Inverse problems span across diverse fields. In medical contexts, computed tomography (CT) plays a crucial role in reconstructing a patient's internal structure, presenting challenges due to artifacts caused by inherently ill-posed inverse problems. Previous research advanced image quality via post-processing and deep unrolling algorithms but faces challenges, such as extended convergence times with ultra-sparse data. Despite enhancements, resulting images often show significant artifacts, limiting their effectiveness for real-world diagnostic applications. We aim to explore deep second-order unrolling algorithms for solving imaging inverse problems, emphasizing their faster convergence and lower time complexity compared to common first-order methods like gradient descent. In this paper, we introduce QN-Mixer, an algorithm based on the quasi-Newton approach. We use learned parameters through the BFGS algorithm and introduce Incept-Mixer, an efficient neural architecture that serves as a non-local regularization term, capturing long-range dependencies within images. To address the computational demands typically associated with quasi-Newton algorithms that require full Hessian matrix computations, we present a memory-efficient alternative. Our approach intelligently downsamples gradient information, significantly reducing computational requirements while maintaining performance. The approach is validated through experiments on the sparse-view CT problem, involving various datasets and scanning protocols, and is compared with post-processing and deep unrolling state-of-the-art approaches. Our method outperforms existing approaches and achieves state-of-the-art performance in terms of SSIM and PSNR, all while reducing the number of unrolling iterations required.

翻译：逆问题广泛存在于多个领域。在医学背景下，计算机断层扫描（CT）在重建患者内部结构中发挥着关键作用，但由于固有不适定逆问题产生的伪影而面临挑战。先前研究通过后处理与深度展开算法提升了图像质量，但存在超稀疏数据下收敛时间延长等问题。尽管有所改进，所得图像常呈现显著伪影，限制了其在实际诊断应用中的有效性。本文旨在探索用于求解成像逆问题的深度二阶展开算法，强调其相较于梯度下降等常见一阶方法具有更快的收敛速度和更低的时间复杂度。我们提出QN-Mixer——一种基于准牛顿方法的算法。通过BFGS算法学习参数，并引入Incept-Mixer这一高效神经架构作为非局部正则化项，捕捉图像内的长程依赖关系。针对准牛顿算法通常需要完整海森矩阵计算的高计算需求，我们提出一种内存高效替代方案：智能下采样梯度信息，在保持性能的同时显著降低计算量。通过稀疏视角CT问题实验（涵盖多种数据集与扫描协议），与当前最先进的后处理及深度展开方法进行比较，验证了所提方法的有效性。本方法在SSIM和PSNR指标上超越现有方法，达到最优性能，同时减少了所需的展开迭代次数。