Optically based measuring techniques are currentlyused for many types of measurement, including flowfield,1,2 displacement,3,4 surface roughness,5 and tem-perature.6,7 In many instances, optical techniques arepreferred over traditional methods because they aretypically noninvasive and provide whole-field mea-surement data. Projection moire interferometry(PMI) is a low-cost, nonintrusive, whole-field dis-placement measuring technique. PMI has proved auseful tool in numerous applications such as windtunnel tests of large-scale structures8,9 and surfacecontour measurements.10
A typical PMI configuration is shown in Fig. 1: Alight source is projected through a grid pattern and afocusing lens onto the structure of interest. When thestructure is in a reference state, the grid lines have aspatially uniform distribution. Under load conditionsthe object deforms and changes the distribution of thegrid lines. When the deformed grid lines are sub-tracted from the reference grid lines, a fringe patternis generated. This fringe pattern is an interferogram,which contains information (moire fringes) fromwhich the magnitude of the structural displacementscan be extracted. The magnitude of the displace-
ments is computed by use of a fringe sensitivity co-efficient (FSC).11 The FSC values are computedthrough a calibration procedure that consists of an-alyzing fringe patterns generated from known dis-placement fields.
As is common with most measurement techniques,PMI has inherent characteristics that facilitate orlimit its use in certain types of applications. The lim-itation addressed in this paper is the tedious andtime-consuming process of obtaining FSC valuesfrom standard calibration procedures. The currentcalibration procedure requires the use of a planarsurface and the ability to rotate or translate thissurface known amounts within a certain tolerancerange. This process can be especially difficult inlarge-scale applications, wind tunnel testing, ane-choic chamber testing, etc., for which it is difficult todesign and implement a precise test object rotation ortranslation system. An automated, self-calibrating,whole-field projection measuring system wouldgreatly increase testing efficiency and save test setupand data acquisition time, which can be expensive forall objects, particularly those of large scale.
In this paper we continue first by presenting anoverview of the conventional PMI method. The pro-cedure and theory of the virtually calibrated projec-tion moire interferometry (VCPMI) technique arethen presented. FSC values computed by the VCPMIand the PMI methods are then computed and com-pared by use of a flat plate test object. Finally, theVCPMI technique is applied to the leading edge of anairfoil test object and the results are compared withresults obtained from a coordinate measuring ma-chine.
The authors are with the Department of Mechanical Engineer-ing, Brigham Young University, Provo, Utah 84602. J. Blotter’se-mail address is [email protected].
Received 13 September 2004; revised manuscript received 16November 2004; accepted 24 November 2004.
2. Overview of the Projection Moire InterferometryProcess
Typical procedures for obtaining out-of-plane dis-placement measurements by means of PMI consist ofsix basic steps. The first step is to establish a refer-ence plane from which a normal vector represents theout-of-plane displacements under investigation. Thesecond step consists of capturing an image of a dotcard placed in the reference plane to remove cameraperspective from all images.12 In the third step asubsequent image, referred to as the reference image,is captured of the projected grid on the test object ora flat surface placed in the reference plane. Step fourrequires that this flat surface be given a number ofknown rotations or translations, and images are cap-tured of the projected grid on the reference plane ateach new position. In the study reported here, theseimages are termed calibration images and are differ-enced from the reference image to generate fringepatterns. The fifth step consists of analyzing the cal-ibration fringe patterns. With knowledge of the dis-placement field for each calibration image, the FSCvalues can then be extracted. The sixth and final stepis to capture an image of the projected grid on adeformed object for which the displacement field isunknown. This image is differenced from the refer-ence image to generate a deformed interferogram. Byapplying the FSC values computed in step 5 to thefringe pattern obtained in step 6, one can compute thedisplacement field of the deformed object.
3. Virtual Calibration Process
Here we address step 4 of the typical PMI procedure,i.e., is the process of capturing the calibration images.With previously used techniques this process can betime consuming and, in large-scale applications, ex-tremely difficult to accomplish. However, this step isnecessary and must be performed carefully to extractan accurate measure of the displacements. Here weshow that this process can be effectively replaced byuse of virtual calibration images.
The virtual calibration consists of four steps.The first step makes use of camera calibrationroutines13–16 to determine the internal characteristicparameters of the camera as well as the precise po-sition and orientation of the reference plane withrespect to the camera. The second step uses the re-
sults obtained in the first step and similar cameracalibration routines to determine likewise the char-acteristic parameters of the projector, including po-sition and orientation with respect to both thereference plane and the camera. In the third step acomputer-simulated PMI setup is created. Using theresults acquired in steps 1 and 2, one places each ofthe three major components (camera, projector, andtest object) in the appropriate position. The fourthstep uses ray-tracing techniques to acquire the vir-tual calibration images from the computer-simulatedPMI system. It is much easier and in most cases moreaccurate to give known displacements in a computermodel than in an actual test situation. These imagesare then used in place of the traditional calibrationimages to extract the FSC values. This procedureeliminates the need to perform the task of rotating ortranslating the reference surface by known amountsto calculate the FSC values. These four steps are nowdiscussed in more detail.
A. Camera Calibration
The first step of the virtual calibration process is toperform the camera calibration. The purpose of thisstep is twofold: to determine internal (intrinsic) pa-rameters of the actual camera and to estimate theposition and orientation (extrinsic) parameters of thecamera relative to the object in its reference position.Camera models typically account for five intrinsicparameters that describe the internal properties, sixextrinsic parameters that represent the camera’s geo-metric degrees of freedom relative to an object, andfive distortion coefficients that describe both radialand tangential distortions. Many techniques exist forextracting calibration parameters.13–16 For thepresent study, Matlab-based camera calibration rou-tines are used.17 These routines are based on captur-ing multiple (as many as 30 or more) images of aplanar checkerboard pattern with known checker di-mensions. From the coordinate system of the check-erboard pattern, the corner locations for each checkerare known. Also, for each image captured, the corre-sponding pixel coordinates for those corners are lo-cated by use of a corner-extracting algorithm.17 Thetask then becomes estimating the intrinsic, extrinsic,and distortion parameters. The intrinsic parametersand distortion coefficients are independent propertiesof the camera and remain constant for all images. Theextrinsic parameters, however, change for eachunique position and orientation, but the use of mul-tiple corners for each image assists in their estima-tion.
Equation (1) below illustrates the general relation-ship between a three-dimensional point �X, Y, Z� onan object (in this case a checkerboard) in its owncoordinate system and a two-dimensional pixel coor-dinate �U, V� on the image plane of the camera, whichat this point assumes no distortion. In Eq. (1) theintrinsic transformation is described by five param-eters, or degrees of freedom. They are the pixel focallengths in two dimensions (fx and fy), the image centerin two dimensions (cx and cy), and the pixel skew
angle (�). An illustration of these five parametersappears in Fig. 2, where f is the focal length and sx
and sy describe the pixel size in two dimensions. Theextrinsic transformation is described by the standardhomogenous transformation matrix,18 consisting of a3 � 3 rotational submatrix (R) and a 3 � 1 transla-tional vector (T). The rotational submatrix is com-posed of three separate 3 � 3 rotational matrices thatrepresent the three rotations ��x, �y, �z� about each ofthe three axes. The translational vector is composedof the three translations �tx, ty, tz� along each axis.Naturally, there are six extrinsic parameters, repre-senting the six degrees of freedom from the cameracoordinate system to that of the test object. A detaileddiscussion of both the intrinsic and extrinsic param-eters is available in the literature17,18:
�UV1�� �fx afy cx
0 fy cy
0 0 1�È
Intrinsic Transformation
�1 0 0 00 1 0 00 0 1 0
�� � R T
�3 � 3� �3 � 1�0 1
�È
Extrinsic Transformation
�XYZ1. (1)
To account for distortion we find the normalizedimage projection point �Pn� shown in Eq. (2) belowfrom the last three matrices of Eq. (1) and then dis-tort it with the five-element distortion vector (D) ac-cording to Eq. (3), where r2 � xn
2 � yn2. The distorted
image point �Pd� is then converted into a final dis-torted pixel coordinate �Pp� by use of the intrinsictransformation shown in Eq. (4):
Pn ��xn
yn
1���1 0 0 0
0 1 0 00 0 1 0
� �R T0 1� �
XYZ1, (2)
Pd � (1 � D1r2 � D2r
4 � D3r6)
È
Radial Distortion
�xn
yn�
� �2D4xnyn � D5(r2 � 2xn
2)D4(r
2 � 2yn2) � 2D5xnyn
�È
Tangential Distortion
, (3)
Pp ��fx �fy cx
0 fy cy
0 0 1� �Pd
1 �. (4)
It is worth noting that the intrinsic parameters anddistortion coefficients are independent properties ofthe camera and remain constant for any capturedimage. However, the intrinsic calculations assumethat the distance from lens to image plane remainsfixed, which is not the case if the focus is adjusted or
if a different zoom configuration is used. Thereforeone must take care in calibrating the camera to en-sure that this distance remains fixed. The extrinsicproperties change for each unique image captured bythe camera, but the use of multiple images assists intheir estimation. In the camera calibration routinesused for the study reported here, multiple images arecaptured of a planar checkerboard pattern. Everychecker corner represents a point on the object thatcan be used in conjunction with the correspondingcorner found in the image. As the checker dimensionsare known, the corner location of each checker in theobject coordinate system is also known. In addition,for each image captured, the corresponding pixel co-ordinates for those corners are located by use of acorner-extracting algorithm17 on the image. The taskthen becomes estimating the intrinsic, extrinsic, anddistortion parameters. Optimization routines basedon an iterative gradient descent with an explicit com-putation of the Jacobian matrix19 are used to performthis task. The two purposes of camera calibration, asit relates to virtual calibration of the PMI setup, canthen be accomplished. The first purpose is to deter-mine the intrinsic parameters, including distortioncoefficients for the camera, and the second is to esti-mate the extrinsic parameters of the camera relativeto the reference plane established in the PMI setup.These results are used both to calibrate the projectorand to model a virtual camera used in a simulation ofthe PMI setup.
B. Projector Calibration
The second step is to determine the intrinsic param-eters of the projector. The same model used for thecamera can be adapted to model the projector. Theprojector’s five intrinsic parameters, distortion coef-ficients, and six extrinsic parameters can be esti-mated by the techniques described in Subsection 3.A.In this step, however, a projected checkerboard pat-tern is used in place of the planar checkerboard usedin camera calibration. With the camera and the ref-erence plane remaining in fixed positions, the projec-tor can then be placed in numerous positions andimages can then be captured of the projected check-erboard on the reference plane.
During camera calibration, the size of the planarcheckers is used to determine a point �X, Y, Z� on theobject, while the corresponding image coordinate
�U, V� is found by use of a corner-extracting algo-rithm on the image captured by the camera. For pro-jector calibration the projected checker corners on thereference plane are first extracted from an imagecaptured by the camera. The result is the camerapixel locations �U, V� of the projected checker corners.By use of the intrinsic parameters, distortion coeffi-cients, and extrinsic parameters of the camera, thoseprojected checker corners can be transformed intothree-dimensional points �X, Y, Z� on the object. Justas the points on the object for a planar checkerboardare known, so are the projector pixel coordinates ofthe projected checkerboard. In both cases the coordi-nates are an equally spaced grid of points. The pro-jector parameters can then be estimated with thesame optimization routines used in camera calibra-tion. It should be noted that, for projector calibrationto occur, the camera must first be properly calibratedto transform the camera pixel coordinates of the pro-jected checkerboard into object coordinates in the ref-erence plane.
C. Virtual Projection Moire Interferometry Setup
After calibration of the actual camera and projector iscompleted, the next step is to create a simulated PMIsetup. This step is accomplished by creation of twovirtual components that represent the camera andthe projector. By use of the intrinsic parameters forboth the camera and the projector and of the extrinsicparameters that relate each component to the refer-ence plane, this can be done in a computer program.In the study reported here, we chose the ray tracingand modeling software Zemax20 to accomplish thetasks associated with this step. We recognize that anumber of software packages exist that could performsimilar tasks and assist in creating a simulated PMIsetup. Zemax was ultimately chosen for this researchbecause of its reliable ray-tracing engine, the avail-ability of several key user-defined functions, and itsrelatively low cost. The computer simulation of a PMIsystem is shown in Fig. 3, where the rays from avirtual projector, represented by a point light source,travel through a slide containing the projected grid orcheckerboard pattern. The rays then strike the objectof interest, scatter to a single point, and are finallycollected by the image plane of the virtual camera.
D. Virtual Camera Modeling
The virtual camera, as shown in Fig. 4, is accuratelyrepresented by a pinhole camera, with which no dis-tortion occurs. The rays striking an object can berefracted with user-defined functions21 to a singlepoint. For the current setup, this point represents theposition in space of the camera with respect to thereference plane. The virtual camera is designed tomatch all the intrinsic parameters of the actual cam-era, minus distortion effects. Focal distance, size ofthe image plane, and number of pixels are all vari-ables that the user defines in the software. One smalllimitation is that only square pixels are allowed�� � 0º�, but, for most cameras, it is a safe assumptionthat only square pixels will be found. It should benoted that camera calibration yields the pixel focallengths (ratio of focal length to pixel width andheight) as a unique solution, but there is an infinitenumber of solutions for focal length given differentpixel sizes. Each solution, however, yields the sameresults as long as the ratio remains fixed.
The virtual projector, as shown in Fig. 5, is repre-sented by a point light source and like the virtualcamera contains no distortion. The light rays travelthrough a slide containing the grid or checkerboardpattern toward the object. The slide is simply a bit-map image for which size and aspect ratio are user-defined variables. The intrinsic parameters of theactual projector can then be used to assign values tothe virtual projector to ensure equality between thetwo. The focal length and the pixel size are treatedsimilarly, as was done with the virtual camera mod-eling. For the projected checkerboard, one value (focallength or pixel size) is chosen, thereby fixing the othervalue. If the information regarding the relative sizesof the checkerboard and grid patterns used in theactual projector is known, it can be used to determinethe relative sizes needed between the two slides inthe simulated PMI setup. With this accomplished,the virtual projected pattern can be refracted towardthe virtual camera and captured on the image plane.Using the reference plane as the object in both thesimulated and the actual PMI setups, one can thencompare a virtual projected pattern with the actualprojected pattern seen by the camera in both cases.These two images, when they are differenced, shouldshow no sign of interference. A phase differencemight be present, depending on the location of the
slide center, but, by using phase-shifted referenceimages,11 one may take this discrepancy into account.
F. Virtual Calibration Images
With the simulated PMI setup mimicking the actualsetup, the reference plane can be given known dis-placements and the calibration images can be cap-tured within the computer simulation. A typicaldisplacement is a rigid body rotation about the localvertical axis of the reference plane. Images are cap-tured in the virtual PMI setup for each known dis-placement and are differenced with the referenceimage to generate fringe patterns. Then one may an-alyze these fringe patterns, using typical PMI meth-ods to extract the FSC values.
4. Experimental Setup and Validation
To validate the developed methods and obtain a bet-ter understanding of the sources and magnitudes oferror, we performed a flat plate experiment. In thisvalidation, the FSC values were calculated by boththe PMI and the VCPMI techniques and then com-pared. A known displacement (rigid body rotation)was given to the flat plate, and a displaced fringepattern was generated. This displaced fringe patternwas then used in conjunction with the FSC valuesobtained to yield the displacement field. A compari-
Fig. 6. Checkerboard plane used for camera calibration. Fig. 7. Checkerboard image used for projector calibration.
son was then made of the PMI and the VCPMI ex-tracted displacement fields.
The experimental setup is represented in Fig. 1and consisted of a digital projector (Epson Model7700p) with a native display format of 1024� 768 pixels for the light source and the focusinglens. The grid pattern used was a bit-map image�640 � 480 pixels� with 0.25 line/pixel. The CCDcamera (Hitachi Model KP-M1U) with 640� 480 pixel resolution was controlled by a computer,which had a frame grabber to digitize the pixel-sensed voltage to an 8-bit intensity pattern. A flataluminum plate �15 cm � 15 cm� was used as a ref-erence to measure displacements and was mountedon a rotary turntable with 1�60° resolution.
A. Camera Calibration
Camera calibration was performed with a 25 � 17checkerboard pattern of 1 cm � 1 cm square checkersshown in Fig. 6. A total of 30 checkerboard imageswere initially taken, and the results were optimizedto produce the intrinsic camera parameters and dis-tortion coefficients. Through the process of optimiza-tion, five images were discarded because theycontained irregular data. The remaining imagesyielded the results given in Table 1. After the distor-tion effects were removed, we then used the intrinsicparameters to find the six extrinsic parameters forthe checkerboard placed in the reference plane. Theseresults are also given in Table 1.
B. Projector Calibration
The bit-map checkerboard image shown in Fig. 7 wasused for the projected checkerboard pattern. It con-sisted of 17 � 17 square checkers of 50 pixels each,with a total image resolution of 850 � 850. With thecamera and the plate left in the reference positions,30 images were captured of the projected checker-board on the plate as seen by the camera. We thenused the camera calibration results to calculate theplaces where the checker corners intersected the ob-ject. We used these values and the prior knowledgethat each checker before projection is 50 pixelssquare to optimize the intrinsic parameters and dis-tortion coefficients. During this process, two imageswere discarded because they contained irregular
data. The results from the projector calibration aregiven in Table 2.
C. Virtual Projection Moire Interferometry Setup
Once the camera and the projector were calibrated,the virtual PMI setup was created. The intrinsic pa-rameters of the virtual camera were developed tomatch those of the actual camera. The focal lengthwas chosen to be a fixed parameter �25 mm� and thenused in conjunction with the focal pixel lengths of theactual camera to yield the pixel size for the imageplane of the virtual camera. The pixel size was cal-culated to be 12.64 �m � 12.73 �m to yield an imageplane of 8.092 mm � 6.110 mm. By matching theextrinsic values for the reference plane in both theactual and the virtual PMI setups, we made a com-parison of the virtual and the actual checkerboardsplaced in the reference plane. The results are shownin Fig. 8, in which we have differenced the two imagesto judge their correlation. The corners for each imagewere also extracted at each corner location and dif-ferenced in both the horizontal and vertical direc-tions. The average absolute difference between thetwo images is 0.1556 pixel in the horizontal and0.0854 pixel in the vertical direction. It should benoted that, because the virtual camera does not in-clude any distortion effects, those effects must first beremoved from any image to be compared with animage captured by the virtual camera.
We likewise created the virtual projector model byusing the intrinsic parameters of the actual projector.Again, the focal length was chosen to be a fixed pa-rameter �100 mm� and used to determine the pixelsize. The pixel size was calculated to be 35.16 �m� 35.16 �m, yielding a total length of 29.888 mm� 29.888 mm for the checkerboard slide. Aftermatching the extrinsic parameters of the virtual pro-jector to those of the actual projector, we capturedimages of the projected checkerboard pattern on thereference plane as seen by the camera in both setups.These two images and the differenced image areshown in Fig. 9. The corners were extracted, and theaverage absolute pixel error was calculated to be0.1362 pixel in the horizontal and 0.1126 pixel in thevertical direction.
In the experiment the two bit-map images used for
Table 2. Projector Calibration Resultsa
Intrinsic Parameters Extrinsic Parameters
Parameter Value Parameter Value
fx 2843.982 � 25.520 �x 178.784°fy 2843.982 � 24.520 �y 22.967°� 0b �z 0.605°cx 439.286 � 8.533 tx �215.935 mmcy 897.870 � 3.855 ty �17.869 mm
tz 654.192 mm
aIn all cases the value of distortion coefficients D1–D5 was 0.bNot estimated owing to square pixels of image.
the grid pattern and the checkerboard slides were ofdifferent sizes (lengths). This does not change thelocation of the projector, but it does mean that the twoimages cannot simply be interchanged within thesimulation. The size of each slide before projectionwas 27.94 cm for the grid pattern and 17.78 cm forthe checkerboard pattern. Care was taken to adjust
positions such that the center of each projected pat-tern was the same point. The value calculated for thelength of the checkerboard �29.888 mm� can be mul-tiplied by the ratio of the actual grid slide to theactual checkerboard slide (27.94�17.78) to yield thesize needed for the virtual grid slide within the sim-ulation if the focal length for the virtual projectionremains fixed at the previously determined value
Fig. 8. Checkerboard images in the reference plane: (a) actualundistorted, (b) virtual, and (c) differenced (actual � virtual).
Fig. 9. Projected checkerboard images: (a) actual undistorted, (b)virtual, and (c) differenced (actual � virtual).
�100 mm�. The calculated length for the virtual gridslide was 46.967 mm.
D. Virtual Calibration Images
The flat plate was given known displacements in boththe actual and the virtual PMI setups. All displace-ments were rigid body rotations about the verticalaxis (y axis) of the plate. These displacements werecompleted in 1° increments from �11º to �20º, andan image was captured at each calibration position. Itshould be noted that the range of known displace-ments should typically cover the range of expecteddisplacements. In this case the range �11º to �20ºwas selected simply for demonstration. These imageswere then differenced from the reference image of therespective setup, whether actual or virtual. We thenused the fringe patterns generated from these proce-dures with standard PMI techniques to determinethe FSC values.
5. Fringe Sensitivity Coefficient Calculations
Techniques employed for determining the FSC val-ues for this research make use of phase-shifted ref-erence images,11 which are used to create a wrappedphase map of the fringe pattern. This map is thenunwrapped, yielding continuous phase informationover the entire image. A pointwise FSC value is com-puted for every pixel of each calibration image, giventhe computed phase and known displacement at thatpixel. A polynomial of the quadratic form ax2 � bx� c is fitted to each row of the image relating the FSCto a horizontal position where the coefficients a, b,and c are determined by a least-squares approachand x is the distance in units of length along thereference plane. An array is generated that is threecolumns wide with as many rows as the image, rep-resenting the three coefficients of the quadratic poly-nomial for each row. The FSC arrays that weregenerated by use of only virtual calibration imageswere then compared with those found by use of actualcalibration images. The discrepancies in the FSC val-ues introduced from virtual calibration were calcu-lated as a percent difference from those of the actualcalibration. The percent difference for each term(quadratic, linear, constant) is shown in Figs. 10–12.The average percent difference for each coefficientwas �6.60% for the quadratic, �1.34% for the linear,and 3.30% for the constant term. To determine dis-placements of an unknown fringe pattern we usedthese FSC values according to Eq. (5) below, where Dis the displacement and phase�x� is the change inphase from the current pixel to the pixel of zero dis-placement. Ultimately, the effect of FSC discrepan-cies on displacement was investigated:
D(x) �phase(x)FSC(x) . (5)
Assuming that the change in phase was correct for agiven fringe pattern, we calculated the whole-fieldpercent difference in displacement, using the two sets
of coefficients, and this difference is shown in Fig. 13.The average percent difference found was �2.68%.Figure 13 also clearly shows that, where displace-ments are expected to be small, the FSC values havethe largest percent discrepancy. This implies that an
Fig. 10. Actual versus virtual FSC percent difference: quadraticcoefficient.
Fig. 11. Actual versus virtual FSC difference: linear coefficient.
Fig. 12. Actual versus virtual FSC difference: constant coeffi-cient.
8.0% discrepancy for which the displacements aresmall may not result in a large error in the displace-ment map. In Fig. 13, where the displacements arelarger, the percent error is considerably less. Thisindicates that translations may provide better FSCvalues because zero displacement values will not bein the field.
Further analysis was performed on experimentaldisplacements of the flat plate, given a known dis-placement field. This analysis served as an additionalvaluable link between PMI and VCPMI setups toestimate and characterize the errors introduced inthe overall process. We used the virtual FSC valuesin conjunction with phase maps generated from im-ages captured in the PMI setup to compute displace-ments. These displacements were then comparedwith true displacements from five separate rigid bodyrotations (1° increments from 11° to 15°). We madethese comparisons by averaging each column fromthe whole-field displacement for each rotation. There-fore, using a sample size of five, we computed dis-placement errors for each column along the length ofthe flat plate. A 95% confidence interval describes theerror with a magnitude of 0.0032 cm � 0.0464 cm(based on a standard deviation of 0.0374 cm). To bet-ter visualize this error, we applied it to displacementsfrom a 13° rigid body rotation. The result is shown in
Fig. 14, with error bars at discrete data points alongthe length of the beam.
In conclusion, the virtual calibration techniquesdeveloped have been shown to yield whole-field dis-placement results within 3% of their traditional tech-nique counterparts. In addition, experimentaldisplacements that use virtual FSC values with ac-tual unwrapped phase maps show a high degree ofcorrelation (roughly within �0.05 cm) with the ac-tual displacements given.
6. Test Case: Section of Airfoil
After the VCPMI method of computing FSC values ona flat plate was validated, a test case was performed.The test object was the leading edge section of anairfoil (approximately 27 cm long and 12 cm wide).The images used for testing represented a small por-tion of the airfoil (approximately 10 cm by 9 cm).During the assembly process the airfoil is pressedinto a fixture and riveted to an elastic frame. When itis removed from the fixture, the airfoil experiences
Fig. 13. Percent displacement error introduced from virtualcalibration.
Fig. 14. Displacement with average errors and error bars.
Fig. 15. Position for airfoil insertion in the PMI setup.
some springback and does not maintain the exactshape of the fixture. Therefore discrepancies existbetween the internal surface of the fixture and theexternal surface of the assembled airfoil. The goal ofthis test was to determine the shape differences be-tween the airfoil fixture and the assembled airfoil.We measured these differences by comparing the dis-placement fields of the assembled airfoil and acomputer-aided design model representing the inter-nal surface of the fixture. This experimental displace-ment field was then compared with displacementsobtained by use of a coordinate measuring machine(CMM), which yielded an accuracy of �5 �m. Scanswere performed with the CMM on both the assembledairfoil and the internal surface of the fixture and werethen differenced to produce the displacement fieldunder investigation. We continue by presenting anddiscussing the results of this test.
7. Virtual Calibration
The experimental setup was identical to that de-scribed above, and the FSC results presented as part
of the verification process for virtual calibration wereused in the airfoil test case as well. The airfoil waspositioned as shown in Fig. 15: Parallelism wasforced between the flat bottom edge of the airfoil andthe reference plane. The center of rotation was usedas the origin, with displacements measured along thez axis as shown. The virtual FSC values were thenapplied to a fringe pattern generated by differencingof the projected grid patterns of the airfoil and refer-ence plane to create a displacement field. The offsetshown in Fig. 15 was then added to each displace-ment value, resulting in a surface topology of theassembled airfoil. This experimental displacementfield is shown in Fig. 16.
We then determined the variations between theassembled airfoil and the computer-aided designmodel of the fixture by differencing the displacementfields, and the result is shown in Fig. 17. This resultwas then compared with the displacements found byuse of the CMM. The CMM displacement field isshown in Fig. 18 and for the study described here wasused as a benchmark with which to judge the accu-racy of the methods developed. Differences between
Fig. 17. PMI versus fixture displacements.
Fig. 18. CMM displacement field.
Fig. 19. VCPMI percent difference.
Fig. 20. Cross-sectional slice of displacement data.
the VCPMI and the CMM methods are considered inthe VCPMI process to be introduced errors. The per-cent error was computed for each pixel, and the re-sults are shown in Fig. 19. This error variesthroughout the entire measurement domain from�14.39% to 14.31%. To better illustrate the overallresults, we show in Fig. 20 a cross-sectional slice ofdata. The errors previously computed are also shown.
8. Discussion of Results and Summary
A method has been developed for determining out-of-plane deformations that makes use of virtual calibra-tion techniques in the measurement process andovercomes the need to use a rotation stage in thecalibration process. The method requires less than10-min computer time on a 1MHz processor. The testcase performed has shown errors of �14.4%. For thepresent study there are two recognized sources oferror. One source of error is introduced when oneattempts to use a common insertion point betweenthe planar checkerboard in the reference plane andthe flat plate. Small differences here contribute to theoverall error. Methods for lessening or eliminatingthis effect could possibly be developed. The secondsource of error is attributed to the camera resolution.A higher-resolution camera will assist in the effort todiminish this effect, but it is also worthwhile to in-vestigate any image analysis algorithms that mightenhance image quality without requiring higher res-olution.
In conclusion, the measurement technique that wehave described overcomes two limitations of tradi-tional PMI methods, and it is anticipated that furtherresearch will yield even more desirable results.
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