Fast Voxelwise SNR Estimation for Iterative MRI Reconstructions

Purpose: To develop a fast, general-purpose framework for voxelwise noise characterization in linear and nonlinear iterative MRI reconstructions, recovering the image-domain noise variance from which SNR, $g$-factor, and related image-quality metrics are derived. The framework addresses both the intractability of closed-form formulas beyond Cartesian sampling and the long runtime of Pseudo Multiple Replica (PMR) methods. Methods: We propose PICO (Probing Image-space COvariance), an estimator that operates in the image domain by probing the image-domain noise covariance operator -- or, for nonlinear compressed-sensing reconstructions, the Jacobian of the converged solution -- with random probe images. Complex random-phase probes are shown theoretically and empirically to minimize estimator variance compared with Gaussian or real-valued alternatives. PICO was validated against analytical benchmarks and high-replica PMR references using retrospective Cartesian knee data ($R=2$), prospective non-Cartesian spiral brain phantom data ($R=2,3,4$), and compressed-sensing knee reconstructions ($R=2$). Results: In Cartesian experiments, PICO accurately reproduced analytical SENSE $g$-factor maps. In non-Cartesian spiral imaging ($R=2$), it achieved 1% estimation error in 64 s compared with 462 s for PMR (approximately 7.2x speedup), with the efficiency advantage persisting at higher acceleration. For nonlinear compressed sensing, the Jacobian-based estimator produced noise maps consistent with PMR while converging faster (52 s vs. 95 s; approximately 1.8x speedup). Conclusion: PICO provides a computationally efficient alternative to PMR for voxelwise noise and $g$-factor estimation across generalized iterative MRI reconstructions. By reusing existing reconstruction primitives, it enables voxelwise noise maps to be produced as a routine by-product of the reconstruction pipeline.