By Jillian Kunze
Determining the conditions under which a material was processed based solely on its texture is an intriguing inverse prediction problem. In classical regression problems, one already knows the covariates—the observed variables that can affect the outcome—and can utilize them to construct a statistical model that estimates their relationship with different results. But in inverse prediction problems, this structure is flipped: the goal is to use an observed response to estimate the covariates.
Figure 1. Different processing conditions result in different particle textures, as shown in these scanning electron microscope images. 1a. A rough particle with numerous edges, dips, and valleys. 1b. A flat particle that is more rectangular and box-like. Figure courtesy of Adah Zhang.
This is an especially important question in nuclear forensics, where the covariate is the conditions under which nuclear materials of unknown origin were made. “Specifically, we want to identify and attribute a set of processing conditions to interdicted special nuclear materials,” Adah Zhang of Sandia National Laboratories said. “We’re curious about processing conditions because knowledge about such is helpful in determining where the materials originated.”
During a minisymposium presentation at the recent 2022 SIAM Conference on Imaging Science, Zhang described her team’s recent efforts to use the particle texture in scanning electron microscope (SEM) images to infer the conditions under which the materials were processed. They were motivated by a dataset comprised of SEM images of particles that were intentionally produced with different processing conditions, which significantly impacted the particle surfaces (see Figure 1). Figure 1a depicts what the team defined as a “rough” particle with numerous stacked plates protruding at different angles and numerous edges, while Figure 1b depicts a “flat” particle with more rectangular shapes and fewer edges.
Figure 2. The steps in the Functional Inverse Prediction framework for estimating material processing conditions. Figure courtesy of Adah Zhang.
Zhang and her group established an approach for estimating material processing conditions called the Functional Inverse Prediction (FIP) framework (see Figure 2). “It’s a general process that uses some form of functional response data to perform inverse predictions on scalar independent variables,” she said. “This does not necessarily rely on any specific conditions or models, and it uses a minimal set of assumptions.”
The FIP model has two stages: a training phase and an inference phase. The first step is to generate forward models of functional responses that are fit to the training data, then perform uncertainty quantification and validate the model. This is followed by the inference phase, which uses relationships that were found from the forward model to estimate unknown covariates from new data. “We can minimize the difference between some new observed dependent variable and the functional prediction we received from the training phase of our model,” Zhang explained. “Then we can estimate our independent variables.” They obtain a distribution of predictions using a modified Monte Carlo simulation.
The researchers first experimented with this approach using a simulated functional dataset that was produced using two covariates, \(\textrm{x}_1\) and \(\textrm{x}_2\). Assuming that \(\textrm{x}_1\) and \(\textrm{x}_2\) were unknown, they then compared the application of a functional linear forward model to a multivariate adaptive regression splines (MARS) forward model within their FIP framework. The simulation study indicated that the framework was indeed able to recover \(\textrm{x}_1\) and \(\textrm{x}_2\), with the MARS model performing better than the linear model.
Figure 3. The results of applying a local standard deviation filter to the scanning electron microscope images from Figure 1. The darker areas have higher standard deviations, which generally occur at peaks and edges. 3a. The rough particle. 3b. The flat particle. Figure courtesy of Adah Zhang.
Zhang then described how her group used the FIP framework with actual SEM imaging to infer processing conditions. This first required applying a local standard deviation filter to the raw intensity values from the SEM images to identify areas with high variance (see Figure 3). The researchers also removed the edges of the particle along with the background, as these were inherent features of the imaging technology rather than part of the texture. To represent this information in a functional way that could work with the inverse prediction framework, they used the particles’ cumulative distribution functions as the functional representations in the FIP framework (see Figure 4b). Figure 4a shows the equivalent probability density function; the flat particle has a higher peak at lower standard deviation values, while the rough particle exhibits a larger tail at higher standard deviation values.
The researchers once more used a linear forward model and a MARS model in their framework and chose five functional principal components of the cumulative distribution functions to use as model response. This captured about 90 percent of the variability in the data for those particles, which was a sufficient amount. MARS performed better at recovering some conditions than the linear forward model, though the latter also had decent performance. In either case, it was clear that the processing framework had to deal with much more noise and error in this situation with real data than in the simulated case.
Figure 4. Functional representations of particle textures to use for inverse prediction. 4a. The probability density functions for the rough and flat particles. 4b. The cumulative distribution functions for the rough and flat particles. Figure courtesy of Adah Zhang.
“In general, models that perform well within this functional inverse prediction framework can suggest new scientific hypotheses about relationships that can be seen empirically in strong inverse prediction results,” Zhang said. And promisingly, this modeling approach is more well-founded in statistical models than some existing black box approaches for image analysis, which could make it more interpretable — a very appealing prospect for stakeholders and decisionmakers that might rely on this technique in nuclear forensics situations. Future work may further improve the inference of processing conditions through a more flexible modeling approach that enables the combination of texture with information about other features, such as particle shape.
Acknowledgements: This work was done jointly with Kurtis Shuler, Daniel Ries, Gabriel Huerta, James D. Tucker, and Katherine Goode (Sandia National Laboratories) as well as Madeline Ausdemore (Los Alamos National Laboratory).
Further Reading
[1] Ries, D., Zhang, A., Tucker, J.D., Shuler, K., & Ausdemore, M. (2022). A framework for inverse prediction using functional response data. ASME. J. Comput. Inf. Sci. Eng., JCISE-21-1393.
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Jillian Kunze is the associate editor of SIAM News. |