A preprint reviewing learned image-reconstruction methods reports a higher score in one denoising test when the smoothing strength was allowed to change less smoothly across the image. The reconstruction using the low-regularity map reached a peak signal-to-noise ratio, or PSNR, of 27.92, compared with 27.42 for a high-regularity map. In paired MRI examples, however, swapping maps between inputs with different noise or sampling patterns produced broadly similar image-quality scores. The examples report image-quality metrics rather than human or clinical outcomes.
The maps behind the reconstruction
The work examines spatially varying regularisation weights, maps that set how strongly an image reconstruction is regularised at different locations. It reviews constant, continuous and piecewise-constant weights, then illustrates them in grayscale denoising and accelerated brain MRI reconstruction. Its hybrid learned method uses a convolutional neural network to infer the map inside an unrolled reconstruction algorithm. The model is trained end to end in supervised fashion on data-ground-truth pairs with a pairwise loss; labels containing the optimal parameters are not required.
A mathematical case for rougher maps
One of the review's theoretical conclusions is that using regularisation maps that are lower-semicontinuous but not continuous does not change the well-posedness of the underlying variational problems. In ordinary terms, the mathematical reconstruction problem remains well behaved even when the weight map is not fully continuous. A cited one-dimensional weighted-TV result also links weight changes to the shape of the solution: positive point-mass changes in the derivative of the weight coincide with jumps in the solution, while negative changes coincide with no new jump and local constancy. This result describes solution structure; it is not a universal claim about image quality.
The review also discusses piecewise-constant maps built on dyadic partitions, a hierarchy that repeatedly divides regions into smaller parts. When each local weight is restricted by the reciprocal bounds [c, 1/c], with c greater than 0, the reviewed result gives an optimal weight for approximating the target and says that further refinement eventually stops improving the approximation. That result does not settle existence for the modified full-domain problem in which the dyadic weights are selected jointly.
The denoising signal
The displayed denoising comparison was an illustration. The numerical section examines the regularity and structure of the learned weights alongside reconstructed-image quality. Using total variation, or TV, a model that controls local image roughness, the low-regularity map scored PSNR 27.92, against 27.42 for the high-regularity map. That comparison favors the more flexible map, but it does not show that the same result holds across all images.
An additional test examined whether each map was tied to the particular noise in its input. One clean DIV2K image was corrupted with two Gaussian-noise instances of variance 0.04. When each map was used with its matching input, PSNR was 27.16 and 27.18, with SSIM, a structural-similarity score, of 0.77 for both. Swapping the maps between the two inputs gave PSNR 24.76 and 24.75, while SSIM was 0.54. The authors interpret this contrast as evidence that the maps adapted to the particular noise realization. The test used two noise instances, so it is a focused illustration rather than an uncertainty estimate.
MRI tells a different story
The MRI setup used a fixed acceleration factor of 4, zero-mean Gaussian noise with variance 0.0025 and Gaussian variable-density masks implemented in MRpro. The implementations were adapted and retrained for accelerated MRI using the fastMRI brain dataset. The reported MRI comparisons showed less separation between same-map and cross-map results.
When the sampling mask was held fixed and the noise changed, cross-application was described as comparable to using the matching map. In one comparison, both applications reported mean squared error, or MSE, of 0.022 and structural similarity, or SSIM, of 0.72. In the other, same-map reconstruction reported 0.022/0.72 and cross-map reconstruction 0.022/0.73 for MSE/SSIM.
With noise held fixed and the mask changed, same-map results were 0.018/0.76 and 0.019/0.74 for MSE/SSIM. Cross-map results were 0.017/0.77 and 0.018/0.76. The reported pairs again stayed close, and cross-application was not uniformly worse.
When both noise and mask changed, same-map results were 0.025/0.76 and 0.023/0.72, while cross-map results were 0.020/0.75 and 0.028/0.73. The ordering changed between the two comparisons. The authors describe the MRI maps as more regular and less overfitted to particular noise and masks than the denoising maps. They suggest that learning from adjoint data may damp fine-scale noise and leave less for the map to adapt to, but explicitly leave that explanation for further investigation.
How far the evidence goes
The contrast between denoising and MRI is the main unresolved point, but the evidence remains illustrative. The supplied review reports no total image count, training-pair composition, train/test split, repetitions, random seeds or uncertainty estimates. Network architecture, optimisation and training details are deferred to earlier publications. TV is the model used for the quantitative comparisons; TGV is reviewed as an extension but is not quantitatively shown here. The examples do not establish generalisation to unseen images, noise distributions, masks or acquisition protocols, or universal superiority over scalar or high-regularity maps.
The document is an arXiv preprint, version 1, dated 25 August 2026. It reports support for L. Calatroni's work from ERC Starting project MALIN under the European Union's Horizon Europe programme, grant 101117133, and from project A4IM under the European Partnership on Metrology.
Paper data and sources
Original title: Learning spatially varying regularisation parameters of low regularity for image reconstruction
Authors: Kostas Papafitsoros, Luca Calatroni, Andreas Kofler
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text