Yu Y. Li^{1,2}

A new parallel imaging reconstruction framework is proposed to accelerate MRI using both coil sensitivity and data sparsity. This framework uses random undersampling and performs parallel imaging reconstruction with a k-space variant constraint. No calibration data are needed. It is demonstrated that this new approach offers a gain over conventional parallel imaging in imaging acceleration.

Figure 1 illustrates the correlation
imaging model^{1,2}, which provides a mathematical description for
parallel imaging. This model may lead to different approaches based on how correlation
functions are estimated. For example, if correlation functions are calibrated directly
from auto-calibration signals (ACS), the reconstruction is equivalent to
GRAPPA. The presented work introduces a new approach to estimating correlation
functions from randomly undersampled data with k-space variant constraints. The
whole k-space is reconstructed in a region-by-region fashion. Each regional
reconstruction follows the steps detailed below:

1).
Estimate correlation functions from randomly undersampled data: Correlation
function is a statistical parameter that may be estimated from a subset of k-space data samples. Ideally, the ensemble
summation for every **k** in Figure 1 (Equation 1) should use the
same number of data samples, which is not the case with uniform undersampling. With
random undersampling, as shown in Figure 2 (top), the ensemble summation at
every **k**≠0 for a local region uses
almost the same number of samples, providing an approach to estimating
correlation functions without fully-sampled calibration data.

2). Improve correlation functions with k-space variant sparsity constraints: MRI relies on gradient encoding that decomposes an image into different signal components in a spatial-frequency domain (k-space). The high-frequency components (outer k-space data) arise mostly from tissue boundary that is naturally sparse in the human anatomy. For this reason, k-space data have tissue boundary sparsity increasing with the distance to the k-space center (the increase of spatial frequencies). As a result, if correlation functions are estimated from different k-space regions, they have tissue boundary sparsity that increases from the center to the outer k-space. This tissue-boundary sparsity variation in k-space is used as a constraint in the estimate of correlation functions (Figure 2 center).

3). Calculate
missing samples from neighboring data with parallel imaging reconstruction: As
illustrated in Figure 2 (bottom), every missing sample is calculated from the
linear combination of its neighboring collected samples. The linear
coefficients for reconstruction can be resolved from the linear equations determined
by estimated correlation functions. It should be noted that these linear
equations are not dependent on correlation functions at the k-space center (**k**=0).
For this reason, although the estimate of correlation functions at **k**=0
may be biased due to the use of more samples in ensemble summation (Figure 2
top), the final parallel imaging reconstruction is not affected.

To validate the approach, a set of 3D brain imaging data were collected using a T1-weighted fGRE sequence (FOV 240×240×240 mm, matrix 240×240×154, TR/TE 9.3/4.6 ms, flip angle 8°, ~6 minutes) on a 3T MRI scanner. The coil array for data collection had 8 elements uniformly positioned around the anatomy. The fully-sampled data were manually undersampled to simulate imaging acceleration in the left-right and anterior-posterior directions. SENSE, GRAPPA, and SPIRiT were used as reference approaches.

1. Li, Y et al., MRM 2012, 68:2005-2017.

2. Li, Y et al., MRM 2015, 74(6): 1574-1586.

Figure
1. Correlation imaging model^{1,2} gives a mathematical description for
general parallel imaging. The model may generate either SENSE-like or a
GRAPPA-like reconstruction depending on how correlation functions are
estimated.

Figure
2. The new parallel imaging reconstruction framework can accelerate data
acquisition with random undersampling and without calibration. The whole
k-space is divided into multiple local k-space regions. In each region,
correlation functions are estimated from randomly undersampled data with a data
sparsity constraint depending on the k-space location. The estimated
correlation functions are used to calculate the missing data using parallel
imaging reconstruction based on correlation imaging model.

Figure 3. In
reference to the real image (leftmost), the new parallel imaging reconstruction
method (rightmost) gives better performance than SENSE, GRAPPA, and SPIRiT with
the same acceleration factor (~10 fold) in a 3D brain imaging experiment with
an 8-channel coil array.