Learning spatially adaptive sparsity level maps for arbitrary convolutional dictionaries

Joshua Schulz; David Schote; Christoph Kolbitsch; Kostas Papafitsoros; Andreas Kofler

eess.IVFeb 25, 2026

Learning spatially adaptive sparsity level maps for arbitrary convolutional dictionaries

Authors: Joshua Schulz, David Schote, Christoph Kolbitsch, Kostas Papafitsoros, Andreas Kofler

Organizations: Physikalisch-Technische Bundesanstalt (PTB), Braunschweig and Berlin, Germany · School of Mathematical Sciences, Queen Mary University of London, UK

Abstract

State-of-the-art learned reconstruction methods often rely on black-box modules that, despite their strong performance, raise questions about their interpretability and robustness. Here, we build on a recently proposed image reconstruction method, which is based on embedding data-driven information into a model-based convolutional dictionary regularization via neural network-inferred spatially adaptive sparsity level maps. By means of improved network design and dedicated training strategies, we extend the method to achieve filter-permutation invariance as well as the possibility to change the convolutional dictionary at inference time. We apply our method to low-field MRI and compare it to several other recent deep learning-based methods, also on in vivo data, where the benefit of using a different dictionary is demonstrated. We further assess the method's robustness when tested on in- and out-of-distribution data. When tested on the latter, the proposed method suffers less from the data distribution shift compared to the other learned methods, which we attribute to its reduced reliance on training data due to its underlying model-based reconstruction component.

Figures & tables

Figure 1: The reconstruction pipeline of the CDL- $\boldsymbol{\Lambda}$ method and the three different versions of $\mathrm{NET}_{\Theta}$ ( V1 – V3 ) (orange blocks) investigated in this study. First, a learned high-pass filtering step solving ( PH ) is performed (green block) . Then, $\mathrm{NET}_{\Theta}$ estimates spatially varying sparsity level maps from an input (blue block) . Last, a FISTA algorithm is unrolled to obtain an approximate solution of ( PR ) (ocher block) . An estimate of the solution is obtained by adding the low-frequency component that was extracted in the first block. In contrast to V1 and V2, the proposed improved V3 is permutation invariant and allows for the use of dictionaries with a different number of filters $K$ .

		V1		V2		V3
Base	SSIM	$0.85$ $\pm$	$0.05$	$0.82$ $\pm$	$0.05$	$0.82$ $\pm$	$0.05$
	MSE	$0.11$ $\pm$	$0.04$	$0.14$ $\pm$	$0.06$	$0.14$ $\pm$	$0.06$
$\pi_{1}$	$\Delta$ SSIM	$-0.06$ $\pm$	$0.01$	$-0.08$ $\pm$	$0.03$	0
	$\Delta$ MSE	$0.12$ $\pm$	$0.04$	$0.14$ $\pm$	$0.10$	0
$\pi_{2}$	$\Delta$ SSIM	$-0.05$ $\pm$	$0.01$	$-0.06$ $\pm$	$0.02$	0
	$\Delta$ MSE	$0.12$ $\pm$	$0.04$	$0.12$ $\pm$	$0.08$	0

Table 1: Results for

\mathrm{NET}_{\Theta}

as in ( V1 ), ( V2 ) and ( V3 ) for three filter permutations

\pi_{1},\pi_{2},\pi_{3}

, shown for

K=32,11\times 11

-kernels. The improved V3 is filter permutation invariant by construction. Note that differences between the baselines of V1, V2, and V3 can be attributed to the deviations of the three versions in terms of the number of trainable parameters, which can be easily addressed by increasing them.

Figure 2: MSE obtained by CDL-

\boldsymbol{\Lambda}

with

\mathrm{NET}_{\Theta}

V3 over the brain MR test set for 16 different choices of dictionaries. Note that, for training, we did not use the

K=128

-dictionaries.

Figure 3: Eight out of $K=32$ $\boldsymbol{\Lambda}$ -maps with the largest variance, seen as indicative of the filter’s importance in the image representation, for each $\mathrm{NET}_{\Theta}$ V1, V2, V3, and for filter size $k_{f}\times k_{f}=11\times 11$ .

Panel of Figure 4 — Figure 4: A comparison of MoDL [ 1 ] , E2E VarNet [ 19 ] , SRDenseNet [ 7 ] and CDL- $\boldsymbol{\Lambda}$ ( $K=64,11\times 11$ -kernels) on in-distribution brain MR data (left, $\sigma^{2}=0.3$ ) and out-of-distribution knee MR data (right, $\sigma^{2}=0.2$ ).

Figure 5: SSIM scores on the brain and knee MR test datasets for noise variances

\sigma^{2}=0.2

and

\sigma^{2}=0.3

. The dashed lines denote the median SSIM value of the respective methods on the brain dataset. Note how for MoDL, E2E-VarNet, and SRDenseNet, the median brain-SSIM lies outside the interquartile range of the corresponding knee-SSIM distributions.

Figure 6: In vivo T2-weighted brain MR image: MoDL, E2E VarNet and SRDenseNet, and the CDL- $\boldsymbol{\Lambda}$ with the improved $\mathrm{NET}_{\Theta}$ ( V3 ), once with $K=16$ filters of size $k_{f}\times k_{f}=9\times 9$ , and once with $K=128$ , $k_{f}\times k_{f}=11\times 11$ . Note how the use of a larger dictionary at inference, which was not used in training, leads to a sharper result.

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