This paper focuses on the development of a space-variant regularization model for solving an under-determined linear inverse problem. The case study is a medical image reconstruction from few-view tomographic noisy data. The primary objective of the proposed optimization model is to achieve a good balance between denoising and the preservation of fine details and edges, overcoming the performance of the popular and largely used Total Variation (TV) regularization through the application of appropriate pixel-dependent weights. The proposed strategy leverages the role of gradient approximations for the computation of the space-variant TV weights. For this reason, a convolutional neural network is designed, to approximate both the ground truth image and its gradient using an elastic loss function in its training. Additionally, the paper provides a theoretical analysis of the proposed model, showing the uniqueness of its solution, and illustrates a Chambolle-Pock algorithm tailored to address the specific problem at hand. This comprehensive framework integrates innovative regularization techniques with advanced neural network capabilities, demonstrating promising results in achieving high-quality reconstructions from low-sampled tomographic data.

翻译：本文致力于开发一种空间变正则化模型，用于求解欠定线性逆问题。以少视角含噪断层数据的医学图像重建为案例研究，所提优化模型的主要目标是在去噪与精细细节及边缘保持之间实现良好平衡，通过应用适当的像素依赖权重，超越广泛使用的全变分（TV）正则化方法的性能。所提策略利用梯度近似来计算空间变TV权重。为此，本文设计了一个卷积神经网络，在训练过程中采用弹性损失函数来逼近真实图像及其梯度。此外，本文对所提模型进行了理论分析，证明了其解的唯一性，并阐述了一种针对该特定问题定制的Chambolle-Pock算法。这一综合框架将创新的正则化技术与先进的神经网络能力相结合，在从低采样断层数据实现高质量重建方面展现出令人瞩目的效果。