Pre-Contrast Conditioned DDPMs

• The paper demonstrates that integrating pre-contrast information into the denoising process accelerates convergence and improves image fidelity without relying on expensive paired data.
• Pre-contrast conditioned DDPMs are generative models that use auxiliary observations, such as low-dose images, to adaptively shape the noise reduction process for realistic synthesis.
• These models employ adaptive priors, residual conditioning, and score-matching to efficiently solve inverse problems, notably in medical imaging and signal restoration.

Pre-contrast conditioned denoising diffusion probabilistic models (DDPMs) are a class of generative models in which information from a "pre-contrast" or otherwise auxiliary observation (e.g., pre-contrast MRI, low-dose CT, mel-spectrogram, degraded image) conditions or structurally guides the probabilistic denoising trajectory. These models have found substantial traction in inverse problems, medical image synthesis, signal restoration, and robust conditional generation, as they enable realistic sample generation or image recovery without reliance on expensive paired ground truth or aggressive domain-specific supervision.

1. Foundations of Pre-Contrast Conditioning in Diffusion Models

In standard DDPMs, a forward Markov process transforms data into a progressively noised latent variable using a known prior—typically an isotropic Gaussian 𝒩(0, I). The learned reverse process reconstructs the original data from this noise, estimating the score function (gradient of log-density). Pre-contrast conditioning modifies this paradigm by integrating additional information—usually a non-contrast (e.g., baseline, low-dose, aliased, degraded) observation—into the forward and/or reverse processes.

Conditioning is implemented in multiple forms:

• As an explicit conditional input to the noise-predictor network: e.g., concatenating a pre-contrast image or degraded signal with the current noisy sample in the reverse process (Mayo et al., 29 Oct 2024, Osuna-Vargas et al., 18 Sep 2024, Liu et al., 2023, Asgariandehkordi et al., 2023).
• Via construction of a data-dependent adaptive prior: replacing the isotropic prior with a modality/statistic-dependent Gaussian prior (mean and covariance derived from the pre-contrast or conditional side information), as in PriorGrad (Lee et al., 2021) or by jointly learning the prior as in RestoreGrad (Lee et al., 19 Feb 2025).
• Through an explicit score-combination or product-of-experts across modalities (including the pre-contrast channel) (Nair et al., 2022).

This approach addresses inherent mismatch between the standard Gaussian prior and data distribution, especially when nontrivial structure is present in the pre-contrast modality, by aligning the stochastic process with the actual distributional characteristics of the problem.

2. Mathematical Formalism and Training Objectives

The pre-contrast conditioned DDPM framework departs from vanilla models in both the forward noising and training objectives:

• Forward Process:

Rather than $$x_t = \sqrt{\bar\alpha_t} x_0 + \sqrt{1-\bar\alpha_t} \epsilon, \quad \epsilon\sim\mathcal{N}(0,I)$$ conditioning yields $$x_t = \sqrt{\bar\alpha_t}(x_0 - \mu(c)) + \sqrt{1-\bar\alpha_t}\,\epsilon, \quad \epsilon\sim\mathcal{N}(0,\Sigma(c))$$ where $\mu(c)$ and $\Sigma(c)$ are derived (analytically or via learned encoders) from pre-contrast information $c$ (Lee et al., 2021, Lee et al., 19 Feb 2025). Alternatively, the forward process may incorporate a residual (e.g., $R = \hat{x}_0 - x_0$ in Resfusion (Shi et al., 2023)), or the noisy step may be conditioned at an intermediate point determined via smooth equivalence transformations.

• Training Loss:

With a data-dependent covariance, the loss takes the form $$-\mathrm{ELBO} = C(\Sigma) + \sum_t \gamma_t\,\mathbb{E}_{x_0,\epsilon}\bigl\|\epsilon - \epsilon_t(\sqrt{\bar\alpha_t}(x_0-\mu) + \sqrt{1-\bar\alpha_t} \epsilon, t) \bigr\|^2_{\Sigma^{-1}}$$ This reinforces the network to predict denoising steps adapted to the variance of the pre-contrast distribution. In variational frameworks (RestoreGrad), KL terms regularize the learned prior and posterior (Lee et al., 19 Feb 2025). In unconstrained inverse problems, the loss may combine a data fidelity or MAP term with the diffusion-model prior, often solved via projection gradient descent with auxiliary variables (Zhang et al., 11 Jun 2024).

• Score-Matching Perspective:

In general state spaces (continuous, discrete, or manifold), an extended score-matching loss of the form $$I_\mathrm{DSM}(\theta) = \frac{1}{2}\int_0^T \mathbb{E}_{q(x_0, x_t, c)}\left[\|\nabla_x\log q_{t|0}(x_t|x_0) - s_\theta(x_t, c, t)\|^2\right]\,dt$$ characterizes the optimality for conditioning on auxiliary observations (Benton et al., 2022).

3. Key Variants, Sampling Strategies, and Efficiency Gains

Several structural and algorithmic advances enable strong quantitative improvements:

• Data-Dependent Adaptive Priors: As in PriorGrad and RestoreGrad, incorporating pre-contrast statistics into the prior (mean, covariance, or more complex learned latent structures) leads to simplified denoising, faster optimization convergence (via better Hessian conditioning), parameter efficiency, and robustness to reduced network capacity (Lee et al., 2021, Lee et al., 19 Feb 2025).
• Residual and Late-Start Approaches: Resfusion conditions the forward process on the residual between degraded and clean images, launching the reverse process from a noisy version of the degraded input, accelerating sampling (e.g., only 5 steps required) and preserving low-frequency structure (Shi et al., 2023).
• Multimodal Product-of-Experts: Effective conditional scores are combined from modality-specific (including pre-contrast) diffusion models, facilitating robust image synthesis even with missing or incomplete modalities (Nair et al., 2022).
• Auxiliary/Projection Variables for Inverse Problems: ProjDiff introduces an auxiliary variable at a chosen diffusion time to enforce observation constraints (e.g., matching observed data in measurement space), with constraints handled via projection steps and an ELBO that separately regularizes the clean and noisy variables (Zhang et al., 11 Jun 2024).
• Conditional Regularization and Uncertainty Quantification: In hybrid frameworks, pretrained unconditional models regularize the sampling trajectory, “projecting” conditional samples back to the learned data manifold to prevent divergence and enhance perceptual realism (Mei et al., 2022, Graikos et al., 2023). Probabilistic outputs allow uncertainty maps and improved signal-to-noise via ensembling (Osuna-Vargas et al., 18 Sep 2024).

4. Applications in Medical Imaging, Restoration, and Beyond

Pre-contrast conditioned DDPMs are prominently applied to medical and scientific imaging:

• Contrast-Enhanced MRI Synthesis: Models synthesize dynamic contrast-enhanced (DCE) MRI from pre-contrast images. Subtraction-based approaches, tumor-aware ROI losses, and mask-conditioning improve the fidelity of synthetic contrast uptake especially in lesions. Quantitative benchmarks demonstrate these models outperforming non-contrast-based approaches across MAE, SSIM, LPIPS, FID, and FRD, with expert reader studies confirming their clinical realism (Ibarra et al., 19 Aug 2025).
• PET and MRI Reconstruction: Conditioning on anatomical priors (e.g., T1w MR for PET or undersampled gridding for MR Fingerprinting) enables higher PSNR/SSIM, reduced uncertainty, and mitigates domain-specific artifacts. Data-consistency constraints further ensure results remain faithful to measured counts or k-space (Gong et al., 2022, Mayo et al., 29 Oct 2024).
• Low-Dose CT, Optical Coherence Tomography, Ultrasound: Zero-shot denoising is enabled by conditioning on, or integrating, low dose or low-resolution pre-contrast images, even in the absence of high-quality paired training samples. Such models preserve clinically relevant textures (e.g., speckles in ultrasound), details, and improve measures like PSNR and GCNR relative to both supervised CNNs and classical restoration (Liu et al., 2023, Hu et al., 2022, Asgariandehkordi et al., 2023, Osuna-Vargas et al., 18 Sep 2024).
• Wireless Communications and General Inverse Problems: DDPMs conditioned on degraded digital signals (e.g., after transmission) can outperform both conventional and DNN-based methods in reconstructing clean signals, with improvements exceeding 10 dB in low-SNR regimes in image transmission (Letafati et al., 2023).

5. Comparative Analysis with Related Approaches

Empirical and theoretical comparisons across works yield several conclusions:

Model	Conditioning Mode	Efficiency/Convergence	Performance Domains
PriorGrad (Lee et al., 2021)	Data-dependent Gaussian prior	Faster convergence, simpler model	Speech synthesis, robust to small models
RestoreGrad (Lee et al., 19 Feb 2025)	Learned latent prior (VAE-style)	5–10× faster convergence, robust	Speech/image restoration
Resfusion (Shi et al., 2023)	Residual-conditioned start	Drastic reduction in sampling steps	Shadow removal, low-light, deraining
Multimodal PoE (Nair et al., 2022)	Product-of-expert on modalities	No retraining for new modalities	Multimodal image synthesis
ProjDiff (Zhang et al., 11 Jun 2024)	Auxiliary var. for constraint	Solves linear/nonlinear inverse	Image restoration, source separation
DDPM-MR(-PETCon) (Gong et al., 2022)	Anatomy prior + data constraint	Higher PSNR/SSIM, less bias	PET denoising
MRF-IDDPM (Mayo et al., 29 Oct 2024)	Aliased gridding as input	Fewer artifacts, uncertainty maps	Quantitative MRI/fingerprinting

The consistent theme is that explicit exploitation of pre-contrast or degraded information—in priors, initialization, loss functions, or constraint projections—not only accelerates training/convergence but also improves fidelity, generalization, and robustness, particularly in high-noise, scarce data, or inverse problem settings.

6. Practical Impact, Limitations, and Future Outlook

Pre-contrast conditioning in DDPMs enables:

• High quality synthesis and restoration even when target (clean/contrast) ground truth is unavailable or sparse, through leveraging information-rich noncontrast or degraded inputs.
• Superior mode coverage and context preservation compared to GANs, especially in medical imaging, as measured by FID, contextual error rates, and domain-specific error statistics (Deshpande et al., 2023).
• Enhanced sample efficiency: models can converge in a fraction of the training epochs and sampling steps versus fixed-prior DDPMs (Lee et al., 19 Feb 2025, Shi et al., 2023).
• Robust handling of out-of-distribution and challenging real clinical data, and uncertainty quantification for downstream decision-making.

Limitations and open areas:

• Dependence on pre-contrast or prior information: when such data lack relevant structural cues or are missing, model performance may degrade.
• Uncertainty in ROI-focused and mask-conditioning methods if segmentation masks are not reliably available or generalizable in screening contexts (Ibarra et al., 19 Aug 2025).
• Computational demands for high-dimensional inputs can remain prohibitive; further improvements in patch-based or low-rank model architectures are under exploration (Mayo et al., 29 Oct 2024, Zhang et al., 11 Jun 2024).
• Formal integration of physics-based constraints in the stochastic process (e.g., Bloch k-space in MRI) is not yet universally handled.

Current research is pursuing extension to nonlinear inverse problems, adaptation to 3D volumetric reconstructions, and seamless integration between generative priors and data-consistency or physics-based constraints.

7. Summary Table of Representative Pre-Contrast Conditioned DDPM Architectures

Approach	Pre-Contrast Modality / Input	Conditioning Mechanism	Performance Benefit
PriorGrad	mel-spectrogram / speech features	Adaptive Gaussian prior (μ, Σ)	Faster, robust speech synth.
RestoreGrad	degraded input (speech/image)	Jointly learned VAE prior	Few steps, fast convergence
Resfusion	degraded image	Residual-initialized forward proc.	Only 5 sampling steps needed
Dn-Dp	low-dose CT	Posterior guided by prior and MAP	Unsupervised, high PSNR
PET DDPM	MR prior, low-dose PET	Multimodal input, data consistency	Reduced bias, higher SSIM
MRF-IDDPM	aliased MRI (gridding reconstr.)	Conditional on pre-contrast image	Sharper, fewer artifacts
Breast DCE-MRI	pre-contrast MRI	Concatenation/mask, ROI-aware loss	Lesion fidelity, clinical

• “PriorGrad: Improving Conditional Denoising Diffusion Models with Data-Dependent Adaptive Prior” (Lee et al., 2021)
• “RestoreGrad: Signal Restoration Using Conditional Denoising Diffusion Models with Jointly Learned Prior” (Lee et al., 19 Feb 2025)
• “Resfusion: Denoising Diffusion Probabilistic Models for Image Restoration Based on Prior Residual Noise” (Shi et al., 2023)
• “PET image denoising based on denoising diffusion probabilistic models” (Gong et al., 2022)
• “Denoising diffusion probabilistic models for magnetic resonance fingerprinting” (Mayo et al., 29 Oct 2024)
• “Denoising diffusion models for high-resolution microscopy image restoration” (Osuna-Vargas et al., 18 Sep 2024)
• “Comparing Conditional Diffusion Models for Synthesizing Contrast-Enhanced Breast MRI from Pre-Contrast Images” (Ibarra et al., 19 Aug 2025)
• “Image Generation with Multimodal Priors using Denoising Diffusion Probabilistic Models” (Nair et al., 2022)
• “Unleashing the Denoising Capability of Diffusion Prior for Solving Inverse Problems” (Zhang et al., 11 Jun 2024)
• “Unsupervised Denoising of Retinal OCT with Diffusion Probabilistic Model” (Hu et al., 2022)
• “Conditional Denoising Diffusion Probabilistic Models for Data Reconstruction Enhancement in Wireless Communications” (Letafati et al., 2023)

This corpus illustrates the theoretical underpinnings, algorithmic diversity, and domain-specific implementation pathways for pre-contrast conditioned DDPMs, substantiating their central role in modern conditional generative modeling and robust signal restoration.