Fast Implementation of Iterative Image Reconstruction

Iterative reconstruction is becoming an increasingly important topic in medical imaging. In computed tomography (CT) iterative reconstruction algorithms are becoming standard and they are implemented on current CT-scanners to reduce radiation dose [1]. This is not a new trend; the very first CT machines used an iterative technique, the algebraic reconstruction technique (ART) [2]. In emission tomography in nuclear medicine iterative reconstructions were used in the early days to integrate prior knowledge about the noise statistics into the reconstruction [3] and more advanced iterative algorithms have become a standard in nuclear medicine [4]. Also in MRI there is a trend for advanced reconstruction algorithms to integrate knowledge about coil sensitivities in parallel MRI [5], about motion [6-9] and other prior information into the reconstruction process. All these approaches result in large set of linear equations Ax=b, which need to be solved. A direct inversion of this equation system is impractical, except in the most trivial cases, since it requires a large amount of memory and a prohibitive number of operations. However, this complexity can be reduced by using iterative reconstruction algorithms, where direct inversion of the system matrix is not needed. These algorithms start with an initial guess for the solution and then iteratively progress towards an approximate solution. A variety of such techniques exist for the treatment of large linear systems. For many cases, the so-called conjugate gradient (CG) method [11] particularly suited. It converges to a minimum in N steps (where N is the rank of the matrix) if the matrix A is symmetric positive definite. The rate of convergence can be improved if preconditioning of the matrix A is performed. However, also in well-conditioned systems still around 10-20 iterations need to be performed. Therefore one of the main limitations of introducing iterative algorithms in image reconstruction has been the long computation time so far hindering the clinical translations of many new reconstruction algorithms. Modern algorithms can make use of acceleration techniques and parallelization to speed up the reconstruction. These techniques make use of the on-going developments in computer processors (CPU) and especially in commodity graphics hardware (GPU). General purpose computation on the GPU is still an emerging research area, but many algorithms have been accelerated significantly compared to their CPU counterparts [12, 13]. One of the most recent GPUs, the Geforce 560 (Nvidia, US), has 352 “stream processors” for these computations allowing significant acceleration if the computational problem can be solved in parallel. More importantly, both major hardware manufacturers (AMD and Nvidia, US) have released high-level programming environments requiring less knowledge of graphics programming in general. The GPU implementation of MR-reconstruction algorithms has been demonstrated for a number of applications [14-20]. In this presentation the GPU-implementation of an iterative reconstruction algorithm is presented for the example of non-Cartesian parallel imaging. However, the basic concepts can be applied to other applications.