefore, at such region, the fringe value is unreliable. By this, the
wrong seed pixel might be chosen and the error propagated to other pixels. For the second cause, it might be that the first
pixel to be unwrapped in the 8-neighbors mask window was incorrectly selected, then, the error occurred propagated to other
pixels in the mask window. As seen, the worst case was that those two causes happened together. However, for the lower
load application point, there was no such erroneous region but the maximal fringe order value was limited to a certain value.
The second cause might make this. Figure 4 shows profiles of absolute fringe orders along the horizontal line passing through
the center of the disk. The obtained values at the disk center given from the PU algorithm were of 2.0766-, 2.3234- and
2.9083-order fringe for N Ru , N Gu and N Bu. The theoretical values of these absolute fringe orders can be calculated using [11]
4P r
πN λ f σ , λ
u
=
(r 2 + x 2 + y 2 )2 − 4r 2 y 2
r2 − x2 − y2
(25)
where r is the disk radius, x and y are the distance measured horizontally and vertically from the disk center, respectively,
Then, by Eq. (25), for the center point at which x = 0 and y = 0, N Ru = 2.0787-, N Gu = 2.3260 - and N Ru = 2.9112 -order fringe.
It is seen that the absolute fringe orders rendered from the PU algorithm are very close to those of theory. Further, the profiles
shown in Fig. 4 are considerably coincident with those of the theoretical map (Fig. 3) (profiles are not shown). Figure 5 shows
the topographical map of Fig. 3a. As previously described that at the upper load application point, the wrong absolute fringe
order values were given by the PU algorithm. This is evident as shown in Fig. 5b for the left lobe.
Conclusion
In this present paper, a new PU algorithm for an automatic determination of the isochromatic parameter (absolute fringe order)
by using three RGB wavelengths is developed. The method is theoretically examined by applying to the simulated photoelastic
fringe images of the circular disk subjected to diametrically compressive load generated based on the phase shifting technique.
According to the numerical interrelation between theoretical and experimental 3D planes, the PU provides good absolute fringe
order map for all wavelengths used except the region near the load application point and the support.
The method overcomes the interaction between the user and the PU algorithm for the need of externally supplied absolute
fringe order by automatically determining the seed point in the range of 0 to 0.5. However, extension of this seed point range
could improve the performance of the method to handle other problems possessing the absolute fringe order out of this range.
Also, improvement of the seed point ordering near the high fringe density zones could solve the unreliable absolute fringe
order. The use of the already proposed method [10], i.e., masking out the region around the load application and the support
of the model, could also provide more reliable values of the absolute fringe order.
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