Noisy MRI Reconstruction via MAP Estimation with an Implicit Deep-Denoiser Prior

Accelerating magnetic resonance imaging (MRI) remains challenging, particularly under realistic acquisition noise. While diffusion models have recently shown promise for reconstructing undersampled MRI data, many approaches lack an explicit link to the underlying MRI physics, and their parameters are sensitive to measurement noise, limiting their reliability in practice. We introduce Implicit-MAP (ImMAP), a diffusion-based reconstruction framework that integrates the acquisition noise model directly into a maximum a posteriori (MAP) formulation. Specifically, we build on the stochastic ascent method of Kadkhodaie et al. and generalize it to handle MRI encoding operators and realistic measurement noise. Across both simulated and real noisy datasets, ImMAP consistently outperforms state-of-the-art deep learning (LPDSNet) and diffusion-based (DDS) methods. By clarifying the practical behavior and limitations of diffusion models under realistic noise conditions, ImMAP establishes a more reliable and interpretable

💡 Research Summary

The paper introduces Implicit‑MAP (ImMAP), a diffusion‑based MRI reconstruction framework that explicitly incorporates the MRI acquisition model and realistic measurement noise into a maximum‑a‑posteriori (MAP) formulation. Building on the stochastic ascent algorithm of Kadkhodaie et al., the authors generalize it to handle the large‑scale linear encoding operator A (undersampled multi‑coil Fourier transform) and additive Gaussian noise ν∼N(0,Σ_y).

The key insight is to use a trained MMSE denoiser f(z;σ) as an implicit prior. By Tweedie’s formula, the score of the noisy image distribution p_σ(z) can be approximated as σ²∇_z log p_σ(z)≈f(z;σ)−z, giving a tractable gradient of the log‑prior. For the likelihood term, the authors model the current iterate z_t as a Gaussian perturbation of the unknown ground‑truth x, and approximate the conditional mean E