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Low-resolution Prior Equilibrium Network for CT Reconstruction

Yijie Yang, Qifeng Gao, Yuping Duan

TL;DR

This work tackles ill-posed CT reconstruction under sparse-view and limited-angle data by introducing a low-resolution image prior integrated into a deep equilibrium unrolled framework. The proposed LRPE network uses a shared-weight, multi-stage gradient-descent scheme with two CNNs learning data fidelity and regularization gradients, and it provides convergence guarantees under Lipschitz and step-size conditions. Empirical results on synthetic and clinical datasets demonstrate superior noise suppression, contrast-to-noise ratio, and edge preservation compared with state-of-the-art methods, highlighting practical gains for dose-reduced CT. The approach reduces reliance on re-scans and offers a theoretically grounded, end-to-end reconstruction paradigm suitable for challenging incomplete-data scenarios in CT imaging.

Abstract

The unrolling method has been investigated for learning variational models in X-ray computed tomography. However, it has been observed that directly unrolling the regularization model through gradient descent does not produce satisfactory results. In this paper, we present a novel deep learning-based CT reconstruction model, where the low-resolution image is introduced to obtain an effective regularization term for improving the network`s robustness. Our approach involves constructing the backbone network architecture by algorithm unrolling that is realized using the deep equilibrium architecture. We theoretically discuss the convergence of the proposed low-resolution prior equilibrium model and provide the conditions to guarantee convergence. Experimental results on both sparse-view and limited-angle reconstruction problems are provided, demonstrating that our end-to-end low-resolution prior equilibrium model outperforms other state-of-the-art methods in terms of noise reduction, contrast-to-noise ratio, and preservation of edge details.

Low-resolution Prior Equilibrium Network for CT Reconstruction

TL;DR

This work tackles ill-posed CT reconstruction under sparse-view and limited-angle data by introducing a low-resolution image prior integrated into a deep equilibrium unrolled framework. The proposed LRPE network uses a shared-weight, multi-stage gradient-descent scheme with two CNNs learning data fidelity and regularization gradients, and it provides convergence guarantees under Lipschitz and step-size conditions. Empirical results on synthetic and clinical datasets demonstrate superior noise suppression, contrast-to-noise ratio, and edge preservation compared with state-of-the-art methods, highlighting practical gains for dose-reduced CT. The approach reduces reliance on re-scans and offers a theoretically grounded, end-to-end reconstruction paradigm suitable for challenging incomplete-data scenarios in CT imaging.

Abstract

The unrolling method has been investigated for learning variational models in X-ray computed tomography. However, it has been observed that directly unrolling the regularization model through gradient descent does not produce satisfactory results. In this paper, we present a novel deep learning-based CT reconstruction model, where the low-resolution image is introduced to obtain an effective regularization term for improving the network`s robustness. Our approach involves constructing the backbone network architecture by algorithm unrolling that is realized using the deep equilibrium architecture. We theoretically discuss the convergence of the proposed low-resolution prior equilibrium model and provide the conditions to guarantee convergence. Experimental results on both sparse-view and limited-angle reconstruction problems are provided, demonstrating that our end-to-end low-resolution prior equilibrium model outperforms other state-of-the-art methods in terms of noise reduction, contrast-to-noise ratio, and preservation of edge details.
Paper Structure (18 sections, 2 theorems, 26 equations, 12 figures, 6 tables, 1 algorithm)

This paper contains 18 sections, 2 theorems, 26 equations, 12 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Convergence studies of learning-based methods
  3. Low-resolution Prior for CT reconstruction
  4. Our contributions
  5. Our Low-resolution Prior Reconstruction Model
  6. Algorithm Implementation
  7. Network architecture
  8. Network optimization
  9. Convergence Analysis
  10. Numerical Results
  11. Comparison algorithms
  12. Datasets and settings
  13. Test Settings and Parameter Choice
  14. Empirical data fidelity or learned data fidelity?
  15. Experiments on the sparse-view reconstruction
  16. ...and 3 more sections

Key Result

Theorem 1

(Convergence of LRPE with empirical data fidelity). Assume that the observed data is without noise, there is $\mathcal{S}=\frac{1}{2}||\bm{Au}-\bm{b}||_2^{2}$ in lowlevel. Let $L=\lambda_{\max}\left(\bm{A}^{\top} \bm{A}\right)$ and $\mu=\lambda_{\min }\left(\bm{A}^{\top} \bm{A}\right)$, where $\lamb for all $\bm{u}, \bm{u}^{\prime} \in \mathbb{R}^{N^{2}}$. The coefficient $\gamma$ is less than 1 i

Figures (12)

  • Figure 1: Depiction of a typical bilevel problem for image reconstruction. The left box represents the training process, which includes an upper-level loss and a lower-level cost function. During training, the objective is to minimize the upper-level loss function. Once the parameters $\bm{\theta}$ are learned, it is employed in the same image reconstruction task, as depicted in the right box.
  • Figure 2: Depiction of a typical unrolling method using the gradient decent.
  • Figure 3: Depiction of a typical deep equilibrium method using the gradient decent.
  • Figure 4: Illustration of the virtual CT scanning system, where (a) a fine CT scanning system; (b) a coarse CT scanning system.
  • Figure 5: The network structure of our LRPE: Low-Resolution Prior Equilibrium network.
  • ...and 7 more figures

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof