Learning spatially adaptive sparsity level maps for arbitrary convolutional dictionaries

While highly accelerated non-Cartesian acquisition protocols significantly reduce scan time, they often entail long reconstruction delays. Deep learning based reconstruction methods can alleviate this, but often lack stability and robustness to distribution shifts. As an alternative, we train a rotation invariant weakly convex ridge regularizer (WCRR). The resulting variational reconstruction approach is benchmarked against state of the art methods on retrospectively simulated data and (out of distribution) on prospective GoLF SPARKLING and CAIPIRINHA acquisitions. Our approach consistently outperforms widely used baselines and achieves performance comparable to Plug and Play reconstruction with a state of the art 3D DRUNet denoiser, while offering substantially improved computational efficiency and robustness to acquisition changes. In summary, WCRR unifies the strengths of principled variational methods and modern deep learning based approaches.

Sparse dictionary coding represents signals as linear combinations of a few dictionary atoms. It has been applied to images, time series, graph signals and multi-way spatio-temporal data by jointly employing temporal and spatial dictionaries. Data-agnostic analytical dictionaries, such as the discrete Fourier transform, wavelets and graph Fourier, have seen wide adoption due to efficient implementations and good practical performance. On the other hand, dictionaries learned from data offer sparser and more accurate solutions but require learning of both the dictionaries and the coding coefficients. This becomes especially challenging for multi-dictionary scenarios since encoding coefficients correspond to all atom combinations from the dictionaries. To address this challenge, we propose a low-rank coding model for 2-dictionary scenarios and study its data complexity. Namely, we establish upper and lower bounds on the number of samples needed to learn dictionaries that generalize to unseen samples from the same distribution. We propose an alternating convex optimization solution, called AODL, which employs alternating optimization between the sparse coding matrices and the learned dictionaries. We demonstrate its quality for data reconstruction and missing value imputation in both synthetic and real-world datasets. For a fixed reconstruction quality, AODL learns up to $90\%$ sparser solutions compared to non-low-rank and analytical (fixed) dictionary baselines. In addition, the learned dictionaries reveal interpretable insights into patterns from training samples.

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.