GSURS: Generalized sparse uniform resampling with application to MRI
2015 International Conference on Sampling Theory and Applications (SampTA)
We present an algorithm for resampling data from a non-uniform grid onto a uniform grid. Our algorithm termed generalized sparse uniform resampling (GSURS) uses methods from modern sampling theory. Selection of an intermediate subspace generated by integer translations of a compactly supported generating kernel produces a sparse system of equations representing the relation between the nonuniformly spaced samples and a series of generalized samples. This sparse system of equations can be solved
... efficiently using a sparse equation solver. A correction filter is subsequently applied to the result in order to attain the uniformly spaced samples of the signal. We demonstrate the application of the new method for reconstructing MRI data from nonuniformly spaced k-space samples. In this scenario, the algorithm is first used to calculate uniformly spaced k-space samples, and subsequently an inverse FFT is applied to these samples in order to obtain the reconstructed image. Simulations using a numerical phantom are used to compare the performance of GSURS with other reconstruction methods, in particular convolutional gridding and the nonuniform FFT. I. INTRODUCTION Medical imaging systems such as magnetic resonance imaging (MRI) and computerized tomography (CT) sample signals in k-space, namely the spatial frequency domain. A non-Cartesian sampling grid in k-space is often used to improve acquisition time and efficiency. A popular approach to recover the original image is to resample the signal on a Cartesian grid and then use the inverse fast Fourier transform (IFFT) in order to transform back into the image domain. It has been shown  that this is advantageous in terms of the computational complexity involved. In MRI, the most widely used resampling algorithm is convolutional gridding (CG)  , which consists of four steps: 1) precompensation for varying sampling density; 2) convolution with a Kaiser-Bessel window onto a Cartesian grid; 3) IFFT; and 4) postcompensation by dividing the image by the transform of the window. Two other notable classes of resampling methods employed in medical imaging are the least-squares (LS) and the nonuniform-FFT (NUFFT) algorithms. LS methods, in particular URS/BURS , , exploit the relationship between the acquired nonuniformly spaced k-space samples and their uniformly spaced counterparts, as given by the standard sincfunction interpolation of the sampling theorem. These methods invert this relationship using the regularized pseudo-inverse.