APPLICATION OF THE GENERALIZED MAXIMUM LIKELIHOOD CRITERION TO EVALUATION OF PARTIAL FRINGE ORDERS IN PHOTOELASTICITY

by I. Miskioglu and N.M. N m a z i

Photoelasticity in stress analysis has long been a powerful full-field technique that helps researchers and designers find solutions to complex prob- lems. In most applications of this technique, fringe patterns are recorded on photographic film for data analysis. The resolution of fringe orders recorded on a photographic negative is, in general, limited to whole- or half-order fringes which, in turn, requires a combination of high loads and optically sensitive birefringent materials to obtain enough data points to reliably define the stress field. This limitation in resolution is due to the fact that the light intensities recorded on the photographic negative are affected by film grain noise, exposure time, sensitivity of film and developing procedures. 1,2 Consequently, recorded intensities do not exhibit the well-behaved sinusoidal trend as would be e ~ p e c t e d . ~ Unless there are whole- or half- order fringes present in the stress field, the recording does not yield a light intensity distribution that results from the birefringence of the material only.

Measurement of partial fringe orders in order to increase the resolution in the photoelastic data has been achieved by automated polar i~cope~,~ and by half-fringe photo- elasticity.6 Both of these methods require ‘live’ models and they cannot be applied to photographic negatives.

In this paper application of the generalized maximum likelihood (GML) principle along with a calibration procedure is described to extract partial fringe orders from photogra- phic negatives of photoelastic fringes.

I Saturation

”

0 Log Exposure Fig. 1 -Typical film response curve

GML FILTERING

Photographic film relies upon the properties of silver halides to record images. When the film is exposed to light, silver halides undergo a latent change that results in the deposition of metallic silver. The amount of darkening produced by the formation of metallic silver is called the optical density. The density of a given type of film is a function of the exposure (light intensity times time) to which the film was subjected. The characteristics of a density-exposure function were first established by Hurter and Driffield7 and a typical curve is shown in Fig. 1.

Optical density D is defined as2

I. Miskioglu (SEM Mrmhrr) is Assistant Priflrssor, Deparlnirnt of Mechanical Enginrrring and N. M. Namnzi is Assistant Profrssor, Departmrnt of Electrical Enginrrring, Mirhignn Trchnological Uniurrsitjl, Houghton. MI where

Experimental Techniques 49

I , = intensity of a reference source of light I2 = intensity of light transmitted through the negative

when illuminated by the reference source.

From Fig. 1, in the linear region

D = y log,, E - Do (2)

where y is the slope of the linear region. Assuming I., the original intensity-creating image is a constant during ex- posure, then E = I . ( A t ) and without loss of generality, letting A t = 1, eq (2) with eq (1) becomes

where K 2 is a constant. Equation (3) shows that representa- tion of intensity in the recorded image is a nonlinear mapping of the original intensity. Also, the recorded image is subject to noise due to the randomness in silver grain formation. This noise is called film grain noise and has been studied extensively. Film grain noise can be approximated by an additive Gaussian random variable distributed about the mean density.

Film grain noise is additive in optical density, so the measurement model can be assumed to be

where

5 = position vector d(z ) = optical density without film grain noise n ( x ) - = film grain noise

In this project, a filter based on the generalized maximum likelihood principle (GML) was implemented to filter the noise in the photoelastic fringe pattern recorded on a negative. The GML algorithm in the discrete form is as follows:8

A (5) z' = 1 , 2 . . . , and d.(x_)- = 0.

E = adaption coefficient

Kd(x ,g ) = covariance function of d(xJ a: = noise variance

d ; ( x ) = estimate of d ( x ) at the ith iteration -

The algorithm proposed in eq (5) uses the covariance function of d ( ~ ) as a priori knowledge. Since this function is not usually available, the image is assumed to be locally Markovian. While there is no restriction on the form of the

covariance function, the separable Markovian covariance function appears to be a feasible choice. Since the image is assumed to be locally Markovian, it is partitioned into sub- blocks of h x k pixels with equal correlation coefficients and each subblock is treated as a different Markov field superimposed by Gaussian noise. Hence K, is assumed in the form

where

IxI - zI I and increments in the horizontal and vertical I x2 - z2 I = directions respectively

u: = optical density variance of the block e l , el = correlations of the adjacent points in the

horizontal and vertical directions, respectively.

Equations (5) and (6) are combined to obtain the algorithm for filtering procedures as A

d,(x) = d , - , ( ~ ) + K . Z C el lXI - ' I1 e 2 " 2 - ' 2 1 [ D(z_)_

-d,-i(z)J (7) I -

A i = 1 ,2 . . . , and d.(x_) = 0

with

The optical density and the light intensities are measured by a digital image analyzer which digitizes light intensity according to the relationship

where

2 = digitized light intensity (0 -black, 255 - white) I = light intensity

K , , yl = constants for the scanner's sensitivity curve

CALIBRATION

After filtering of the film grain noise is achieved, the resulting fringe pattern is used for calibration to evaluate partial fringe orders as follows: for a photoelastic model under load, when viewed through a circular polariscope the light intensity is given by3

I. = Lax cos' ( N T ) (light field) (9)

when eq (10) is substituted into eq (4), with eq (9), the digitized intensity is

where

A h

P

I P 25.4 mm

J L

25.4 mm 101.6 mm 25.4 mm I

P

I P The necessary calibration to obtain the constants B and i3 is done by a beam under four-point bending.

Once the constants B and /3 are determined, the measured intensity 2, will yield a partial fringe order between 0.00 and 0.50 as

(11) 1 N = - COS-I (BIZ2)”20 lr

This partial fringe order is used in conjunction with data on whole- and half-order fringes to determine the fringe order at a point.

EXPERIMENTAL PROCEDURE

A beam in four-point bending was used to obtain the calibration constants B and @. Then these Constants were used to estimate fringe orders on a disk in compression. Both specimens were machined from 6.4-mm thick H-911 material (f. = 19 kN/m). The dimensions of both specimens are depicted in Fig. 2. In the bending specimen the con- stant moment was 2.49 N-m and for the disk P = 383 N. Light-field isochromatic fringe patterns of both specimens were recorded on 400 ASA black and white film and developed simultaneously. Then images were digitized and first the GML algorithm was implemented using KO = 1.0 and 8 pixels by 16 pixels subblock. The filtering was performed on a strip that was 400 pixels by 16 pixels long. An example of results of filtering is shown in Fig. 3. Data were collected from the filtered area and used in eq (10) to evaluate B and P using a log-linear curve fitting procedure. Since eq (10) is only valid in the linear region of D-log,,,E curve (Fig 11, care was taken to stay out of the fogged and saturated regions on the negative.

After the calibration was performed, fringe orders along the horizontal diameter of the disk were estimated.

RESULTS

Figure 4 shows the beam and disk specimens with the light-intensity distribution used as data in this study. For calibration, a total of 997 points were used and the degree of fit obtained was 0.82. The constants evaluated were /3 = -0.911 and B = 177.8.

Figure 5 depicts a comparison of measured versus theoretical values of the fringe pattern in the beam under four-point bending and Fig. 6 is a similar plot for the disk in compression. For both cases agreement was not good

Diameter: 50.8 mm

(b) Fig. 2-Specimens used in the experiments (a) beam specimen; (b) disk specimen

Fig. 3-Example of GML filtering

Experimental Teohniqueo 61

I I

- 0.9

0.8 0 7 0.7 0

0.6 0 E 0.5

0.4

0.3

0.2

0.1

t

I;

(a) beam specimen

- - - - - - -

- -

I I I I I I I I I

(b) disk specimen

Fig. 4-Light-intensity distributions used as input data

1 1.lh .o - Theory n Eatlmaled

- Theory n Esllrn.td

0.8

- 0.8

0.5

0.4

0.3

I I I I 1 I I I ‘ -20.32 -15.24 -10.16 -5.08 -0.0 5.08 10.16 15.24 20.32

Dlstance from the Center of Disk (mm)

Fig. 6-Comparison of the theoretical and estimated partial fringe orders along the horizontal diameter of the disk specimen

(errors up to 35 percent) when measurements were taken in areas close to whole- or half-fringe orders since the response of the negative is nonlinear in those areas. Agreement was very good (less than 5 percent error) for regions where the response of the negative is linear.

CONCLUSION

Partial fringe orders can be extracted from photographic negatives of photoelastic fringes with reasonable accuracy. The accuracy can be improved by prefogging the film and adjusting the exposure so that saturation does not occur.

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3. Dally, J. W. and Riley, W.F., Experimental Stress Analysis, 2nd ed., McGraw-Hill (197%). 4. Redner, S.. “New A u t m t i c Polariscope System, ’ I EXPERIMENTAL

5. Mueller, R.K. and Saackel, L., “Complete Automatic Analvsis o/ Photoelastic Fringes,” EXPERIMENTAL MECHANICS, 19 (7).

6. Voloshin. A.S. and Burger, C. P., “Half-frp‘nge Photoelasticity: A New Approach to Whole field Stress Analysis,” EXPERIMENTAL

7. Mees, C.E.K., The Theory of the Photographic Process, Mac- Millan (1954). 8. Namati, N.M. and Sfuller, J.A., “A New Approach to Signal Registration with the Emphasis on Variable Time Delay Estimation,” IEEE Transactions on Acoustics, Speech and Signal Prowssing,

693-708 (1975).

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62 Jmuary/February 1991