| In recent years, a considerable amount of research in the area of computed tomography (CT) has been directed to the incorporation of the energy dependency of X-ray attenuation into the reconstruction scheme. Considering energy dependency is crucial in order to characterize the chemical composition of materials under investigation rather than simply providing relative attenuation images as is done in conventional tomography. In this thesis, novel iterative reconstruction techniques for polychromatic dual energy and multi-energy CT, which incorporate energy dependency in different ways, are proposed. Dual energy CT uses two different spectra at the source side to obtain energy selective information, whereas multi-energy employs energy discriminating photon counting detectors.;The proposed dual energy algorithm has an emphasis on detection and characterization of piecewise constant objects embedded in an unknown, cluttered background. Physical properties of the objects, specifically the Compton scattering and photoelectric absorption coefficients, are assumed to be known with some level of uncertainty. Our approach is based on a level-set representation of the characteristic function of the object and encompasses a number of regularization techniques for addressing both the prior information we have concerning the physical properties of the object as well as fundamental, physics-based limitations associated with our ability to jointly recover the Compton scattering and photoelectric absorption properties of the scene. In the absence of an object with appropriate physical properties, our approach returns a null characteristic function and thus can be viewed as simultaneously solving the detection and characterization problems. Unlike the vast majority of methods which define the level set function non-parametrically, (i.e., as a dense set of pixel values), we define our level set parametrically via radial basis functions (RBF's) and employ a Gauss-Newton type algorithm for cost minimization. Numerical results show that the algorithm successfully detects objects of interest, finds their shape and location, and gives an adequate reconstruction of the background.;The development of energy selective, photon counting X-ray detectors makes possible a wide range of new and exciting possibilities in the area of multi-energy CT image formation. Under the assumption of perfect energy resolution, here we propose a tensor based iterative algorithm that simultaneously reconstructs the X-ray attenuation distribution for each energy level. We use a multi-dimensional image model rather than a vector representation in order to develop a novel tensor-based regularizer. Specifically, we model the multi-spectral unknown as a 3-way tensor where first two dimensions are in space and the third dimension is in energy. This approach allows for the design of a tensor nuclear norm regularizer, which like its two dimensional counterpart, is a convex function of the multi-spectral unknown. Additionally, we introduce a Tikhonov type regularization method called adaptively weighted ℓ 2 (AWL2), which penalizes the weighted quadratic sum of the differences between neighbouring pixels, where the weights are updated at each iteration using a multiplicative update formula adapted to edge information. The solution to the resulting convex optimization problem is obtained using the alternating direction method of multipliers (ADMM). Simulation results shows that the generalized tensor nuclear norm can be used as a stand alone regularization technique for the energy selective (spectral) computed tomography (CT) problem. When combined with total variation (TV) regularization of AWL2 it enhances the regularization capabilities of these techniques especially at low energy images where the effects of noise are most prominent. Moreover, AWL2 provides excellent edge preserving and noise reduction capabilities with a simple quadratic formula that are superior to TV in the spectral CT set-up. |
