## 1 Introduction
In the majority of cases, both sets of spectra have some degree of overlap which can impact the effectiveness of materials separation. Dual-energy CT has previously been used for materials separation in the absence of an absorption edge. By creating an in vitro'stone library' comprising 37 stones of known chemical composition and attenuation ratios, separation and chemical characterisation of kidney stones in twenty seven patients was achieved via the use of a multi-layer X-ray detector [27]. Furthermore, dual energy CT has also been used to measure mass density [28] and to extract materials specific density for two different types of tissues [26]. Whilst dual-energy CT has been successfully applied to soft tissue differentiation, some of the known disadvantages include limitations in terms of flexibility and the potential need for beam-hardening correction [29]. Another approach that was recently demonstrated for both two and three dimensional X-ray imaging uses Ross filter pairs to obtain elemental maps of the sample with a polychromatic source [2]. The basic principle of this method relies on the selection of pairs of filters which have an absorption edge located above and below the absorption edge of the elemental of interest. The Ross filter pair method for elemental contrast provides a very well-defined and sharp energy window which enables elemental enhancement for materials with an absorption edge within the defined energy range. In addition, X-ray fluorescence computed tomography (XFCT) was also used with laboratory-based X-ray sources to characterize the different elements within the sample or tissue [30]. For biological materials this approach generally provides excellent image contrast by incorporating contrast-enhancing elements (often in the form of nanoparticles) such as Mo and Au. The main limitation associated with XFCT in the context of the present work is the need to use these additional agents and ensure they perfuse homogeneously throughout the sample. Other issues include the contamination of the CT images due to Compton scattering and the longer data acquisition times required to raster scan a small pencil beam across the sample, which is particularly challenging for X-ray CT. Some of these issues have been addressedfor example a recent study conducted by [PERSON] et al. with the pencil-beam geometry showed improvements in the quality and speed of XFCT using a ring of spectrometers to detect the fluorescent signal [31]. Recent studies of elemental mapping using polychromatic laboratory sources include the use of polycapillary lenses in a confocal geometry [32; 33]. The authors of this work were able to achieve both 2D and 3D elemental mapping via colour tomography. Their method enabled depth profile analysis for a variety of sample types and had excellent sensitivity due to the simultaneous use of two separate detectors optimized to measure the X-ray spectrum independently at both high and low energy. Another approach to improving elemental selectivity using polychromatic lab-based X-ray sources is to use energy-discriminating X-ray detectors. These types of detectors have undergone rapid development and provide a high degree of spectral filtering via energy thresholds which can be easily defined in software. Detectors capable of full energy resolution imaging with a limited degree of pixelation first became commercially available well over a decade ago, however, the limited number of pixels and large pitch currently associated with these types of detectors [34] limits their application to high-resolution X-ray \(\upmu\)CT. A large number of different types of energy-discriminating detector are now commercially available and could potentially be used with the matrix method presented here, including the Medipix [35], Pilatus [36], XPAD [37], PiXirad [15], HEXITEC detector [38] and more recently, the hyperspectral X-ray detector [39].