K-space reconstruction with anisotropic kernel support (KARAOKE) for ultrafast partially parallel imaging

Abstract

Purpose: Partially parallel imaging (PPI) greatly accelerates MR imaging by using surface coil arrays and under-sampling k-space. However, the reduction factor (R) in PPI is theoretically constrained by the number of coils (N_C). A symmetrically shaped kernel is typically used, but this often prevents even the theoretically possible R from being achieved. Here, the authors propose a kernel design method to accelerate PPI faster than R = N_C.

Methods: K-space data demonstrates an anisotropic pattern that is correlated with the object itself and to the asymmetry of the coil sensitivity profile, which is caused by coil placement and B₁ inhomogeneity. From spatial analysis theory, reconstruction of such pattern is best achieved by a signal-dependent anisotropic shape kernel. As a result, the authors propose the use of asymmetric kernels to improve k-space reconstruction. The authors fit a bivariate Gaussian function to the local signal magnitude of each coil, then threshold this function to extract the kernel elements. A perceptual difference model (Case-PDM) was employed to quantitatively evaluate image quality.

Results: A MR phantom experiment showed that k-space anisotropy increased as a function of magnetic field strength. The authors tested a K-spAce Reconstruction with AnisOtropic KErnel support (“KARAOKE”) algorithm with both MR phantom and in vivo data sets, and compared the reconstructions to those produced by GRAPPA, a popular PPI reconstruction method. By exploiting k-space anisotropy, KARAOKE was able to better preserve edges, which is particularly useful for cardiac imaging and motion correction, while GRAPPA failed at a high R near or exceeding N_C. KARAOKE performed comparably to GRAPPA at low Rs.

Conclusions: As a rule of thumb, KARAOKE reconstruction should always be used for higher quality k-space reconstruction, particularly when PPI data is acquired at high Rs and∕or high field strength.

INTRODUCTION

In parallel MR k-space reconstruction, a missing datum is estimated from selected sampled data points. The selection of the sampled data points, or kernel, is of vital importance to the success of k-space reconstruction algorithms like GRAPPA (Ref. ¹) and PARS.² Very commonly, a symmetric (e.g., squared and circular) or rectangular kernel is used. However, the kernel shape can be varied to achieve more optimal reconstruction, especially when the k-space is highly undersampled.³ Some recent studies proposed kernels based on error minimization,^{3, 4} but justification for such kernel design is lacking, and the computational cost of these methods is too large for practical use. Thus, a more practical method for designing kernels for improved k-space reconstruction is desired, particularly for highly accelerated, or so-called ultrafast partially parallel imaging (PPI). Due to the object itself and the asymmetry of coil sensitivity profile, we believe most MR k-space signals have an anisotropic pattern that skews the power of the k-space signals. Thus, based on the theory of spatial analysis,⁵ a kernel with signal-dependent anisotropic shape is preferred for an ideal k-space reconstruction. In this letter, we propose a method to extract k-space signal anisotropy to design kernels for use with a GRAPPA-like reconstruction. We also study the influence of field strength on kernel shape design. A perceptual difference model (Case-PDM) (Ref. ⁶) was used to quantitatively evaluate image quality. Experiments were performed on both in vivo and phantom data with high reduction factors (Rs).

METHODS

The full k-space data is obtained from the Fourier transform of the imaged object scaled by the coil sensitivities. The imaged object usually has a predominant orientation due to the internal structure of the object, on top of which the sensitivity profile of an individual coil element is usually asymmetric due to both coil placement and B₁ inhomogeneity, particularly at high magnetic fields.⁷ Since rotations in image space result in rotations in k-space, according to the properties of Fourier transform,⁸ the k-space also has a predominant orientation (anisotropy).

We use “k-space anisotropy” to refer to the pattern of high power signals in the low-frequency region of k-space. As can be seen in Fig. 1, the high signal power coefficients are often not circularly symmetric. Although complex patterns may be seen, a bivariate Gaussian function is usually sufficient to capture the k-space anisotropy. Therefore, we quantify the k-space anisotropy by first fitting the bivariate Gaussian function (by using a least-squares fitting technique) to the local k-space signal magnitude, channel-by-channel, and then calculating the anisotropy index (AI) defined by

$$\begin{array}{c}A{I}_n=|{\sigma }_1-{\sigma }_2|,\end{array}$$ (1)

where σ₁ and σ₂ are the two variances of the bivariate Gaussian, n is the index for channel with n∈[1,N_C] where N_C is number of channels. A value of zero signifies isotropy, while a larger value indicates a higher degree of anisotropy. To investigate the anisotropy of k-space signals in relation to field strength, three transverse sections of a cylindrical water phantom were acquired at three different field strengths. The details of the data sets (datasets 1-3) can be found in Table TABLE I..

Figure 1.

The GRAPPA kernel has a fixed kernel size of 20 elements per channel when used for a 4-channel k-space data. (Dashed line: unsampled signals and solid line: sampled signals.) Conversely, the anisotropic kernel adapts to the anisotropic nature of k-space signal magnitude.

Table 1.

Descriptions of MR data sets.

Dataset	Scanner	Imaged object	Filed strength	RF coils	B1 field measure
1	Siemens Magnetom Espree	Cylindrical water phantom	1.5 T	4-ch head	Close-to-homogeneous
2	Philips Achieva x-series	Cylindrical water phantom	3.0 T	8-ch head	Inhomogeneous
3	Philips Achieva research system	Cylindrical water phantom	7.0 T	11-ch head	Inhomogeneous
4	GE Discovery	Human volunteer’s heart	1.5 T	8-ch chest	N∕A
5	GE Discovery	Human volunteer’s brain	1.5 T	4-ch head	N∕A
6	Siemens Magnetom Espree	Human volunteer’s brain	1.5 T	4-ch head	N∕A
7	Siemens Magnetom Espree	Human volunteer’s heart	1.5 T	12-ch chest	N∕A
8	Philips Achieva x-series	Human volunteer’s brain	3.0 T	8-ch head	N∕A

GRAPPA is a popular k-space reconstruction method in PPI.¹ In standard GRAPPA, the first step is to create a set of over-determined linear equations from the auto-calibrated signal (ACS) lines and estimate the kernel weights by solving the equations with a least-squares algorithm. The GRAPPA kernel has a set rectangular shape (Fig. 1). To create the linear equations, blocks of ACS signals from all coils are used to fit a single ACS line in one coil. Each block is composed of one line of measured signal and R lines of target signal. Once the kernel weights are obtained, missing samples can be reconstructed in a fashion similar to convolution. More details about GRAPPA reconstruction can be found in Ref. 1. The reduction factor of standard GRAPPA is limited to the number of coils, N_C. However, our proposed method, named K-spAce Reconstruction with AnisOtropic KErnel support (KARAOKE), can go beyond this limitation.

Unlike GRAPPA, we allowed the kernel shapes to vary to better reflect the relationship between the missing signals and sampled signals. The simplest way to construct an anisotropically shaped kernel is to threshold the signal magnitudes of the ACS region, and to choose pixels that follow the pattern of undersampling as kernel elements. However, the fluctuation of signal magnitudes across k-space would leave holes in the kernel. As a result, instead of thresholding ACS magnitudes, we first fit a bivariate Gaussian function to the magnitudes to capture the anisotropic pattern. This bivariate Gaussian can model the spread of high power signals along both the phase encoding (PE) and frequency encoding (FE) directions. Then, sorting the vectorized Gaussian at sampled locations (i.e., at one of every R phase encodes) in a descending order and thresholding their ranks with a value of kernel size yields the kernel elements. Thus, the kernel extracted at high reduction factor can capture the anisotropic pattern better than at low reduction factor. Data employed in the equation system on the left-hand-side (LHS) are more spatially correlated to the right-hand-side (RHS) in such a anisotropic neighbourhood. Therefore, KARAOKES can go further than the limit of GRAPPA. Figure 1 compares the anisotropic kernel to a rectangular one. KARAOKE calibrates kernel weights and reconstructs missing data in the same way as GRAPPA does, but with an additional degree of freedom: the shape of the kernel.

To demonstrate the robustness of our algorithm, all MR images were reconstructed without using the ACS in the final reconstructions. The size of the anisotropic kernel is defined by the number of its elements, and we used a size of 20 elements toward the preliminary result presented in this letter. All MR data sets are summarized in Table TABLE I.. We tested our algorithm on 5 MR raw data sets: 2 brain images (datasets 5 and 6), 2 cardiac images (datasets 4 and 7), and 1 phantom image (dataset 2). Different kernel sizes, reduction factors, and ACS sizes were tested.

RESULTS

Figure 2 shows the variation of k-space anisotropy, measured by AI, across different channels of the phantom data and at different field strengths. Mean values of AI calculated across all channels for the 1.5 T, 3 T, and 7 T phantom data sets are 0.42, 0.89, and 1.57. K-space anisotropy increased as a function of field strength since coil sensitivity profile is more asymmetric at higher field strengths as also demonstrated by published results on MR experiments and simulations on surface coils at high magnetic field.^{7, 9}

Figure 2.

Plot for AI of each channel for dataset 1 (1.5 T), dataset 2 (3 T), and dataset 3 (7 T). Averaged AI’s across different channels for these 3 datasets are 0.42, 0.89, and 1.57.

Figure 3 demonstrates the difference between KARAOKE and GRAPPA reconstructions on dataset 2 at different Rs. At a low R of 4, KARAOKE performed on par with GRAPPA (KARAOKE had 13% lower summed residual magnitudes than GRAPPA), at an R of 6, KARAOKE performed significantly better than GRAPPA (KARAOKE had 49% lower summed residual magnitudes), and at an R of 8, GRAPPA algorithm failed but KARAOKE was still able to reconstruct a correct profile of the cylindrical phantom (KARAOKE had 82% lower summed residual magnitudes). KARAOKE reconstructions showed a lower residual magnitude than GRAPPA in a majority of k-space locations (in the third row of Fig. 3). Figure 4 demonstrates the comparison on an in vivo 8-channel cardiac data (dataset 4). PDM (Ref. ⁶) assessment showed that both KARAOKE and GRAPPA produced identical image quality at an R of 4. However, at higher Rs of 6 and 8, KARAOKE significantly outperformed GRAPPA, giving much less noise and aliasing, and having many fewer errors in both high- and low-frequency regions. The standard GRAPPA algorithm failed at an R of 6, but KARAOKE was still able to produce a meaningful result with most of edges being preserved, until it failed at an R of 18, which exceeds twice N_C. We observed a similar relative result when ACS data was integrated into final reconstructions.

Figure 3.

Compared to GRAPPA, KARAOKE obviously improves the reconstructed images quality at high acceleration factor (R > 4). Reference image is from an 8-channel full sampled data (256 × 256 × 8). ACS size of 51 PE lines was used for complex-valued kernel weight calibration. The first row is for GRAPPA reconstructions with kernel size (6 × 5, PE × FE) at different Rs. The second row is for KARAOKE reconstructions. A reconstruction error analysis suggests that KARAOKE generally reconstructs signals with lower residuals than GRAPPA, as demonstrated by a binary map with size equal to recovered data (third row), which labels a white pixel if KARAOKE gave a lower magnitude residual value. More white pixels indicate a better reconstruction is.

Figure 4.

Compared to GRAPPA, KARAOKE obviously improves image reconstruction quality, especially at high acceleration factors, as indicated by PDM scores in plot (a) in which letters index sub-images. A lower PDM score indicates lower perceptual difference between the reconstruction and the reference image. (b) is a reference image from 8-channel full sampled data. (c) and (d) are images reconstructed by GRAPPA at Rs of 6 and 8, respectively, and (e) and (f) are the corresponding images reconstructed by KARAOKE. Visually, the image quality improvement by the proposed method is obvious. Images (g)-(i) were reconstructed by KARAOKE at beyond-channel-number Rs of 10, 12, and 14, respectively.

DISCUSSION AND CONCLUSION

Our method improves PPI k-space reconstruction by adding kernel anisotropy as a degree of freedom. KARAOKE, compared to GRAPPA, was able to reconstruct with comparable image quality at low reduction factors and significantly better image quality at high reduction factor. Both PDM scores and visual inspection in all head-to-head comparisons confirmed these findings. We believe that this anisotropy is correlated to coil sensitivity, coil placement, and the object itself. Such information is not utilized in standard GRAPPA, heuristic searches for GRAPPA reconstruction kernels minimizing total error,³ or kernel optimization based on approximation error of a sparse matrix’s pseudoinverse.⁴ KARAOKE can extract the useful information, like the distribution pattern of the whole k-space data, from either fully sampled ACS regions or undersampled k-space data only (results not shown here). Thus, PPI can be further sped up by the proposed method even without fully sampling the k-space center. This is particularly useful for dynamic imaging applications like k-t GRAPPA (Ref. ¹⁰) because kernel design can be customized for each undersampled time frame.

Since the equation system in KARAOKE is over-determined, the reconstruction is still not perfect and there is room for further improvement. As the implementation only affects the kernel shape, any regularization methods¹¹ and robust regression¹² can be incorporated into the algorithm to give further improvement in image quality. We have recently demonstrated the success of MONKEES,¹³ a k-space reconstruction algorithm that models kernel weight nonstationarity in very high quality image reconstruction for PPI. KARAOKE can be combined with MONKEES to have an asymmetric kernel with kernel weights varying across k-space locations. Kernel size can be varied across different coils too, according to their powers and contributions. Thus, a robust KARAOKE can further boost the reconstruction quality with fewer samples. We plan to study this in more detail in the future.

Many recent developments in image domain MR motion correction techniques are dependent on an accurate image reconstruction from very few k-space samples of ultrafast PPI.¹⁴ Motion correction is performed by registering all the motion-free frames from accelerated data. However, a successful image registration very often depends on the edge information and it is very challenging to recover correct edge information when the data is highly undersampled. KARAOKE is especially helpful in the image domain motion correction because it can reconstruct correct edge information from very few acquired k-space samples while standard method fails. In one experiment, we applied KARAOKE to reconstruct brain skull profiles for 16 ultra-short-time interleaves, each of which has only 1∕16 of the full samples. The Turbo Spin Echo data set imaged a rotating human head (dataset 8). The rigid-body rotation of the head can be corrected nicely by registering these 16 reconstructed images using mutual-information based techniques.

In conclusion, the KARAOKE method has been proposed in this letter to accurately reconstruct images from highly undersampled PPI data sets. Preliminary experimental studies have been conducted to confirm the existence of an anisotropic k-space pattern, and to test the proposed KARAOKE algorithm. Anisotropy becomes more pronounced at higher field strengths as coil sensitivity profiles get more asymmetric.⁷ KARAOKE reconstruction can preserve most of edges at high reduction factors where GRAPPA fails: it is able to yield useful results with reduction factors beyond the limit of coil number (N_C). KARAOKE can replace GRAPPA because, no difference exists between the reconstructions at low Rs, and KARAOKE is better at high Rs and high field strengths. The computational complexity of our method is comparable to that of GRAPPA.