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Cascaded Posterior Sampling With Diffusion Priors for highly Accelerated MRI

Jyothi Rikhab Chand, Sahil Murtaza, Patrick Nicholas Nowak, Mathews Jacob

arXiv:2610.34170v1eess.IV

TL;DR

Full-resolution diffusion priors for accelerated MRI require large networks, and posterior-sampling guidance is least accurate when coarse anatomy is formed. The paper introduces cascaded posterior sampling: a low-resolution estimate from central k-space conditions a high-resolution prior combined with all measurements. At 15× acceleration, it improves on the best baseline by 4–6 dB in PSNR, with each network about 10× smaller.

  • Problem

    Full-resolution diffusion priors require large networks to capture long-range structure, while measurement guidance is least accurate at high noise, where coarse anatomy is determined.

    Source: §1 · p. 1

  • Method

    Cascaded posterior sampling recovers a coarse image from central k-space samples, then uses it to condition a high-resolution diffusion prior combined with all measurements.

    Source: §1 · p. 1

  • Results

    4–6 dB in PSNR at 15× acceleration: the method improves on the best baseline, with each separately trained network about 10× smaller.

    Source: §1 · p. 2, §5 · p. 4

  • Takeaways & Limitations

    The image-only priors apply to any sampling pattern, and data consistency remains linear at every scale.

    Source: §5 · p. 4

    Stage 1 approximates p(yLR | x) by p(yLR | xLR) and treats Z as constant, assumptions considered reasonable when the k-space periphery is sparsely sampled and carries little information about xLR.

    Source: §3.1 · p. 2

Abstract

from arXiv · show

Diffusion models are powerful priors for inverse problems, including accelerated MRI. However, modeling the distribution of high-resolution images requires large networks with high computational cost. We propose a posterior sampling framework that uses cascaded diffusion models as multiscale priors. Conditioned on a coarse image, the distribution of fine-scale detail is local and well conditioned, and it can therefore be captured by a small network. Similarly, a small unconditional network is sufficient to represent low-resolution images, and it is used to recover the coarse image from the k-space center. This estimate then conditions the high-resolution prior, which is combined with all measurements to recover the final image. Both priors are trained on images alone, so they apply to any sampling pattern, and data consistency remains linear at every scale. In multi-coil brain MRI, the proposed method improves on the best single-scale diffusion baseline by 4-6 dB in PSNR at 15x acceleration, while each of its separately trained networks is about 10x smaller. The gain grows as the problem becomes more ill-posed, and ablations show that it arises from informative low-resolution conditioning.

1. INTRODUCTION

Full-resolution diffusion priors require large networks to capture long-range structure, and posterior-sampling guidance is least accurate at the high-noise stages where coarse anatomy is determined. Cascaded posterior sampling instead recovers a measurement-informed coarse image to condition a fine-scale prior, improving on the best baseline by 4–6 dB in PSNR at 15× acceleration.

1. INTRODUCTION

  • Diffusion priors can be combined with different forward models in a plug-and-play manner, unlike end-to-end methods trained for a particular acquisition setting.This flexibility follows from a trained diffusion model not being tied to a specific acquisition operator.Source: §1 · p. 1
  • A full-resolution diffusion prior needs a large network with global receptive fields, and measurement guidance is least accurate at high noise, risking incorrect coarse anatomy even with a densely sampled k-space center.Latent diffusion reduces computational cost but reconstruction accuracy is bounded by its autoencoder, whose decoder also makes data-consistency updates nonlinear.Source: §1 · p. 1
  • Coarse-to-fine factorization makes fine-scale detail more local and better conditioned, so small networks can model it while a low-resolution global model captures long-range structure.Dense sampling of the k-space center allows the coarse image to be recovered as a separate, better-posed inverse problem; it then constrains the fine-scale prior where ambiguity is greatest.Source: §1 · p. 1
  • Cascaded posterior sampling recovers a coarse image from central k-space samples, then conditions a high-resolution prior that uses all measurements; each prior is about 10× smaller than a single-scale model.The two priors are trained and used separately, so peak memory and computational complexity are governed by one small network.Source: §1 · p. 1
  • 4–6 dB in PSNR at 15× acceleration: the cascade improves on the best single-scale baseline, while matching single-scale diffusion baselines at moderate acceleration.Denoising experiments attribute the gain to informative low-resolution conditioning rather than the additional network input.Source: §1 · p. 2

2. BACKGROUND

The paper formulates accelerated multi-coil MRI reconstruction as recovering an unknown high-resolution image from undersampled k-space measurements. In a Bayesian view, diffusion-based posterior sampling combines an image prior with the measurement likelihood.

2. BACKGROUND

  • The unknown is a high-resolution MR image, and the acquired data are undersampled multi-coil k-space measurements.The framework is presented as broadly applicable to computational imaging inverse problems, while this work focuses on accelerated MRI.Source: §2 · p. 2
  • Bayesian reconstruction samples from the image posterior given the measurements, proportional to the product of the measurement likelihood and the image prior.Diffusion-based posterior-sampling algorithms combine a learned diffusion prior with the measurement likelihood.Source: §2 · p. 2
  • The multi-coil forward operator is A = SFC, where S is the sampling mask, F the Fourier matrix, and C the coil sensitivity map; measurement noise is Gaussian.The model writes the acquired measurements as a forward operation on the unknown image plus noise.Source: §2 · p. 2

3. CASCADED POSTERIOR SAMPLING

The framework factorizes the image prior into low-resolution and conditional high-resolution components, then samples their measurement-informed posterior in two stages. The coarse estimate comes from central k-space samples and conditions a fine-scale reconstruction using the full measurements.

3. CASCADED POSTERIOR SAMPLING

  • A hierarchical prior factorizes the image distribution into a low-resolution prior and a conditional high-resolution prior; the coarse image is sampled first, followed by fine-scale details conditioned on it.The low-resolution model has 16× fewer pixels, and conditioning makes the fine-scale distribution more local and better conditioned, enabling a small network with limited receptive field.Source: §3 · p. 2
  • The two stages implement posterior ancestral sampling: first draw the coarse estimate from its measurement-informed marginal, then draw the high-resolution image conditioned on both measurements and that estimate.Figure 1 depicts the coarse reconstruction from central samples and its use as the conditioning input for reconstruction with full-resolution measurements.Source: §3 · p. 2, Figure 1 · p. 2

3.1. Stage 1: Low-resolution image reconstruction

  • Stage 1 samples a low-resolution image from the posterior marginal using central k-space samples yLR = Py.The marginal approximation replaces p(yLR | x) with p(yLR | xLR) and treats Z as constant, justified in accelerated MRI because sparsely sampled k-space periphery carries little information about xLR.Source: §3.1 · p. 2
  • The low-frequency measurements and coarse images are linked by a low-resolution forward operator using the central mask crop and coil sensitivity maps pooled by 4×4 average pooling.The low-resolution inverse problem has 16× fewer unknowns than the full-resolution problem, reducing computational cost.Source: §3.1 · pp. 2–3

3.2. Stage 2: High-resolution image reconstruction

  • Stage 2 conditions a high-resolution diffusion prior on the coarse estimate and combines it with the likelihood enforcing consistency with all undersampled multi-coil measurements.The coarse condition supplies anatomical and structural information, restricting plausible high-resolution images and reducing reconstruction ambiguity; the paper samples this posterior using ALPS.Source: §3.2 · p. 3

4. RESULTS

The cascaded diffusion model was evaluated on fastMRI brain data across 1D and 2D sampling patterns and acceleration factors. At 15× acceleration, it outperformed the best baseline, while denoising tests showed that informative low-resolution conditioning benefits performance increasingly at high noise.

4.1. Dataset

  • The evaluation uses sixteen-channel fastMRI brain data with 384 × 384 complex images and 1D Cartesian and 2D undersampling patterns.The dataset is publicly available.Source: §4.1 · p. 3

4.2.1. Cascaded Diffusion Model

  • Each cascaded prior has around 6.7 million parameters; the low-resolution prior is trained unconditionally and the high-resolution prior is conditioned on low-resolution images.The low-resolution training images are 96 × 96, pooled from 384 × 384 images; conditioning augmentation improves robustness to imperfect inference inputs.Source: §4.2.1 · p. 3

4.2.2. State-of-the-art (SOTA) methods for comparison

  • The comparison evaluates the cascade against ALPS, DAPS, and DPS, using each method's recommended settings and a 65.5-million-parameter unconditional DDPM++ baseline.The baseline operates at 384 × 384 image size and is around 10× the size of the cascaded networks.Source: §4.2.2 · p. 3

4.3. Denoising Performance

  • At high noise levels, ground-truth low-resolution conditioning improves denoising more than random or zero conditions, while conditioning has little effect at low noise.The advantage grows as the denoising problem becomes more ill-posed.Source: §4.3 · pp. 3–4, Figure 2 · p. 3

4.4. Reconstruction performance

  • Fig. 3 displays reconstructions, magnified regions, and error maps for 2D and 1D 15× acceleration.The comparison is between the cascade conditioned on a sampled low-resolution image and the unconditional diffusion model.Source: Figure 3 · p. 4
  • At 15× acceleration, the cascade beats the best baseline by 4.2 dB (1D) and 6.0 dB (2D) in PSNR; at 2D 10×, it matches ALPS.Table 1 compares reconstruction performance across acceleration factors.Source: §4.4 · p. 4, Table 1 · p. 4
  • Fig. 4 reports approximately 10× lower uncertainty for 2D 15× sampling than for 1D 15× sampling.The uncertainty maps are low across most of the image.Source: Figure 4 · p. 4, §4.4 · p. 4

5. CONCLUSION

The paper concludes that measurement-informed multiscale diffusion priors can improve accelerated multi-coil MRI reconstruction while using smaller networks than a single-scale baseline. The authors identify 3D and high-resolution imaging as potential applications and propose extending the cascade to more scales and joint inference across scales.

5. CONCLUSION

  • In multi-coil MRI, the method improves on the best baseline by 4–6 dB in PSNR at 15× acceleration, with networks that are each about 10× smaller and faster to train.The priors are trained on images alone, apply to any sampling pattern, and preserve linear data consistency at every scale.Source: §5 · p. 4

6. COMPLIANCE WITH ETHICAL STANDARDS

The study used a publicly available human subject dataset, and ethical approval was not required under its open-access license.

6. COMPLIANCE WITH ETHICAL STANDARDS

  • The study used a publicly available human subject dataset; its open-access license meant ethical approval was not required.The work acknowledges support from three NIH grants.Source: §6 · p. 4

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