$ℓ_1$DecNet+: A New Architecture Framework by $ℓ_1$ Decomposition and Iteration Unfolding for Sparse Feature Segmentation (CSIAM Transactions on Applied Mathematics, 2025)
$ℓ_1$ based sparse regularization plays a central role in compressive sensingand image processing. In this paper, we propose $ℓ_1$DecNet, as an unfolded networkderived from a variational decomposition model, which incorporates $ℓ_1$ related sparseregularizations and is solved by a non-standard scaled alternating direction methodof multipliers. $ℓ_1$DecNet effectively separates a spatially sparse feature and a learnedspatially dense feature from an input image, and thus helps the subsequent spatiallysparse feature related operations. Based on this, we develop $ℓ_1$DecNet+, a learnable architecture framework consisting of our $ℓ_1$DecNet and a segmentation module which operates over extracted sparse features instead of original images. This architecture combines well the benefits of mathematical modeling and data-driven approaches. To our best knowledge, this is the first study to incorporate mathematicalimage prior into feature extraction in segmentation network structures. Moreover, our $ℓ_1$DecNet+ framework can be easily extended to 3D case. We evaluate the effectiveness of $ℓ_1$DecNet+ on two commonly encountered sparse segmentation tasks: retinalvessel segmentation in medical image processing and pavement crack detection in industrial abnormality identification. Experimental results on different datasets demonstrate that, our $ℓ_1$DecNet+ architecture with various lightweight segmentation modules can achieve equal or better performance than their enlarged versions respectively.This leads to especially practical advantages on resource-limited devices.