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.

该研究聚焦于开发一种用于解决欠定线性反问题的空间变异正则化模型。研究案例为从少视角层析噪声数据中重建医学图像。通过应用适当的像素相关权重，该优化模型的主要目标是在去噪和保留细节和边缘之间取得良好平衡，克服了广泛使用的总变差（TV）正则化方法的性能。提出的策略利用梯度逼近来计算空间变异 TV 权重。为此，设计了一个卷积神经网络，使用其训练中的弹性损失函数来逼近真值图像及其梯度。此外，本文对所提出的模型进行了理论分析，展示了其解的唯一性，并展示了针对特定问题的 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，展示出在从低采样层析数据中实现高质量重建方面的有 promising 结果。