  Ultra low radiation dose in X-ray Computed Tomography (CT) is an important
clinical objective in order to minimize the risk of carcinogenesis. Compressed
Sensing (CS) enables significant reductions in radiation dose to be achieved by
producing diagnostic images from a limited number of CT projections. However,
the excessive computation time that conventional CS-based CT reconstruction
typically requires has limited clinical implementation. In this paper, we first
demonstrate that a thorough analysis of CT reconstruction through a Maximum a
Posteriori objective function results in a weighted compressive sensing
problem. This analysis enables us to formulate a low dose fan beam and helical
cone beam CT reconstruction. Subsequently, we provide an efficient solution to
the formulated CS problem based on a Fast Composite Splitting Algorithm-Latent
Expected Maximization (FCSA-LEM) algorithm. In the proposed method we use
pseudo polar Fourier transform as the measurement matrix in order to decrease
the computational complexity; and rebinning of the projections to parallel rays
in order to extend its application to fan beam and helical cone beam scans. The
weight involved in the proposed weighted CS model, denoted by Error Adaptation
Weight (EAW), is calculated based on the statistical characteristics of CT
reconstruction and is a function of Poisson measurement noise and rebinning
interpolation error. Simulation results show that low computational complexity
of the proposed method made the fast recovery of the CT images possible and
using EAW reduces the reconstruction error by one order of magnitude. Recovery
of a high quality 512$\times$ 512 image was achieved in less than 20 sec on a
desktop computer without numerical optimizations.
