The Energy Integrating Detector Model. ∫ ∫ The computed tomography (CT) process mainly consists of three parts: x-ray beams are emitted from a source with specific energies, an object is illuminated by x-ray beams and attenuated x-ray beams are received by a detector. In this process, the intensities of x-ray beams are reduced and using Beer’s law [18], the energy integrating detector model can be written as yi = S(e) exp − µ (→r (t) , e) d t d e + ηi, i = 0, 0, · · · , Xx × Np, (3.2) E t∈l where • yi is the x-ray intensity of the i-th pixel in the detector. • E is the photon flux density. Figure 3.1 shows a curve of E versus photon energy with relative low potential (26 keV). • Nd is the number of detector pixels. For a material map of size n by n, we assume Nd = n and the number of projection rays for each angle is equivalent to [√2Nd♩. • Np is the number of projections. For cone/fan beam CT, projections are dis- tributed equally from 0 to 360 degrees. • S(e) represents the system spectral response, which is a product of x-ray energy with the number of incident photons at that energy. • The outer integral is over all x-ray energies emitted from the source, and the inner integral is along lines that follow the x-xxx xxxx paths through the object. • µ (→r (t) , e) denotes the attenuation coefficient, which is related to the position function →r (t) and the energy level e. • ηi represents unknown errors in the measurements, which can include x-ray scatter and electronic noise. The traditional methods to solve this inverse problem are mostly based on filtered back-projection (FBP) [49]. If we assume the source to be monoenergetic and the source energy is se, then we can build a linear inverse problem by dividing se on both sides and by applying the natural logarithm function to the data and to the model. Other approaches can be used to solve this linear inverse problem, such as incorporating different regularization schemes. These usually involve applying appropriate iterative optimization methods, such as the conjugate gradient method [45]. However, the images obtained from traditional methods on the simplified linear model might lead to significant beam hardening artifacts when the object is made up of several very distinct materials, such as bone and soft tissue. In addition, the linear models cannot be used to recognize the the actual types of materials from the results, nor can they separate different materials when they are mixed [33]. Moreover, the traditional methods are unstable when it ...
