C-MSM: Calibration-Free Diffusion MRI Reconstruction

• The paper introduces C-MSM, which eliminates pre-calibrated coil sensitivity maps by jointly learning measurement scores and automatic CSMs from undersampled k-space data.
• It employs a stochastic diffusion strategy that approximates full measurement scores by aggregating randomized partial scores, enabling reconstruction without ground-truth images.
• C-MSM demonstrates competitive PSNR, SSIM, and LPIPS metrics on fastMRI data, highlighting its potential for calibration-free parallel MRI reconstruction.

Searching arXiv for the primary papers and any necessary bibliographic verification. Calibration-free Measurement Score-based diffusion Model (C-MSM) is a self-supervised diffusion framework for compressed-sensing parallel MRI reconstruction that jointly performs automatic coil sensitivity map (CSM) estimation and learning of measurement scores directly from undersampled k-space data. It was introduced as an MRI-specific extension of the Measurement Score-Based Diffusion Model (MSM), which learns partial measurement scores from noisy and subsampled measurements and represents the score of the full measurement distribution as an expectation over randomized partial scores (Liu et al., 22 Sep 2025, Park et al., 17 May 2025). In this formulation, reconstruction is performed by stochastic sampling over partial measurement posterior scores while simultaneously estimating CSMs, with the explicit goal of avoiding both clean ground-truth training images and pre-calibrated CSMs (Liu et al., 22 Sep 2025).

1. Terminology, scope, and relation to MSM

The name Measurement Score-Based Diffusion Model (MSM) refers to the framework introduced in 2025 for learning diffusion models without clean ground-truth images by training directly on noisy and subsampled measurements (Park et al., 17 May 2025). The later name Calibration-free Measurement Score-based diffusion Model (C-MSM) refers to the MRI reconstruction method that extends this measurement-space score-learning perspective by adding automatic CSM estimation from undersampled k-space (Liu et al., 22 Sep 2025).

This distinction matters because the earlier MSM paper does not use the term “Calibration-free Measurement Score-based diffusion Model” as a formal name. It consistently refers to the method as MSM. By contrast, the later MRI paper explicitly introduces C-MSM as a method that “eliminates these dependencies by jointly performing automatic CSM estimation and self-supervised learning of measurement scores directly from k-space data” (Liu et al., 22 Sep 2025).

A recurring misconception is to treat “C-MSM” as the official name of the original MSM framework. The published record does not support that usage. A more precise reading is that C-MSM is an MRI-specific extension of MSM, and that “calibration-free” refers to the removal of pre-calibrated coil sensitivity maps and ground truth images in the later reconstruction setting. Another related misconception is to read “calibration-free” as “ACS-free.” The later paper does not support that interpretation: it assumes that some ACS region is available at test time for estimating CSMs (Liu et al., 22 Sep 2025).

2. Measurement-space score modeling in the original MSM framework

MSM starts from a fully sampled measurement $z \in \mathbb{R}^n$, but during training only observes a partial measurement

$s = S z \in \mathbb{R}^m,$

where $S \in \{0,1\}^{m \times n}$ is a random subsampling operator drawn from $p(S)$, with $m<n$ (Park et al., 17 May 2025). The paper gives application-specific examples: for inpainting, $z=x$; for MRI, $z = F C x$; and, more generally, $z = T x$ for an invertible transform $T$.

The forward diffusion on the partial measurement is

$$s_t = s + \sigma_t n, \quad n \sim \mathcal{N}(0, I),$$

with $s = S z \in \mathbb{R}^m,$0. The denoiser is

$s = S z \in \mathbb{R}^m,$1

trained with the MSE objective

$s = S z \in \mathbb{R}^m,$2

Using Tweedie’s formula, MSM defines the learned partial measurement score as

$s = S z \in \mathbb{R}^m,$3

The central representation is that the score of the full measurement distribution is written as an expectation over randomized partial scores: $s = S z \in \mathbb{R}^m,$4 Here, $s = S z \in \mathbb{R}^m,$5 is a randomized partial view of the noisy full iterate, $s = S z \in \mathbb{R}^m,$6 maps the partial score back into the full measurement space, and $s = S z \in \mathbb{R}^m,$7 is a reweighting vector compensating for overlapping contributions: $s = S z \in \mathbb{R}^m,$8

Because the exact expectation over masks is expensive, MSM replaces it with a stochastic approximation using $s = S z \in \mathbb{R}^m,$9 sampled masks $S \in \{0,1\}^{m \times n}$0: $S \in \{0,1\}^{m \times n}$1 with corresponding denoised estimate

$S \in \{0,1\}^{m \times n}$2

This score estimate is inserted into the reverse diffusion update

$S \in \{0,1\}^{m \times n}$3

The original MSM paper also provides a theoretical analysis of this stochastic approximation. Under the bounded-variance assumption

$S \in \{0,1\}^{m \times n}$4

the induced KL divergence is bounded by

$S \in \{0,1\}^{m \times n}$5

where $S \in \{0,1\}^{m \times n}$6 is finite and independent of $S \in \{0,1\}^{m \times n}$7. The proof uses Girsanov’s theorem and a chain-rule argument for KL divergence between path measures (Park et al., 17 May 2025). This suggests that the stochastic partial-score mechanism inherited by C-MSM becomes more accurate as $S \in \{0,1\}^{m \times n}$8 increases, while also increasing computation.

3. Parallel MRI formulation and the calibration-free extension

In parallel MRI, the unknown image is reconstructed from measurements acquired by multiple receiver coils, each with its own spatial sensitivity profile. C-MSM adopts the coil-wise forward model

$S \in \{0,1\}^{m \times n}$9

or, equivalently,

$p(S)$0

The inverse problem is often written as

$p(S)$1

with Gaussian-noise data fidelity

$p(S)$2

(Liu et al., 22 Sep 2025).

The specific difficulty targeted by C-MSM is that diffusion-based inverse problem solvers in compressed-sensing parallel MRI typically rely on pre-calibrated coil sensitivity maps and ground truth images. The later paper characterizes both dependencies as impractical: CSMs are difficult to estimate accurately under heavy undersampling, and ground-truth images are often unavailable. C-MSM departs from standard methods because it does not assume known coil sensitivities or clean ground-truth training images. Instead, it learns from undersampled k-space alone (Liu et al., 22 Sep 2025).

To remove the need for pre-calibrated CSMs, C-MSM adds an automatic CSM prediction network trained self-supervised from the same undersampled data used for reconstruction. From the undersampled measurement $p(S)$3, the method extracts the ACS region $p(S)$4, transforms it to image space, and feeds it to a CSM network $p(S)$5: $p(S)$6 The predicted CSMs are then normalized by their root-sum-of-squares (RSS) magnitude to stabilize the forward and adjoint operators and balance coil contributions. The paper explicitly notes that this means C-MSM does not require any external sensitivity calibration stage such as ESPIRiT-style preprocessing (Liu et al., 22 Sep 2025).

4. Joint self-supervised learning of measurement scores and coil sensitivities

Like MSM, C-MSM assumes access only to partial observations of a fully sampled measurement, but here the measurements are complex-valued multi-coil k-space data: $p(S)$7 with $p(S)$8. No clean image $p(S)$9 is required for training (Liu et al., 22 Sep 2025).

The diffusion corruption is applied directly in measurement space: $m<n$0 where $m<n$1 is the diffusion noise level. Because multi-coil k-space is high-dimensional, C-MSM converts the noisy measurement into the image domain using the predicted CSMs, denoises there, and maps back to measurement space with

$m<n$2

Operationally, $m<n$3 is transformed back into image space, coil information is combined using $m<n$4, the diffusion denoiser $m<n$5 removes noise at level $m<n$6, and the result is transformed back to measurement space (Liu et al., 22 Sep 2025).

The main measurement score-based objective is an MSE on measurements: $m<n$7 Since physical coil sensitivities are spatially smooth, C-MSM adds the explicit regularizer

$m<n$8

The total training objective is

$m<n$9

with $z=x$0 in the experiments (Liu et al., 22 Sep 2025).

The paper’s own summary of this construction is that the joint loss is what makes C-MSM “calibration-free”: no pre-calibrated CSMs are needed, no fully sampled ground-truth images are needed, and both the sensitivity estimator and the diffusion prior are trained from undersampled k-space alone. A plausible implication is that the calibration burden is shifted from an external preprocessing step to a learned component optimized together with measurement-space denoising.

5. Stochastic posterior sampling and reconstruction

At inference time, C-MSM addresses the conditional reconstruction problem

$z=x$1

where $z=x$2 is the fully sampled measurement and $z=x$3. As in training, the ACS region $z=x$4 is passed through the pretrained CSM network,

$z=x$5

followed by RSS normalization (Liu et al., 22 Sep 2025).

The reconstruction mechanism is a stochastic approximation of the full posterior based on multiple random subsampling operators $z=x$6, $z=x$7. These define partial diffusion iterates

$z=x$8

and each partial measurement is denoised using the learned model

$z=x$9

The paper describes this as the “partial posterior score” idea: instead of trying to evaluate one full high-dimensional posterior directly, the method samples and denoises several partial views and then combines them (Liu et al., 22 Sep 2025).

To enforce consistency with the observed data, C-MSM applies a measurement consistency update on overlapping coordinates: $z = F C x$0 where

$z = F C x$1

The denoised partial measurements are merged using

$z = F C x$2

and the reconstructed full measurement is

$z = F C x$3

The next iterate is then sampled from

$z = F C x$4

This stochastic posterior construction is directly aligned with the posterior formulation already present in MSM, where the posterior score is approximated by combining an MSM prior term and a likelihood term, and where compressed-sensing MRI admits a specialized form in the subsampled measurement domain (Park et al., 17 May 2025). The later C-MSM paper extends that idea by conditioning the denoiser on automatically estimated CSMs. The paper’s own interpretation is that the method is “a stochastic, score-based analogue of using overlapping mini-batches of k-space to reconstruct a complete signal” (Liu et al., 22 Sep 2025).

6. Experimental profile, comparative results, and limitations

C-MSM is evaluated on the multi-coil brain fastMRI dataset with 80 held-out test images, acceleration factors $z = F C x$5 and $z = F C x$6, measurement noise $z = F C x$7, and 20 ACS lines provided to all methods at test time (Liu et al., 22 Sep 2025). Training uses 2,000 center-cropped T2-weighted slices of size $z = F C x$8, runs for 1 million iterations on a single NVIDIA RTX 3090 GPU, and uses the same diffusion backbone for all models. The reported baselines are TV, DPS, DDNM, MSM, and A-DPS. In the paper’s comparison, DPS and DDNM are diffusion-based inverse solvers using clean diffusion priors and pre-calibrated CSMs, whereas MSM and A-DPS do not use fully sampled images for training but still rely on pre-calibrated CSMs (Liu et al., 22 Sep 2025).

Quantitatively, at $z = F C x$9, C-MSM reports PSNR 32.7, SSIM 0.853, and LPIPS 0.144. The best PSNR and SSIM at this acceleration are DDNM 32.9 and DDNM 0.864, while the best LPIPS is C-MSM 0.144. At $z = T x$0, C-MSM reports PSNR 28.8, SSIM 0.760, and LPIPS 0.265. The best PSNR at $z = T x$1 is C-MSM 28.8, the best SSIM is DDNM 0.764, and the best LPIPS is DPS 0.247 (Liu et al., 22 Sep 2025). The paper’s stated conclusions are that C-MSM consistently improves over self-supervised baselines such as MSM and A-DPS, and that it achieves performance close to clean-prior diffusion solvers such as DPS and DDNM despite not using clean training images or pre-calibrated coil sensitivities. The qualitative comparison is reported to show reconstructions close to DPS and DDNM, with good detail preservation and reduced artifacting.

The main practical advantages stated for C-MSM are that it needs no ground-truth training images, no pre-calibrated CSMs, and is trained directly on undersampled k-space. The paper also identifies several assumptions and limitations. Some ACS region must be available at test time for estimating CSMs; coil sensitivities are assumed to be spatially smooth, which motivates the gradient penalty; and the diffusion backbone and noise schedule must be suitable for the k-space statistics of the dataset (Liu et al., 22 Sep 2025). The method is therefore calibration-free, not ACS-free. Additional limitations reported in the paper are that stochastic sampling over multiple partial measurements may increase computational cost, that performance is not uniformly better than the best clean-prior solvers on every metric, and that the demonstration is on fastMRI brain data rather than a broader set of anatomies and scanners.

Taken together, MSM and C-MSM define a measurement-space alternative to clean-image-trained diffusion priors. MSM supplies the core representation of full measurement scores as expectations over randomized partial scores, together with stochastic sampling and a KL-divergence analysis (Park et al., 17 May 2025). C-MSM specializes that framework to compressed-sensing parallel MRI by coupling partial-measurement posterior sampling with automatic CSM estimation from ACS data, thereby removing the need for both pre-calibrated sensitivity maps and clean training images in the reported fastMRI setting (Liu et al., 22 Sep 2025).