K-space Cold Diffusion for Accelerated MRI

• K-space cold diffusion is a deterministic computational framework that inverts progressive k-space masking to restore fully-sampled images from undersampled MRI data.
• It uses iterative neural networks, typically U-Nets, to learn inversion of information loss induced by structured degradation schedules in the Fourier domain.
• Empirical results demonstrate state-of-the-art performance in PSNR and SSIM across various acceleration factors and k-space mask types, highlighting its clinical relevance.

K-space cold diffusion refers to a family of computational frameworks that invert information-reducing transformations in Fourier (k-space) domains, typically for applications such as accelerated MRI reconstruction. In contrast to classical (hot) diffusion models, which utilize explicit stochastic noise (usually Gaussian) as the degradation mechanism, k-space cold diffusion employs entirely deterministic, structured operations, such as progressive undersampling or masking of k-space coefficients. Restoration is achieved through iterative inversion steps employing learned neural networks, yielding state-of-the-art image reconstruction fidelity under severe measurement constraints and providing a principled alternative to stochastic score-based generative approaches (Shen et al., 2023, Bansal et al., 2022, Huang et al., 2023).

1. Mathematical Formulation and the Cold Diffusion Principle

K-space cold diffusion generalizes diffusion-type iterative inversion to arbitrary, typically deterministic, k-space (or other frequency-domain) degradations. Prefixing the forward process, a fully-sampled complex image $x_0\in\mathbb C^{n\times n}$ is transformed to k-space by the Fourier operator $k=\mathcal F(x_0)$. Undersampling is modeled by binary masks $M_t\in\{0,1\}^{n\times n}$ at each discrete step $t=0,\dots,T$, with $M_0$ corresponding to fully-sampled and $M_T$ to the desired target undersampling pattern.

The cold diffusion forward operator becomes

$$X_t = D(x_0, t) = \mathcal F^{-1}\left[ M_t \odot \mathcal F(x_0) \right]$$

where $\odot$ denotes pointwise multiplication. The mask schedule $M_t$ is constructed by incrementally removing k-space lines or regions, typically at each $t$ adding a randomly drawn submask $M_e$ from the set of yet-unremoved indices $M^c=1-M$, with $|M_e|$ proportional to $(T-t)/T$.

Unlike standard diffusion models, which iterate stochastically via $q(x_t|x_{t-1})\sim\mathcal N(\ldots)$, k-space cold diffusion’s forward process is purely deterministic; no explicit noise is added, and the restoration problem is not denoising per se, but inpainting or de-aliasing via learned inversion of information loss induced by k-space masking (Shen et al., 2023, Bansal et al., 2022, Huang et al., 2023).

For the inverse process, a restoration network $R_\theta(X_t, t)$, typically parameterized as a U-Net, is trained to approximate $x_0$ from $X_t, t$ at random $t$. The sampling process proceeds via

$$\begin{aligned} \hat x_0 &= R_\theta(X_t, t) \ X_{t-1} &= X_t - D(\hat x_0, t) + D(\hat x_0, t-1) \end{aligned}$$

This update corrects first-order errors in the learned restoration and recovers the fully-sampled image in $T$ iterations (Shen et al., 2023, Bansal et al., 2022).

2. Network Architectures and Training Regimes

Implementation predominantly adopts image-domain U-Nets with explicit support for complex-valued data via dual-channel (real/imaginary) representation. Resolution levels and channel progression (e.g., encoder: 64–128–256–512) mimic architectures found in other generative inverse problems.

The loss function is typically the $\ell_1$ norm between the network prediction and ground truth,

$$L(\theta) = \mathbb E_{x_0\sim p_\text{data}} \mathbb E_{t\sim \text{Uniform}[0,T]} \|R_\theta(D(x_0,t), t) - x_0\|_1$$

Important training hyperparameters include:

• Number of cold diffusion steps, $T$ (e.g., 100–125; ablations show robustness up to $T=1000$).
• Batch size (6–24), number of iterations ($10^5$–$7\times10^5$), and learning rates (typically $2\times10^{-5}$ with Adam optimizer).
• Randomization of degradation mask $M_e$ per mini-batch to enhance generalization and preclude overfitting to specific k-space undersampling patterns (Shen et al., 2023, Huang et al., 2023).

In k-space domain variants, direct prediction of $\Delta K_s$ (the new frequency components) is possible, though the widespread approach is to operate on the image domain and perform masking/inversion inside the algorithmic loop (Bansal et al., 2022).

3. Mask Schedules and Degradation Strategies in K-space

Mask schedule design is central to k-space cold diffusion efficacy. Appropriate schedules must ensure:

• Invertibility: Each subsequent mask $M_{t-1}$ must reveal strictly more k-space information than $M_t$.
• Diversity: Varying mask structures (Cartesian, radial, Poisson-disk, 2D Gaussian) encourage the network to learn reconstruction priors across a manifold of lost frequency scenarios.
• Smoothness: Changes between $M_t$ and $M_{t-1}$ should be minimal (add or remove a small set of lines/regions per iteration) to increase conditioning and facilitate stable inversion (Shen et al., 2023, Bansal et al., 2022, Huang et al., 2023).

Acceleration factors (e.g., 4×, 8×, 16×) are modeled by setting the final mask $M$ to retain a set fraction of central or otherwise prioritized k-space regions, with intermediate masks constructed by sequentially adding lines back in the reverse diffusion direction.

4. Performance, Metrics, and Comparative Evaluation

Empirical results on the fastMRI knee dataset (single-coil) demonstrate that k-space cold diffusion achieves or surpasses the performance of strong deep learning baselines and end-to-end variational networks, as summarized in the following table for key PSNR/SSIM benchmarks (Shen et al., 2023):

Model	4× Cartesian	8× Cartesian	4× Gaussian	8× Gaussian
U-Net	28.21/0.6001	27.03/0.5400	29.07/0.6244	28.24/0.5577
W-Net	29.34/0.6464	28.42/0.6149	29.76/0.6560	29.23/0.6071
E2E-VarNet	30.29/0.6850	29.16/0.6338	29.90/0.6621	29.42/0.6151
K-space Cold Diffusion	30.58/0.7150	29.51/0.6414	30.31/0.7059	29.59/0.6416

Qualitatively, the model recovers fine textures and edges with fewer residual artifacts than competing approaches, equally effective under both Cartesian and spatially random (Gaussian) mask placements (Shen et al., 2023, Huang et al., 2023).

Repeated sampling with randomized mask schedules allows estimation of pixel-wise uncertainty and small additional gains in PSNR (e.g., $\sim0.2\,\text{dB}$) upon averaging, with the option to trade sharpness versus smoothness (Shen et al., 2023).

CDiffMR (Huang et al., 2023) further demonstrates that a single pre-trained cold diffusion model operates effectively across a range of acceleration factors (×4, ×8, ×16) without retraining, owing to exposure during training to a continuum of sampling rates.

5. Practical Implementation and Computational Considerations

Each cold diffusion reconstruction requires $T$ unrolled network passes (e.g., $T=125$), resulting in a computational cost of $\mathcal O(0.5$--$1\,\text{s})$ per $320\times320$ slice on a modern GPU (NVIDIA V100). Memory requirements match those of a conventional single U-Net plus a pair of image buffers, compatible with 16GB single-GPU setups (Shen et al., 2023).

Cold diffusion demonstrates stable training under linear mask scheduling; task- or dataset-specific reweighting of $t$ (e.g., bias towards restoration of high-frequency k-space late in the process) is supported. While $\ell_1$ image-domain loss is generally robust, perceptual or adversarial losses can be explored. Fixing random seeds across Python/NumPy/PyTorch, preserving linear mask schedules, and keeping strict input/output formatting are necessary for reproducibility (Shen et al., 2023, Bansal et al., 2022).

Data consistency projection—imposing that known k-space samples at each step match acquired measurements—can easily be incorporated as a post-processing step after network output, using

$$\operatorname{Proj}_\text{DC}(u) = \mathcal F^{-1}\left[ (1-M)\cdot \mathcal F(u) + M\cdot y \right]$$

as shown in CDiffMR (Huang et al., 2023).

6. Extensions, Related Paradigms, and Physical Analogues

Cold diffusion extends beyond MRI reconstruction. The framework accommodates any invertible, information-reducing transformation, such as blurring, downsampling, inpainting, or low-pass filtering, with the same restoration principles applied (Bansal et al., 2022). The formalism also maps to other domains, for example, kinetic theory treatments of diffusion and drift coefficients in k-space or momentum space for cold Fermi systems, as in Lukyanov (Lukyanov, 2022). In these cases, the relevant stochastic process is encoded by Fokker–Planck dynamics, but the diffusion operator need not be noisy in the classical sense—momentum space diffusion coefficients can remain nonzero at zero temperature, reflecting underlying deterministic quantum processes.

A plausible implication is that the cold diffusion paradigm enables flexible, modular frameworks for inverse problems across physical and imaging domains, combining deterministic, loss-driven forward processes with learned, data-driven restoration, and circumventing reliance on explicit stochasticity.

7. Significance and Future Directions

K-space cold diffusion provides a principled, extensible alternative to conventional (noisy) diffusion models for inverse problems under sampling constraints. Its main strengths are:

• Generalization to non-noise degradations: restoring images with arbitrary structured corruption beyond Gaussian noise (Bansal et al., 2022, Shen et al., 2023).
• Flexibility to adapt to any mask or schedule: supporting diverse k-space undersampling strategies (Huang et al., 2023).
• Simplicity of training and inference: leveraging $\ell_1$ losses, deterministic updates, and established deep learning architectures with minor modifications.

Open directions include optimization of mask schedules for specific clinical or physical tasks, improved uncertainty quantification using multi-sample averaging, network topologies for direct k-space restoration, and extension to multi-coil and three-dimensional acquisitions. The close ties between deterministic cold diffusion and physical diffusion processes in quantum or kinetic contexts are subjects of ongoing theoretical exploration (Lukyanov, 2022).