Frequency Error Prior (FEP) in MC-MRI

• Frequency Error Prior (FEP) is a frequency-domain metric that quantifies the k-space discrepancy between synthesized and fully-sampled MRI images, enabling targeted acquisition of challenging frequency components.
• The method integrates conditional diffusion models with deep unfolding networks to optimize k-space sampling patterns and improve reconstruction metrics such as PSNR and SSIM.
• Empirical results demonstrate significant gains in image quality, with improvements up to 6 dB in PSNR across various acceleration regimes and datasets.

The Frequency Error Prior (FEP) is a frequency-domain metric introduced in the context of multi-contrast magnetic resonance imaging (MC-MRI) reconstruction to optimize $k$-space under-sampling patterns and guide deep unfolding networks toward maximal image fidelity under aggressive acquisition acceleration. FEP quantifies the per-coefficient discrepancy in $k$-space between the target contrast synthesized from a reference by a conditional diffusion model and the fully sampled ground truth, enabling targeted sampling of frequency components least recoverable from reference data. This framework integrates model-driven and data-driven strategies to address the limitations of earlier MC-MRI reconstruction methods relating to suboptimal reference fusion and rigid sampling patterns, delivering significant improvements in peak signal-to-noise ratio (PSNR) and structural similarity across multiple datasets and acceleration regimes (Fang et al., 14 Jan 2026).

1. Definition and Motivation

The Frequency Error Prior $r$ is defined as the pointwise magnitude difference in $k$-space between a synthesized target-contrast image $X_\mathrm{sys}$, produced by a conditional diffusion model (CDM) conditioned on a reference image $Y$, and the fully sampled ground-truth image $X_\mathrm{gt}$:

$$r = \left| \mathcal{F}(X_\mathrm{sys}) - \mathcal{F}(X_\mathrm{gt}) \right|, \qquad r \in \mathbb{R}^{n \times n}.$$

Here, $\mathcal{F}(\cdot)$ denotes the 2D discrete Fourier transform. The entry $r_{i,j}$ quantifies the local failure of the diffusion model to accurately synthesize the frequency coefficient at $(i,j)$; higher $r_{i,j}$ values flag spatial frequencies that the reference contrast cannot adequately predict, typically reflecting high-frequency, modality-specific details.

MC-MRI typically exploits a fast-acquired reference contrast $Y$ to reconstruct an accelerated target contrast $X$. While $Y$ robustly determines global structure and low-frequency content, it alone cannot precisely recover the full target contrast’s high-frequency detail. FEP provides explicit, data-adaptive guidance on which $k$-space locations provide maximal reconstruction utility, both for mask design and network-driven image synthesis.

2. Mathematical Formulation

2.1. Diffusion-Based Modelling of FEP

The FEP arises from the conditional distribution:

$$p_f(\Delta k \mid Y) = \mathbb{P}\left( \Delta k = \mathcal{F}(X_\mathrm{sys}) - \mathcal{F}(X_\mathrm{gt}) \mid Y \right),$$

where $X_\mathrm{sys} \sim \mathrm{CDM}(Y)$. In practice, FEP is used as a pointwise magnitude estimate rather than a fully parameterized density:

$$r = \left| \mathcal{F}(X_\mathrm{sys}) - \mathcal{F}(X_\mathrm{gt}) \right|.$$

2.2. Conditional Diffusion Model Specifications

The conditional diffusion model adopts a standard Gaussian diffusion formulation:

• Forward process: Adds Gaussian noise to $X_0 = X_\mathrm{gt}$ via

$$q(X_t \mid X_{t-1}) = \mathcal{N}\left( X_t; \sqrt{\alpha_t}X_{t-1}, (1-\alpha_t)\mathbf{I} \right)$$

and can be reparameterized as

$$X_t = \sqrt{\bar\alpha_t} X_0 + \sqrt{1-\bar\alpha_t} \epsilon, \quad \epsilon \sim \mathcal{N}(0,\mathbf{I})$$

where $\bar\alpha_t = \prod_{s=1}^t \alpha_s$.

• Reverse process: Denoised iteratively via

$$p_\theta(X_{t-1} \mid X_t, X_\mathrm{cond}) = \mathcal{N}\left( X_{t-1}; \mu_\theta(X_t, t, X_\mathrm{cond}), \sigma_t^2 \mathbf{I} \right)$$

with the parameterization implemented as a U-Net.

• Loss function: The simplified denoising loss, per Ho et al. (2020),

$$\mathcal{L}_\mathrm{diff} = \mathbb{E}_{X_0, t, \epsilon}\left[ \left\| \epsilon - \epsilon_\theta(X_t, t, X_\mathrm{cond}) \right\|_2^2 \right ].$$

2.3. Joint Mask and Network Optimization

Sampling is modulated via a learnable $P_\mathrm{mask} \in [-1,1]^{n\times n}$ combined with the normalized FEP, yielding a continuous mask:

$$M_c = \sigma_\beta\Bigl( S_\gamma\bigl(\sigma_\alpha(\mathrm{norm}(r) + P_\mathrm{mask})\bigr) - U \Bigr),$$

where

• $\sigma_x(z)$: sigmoid with sharpness $x$,
• $S_\gamma(\cdot)$: sparsifies to mean sampling rate $\gamma$,
• $U \sim \mathcal{U}(0,1)$: stochastic binarization. The reconstruction target is the minimization

$$\{\hat{P}_\mathrm{mask},\,\hat{\theta}\} = \arg\min_{P_\mathrm{mask},\,\theta} \sum_{i=1}^N \left\| \varphi_\theta(X_u^{(i)}, Y^{(i)}, M_c^{(i)}) - X_\mathrm{gt}^{(i)} \right\|_1,$$

where $\varphi_\theta$ denotes the deep-unfolding reconstruction network and $X_u$ is the under-sampled image.

Discretized masks $M_d$ are derived as $M_c \to M_d$ for practical deployment, followed by final fine-tuning of $\theta$ (Fang et al., 14 Jan 2026).

3. Integration with Reconstruction Pipelines

3.1. Mask Differentiability and Learning Dynamics

The use of a continuous, differentiable mask facilitates gradient-based updates to $P_\mathrm{mask}$, focusing sampling density onto frequencies with persistently high FEP values. During training, gradients propagate through $M_c$ but not $X_\mathrm{sys}$, which remains fixed after pretraining the CDM. Budget constraints ($E[M_c] = \gamma$) are enforced via $S_\gamma$.

3.2. Deep Unfolding and Feature Fusion

The reconstruction network performs a block-majorization minimization (TITAN) protocol with $T'$ unrolling stages for the composite objective:

$$\begin{aligned} h(X, K, S, D, \phi) &= \tfrac{1}{2} \| M\mathcal{F}(X) - \tilde K \|_F^2 + \tfrac{\gamma}{2} \| S + D - Y_\mathrm{SA} \|_F^2 \ &\quad + \tfrac{\alpha}{2} \| K - \mathcal{F}(X) \|_F^2 + \tfrac{\beta}{2} \| AX - BS \|_F^2 + \sum_{i=1}^{5} \lambda_i \mathcal{R}_i(\cdot), \end{aligned}$$

incorporating spatial alignment, reference decomposition, and ResNet-based (ProxNet) subproblem updates. The FEP term operates exclusively in mask computation; after mask finalization, inference incurs no further diffusion model cost.

4. Experimental Validation and Performance

Empirical evaluation across the IXI, BraTS2018, and FastMRI datasets at acceleration factors 4×, 8×, 10×, and 30× demonstrates that frequency error-guided sampling confers PSNR increases of 0.3–6 dB over the best previously published under-sampled MC-MRI methods. For example, on IXI at 4× with a learned mask, FEP-informed sampling yields PSNR = 54.23 dB compared to 52.62 dB for MC-DuDoN(LOUPE); at 30×, the improvement is from 39.51 dB to 41.62 dB. SSIM remains above 0.99 at moderate acceleration and above 0.97 at 30× (Fang et al., 14 Jan 2026).

Ablations comparing masks learned with and without the FEP establish that FEP shifts sampling density toward mid- and high-frequency regions that are undersampled by LOUPE-based approaches, yielding sharper reconstructions and reduced spatial error, with PSNR advantages verified as statistically significant (paired $t$-tests, $p < 10^{-3}$).

Metric	JUF-MRI (FEP-aware)	MC-DuDoN(LOUPE)
PSNR, IXI 4×	54.23 dB	52.62 dB
PSNR, IXI 30×	41.62 dB	39.51 dB

5. Key Implementation Details

• Diffusion Model: U-Net backbone with time-step embeddings, channel widths 64$\to$128$\to$256, 4 hierarchical scales, trained using Adam (learning rate $2\times10^{-5}$, batch size 8, 300 epochs, EMA=0.999), noise schedule $\alpha_t$ linearly annealed over $T=1000$ steps, $\bar\alpha_T \approx 0$ at horizon.
• Mask/Reconstruction: Adam optimizer at $10^{-4}$, batch size 2, 30 epochs, 4 unfolding stages, 12-block ProxNet modules, sigmoid slopes $\alpha=4$, $\beta=8$, target sparsity $\gamma$ selected per acceleration, $P_\mathrm{mask}$ initialized from $\mathcal{U}(-1,1)$, inertial factor $\xi=0.25$.
• Training Protocol: Pretrain CDM to convergence before initiating mask learning; hold CDM parameters fixed during mask/network optimization; normalize $r$ on a per-volume basis to $[0,1]$ prior to use in $M_c$; anneal $\alpha$, $\beta$ during initial mask learning stages to stabilize gradients; fine-tune reconstruction for 5–10 epochs after mask binarization.

Best practices suggest freezing CDM weights post-pretraining and employing per-volume normalization of the FEP to ensure generalizable and stable mask learning (Fang et al., 14 Jan 2026).

6. Context, Limitations, and Implications

The Frequency Error Prior addresses key challenges in the MC-MRI domain by enabling adaptive, frequency-aware mask design that prioritizes acquisition of frequency coefficients least recoverable through reference-based synthesis, thus resolving the deficits of uniform, random, or purely data-driven sampling. By integrating FEP within a deep-unfolding framework, the approach leverages both analytic and learned priors, yielding high-fidelity reconstructions under substantial acceleration.

A plausible implication is that similar frequency error-guided strategies could generalize to other multi-modal inverse problems where reference data is available but imperfectly predictive, although validation is currently limited to MC-MRI contexts in (Fang et al., 14 Jan 2026). The approach’s reliance on a well-trained diffusion prior further suggests sensitivity to diffusion model capacity and training dynamics, and the practical overhead of model pretraining must be considered, though inference during deployment incurs no additional diffusion cost.

In summary, the Frequency Error Prior provides a robust frequency-domain signal, mediating both under-sampling strategy and signal recovery, and demonstrably advances the state of the art in accelerated multi-contrast MRI reconstruction (Fang et al., 14 Jan 2026).