Off-axis Two-dimensional Digital Image Correlation

Off-Axis Two-Dimensional Digital Image Correlation Jeffrey D. Helm, Jeremy R. Deaner Lafayette College, Dept. of Mechanical Engineering, Acopian Engineering Center, Easton, PA 18042

ABSTRACT A calibration process is developed that allows single-camera two-dimensional image correlation to be performed using cameras that are not perpendicular to the surface of the specimen. Numerical studies show that the calibration process should be able to compensate for off-axis angles as high as 30° even under the presence of moderate lens distortion. Experimental studies show that the current application of the calibration process results in a systematic error in the direction of camera swing. The experimental data also shows that the distortion present after calibration, while large enough to effect strain measurement, is significantly smaller than the distortions in uncalibrated data and may be used for displacement measurement. This was shown to be true for specimens that undergo rigid body rotations, including rotations of 180° or larger. In rotation tests, calibrated values for a camera 30° off-axis are approximately 40 times better than the uncalibrated displacement data and resulted in a maximum strain error of 4000 µstrain at 180° of rotation. INTRODUCTION Two-dimensional digital image correlation is an increasingly popular method for measuring in-plane deformation and strain on flat specimens that experience only in-plane deformations. Currently the method is limited to experiments where the camera can be positioned perpendicular to the surface of the specimen. This work introduces a simple calibration method that allows the use of 2D digital image correlation for displacement measurement in situations where the direct view of the specimen may be obscured, a situation that typically occurs due to the load frame configuration, fixturing or other measurement apparatus. Moving the camera to a non-perpendicular position effects the perception of the viewed surface area in two related ways. First, the shape of surface features changes. For example, a square is distorted into a keystone shape. Second, the magnification of the surface is no longer constant, and continually changes in both the X and Y directions depending on the location in the image. Both of these effects will adversely affect the measurement accuracy of traditional 2D image correlation. These effects are illustrated in Figure 1. The proposed calibration technique uses a bi-cubic correction function to map the off-axis displacement data back to its original coordinate system. The method does not directly model the camera parameters but relies on a calibration standard to provide the needed information to obtain the mapping parameters. The technique is simple to apply and requires only a single image of a grid standard to compensate for both the shape and magnification changes introduced by an off-axis camera. CALIBRATION METHOD

Figure 1 - Distortions typical of off-axis projection

The calibration method is based on a single image of a known measurement standard. In this work, images of grids were used for the calibration. The goal of the calibration is to map the measured locations of the grid points (Ximgi, Yimgi) to the physical grid coordinate (Xgrdi, Ygrdi). The mapping function adopted for this work was a simple sixteen parameter bi-cubic surface. Separate functions are used to map locations from the image to the grid X locations and the grid Y locations as shown in Equations 1 and 2. Xgrd

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Because the bi-cubic surface equation is linear, the sixteen parameters for each correction equation can be determined through a least squares analysis between the image coordinates and the corresponding grid coordinates. This approach to calibration offers several advantages. First, the determination of the sixteen parameters is simplified by using a linear equation. Secondly, mapping the points onto a regular grid corrects for both the change in perceived shape and the changes in magnification. Thirdly, the calibration compensates for cameras with non-unity aspect ratios and can compensate for some lens distortion. Finally, the output of the calibration is in a coordinate system aligned with the grid and the output is given in dimensional units. It is noted that this calibration process can be, at best, an approximation of the more complex and non-linear mapping that results from viewing the surface from an off-axis position. For this reason, the range of camera positions where the calibration procedure will be valid must be assessed. CALIBRATION ASSESSMENT The ability of the calibration process to compensate for the non-linear nature of the perspective distortion was assessed through numerical simulations. The camera system was modeled as a pinhole camera with lens distortion. This model was used to evaluate the suitability of the calibration process. The procedure was as follows: 1) Using the projection equations for a pinhole camera, grid locations in space were projected onto the model’s image sensor. 2) The projected locations (image locations) were used to determine the two sets of calibration constants by performing a least squares fit of the mapping functions. 3) The calibration functions were then used to map the image locations back to the grid. 4) The mapped points were then projected back to the sensor coordinates using the same camera parameters as in step 1. This step was done so the error analysis could be performed using sensor coordinates, which are largely independent of the system magnification. The error in the base calibration process was then calculated from the distances between the image coordinates obtained in step 1 and the calibrated sensor positions determined in step 4. The error assessment described above provides information on how well the calibration process maps the image coordinates of the grid back to the original grid locations. For the calibration to be applicable to general measurement problems the accuracy of the method for points in between grid points must be assessed, as must the accuracy of the method for points outside the calibration grid area. To investigate the error between the points used to determine the calibration parameters a finer grid was projected onto the sensor using the same camera parameters used in step 1. The points were then mapped to the grid coordinates using the constants calculated from the grid locations. Finally, the calibrated points were projected back to the sensor and the error calculated as described in step 4. A second grid of finely spaced points 40% larger than the calibration grid was analyzed in a similar manner. In this way the errors encountered when using the calibration outside the initial grid area could also be assessed.

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For all the simulations, a 10mm x 10mm grid was modeled 40mm from the pinhole of the camera. The pinhole distance (magnification factor) for the lens was set to 2800 pixels. The projected grid shown previously in Figure 1, was developed -8 using these base parameters, a camera pan angle of 30° and a distortion coefficient of 1x10 . The effect of the pan angle of the camera on the calibration is shown in Figure 2. The figure presents the average error, the maximum error and the error’s standard deviation for camera pan angles from 0°to 30°. Figure 3 presents the average error, maximum error and standard deviation for the same pan angles for the finer grid and Figure 4 presents the same information for the oversize grid.

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Figure 2 – Error in base grid calibration error for various pan angles

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Figure 3 - Fine grid error for various pan angles

The error for the base calibration and the fine grid pattern does not exceed 0.02 pixels for any angle between 0 and 30 degrees, as shown in Figures 2 and 3. It can be seen from the errors reported in Figure 4, and the error map in Figure 5, that the calibration quickly deviates from the desired values for points outside of the base calibration. This suggests that the calibration can be used within these angles, but the points should be restricted to the initial calibration area. Figures 6, 7 and 8

present the same information for a system where both the pan and tilt angles change by equal amounts. Once again the calibration system produces errors under 0.02 pixels for any of the angles provided the points stay within the initial calibration area. Figures 9 through 13 present the same error information for a camera system 15° off-axis with lens distortion -7 -9 -7 -9 coefficients ranging from 1x10 to 1x10 for pincushioning and -1x10 to -1x10 for barreling distortion. These simulations show the calibration process should be able to correct the perspective distortions encountered with camera setups with pan/tilt angles from 0 to 30 degrees and in the presence of the moderate lens distortions commonly found in modern lens systems. 2.00E-01 1.75E-01

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Figure 6 – Error in base grid calibration for changing pan and tilt angles

Figure 7 – Fine grid error for changing pan and tilt angles

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Figure 8 - Oversized grid error for changing pan and tilt angles

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Figure 9 – Error in base grid calibration with pincushioning distortion

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Figure 11 – Oversized grid error with pincushioning distortion

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Figure 10 – Fine grid error with pincushioning distortion

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Figure 12 - Error in the base grid calibration with barreling distortion

Figure 13 – Fine grid error with barreling distortion

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Figure 14 - Oversized grid error with barreling distortion EXPERIMENTAL EVALUATION The calibration process was also evaluated experimentally. Two types of experimental setups were used for the evaluation. In the first experiment, strain in an axially-loaded dog bone specimen of 1mm thick 2024-T3 aluminum sheet was measured using cameras placed both perpendicular to and off-axis from the specimen. The calibration method was also tested by

measuring the strain in a non-loaded specimen subject to 360° of rigid body rotation. The two experimental methods are presented in the following sections: TENSION TESTS – EXPERIMENTAL PROCEDURE For the first experiment, dog bone specimens of 1 mm thick, 2024-T3 aluminum, conforming to ANSI – E standards, were loaded in tension. As shown in the photograph in Figure 15, two cameras imaged an area of approximately 12 mm x 12 mm of the gage section. One camera was positioned perpendicular to the surface of the object and the other camera was positioned off-axis from the specimen. The second camera was placed 30° off-axis to the normal of the surface and in line with the vertical axis. A 10 mm x 10 mm grid of 0.5 mm squares was attached to the specimen at the area of interest and calibration images were acquired from each camera. The grid was removed and the specimens were loaded in 2 kN increments until plastic deformation was evident, after which the specimens were loaded in 0.3 mm increments until an overall grip displacement of 5.00 mm was achieved. Images were acquired at each load/displacement step. TENSION TESTS - RESULTS The images acquired using the procedure described above were analyzed in the following manner: Grid intersection points were extracted from the images taken of the calibration grid. Calibration constants for both the on-axis and off-axis cameras were calculated from the extracted grid points. The images of the specimen were correlated using standard image correlation techniques. The calibration process was used to transform the correlation data from both cameras. Two strain values were calculated for each image, one from the raw correlation data and one from the calibrated data.

Figure 15 – Experimental setup for tension tests

The results for three experimental runs with the off-axis camera at 30° are presented in Figures 16 and 17. The graphs compare the strain values from the calibrated cameras to those of the uncalibrated on-axis camera. Figures 18 through 20 0.0E+00

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Figure 16 – Tension tests: measured εxx strain

Figure 17 - Tension tests: measured εyy strain

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Figure 18 - Tension tests: εxx difference from onaxis uncalibrated strains

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Figure 19 - Tension tests: εyy difference from onaxis uncalibrated strains

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Figure 21 – Rotation experiment camera placement

Figure 20 - Tension tests: εxy difference from onaxis uncalibrated strains

show the difference between the strain measured by a) the calibrated 0° camera, b) the uncalibrated 30° camera, c) the calibrated 30° camera and the uncalibrated 0° camera. The data shows that the calibration process did not significantly affect the on-axis strain values. These values changed by an average of less than 40µS for all three strains. The calibration process was also able to correct the εxx and εxy strains to values comparable with the on-axis camera. The strain in the direction of the camera tilt shows a definitive bias of approximately 1000 µS. While this deviation is less than that from the uncalibrated camera, the magnitude of the bias indicates a systematic error in either the calibration or the correlation process when subject to large perspective distortions. While the error would not have a significant effect on displacement measurements, the bias will need to be corrected if the off-axis correlation is to be used for small strain measurements. ROTATION TESTS – EXPERIMENTAL PROCEDURE Figure 21 is a photograph of the experimental test setup for the rotation tests. A square specimen was attached to a plate that could be rotated about its central point. It was positioned so the specimen was perpendicular to the on-axis camera. A small 10 mm x 10 mm section from the center of the specimen was imaged and the specimen mount was rotated 360° in approximately 22.5° increments with images taken by from the on-axis and off-axis at each step. Before moving the specimen the grid was attached to the rotation plate and calibration images were taken. The procedure was repeated for off-axis camera positions of 10, 20 and 30 degrees. ROTATION TESTS - RESULTS The grid images from the rotation tests were used to calibrate the on and off-axis cameras for each experimental setup. The images for the complete rotation were correlated using standard 2D correlation techniques. As with the tension tests, a single strain value was determined from each set of correlation data. The results of the tests are shown in Figures 21-23. Figures 24–26 present only the data from the calibrated data sets and the data from the on-axis camera. As seen in Figure 24, The strain error in the camera tilt direction, seen in the tension tests, is also apparent in the rotation tests. This strain error increases with both the degree of specimen rotation and the angle of the camera. It should be noted that the calibration process reduced the strain errors from 155,000 µS to 4,000 µS. It should also be noted that the 4,000 µS 1.80E-01

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Figure 21 - Rotation tests: εxx strain

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Figure 22 - Rotation tests: εyy strain

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Figure 23 - Rotation tests: εxy strain

Figure 24 - Rotation tests: exx calibrated measurements

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Figure 25 - Rotation tests: εyy calibrated measurements

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Figure 26 - Rotation tests: εxy calibrated measurements

error represents an error in the displacement measurement of 0.028 mm compared with 1.12mm for the uncalibrated results. CONCLUSIONS A calibration process was developed to expand the application of standard 2D image correlation to situations where the camera is placed off axis from the surface of the specimen. Numerical studies of the effectiveness of the calibration technique were performed using a pinhole camera model that included the effect of lens distortion. From the numerical studies the calibration technique should be able to correct for the perspective distortions for camera angles up to 30° in combined pan and tilt angles even with a moderate degree of lens distortion. The calibration was then evaluated experimentally using two types of tests, a simple tension test and a rigid body rotation test. While the calibration was successful in significantly reducing the effect of the camera angle, both tests indicate a systematic error in the direction of the camera tilt. The effect was consistent for all three tension tests performed with the off-axis camera at 30°. The rotation tests showed that the magnitudes of the errors were related to the degree of camera tilt. These tests indicate that the method requires further refinement before it can be applied to strain measurements. The systematic nature of the error does suggest that a correction for the calibration/correlation method may be possible. While it is disappointing that the strain measurements were adversely effected, the tests also indicate that the correction would be sufficient for many, if not most, applications where displacement measurements are needed.