Accelerating Iterative SENSE-based Algorithms for Cartesian Trajectories using the Point Spread Function and Coil Compression
Malte Steinhoff1, Kay Nehrke2, Alfred Mertins1, and Peter Börnert2,3

1Institute for Signal Processing, University of Lübeck, Lübeck, Germany, 2Philips Research Hamburg, Hamburg, Germany, 3Dept. Radiology, LUMC, Leiden, Netherlands


For multi-shot diffusion-weighted imaging, iterative SENSE-based algorithms like POCSMUSE are boosting SNR allowing for higher image resolution. These advantages are achieved at the cost of higher computational load, thereby narrowing the clinical use case. The Cartesian implementation of such SENSE algorithms iteratively involves time-consuming 1D-Fast Fourier Transforms. In this abstract, the well-known point spread function for regular Cartesian undersampling is exploited to accelerate gradient- and projection-based SENSE updates. Accelerations of approximately 45% were achieved. Furthermore, coil compression is evaluated for these algorithms.


In recent years, powerful iterative reconstruction schemes have been presented to solve image reconstruction problems, but unfeasible computation times remain a major obstacle regarding clinical adoption. For Cartesian SENSE1 applications, the point spread function (PSF) and coil compression2 (CC) have a long history for reconstruction acceleration. Nevertheless, these Cartesian techniques have not yet been tested for iterative multi-shot diffusion algorithms like POCSENSE3 and POCSMUSE4. In these algorithms, the Cartesian processing is based on the 1D-Fast Fourier Transform (FFT).


Standard SENSE reconstructions enforce data consistency between the model and the undersampled data within an L2-norm1. Two main iterative approaches exist to minimize SENSE-based functionals. Firstly, gradient-based methods gradually decrease the functional by evaluating the gradient of the functional5. Secondly, projection-based methods approach a solution by enforcing data consistency in k-space6.

For Cartesian trajectories, both approaches involve forward and backward 1D-FFTs as well as regular k-space undersampling which can be expressed by three image-space operations. Firstly, a phase ramp7 $$$\phi$$$ is multiplied in image space to account for optional shot trajectory offsets before undersampling. These offsets from k-space center occur, for example, in interleaved acquisitions. Secondly, regular Cartesian undersampling is achieved by convolution in image space $$$PSF*$$$. Third, the conjugate phase ramp $$$\phi^H$$$ is applied to reverse the initial trajectory shift. Hence, k-space undersampling comprising 1D-FFT $$$F$$$ and the sampling mask $$$M$$$ can be rewritten as


This identity allows us to reformulate gradient-based approaches like CG-SENSE by

$$grad\;J(x)=S^H\phi^HPSF*\; \phi Sx-S^H \phi^Hd,$$

where $$$d$$$ is the undersampled full-FOV image space data, $$$S$$$ the SENSE operator and $$$x$$$ the image.

Projection-based methods like POCSENSE and POCS-MUSE can be rewritten as

$$P_c x=S_cx+d_c-\phi^HPSF*\;\phi S_c x,$$

where $$$P$$$ is the projection operator and $$$c$$$ the coil index. Note that the FFTs have vanished. Furthermore, the image space-based operation requires the image matrix size to be divisible by the reduction factor.

Moreover, coil compression (CC) can be performed to further reduce computational load2. For this purpose, the coil set is reduced using the principal component analysis and the data is transformed to the new basis. All SENSE-related operations scale linearly with the coil number so that computations are expected to be accelerated by this factor. Note that this operation is not an identity but an approximation.


For numerical evaluation, three multi-shot EPI diffusion reconstructions were implemented based on the POCSMUSE algorithm:

1) The k-space-based method is the direct reimplementation of Cartesian POCSMUSE using 1D-FFTs.

2) The PSF-based method uses the aforementioned image space-based projection operator.

3) The PSF-based + CC method further uses coil compression with a threshold of 99%.

The approaches were compared in BrainWeb8 simulations and in-vivo. For simulations, 12 2D-Gaussian sensitivities were placed circularly around the head. Phase maps were created9, $$$N_{shots}=\{2, 3, 4, 5, 6\}$$$ and $$$SNR=\{10, 20, 30, 40\}$$$. Performance was averaged over 10 random cases. For in-vivo evaluation, multi-shot EPI DWI brain data was attained from 5 healthy volunteers using a 13-channel head coil, 3T Philips Ingenia, $$$b=\{0, 1000\} \, \dfrac{s}{mm^2}$$$ with three orientations and $$$1 \times 1 \times 4 \, mm^3$$$ resolution. Informed consent was attained according to the rules of the institution. Non-Cartesian EPI ramp sampling was gridded in advance to enable PSF-based methods.

The algorithms were executed with a 2.7GHz Intel Core i7 4-core CPU and 16 GB RAM using Python 3.6.5. Convergence was assumed when the residual error6 of subsequent iterations fell below 10-3 or the maximum iteration number of 300 was exceeded.


Figure 1 contains the normalized root-mean-square errors and durations of the simulations for varying segmentation. Figure 2 shows the convergence behavior over time for four and six shots. The final difference in nRMSE averaged over all simulations was $$$3 \cdot 10^{-5}$$$ and $$$2 \cdot 10^{-3}$$$ for method 1-2 and 1-3 respectively. In-vivo reconstructions and convergence criterions are compared in Figure 3 and 4 for four and six shots respectively.


The PSF formulation of projections and gradients is computationally beneficial without compromising image quality. Coil compression further accelerates image reconstructions, but this approximation can deteriorate image quality especially under ill-posed conditions as for high segmentation. The coil compression needs to be evaluated for the specific coil setup.


In conclusion, PSF-based iterative optimization reduces computation times for Cartesian trajectories and EPI by approximately 45% without significant loss of accuracy. Coil compression is a valuable approximation tool which has to be examined for the intended application. In this way, efficient iterative algorithms become more feasible paving the way for clinical adoption.


No acknowledgement found.


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8. Brainweb: Simulated Brain Database, http://www.bic.mni.mcgill.ca/brainweb/. Accessed July 27, 2017.

9. Hu et al. Phase-Updated Regularized SENSE for Navigator-Free Multishot Diffusion Imaging. MRM. 2017;78(1):172-181.


Figure 1: nRMSE and computation times of the three iterative Cartesian algorithms for multi-shot echo-planar diffusion BrainWeb7 simulations. The performance was evaluated with respect to the ground truth. The results were averaged over 10 cases for each segmentation. The whiskers of the boxplots contain the full range of results. The error increases for higher segmentation, but is equal for the k-space- and PSF-based methods and slightly increased including coil compression. Computation times are approximately halved using the PSF-based approach. Coil compression further accelerates computations.

Figure 2: nRMSE convergence of the three iterative Cartesian algorithms for a four- and a six-shot case for multi-shot echo-planar diffusion BrainWeb7 simulations. nRMSE was evaluated with respect to the ground truth BrainWeb phantom. The PSF-based method nearly halves the k-space-based durations without changing the convergence shape. The coil compression in PSF-based + CC further accelerates convergence, thereby changing the convergence characteristic.

Figure 3: Multi-shot Echo-planar DWI reconstructions for different b-values and orientations of a 4-shot in-vivo brain dataset. The upper row in a) contains the k-space-based image reconstructions. The middle and the lower row show the error of the corresponding methods scaled by a factor of 5. b) shows the convergence of the stopping criterion for the three methods.

Figure 4: Multi-shot Echo-planar DWI reconstructions for different b-values and orientations of a 6-shot in-vivo brain dataset. The upper row in a) contains the k-space-based image reconstructions. The middle and the lower row show the error of the corresponding methods scaled by a factor of 5. b) shows the convergence of the stopping criterion for the three methods.

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)