Spectral computed tomography with linearization and preconditioning

In the area of image sciences, the emergence of spectral computed tomography (CT) detectors highlights the concept of quantitative imaging, in which not only are reconstructed images offered, but weights of different materials that compose the object are also provided. If a detector is made up of several energy windows and each energy window is assumed to detect a specific range of energy spectrum, then a nonlinear matrix equation is formulated to represent the discretized process of attenuation of x-ray intensity. In this paper, we present a linearization technique to transform this nonlinear equation into an optimization problem that is based on a weighted least squares term and a nonnegative bound constraint. To solve this optimization problem, we propose a new preconditioner that can significantly reduce the condition number, and with this preconditioner, we implement a highly efficient first order method, the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA), to achieve substantial improvements on convergence speed and image quality. We also use a combination of generalized Tikhonov regularization and \ell ₁ regularization to stabilize the solution. With the introduction of new preconditioning, a linear inequality constraint is introduced. In each iteration, we decompose this constraint into small-sized problems that can be solved with fast optimization solvers. Numerical experiments illustrate convergence, effectiveness, and significance of the proposed method.