In parallel magnetic resonance imaging (pMRI), *k*-space data are acquired from multiple channels simultaneously such that they can be sampled at a rate lower than the Nyquist sampling rate. Standard pMRI reconstruction methods include sensitivity encoding (SENSE) (1, 2), simultaneous acquisition of spatial harmonics (SMASH) (3), and generalized autocalibrating partially parallel acquisitions (GRAPPA) (4). Among them, SENSE is known to theoretically be able to give the exact reconstruction of the imaged object in the absence of noise. However, in practice, a well-known issue with SENSE is noise amplification due to the ill-conditioned nature of the inverse problem. The issue is especially serious when a large reduction factor is employed.

Regularization (5–15) has been employed as one of the simplest techniques to address the ill-conditioning issue in SENSE by solving an unconstrained minimization problem as:

where **d** is the vector formed from all *k*-space data acquired in all channels, **f** is the unknown *n*-dimensional vector defining the desired full field of view (FOV) image to be computed, and **E** is the sensitivity encoding matrix comprising Fourier encoding and sensitivity weighting, *G*(**f**) is the regularization term to describe the prior information of **f** and λ > 0 is the regularization parameter chosen to balance the data consistency (first term) and the deviation (second term) from the prior information of the image. The parameter λ can be set a global value heuristically or automatically using the L-curve method, variance partitioning method, or maximum likelihood estimation (7–9). It can also be set multivariate values based on different noise levels in wavelet domain (10, 11) or the g-factor map (16).

Most commonly used regularization techniques in SENSE include Tikhonov regularization (7, 12), total variation (TV) regularization (13, 14), and regularization with a Markov random field (MRF) model (9, 15). A common issue with Tikhonov regularization is the smoothing effect on edges due to assuming intensities vary smoothly over the entire image. To overcome this issue, some edge-preserving regularization techniques have attracted a lot of attention. TV regularization preserves edges by imposing the constraint that the image is piecewise smooth, where the regularization term is the TV norm of an image defined as a function of the image gradient (17):

∇_{x} and ∇_{y} denote the gradient along horizontal and vertical directions respectively, and |…| denotes the complex modulus. Other edge-preserving regularization techniques use a MRF model with edge-preserving priors (9, 15). For example, in EPIGRAM (15), a truncated Gibbs prior is used with the regularization function in Eq. [ 1] being

where ∇ denotes the local gradient between the nearest neighboring pixels and *T* is a constant. The method penalizes the intensity differences between neighbors only when they are below a predefined threshold *T* such that edges with the difference above the threshold are preserved. A drawback of TV and MRF-based regularization methods is that they both use local information only, which may cause blocky effects with a loss of fine structures while preserving edges in reconstruction.

Nonlocal regularization methods (18–24) such as nonlocal TV (NLTV) and nonlocal H^{1} have recently been studied in image denoising to address the issue of blocky effect by employing nonlocal pixels for calculating the gradients in the regularization term. Nonlocal H^{1} is regarded as a variational equivalent of the nonlocal mean filter and has been used in SENSE reconstruction for improving the signal-to-noise ratio (25). In this study, we investigate NLTV for SENSE regularization to address the issue of blocky effect with TV-regularized SENSE. NLTV has demonstrated its superior performance to TV (21–24) and also nonlocal H^{1} (18, 20) in other regularization applications. In NLTV, the gradient for the regularization term is calculated with pixels belonging to the whole image, instead of only the nearest neighboring pixels as used in TV regularization. In addition, a weighted graph between the current pixel and all image pixels is used in calculating the gradient. These differences allow the NLTV regularization to effectively remove noise without destroying the salient features of the original image. Our in vivo results demonstrate that the proposed NLTV regularization is able to preserve more details and fine structures than the existing regularization methods while suppressing noise.