RELAY: Cross-Step Data-Consistency Reuse for Few-Step Medical Image Reconstruction

Sparse-view computed tomography (CT) and accelerated magnetic resonance imaging (MRI) reduce radiation dose and scan time but yield severely ill-posed inverse problems. Plug-and-play reconstruction avoids protocol-specific retraining by combining a pretrained image prior with operator-specific data consistency (DC), while consistency models reduce prior evaluation to only a few neural function evaluations (NFEs). For sparse-view Radon and multi-coil encoding, however, DC typically requires iterative conjugate gradients (CG), making its truncation a critical source of error under tight computational budgets. We show that this inner solve becomes the primary bottleneck as operator conditioning worsens, while noise injection and momentum provide little benefit. To address this challenge, we introduce RELAY, a training-free solver that transfers the DC solution across outer steps and restarts standard CG using the exact residual of the current system. RELAY improves DC accuracy under unchanged inner-iteration and NFE budgets while preserving the exact DC solution. We further establish non-asymptotic guarantees that characterize its cross-step error propagation and conditioning-dependent iteration savings. Across sparse-view CT and accelerated multi-coil low-field MRI, RELAY achieves the best PSNR among learned baselines across all tested settings, uses substantially fewer CG iterations than a cold start, and on CT surpasses compressed sensing run to convergence.