## Jessica Seeger and Dr. Candace Berrett, BYU Department of Statistics

### Introduction

Hyperspectral imaging (HSI) is a technology that provides a dense set of previously un- available data{o ering the opportunity for use in a variety of applications such as food safety, ecology, and non-proliferation research. HSI stores measurements across three dimen- sions (two-dimensional space and the electromagnetic spectrum), resulting in large, three- dimensional data cubes. This additional amount of information can be used to identify materials in the images remotely. However, due to the many possibilities for measurement and other errors, it is hard to distinguish between the signal (the material spectra) and these sources of noise. We combine a known physical model within a statistical model to sepa- rate out the material-speci c emissivity spectra to better identify materials in these images. Speci cally, we designed a Bayesian nonparametric statistical model, and tested its eciency using simulation studies.

The dataset used for this project, collected using the Denali long-wave infrared region (LWIR) HSI sensor, was designed speci cally for the exploration of hyperspectral datasets 1. The HSI are of a wooden plank displaying 28 materials (e.g., granite and pine wood) one square foot each and images of this scene were captured approximately every ten minutes through- out a three month period. We analyzed a small subset of this data for model development. We compared the resulting estimated emissivity spectra to known emissivity spectra of sim- ilar materials from the Advanced Spaceborne Thermal Emission and Re ection Radiometer (ASTER) spectral library.

### Methodology

In this analysis, we used a Bayesian nonparametric statistical model, a method well-known for its ability to cluster similar items and obtain distributions for unknown parameters. We built our model to cluster on material-speci c emissivities { an approach not previously taken. Our model is:

where Yijt is the observed radiance at a given pixel (i = 1;…;N), wavelength (j = 1;…;W), and time (t = 1;…;M), Eij is the emissivity at a given pixel and wavelength, and B(\j ; Tit) is Planck’s law for blackbody radiation at wavelength \j (the known value of the jth wave- length) and temperature Tij , and Eijt is a zero-mean white noise term. We also set prior distributions on unknown parameters: Eij follows a Dirichlet Process with parameters and Tit follows a normal distribution with parameters UT and O2T, and eijt follows a zero-mean normal distribution with variance O2e.

From equation (1), it is clear that the observed radiance spectra are a ected both by temper- ature and emissivity. However, their e ect on radiance is somewhat confounded. As either variable increases, radiance increases, making it dicult to tell their individual contributions to the changes in radiance spectra. Because of this e ect, we conducted a simulation study using various prior parameter values to determine appropriate values so that we can identify parameters but still allow the data to be well represented.

### Results

Using our model, we generated synthetic data sets with 10 observations (N = 10), 2 wave- lengths (W = 2), and 2 times (M = 2). Keeping all other variables constant, we simulated three data sets, where O2e = 0:001; 0:01; 0:1, and t each of these data sets with varying prior values of (= 1, 10, 1000) and (= 0.001, 0.01, 0.1). We found that as O2e increases, a had to be low and E needed to be relatively large to obtain posterior distributions that closely matched the truth but did not cluster too strongly (i.e., the prior distributions were not too informative). For example, the posterior distributions when O2e = 0:1; a = 1; and E = 0:1 included the true parameter values and represented the data well by not being overpowered by the priors (Figure 1 on the left). When these criteria are not met, the clustering mecha- nism performed poorly and did not accurately capture true parameter values, as seen in the posterior distributions of Figure 1 on the right.

### Discussion

The simulation study allowed us to e ectively t our model to the Denali data described in the Introduction. As seen in Figure 2, the model clustered well for the selected four materials. While noticeable di erences exist between our results and true values, it is important to note that radiance is heavily a ected by location; because our data was not collected in the same area as the ASTER library data, we expect some di erences. In spite of this, the spectral trends in the posterior means follow those of the ASTER library spectra.

### Conclusion

We developed statistical methodology for temperature-emissivity separation of LWIR HSI { a problem not easily solved. By combining physical and statistical models, and analyzing the impact of the associated model parameters, we are able to build a robust model to identify pixels of matching materials and pull out the material-speci c emissivity spectra.