# Learning spatially varying regularisation parameters of low regularity for image reconstruction

arXiv: https://arxiv.org/abs/2608.25127 · HTML: https://mlsanity.com/paper/2608.25127/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 28 of its 62 references that are in the map.
- First posted on arXiv 25 Aug 2026.

## Builds on (most cited first, 28 of 28)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [fastMRI: An Open Dataset and Benchmarks for Accelerated MRI](https://mlsanity.com/paper/1811.08839/) (2018)
- [Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal and Image Processing](https://mlsanity.com/paper/1912.10557/) (2019)
- [On instabilities of deep learning in image reconstruction - Does AI come at a cost?](https://mlsanity.com/paper/1902.05300/) (2019)
- [Learned reconstruction methods with convergence guarantees](https://mlsanity.com/paper/2206.05431/) (2022)
- [Bilevel methods for image reconstruction](https://mlsanity.com/paper/2109.09610/) (2021)
- [Learning Regularization Parameters of Inverse Problems via Deep Neural Networks](https://mlsanity.com/paper/2104.06594/) (2021)
- [Bilevel approaches for learning of variational imaging models](https://mlsanity.com/paper/1505.02120/) (2015)
- [Learning Regularization Functionals for Inverse Problems: A Comparative Study](https://mlsanity.com/paper/2510.01755/) (2025)
- [Higher-order total variation approaches and generalisations](https://mlsanity.com/paper/1912.01587/) (2019)
- [Neural-network-based regularization methods for inverse problems in imaging](https://mlsanity.com/paper/2312.14849/) (2023)
- [Analytical aspects of spatially adapted total variation regularisation](https://mlsanity.com/paper/1609.01074/) (2016)
- [Dualization and Automatic Distributed Parameter Selection of Total Generalized Variation via Bilevel Optimization](https://mlsanity.com/paper/2002.05614/) (2020)
- [Whiteness-based bilevel learning of regularization parameters in imaging](https://mlsanity.com/paper/2403.07026/) (2024)
- [Deep unrolling for learning optimal spatially varying regularisation parameters for Total Generalised Variation](https://mlsanity.com/paper/2502.16532/) (2025)
- [Asymptotic behaviour of total generalised variation](https://mlsanity.com/paper/1502.06933/) (2015)
- [A function space framework for structural total variation regularization with applications in inverse problems](https://mlsanity.com/paper/1710.01527/) (2017)
- [Dyadic partition-based training schemes for TV/TGV denoising](https://mlsanity.com/paper/2305.07150/) (2023)
- [Inclusion and estimates for the jumps of minimizers in variational denoising](https://mlsanity.com/paper/2312.01900/) (2023)
- [Whiteness-based bilevel estimation of weighted TV parameter maps for image denoising](https://mlsanity.com/paper/2503.07814/) (2025)
- [Learning Spatially Adaptive ℓ1-Norms Weights for Convolutional Synthesis Regularization](https://mlsanity.com/paper/2503.09483/) (2025)
- [MRpro: open framework for model-based, learned, and quantitative MR imaging](https://mlsanity.com/paper/2507.23129/) (2025)
- [A study of the one dimensional total generalised variation regularisation problem](https://mlsanity.com/paper/1309.5900/) (2013)
- [Discontinuities of the minimizers of the weighted or anisotropic total variation for image reconstruction](https://mlsanity.com/paper/1402.0026/) (2014)
- [The jump set under geometric regularisation. Part 1: Basic technique and first-order denoising](https://mlsanity.com/paper/1407.1531/) (2014)
- [The jump set under geometric regularisation. Part 2: Higher-order approaches](https://mlsanity.com/paper/1407.2334/) (2014)
- [Monotonicity of the jump set and jump amplitudes in one-dimensional TV denoising](https://mlsanity.com/paper/2502.11714/) (2025)
- [Learning spatially adaptive sparsity level maps for arbitrary convolutional dictionaries](https://mlsanity.com/paper/2602.21707/) (2026)

## Abstract (arXiv)

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.