Computerized Image Processing for Whole-field Determination of Isoclinics and Isochromatics

by A.V.S.S.S.R. Sarma, S.A. Pillai, G. Subramanian and T.K. Varadan

ABSTRACT--A new computerized method for the whole-field determination of isoclinic and isochromatic parameters in a stressed photoelastic model is presented. The intensity data obtained from the fringe pattern at three analyzer positions (0, 45, and 90 degrees) in a plane polariscope, are used for the computations. The relationships between the intensity values and the photoelastic parameters are derived using Stokes representation of polarized light. The experiments were carried out using a computer-based image-processing system. The accuracy achieved is assessed by comparing the theoretical and measured values. Accuracies of within +-3 degrees for isoclinics and +- 0.05 fringe for isochromatics are shown to be possible.

Introduction

The photoelastic technique is one of the most elegant whole-field techniques for stress analysis. Both the magni- tude and direction of the principal stresses are obtained, using models made of birefringent materials. It is capable of providing the stresses/strains at each point (zero-gage- length resolution) in the area under consideration. The magnitude and direction of the principal stresses are given by the isochromatic and isoclinic fringe orders. Tra- ditionally these data are measured manually by the con- ventional polariscope setups.

Obtaining the whole-field isochromatic and isodinic fringe orders requires measurement at a large number of points and tedious compensation methods are needed to obtain the fractional fringe orders. This is quite involved and requires skill in the identification and measurement of isochromatic and isoclinic fringe orders. Automation of the photoelastic data acquisition and analysis to minimize

A.V.S.S.S.R. Sarma is Head, Structures Laboratory, and S.A. Pillai is Engineer, Vikram Sarabhai Space Center, Trivandrum - 695 022, India. G. Subramanian (SEAl Member) is Professor and Head, and T.K. Varadan is Professor, Department o f Aerospace Engineering, Indian Institute o f Technology, Madras - 600 036, India. Original manuscript submitted: April 1991. Final manuscript received." November 4, 1991.

these problems and to provide faster and more accurate results, has become essential.

Many of the studies reporting on this aspect include methods using the spinning of the polarizers, automated compensation, intensity measurement, spectral-content analysis, etc. Robert I proposes a method using the spinning analyzer. In this method retardation less than one wave- length only can be measured and requires the manual recognition of the direction of the light ellipse. Young ~ describes a procedure based on the fractional fringe-order measurement using Tardy methods. Young uses servo motors to drive the polarizers until the minimum intensity is achieved. He uses two wavelengths to determine the total fringe order. Cernosek 3 proposes a method using a compensator and spinning polarizer for the measurement of fractional fringe orders. Pillai e t a l . 4 use a servo- motor for driving the Babinet-Soleil compensator until null compensation is achieved. They use a photometer to measure the intensity. The output of the photometer is used to drive the servomotor. In this method the com- pensator has to be manually aligned along the principal stress direction.

Redner 5 proposes a method for complete measurement of photoelastic data using a spinning polarizer and analyzer. This method utilizes the phase difference of the light intensity measured using two photocells to extract the isoclinic and isochromatic information. The spectral content analysis method by Redner 6. 7 is one of the recent developments in automated measurement of birefringence. This method deals with the measurement of principal- stress difference only. Yao 8 uses image division and image differentiation methods to extract the isoclinics. This method uses both plane and circular polariscope setups and isoclinic images at many angles.

Most of the above methods use quarter waveplates. Errors due to dispersion in the quarter waveplates for polychromatic light exists in these methods. Also, these methods are point by point methods requiring scanning of the whole field with sensors and the use of many frames for isoclinics.

A new approach for obtaining the isoclinic and iso- chromatic fringe data using intensity measurement is proposed in this paper. Evaluation of the isoclinic and

24 �9 March 1992

isochromatic parameters through intensity measurements is generally avoided due to the associated fluctuations in intensity, noise problems, difficulties in recording and analysis of a huge amount of intensity data, etc. With the advent of microprocessor-based image-processing systems and high-quality recording devices these problems are now surmountable. The proposed method makes use of the whole-field intensity data for three analyzer positions in a plane-polariscope setup, thus avoiding the errors due to dispersion of the quarter waveplates.

Theory

Light Ellipse and Stokes Parameters The relationships among the isoclinic and isochromatic

parameters and the intensity levels can be conveniently derived using the Stokes parameter representation of polarized light?

The most general type of polarized light is elliptically polarized light, which can be represented by a light ellipse 9 with two simple harmonic vibrations along two ortho- gonal reference axes (Fig. 1) ox arid oy as

E, = a, cos cot

and

E, = ay cos (o~t + e)

where ax, ay are the amplitudes of the components of the light vectors along the two axes and e is the relative phase difference.

The major axis of the light ellipse makes an angle c~ with the y axis such that

tan 2c~ = tan 20 cos

where

a. tan 0 = - -

a,

e = relative phase difference between these two vibrations a = the azimuth of the light ellipse with respect to the reference axis oy

~' ax Y

Fig. 1--Representation of polarized light by light ellipse

b Tan o~ = - - = e, is the ellipticity where a and b are the a semi major and semi minor axes of the light ellipse.

The four quantities (ay 2 + a,2), (ay 2 - ax2), 2ax ay cos e and 2a, a, sin e are referred to as the Stokes parameters. They are represented as Stokes vector V = (SO, S1, $2, $3). The first parameter SO = (a,=+ a, 2) which refers to the total light intensity at a point is not of relevance in the present discussion since this can be assumed to be constant. The other three parameters are represented as

is1] ] rcos20 ] $2 = 2ax ay cos = | s i n 20 cos $3 2a= ay sin Lsin 20 sin e

= cos 2~o sin (1) sin 2c0

Taking ay ~ + a~ 2 = 1 it can be seen that S12+ $2 ~+ $3 2 = 1 .

These parameters are with reference to a set of perpen- dicular axes (polarizer axes in this case) which make an angle c~ with the major and minor axes of the light ellipse. The Stokes parameters change when either the reference axes are rotated or a certain amount of phase difference is introduced along a set of perpendicular axes.

Transformation of Stokes Vector

For vertically polarized light the Stokes vector with respect to the polarizer axes is represented by

$2 = (2) S3

At L1 (Fig. 2) the plane-polarized light is represented by the above eq (2). When this plane-polarized light enters a stressed model it is resolved along the major and minor principal-stress directions. The light vectors emerging along the principal directions will have a relative retarda- tion ~5 proportional to the magnitude of the principal- stress difference. Counter-clockwise rotations are treated as positive, looking towards the source. Rotating the reference axes by/3 in the clockwise direction to coincide with the major principal stress axis al (Fig. 2), the Stokes vector is represented by

[ ~ S l c ~

S-2 = $2 cos 2/3 - S1 sin 2/3 (3)

S3

POLARIZER AXIS HODEL ANALYZER ,4, (VERTICAL)

V ] V / 0-1 A1 I A31/'5~

0~

L1 L2 A2(90 o)

AI, A2,A3-ANALYZER POSITIONS COUNTER CLOCKWISE DIRECTION IS POSITIVE

Fig. 2--Schematic of the plane-polariscope setup

Experimental Mechanics �9 25

By addition of a retardation, b, the Stokes vector becomes

~ 1 ] F S1 cos 2/3 + $2 sin 2/3 /

gg, j

($2 cos 23 - S1 sin 2/3) cos 6 - $3 sin 6

$3 cos 6 + ($2 cos 2/3 - S1 sin 23) sin 6

(4)

At L2, the light ellipse is represented by the Stokes parameters SI , , $2, and $31 in eq (4), the reference axis being the principal-stress axis.

Using the values of S1, $2 and $3 in eq (2) for plane- polarized light, we get i cos21

~ 1 = - sin 2/3 cos 6

S31_1 sin 2/3 sin 6

(5)

Since /3 is an unknown, the reference axis is rotated back to the original vertical axis (polarizer axis), i.e., the reference axis is rotated by +/3. So the Stokes parameters at L2 with reference to the vertical axis are given by

E c~ sin22 c~ ~ s / = sin 23 cos 2/3 cos 6 + cos 2/3 sin 2 (6)

/ S"3s _J sin 2/3 sin

Light-intensity Considerations

When the analyzer axis is along the polarizer axis the intensity of the light coming from the analyzer is propor- tional to a/. , i.e., /1 oc a / or /1 = K a / where K is a constant. Similarly, when the analyzer is at 90 degrees to the polarizer axis, Is oc a, s or /2 = K a~ z

11 + Is = K ( a / + a. s)

K = (L +/2) because a / + a / = 1

When the analyzer axis is rotated clockwise by 45 degrees from the polarizer axis (which is the reference axis) tile light emerging from the analyzer is represented, using eqs (6) and (3), as

LS--3s J sin 23 sin

The intensity of light measured at the analyzer position of 45 degrees is proportional to ays,2 i.e., /3 oc aysS or Is = g a,]

2 L 2 L - L - L S l s = ay z - a / - K 1 - / 1+ /2 (8)

From eq (7),

- sin 23 cos 23 cos 6 + cos2/3 sin 23 - L +I2

From eq (6),

(cos s 2/3 + sin ~ 2/3 cos 6) = a / - a / - L - Is I, +/2

(10)

Determination of Isoclinic Parameter

From eqs (9) and (10), eliminating cos 6 and simplifying we obtain

tan 2/3 = 2/2 2/3 - / 1 - I z

o r

1 2/2 /3 = ~- tan- ' [ 2 1 3 - L - L ] (11)

where I , , 12 and/3 are the intensities measured at analyzer positions of 0 degree, 90 degrees and 45 degrees respec- tively with respect _to the polarizer axis, and /3 is the inclination of the principal-stress direction with the reference axis. /3 is the isoclinic parameter with respect to the axis of the incident plane-polarized light. Thus /3 is evaluated using eq (11).

The value of/3 is obtained from eq (11) with ambiguity of 90 degrees, since tan 2/3 = tan (2/3+ 180). This is acceptable for the isoclinic parameter since it can represent either of the two principal-stress directions which are mutually perpendicular.

Determination of the Isochromatic Fringe Order Using eqs (6), (7), (9) and (10), we obtain

cos z 2 3 + sin z 2/3 cos 6 = A

cos 2/3 sin 2/3 (1 - cos tS) = B

Simplifying these equations and eliminating /3, we obtain

A - A 2 - B 2 cos 6 -

(1 - A )

o r

6 = cos -1 [.A-(1A2-_A) B2 ] (12)

t"~OtL ! : ST.kSILIZED HUNO CHROHATIC LENS P A

DIFFUSER POLARIZER ANALYZER POWER SUPPLY LIGHT SOUR rF He - He LASER

P C- AT BA.~ED ItAC, E PROCESS!NIl

SYSTEH

(9) Fig. 3--Schemat ic of the experimental setup

YF~O CA~RA (ULTRICON TUGE)

26 * March 1992

where

1 1 - 1 2 A - - -

I, + I2

2 h - I 1 - h B - L +I2

N = 27r

(13)

The fractional fringe orders are determined using eq (13).

Experimentation A circular disk stress frozen under diametrical compres-

sion.is used to demonstrate the method. The whole-field data of the intensity levels are grabbed using a micro- processor-based image-analyzer system. The schematic of the arrangement is shown in Fig. 3. The system and the experimental setup are shown in Figs. 4 and 5.

Images were captured using an RCA ultracon tube black and white camera connected to the PC-EYE frame grabber installed in the microcomputer-based image- processing system. The images captured were found to have time-dependent noise. For the same input light, the intensity level at any given pixel was observed to vary with respect to time. Averaging of frame-wise intensity levels recorded at different instances of time, revealed that the pixel-intensity values converge after averaging

Fig. 4--Photograph of the experimental setup

about seven to eight frames. For the experiments reported in this paper, the averaging was done with 10 frames. The images captured for each of the three analyzer positions were averaged before further processing. These time- averaged images were stored in image data files. Each image frame is represented by 512 x 512 pixel values. The area covered by each pixel depends on the size of the image. The area covered by each pixel is around 0.01 mm 2 for the image used for the present study. The influence of general background light was determined using an un- stressed model of the same material and thickness, viewed in the crossed-analyzer position. This value was sub- tracted from the image-pixel intensities before they were space averaged. Spatial averaging of the pixel values with the intensities of the surrounding eight pixels showed stable values and is assigned to the central pixel. The image data were thus space averaged before they were used for further computations. The isoclinic and iso- chromatic fringe orders were determined using eqs (11) and (13) respectively.

The images captured for the three analyzer positions are presented in Figs. 6(a), 6(b) and Fig. 6(c).

Discussion The intensity values at the three analyzer positions of

0 degree, 90 degrees and 45 degrees are used for the determination of the isoclinics and isochromatics. Any three independent analyzer positions are sufficient for obtaining these parameters, but the equations corre- sponding to these analyzer positions are to be used. The intensity data for the three analyzer positions considered in the present case (0, 45, and 90 degrees) are related to the isoclinic and isochromatic parameters through simple equations.

During recording of these images, the polarizer axis was kept at 45 degrees to the loading axis of the disk. The isoclinic angles obtained from computations (using the present method) have the polarizer axis as the reference axis. For better presentation and comparison of the results with the theoretical and measured values, the reference axis is rotated by 45 degrees to coincide with the vertical axis (i.e., loading axis of the disk). The computed, theoretical and measured values of the isoclinics with respect to the vertical (loading) axis of the disk are given in Table 1. The measured values of isoclinics were deter- mined from the plot of the isoclinics at 5-deg intervals and interpolating the values corresponding to the required

Fig. 5--Photograph of the image- processing system used for the experiment

ExperimentaI Mechanics �9 27

(a~

points. The isoclinic angles presented in Table 1 are with respect to the vertical axis, measured in the counter- clockwise direction looking towards the source of light.

The isochromatic fringe orders at typical locations are presented in Table 1 along with theoretical and measured values. It is to be noted from eq (12) that the retardation is obtained in terms of cos/~. Only fractional fringe orders could be obtained from direct use of eq (13). The value of ~i is determined with 'sign ambiguity', i.e., _+ 5. The ambiguity of the sign of/~ can be resolved by deter- mination of the integral fringe order with the data from two or three wavelengths or two or three loads and using the procedure described in Ref. 10.

The isoclinic and isochromatic values at any point of interest can be obtained by using the corresponding pixel values at that point. The images captured have an aspect ratio of 1.4. This factor is taken into consideration in determining the pixel locations corresponding to the desired coordinates.

It can be seen from Table 1 that the computed values of the isoclinics and isochromatics are in close agreement with the theoretical and measured values (using the Tardy compensation method). The computed values from the present method are closer to the values directly measured. At regions very close to the loading point, which is a singular point, the isoclinic parameters are not very relevant because all the isoclinics pass through this point. The isoclinic fringes in this region will be very close. The

(b) (c)

Fig. 6--Damage of the disk stress frozen under diametrical compression for analyzer positions (a) 90 deg, (b) 0 deg and (c) 45 deg

28 �9 March 1992

T A B L E 1 - - C O M P U T E D (USING THE PRESENT METHOD) , T H E O R E T I C A L A N D M E A S U R E D V A L U E S OF I S O C L I N I C S A N D I S O C H R O M A T I C S AT T Y P I C A L L O C A T I O N S . CENTER OF THE D ISK T A K E N AS THE ORIGIN. D I A M E T E R OF THE D ISK IS 40 mm

Coordinate Isoclinic Data (~) Isochromatic Data (N)

Y X /3 /~ /~ Difference AN 1-AN N N Difference mm mm Computed Theoretical Measured (Comp-measured) Computed Computed Theoretical Measured (AN Computed/

AN Measured)

0 0 89 0 0 1 0.22 0.78 3.22 3.22 0.00 0 - 4 2 0 0 2 0.23 0.77 2.86 2.82 - 0.05 0 - 6 89 0 0 1 0.43 0.57 2.47 2.48 - 0.05 0 - 8 2 0 0 2 0.14 0.86 2.01 2.09 0.05 0 - 14 88 0 0 2 0.30 0.70 0.74 0.72 - 0.02 2 0 1 0 0 1 0.25 0.75 3.25 3.21 0.04 2 - 4 5 3 3 2 0.22 0.78 2.88 2.82 -0 .04 2 - 8 7 4 6 1 0.15 0.85 2,01 1.90 -0 .05 2 - 1 2 4 6 7 - 3 0,16 0.84 1.10 1.12 0.04 2 - 14 5 6 7 - 2 0.30 0.70 0.73 0.68 0.02 4 0 89 0 0 1 0.35 0.65 3.35 3.32 0.03 4 - 4 9 5 7 2 0.19 0.81 2.94 2.88 -0 .07 4 - 14 11 11 12 - 1 0.33 0.67 0.69 0.65 0.02 8 0 3 0 0 3 0.24 0.76 3.83 3.77 -0.01 8 - 12 27 22 27 0 0.20 0.80 0.93 0.85 - 0.05

10 0 5 0 0 5 0.24 0.76 4.29 4.19 0.05 10 - 4 21 15 20 1 0.27 0.73 3.44 3.32 -0 .05 10 - 1 2 32 29 34 - 2 0.47 0.53 0.79 0.59 -0 .06 12 - 4 25 20 26 - 1 0.39 0.61 3.72 3.58 0.03

intensity values I, , /2 and/3 used for the computation are the average values of nine pixels (i.e., on an area of approximately 0.1 mm 2 in the present case). The area over which the intensities are averaged can be reduced by using a premagnified image for capture. The computed values of the isoclinics are within +3 degrees of the measured values except at regions very close to the loading point.

After time averaging and space averaging an intensity variation up to one unit is still observed at certain pixel locations. This value is independent of the absolute value of the pixel intensity. So at lower pixel values the error due to this can be larger. From the consideration of various possible combinations, it is seen that the maximum error due to this variation even at lower pixel values is within +2 degrees in the isoclinic angle and +0.05 in the fringe order and represents the accuracy obtainable using this technique. The other sources of error could be related to the loading errors, thickness variation in the model and errors in manual measurements, etc. It may be seen from Table 1, that the isoclinic angles are within _+ 3 degrees of the measured values and the isochromatic within +0.05 fringe.

Conclusion A computerized whole-field method for determination

of isoclinics and isochromatic parameters in a stressed photoelastic model is presented. The photoelastic data are derived from the intensity data for three analyzer positions in a plane polariscope setup. Since only polarizers are used, no errors due to dispersion in the quarter wave- plates are introduced. The relationships between the intensity values at three analyzer positions and the photo- elastic parameters are derived using the Stokes vector representation of polarized light and its transformations. The accuracy achievable is of the order of _+ 2 degrees for isoclinics and +0.05 fringe for isochromatics. Only frac-

tional fringe orders are obtained directly from the proposed method. Integral fringe orders can be obtained by the method using data obtained at two or three wavelengths as described in Ref. 10.

Acknowledgments

The authors acknowledge R. Vijaya Kumar, STL, SEG, Vikram Sarabhai Space Centre, Trivandrum, India for his support in the experimentation and photography. This research work was supported by I.S.R.O. under advanced R&D program. Their support is gratefully acknowledged.

References

1. Robert, J., "'New Methods in Photoelasticity, "" EXPERIMENTAL MECHANICS, '7 (5), 224-232 (1967).

2. Young, W.C., "'An Automated Process for Three-dimensional Photoelastie Analysis, '" EXPERIMENTAL MECHANICS, 9 ((6), 275-280 (1967).

3. Cernosek, J , "New Compensation Method in Photoelasticity, '" EXPERIMENTAL MECHANICS, 16 (7), 263-266 (1976).

4. PillaL S.A. and Satyanarayana, K.R., "'An Automation System for the Measurement o f Fringe Orders in Photoelastic Models, "" Proc. Nat. Conf. on lnstr, hem at CSIO, Chandigarh, India (1983).

5. Redner, A.S., "'New Automatic Polariscope System," EXPERI- MENTAL MECHANICS, 14 (2), 486-491 (1974).

6. Redner, A.S., "'Photoelastic Measurement by Means o f Computer- assisted Spectral-content Analysis, "" EXPERIMENTAL MECHANICS, 148-153 (Jane 1985).

7. Voloshin, A.S. and Redner, A.S., "'Automated Measurement o f Birefringence: Development and Experimental Evaluation o f the Tech- nique," EXPERIMENTAL MECHANICS, 29 (3), 252-257 (1989).

8. Yao, J.Y., "'Digital Image Processing and Isoclinics, "" EXPERI- MENTAL MECHANICS, 30 (3), 264-269 (1990).

9. Sarma, A.V.S.S.S.R. and Srinath, L.S., "Photoelastic Analysis with Stokes Vector and New Methods for the Determination o f the Characteristic Parameters in 3D Photoelasticity, "" J. Aero. Soc. o f India, 24 (2), 300-306 (May 1972).

10. Srinath, L.S. and Sarma, A.V.S.S.S.R., "'Determination o f Integral Fringe Orders in Photoelastt~ity, "" EXPERIMENTAL MECHANICS, 13 (3), 138-141 (1973).

11. Shurcliff, W.A., "'Polarised Light: Production and Use,'" Cam- bridge Harvard University Press (1962).

Experimental Mechanics �9 29