OPERATING INSTRUCTIONS FOR. COPA (Computer Aided Photoelastic Analysis) PROGRAMME: VERSION 3. Philip Siegmann & Eann Patterson

Transcription

1 OPERATNG NSTRUCTONS FOR COPA (Computer Aie Photoelasti Analysis) PROGRAMME: VERSON 3 Philip Siegmann & Eann Patterson ntroution This oument aompanies a new piee of software initially proue in the Experimental Stress Analysis Laboratory in the Department of mehanial Engineering at the University of Sheffiel. COPA version 3 is written in matlab p-files an uses an improve unwrapping algorithm with respet to previous versions. t is base on an algorithms whih are esribe in [Siegmann P., Bekman D. an Patterson E.A. A robust approah to emoulate an unwrapping phase-steppe photoelasti ata, Exp. Meh. 45(3), pp , 005] an are exeute in MatLab. The objetive is to proess phase-steppe photoelasti images aquire in either a transmission or refletion polarisope. The input for the software is.six files generate by the Cathsix software also proue by the Experimental Stress Analysis Laboratory. Data output is in several forms as esribe below inluing a new file suffixe pa. A paper esribing the algorithms an proviing examples an valiation is in preparation for submission for publiation in the open literature. For more information ontat Philip Siegmann (philip.siegmann@uah.es). nstallation The urrent COPA program is written in Matlab, therefore you nee to have the Matlab program installe in your omputer. Copy the COPA iretory into any iretory of your hoie in your omputer. Open Matlab an selet as Current Diretory: the COPA iretory, or set a path to it (press File, Set path, ). Write COPA in the Matlab omman line an press enter. Operation The COPA winow will isplay a front page an then move the main winow. To proess photoelasti images, start from the top left an work leftwars an own using at least one button in eah ark grey zone. The results of eah operation will be shown in the figure spae 1

2 in the bottom left of the winow an a re aption below the figure will give an instrution on what to o next. Data Aquisition Data shoul be aquire using the Cathsix program to apture six phase-steppe images an pakage them in a.six file. Use the Open button to ientify the appropriate.six file. Note that other file formats an be rea, please onsult the authors for more information. After seleting the file to be opene the six images will be isplaye in a new winow labelle as shown in table 1 in the Appenix using pseuo-olour to represent intensity. This winow is remove automatially after a few seons, an the wrappe isolini map orresponing to equation (1) in the Appenix is isplaye in the main winow. Saving ata One a file has been opene the ata isplaye in the figure in the main winow an be save using the Save button after seleting the appropriate file format using the rop-own menu to the left of the button. f no seletion is mae the efault file type is txt. Bakgroun Removal / Applying a Mask The ark grey box in the top right orner of the main winow offers two hoies: importing a mask or reating a new one. A mask that has been reate by Cathsix an be importe by opening a.bng file via the mport mask button. The mask is applie automatially an the bakgroun in the image will be remove in the figure isplaye in the winow. The ata save as.pa save the six phase shifte image an the reate mask in Crate Mask option. By open a.pa file also the orresponing mask is loae automatially. (The pa file, like.six, is an aapte.bmp file, but with aitional mask inlue). Alternatively, a new mask an be reate by using the Create Mask button. A new winow will appear with the image orresponing to a ark fiel irular polarisope on the left an a series of buttons of on the right. You an hoose to efine a mask using a irle, a polygon or any ombination of these shapes. Selet the require shape an follow the on-sreen instrutions. t is possible to ut out an enlarge an area of the image using the Cut out button to efine the area of interest. One a shape has been efine you must iniate whether the area to be maske is insie or outsie the shape using the buttons: Mask insie or Mask outsie respetively. Press Finishe Mask to return to the main winow. t is possible to use the funtions repeate to buil a omplex shape for the mask. f a mistake has been one by

3 reating mask you an either lik the right mouse button uring the input of points, or lose the reate mask winow without pressing Finishe Mask. solini Demoulation This funtion is not essential but is reommene when the isohromati fringe range is greater than half a fringe, beause the half orer fringes will ause errors in the isolini map as a result of moulation. Also 45º isolinis ause errors in the isohromati map via the same moulation. The proess esribe in the Appenix in lines 43 to 14 is applie when the solini emoulation button is use. This proess is almost entirely automati exept the operator being aske to selet the loation from whih to start the unwrapping of the areas esribe starting at line 64 in the Appenix. t is best to hoose the largest area isplaye in the temporary winow beause this will provie the fastest solution. The emoulate isolini map is isplaye in the figure in the main winow when the proess is finishe. t is possible to ajust the winow size, w use in the emoulation proess using the slier bar below the button. However, the efault value is usually satisfatory. More etails about this winow size are given in lines 108 to 118 in the Appenix. The efault is for a Wiener filter to be applie to the isolini ata prior to emoulation as esribe in lines 91 to 99 of the Appenix. This filtering an be omitte by removing the tik from the box beneath the Demoulation button. t shoul be note that regarless of the use or otherwise of this filter, unfiltere isolini ata is always use in the alulation of the relative retaration ata. Relative retaration/sohromati Phase Map On seleting the sohromati Phase Map button the wrappe relative retaration is alulate using equation (4) in the Appenix an isplaye in the figure. The efault alulation is to use the emoulate isolini map, but the raw isolini ata an be employe instea by removing the tik in the box beneath the sohromati Phase Map button. sohromati Fringe Orer The relative retaration or isohromati phase map (same ata but alternative names) is perioi or wrappe an must be unwrappe to generate a ontinuous map of fringe orer. This proess is performe when the Unwrap isohromati button is presse an is esribe in lines 15 to 137. The proess an be performe ompletely automatially with the see point being selete as the pixel having the highest quality. To selet the see point manually 3

4 tik the box uner the Unwrap isohromati button an a temporary winow will appear. n this winow the quality map is isplaye in grey sale for eah pixel, arker pixels have less quality value. The unwrappe (ontinuous) map of isohromati fringe orer is isplaye in the figure in the main winow at the en the proess. The unwrapping proess automatially sets the lowest fringe orer in the map to zero so that no alibration maybe neee, otherwise see below. There is some ambiguity in the efinition of isolini angle an hene in the solution for the isohromati fringe orer, suh that the resultant map maybe inverte as explaine in line 140 to 144 of the Appenix simply press the nvert button to reverse the istribution. Selet 5 bins Unwrapping for a faster unwrapping, for more preise but slower unwrapping selet 10 or 100 bins Unwrapping. The respetive quality maps an be seen when the option of Selet nitial Pixel is ativate. For noisier ata, more bins shoul be onsiere for the unwrapping. Calibration The unwrappe isohromati map may require alibration if there is no zero orer fringe present. Enter the value of the fringe orer you inten to use for alibration in the box in the ark grey zone in the bottom right orner of the main winow. The use of half orer or integer fringes is reommene. Then, press the Calibration in N button an a temporary winow will appear with a isplay of ¼ orer ontours. Selet a point on an appropriate ontour with the mouse an right lik. The proess an be performe as many times as neessary. Subsequently a map of maximum shear stress or strain (or ifferene in prinipal values) an be generate by performing a further alibration. Again enter the value of the stress or strain metri in the same box, but also type in the appropriate units in the box below, then press the Calib. Stress/strain button an selet the point at whih the stress or strain is known. This might the same or a ifferent point to the one use for the fringe orer alibration. Data Display The figure (inluing the olour bar) an be save using the Save urrent figure as.eps,.tif,.jpg button. The ata in the urrent figure an be save using Save urrent figure as.txt (asii file), the ata will be save in matrix format (same size as the figure) an in the hea line of the 4

5 reate asii file are given the units. The button Save *.pa file o not save the proesse image, only the six phase shifte images an the mask. Profiles aross the ata isplaye in the figure an be obtaine by pressing the Profile button an then left liking on a start point in the map followe by a left lik of the en point of the esire profile followe by pressing the Enter key. A temporary winow is isplaye with the plot of the ata as a funtion of pixel loation along the profile. After ata proessing the map isplaye in the figure an be selete using the rop-own menu marke Show below the Profile button. Similarly, the units of the ata an be hange using the ajaent rop-own menu marke Units. The olour sheme for the figure an be selete using the rop-own menu marke Color Map. 5

6 Appenix: Desription of the algorithm Nomenlature Α(i) Α(i+1) h nitial area for proessing Next area for proessing Threshol height use in spike removal proess applie to partial erivative ata i 1...i 6 ntensity measure in polarisope orientate as shown in Table 1 i m i v Ι = i 4 i 6 Ι = i 1 i Ι s = i 5 i 3 n,m N,M Q θ (n,m) w 1 w ψ x (i,j), ψ y (i,j) ψ x m), ψ y m) θ ρ σ 1, σ ntensity term to aount for stray light entering the polarisope ntensity observe in a polarisope when the fast axes of all the elements are aligne Co-orinates of pixel of interest Number of pixels in x, y iretion in ata map Quality value for the pixel loate at (n,m) Winow imension use in evaluating quality map Winow imension use in reuing number of areas to be proesse. Relative retaration Differene in partial erivative of the phase ata with respet to the x, y iretions Mean of the ifferenes in partial erivative of the phase ata with respet to the x, y iretions within the winow solini angle Perentage area left unwrappe Maximum an minimum prinipal stresses 5 10 ntroution The aim is to generate a ontinuous map of isohromati fringe orer that is inepenent of the isolini angle an similarly to proue a moulo π map of isolini angle efining the iretion of one ientifie prinipal stress that is inepenent of the isohromati fringe orer. Possession of the latter enables a relatively straightforwar alulation of the former using the phase-stepping relationships. Ramesh an Ganapathy [1996] ientifie the phase-stepping algorithm given by Patterson an Wang [1991] as the most robust so it is use here as the funamental step to obtain phase ata. Six phase steps are employe in a irular polarisope as provie in Table 1 suh that the isolini angle is obtaine as: 6

7 15 1 s θ = artan (1) where s = i 5 - i 3 an = i 4 - i 6. The relative retaration an be obtaine using many alternative formulae but will always ontain the moulation introue by the ambiguity in the isolini ata. The relative retaration is usually efine as: = artan () osθ 0 5 where = i 1 - i. Patterson an Taroni [003] have reently shown that if the usual artangent operator is replae by an arotangent funtion by inverting expression () then the perio bounaries in the relative retaration are translate from -π/4, π/4, 3π/4 to 0, π/, π This is more onvenient for unwrapping an interpretation of the ata beause the perio bounaries orrespon to half orer an integer fringes whih are reaily observe in a manual polarisope; an beause fewer unwrapping operations may be neessary. To further reue the number of unwrapping operations, a simple logi operation has been employe to effetively exten the range of relative retaration from π/ to π [Sparling et al], using: osθ = π + ot for i 1 > i & i 4 < i 6 (3a) ar 30 osθ = π + arot for i 1 < i (3b) osθ = arot else (3) A further refinement was introue to ensure high moulation over the entire ata fiel an hene to reue noise by merging the ieas given above with the expression given by Quiroga an Gonzalez [1997] to give: 35 osθ + osθ s = π + artan for i 1 > i & i 4 < i 6 (4a) osθ + osθ s = π + artan for i 1 < i (4b) osθ + osθ s = artan else (4) 7

8 Maps of the isolini angle an relative retaration obtaine using expressions (1) an (4) are shown in figures 1a an b. The influene of the isolini angle on the relative retaration map is learly seen at loations where the isolini angle is 45 egrees an similarly the influene of the relative retaration on the map of isolini angle is learly seen at the loations of half orer an integer fringes. n orer to emoulate this ata, the map of isolini angle is examine an the bounaries of the regions assoiate with eah prinipal stress are ientifie using a zero-rossing bounary etetion tehnique. This tehnique is similar to ege etetion methos in image proessing. Briefly, in this tehnique two matries, terme positive an negative are reate that are equal in size to the ata array being onsiere an initially populate with zero values. A raster san of the isolini map is performe pixel by pixel with a two by two winow forme by the pixels to the right, below an iagonal to the right of the pixel being onsiere. The sign of the isolini angle at eah pixel in the winow is examine, if the sign of all the values are the same then no ation is taken an the san moves on the next pixel, otherwise for those pixels in the winow with a negative values of isolini angle the orresponing zero in the negative matrix is hange to one an similarly for the pixels with positive values the orresponing zero in the positive matrix is hange to one. The san ontinues so that when it is omplete the loation of all the zero-rossings is ientifie by unity values in the positive an negative matries. Negative areas are efine as those enlose by a isontinuity represente by a positive or upwar step in the isolini angle whih will our at pixel loations efine by the unity values in positive matrix, similarly positive areas are enlose by a negative or ownwar step in the isolini angle efine in the negative matrix. The zero-rossing bounaries an orresponing areas for the ata in figure 1a are shown in figure 1 with the positive areas outline in white an the negative ones in blak. These areas are subsequently unwrappe ommening from A(i), whih is best assume to be the largest, an employing a quality map, base on the phase erivative variane proeure propose by Pritt [1996]. The quality of a pixel at loation (n,m) was ientifie using Q θ ( ) + ( ψ ( i, j) ψ m) ) 1 = x x w1 i, j w1 xw1 i, j 1 (5) m) ψ ( i, j) ψ m) w xw1 y y 8

9 where ψ x ( i, j) is the ifferene in the partial erivative with respet to x of the isolini angle obtaine from equation (1) for the pixel loate at (i,j) in a winow w 1 xw 1 entre on the pixel (n,m). Prior to applying the ifferene operation, the partial erivative is smoothe in a wrapping operation that involves sanning the partial erivative ata for spikes of height h/ an translating the values in the spike by h, where h is π/ for the isolini ata. ψ x m) is the mean value of the ifferene in the partial erivative over the winow. ψ y ( i, j) an ψ y m) are orresponing parameters in the y iretion. t is worthy of omment the ψ ( i j) x, y, term is effetively a seon orer partial erivative of the phase map.. The bounary of A(i) is searhe for the highest quality pixel an the area on the other sie of the bounary at this point is A(i+1). When a isontinuity exists between areas A(i) an A(i+1) of magnitue between π/4 an 3π/4 then value of the isolini angle for all points within A(i+1) are translate by -π/; similarly when isontinuities between areas A(i) an A(i+1) of magnitue between -π/4 an -3π/4 are ientifie then all the points with A(i+1) are translate by π/. Then, the area A(i+1) is inorporate into A(i) an the pixel with the highest quality value on the new bounary is ientifie an the proess repeate until all the areas have been onsiere. t shoul be note that only the isontinuity aross the bounary at the pixel with the value is Q θmax is onsiere at eah step. The impliations of the above proess is that isontinuities of height π are allowe to remain in the map whih is onsistent with the lassial efinition of the isolini angle [Froht, 1940]; an seonly that zero-rossings ourring beause the isolini is zero value are ignore. Real ata, partiularly from refletion photoelastiity, tens to ontain spekle whih auses an enormous number of small areas to be ientifie by the bounary etetion tehnique an the subsequent analysis of these areas extens the proessing time unaeptably. The proessing time an be reue by an orer of magnitue if the isolini map is filtere prior to applying the bounary etetion tehnique an if the smallest areas are left wrappe. The filtering proess employe was a Wiener filter [Press et al, 199] whih is an aaptive filtering tehnique that employs a small amount of smoothing when large varianes an vie versa. The filter was applie to the s an ata prior to their use in equation (1) to generate the isolini angle. 9

10 The spekle generate by the iffuse nature of the refletive layer in a photoelasti oating an ause many very small areas to be ientifie by the zero-rossing bounary etetion tehnique. However, these small areas were foun to not signifiantly effet the form of the isohromati map of when a quality-guie unwrapping routine was employe although their influene oul be very severe when a raster or floo-fill routine is use. Hene it is possible to leave the smallest areas unproesse an ahieve a signifiant reution in the proessing time for emoulation. Typially, between 5% an 10% of the area in the map of isolini angle is left unproesse. n the program written to implement the new approah the size of the winow (w x w ) for the filter an the perentage (ρ) of area left wrappe were linearly relate as: 10 ρ = w (6) ( w ) max where w max is epenent on the size of the ata array suh that 0MN w = (7) 56 ( ) max where N an M are the number of pixels in the x an y iretion of the ata map an w an be set by the operator to be 0 w ( w ) max < but has a efault value of 0.5(w ) max. The seletion of w allows the algorithm to be ajuste base on experiene to optimise its performane with ata sets from ifferent experimental environments suh as transmission an refletion photoelasti arrangements. Another important feature of the implementation of the above proess is that the filtere isolini ata is only use to generate the zero-rossing bounaries an the unfiltere isolini ata is unwrappe so that any errors introue by the filtering proess are not propagate into the maps of stress iretion an magnitue. For the ata shown in figure 1 the winow size, w was 0 pixels an the emoulation proess results in the isolini map shown in figure 1. The emoulate isolini angle is substitute into expression (4) to obtain a map of the relative retaration whih is also emoulate from the influene of the isolini angle (figure 1e). A quality-guie approah [Pritt, 1996] is use to unwrap the relative retaration using a phase erivative variane quality map [Ghigli & Pritt, 1998] whih is similar to the one use in the emoulation proess esribe above 10

11 exept that the partial erivatives of the relative retaration are employe in equation (5) an the threshol height, h for the spike removal is π. n this ase an unwrapping algorithm prepare for the fringe projetion tehnique [Hereia & Patterson, 003] has been use beause it was reaily available an alreay been optimise for spee of proessing. Proessing times were foun to be similar to those require by the Wang & Patterson [1995] algorithm. At the en of the unwrapping proess the results are translate along the fringe orer axis so that the minimum fringe orer is assigne as zero. f a zero orer fringe is present in the ata fiel, then no alibration of the unwrappe map will be neessary, otherwise the map of isohromati fringe orer an be alibrate by proviing a known fringe orer at any point in the ata fiel. The assoiation with either the maximum or minimum prinipal stress iretion of the see or initial area A(i) in the emoulation proess ontrols whether the isohromati map that results from the unwrapping proess is upright or requires inversion. n the latter ase, the isohromati fringe orer istribution is reflete about its meian value. A typial result is shown in figure 1f. Referenes Froht, M.M., 1940, Photoelastiity, vol.1, John Wiley & Sons n., New York, Ghiglia, D.C., Pritt, M.D., 1998, Two-imensional phase unwrapping; Wiley- ntersiene, New York. Hereia-Ortiz, M., Patterson, E.A., 003, On the inustrial appliations of moiré an fringe projetion tehniques, Strain, 39: Patterson, E.A., an Wang, Z.F., Towars full-fiel automate photoelasti analysis of omplex omponents. Strain, 7: Patterson, E.A., & Taroni, M., 003, High frequeny quantitative photoelastiity applie to jet engine omponents, Pro SEM Ann Conf. Experimental & Applie Mehanis, Charlotte, NC. paper no. 5, also submitte for publiation in Experimental Mehanis. Press, W.H.; Flannery, B.P.; Teukolsky, S.A.; an Vetterling, W.T., 199, Optimal (Wiener) Filtering with the FFT in Numerial Reipes in FORTRAN: The Art of Sientifi Computing, n e. Cambrige, Englan: Cambrige University Press, pp Pritt, M.D., 1996, Phase unwrapping by means of multi-gri tehniques for interferometri SAR. EEE Trans. Geo. & Remote Sensing, 34(3): Quiroga, J.A., an Gonzalez-Cano, A., 1997, Phase measuring algorithm from extration of isohromatis of photoelasti fringe patterns, Appl. Optis 36(3):

12 170 Ramesh, K., an Ganapathy, V., Phsae-shifting methoologies in photoelasti analysis appliation of Jones alulus. J. Strain Analysis 31: Sparling, S., Gagnon, J., Komorowski, J., Photoelasti analysis by 6-image phasestepping with RGB input. NRC nternal report LTR-St-154 4, NRC Ottawa Wang, Z.F., an Patterson, E.A., 1995, Use of phase-stepping with emoulation an fuzzy sets for birefringene measurement. Optis an Lasers in Engineering, : Table 1 Six-step shemes for igital transmission photoelastiity [Patterson & Wang, 1991] mage No φ β Light intensity i 1 0 π/4 i = i m + i os 1 v i 0 -π/4 i = i m i os v i i = i m i sin sin θ 3 v i 4 π/4 π/4 i = i m + i os sin θ 4 v i 5 π/ π/ i = i m + i sin sin θ 5 v i 6 3π/4 3π/4 i = i m i os sin θ 6 v 1

13 Figure 1: Simulation results from a is loae in iametral ompression (from top by rows): (a) Map of the isolini angle obtaine using expression (1) from six images base on those esribe in Table 1. The influene of half an integer fringe orers an be seen as isontinuities an the isolini angle swithes allegiane between the prinipal stresses aross these isontinuities. The map is wrappe with perios of π/. (b) Map of the relative retaration obtaine using expression (3) from six images as in (a). The influene of the isolini angle an be seen at loations with an isolini angle of 45 egrees. The map is wrappe with perios of with perio ens at 0, π, π () Zero-rossing bounaries. Areas within the bounaries are assoiate with a single prinipal stress. Positive areas are enlose by white lines an negative ones by blak lines. Positive areas are protrusions surroune by a step own to their neighbours whilst negative areas are inents surroune by a step up to their neighbours. The winow size, w use to ompute this map was 0 pixels. () Demoulate map of isolini angle generate using the zero-rossing bounaries in () to unwrap the ata in (a). Some noise remains at the loations of half an integer fringe orers. The perioiity of the ata is orresponing to the lassial efinition of isolini angle, an it is everywhere assoiate with the same prinipal stress. (e) Demoulate map of relative retaration proue using the ata in (b) an (). (f) Unwrappe map of relative retaration generate from the ata in (e) using the quality guie algorithm. 13