Projection Embedded Diffusion Bridge for CT Reconstruction from Incomplete Data

TL;DR

The paper tackles CT reconstruction from incomplete projection data by introducing the Projection Embedded Diffusion Bridge (PEDB). PEDB adds data consistency by embedding the projection data into the posterior score of a reverse SDE, conditioning sampling on both the FBP reconstruction and observed projections. Under Gaussian assumptions, the authors derive a tractable posterior score and a discretization strategy that reduces sampling error, plus a stochasticity parameter $oldsymbol{gamma}$ to adapt to domain shifts. Extensive experiments across sparse-view, limited-angle, and truncated projections show PEDB consistently outperforms state-of-the-art diffusion-bridge methods in standard, noisy, and domain-shift scenarios, with improved fidelity and perceptual quality. The work offers a principled, data-consistent diffusion-bridge framework that could extend to 3D CT and other ill-posed inverse problems in medical imaging.

Abstract

Reconstructing CT images from incomplete projection data remains challenging due to the ill-posed nature of the problem. Diffusion bridge models have recently shown promise in restoring clean images from their corresponding Filtered Back Projection (FBP) reconstructions, but incorporating data consistency into these models remains largely underexplored. Incorporating data consistency can improve reconstruction fidelity by aligning the reconstructed image with the observed projection data, and can enhance detail recovery by integrating structural information contained in the projections. In this work, we propose the Projection Embedded Diffusion Bridge (PEDB). PEDB introduces a novel reverse stochastic differential equation (SDE) to sample from the distribution of clean images conditioned on both the FBP reconstruction and the incomplete projection data. By explicitly conditioning on the projection data in sampling the clean images, PEDB naturally incorporates data consistency. We embed the projection data into the score function of the reverse SDE. Under certain assumptions, we derive a tractable expression for the posterior score. In addition, we introduce a free parameter to control the level of stochasticity in the reverse process. We also design a discretization scheme for the reverse SDE to mitigate discretization error. Extensive experiments demonstrate that PEDB achieves strong performance in CT reconstruction from three types of incomplete data, including sparse-view, limited-angle, and truncated projections. For each of these types, PEDB outperforms evaluated state-of-the-art diffusion bridge models across standard, noisy, and domain-shift evaluations.

Figures (7)

• Figure 1: Framework of the proposed PEDB. PEDB establishes a stochastic transformation between $q_{\text{data}}\left(X_0|X_{\text{FBP}},y\right)$ and $\delta\left(X_T-X_{\text{FBP}}\right)$. The forward process follows the same forward SDE as image-domain diffusion bridge models. In the reverse process, we propose a novel reverse SDE in which the incomplete projection data $y$ is embedded into the score function, and a free parameter $\gamma$ is introduced to control the level of stochasticity. At each reverse step from $t$ to $t-\Delta t$, the trained image-domain data predictor is used to predict the image-domain expected-mean $\hat{X}_0^{\left(t\right)}$. Data consistency is then incorporated by solving a quadratic optimization, producing the projection-embedded expected mean $\hat{X}_{0,\text{p}}^{\left(t\right)}$ from $\hat{X}_0^{\left(t\right)}$ and $y$. Using $\hat{X}_{0,\text{p}}^{\left(t\right)}$, $X_t$ and $X_{\text{FBP}}$, the next intermediate image $X_{t-\Delta t}$ is computed by discretizing the reverse SDE.
• Figure 2: Representative visualization results for CT reconstruction from sparse-view projections. The regions enclosed by blue and yellow boxes are magnified to highlight structural details. The display window is set to [-1000HU, 800HU] for full chest CT images, [-160HU, 240HU] for full pelvic CT images and zoomed-in soft tissue regions, and [-1300HU, 200HU] for zoomed-in lung regions. Our method PEDB uses the I$^2$SB-specified image-domain data predictor with 50 NFEs. I$^2$SB and CDDB are also evaluated with 50 NFEs.
• Figure 3: Representative visualization results for CT reconstruction from limited-angle projections. The regions enclosed by blue and yellow boxes are magnified to highlight structural details. The display window is set to [-1000HU, 800HU] for full chest CT images, [-160HU, 240HU] for full pelvic CT images and zoomed-in soft tissue regions, [-1300HU, 200HU] for zoomed-in lung regions, and [-500HU, 1500HU] for zoomed-in bone regions in real experiments. Our method PEDB uses the I$^2$SB-specified image-domain data predictor with 50 NFEs. I$^2$SB and CDDB are also evaluated with 50 NFEs.
• Figure 4: Representative visualization results for CT reconstruction from truncated projections. The regions enclosed by blue and yellow boxes are magnified to highlight structural details. The display window is set to [-1000HU, 800HU] for full chest CT images, [-160HU, 240HU] for full pelvic CT images and zoomed-in soft tissue regions, and [-500HU, 1500HU] for zoomed-in bone regions in real experiments. Our method PEDB uses the I$^2$SB-specified image-domain data predictor with 50 NFEs. I$^2$SB and CDDB are also evaluated with 50 NFEs.
• Figure 5: Ablation study on the hyperparameter $k_x$, showing RMSE–$k_x$, SSIM–$k_x$, and LPIPS–$k_x$ curves across three types of incomplete data, including sparse-view, limited-angle, and truncated projections. Evaluations were conducted in the low-noise simulation scenario. I$^2$SB-specified image-domain data predictors were used, with NFE set to 10 and the number of CG iterations $m$ set to 20. The hyperparameter $k_x$ was varied from 0.001 to 100.

Theorems & Definitions (2)

• Theorem 1
• Corollary 1.1