Rician Denoising Diffusion Model (RDDPM)

• RDDPM is an advanced generative denoising framework that converts Rician noise in MRI magnitude data into Gaussian latent variables for improved signal fidelity.
• It employs a corrective network and a pre-trained DDPM denoiser to preserve fine image structures, crucial for accurate sodium breast MRI analysis.
• RDDPM consistently outperforms standard DDPM and CNN-based methods in IQA metrics, demonstrating superior detail preservation and noise correction.

The Rician Denoising Diffusion Probabilistic Model (RDDPM) is an advanced generative denoising framework designed to address the statistical mismatch between conventional Denoising Diffusion Probabilistic Models (DDPM) and Rician-corrupted MRI magnitude data, specifically in the context of sodium breast MRI. Sodium MRI provides quantitative in vivo insights into tissue sodium concentration for oncology and related applications but is highly susceptible to low SNR and Rician-distributed noise, complicating postprocessing and analysis. RDDPM introduces a corrective mechanism that inverts Rician magnitude statistics to underlying Gaussian latent variables at each diffusion timestep, enabling more effective denoising and fine-structure preservation compared to standard DDPM and CNN-based baselines (Yuan et al., 2024).

1. Transition from DDPM to RDDPM: Foundations and Motivation

Standard DDPM operates under the assumption of Gaussian-distributed noise, with a forward Markov chain implemented as

$$q(x_t|x_{t-1}) = \mathcal{N}(x_t; \sqrt{1-\beta_t}x_{t-1}, \beta_t I)$$

where $\{\beta_t\}$ follows a specified schedule, and $I$ is the identity. The closed-form relation

$$x_t = \sqrt{\bar{\alpha}_t}x_0 + \sqrt{1-\bar{\alpha}_t}\epsilon, \quad \epsilon \sim \mathcal{N}(0, I)$$

facilitates both sampling and training. Training minimizes the MSE between true noise $\epsilon$ and predicted noise $\epsilon_\theta(x_t, t)$. However, in MRI, and especially sodium MRI, magnitude data are contaminated by Rician—not Gaussian—noise. The Rician-distributed observation,

$$p_R(R|S; \sigma) = \frac{R}{\sigma^2}\exp\left[-\frac{R^2 + S^2}{2\sigma^2}\right] I_0\left(\frac{RS}{\sigma^2}\right), \quad R\geq0,$$

exhibits nonlinear mean and variance dependencies on the underlying signal. Applying DDPM directly often results in suppressed fine detail, blurring, and artifacts (Yuan et al., 2024).

RDDPM addresses this by embedding an auxiliary network $\theta$ that, at each diffusion step, learns to invert the squared Rician magnitude $A_t^2$ to an estimate of the latent squared Gaussian $x_t^2$. The estimated $\{\beta_t\}$0—obtained as $\{\beta_t\}$1—is then processed by a pre-trained DDPM denoiser. At inference, consistent Rician noise statistics are reintroduced by combining two independent Gaussians in magnitude form.

2. Forward Process: Rician Noise Mapping and Gaussian Latents

The RDDPM generative process begins by defining a latent Gaussian diffusion chain,

$\{\beta_t\}$2

The observable at each step is the Rician magnitude,

$\{\beta_t\}$3

Each $\{\beta_t\}$4 is thus Rician-distributed with mean $\{\beta_t\}$5 and noise level $\{\beta_t\}$6. The forward density becomes

$\{\beta_t\}$7

RDDPM leverages the property

$\{\beta_t\}$8

to approximate $\{\beta_t\}$9 for high-SNR or large-sample regimes, but learns a data-driven correction by training the network $I$0 to regress $I$1.

3. Reverse Process and Training Protocol

The RDDPM denoising process (reverse process) operates iteratively:

• At each $I$2, two independent noise vectors $I$3 are sampled.
• The corrective network $I$4 predicts $I$5, with $I$6.
• The pre-trained DDPM network yields $I$7.
• The next latent is computed as

$I$8

• If $I$9, noise is re-injected: $$x_t = \sqrt{\bar{\alpha}_t}x_0 + \sqrt{1-\bar{\alpha}_t}\epsilon, \quad \epsilon \sim \mathcal{N}(0, I)$$0, $$x_t = \sqrt{\bar{\alpha}_t}x_0 + \sqrt{1-\bar{\alpha}_t}\epsilon, \quad \epsilon \sim \mathcal{N}(0, I)$$1.

Training proceeds as follows:

• A pre-trained CNN (U-Net) supplies “clean” targets by denoising images simulated with Rician statistics.
• For each training step, a timestep $$x_t = \sqrt{\bar{\alpha}_t}x_0 + \sqrt{1-\bar{\alpha}_t}\epsilon, \quad \epsilon \sim \mathcal{N}(0, I)$$2 and Gaussian noise are sampled; $$x_t = \sqrt{\bar{\alpha}_t}x_0 + \sqrt{1-\bar{\alpha}_t}\epsilon, \quad \epsilon \sim \mathcal{N}(0, I)$$3 is computed; multiple synthetic Rician corruption realizations $$x_t = \sqrt{\bar{\alpha}_t}x_0 + \sqrt{1-\bar{\alpha}_t}\epsilon, \quad \epsilon \sim \mathcal{N}(0, I)$$4 are generated.
• The loss

$$x_t = \sqrt{\bar{\alpha}_t}x_0 + \sqrt{1-\bar{\alpha}_t}\epsilon, \quad \epsilon \sim \mathcal{N}(0, I)$$5

is minimized using MSE over 50 inner steps per timestep sample.

4. Algorithm Summary and Implementation

Training

• Pre-train a U-Net denoiser on Rician-corrupted sodium MRI to estimate $$x_t = \sqrt{\bar{\alpha}_t}x_0 + \sqrt{1-\bar{\alpha}_t}\epsilon, \quad \epsilon \sim \mathcal{N}(0, I)$$6 targets.
• Set diffusion length $$x_t = \sqrt{\bar{\alpha}_t}x_0 + \sqrt{1-\bar{\alpha}_t}\epsilon, \quad \epsilon \sim \mathcal{N}(0, I)$$7.
• For randomly chosen $$x_t = \sqrt{\bar{\alpha}_t}x_0 + \sqrt{1-\bar{\alpha}_t}\epsilon, \quad \epsilon \sim \mathcal{N}(0, I)$$8, form $$x_t = \sqrt{\bar{\alpha}_t}x_0 + \sqrt{1-\bar{\alpha}_t}\epsilon, \quad \epsilon \sim \mathcal{N}(0, I)$$9, corrupt to $\epsilon$0, and train $\epsilon$1 for $\epsilon$2 inner loop steps (Adam optimizer, $\epsilon$3 learning rate).

Sampling/Inference

• Initialize from observed $\epsilon$4 ($\epsilon$5).
• For $\epsilon$6 down to 1: apply $\epsilon$7 to $\epsilon$8 to estimate $\epsilon$9, predict noise with frozen DDPM $\epsilon_\theta(x_t, t)$0, perform one reverse step, inject noise, and recalculate $\epsilon_\theta(x_t, t)$1. Output final $\epsilon_\theta(x_t, t)$2.

Architecture and Hardware

• Both $\epsilon_\theta(x_t, t)$3 and $\epsilon_\theta(x_t, t)$4 are U-Net style models.
• Training uses a batch size of 10 on an Nvidia A100 GPU.

5. Experimental Results

Experiments were conducted on 540 training slices from 13 patients; evaluation used 270 slices from 9 patients, all validated with U-Net denoised images as surrogates for ground truth. Performance was measured using no-reference image quality assessment (IQA) metrics BRISQUE (lower is better), MUSIQ (lower is better), and PaQ2PiQ (higher is better):

Method	BRISQUE	MUSIQ	PaQ2PiQ
BM3D	72.4820	5.0743	5.1764
DnCNN	60.9133	4.4017	4.6366
Unet	58.7705	3.7178	4.3825
ResUnet	52.7632	3.1747	4.2747
ADNet	40.6889	3.6877	4.4727
DDPM	46.7097	3.4669	4.3865
RDDPM	34.4638	2.7866	4.3806

RDDPM achieved the best BRISQUE and MUSIQ scores, and near-best PaQ2PiQ, consistently outperforming DDPM and other CNN-based methods. Qualitatively, DDPM exhibited excessive blurring in low-SNR regions, whereas RDDPM preserved fine structures and maintained high-SNR peaks due to explicit Rician handling (Yuan et al., 2024).

6. Broader Applications, Limitations, and Prospects

The Rician noise model is relevant to all magnitude MRI modalities operating at low SNR, including proton and phosphorus MRI. Adaptations of the RDDPM approach could extend to other non-Gaussian observation processes, such as Rice-to-Gaussian corrections for ultrasound envelope images. Current RDDPM requires a pre-denoiser for target generation and operates with MSE on squared magnitudes; future work may involve joint end-to-end optimization of $\epsilon_\theta(x_t, t)$5 and $\epsilon_\theta(x_t, t)$6, adaptively learned or anatomically informed $\epsilon_\theta(x_t, t)$7 schedules, or explicit modeling of heteroscedastic or coil-dependent noise. This suggests a plausible expansion to heteroscedastic or adaptive-noise denoising frameworks for broader biomedical signal domains.

By incorporating an explicit inversion of Rician distributional effects at each diffusion step, RDDPM resolves the fundamental misalignment present in standard diffusion models for magnitude MRI, enabling superior denoising while preserving diagnostically relevant fine structure (Yuan et al., 2024).