The recent development of parallel MRI acquisition can enhance the spatiotemporal resolution of MRI in both anatomical and functional scans. Parallel MRI utilizes an RF coil array to simultaneously acquire data from multiple receivers, and acceleration is achieved by a reduced phase-encoding *k*-space trajectory. The nature of the subsampled *k*-space trajectory requires the use of a reconstruction algorithm to restore aliased images into full field-of-view (FOV) images. Previously proposed reconstruction algorithms include the *k*-space-based SMASH method (1) and the image domain-based SENSE approach (2). In addition to improving the spatiotemporal resolution, parallel MRI can reduce the image distortion in echo-planar imaging (EPI) (3) or diminish the acoustic noise by lowering gradient switching rates (4). Nevertheless, these advantages come at the cost of lowered signal-to-noise ratio (SNR) because the number of acquired data samples in parallel MRI is reduced. In addition, reconstructing parallel MRI acquisitions heavily depends on the independent information from each channel in an RF coil array. Correlations in spatial information caused by the geometric arrangement of the array element can deteriorate image quality. To mitigate this issue, researchers have previously optimized coil geometry (5) or improved the stability of the reconstruction algorithm. The increased noise originating from correlated spatial information in the array elements can be estimated based on the array geometry and quantified by a geometric factor (g-factor) (2).

Previously, we proposed to minimize the g-factor by using a full-FOV reference scan to provide prior information and to stabilize image reconstruction. The method was based on the Tikhonov regularization framework (6) to reduce the noise amplification of parallel MRI reconstruction when the encoding matrix is ill-conditioned. Applying regularized SENSE imaging to anatomical and dynamic functional MRI of human brain reduced the noise level and enabled higher detection power in acquisitions with high acceleration rates (7, 8). The importance of choosing appropriate regularization parameters has also been reported for different parallel MRI reconstruction algorithms with versatility in the choice of prior information and a regularization parameter (9–13). However, one disadvantage of regularized parallel MRI reconstruction is the computational time associated with regularization parameter estimation. In order to estimate a regularization parameter, the L-curve approach requires an iterative search for each set of aliased image pixels; each iteration step involves calculations of prior error and model error terms (7). As clearly demonstrated in this study, another challenge of the L-curve method is instability. Given a fixed noise level, the regularization parameter estimated by L-curve may vary significantly, potentially due to an ill-defined “elbow” region in L-curve calculation. This problem is particularly more prominent in low SNR acquisitions, where regularization is more crucial for suppressing noise in reconstruction.

The purpose of this study is to propose an alternative method to calculate a regularization parameter in parallel MRI reconstruction with improved computational efficiency and reduced variability in regularization parameter estimation. In this article we present a fast and robust method to estimate regularization parameters without an iterative search. The SNR of the set of aliased pixels in parallel MRI data is estimated and used to partition the singular value spectrum of the encoding matrix for regularization parameter estimation. Unlike other implementations of regularization parameter estimation algorithms that utilize minimal heuristic intervention (12, 13), the Variance Partitioning Regularization (VPR) method introduced here is fully automatic and adapts to the noise level of the acquired data. We evaluated the performance of VPR in terms of the variability of the estimated regularization parameter, the quality of the reconstructed image, and computational time using anatomical and functional MRI data.