Three fringe photoelasticity - use of colour image processing hardware to automate ordering of isochromatics
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Three fringe photoelasticity - use of colour image processing hardware to automate ordering of isochromatics by K. Ramesh and Sanjeev S. Deshmukh, Dept of Mechanical Engineering, Indian Institute of Technology, Kanpur, India 208016.
The early application of digital image processing (DIP) technique to automate photoelastic analysis is based on black and white ( B & W ) image processing systenzs. The various methodologies reported can be traced to rely upon one or several features of u B & W DIP system. With the advancements in computer technology, nowadays colour image processing systems are available at affordable prices. A systematic study has been carried out on how to use the red green blue (RGB) value recorded using a colour image processing system for determining fringe orders up to 3 in afringefield. The study has revealed that instead of directly using RGB values corresponding to a dark field image, ifthe difference of RGB values between bright and darkfield images is used, the number of noise points is less. Experiments have been carried out for several arbitrary lines in a circular disk under diametral compression and the results are found to be good.
Key words: Isochromatics, fringe ordering, photoelastic analysis, colour code, pixel, RGB colour model, colour image processing
B, e Least square error F, Material stress fringe value G, N Fringeorder R, B &W Black and white CCD Charge coupled device DIP Digital image processing Pixel Picture element representing a point RGB Red green blue SCA Spectral content analysis
Grey scale value of the B plane
Grey scale value of the G plane
Grey scale value of the R plane
Among the experimental techniques, optical methods have the advantage of yielding full field infermation in the form of fringes. Fringe ordering is one of the crucial steps in the process of determining quantitative information from the fringe field. The method of photoelasticity provides the information of difference in principal stresses (isochromatics) and their orientation (isoclinics) in the form of fringes. A large class of problems such as determination of stress concentration factor(SCF), stress
intensity factor (SIF) and also evaluation of contact stress parameters require only the information of isochromatic fringe orders from the field. For several problems in which an evaluation has to be made between different designs, it is enough ifoneknows theisochromaticfringeorder'. Only in special cases where one is interested in determining individual components of stress tensor, one requires both isochromatic and isoclinic fringe orders.
Automation of the ordering of photoelastic fringes is one of the challenging tasks even today. With the advent of digital image processing (DIP), several attempts have been made to obtain fringe orders for the entire field. The earlier investigators have essentially usedB & W image processing systems in view of their easy availability and lower cost to devise their algorithms. The approach can be broadly classified into, (i) fringe thinning followed by fringe cl~stering~.~, (ii) intensity based measurement technique in (a) spatial d ~ m a i n ~ - ~ and (b) frequency d ~ m a i n . ~
The use of fringe thinning followed by fringe clustering is applicable to both static and dynamic problems as only one image needs to be processed. It is shown elsewhere3 that among the fringe thinning techniques, intensity based algorithms are better in general and it has been established that the algorithm of Ramesh and Pramod is the best to extract the fringe skeletons. Though a class of fringe thinning algorithms including that of Ramesh and Pramod uses intensity data, the intensity values are used to extract relative minimum intensity points forming the fringe skeleton. On theotherhand, inintensity basedmeasurement techniques, the intensity values are directly used to find out the fractional fringe order in one way or the other. One of theearliestapproaches in thisdirection was theintroduction of hayfringe photoelasticity by Voloshin and Burger"' in 1985. In this approach, the model material and the loading are so chosen that the maximum fringe order in the fringe field does not exceed 0.5. They digitised the image and establishedone toone correspondencebetween the intensity value and the fractional fringeorder by properly calibrating the system.
The restriction that maximum fringe order in the field should be only 0.5 while using half fringe photoelasticity is overcome with the introduction of phase shifting methodologies in photoelasticity6. These methods are basically restricted to static problems as one has to record up to six images for a given situation. Ramesh et a16 has brought out the elegance of the algorithm of Patterson and
'Strain', August I996 79
Wang'. The phase shifting technique can yield both fractional fringe order and isoclinic angle (with some restriction) at every point in the fringe field. To find the absolute fringe order at any point one needs the total fringe order for at least one point in the fringe field to be supplied by auxiliary means.
Automated evaluation of fractional fringe order for every point in the fringe field by processing one photograph has also received attention in the literature. Quan et al' have shown that by using carrier fringes and operating in the frequency domain, it is possible to evaluate the fractional fringe orders using one photograph. The approach is computationally intensive as direct and inverse Fourier transforms have to be computed. The utility of the method is restricted by the availability of high density carrier fringes.
Sanford'reintroducedtheuseofwhitelightin transmission photoelastic analysis in 1986 and proposed a method of identifying fringe orders by analysing the spectral content of the transmitted light using a spectroscope. The method is based on the premise that each fringe order has a distinct spectral signature. For each data point, the transmission spectrum as a function of wavelength is experimentally obtained. A theoretical equation has been developed for the transmission spectrum as a function of retardation. The valueofretardation in the theoretical equation is iteratively changed until the theoretical and experimental curves are close in a least squares sense. It is to be noted that the retardation at each point is determined iteratively. Further, this approach is essentially a point by point technique.
Voloshin and Redner' came up with commercial equipment based on spectral content analysis (SCA). Carazo-Alvarez et allo combined SCA and phase shifting technique to completely automate the photoelastic analysis. They used SCA to determine total fringe order at just one point in the fringe field which is used as an input to unwrap the phases in phase shifting technique. They have also conducted some studies on how precisely one has to measure the s p e c m experimentally. The white light source used, in general, emits wavelengths in the range of 450 to 750 nm. They compared the use of collecting experimental data in steps of 40 nm and also in steps of 5 nm. They established that collecting dataat 40nm intervals is reasonably accurate for problems where fringe order gradient is less than 1.4 fringe/mm. Extending this idea, Haake and Patterson1I proposed a new approach wherein the use of a spectroscope is replaced by using eight high quality optical filters in conjunction with a B & W CCD camera. This has made the method wholefield. However, unlike phase stepping, here for each pixel, the fringe order has to be determined iteratively in a least squares sense and thus is computationally intensive. The error introduced due to mismatch of quarter wave plates in SCA was studied by Ajovalasit et all2 in 1995. They have used colour image processing system for their analysis.
A colour image processing system can be configured to
capture images in RGB mode. In RGB mode, the image is identified as a superposition of the image planes of red, green and blue. Similar to a B & W image processing system, the intensity is quantifiedin the range of 0-255, but for each image plane of R,G and B. Thus, for each point on the colour image, identified as a picture element (pixel), three numbers are specified to represent the colour at that point (pixel).
Voloshin and Burgefl exploited the hardware feature of the B & W image processing system to identify 256 grey level shades between pitch black and pure white to directly find the fractional fringe order between 0 to 0.5 or any fringe field in which the difference between maximum and minimum fringe order is 0.5. This imposed severe restrictions on selection of model material and loading. Despite this, several successful applications for solving practical problems are reported in the literature. While using colour images, in principle, one can identify fringe orders up to three using a colour code.
In this paper, a systematic study is undertaken to explore the use of RGB values of acolour image to unambiguously identify fringe orders up to three. Ajovalasit et a l l z have reported to a limited extent the use of RGB values to identify fringeorders. However, they weremore concerned with the error introduced in the theoretical estimation of intensity of light transmitted due to mismatch of quarter wave plates while using white light.
System con figuration
The system consists of a colour CCD camera (TMC-76 RGB PULNiX) giving RGB output. The camera has a pixel resolution of 512 x 512 pixels and it digitises the image at video rates. The camera is connected to a PC based image processing system equipped with a colour image processing card (MVP-AT, Matrox corporation) connected to a high resolution video monitor. The system can be configured to have four 512 x 512 pixels x 8 bits monochrome frame buffers or two 512 x 512 pixels x 16 bits colour frame buffers or one 5 12 x 5 12 pixels x 24 bits colour frame buffer. The 24 bit colour frame buffer is generated by overlapping of three 8 bit monochrome frame buffers.
Determination of RGB calibration table
The basic methodology proposed here is to compare the RGB values of a point with unknown fringe order with the calibrated RGB values assigned with known fringe orders so as to determine the fringe order at a given data point. The calibration table containing RGB values associated with known fringe orders is prepared using a beam under four point bending. The beam under four point bending was chosen because the fringe order variation is linear over its depth. The calibration table is generated upto fringe order three, as beyond this, the colours merge and it is difficult to use a colour code. Figure 1 shows abeam under bending and it is clearly seen that the fringes are exactly horizontal.
80 'Strain', August 1996
It may be noted that the digitised colour image of the beam is photographed using a B & W negative. In order to account for the fringe gradient, the calibration tables corresponding to 0-1 (low fringe gradient), 0-2 (medium fringe gradient) and 0-3 (high fringe gradient) fringe orders are obtained. Initially, the load is applied such that, the farthest fringe order seen is one. Data of RGB values were collected along a vertical line starting from fringe order zero to fringe order one. In view of the linear variation of the fringe gradient, the fractional fringe order is easily calculated for each pixel, which is then used to construct the calibration table. Care was taken while preparing a calibration table, so that the table would not include a noise point with sudden change in the RGB values. This was ensured by collecting data from the central region of the beam and by taking the average of 40 pixels horizontally for each fringe order. Similarly, load was increased in steps such that the farthest fringe order seen is two and then three. Data was collected in a similar fashion for fringe order variations of 0 to 2 and 0 to 3. Tables I and I1 show the calibration tables for fringe order variation of 0 to 1 and 0 to 3 respectively.
86 1 0 4 208 209 85 106
Fig.1 Beam under four point bending (Please note that, the colour image on the video monitor is captured using a 6 B W film.)
. . . ~ ~ ~ 1.0000
The calibration tables were prepared initially for the dark field arrangement of a circular polariscope. Other combinations were also prepared so as to minimise the effect of non-uniform illumination in actual experimental situations. Two approaches were adopted. In one, the RGB values of the unloaded model were subtracted from RGB values of the loaded model. In another approach, the RGB values of dark field are subtracted from the bright field intensity values. Tables 111 and IV show the calibration tables for a bright-dark combination for fringe order variations of 0-1 and 0-3.
Evaluation of fringe order at a point in a least squares sense
By comparing RGB values of any data point with Ihe RGB values in the calibration table, fringe order at that point can be determined. Ideally, RGB values have to be unique for any fringe order. However, in view of experimental difficulties, the RGB values corresponding to a data point may not exactly coincide with the RGB values in the calibration table. For any test data point, an error term e is defined as,
e= (R,-Rc) + (Gc-Gc) + (Be-Bc)
RGB Table: Dark field 10-1)
R G B N
68 70 67 67 72 73 76 76 81 80 84 89 90 96 98
103 109 110 1 1 4 121 124 1 3 0 130 138 14 3 147 147 152 155 158 158 1 6 1 160 162 162 163 165 167 166 168 167 169 170 1 7 1 169 169 168 170 170 170 169 170 168 165 163 164 1 6 1 162 159 156 156 153 150 145 147 143 137 137 138 12 8 129 123 1 2 1 126 112 106 100 106
95 92 94 88 83 84 86
91 86 90 93 90 97
1 0 1 108 1 1 3 122 128 134 142 148 157 1 6 1 164 172 180 1 8 1 190 192 202 203 208 212 219 2 2 1 225 227 235 234 24 0 242 246 247 247 248 2 5 1 2 5 1 2 5 1 2 5 1 2 5 1 24 9 2 5 1 250 250 245 243 24 0 239 236 233 230 228 222 2 2 0 213 210 206 200 192 189 185 173 170 166 158 149 146 135 1 3 0 123 108 113 108 107
94 98 95 93 94
103 102 103
77 82 86 83 95 89 94 93
108 100 117 113 124 120 125 1 3 0 14 9 1 4 5 145 157 158 160 1 6 1 168 173 172 177 178 180 184 171 185 173 173 166 165 167 166 165 156 158 148 1 5 1 146 138 132 127 130 128 120 111 102
99 9 1 83 73 64 62 58 42 37 52 36 26 45 28 34 34 4 2 37 63 58 64 99 88
100 106 134 134 147 160 165 169 178 194
0 .00000 0 .01163 0 .02326 0 .03488 0 . 0 4 6 5 1 0.05814 0 .06977 0 .08140 0 .09302 0.10465 0.11628 0 . 1 2 7 9 1 0.13953 0 .15116 0 .16279 0 .17412 0 .18605 0 .19767 0 .20930 0 .22093 0.23256 0 .24419 0 .25581 0.26744 0 .27907 0.29070 0.30233 0.31395 0.32558 0 .33721 0.34884 0.3604 7 0.37209 0.38372 0 .39535 0 .40698 0.41860 0.43023 0.44186 0.45349 0 .46512 0.47674 0 .48837 0 .50000 0.51163 0 .52326 0.53488 0 . 5 4 6 5 1 0.55814 0 .56977 0 .58...