Multiple receiver coils have been used since the beginning of MRI (1), mostly for the benefit of increased signal- to-noise ratio. In the late 1980's, Kelton et al.(2) proposed in an abstract to use multiple receivers for scan acceleration. However, it was not until the late 1990s when Sodickson and Manning (3) presented their method simultaneous acquisition of spatial harmonics (SMASH) and later Pruessmann et al.(4) presented SENSE, that accelerated scans using multiple receivers became a practical and viable option.

Multiple receiver coil scans can be accelerated because the data obtained for each coil are acquired in parallel and each coil image is weighted differently by the spatial sensitivity of its coil. This sensitivity information in conjunction with gradient encoding reduces the required number of data samples that are needed for reconstruction. This concept of reduced data acquisition by combined sensitivity and gradient encoding is widely known as parallel imaging.

Over the years, a variety of methods for parallel imaging reconstruction has been developed. These methods differ by the way the sensitivity information is used. Methods like SMASH (3), sensitivity encoding (SENSE) (4, 5), sensitivity profiles from an array of coils for encoding and reconstruction in parallel (SPACE-RIP) (6), parallel magnetic resonance imaging with adaptive radius in *k*-space (PARS) (7), and parallel imaging reconstruction for arbitrary trajectories using *k*-space sparse matrices (kSPA) (8) explicitly require the coil sensitivities to be known. In practice, it is very difficult to measure the coil sensitivities with high accuracy. Errors in the sensitivity are often amplified and even small errors can result in visible artifacts in the image (9). On the other hand, autocalibrating methods like auto-SMASH (10, 11), partially parallel imaging with localized sensitivities (PILS) (12), generalized autocalibrating partially parallel acquisitions (GRAPPA) (13), and anti-aliasing partially parallel encoded acquisition reconstruction (APPEAR) (14) implicitly use the sensitivity information for reconstruction and avoid some of the difficulties associated with explicit estimation of the sensitivities. Another major difference is in the reconstruction target. SMASH, SENSE, SPACE-RIP, kSPA, and AUTO-SMASH attempt to directly reconstruct a single combined image. Coil-by-coil methods, PILS, PARS, and GRAPPA directly reconstruct the individual coil images, leaving the choice of combination to the user. In practice, coil-by-coil methods tend to be more robust to inaccuracies in the sensitivity estimation and often exhibit fewer visible artifacts (9, 14, 15).

SENSE is an explicit sensitivity-based, single-image reconstruction method. Among all methods, the SENSE approach is the most general. It provides a framework for reconstruction from arbitrary *k*-space sampling and to easily incorporate additional image priors. When the sensitivities are known, SENSE is the optimal solution (14, 15). To the best of the authors' knowledge, none of the coil-by-coil autocalibrating methods are as flexible and optimal as SENSE. Some proposed methods (16–19) adapt GRAPPA to reconstruct some non-Cartesian trajectories, but these require approximations and therefore lose some of the ability to remove all the aliasing artifacts.

Here, we propose a new approach to parallel imaging reconstruction called SPIRiT (iterative self-consistent parallel imaging reconstruction). It is a coil-by-coil autocalibrating reconstruction. It is heavily based on the GRAPPA reconstruction but also draws its inspiration from SENSE in the sense that the reconstruction is formulated as an inverse problem in a very general way. The result is that the reconstruction is the solution for a least-squares optimization. SPIRiT is based on self-consistency with the calibration and acquisition data. It is flexible and can reconstruct data from arbitrary *k*-space sampling patterns and easily incorporates additional image priors.

In this paper, we first review the foundations of SPIRiT, e.g., the GRAPPA method. We then define the SPIRiT consistency constraints and show that the reconstruction is a solution to an optimization problem. We then extend the method to arbitrary sampling patterns with *k*-space-based and image-based approaches. Finally, we show that the method can easily introduce additional priors such as off-resonance correction and regularization.