Ali Pour Yazdanpanah^{1}, Onur Afacan^{1}, and Simon K. Warfield^{1}

Conventional parallel imaging exploits coil sensitivity profiles to enable image reconstruction from undersampled data acquisition. The extent of undersampling that preserves

The discretized version of MR imaging model given by $$d=Ex + n (1)$$

where $$$x$$$ is the unknown MR image, $$$d$$$ is the undersampled $$$k$$$-space data. $$$E=FS$$$ is an encoding matrix, and $$$F$$$ is an undersampled Fourier matrix. $$$S$$$ represents the sensitivity information. Assuming that the interchannel noise covariance has been whitened, the imaging model can be solved and reach the optimal maximum likelihood estimate when $$$E$$$ has full column rank. This can be done by solving the least squares problem

$$\hat{x} =(E^{H}E)^{-1}E^{H}d (2)$$

In a case of undersampled $$$k$$$-space data, Eq.2 is ill-posed. If we consider Cartesian-type sampled $$$k$$$-space, we end up with foldover artifacts in the coil images. The unfolding process in Eq.2 is possible as long as the matrix inversion can be performed. Here, based on the proposed reconstruction framework, we extend the SENSE equations using extra information extracted from first-order derivatives of Eq.1 to improve the condition number of unfolding matrix. Using the additional information, we can extend SENSE equation in the formulation after Fourier transform of the undersampled data: $$\binom{d_{I}}{Dd_{I}} = \binom{S}{DS + SD} \left(x\right) with \ \hat{E} = \binom{E_1}{E_2}$$

where $$$D$$$ represents first-order finite difference operator, and $$$d_{I}=F^H d$$$. Compared with standard SENSE, the number of equations in our reconstruction is doubled. The extended encoding matrices $$$\hat{E}$$$, is too big regarding the size occupied in memory, so instead of solving it through Eq.2, We formulate the problem as an optimization problem T and consider the additional equation, as equality constraint and solve it iteratively through the proposed solution using AL outline [3,4]. $$T: \underset{x,u_0} {minimize} \ \frac{1}{2}\|d-FSx\|_{2}^{2}+\beta\|u_0\|_p \ \ subject \ to\ \ u_0=D d_{I}-DSx-SDx$$

[1] Pruessmann KP,Weiger M, Scheidegger MB, Boesiger P, et al. SENSE: sensitivity encoding for fast MRI. Magnetic resonance in medicine 1999;42(5):952–962.

[2] Griswold MA, Jakob PM, Heidemann RM, Nittka M, Jellus V,Wang J, et al. Generalized autocalibrating partially parallel acquisitions (GRAPPA). Magnetic resonance in medicine 2002;47(6):1202–1210.

[3] Ramani S, Fessler JA. Parallel MR image reconstruction using augmented Lagrangian methods. IEEE Transactions on Medical Imaging 2011;30(3):694–706.

[4] Bertsekas DP. Multiplier methods: a survey. Automatica 1976;12(2):133–145.

Figure 1. The first row (left to right): Gold standard reconstruction result using fully sampled data, our reconstruction result with undersampling factor of 4, and GRAPPA reconstruction result with undersampling factor of 4 for 2D T2 Fast Spin Echo data. The second row includes reconstruction zoomed areas and error maps correspond to each reconstruction results for comparison.

Figure 2. Reconstruction NRMSE vs acceleration factor for 2D T2 Fast Spin Echo data.

Figure 3. The first row (left to right): Gold standard reconstruction result using fully sampled data, our reconstruction result with undersampling factor of 2x2 along both phase encoding dimensions, and GRAPPA reconstruction result with undersampling factor of 2x2 along both phase encoding dimensions for 3D MPRAGE data. The second row includes error maps correspond to each reconstruction results for comparison.

Figure 4. Reconstruction NRMSE vs acceleration factor for 3D
MPRAGE data.