Afaq Ashfaq^{1}, Muhammad Adil Khalil^{1}, Hassan Shahzad^{2}, Sohaib Ayyaz^{1}, and Hammad Omer^{1}

The estimation of receiver coil sensitivity information is important for SENSE (Sensitivity Encoding) reconstruction. Inaccurate sensitivity profiles degrade the reconstructed image quality. However, the methods to estimate the receiver coils sensitivity information are computationally intensive. This work proposes a parallel framework (for GPU implementation) for a recently proposed method of sensitivity estimation (which uses Eigen value decomposition of the multi coil low resolution images). The results show that the proposed method provides a 3.5x speed in our experiments while maintaining the reconstructed image quality.

Eigen-value
method for receiver coil sensitivity estimation is shown in Figure 1. It uses a
series of eigenvalue decompositions in *k*-space and provides receiver
coil sensitivities which appear corresponding to the main eigenvector. The
input to the algorithm, shown in Figure 1, is the low resolution multi coil
images. The processes shown with grey blocks in Figure 1 are the ones which have
been identified to be highly intensive in terms of computation times. However, they
contain inherit parallelism, which can be exploited using Graphics Processing
Units (GPUs). After analyzing the parallelism in these processes, CUDA kernels
were designed for all the processes (grey blocks in Figure 1) to exploit their
inherent parallelism on GPU. These kernels include “Fast 3D Matrix Augmentation”
of Eigen values and Eigen vectors, efficient data processing from GPU memories
and the synchronization of the GPU kernels to avoid race conditions. The
synchronization of all the parallel processes also helped to remove the chance
of corrupted output. The most
computationally intensive task is the computation of Singular Value
Decomposition (SVD)^{3}, for which we used the Jacobi SVD algorithm
proposed by S.A. Qazi^{7}.

Once the receiver coils sensitivity information is obtained using the proposed architecture, image reconstruction is performed with SENSE using the sensitivity maps estimated by the proposed framework (on GPU) and the conventional Eigen Value method on Central Processing Unit (CPU).

To test the proposed
method, a fully-sampled human brain data was acquired using 1.5T scanner with eight-channel
receiver coils^{2} having dimensions 200 × 200. The GPU used
for this research is NVIDIA’s GeForce GTX 1060 having 6GB memory and 1280 cores
with a clock speed of 1.7MHz. While the CPU used is Intel’s Core i-7 having
16GB RAM with 3.6GHZ clock speed.

Figure 2 shows the receiver coil sensitivity estimates obtained using the conventional implementation of Eigen-value method on CPU (Figure 2a) and the sensitivity maps using the proposed method (Figure 2b). The difference between the sensitivity maps estimations of both the methods is shown in Figure 2c.

The difference images in Figure 2(c) show that there is no difference between the receiver coil sensitivity maps obtained using the proposed method and the conventional implementation on CPU. Figure 3 shows the reconstructed images obtained from both (conventional and proposed) the methods at different acceleration factors (2 and 4). Visually there is no difference between both the images, which is also confirmed using artifact power at different acceleration factors. Both methods (conventional and proposed) provide same artifact power at a given acceleration factor; which is 0.0053 and 0.0067 at acceleration factor 2 and 4 respectively.

Figure 4 shows the computation time comparison of the proposed method and the conventional Eigen-value method. The conventional method takes 6240 milliseconds while the proposed method takes only 1621 milliseconds to get the reconstructed images. The results show that the proposed framework of Eigen-value method on GPU significantly reduces the computation time for sensitivity map estimation, thus making SENSE reconstruction more proficient.

[1] Pruessmann KP, Weiger M, Scheidegger MB, Boesiger P. SENSE: sensitivity encoding for fast MRI. Magnetic resonance in medicine. 1999 Nov 1;42(5):952-62.

[2] Uecker M, Lai P, Murphy MJ, Virtue P, Elad M, Pauly JM, Vasanawala SS, Lustig M. ESPIRiT—an eigenvalue approach to autocalibrating parallel MRI: where SENSE meets GRAPPA. Magnetic resonance in medicine. 2014 Mar;71(3):990-1001.

[3] Golub G, Kahan W. Calculating the singular values and pseudo-inverse of a matrix. Journal of the Society for Industrial and Applied Mathematics, Series B: Numerical Analysis. 1965;2(2):205-24.

[4] Drmač Z, Veselić K. New fast and accurate Jacobi SVD algorithm. I. SIAM Journal on matrix analysis and applications. 2008 Jan 4;29(4):1322-42.

[5] Nvidia Corp. CuSolver: CUDA toolkit documentation. [https://docs.nvidia.com/cuda/cusolver/index.html]. Accessed [Jan 2018].

[6] NVIDIA Corp. [Website] https://developer.nvidia.com/cuda-downloads. Accessed [2018].

[7] Sohaib.A.Qazi, Abeera S, Saima N, Hammad O. Singular Value Decomposition Using Jacobi Algorithm in pMRI and CS. Appl Magn Reson (2017) 48:461–471.

Flow chart of Eigen-value method with initialization
parameters and outputs. Whereas the highlighted parts are implemented on GPU.

Images show the magnitude component of the receiver
coil sensitivity maps estimated (a) using conventional approach on CPU,(b) by the proposed method on GPU and (c) the difference
images

Reconstruction results of 1.5T human head data set using
conventional method and the proposed method at Acceleration Factor (AF) 2 and 4.

Computation time comparison
of proposed method (on GPU) with the conventional Eigen-value method (on GPU) along
with Jacobi SVD on CPU (Conventional) and GPU (Proposed).