Path-Integrated Stereo X-Ray Digital Image Correlation: Resolving the Violation of Conservation of Intensity

Abstract

Background

X-ray imaging addresses many challenges with visible light imaging in extreme environments, where optical diagnostics such as digital image correlation (DIC) and particle image velocimetry (PIV) suffer biases from index of refraction changes and/or cannot penetrate visibly occluded objects. However, conservation of intensity—the fundamental principle of optical image correlation algorithms—may be violated if ancillary components in the X-ray path besides the surface or fluid of interest move during the test.

Objective

The test series treated in this work sought to characterize the safe use of fiber-epoxy composites in aerospace and aviation industries during fire accident scenarios. Stereo X-ray DIC was employed to measure test unit deformation in an extreme thermal environment involving a visibly occluded test unit, incident surface heating to temperatures above 600^oC, and flames and soot from combusting epoxy decomposition gasses. The objective of the current work is to evaluate two solutions to resolve the violation of conservation of intensity that resulted from both the test unit and the thermal chamber deforming during the test.

Methods

The first solution recovered conservation of intensity by pre-processing the path-integrated X-ray images to isolate the DIC pattern of the test unit from the thermal chamber components. These images were then correlated with standard, optical DIC software. The second solution, called Path-Integrated Digital Image Correlation (PI-DIC), modified the fundamental matching criterion of DIC to account for multiple, independently-moving components contributing to the final image intensity. The PI-DIC algorithm was extended from a 2D framework to a stereo framework and implemented in a custom DIC software.

Results

Both solutions accurately measured the cylindrical shape of the undeformed test unit, recovering radii values of $R = 76.20 \pm 0.12$ mm compared to the theoretical radius of $R_{theor}=76.20$ mm. During the test, the test unit bulged asymmetrically as decomposition gasses pressurized the interior and eventually burned in a localized jet. Both solutions measured the heterogeneous radius of this bulge, which reached a maximum of approximately $R=91$ mm, with a discrepancy of 2–3% between the two solutions.

Conclusions

Two solutions that resolve the violation of conservation of intensity for path-integrated X-ray images were developed. Both were successfully employed in a stereo X-ray DIC configuration to measure deformation of an aluminum-skinned, fiber-epoxy composite test unit in a fire accident scenario.

Appendix: Initializations of 2D PI-DIC Algorithm

Determining appropriate initializations for DIC in general is non-trivial and the subject of many research papers in the DIC community, e.g. [37,38,39,40,41,42,43,44,\(P_{xx, F_1}\), $P_{xy, F_1}$, $P_{yx, F_1}$, $P_{yy, F_1}$, $P_{xx, F_2}$, $P_{xy, F_2}$, $P_{yx, F_2}$, and $P_{yy, F_2}$) were initialized at zero for every subset in every image.

Initializations of the Path-Integrated Images

First, either 15 or 25 seed points, for Cam0 and Cam1, respectively, were manually identified in the full-resolution versions ($2560 \times 2160$ px²) of the synthetic reference image of the DIC pattern, $F_1$, and the undeformed, path-integrated image, $G_{PI}^1$. Next, a two-dimensional cubic polynomial was fit to the locations of these seed points and evaluated at 50 px intervals to provide a finer grid of seed points. The coordinates of the seed points were then divided by 5 to align with the binned image size ($512 \times 432$ px²).

As the correlation progressed through the time series of images, a bilinear interpolant was created for the displacements of the previous path-integrated image ($P_{u,G_{PI}}^{j-1}$ and $P_{v,G_{PI}}^{j-1}$). The interpolant was evaluated to provide initializations for displacements of the current path-integrated image ($P_{u,G_{PI}}^{j}$ and $P_{v,G_{PI}}^{j}$). Thus, if a point failed to correlate in one image, an initialization was still provided for that point in future images by interpolating the displacements from neighboring points.

Initializations of the Pseudo Light-Field ReferenceIimage

The pseudo light-field reference image of the thermal chamber, $F_2$, lacked sufficient intensity gradients to be well tracked on its own. As a result, subsets in this image were frequently poorly identified, with unrealistically large war** parameters ($P_{xx, F_2}$, $P_{xy, F_2}$, $P_{yx, F_2}$, and $P_{yy, F_2}$) and large vertical displacements ($P_{v,F_2}$). To correct this behavior, two additional constrictions were placed on the initializations for image $F_2$.

First, because the heater rods remained nominally stationary while the test unit deformed, the DIC pattern appeared to deform over a nominally fixed background image. This behavior is illustrated in Fig. 8 in the main text, where both the purple subset of the path-integrated image and the blue subset of the pseudo light-field image started centered over one of the heater rods at time step 1 and translated to the edge of the heater rod in time step j. Therefore, the initializations for the displacement parameters of the thermal chamber subsets ($P_{u,F_2}^j$ and $P_{v,F_2}^j$) were set to the same values as the initializations of the displacement parameters of the path-integrated subsets ($P_{u,G_{PI}}^j$ and $P_{v,G_{PI}}^j$). Second, bounds were placed on the optimization parameters, relative to the initializations, as reported in Table 5.

Table 5 Bounds for the affine subset shape function parameters for PI-DIC, relative to the initializations