A simple iterative algorithm, termed deconvolution-interpolation gridding (DING), is presented to address the problem of reconstructing images from arbitrarily-sampled k-space. The new algorithm solves a sparse system of linear equations that is equivalent to a deconvolution of the k-space with a small window. The deconvolution operation results in increased reconstruction accuracy without grid subsampling, at some cost to computational load. By avoiding grid oversampling, the new solution saves memory, which is critical for 3D trajectories. The DING algorithm does not require the calculation of a sampling density compensation function, which is often problematic. DING's sparse linear system is inverted efficiently using the conjugate gradient (CG) method. The reconstruction of the gridding system matrix is simple and fast, and no regularization is needed. This feature renders DING suitable for situations where the k-space trajectory is changed often or is not known a priori, such as when patient motion occurs during the scan. The reconstruction of images from nonuniform samples of their spatial frequency domain (k-space) is an important problem in fields as diverse as computed imaging and radioastronomy. A convolution-interpolation algorithm known as the "gridding algorithm" is widely used to reconstruct images from such data. Basically, this conventional gridding algorithm (1,2) convolves the nonuniform samples with a small-width window, and samples the result onto a rectilinear grid. It consists of the following four steps: First, the nonuniform samples are compensated for the nonuniform sampling density in k-space by multiplication with a weighting function the density compensation function (DCF), which has small values in areas of high sampling density and large values in sparsely sampled areas. Second, the density-compensated data are interpolated to a uniform grid using a small convolution window. Third, the data are Fourier transformed (FT) into the image domain. Fourth, apodization caused by the convolution step is compensated for by dividing the result by the FT of the convolution window. The conventional gridding algorithm is efficient and stable because the convolution step has a smoothing effect, but it has been shown to be non-optimal and artifact-prone (3,4). Density compensation is necessary to adequately approximate the convolution operation (3), and several studies have attempted to determine the optimum DCF (2,5,6). The final step in the conventional gridding algorithm, which compensates for the convolution roll-off, results in large signal at the image periphery (wings) due to the side lobes of the convolution window, and the aliasing caused by the sampling process. Sampling the result of the convolution onto a finer grid reduces aliasing and wing artifacts in the reconstructed image. A twofold subsampling ("twice-finer" grid) is commonly used to achieve acceptable image quality. Other methods used to overcome limitations in the conventional gridding algorithm commonly exploit the relationship between the acquired k-space samples and their rectilinear counterparts, as given by the standard sincfunction interpolation of the sampling theorem (7). This relationship is strictly valid only if the signal is of infinite length in k-space, but it does provide a good approximation if k-space coverage is large. Discrete sampling of the interpolation results in a linear system with a dense sincinterpolation matrix, the inversion of which results in the uniform resampling (URS) solution (8). While inverting the sinc-coefficients matrix is practical for one-dimensional (1D) signals, it becomes impractical for higher dimensions due to the tremendous matrix size, except for special cases (9). Nevertheless, the conventional gridding algorithm has been shown to be an approximation of the URS solution, and an optimal DCF that minimizes the difference between the gridding and the URS solution is obtained with matrix approximation techniques for structured matrices (7). However, these techniques are computationally exhaustive. The block URS (BURS) method is an approximation to the URS solution that aims to reduce computational effort (8). A small block of the sinc-matrix around each grid point is isolated and inverted to obtain a spatially varying convolution window. Attention to regularization is critical, and poor estimates are reported for Lissajous k-space trajectories (9 -11). The computation required to reconstruct the gridding matrix is still huge, although it only has
Alternatively, the authors in REF provide a simple iterative algorithm, termed deconvolution-interpolation gridding (DING) for reconstructing MRI image from arbitrarily-sampled k-space.
Deconvolution-interpolation gridding (DING): Accurate reconstruction for arbitrary k-space trajectories
Magnetic resonance in medicine
Magnetic resonance in medicine
Computer Science Medicine