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Volume 6, Issue 2
$ℓ_1$DecNet+: A New Architecture Framework by $ℓ_1$ Decomposition and Iteration Unfolding for Sparse Feature Segmentation

Yumeng Ren, Yiming Gao, Xue-Cheng Tai & Chunlin Wu

CSIAM Trans. Appl. Math., 6 (2025), pp. 250-271.

Published online: 2025-05

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  • Abstract

$ℓ_1$ based sparse regularization plays a central role in compressive sensing and image processing. In this paper, we propose $ℓ_1$DecNet, 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. $ℓ_1$DecNet 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 $ℓ_1$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 mathematical image 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: retinal vessel 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.

  • AMS Subject Headings

94A08, 68T01, 68U10

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COPYRIGHT: © Global Science Press

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@Article{CSIAM-AM-6-250, author = {Ren , YumengGao , YimingTai , Xue-Cheng and Wu , Chunlin}, title = {$ℓ_1$DecNet+: A New Architecture Framework by $ℓ_1$ Decomposition and Iteration Unfolding for Sparse Feature Segmentation}, journal = {CSIAM Transactions on Applied Mathematics}, year = {2025}, volume = {6}, number = {2}, pages = {250--271}, abstract = {

$ℓ_1$ based sparse regularization plays a central role in compressive sensing and image processing. In this paper, we propose $ℓ_1$DecNet, 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. $ℓ_1$DecNet 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 $ℓ_1$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 mathematical image 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: retinal vessel 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.

}, issn = {2708-0579}, doi = {https://doi.org/10.4208/csiam-am.SO-2024-0033}, url = {http://global-sci.org/intro/article_detail/csiam-am/24086.html} }
TY - JOUR T1 - $ℓ_1$DecNet+: A New Architecture Framework by $ℓ_1$ Decomposition and Iteration Unfolding for Sparse Feature Segmentation AU - Ren , Yumeng AU - Gao , Yiming AU - Tai , Xue-Cheng AU - Wu , Chunlin JO - CSIAM Transactions on Applied Mathematics VL - 2 SP - 250 EP - 271 PY - 2025 DA - 2025/05 SN - 6 DO - http://doi.org/10.4208/csiam-am.SO-2024-0033 UR - https://global-sci.org/intro/article_detail/csiam-am/24086.html KW - Variational model, $ℓ_1$ regularization, $ℓ_1$ decomposition, ADMM, deep unfolding, sparse feature extraction, sparse feature segmentation. AB -

$ℓ_1$ based sparse regularization plays a central role in compressive sensing and image processing. In this paper, we propose $ℓ_1$DecNet, 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. $ℓ_1$DecNet 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 $ℓ_1$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 mathematical image 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: retinal vessel 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.

Ren , YumengGao , YimingTai , Xue-Cheng and Wu , Chunlin. (2025). $ℓ_1$DecNet+: A New Architecture Framework by $ℓ_1$ Decomposition and Iteration Unfolding for Sparse Feature Segmentation. CSIAM Transactions on Applied Mathematics. 6 (2). 250-271. doi:10.4208/csiam-am.SO-2024-0033
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