Auto-calibration in the image reconstruction of partially-acquired

**k**-space data has shown some favorable properties in previous studies of parallel imaging methods [

5], [

14], [

16], [

17], [

30], [

31]. We have demonstrated that auto-calibration schemes can also be incorporated into the kSPA algorithm. In addition, we have shown that the calibration data can be acquired, in principle, at any region in

**k**-space.

The proposed auto-calibrated kSPA shares the same characteristics as AUTO-SMASH [

14], [

30] and GRAPPA in which calibration steps are performed directly on acquired extra

**k**-space calibration data. Although the popular GRAPPA algorithm was originally proposed for Cartesian trajectory, a number of recent works have extended GRAPPA for spiral and radial trajectories. For example, Heidemann

*et al.* presented a method for constant angular-velocity spiral trajectory [

17]. In that method, samples on a constant angular-velocity spiral trajectory are reordered and placed on a hybrid Cartesian grid where the original GRAPPA algorithm is applicable. More recently, Heberlein and Hu introduced a method to interpolate missing spiral interleaves using GRAPPA interpolation kernels by dividing the

**k**-space into angular sectors and assuming that the kernel within each sector is radially invariant [

16]. While a GRAPPA based kernel provides the weighting factors to directly synthesize missing data in

**k**-space, auto-calibrated kSPA uses the calibration data to estimate the sparse matrix inversion. In addition, auto-kSPA does not attempt to fill in missing data points in

**k**-space, instead, it computes the whole

**k**-space based on measured samples. Auto-calibrated kSPA is a general reconstruction algorithm that applies to all sampling patterns. Some recent studies have also extended GRAPPA auto-calibration to non-Cartesian trajectories. For example, Seiberlich

*et al.* used a procedure with GRAPPA operator gridding (GROG) to reconstruct non-Cartesian data [

15], [

32]. In addition, Samsonov established a connection between the technique of parallel magnetic resonance imaging with adaptive radius in

**k**-space (PARS) [

33] and GRAPPA by using the concept of

*in vivo* coil sensitivity which is the true sensitivity modulated by the object [

34]. Such a connection should also allow auto-calibration for PARS.

The proposed algorithm has been demonstrated with both simulated and

*in vivo* data. The quality of reconstructed images is good. The average RMSE is 2.1% for simulated data and 4.0% for

*in vivo* data as shown in and . In the simulated data, the residual artifacts mainly originate from the edges of the Shepp-Logan phantom where high spatial frequency components exist. Such artifacts are consistent with the interpolation nature of the kSPA algorithm and the non-Cartesian sampling in

**k**-space. On the other hand, there is noticeable noise amplification in the center of the

*in vivo* images. This noise enhancement is explained by the increased g-factor and is consistent with that observed commonly in SENSE reconstruction [

3], [

4].

We show, in theory, that calibration data can be acquired at any location in

**k**-space. This result applies to all sampling trajectories and is consistent with the well-known shift-invariant feature of the GRAPPA kernel for Cartesian trajectory [

5] and the technique of parallel magnetic resonance imaging with adaptive radius in

**k**-space (PARS) [

33]. This result implies that the reconstruction matrix or the interpolation weights in

**k**-space are solely determined by the sampling pattern and the coil configurations, and they are independent of the absolute location in

**k**-space. This observation is also verified with both simulated and

*in vivo* data as shown in . While there are no observable artifacts in the simulated data without noise, the quality of the simulated images with noise and the

*in vivo* images appears to deteriorate as the calibration region shifts towards outer

**k**-space. This disparity demonstrates that although the location of the calibration region does not affect the algorithm in the noiseless situation, image quality may decrease due to the higher noise level at outer

**k**-space.

Computing the reconstruction matrix requires solving ill-conditioned systems of linear equations. The system of linear equations shown in (

12) is generally poorly conditioned as a result of both the irregular sampling pattern and noisy

**k**-space data that appear in both sides of (

21). Because data are sampled on a non-Cartesian trajectory, data values on both sides of the equation have to be computed with interpolation. Errors resulting from imperfect sampling density correction or limited width of interpolation kernel will worsen the condition of (

18). Finding the solution using matrix pseudo-inversion seems to work well for the simulated data without noise but not for the simulated data with noise and the

*in vivo* data especially at relatively high reduction factors as shown in . This issue is resolved effectively by using the LSQR algorithm that has been shown to be more robust for solving ill-posed linear problems.

There are two parameters that need to be optimized for the auto-calibrated kSPA algorithm: the cutoff bandwidth of sensitivity *w*_{s} and the half-width of reconstruction kernel *w.* At first thought, it seems that larger *w*_{s} and larger *w* should improve image reconstruction quality. Although our limited data seem to support the idea that RMSE decreases monotonically as the calibration region grows, there does not seem to be a monotonic relationship between RMSE and the width of the reconstruction kernel *w* as shown in . At *R* = 3, it appears that a *w* of 3 or 4 performs consistently better that other values. One reason that further increasing w does not reduce RMSE is that the number of independent equations is limited given a value of *w*_{s}. On the other hand, increasing the value of *w* increases the total number of unknowns. Further evaluation is necessary to investigate whether there is an optimal combination of these two parameters that achieves the best reconstruction quality.

In comparison to the original kSPA algorithm, auto-calibrated kSPA reconstructs one image for each coil instead of one final image without sensitivity weighting. As a result, the computation for auto-kSPA is approximately slowed by a factor equal to the total number of coils. Therefore, for massively parallel imaging, kSPA would still be the method of choice. For a small number of coils which is what is currently used by a vast majority of scanners, auto-kSPA offers the advantage of robustness and immunity to coil sensitivity estimation error. A further improvement to auto-kSPA would be reconstructing one final image instead of reconstructing one image for each coil, which is currently under investigation. Furthermore, auto-calibrated kSPA provides a solution for cases where an accurate sensitivity map is difficult or impossible to obtain. These scenarios include, for example, calibration data being too noisy, calibration data being off center and image being sparse such that it does not support a sensitivity map over the whole field-of-view. In the specific examples shown in , the original kSPA algorithm is certainly capable of reconstructing images of similar quality when the calibration data are located around the center of **k**-space and are of high SNR. However, it is not applicable for the two cases in .