Coherence-Minimized Sensing Matrix Design for MRI Reconstruction via Dual-Space Projection Optimization

Compressed sensing magnetic resonance imaging (CS-MRI) heavily relies on the low mutual coherence between the measurement matrix and the sparsity basis. However, under highly accelerated Cartesian undersampling, the severe structural coherence between Fourier measurements and spatial bases, discrete cosine transform (DCT) for example, fundamentally violates this requirement, causing classical sparse recovery algorithms to stagnate. To mitigate this fundamental bottleneck, we propose a synergistic dual-space projection framework, denoted as $\mathbf{PAQ}$. Instead of merely designing heuristic sampling masks, our method directly reshapes the equivalent dictionary. Specifically, we introduce a diagonal-dominant random rotator $\mathbf{Q}$ in the feature space to probabilistically disrupt structural alignment, and an active orthogonalization projector $\mathbf{P}$ in the measurement space to deterministically whiten the residual correlations. We theoretically demonstrate that this dual-space mechanism bounds the mutual coherence with an exponentially decaying tail via sub-exponential distribution properties. Experimental validations on clinical MRI datasets under 20\% Cartesian sampling demonstrate that plugging the $\mathbf{PAQ}$ preconditioners into the standard ISTA solver significantly suppresses aliasing artifacts and yields consistent peak signal-to-noise ratio (PSNR) improvements.