Non-convex joint-sparsity regularization for synergistic PET and SENSE MRI reconstruction

Objectives Simultaneous acquisition of PET and MRI data in integrated PET-MR systems provides opportunity to synergistically and jointly reconstruct PET and MRI images with a quality beyond that obtained via conventional independent reconstructions. In this work, we propose a new, non-convex joint sparsity prior for regularized PET and under-sampled sensitivity encoded (SENSE) MRI reconstruction. An augmented Lagrangian optimization framework is used to improve upon the performance of existing joint priors by enhancing common edges irrespective of their orientation, preserving modality-unique features and allowing for a feasible numerical optimization.

Methods The newly proposed prior promotes the joint sparsity of the discrete gradients of the PET and MRI images compared to the L₁ norm prior used in joint total variation (TV) regularization. The joint reconstruction was formulated as an equality-constrained optimization and solved using the alternating direction method of multipliers (ADMM). In this framework, the master problem was effectively optimized using the well-established MAP-EM one-step-late algorithm for PET, a regularized SENSE conjugate-gradient algorithm for MRI, and an iteratively weighted soft thresholding rule invoked by the linearization of the joint sparsity prior. The dependency of the joint prior on the PET and MRI signal intensities was addressed by novel alternating scaling of the distribution of the gradient vectors.

Results Using simulated PET and T₁-weighted MRI data, it was demonstrated that the proposed regularization substantially outperforms the separate TV and joint TV regularizations as well as the recently proposed linear parallel level set (PLS) reconstruction optimized by the quasi-Newton L-BFGS-B algorithm. The proposed algorithm outperformed its counterparts in terms of i) joint reconstructions which neither induce artifacts nor suppress modality-unique features, and ii) stability and convergence irrespective of initialization.

Conclusions The new, non-convex joint sparsity regularization within the presented joint reconstruction framework is a promising technique to enhance quantitative accuracy of PET-MR studies.