FAST SPIRAL FOURIER TRANSFORM FOR ITERATIVE MR IMAGE RECONSTRUCTION Michael Lustig 1 , Jacob Tsaig 2 , Jin Hyung Lee 1 , David Donoho 3 1 Department of Electrical Engineering, Stanford University 2 Department of Computer Science (SCCM), Stanford University 3 Department of Statistics, Stanford University ABSTRACT We present a fast and accurate Discrete Spiral Fourier Transform and its inverse. The inverse solves the problem of reconstructing an image from MRI data acquired along a spiral k-space trajectory. First, we define the spiral FT and its adjoint. These discrete operators allow us to efficiently compute the inverse using fast-converging conjugate gradient methods. Next, we developed a fast approximate spiral FT using the pseudo-polar FFT, to enhance the computational performances and numerical accuracy of the algorithm. Preliminary results demonstrate that the proposed algorithm is more accurate than existing iterative methods that use similar interpolation and grid size. 1. INTRODUCTION Spiral MRI has received much attention because of its fast acquisition, efficient use of the hardware, low motion and flow sensitivity. One can regard the spiral MRI as a physical device, which computes a spiral FT of an image. Formalizing this mathematically, we define the spiral FT as the operator that takes the digital image into the frequency domain by evaluating the FT on a family of spirals. The problem of reconstructing the image is then the problem of inverting the spiral FT. Heuristic approaches to approximately reconstructing and image from spiral MRI data have been well addressed in literature [1-4], though often without specifically identifying the problem as one of inverting the spiral FT operator. Most of the approaches rely on interpolating the non-uniform data samples from the spiral grid onto a usually over sampled Cartesian grid and then applying a 2D FFT to reconstruct the data (i.e. gridding algorithms). These methods differ mostly by the choice of interpolation kernel and grid size. A weighting is usually applied on the spiral data before the interpolation to compensate for the difference in the grids’ sampling density. To obtain a more accurate inverse spiral FT, iterative methods have been developed [5]. As we will discuss later, inverting the spiral FT can be achieved iteratively by combined application of the forward spiral FT and its adjoint operations. Existing iterative approaches can be viewed as computing approximations to the forward and adjoint transforms using gridding. One can think of the gridding algorithm as approximating the adjoint of the spiral transform, and the density compensation in the gridding algorithm as preconditioning. Another way to think about it would be as a single iteration of the iterative approach. The above methods are fast and efficient in providing an approximate operation for the inverse spiral FT. However, the interpolations in the methods above are done onto a uniformly sampled Cartesian grid whereas the spiral trajectories have a highly non-uniform density around the center of k-space. Reconstruction then becomes a tradeoff between extensive over sampling to resolve the low frequency components at the expense of complexity, or loosing valuable information in the low frequency region. We have developed fast forward and inverse spiral FT procedures. Specifically, we are able to accurately and rapidly compute the forward spiral FT and its adjoint. In our proposed method we use re-sampling onto a variable density grid (The pseudo-polar grid) that is more natural to the spiral sampling characteristics. By doing so, we are able to accurately reconstruct the low frequency data without the need of extensive over sampling. One can think of the adjoint operation of our proposed algorithm as gridding onto a pseudo-polar grid and then taking the adjoint of the pseudo-polar FFT.