CONSISTENCY MODELS FOR FAST MRI USING REGULARIZATION BY DENOISING

Abstract

Diffusion models (DMs) have shown strong generative performance for MR image reconstruction, but their use is limited by computationally expensive iterative sampling. Consistency models (CMs) provide a fast and compact alternative to diffusion models by learning direct mappings from noisy inputs to clean reconstructions. This reduces the multi-step diffusion sampling process into a single mapping, while preserving powerful learned priors that generalize across scanner types and field strengths. To harness the potential of CMs in MRI reconstruction, we introduce CM-RED, a novel framework that employs a pretrained CM within the regularization by denoising (RED) formulation. Our approach builds upon the accelerated proximal gradient (RED-APG) algorithm, and further introduces noise injection into its update steps to enhance generative performance and improve convergence speed. Results show that CM-RED achieves high-quality reconstructions on the fastMRI knee dataset in only 4 NFEs, outperforming DM- and CM-based methods both quantitatively and qualitatively. These highlight the potential of CM-RED as an efficient generative AI framework for MRI reconstruction.

1. INTRODUCTION

Reconstructing MR images from undersampled k-space data remains challenging, as it must balance acquisition speed, signal-to-noise ratio (SNR), and anatomical fidelity [1]. Classical techniques such as parallel imaging and compressed sensing can shorten scans but often fail at high acceleration rates, causing loss of fine structures or aliasing [1]. Deep learning (DL) based methods overcome these limitations by learning powerful image priors directly from data [2]. Among deep learning-based reconstruction methods, physics-driven unrolled networks [3–8] alternate between data-fidelity and learned proximal steps, enabling high-quality reconstructions within a limited number of iterations. Alternatively, plug-and-play (PnP) priors [9] incorporate pretrained denoisers as implicit regularizers within iterative schemes, while regularization by denoising (RED) [10, 11] makes this connection explicit by formulating a well-defined regularization term based on the denoiser. However, their performance is often tied to specific sampling patterns and degrades under large protocol shifts [12].

Generative models, particularly diffusion models (DMs) [13, 14], offer a promising alternative owing to their strong generalization across acquisition settings and SNR levels [15, 16]. However, their iterative sampling process requires hundreds of denoising steps, leading to high computational burden and slow inference times that are clinically impractical [17]. Consistency models (CMs) [18] bridge this gap by distilling diffusion processes into efficient, one-step mappings that preserve generative quality while offering substantial acceleration. Their denoising-like behavior further allows them to function as learned proximal operators [19].

Motivated by these advances, we introduce CM-RED, a novel framework that leverages a pretrained CM as a proximal prior within the RED formulation [10]. By extending the momentum-accelerated RED-APG scheme [11] with controlled noise injection and iteration-adaptive denoising strength guided by the CM, CM-RED achieves faster convergence while enabling the prior to operate effectively across a wider range of noise conditions. Evaluations on the fastMRI knee dataset [20] demonstrate that CM-RED surpasses DM-based approaches such as diffusion posterior sampling (DPS) [21] and decomposed diffusion sampling (DDS) [16], as well as CM-based methods such as CM4IR [22], achieving state-of-the-art reconstruction quality while delivering a ×25-×250 speed-up.

2. BACKGROUND

2.1. MRI Inverse Problem and End-to-End Training

The MRI reconstruction problem is formulated with a regularized least squares objective under i.i.d. Gaussian noise:

$$\stackrel{\mathbf{ˆ}}{\mathbf{x}}=\mathrm{a}\mathrm{r}\mathrm{g}\underset{\mathbf{x}}{\mathrm{m}\mathrm{i}\mathrm{n}}{‖{\mathbf{y}}_{\mathrm{\Omega }}-{\mathbf{E}}_{\mathrm{\Omega }}\mathbf{x}‖}_2^2+ℛ(\mathbf{x}),$$ (1)

where ${\mathbf{E}}_{\mathrm{\Omega }}$ is the multi-coil encoding operator associated with k-space undersampling pattern $\mathrm{\Omega }$, combining Fourier undersampling and coil sensitivities. First term enforces data-fidelity (DF) with acquired k-space measurements, while the regularizer $ℛ(\mathbf{x})$ imposes prior knowledge about the image.

This optimization is typically solved via iterative schemes alternating between data-fidelity and the proximal operator for $ℛ(\cdot )$. PnP methods [9] generalize this process by replacing the explicit regularizer $ℛ(\cdot )$ with a denoiser $D(\cdot )$ acting as an implicit proximal operator. On the other hand, algorithm unrolling [1, 4] adopts a similar principle but jointly learns the DF and proximal operator modules within a fixed number of trainable iterations. However, end-to-end learned reconstructions often show limited robustness to variations in sampling patterns and SNR [12], reducing generalizability.

2.2. DM-based Inverse Problem Solvers

DMs [13, 14] reverse a gradual noising process to learn data distributions, providing powerful priors that can be harnessed for inverse problems. In this setup, DM-based inverse problem solvers approximate the posterior $p(\mathbf{x}\mid \mathbf{y})$ by coupling generative priors with physics-based data-fidelity terms.

Among these, DPS [21] steers the denoising trajectory using gradients of the approximate measurement likelihood calculated using the Tweedie estimate. Apart from its approximate nature for noisy latent variables, it also typically requires hundreds of neural function evaluations (NFEs) to achieve convergence, making it computationally prohibitive for large-scale or time-sensitive imaging applications. To address this limitation, several follow-up methods [23–25] introduced more efficient sampling and optimization schemes, but still face trade-offs between quality and runtime.

DDS [16] further rethinks this challenge from a geometric perspective, modeling the denoising dynamics along an affine approximation of the data manifold. By replacing explicit gradient steps with conjugate gradient solvers, DDS achieves faster and more stable reconstructions, reducing inference to roughly 100 NFEs but still remaining computationally impractical for real-time deployments.

2.3. CM-based Inverse Problem Solvers

CMs [18] have recently emerged as a fast alternative to DMs, enabling high-quality sampling with only a single or a few NFEs. Unlike DMs that require sequential denoising across the full trajectory (orange arrows in Fig. 1), CMs are trained to directly map a noisy latent ${\mathbf{x}}_t$ at time step $t$ to its corresponding clean estimate ${\stackrel{\mathbf{ˆ}}{\mathbf{x}}}_0$ (blue arrows in Fig. 1):

$${\stackrel{\mathbf{ˆ}}{\mathbf{x}}}_0=f_{\theta }\left({\mathbf{x}}_t,t\right),t\in (0,T],$$ (2)

along with a boundary constraint to ensure mapping to the clean data $f_{\theta }\left({\mathbf{x}}_{ϵ},ϵ\right)={\mathbf{x}}_{ϵ}$. To ensure this mapping remains valid across different noise levels, the model enforces a consistency condition such that predictions from distinct timesteps yield the same output:

$$f_{\theta }\left({\mathbf{x}}_t,t\right)=f_{\theta }\left({\mathbf{x}}_{t^{\prime }},t^{\prime }\right),\forall t,t^{\prime }\in [ϵ,T].$$ (3)

The parameters $\theta$ are optimized using a consistency objective that encourages agreement between outputs corresponding to two perturbed instances of the same clean sample:

$${ℒ}_{\mathrm{C}\mathrm{M}}={\mathbb{E}}_{{\mathbf{x}}_0,t,t^{\prime }}\left[w(t)d\left(f_{\theta }\left({\mathbf{x}}_t,t\right),f_{\theta -}\left({\mathbf{x}}_{t^{\prime }},t^{\prime }\right)\right)\right],$$ (4)

where $w(t)$ is a weighting function, $d(\cdot ,\cdot )$ denotes a distance measure such as ${\ell }_2$ or LPIPS, and $f_{\theta -}$ represents a teacher network. If the CM is being distilled from another DM, ${\mathbf{x}}_{t^{\prime }}$ is predicted from ${\mathbf{x}}_t$ using an ordinary differential equation (ODE) solver [18]. In effect, the CM learns to map to the clean sample from any point on the ODE trajectory. Note this is different from the denoising objective of DM, which learns to denoise on average.

Fig. 1.

Illustration of DM and CM trajectories on MRI data.

Given their efficiency and expressive priors, CMs have recently been explored for solving inverse problems. While these methods were primarily developed for natural image restoration, their direct use in MRI remains limited. The original CM formulation [18] incorporates measurement consistency during sampling but still requires approximately 40 NFEs, which is prohibitively slow for clinical use. CoSIGN [26] extends this idea with a ControlNet that encodes the forward operator, enabling few-step guidance but necessitating retraining for each acquisition model. CM4IR [22] introduces a pseudo-inverse back-projection step to enforce measurement consistency, effectively coupling CMs with linear reconstruction. However, it does not naturally extend to forward operators commonly encountered in MRI, where the pseudo-inverse solution ${\mathbf{E}}_{\mathrm{\Omega }}^{†}{\mathbf{y}}_{\mathrm{\Omega }}$ often leads to residual aliasing and noise amplification.

3. METHODS

3.1. Regularization by Denoising (RED) and RED-APG

RED defines an explicit denoiser-based regularizer as:

$${ℛ}_{\mathrm{R}\mathrm{E}\mathrm{D}}(\mathbf{x})=\frac{\lambda }{2}{\mathbf{x}}^{\mathrm{\top }}(\mathbf{x}-D(\mathbf{x})),$$ (5)

where $D(\cdot )$ is the denoising operator. This formulation enables principled convergence analysis [10]. Its convergence was further improved in [11] using a quadratic majorizer of ${ℛ}_{\text{RED}}(\mathbf{x})$ and introducing a proximal gradient (RED-PG) algorithm and its accelerated variant (RED-APG), which incorporates a momentum term after the data-fidelity step.

Algorithm 1.

Proposed Algorithm: CM-RED

Require: CM $f_{\theta }(\cdot )$, encoding operator ${\mathbf{E}}_{\mathrm{\Omega }}$, number of iterations $N$, momentum coefficient ${\beta }_n$, regularization parameters ${\lambda }_n$, CM output scale ${\alpha }_n$, injected noise levels ${\sigma }_n$, CM noise levels ${\sigma }_n^{\text{CM}}$.
	▶ Initialize with CG-SENSE reconstruction:
1:	Initialization: ${\mathbf{x}}_N={\mathbf{x}}_N^{(m)}={\left({\mathbf{E}}_{\mathrm{\Omega }}^H{\mathbf{E}}_{\mathrm{\Omega }}\right)}^{-1}{\mathbf{E}}_{\mathrm{\Omega }}\mathbf{y}$
2:	for $n$ from $N-1$ to 0 do
	▶ CM-based proximal update with noise injection:
3:	$ϵ\sim 𝒩(0,\mathbf{I})$
4:	${\tilde{\mathbf{x}}}_n={\alpha }_n{f}_{\theta }({\mathbf{x}}_{n+1}^{(m)},+{\sigma }_nϵ,{\sigma }_n^{\mathrm{c}\mathrm{M}})+\left(1-{\alpha }_n\right){\mathbf{x}}_{n+1}^{(m)}$
	▶ Data-fidelity (DF) update, solved with CG-SENSE:
5:	${\mathbf{x}}_n={\left({\mathbf{E}}_{\mathrm{\Omega }}^H{\mathbf{E}}_{\mathrm{\Omega }}+{\lambda }_n\mathbf{I}\right)}^{-1}\left({\mathbf{E}}_{\mathrm{\Omega }}\mathbf{y}+{\lambda }_n{\tilde{\mathbf{x}}}_n\right)$
	▶ Momentum acceleration:
6:	${\mathbf{x}}_n^{(m)}={\mathbf{x}}_n+{\beta }_n\left({\mathbf{x}}_n-{\mathbf{x}}_{n+1}\right)$
7:	end for
8:	Return: ${\mathbf{x}}_0$

3.2. Proposed Algorithm: CM-RED

Building upon the RED-APG framework [11], we propose CM-RED, a reconstruction algorithm that integrates a pretrained CM as a proximal prior. Our approach incorporates noise injection and step-specific parameter adjustment to strengthen the generative effect of the CM prior while accelerating convergence.

At each iteration, we inject small Gaussian perturbations into the CM input, effectively operating the model at a higher noise level. This enables the CM to perform stronger denoising updates, rather than producing near-identity mappings that occur at low noise levels. As a result, the model explores a broader solution space and avoids convergence to suboptimal minima [27]. Such randomization, widely used in both generative modeling and optimization [18, 22], has been shown to enhance convergence robustness and mitigate overfitting. Inspired by recent stochastic RED and PnP formulations [28], this lightweight strategy improves both reconstruction fidelity and convergence speed, even with a limited number of iterations.

Alg. 1 summarizes CM-RED’s iterative process, combining CM-based proximal updates, data-fidelity enforcement, and momentum acceleration. This hybrid design unites the expressive generative prior of CMs with the theoretically grounded RED framework, enabling rapid and high-fidelity MRI reconstruction in just a few steps.

3.3. Experimental Setup

Dataset and preprocessing.

We employ the New York University fastMRI dataset [20], which provides complex-valued multi-coil k-space measurements from fully-sampled coronal proton density (PD) and proton density with fat-suppression (PD-FS) knee MRI scans, acquired under institutional review board approval. The minimum-variance unbiased estimates (MVUE) [15] of the complex-valued data are used to train the DM and CM, with the real and imaginary components stacked along the channel dimension, resulting in images of size 320×320×2. For DM/CM training, we used 973 subjects from the training set of the fastMRI dataset, excluding the first and last five slices of each volume, following the protocol of prior works [16]. For evaluation, 10 distinct subjects from the validation set were used, resulting in 228 PD and 240 PD-FS slices. Corresponding k-space data was retrospectively undersampled using uniform/equidistant patterns at acceleration factors of $R\in \{4,8\}$, with 24 central k-space lines retained.

CM Training details.

Following the publicly available code from [14, 18], we trained a CM from scratch on the fastMRI training dataset in two stages: (1) training an EDM model [14], and (2) distilling it into a CM. The EDM model is trained for 1M iterations with a batch size of 4 and subsequently distilled into a consistency model over an additional 700K iterations using the same batch size. All training processes are conducted using a single NVIDIA A100 GPU.

Implementation details.

We implemented CM-RED, as shown in Alg. 1, using $N=4$ iterations and 10 inner CG iterations. All algorithmic hyperparameters were empirically tuned for PD and PD-FS datasets at acceleration rates $R=4$ and 8, using decaying noise levels and momentum acceleration, an increasing CM output scale, and a constant regularization parameter. The denoising noise level used in CM was slightly higher than the injected noise level in practice, which led to improved reconstruction quality, as observed in previous studies [22].

We compared our method against two DM-based MRI reconstruction approaches, DPS [21] and DDS [16], as well as the CM-based CM4IR [22], using their publicly available implementations. The algorithmic hyperparameters for DDS, DPS and CM4IR were also empirically fine-tuned to achieve the best performance.

4. RESULTS

Representative reconstructions for Coronal PD and PD-FS datasets at accelerations of $R=4$ and $R=8$ are presented in Fig. 2. Our proposed CM-RED effectively mitigates the blurring artifacts that remain in DPS reconstruction and the structured artifacts still present in the other approaches. Note CM4IR reconstructions exhibit visibly high noise, which stems from its backprojection-based data-fidelity formulation that drives the solution toward the conventional CG-SENSE reconstruction, known to be susceptible to noise amplification. Quantitative comparisons are summarized in Tab. 1. Across all configurations, the proposed CM-RED method with NFE = 4 consistently achieves higher PSNR and SSIM values, while requiring substantially fewer iterations than the DM-based DPS (NFE = 1000) and DDS (NFE = 100).

Fig. 2.

Qualitative results for DPS, DDS, CM4IR, and CM-RED on coronal PD and PD-FS data (R = 4 and 8). The proposed method CM-RED achieves clearer reconstructions by reducing artifacts (red arrows) and blurring present in the other methods.

Table 1.

Quantitative comparison of reconstruction methods with R = 4 and 8 using 1D equidistant undersampling. Best: BOLD, second-best: Underlined.

Method	NFEs ↓	R	Cor PD		Cor PD-FS
			PSNR↑	SSIM↑	PSNR↑	SSIM↑
DPS [21]	1000	×4	33.59	0.886	30.92	0.784
		×8	30.28	0.833	27.83	0.725
DDS [16]	100	×4	34.02	0.899	30.39	0.777
		×8	30.70	0.834	28.05	0.711
CM4IR [22]	4	×4	29.70	0.754	24.76	0.540
		×8	28.30	0.714	24.14	0.518
CM-RED (ours)	4	×4	35.76	0.926	31.87	0.811
		×8	31.51	0.862	30.33	0.783

5. CONCLUSION

In this paper, we presented CM-RED, a fast and robust MRI reconstruction framework that leverages a pretrained consistency model as a proximal operator within a plug-and-play RED formulation. By integrating noise injection into the CM inputs, we enable the CM to operate at a higher denoising level, which allows larger update steps and leads to faster convergence. From a generative performance, it also helps to bring the samples back to the noisy manifold when the initial estimate falls off the trajectory due to potential discretization errors [18]. Our experiments on the fastMRI knee dataset demonstrate that CM-RED attains high-quality reconstructions within only 4 NFEs, highlighting its potential as a generative AI based inverse problem solvers, combining efficiency, scalability, and strong data-fidelity for high-fidelity MRI reconstruction.

7. COMPLIANCE WITH ETHICAL STANDARDS

This study was performed in line with the principles of the Declaration of Helsinki. Knee MRI data were obtained from the NYU fastMRI initiative database [20]. NYU fastMRI database was acquired with the relevant institutional review board approvals as detailed in [20]. NYU fastMRI investigators provided data but did not participate in analysis or writing of this report.