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This chapter reviews the use of spatially adaptive regularisation weight functions in variational image reconstruction, focusing on their ability to enhance edge and detail preservation by varying regularisation strength according to local image content. By integrating these adaptive weights into classical regularisers like Total Variation (TV) and Total Generalised Variation (TGV), the authors demonstrate significant improvements in reconstruction quality, particularly in image denoising and MRI applications. The discussion is grounded in theoretical insights and practical examples, emphasizing the low regularity of learned weights and their adaptability to both image structure and noise characteristics.
Spatially adaptive regularisation can drastically enhance image reconstruction quality by tailoring the regularisation strength to local content and noise conditions.
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.