Efficient calculation of g-factors for CG-SENSE in high dimensions: noise amplification in random undersampling

Background SENSE [1,2] is one of the most used parallel imaging techniques. In [1], uniform undersampling was employed to efficiently reconstruct an unalised image, whereas in [2], a conjugate gradient-based method (CG-SENSE) was used for reconstruction with arbitrary trajectories. SENSE framework allows the calculation of g-factors, characterizing the noise amplification for a given k-space trajectory and coil configuration [1]. However, calculation of g-factors for arbitrary trajectories in high dimensions is time-consuming [3]. Furthermore, noise characteristics of random undersampling, used in compressed sensing, is not wellunderstood. In this work, we use a Monte-Carlo (MC) method for fast calculation of g-factors for CG-SENSE similar to [4,5] and apply it to random Cartesian undersampling trajectories. Theory: SENSE involves a prewhitening step [1,2], thus without loss of generality, we assume white noise. SENSE reconstruction solves minm ||y Em||2, where E is the system matrix, and y are the undersampled measurements. The g-factor for the k voxel is given by gk = √([E*E] -1 k,k [E*E]k,k). Inverting E*E is not feasible in high dimensions. Instead we note the gk corresponds to the k diagonal of the reconstruction noise covariance matrix (for normalized coil sensitivities), where nrecon = (E*E) E*nmeas, and nmeas is measurement noise with identity covariance matrix. We calculate the sample correlation matrix using a MC approach (since sample mean goes to 0), as 1/(p-1)∑p n p recon (n p recon)* for p instances of nrecon. Note we only calculate and store the diagonal elements of this matrix, significantly increasing efficiency. Methods The MC method was first verified in a numerical simulation, where the g-factor was explicitly calculated for a 2D coil configuration, to determine how many MC simulations suffice. Whole-heart imaging was performed with an isotropic resolution of 1.3 mm using a 32-channel coil array. Two 4-fold accelerated acquisitions were performed, one with uniform undersampling (2 × 2 in the ky-kz plane) and one with random undersampling. Coil sensitivity maps were exported. Images were reconstructed using SENSE (for uniform) and CG-SENSE (for both). g-factors were also calculated with the proposed approach.