## 4 Conclusions  
We have described and demonstrated an approach to material separation based on solution of the linear attenuation matrix equations which can be applied to polychromatic laboratory sources using commercially available energy resolving X-ray detectors. The approach presented here enables a more general application of this method by using the energy thresholding capabilities of an energy-discriminating (PiXirad) detector to separate multiple components. This concept was explored in both simulation and experiments for a phantom sample in 2D and a'real-world' biological sample in 3D. In order to separate two materials two energy bandwidths, need to be defined based on four intensity images in order to obtain two quasi-monochromatic images. Using the PiXirad, this can be achieved in a single exposure. However, as we have shown previously, the number of elements to be separated can be increased be taking multiple images with a different set of energy thresholds defined per exposure [13]. The ability to carry out this type of experiment is  
Figure 9: Tomographic reconstruction of the mouse hand. Left column: result corresponding to the 16 keV energy threshold. Middle column: retrieved thickness from bone and Right column: retrieved thickness from soft tissue after applying the matrix method. now widely available via a variety of different commercial X-ray energy discriminating detectors. Based on the results of this study the matrix method for materials separation combined with commercial energy-discriminating X-ray detection appears to be a promising avenue of research with a wide range of potential applications particularly in the field of medical imaging. An important question is how materials with very similar densities could be separated using the matrix method combined with energy-discriminating X-ray detection. An interesting application of this would be the differentiation of glandular and adipose regions in breast tissue or other samples that contain very similar density materials. The current state-of-the-art for research using energy discriminating detectors for soft tissue differentiation is to address the challenge of separating materials of comparable bulk density but differing \(\mu\)(E) dependence which is not amenable to analysis by traditional methods. In the present case, the key limiting factor is likely be the signal to noise ratio in the data and whether data of sufficient quality can be collected in order to reliably separate materials with almost identical X-ray interaction characteristics. Moreover, in this current setup, options for improving the SNR include increasing the number of frames per projection, binning together pixels at the cost of reduced spatial resolution, or by using an alternate energy-resolving detector with a higher dynamic range (e.g., 32 bit). In order to explore this application, future work will focus on the further optimisation of the energy ranges and bandwidths used for these measurements which, in the case of energy-discriminating detectors, can be easily varied by the user. The following are available online at [[https://www.mdpi.com/article/10.3390/app12063198/s1](https://www.mdpi.com/article/10.3390/app12063198/s1)]([https://www.mdpi.com/article/10.3390/app12063198/s1](https://www.mdpi.com/article/10.3390/app12063198/s1)), Figure S1. Fitting parameter K_(E_th) which characterizes the degree of charge sharing as a function of energy threshold for the HA400 calibration sample, Figure S2. (a) Energy resolution of the PiXirad detector as a function of energy. Data taken from [PERSON]. [PERSON] et al. [5]. The position of the experimental energy thresholds are indicated by the orange circles. (b) Corresponding energy resolution values for each of the four experimental en-ergy threshold values extrapolated from (a), Figure S3. Schematic diagram showing the key steps in the forward simulation and subsequent retrieval of epoxy and HA insert components. All simulations were performed using MATLAB, Table S1.