用于图像重建的低正则性空间变正则化参数学习
Learning spatially varying regularisation parameters of low regularity for image reconstruction
- School of Mathematical Sciences, Queen Mary University of London（伦敦玛丽女王大学数学科学学院）
- Università di Genova（热那亚大学）
- Istituto Italiano di Tecnologia（意大利技术研究院）
- Physikalisch-Technische Bundesanstalt (PTB)（德国联邦物理技术研究院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究探讨变分图像重建中空间自适应正则化权重的低正则性性质，结合深度神经网络学习此类权重，经图像去噪与MRI重建验证可提升重建效果，并指出未来研究方向。
AI中文摘要：
在本章中，我们回顾并讨论变分图像重建中所用空间自适应正则化权重函数的正则性性质。将此类权重融入经典基于模型的正则化器（如全变分（Total Variation, TV）和广义全变分（Total Generalised Variation, TGV）），可使正则化强度在图像中变化并适配局部图像内容。若估计得当，这些权重能显著提升重建中边缘与细节的保留效果。我们回顾了该主题下针对不同正则性类别的现有理论文献，包括常数函数、连续函数及分段常数函数。近期结合基于模型的正则化与深度神经网络以学习高自适应正则化权重的混合图像重建方法，为我们的讨论提供了动机。我们尤其从理论与实践视角探讨了这些权重的结构性质如何影响重建结果。通过图像去噪与磁共振成像（Magnetic Resonance Imaging, MRI）重建中的代表性示例，我们证明学习得到的权重常具有低正则性，不仅能适配图像结构，还能适配特定的噪声实现。最后，我们强调了该主题未来研究的若干方向。
英文摘要：
In this chapter, we review and discuss the regularity properties of spatially adaptive regularisation weight functions used in variational image reconstruction. Incorporating such weights into classical model-based regularisers, such as Total Variation (TV) and Total Generalised Variation (TGV), allows the regularisation strength to vary across the image and adapt to local image content. When appropriately estimated, these weights can thus significantly improve edge and detail preservation in the reconstructions. We review the existing theoretical literature on this topic for different regularity classes, including constant, continuous, and piecewise constant functions. Our discussion is motivated by recent work on hybrid image reconstruction methods that combine model-based regularisation with deep neural networks to learn highly adaptive regularisation weights. In particular, we discuss how the structural properties of these weights influence the reconstruction from both theoretical and practical perspectives. Through representative examples in image denoising and magnetic resonance imaging (MRI) reconstruction, we demonstrate that the learned weights are often of low regularity and can adapt not only to the image structure but also to the specific noise realisation. We conclude by highlighting several directions for future research on this topic.
