---
title: 'C-MSM: Calibration-Free Diffusion MRI Reconstruction'
url: https://www.emergentmind.com/topics/calibration-free-measurement-score-based-diffusion-model-c-msm
type: topic
---

# C-MSM: Calibration-Free Diffusion 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 [2509.18402][2505.11853]. 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 [2509.18402].

## 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 [2505.11853]. 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 [2509.18402].

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” [2509.18402].

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 [2509.18402].

## 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 \(m<n\) [2505.11853]. 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 \(s_0=s\). The denoiser is
\[
\hat{s}_{\theta}(s_t \,;\, \sigma_t) = \mathsf{D}_{\theta}(s_t \,;\, \sigma_t),
\]
trained with the MSE objective
\[
\mathcal{L}(\theta) = \mathbb{E}_{s_t, t} \left[\| s - \hat{s}_{\theta}(s_t \,; \sigma_t) \|_2^2 \right].
\]
Using Tweedie’s formula, MSM defines the learned partial measurement score as
\[
\mathsf{S}_{\theta}(s_t\,;\,\sigma_t)= \frac{1}{\sigma_t^2}(\hat{s}_{\theta}(s_t \,;\, \sigma_t) - s_t).
\]

The central representation is that the score of the full measurement distribution is written as an expectation over randomized partial scores:
\[
\nabla \log q_{\sigma_t}(z_t) = W \, \mathbb{E}_{S\sim p(S)} \left[ S^\top \nabla \log p_{\sigma_t}(s_t) \Big|_{s_t = S z_t} \right].
\]
Here, \(S z_t\) is a randomized partial view of the noisy full iterate, \(S^\top\) maps the partial score back into the full measurement space, and \(W\) is a reweighting vector compensating for overlapping contributions:
\[
W = \left[ \max \left( \mathbb{E}_{S \sim p(S)} \left[ \mathrm{diag}(S^\top S) \right],\, 1 \right) \right]^{-1}.
\]

Because the exact expectation over masks is expensive, MSM replaces it with a stochastic approximation using \(w\) sampled masks \(S^{(i)} \sim p(S)\):
\[
\nabla \log \widehat{q}_{\sigma_t}(z_t) = W \left[\frac{1}{w} \sum_{i=1}^{w} S^{(i)\top} \nabla \log p_{\sigma_t}(s_t^{(i)})\Big|_{s_t^{(i)} = S^{(i)} z_t}\right],
\]
with corresponding denoised estimate
\[
\hat{z}_{\theta} = W  \sum_{i=1}^{w} S^{(i)\top} \hat{s}_{\theta}(s^{(i)}_t\,;\, \sigma_t).
\]
This score estimate is inserted into the reverse diffusion update
\[
x_{t-1} = x_{t} + \tau_t \nabla \log p_{\sigma_t}(x_t) + \sqrt{2\tau_t\mathcal{T}_t}n.
\]

The original MSM paper also provides a theoretical analysis of this stochastic approximation. Under the bounded-variance assumption
\[
\mathbb{E}\left[\|\nabla \log q_{\sigma_t}(z) - \nabla \log \widehat{q}_{\sigma_t}(z)\|_2^2\right] \leq \frac{v^2}{w},
\]
the induced KL divergence is bounded by
\[
D_{\mathrm{KL}}(q(z)\,\|\,\widehat{q}(z)) \leq \frac{v^2}{w} C,
\]
where \(C\) is finite and independent of \(w\). The proof uses Girsanov’s theorem and a chain-rule argument for KL divergence between path measures [2505.11853]. This suggests that the stochastic partial-score mechanism inherited by C-MSM becomes more accurate as \(w\) 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
\[
y_k = S F C_k x + e_k, \qquad k=1,\dots,n_c,
\]
or, equivalently,
\[
y = A x + e.
\]
The inverse problem is often written as
\[
\hat{x} \in \arg\min_x g(x) + h(x),
\]
with Gaussian-noise data fidelity
\[
g(x) = \frac{1}{2}\|y - Ax\|_2^2
\]
[2509.18402].

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 [2509.18402].

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 \(s\), the method extracts the ACS region \(s_{\text{ACS}}\), transforms it to image space, and feeds it to a CSM network \(f_\phi\):
\[
\hat{C}_\phi(s_{\text{ACS}}) = f_\phi(F^H s_{\text{ACS}}).
\]
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 [2509.18402].

## 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:
\[
s = S z \in \mathbb{C}^m,\qquad m < n,
\]
with \(z \in \mathbb{C}^n\). No clean image \(x\) is required for training [2509.18402].

The diffusion corruption is applied directly in measurement space:
\[
s_t = s + \sigma_t n,\qquad n \sim \mathcal{N}(0,I),
\]
where \(\sigma_t\) 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
\[
\hat{s}_\theta(s_t;\sigma_t,\hat{C}_\phi) = F \hat{C}_\phi \, \mathsf{D}_\theta\!\left( F^H \hat{C}_\phi^H s_t;\sigma_t \right).
\]
Operationally, \(s_t\) is transformed back into image space, coil information is combined using \(\hat{C}_\phi^H\), the diffusion denoiser \(\mathsf{D}_\theta\) removes noise at level \(\sigma_t\), and the result is transformed back to measurement space [2509.18402].

The main measurement score-based objective is an MSE on measurements:
\[
\mathcal{L}_{\text{MSM}}(\theta;\hat{C}_\phi) = \mathbb{E}_{s_t,t} \left[ \left\|s - \hat{s}_\theta(s_t;\sigma_t,\hat{C}_\phi)\right\|_2^2 \right].
\]
Since physical coil sensitivities are spatially smooth, C-MSM adds the explicit regularizer
\[
\mathcal{L}_{\text{CSM}}(\phi) = \|\nabla \hat{C}_\phi(s_{\text{ACS}})\|_2^2.
\]
The total training objective is
\[
\mathcal{L}_{\text{total}}(\theta,\phi) = \mathcal{L}_{\text{MSM}}(\theta;\hat{C}_\phi(s_{\text{ACS}})) + \lambda \mathcal{L}_{\text{CSM}}(\phi),
\]
with \(\lambda = 1000\) in the experiments [2509.18402].

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
\[
y = H z + e,
\]
where \(z\) is the fully sampled measurement and \(e \sim \mathcal{N}(0,\eta I)\). As in training, the ACS region \(y_{\text{ACS}}\) is passed through the pretrained CSM network,
\[
\hat{C}_\phi(y_{\text{ACS}}) = f_\phi(F^H y_{\text{ACS}}),
\]
followed by RSS normalization [2509.18402].

The reconstruction mechanism is a stochastic approximation of the full posterior based on multiple random subsampling operators \(S^{(i)}\), \(i=1,\dots,w\). These define partial diffusion iterates
\[
s_t^{(i)} = S^{(i)} z_t,
\]
and each partial measurement is denoised using the learned model
\[
\hat{s}_\theta^{(i)}(s_t^{(i)};\sigma_t,\hat{C}_\phi(y_{\text{ACS}})).
\]
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 [2509.18402].

To enforce consistency with the observed data, C-MSM applies a measurement consistency update on overlapping coordinates:
\[
\tilde{s}_\theta^{(i)} = \hat{s}_\theta^{(i)} - \gamma_t \nabla \left\|y^{(i)} - \tilde{H}^{(i)} \hat{s}_\theta^{(i)}\right\|_2^2,
\]
where
\[
\tilde{H}^{(i)} := S^{(i)} H^H H S^{(i)H}.
\]
The denoised partial measurements are merged using
\[
W := \left[ \max\!\left( \sum_{i=1}^w \operatorname{diag}(S^{(i)H}S^{(i)}),\,1 \right) \right]^{-1},
\]
and the reconstructed full measurement is
\[
\tilde{z}_\theta = W \sum_{i=1}^w S^{(i)H}\tilde{s}_\theta^{(i)}.
\]
The next iterate is then sampled from
\[
z_{t-1} \sim p(z_{t-1}\mid z_t,\tilde{z}_\theta).
\]

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 [2505.11853]. 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” [2509.18402].

## 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 \(\times 4\) and \(\times 8\), measurement noise \(\sigma = 0.01\), and **20 ACS lines provided to all methods at test time** [2509.18402]. Training uses **2,000 center-cropped T2-weighted slices** of size \(256 \times 256\), 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 [2509.18402].

Quantitatively, at \(\times 4\), 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 \(\times 8\), C-MSM reports **PSNR 28.8**, **SSIM 0.760**, and **LPIPS 0.265**. The best PSNR at \(\times 8\) is **C-MSM 28.8**, the best SSIM is **DDNM 0.764**, and the best LPIPS is **DPS 0.247** [2509.18402]. 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 [2509.18402]. 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 [2505.11853]. 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 [2509.18402].

Source: https://www.emergentmind.com/topics/calibration-free-measurement-score-based-diffusion-model-c-msm