## 1 Introduction
Hyperspectral X-ray detectors are able to provide a full X-ray spectrum at each individual detector pixel, typically by separating and classifying the incident X-ray photon energy into a very large number (up to several thousand) of discrete energy bins. A recent example of a hyperspectral detector used for X-ray imaging is the SLcam [40]. For a comparison of the performance and characteristics of these types of detector the reader is referred to the recent review of this field by [PERSON] et al. [41]. Early work using energy-discriminating detectors to generate quasi-monochromatic X-ray images employed sensors which comprised a relatively small number of pixels with a large pixel pitch [42; 43]. Through continuous development higher resolution energy discriminating detectors have now become available commercially. These detectors permit the specification of multiple energy thresholds resulting in the simultaneous production of multiple quasi-monochromatic images, each corresponding to a different characteristic energy range with a pixel resolution in the micron range [44]. One of the first demonstrations of material separation using energy-discriminating X-ray detectors was published by [PERSON] et al. [11]. In this paper, the authors used specific X-ray contrast enhancing agents in the sample and based on their K-edge signature were able to obtain separate images for each of the different elements. Similarly, [PERSON] [45] also used contrast enhancing agents including iodine and CaCO\({}_{3}\) to facilitate soft tissue enhancement in Computed Tomography (CT) using energy-resolving X-ray detectors in a fan-beam geometry. In this work, the author used a total of 5 discrete energy bands to enable separation of the various elements within a phantom test sample. Another study using energy discriminating detectors was also conducted by [PERSON] et al. using a semiconductor matrix direct conversion detector and optimising the energy range in CT to enable differentiation between aluminum and a potassium iodide solution within a phantom test sample [46]. In addition, to enabling elemental separation based on the characteristic K-edge absorption profile of different elements [13] energy-discriminating X-ray detectors also provide a simple and effective means of eliminating or reducing beam-hardening artifacts [24; 47]. In a recent study, a high-resolution energy-discriminating detector (PiXirad) was used in X-ray CT to separate soft-tissue from bone, without employing any contrast enhancing agents. The tomographic reconstructions were then quantitatively analyzed enabling bone densitometry using a polychromatic lab-based X-ray source [24]. In summary, the increasing availability of energy-resolving X-ray detection has driven a significant amount of interest in exploiting these detectors for materials differentiation. In the present study, we have investigated the use of the energy-discriminating PiXirad detector for materials separation in the context where there are no K-edges or contrast enhancing agents. Here we present the first application of the matrix method using multiple quasi-monochromatic images generated with each single exposure to solve the linear attenuation matrix equation resulting in the three-dimensional separation of the sample components. We find that when combining this approach with energy-resolving X-ray detection the method can be generalized to apply to any polychromatic X-ray source and samples where no defined absorption edge is present in the transmission spectrum and no contrast enhancing agent is used. In order to demonstrate this, a sample consisting of a mouse hand is imaged and the 3D distributions of extracted bone and soft tissue.