## 1 Introduction  
Polychromatic laboratory-based X-ray sources are used for imaging experiments on a wide range of life and materials science samples [1, 2]. Since its initial discovery, X-ray imaging has undergone a process of continuous improvement in terms of data collection speed, as well as spatial resolution [3]. A key area of current research is the separation of composite materials into their discrete components based on the acquisition of a single image at each projection angle [4]. Differentiating materials into their constituent components is particularly challenging for weakly interacting biological samples [5, 6], which typically do not produce strong transmission features such as absorption edges. Techniques such as X-ray phase contrast microscopy have been employed to help differentiate biological components, but these are often only applicable with coherent, highly monochromatic sources [7, 8], particularly when very high spatial resolution is required [9]. However, a pathway towards the clinical application of phase contrast imaging using more compact laboratory-based X-ray sources has emerged in recent years [10]. The availability of high-resolution, energy-discriminating, X-ray detectors that enable discretization of the incident polychromatic signal, offer new opportunities for developing approaches that allow for the effective separation of materials [11, 12]. By energy thresholding the polychromatic transmission image at the detector, multiple quasi-monochromatic images can be simultaneously collected [13]. Each image corresponds to a different narrow bandwidthwithout suffering from any spectral overlap [14; 15; 16]. Here, we combine energy discriminating X-ray detection with the application of a matrix-based approach to materials separation using the linear attenuation coefficient. We evaluate the performance of this approach for samples composed of two constituent materials, which together form a composite sample using X-ray CT data collected using an energy-discriminating detector. Elemental contrast or mapping the spatial distribution of an element within samples can be readily achieved using tunable monochromatic X-ray sources, which are typically provided by synchrotrons. For example, by measuring the characteristic peaks associated with X-ray fluorescence spectra, specific elements can be mapped, and their spatial location and concentration determined [17; 18; 19]. This approach typically involves scanning a small X-ray probe across the sample to construct a 2D or 3D elemental distribution map. The requirements of this method include the need for tightly focused X-ray beams in order to achieve high spatial resolution and a sufficient amount of X-ray flux that data can be collected within a practical time frame. Another method for obtaining the distribution of the specific elements which can be used with a full-field imaging setup is Absorption-Edge Contrast Imaging (AECI). AECI exploits the discrete jumps (absorption edges), which are present in the transmission spectra associated with different elements as a function of the incident X-ray energy [20; 21]. The AECI method is based on the subtraction of images collected either side of the absorption edge of the element of interest [21]. Hence, for AECI to work effectively, the tunability of the monochromatic source is critical. For laboratory-based X-ray sources, which typically have a polychromatic spectrum, elemental mapping is more challenging; however, a number of solutions have been proposed to address this. Methods for elemental mapping that have been successfully demonstrated using lab-based X-rays include dual energy CT [22; 23], Ross filter pairs [2] and energy-thresholding using energy-discriminating detectors [13; 24]. Dual-energy CT has previously been used to separate bone and soft tissue by exploiting differences in their attenuation properties as a function of energy [25]. The basic principle of dual-energy CT involves collecting data using two distinct energy spectra peaked around different incident X-ray energies [25; 26].