Parallel magnetic resonance imaging (MRI) utilizes multiple coils to simultaneously receive radio frequency (RF) signals emitted from a scan subject (1–4). Aided by the spatial distribution of the RF coils' reception sensitivity, parallel MRI has been widely used for improving imaging speed or reducing artifacts. A key problem in developing parallel MRI has been the computational difficulty in forming an image from data acquired by multiple coils. Over the past several years, two feasible classes of algorithm have been introduced for parallel imaging reconstruction: image-domain algorithms and *k*-space algorithms. Image-domain algorithms directly compute the image from the *k*-space data acquired by each coil, where as the *k*-space algorithms compute the spectrum of the image.

The most successful image-domain algorithm has been the iterative sensitivity encoding (SENSE) algorithm for arbitrary trajectories by Pruessmann et al. (5, 6). Another image-based algorithm named “sensitivity profiles from an array of coils for encoding and reconstruction in parallel” (SPACE RIP) has also been introduced by Kyriakos et al (7) to reconstruct an image column-by-column along the phase encoding direction. *k*-Space algorithms include, for example, simultaneous acquisition of spatial harmonics (SMASH) by Sodickson and Manning (8) and generalized GRAPPA by Bydder et al. (9), generalized autocalibrating partially parallel acquisitions (GRAPPA) by Griswold et al. (10, 11), and parallel imaging with adaptive radius in *k*-space (PARS) by Yeh et al. (12), among other methods (13, 14). The iterative SENSE algorithm expressed the *k*-space data as a linear combination of the spatially-encoded magnetization in which the multiplicative spatial encoding function is the product of coil sensitivity and the Fourier encoding function. The resulting system of linear equations is solved with the conjugate gradient (CG) or related method in an iterative fashion since the sheer size of the design matrix precludes a direct solution of the inverse problem (15). To improve the reconstruction speed, the matrix-vector multiplication encountered in each iteration is replaced by a gridding and inverse gridding procedure, which is an important innovation that has made iterative SENSE feasible for arbitrary sampling trajectories with a reasonable reconstruction speed (6). With the perfect knowledge of coil sensitivity, the iterative SENSE algorithm is shown to provide an accurate estimation of the true underlying image.

GRAPPA was first introduced to reconstruct an image that is partially acquired on a Cartesian grid (10). The essential idea is to estimate the missing *k*-space data points by linearly combining its acquired neighboring points. The weights used in this estimation are first trained on some calibration lines that are typically acquired near the center of the *k*-space. In addition, there have been some recent efforts to extend this method beyond Cartesian trajectory (16, 17). The PARS algorithm estimates the *k*-space data on a grid using its neighboring data points sampled on arbitrary trajectories similar to the gridding procedure. Contrary to GRAPPA, the combination weights of PARS are calculated using coil sensitivity maps rather than the calibration lines. Both GRAPPA and PARS compute each individual coil image first and combine the resulting coil images through a sum-of-squares reconstruction.

Despite the successes of these existing algorithms, it remains a challenge to rapidly and reliably reconstruct an image from undersampled *k*-space data. One major issue is the difficulty in processing the large amount of data generated by higher spatial resolution, more coil elements, and 3D imaging. In addition, functional studies, such as functional MRI (fMRI) (18–20), perfusion imaging (21, 22), and diffusion tensor imaging (DTI) (23–27), pose an especially difficult challenge, in which thousands of images have to be reconstructed for a single study. For example, a typical whole-brain DTI study generates on the order of 1000 images. Without parallel computing implementation, it typically takes the iterative SENSE algorithm minutes to reconstruct a 256 × 256 image acquired with an eight-channel receiving coil. Even with a speed of one minute per image, it takes over 16 hours to reconstruct every 1000 images. Such long computational time makes parallel imaging essentially impractical for routine clinical functional studies. For dynamic studies or DTI studies, in principle, the design matrix only needs to be inverted once and then can be applied repetitively to reconstruct all subsequent images. However, iterative reconstruction has precluded such approach. Alternative means that allow the precomputing of the complex weights needed for parallel imaging reconstruction are, therefore, highly desirable for these types of applications.

In this work, a systematic parallel imaging reconstruction algorithm in *k*-space, termed *k*-space sparse matrices (kSPA), is proposed that particularly suits such kind of repetitive image reconstruction process. The image reconstruction problem is formulated in the *k*-space as a linear algebra problem. The kSPA algorithm then solves the problem by taking advantage of the sparsity of the resulting matrices. With this new algorithm, a sparse approximate reconstruction matrix (i.e., the inverse of the design matrix) can be directly computed. Multiple images can then be reconstructed in a fraction of the time needed for iterative SENSE by repetitively applying the reconstruction matrix through a matrix and vector multiplication. This algorithm is demonstrated with both simulated and *in vivo* data.