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Deep Inertia $L_p$ Half-Quadratic Splitting Unrolling Network for Sparse View CT Reconstruction

Yu Guo, Caiying Wu, Yaxin Li, Qiyu Jin, Tieyong Zeng

TL;DR

This work tackles sparse-view CT reconstruction as an ill-posed inverse problem by marrying $L_p$ regularization with wavelet sparsity in a half-quadratic splitting framework, augmented with inertial steps to accelerate convergence. The authors develop IHQS$_p$-CG, solving subproblems via a proximal operator and conjugate gradients, and prove global convergence under practical inertia bounds; they further embed a deep learning initializer to accelerate CG, yielding IHQS$_p$-Net with convergence guarantees. Extensive experiments on NIH-AAPM-Mayo and Covid-19 datasets show that IHQS$_p$-CG and its deep-unrolled variant outperform traditional and deep-learning baselines, especially under extreme sparse-view and noisy conditions. The work provides both strong empirical gains and theoretical guarantees, enabling robust, fast, and scalable sparse-view CT reconstruction in clinical settings.

Abstract

Sparse view computed tomography (CT) reconstruction poses a challenging ill-posed inverse problem, necessitating effective regularization techniques. In this letter, we employ $L_p$-norm ($0<p<1$) regularization to induce sparsity and introduce inertial steps, leading to the development of the inertial $L_p$-norm half-quadratic splitting algorithm. We rigorously prove the convergence of this algorithm. Furthermore, we leverage deep learning to initialize the conjugate gradient method, resulting in a deep unrolling network with theoretical guarantees. Our extensive numerical experiments demonstrate that our proposed algorithm surpasses existing methods, particularly excelling in fewer scanned views and complex noise conditions.

Deep Inertia $L_p$ Half-Quadratic Splitting Unrolling Network for Sparse View CT Reconstruction

TL;DR

This work tackles sparse-view CT reconstruction as an ill-posed inverse problem by marrying regularization with wavelet sparsity in a half-quadratic splitting framework, augmented with inertial steps to accelerate convergence. The authors develop IHQS-CG, solving subproblems via a proximal operator and conjugate gradients, and prove global convergence under practical inertia bounds; they further embed a deep learning initializer to accelerate CG, yielding IHQS-Net with convergence guarantees. Extensive experiments on NIH-AAPM-Mayo and Covid-19 datasets show that IHQS-CG and its deep-unrolled variant outperform traditional and deep-learning baselines, especially under extreme sparse-view and noisy conditions. The work provides both strong empirical gains and theoretical guarantees, enabling robust, fast, and scalable sparse-view CT reconstruction in clinical settings.

Abstract

Sparse view computed tomography (CT) reconstruction poses a challenging ill-posed inverse problem, necessitating effective regularization techniques. In this letter, we employ -norm () regularization to induce sparsity and introduce inertial steps, leading to the development of the inertial -norm half-quadratic splitting algorithm. We rigorously prove the convergence of this algorithm. Furthermore, we leverage deep learning to initialize the conjugate gradient method, resulting in a deep unrolling network with theoretical guarantees. Our extensive numerical experiments demonstrate that our proposed algorithm surpasses existing methods, particularly excelling in fewer scanned views and complex noise conditions.
Paper Structure (14 sections, 4 theorems, 20 equations, 5 figures, 2 tables, 1 algorithm)

This paper contains 14 sections, 4 theorems, 20 equations, 5 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Inertial HQS$_p$-CG Algorithm and Convergence Analysis
  3. Inertial HQS$_p$-CG Algorithm
  4. Convergence analysis
  5. Conjugate gradient initialization and deep unrolling networks
  6. Numerical experiments
  7. Datasets and experimental settings
  8. Comparison of quantitative and qualitative results
  9. Ablation experiment
  10. Conclusion
  11. Convergence analysis
  12. Supplementary experimental data
  13. Comparison of MAE and RMSE
  14. Supplementary visualisation results

Key Result

Lemma 2.1

Suppose that the sequences $\left\{v^{k}\right\}$ and $\left\{\bar{v}^{k}\right\}$ generated via Algorithm alg1, $0<\beta<\frac{\sqrt{5}-1}{2}$, then the sets $\left\{v^{k}\right\}$ and $\left\{\bar{v}^{k}\right\}$ are bounded.

Figures (5)

  • Figure 1: IHQS$_p$-Net algorithm framework diagram.
  • Figure 2: 120 sparse views CT image reconstruction results and magnified ROIs from Covid-19. The sinogram is corrupted by $\sigma=0.3$ and $I_0=5\times 10^{5}$.
  • Figure 3: Robustness analysis of all compared algorithms.
  • Figure 4: (a) Regarding the ablation experiment of $p$-value. (b) The impact of inertia step on algorithm convergence.
  • Figure 5: 60 sparse views CT image reconstruction results and magnified ROIs from AAPM. The sinogram is corrupted by $\sigma=0.3$.

Theorems & Definitions (6)

  • Lemma 2.1
  • Theorem 2.1
  • Lemma 6.1
  • proof : Proof
  • Theorem 6.1
  • proof : Proof