(untitled) (CSIAM Transactions on Applied Mathematics, 2025)
ℓ_1 based sparse regularization plays a central role in compressive sensing and image processing. In this paper, we propose ℓ_1DecNet, as an unfolded network derived from a variational decomposition model, which incorporates ℓ_1 related sparse regularizations and is solved by a non-standard scaled alternating direction method of multipliers. ℓ_1DecNet effectively separates a spatially sparse feature and a learned spatially dense feature from an input image, and thus helps the subsequent spatially sparse feature related operations. Based on this, we develop ℓ_1DecNet+, a learnable architecture framework consisting of our ℓ_1DecNet 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 mathematical image prior into feature extraction in segmentation network structures. Moreover, our ℓ_1DecNet+ framework can be easily extended to 3D case. We evaluate the effectiveness of ℓ_1DecNet+ on two commonly encountered sparse segmentation tasks: retinal vessel segmentation in medical image processing and pavement crack detection in industrial abnormality identification. Experimental results on different datasets demonstrate that, our ℓ_1DecNet+ 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.