RN-SDEs: Limited-Angle CT Reconstruction with Residual Null-Space Diffusion Stochastic Differential Equations

TL;DR

Through extensive experiments, it is demonstrated that by leveraging learned MR-SDEs as a prior and emphasizing data consistency using Range-Null Space Decomposition (RNSD) based rectification, this approach can recover high-quality images from severely degraded ones and achieve state-of-the-art performance in most LACT tasks.

Abstract

Computed tomography is a widely used imaging modality with applications ranging from medical imaging to material analysis. One major challenge arises from the lack of scanning information at certain angles, resulting in distortion or artifacts in the reconstructed images. This is referred to as the Limited Angle Computed Tomography (LACT) reconstruction problem. To address this problem, we propose the use of Residual Null-Space Diffusion Stochastic Differential Equations (RN-SDEs), which are a variant of diffusion models that characterize the diffusion process with mean-reverting (MR) stochastic differential equations. To demonstrate the generalizability of RN-SDEs, we conducted experiments with two different LACT datasets, ChromSTEM and C4KC-KiTS. Through extensive experiments, we demonstrate that by leveraging learned MR-SDEs as a prior and emphasizing data consistency using Range-Null Space Decomposition (RNSD) based rectification, we can recover high-quality images from severely degraded ones and achieve state-of-the-art performance in most LACT tasks. Additionally, we present a quantitative comparison of RN-SDE with other networks, in terms of computational complexity and runtime efficiency, highlighting the superior effectiveness of our proposed approach.

Figures (7)

• Figure 1: Illustration of limited angle tomography, with missing angles, $\theta_{\rm miss}$, set to be 90°. (a) Sinogram of an exemplary 2D observation with certain angles missing; (b) Sampling process in the Fourier space; (c) Reconstruction using the FBP algorithm, with distortion and artifacts present.
• Figure 2: Quantitative performance evaluation on the C4KC-KiTS dataset heller2019kits19. This plot demonstrates the relationship between image quality and average processing time per reconstruction in minutes. The quality is measured as the average value of the relative performance of each method in each of the three metrics (PSNR, SSIM, and LPIPS zhang2018unreasonable). Our proposed RN-SDE method obtains, on average, higher quality while having significantly lower computational complexity. More details are provided in Section \ref{['experiment: compare']}.
• Figure 3: Visualization of our RN-SDE denoising diffusion process for LACT reconstruction. (a) Initialize the terminal state $\hat{\mathbf{x}}_T$ of diffusion as the summation of a low-quality reconstruction $\boldsymbol{\mu}$ and a Gaussian noise $\boldsymbol{\epsilon}$. This step positions the starting point of the reverse diffusion closer to $\mathbf{x}_0$ within the distribution space, facilitating a more effective and accurate convergence towards the GT image during the denoising stage; (b) illustrates the rectification mechanism involved in the reverse diffusion process, where we applied range-null space decomposition (RNSD) to the intermediate clean prediction $\hat{\mathbf{x}}_{0|t}$ to enforce data consistency; (c) shows the dynamics of a single transition step in the denoising process. Providing a visual example of how the RNSD-based rectification effectively reduces stochasticity during the reverse diffusion process.
• Figure 4: $\mathbf{A}^{\dagger}_{\eta}(\cdot)$ mainly consists of two blocks: The first is the learnable back-projection block, where we apply a learnable transformation to the Ramp filter and filter the input sinogram $\mathbf{Ax}$. Then, we back-project the image from the sinogram domain to the image domain. The second block uses a non-bias NafNet to perform post-processing on the back-projection results and outputs an estimation of the range-space content$\mathbf{A}_\eta^{\dagger}(\mathbf{y})\in\mathbb{R}^{M}$ that satisfies $\mathbf{A}{\mathbf{A}_\eta^{\dagger}(\mathbf{y})}\approx\mathbf{y}$.
• Figure 5: Visual evaluation of limited angle tomographic reconstruction on the ChromSTEM test set using FBP, NafNet, DPS, DOLCE, and our RN-SDEs. From top to bottom, we present the visualization of a typical reconstruction result for $\theta_{\rm miss} \in$ {60°, 90°, 120°}. The bottom right corner of each example shows the corresponding PSNR metric compared to the Ground Truth (first column).