AN AUTOMATED PHASE UNWRAPPING ALGORITHM FOR ISOCLINIC PARAMETER IN PHASE-SHIFTING PHOTOELASTICITY

P. Pinit1 and E. Umezaki2

1Graduate Student, Nippon Institute of Technology 4-1 Gakuendai, Miyashiro, Saitama 345-8501, Japan

2Department of Mechanical Engineering, Nippon Institute of Technology 4-1 Gakuendai, Miyashiro, Saitama 345-8501, Japan

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 isochromatic-isoclinic 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

B),G,R()()(sin)(sin)()(1)( bp

upper

lower

=+−⎟⎟⎠

⎞⎜⎜⎝

⎛= ∫ λλθφλλδλλ

λ∆λ

λ

λIdIFI mm 2

222 (1)

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 where Elaborately manipulating to Eq. (2) with the trigonometric identity sin2(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 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 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 the light when the crossed Polaroid's is being rotated. Also, can be determined using the following equation.

)()(2sin2

)(sin)()( b22

mod λθφλδλλ III mm +−=

λλδλλλ∆

λλ

λdIFI ∫=

upper

lower2

)(sin)()(1)( 2pmod

0fortan41

8smods

4s2

s3

s11 ≠⎟

⎟⎠

⎞⎜⎜⎝

⎛

−

−−= − I

IIII

wπφ

)B()G()R(smmmm IIII ++=

)B()G()R( modmodmodsmod IIII ++=

2s4

s2

2s3

s1

smod )()( IIIII −+−=

Whitelight source

Personalcomputer

Polarizer

Model

Analyzer

Digital camera

Figure 1. Dark-field plane polariscope system with the circular disk subjected to three radial compressive loads.

(2)

(7)

(6)

(5)

(4)

(3)

smodI

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.

(a) (b) (c) (d)

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)

P

P

Symmetrical line

12

+R10

30

45

+ 50 30

(a) (b)

P

25

0.707P0.707P

V-block

45 45

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/2

+π/4

−π/4

(d) (e)

first order isochromatics

isoclinics

isotropic point

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 (−π/2,+π/2], (d) map (Eq. (7)), (e) green channel of color image separated from Fig. 3c.

smodI

smodI

(a) (b) (c)

+π/2

0

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. These lines occurred because . 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

0smod =I

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.

(a) (b) (c) (d)

+π/2

−π/2

+π/2

−π/2

+π/4

−π/4

+π/2

0

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.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Patterson, E.A. and Wang, Z.F., Towards Full Field Automated Photoelastic Analysis of Complex Components, Strain, 27(2), 49-56 (1991). Buckberry, C. and Towers, D., New Approaches to the Full-field Analysis of Photoelastic Stress Patterns, Opt. Lasers Eng., 24(5-6), 415-428 (1996). Nurse, A.D., Full-Field Automated Photoelasticity by Use of a Three-Wavelength approach to Phase Stepping, Appl. Opt., 36(23), 5781-5786 (1997). Plouzennec, N. and Lagarde, A., Two-wavelength Method for Full-field Automated Photoelasticity, Exp. Mech., 39(4), 274-277 (1999). Barone, S., Burriesci, G. and Petrucci, G., Computer aided photoelasticity by an optimum phase stepping method, Exp. Mech., 42(2), 132-139 (2002). Kihara, T., An Arctangent Unwrapping Technique of Photoelasticity Using Linearly Polarized Light at Three Wavelengths, Strain, 39(2), 65-71 (2003). Pinit P. and Umezaki E., Full-Field Determination of Principal-Stress Directions Using Photoelasticity with Plane Polarized RGB Lights, Opt. Rev., 12(3), 228-232 (2005). Quiroga, J.A. and Gonzalez-Cano, A., Separation of isoclinics and isochromatics from photoelastic data with a regularized phase-tracking technique, Appl. Opt., 39(17), 2931-2940 (2000). Pinit, P. and Umezaki, E., Digitally whole-field analysis of isoclinic parameter in photoelasticity by four-step color phase-shifting technique, Opt. Lasers Eng., to appear. Forcht, M. M., Photoelasticity, Vol. 1, John Wiley & Sons (1941).