DEcomp-MoD: Dual-Energy Diffusion Framework

• DEcomp-MoD is a deep generative framework that fuses dual-energy CT physics with score-based diffusion models for quantitative material decomposition.
• It incorporates explicit forward models, diffusion-based priors, and data consistency constraints to improve reconstruction fidelity under low-dose and sparse-view conditions.
• Experimental benchmarks show significant gains in metrics like PSNR and SSIM, outperforming classical and standard deep learning methods in challenging imaging scenarios.

Dual-Energy Decomposition Model-based Diffusion (DEcomp-MoD) refers to a suite of deep generative algorithms that integrate rigorous physical dual-energy computed tomography (DECT) modeling with state-of-the-art score-based diffusion models for quantitative material decomposition, contrast enhancement, and artifact suppression. DEcomp-MoD algorithms are designed to optimally leverage the measurement physics of DECT (including acquisition-specific forward models), enforce data consistency, and exploit expressive learned distributions over target images, yielding substantial improvements over conventional and standard deep-learning methods under low-dose, limited-view, and challenging SNR regimes (Lyu et al., 2023, Jiang et al., 2024, Peng et al., 16 Apr 2025, Xu et al., 24 Jul 2025).

1. Underlying Physical and Mathematical Models

DEcomp-MoD builds upon an explicit forward model of DECT acquisition, in which the linear attenuation coefficient $\mu(\mathbf{r}, E)$ at position $\mathbf{r}$ and energy $E$ is decomposed as a linear combination (or, for three-material cases, a mass-fraction mixture) of basis material functions:

$$\mu(\mathbf{r}, E) = \sum_{i=1}^B \phi_i(E)f_i(\mathbf{r})$$

where $\phi_i(E)$ denotes the mass-attenuation spectrum for basis $i$, and $f_i(\mathbf{r})$ its spatial density map; $B=2$ is common (e.g., soft tissue and bone, or water and calcium), but three-basis decompositions (water, bone, iodinated contrast) are also studied (Lyu et al., 2023, Xu et al., 24 Jul 2025).

Polychromatic forward projections across $k$ energy channels (e.g., dual-kVp, dual-layer, or monoenergetic reconstructions at $E_L$, $\mathbf{r}$0) are given by:

$\mathbf{r}$1

or, as in the linearized form,

$\mathbf{r}$2

where $\mathbf{r}$3 is the (block-diagonal) projection operator and $\mathbf{r}$4 gathers all basis maps (Jiang et al., 2024, Xu et al., 24 Jul 2025).

Material decomposition in classical approaches typically uses analytical inversion of these models, but this is ill-posed and degrades under noise, limited views, or reduced contrast dose.

2. Diffusion-Model-Based Priors and Conditioning

A core element of DEcomp-MoD is the use of deep score-based diffusion models as priors for material images or dual-energy reconstructions. Both unconditional and conditional Denoising Diffusion Probabilistic Models (DDPMs) are used, with the forward noising process specified via a fixed variance schedule $\mathbf{r}$5:

$\mathbf{r}$6

with the marginal

$\mathbf{r}$7

Training minimizes the expected squared error in noise space: $\mathbf{r}$8

Conditional diffusion (e.g., for post-reconstruction DECT enhancement or dual-domain approaches) is realized by concatenating measurement-derived features or other physics priors onto every residual block of the backbone UNet, with additional timestep embeddings incorporated via FiLM or sinusoidal encodings (Lyu et al., 2023, Jiang et al., 2024, Peng et al., 16 Apr 2025).

3. Model-Based Posterior Sampling and Inverse Solutions

DEcomp-MoD algorithms achieve physically grounded, data-consistent inference by embedding the CT measurement forward model directly within the generative sampling loop, typically via model-based diffusion posterior sampling (DPS) strategies. In the general case, the target is the posterior $\mathbf{r}$9, achieved by augmenting the reverse diffusion SDE (or its discrete counterpart) with data consistency gradients:

$E$0

The data-fidelity term $E$1 enforces projection-domain consistency (typically via projection operators $E$2 and residuals $E$3), while the diffusion (score) term steers sampling toward plausible material images (Jiang et al., 2024, Xu et al., 24 Jul 2025).

Accelerated variants use jumpstarted DPS (JSDPS), initializing the reverse process at $E$4 using approximate analytical or fast deep-learning decompositions for significant speedup and stability improvements, with $E$5 (vs. $E$6) yielding an $E$7 runtime reduction and highest SSIM/PSNR accuracy (Jiang et al., 2024).

Alternatively, inference can employ a hybrid HQS/DDIM update scheme, alternating between DDPM-based denoising and explicit data-consistency (CG-based) updates in the material-image domain (Xu et al., 24 Jul 2025).

4. Representative Architectures and Training Paradigms

All DEcomp-MoD algorithms leverage residual or attention UNet backbones with domain-specific adaptations:

• Volume and Projection Domains: For ultra-sparse-view DECBCT, dual-domain diffusion stacks (Proj-DM and Vol-DM) with differentiable physics modules enforce cycle and spectral consistency (Peng et al., 16 Apr 2025).
• Conditional Enhancement: In contrast reduction tasks, 2D conditional UNets are used, with low-dose DECT inputs concatenated at all feature resolutions; a single self-attention block is commonly present at the $E$8 resolution (Lyu et al., 2023).
• Hyperparameters and Training: Adam optimizer (lr $E$9 $$\mu(\mathbf{r}, E) = \sum_{i=1}^B \phi_i(E)f_i(\mathbf{r})$$0 to $$\mu(\mathbf{r}, E) = \sum_{i=1}^B \phi_i(E)f_i(\mathbf{r})$$1), batch size $$\mu(\mathbf{r}, E) = \sum_{i=1}^B \phi_i(E)f_i(\mathbf{r})$$2, training for $$\mu(\mathbf{r}, E) = \sum_{i=1}^B \phi_i(E)f_i(\mathbf{r})$$3K–$$\mu(\mathbf{r}, E) = \sum_{i=1}^B \phi_i(E)f_i(\mathbf{r})$$4K iterations is standard (Lyu et al., 2023, Jiang et al., 2024, Xu et al., 24 Jul 2025). Noise schedules are typically linear ($$\mu(\mathbf{r}, E) = \sum_{i=1}^B \phi_i(E)f_i(\mathbf{r})$$5 to $$\mu(\mathbf{r}, E) = \sum_{i=1}^B \phi_i(E)f_i(\mathbf{r})$$6); DDPM/score-loss is the only explicit training objective, with no adversarial or perceptual regularizers.

For dual-domain architectures, additional losses enforce cycle-consistency and spectral subtraction-map accuracy:

$$\mu(\mathbf{r}, E) = \sum_{i=1}^B \phi_i(E)f_i(\mathbf{r})$$7

where $$\mu(\mathbf{r}, E) = \sum_{i=1}^B \phi_i(E)f_i(\mathbf{r})$$8 promotes accurate inter-energy difference maps (Peng et al., 16 Apr 2025).

5. Performance Evaluation and Experimental Findings

The DEcomp-MoD framework has undergone extensive simulation and real-data validation across diverse DECT settings.

Quantitative Metrics

• Head-neck DECT angiography: Reader studies under extreme contrast reduction (down to $$\mu(\mathbf{r}, E) = \sum_{i=1}^B \phi_i(E)f_i(\mathbf{r})$$9 of normal dose) show DEcomp-MoD achieves mean quality scores of $\phi_i(E)$0 (10 cc) and $\phi_i(E)$1 (40 cc) vs. UNet ($\phi_i(E)$2, $\phi_i(E)$3), MDE ($\phi_i(E)$4, $\phi_i(E)$5), and low-dose input ($\phi_i(E)$6, $\phi_i(E)$7) (Lyu et al., 2023).
• Sparse-view and Low-dose DECT: On synthetic AAPM data, DEcomp-MoD yields state-of-the-art decomposition accuracy:

Material	FBP PSNR	DIRECT-Net PSNR	DEcomp-MoD PSNR
Water	20.50	23.28	29.14
Bone	27.26	26.67	33.88

SSIM improves to $\phi_i(E)$8 (water) and $\phi_i(E)$9 (bone), the highest among tested approaches (Xu et al., 24 Jul 2025).

• Spectral CBCT from Two Views: On 0°/90° dual-view phantoms, mean absolute error for low/high energy volumes is $i$0 HU, SSIM $i$1, and subtraction-map SSIM $i$2 (Peng et al., 16 Apr 2025).
• Dual-kVp and Dual-layer CT: JSDPS delivers SSIM $i$3 per channel, run-times $i$4100s, outperforming classical model-based MBMD and non-accelerated DPS (Jiang et al., 2024).

Qualitative Findings

• DEcomp-MoD effectively restores subtle vessel structure, preserves sharp anatomical boundaries, and suppresses streak, beam-hardening, and “hole” artifacts, notably outperforming both analytical inversion and direct UNet methods, particularly under ultra-low-dose or highly incomplete-view acquisition (Lyu et al., 2023, Xu et al., 24 Jul 2025, Peng et al., 16 Apr 2025).
• Cycle-consistency and spectral-consistency losses are essential for retaining accurate inter-energy contrasts for subtraction imaging in sparse projection settings (Peng et al., 16 Apr 2025).

6. Limitations and Prospective Extensions

Key limitations and future directions include:

• Dependence on Physics Models and Priors: DEcomp-MoD’s performance for $i$5 basis materials or in presence of strong model mismatch depends on the expressivity of the diffusion prior and accuracy of the physics/forward model (Jiang et al., 2024).
• Data Requirement: Current validations are predominantly simulation- or phantom-based. Direct clinical translation necessitates validation on raw, truly acquired low-dose and contrast-reduced human datasets (Lyu et al., 2023, Peng et al., 16 Apr 2025).
• Speed vs. Fidelity Tradeoff: Standard DDPM samplers require $i$6 network passes; accelerated schemes (DPM-Solver, DDIM, JSDPS) can reduce this to $i$7 with similar performance, but may introduce tunable hyperparameters affecting consistency and stability (Jiang et al., 2024, Xu et al., 24 Jul 2025).
• Extension to Advanced Spectral Modalities: Research is ongoing to extend DEcomp-MoD to photon-counting CT, multi-contrast, and multi-spectral imaging (expanding $i$8), as well as robustification in the presence of scatter and noise via learned likelihood surrogates (Jiang et al., 2024, Peng et al., 16 Apr 2025).
• Uncertainty Quantification: Incorporation of ensemble statistics and uncertainty maps during inference is highlighted as a promising direction for both stopping-criterion adaptation and diagnostic confidence (Jiang et al., 2024).

7. Significance and Methods Comparison

DEcomp-MoD constitutes a paradigm shift in DECT and volumetric spectral imaging, fusing physics-based data-consistency with the generative capacity of score-based diffusion models. It unifies under the same algorithmic umbrella (i) post-reconstruction enhancement for low-dose/low-contrast DECT (Lyu et al., 2023), (ii) direct sinogram-to-material decomposition (Xu et al., 24 Jul 2025), (iii) dual-domain DECBCT from sparse/-view data (Peng et al., 16 Apr 2025), and (iv) model-based DPS for multi-platform spectral CT (Jiang et al., 2024).

In all major settings, DEcomp-MoD outperforms classical model-based decomposition (MBMD/MDE), conventional iterative debiasing, and supervised deep learning baselines (including direct and mutual-domain UNets and prior score-matching generative schemes) in terms of structural similarity, peak SNR, visual fidelity, and artifact suppression under challenging acquisition constraints.

This synthesis reflects content and results exclusively from peer-reviewed and preprint literature on (Lyu et al., 2023, Jiang et al., 2024, Peng et al., 16 Apr 2025), and (Xu et al., 24 Jul 2025).