Fig. 1![\"Fig.](\"https:\/\/www.newsbeep.com\/il\/wp-content\/uploads\/2026\/10\/41598_2026_54738_Fig1_HTML.png\")
<\/p>\n
RegNet-X network architecture.<\/p>\n
RegNet-X architecture and steel-specific adaptations<\/p>\n
The RegNet-X backbone transforms the input image \\(\\textbf{I}_t\\) through hierarchical stages to extract multi-scale features. The overall feature extraction function \\(\\mathcal {R}\\) is defined as a composite of stage transformations:<\/p>\n
$$\\begin{aligned} \\mathcal {R}(\\textbf{I}_t) = \\mathcal {H}_S \\circ \\mathcal {H}_{S-1} \\circ \\cdots \\circ \\mathcal {H}_1(\\textbf{F}_0), \\quad \\textbf{F}_s = \\mathcal {H}_s(\\textbf{F}_{s-1}), \\quad s = 1, 2, \\ldots , S \\end{aligned}$$<\/p>\n
\n (8)\n <\/p>\n
where each stage consists of \\(d_s\\) X-blocks. The X-block transformation integrates group convolution to reduce computational costs by a factor of g while maintaining the residual flow:<\/p>\n
$$\\begin{aligned} \\mathcal {B}(\\textbf{X}) = \\textbf{X} + \\mathcal {C}_{1\\times 1}^{(out)}\\left( \\sigma \\left( \\mathcal{B}\\mathcal{N}\\left( \\mathcal {C}_{3\\times 3}^{(group)}\\left( \\sigma \\left( \\mathcal{B}\\mathcal{N}\\left( \\mathcal {C}_{1\\times 1}^{(in)}(\\textbf{X})\\right) \\right) \\right) \\right) \\right) \\right) \\end{aligned}$$<\/p>\n
\n (9)\n <\/p>\n
To adapt to the subtle boundaries of steel defects, an edge enhancement module is integrated after the stem layer. This module computes horizontal and vertical gradient maps (\\(\\textbf{G}_x, \\textbf{G}_y\\)) via Sobel operators to amplify edge strength:<\/p>\n
$$\\begin{aligned} \\textbf{E} = \\textbf{F}_0 \\odot \\sigma _{sigmoid}\\left( \\mathcal {C}_{1\\times 1}\\left( \\sqrt{\\textbf{G}_x^2 + \\textbf{G}_y^2}\\right) \\right) + \\textbf{F}_0 \\end{aligned}$$<\/p>\n
\n (10)\n <\/p>\n
The resulting enhanced features are fused through a Feature Pyramid Network (FPN) to handle the significant scale variation of defects:<\/p>\n
$$\\begin{aligned} \\textbf{P}_l = \\mathcal {U}_{2\\times }(\\textbf{P}_{l+1}) + \\mathcal {C}_{1\\times 1}(\\textbf{F}_l), \\quad l = S-1, S-2, \\ldots , 1 \\end{aligned}$$<\/p>\n
\n (11)\n <\/p>\n
Theorem 1<\/p>\n
(Feature Representation Sufficiency) For the steel defect detection task with K defect categories, when the RegNet-X network satisfies the capacity constraint \\(w_s \\ge w_{min}\\) for all stages \\(s \\in \\{1, \\ldots , S\\}\\) and total depth \\(D = \\sum _{s=1}^{S} d_s \\ge D_{min}\\), there exists an optimal parameter configuration \\(\\theta ^*\\) such that the feature extraction function \\(\\mathcal {R}_{\\theta ^*}\\) guarantees:<\/p>\n
$$\\begin{aligned} \\max _{\\theta } \\mathcal {I}(\\textbf{D}_t; \\mathcal {R}_{\\theta }(\\textbf{I}_t)) \\ge (1-\\epsilon ) \\cdot \\mathcal {I}(\\textbf{D}_t; \\textbf{I}_t) \\end{aligned}$$<\/p>\n
\n (12)\n <\/p>\n
where \\(\\epsilon \\in (0, 1)\\) is the information loss upper bound dependent on network capacity, and \\(\\mathcal {I}(\\textbf{D}_t; \\textbf{I}_t)\\) represents the maximum achievable mutual information between the original image and defect annotations.<\/p>\n
This theorem establishes the theoretical basis for network capacity selection in ESAD-Net. The design parameters \\(w_0 = 24\\), \\(w_a = 36\\), and stage depths [1,\u00a02,\u00a07,\u00a012] are configured to satisfy the sufficiency condition while minimizing computational overhead. The theorem directly guides the architecture design by ensuring that the selected configuration preserves sufficient discriminative information for the \\(K = 6\\) defect classification task. The complete proof based on information-theoretic analysis is provided in Appendix A.<\/p>\n
Corollary 1<\/p>\n
(Computational Efficiency Guarantee) Under the conditions of Theorem 1<\/a>, the computational complexity of RegNet-X satisfies the real-time constraint:<\/p>\n $$\\begin{aligned} \\text {FLOPs}_{RegNet-X} = \\sum _{s=1}^{S} d_s \\cdot \\left( 2H_s W_s w_{s-1} w_s + \\frac{9 H_s W_s w_s^2}{g} \\right) \\le \\text {FLOPs}_{max} \\end{aligned}$$<\/p>\n \n (13)\n <\/p>\n where the two terms correspond to \\(1\\times 1\\) convolutions and \\(3\\times 3\\) group convolution respectively, g is the group number, and \\(\\text {FLOPs}_{max}\\) is determined by the real-time constraint \\(T_{process} \\le T_{max}\\).<\/p>\n This corollary provides an explicit computational complexity formula that guides the trade-off between detection accuracy and inference speed. The group convolution with \\(g = 8\\) reduces the \\(3\\times 3\\) convolution cost by a factor of 8, enabling the real-time target. The derivation details are provided in Appendix A.<\/p>\n Adaptive detection for steel defectsAdaptive strategy for defect imbalance<\/p>\n Steel defects exhibit extreme geometric diversity, ranging from elongated scratches to compact pitted surfaces. Static IoU thresholding often fails to provide sufficient training samples for these morphologically diverse targets. We employ the Adaptive Training Sample Selection (ATSS) mechanism, which dynamically adjusts the positive sample selection threshold based on the statistical moments of anchor-target overlaps. As illustrated in Fig. 2<\/a>, this ensures that each defect category, regardless of scale or aspect ratio, obtains appropriate supervision during the optimization process.<\/p>\n Fig. 2 To address the positive-negative sample imbalance and the morphological diversity of steel defects, ESAD-Net incorporates the Adaptive Training Sample Selection (ATSS) algorithm31<\/a>. Unlike the anchor-free design of YOLOv8n, we adopt an anchor-based paradigm to leverage predefined geometric priors, which facilitate a more systematic analysis of spatial relationships across multiple scales\u2013a critical requirement for detecting defects ranging from high-aspect-ratio scratches to compact inclusions. The detection architecture is illustrated in Fig. 2<\/a>.<\/p>\n The ATSS head processes the multi-scale feature pyramid \\(\\{\\textbf{P}_l\\}_{l=1}^{5}\\) to generate category, regression, and centerness predictions. For each feature level l, a set of \\(R = 9\\) anchors is generated at each spatial location (i,\u00a0j):<\/p>\n $$\\begin{aligned} \\{(\\textbf{O}_{cls}^{(l)}, \\textbf{O}_{reg}^{(l)}, \\textbf{O}_{cen}^{(l)})\\} = \\{(\\mathcal {F}_{cls}(\\textbf{P}_l), \\mathcal {F}_{reg}(\\textbf{P}_l), \\mathcal {F}_{cen}(\\textbf{P}_l))\\}, \\quad l = 1, 2, \\ldots , 5 \\end{aligned}$$<\/p>\n \n (14)\n <\/p>\n where \\(\\mathcal {F}_{cls}\\), \\(\\mathcal {F}_{reg}\\), and \\(\\mathcal {F}_{cen}\\) represent the prediction branches implemented via \\(3 \\times 3\\) convolutional stacks with \\(C_{fpn} = 256\\) channels. The core mechanism defines a candidate anchor set \\(\\mathcal {A}_{cand}^{(n)}\\) for each ground truth defect \\(g^{(n)}\\) by selecting \\(k_{top}=9\\) anchors based on the \\(L_2\\) distance between anchor centers and the ground truth center at each pyramid level:<\/p>\n $$\\begin{aligned} \\mathcal {A}_{cand}^{(n)} = \\bigcup _{l=1}^{5} \\text {TopK}_{dist}\\left( \\{a_{i,j}^{(l)}\\}, g^{(n)}, k_{top}\\right) \\end{aligned}$$<\/p>\n \n (15)\n <\/p>\n The adaptive selection threshold \\(\\tau ^{(n)}\\) is then derived from the statistical moments of the IoU distribution within the candidate set:<\/p>\n $$\\begin{aligned} \\mu _{IoU}^{(n)} = \\frac{1}{|\\mathcal {A}_{cand}^{(n)}|} \\sum _{a \\in \\mathcal {A}_{cand}^{(n)}} \\text {IoU}(a, g^{(n)}), \\quad \\sigma _{IoU}^{(n)} = \\sqrt{\\frac{1}{|\\mathcal {A}_{cand}^{(n)}|} \\sum _{a \\in \\mathcal {A}_{cand}^{(n)}} \\left( \\text {IoU}(a, g^{(n)}) – \\mu _{IoU}^{(n)}\\right) ^2} \\end{aligned}$$<\/p>\n \n (16)\n <\/p>\n $$\\begin{aligned} \\tau ^{(n)} = \\mu _{IoU}^{(n)} + \\sigma _{IoU}^{(n)} \\end{aligned}$$<\/p>\n \n (17)\n <\/p>\n The threshold \\(\\tau ^{(n)}\\) dynamically adjusts the selection criterion: ground truth boxes with high-quality anchor matches receive higher thresholds, while challenging samples receive lower thresholds. An anchor a is assigned as a positive sample \\(\\mathcal {S}_{pos}^{(n)}\\) if it satisfies both the statistical threshold and the spatial constraint:<\/p>\n $$\\begin{aligned} a \\in \\mathcal {S}_{pos}^{(n)} \\Longleftrightarrow \\text {IoU}(a, g^{(n)}) \\ge \\tau ^{(n)} \\quad \\wedge \\quad \\text {center}(a) \\in g^{(n)} \\end{aligned}$$<\/p>\n \n (18)\n <\/p>\n To achieve joint optimization of accuracy and localization, the total loss function \\(\\mathcal {L}_{total}\\) integrates focal loss (\\(\\mathcal {L}_{cls}\\)), Generalized IoU loss (\\(\\mathcal {L}_{reg}\\)), and centerness loss (\\(\\mathcal {L}_{cen}\\)):<\/p>\n $$\\begin{aligned} \\mathcal {L}_{total} = \\lambda _{cls} \\mathcal {L}_{cls} + \\lambda _{reg} \\mathcal {L}_{reg} + \\lambda _{cen} \\mathcal {L}_{cen} \\end{aligned}$$<\/p>\n \n (19)\n <\/p>\n where weights \\(\\lambda _{cls}=1.0\\), \\(\\lambda _{reg}=2.0\\), and \\(\\lambda _{cen}=1.0\\) emphasize localization precision for industrial quality control.<\/p>\n Theorem 2<\/p>\n (Adaptive Threshold Convergence) As the training process approaches statistical equilibrium, the adaptive threshold \\(\\tau ^{(n)}\\) and the resulting positive sample ratio \\(\\pi ^{(n)}\\) converge to stable values:<\/p>\n $$\\begin{aligned} \\lim _{t \\rightarrow T_{conv}} |\\tau ^{(n)}(t) – \\tau ^{(n)}(t-1)| \\le \\epsilon _{conv}, \\quad \\pi ^{(n)} \\xrightarrow {t \\rightarrow T_{conv}} \\pi _{stable}^{(n)} \\pm \\delta \\end{aligned}$$<\/p>\n \n (20)\n <\/p>\n The formal proof of threshold stability and the convergence of positive sample ratios under the stochastic optimization of feature representations is detailed in Appendix A.<\/p>\n Corollary 2<\/p>\n (Small Target Detection Improvement) The ATSS mechanism provides a systematic mAP gain for small target defects by normalizing the selection criterion relative to the IoU distribution variance:<\/p>\n $$\\begin{aligned} \\Delta mAP^{small} = mAP_{ATSS}^{small} – mAP_{fixed}^{small} \\ge \\kappa \\cdot \\left( \\frac{\\sigma _{IoU}^{small}}{\\mu _{IoU}^{small}}\\right) \\end{aligned}$$<\/p>\n \n (21)\n <\/p>\n The mathematical derivation linking the coefficient of variation (CV) of the IoU distribution to the effective gradient utilization for small-scale targets is provided in Appendix A.<\/p>\n ESAD-net steel defect detection algorithm<\/p>\n The overall architecture of ESAD-Net is illustrated in Fig. 3<\/a>. The framework integrates three main components: a feature extraction module combining the Stem Layer and Edge Enhancement Module followed by four RegNet-X stages; a multi-scale fusion module based on FPN generating five-level feature maps P1\u2013P5; and an ATSS-based detection head with three branches for classification, regression, and centerness prediction. The ESAD-Net framework achieves end-to-end joint optimization through the following training procedure: input images are processed through the stem layer and edge enhancement module, then passed through the four RegNet-X stages to extract hierarchical features, subsequently fused via the FPN to generate multi-scale representations, and finally fed to the ATSS detection head. The ATSS algorithm dynamically computes positive sample assignments based on IoU statistics at each training iteration, and gradients from the total loss \\(\\mathcal {L}_{total}\\) defined in Equation (19<\/a>) backpropagate through both the detection head and backbone network simultaneously. The training employs SGD optimizer with momentum 0.9, weight decay 0.0001, and initial learning rate 0.01 with linear warmup over the first 500 iterations, followed by multi-step decay at epochs 20 and 25. The complete training and inference procedure is summarized in Algorithm\u00a01.<\/p>\n Fig. 3 Overall architecture of ESAD-Net, including the feature extraction, multi-scale fusion, and ATSS-based detection head components.<\/p>\n Algorithm 1 ESAD-Net: Efficient steel adaptive detection network<\/p>\n The computational complexity of ESAD-Net consists of three components. The RegNet-X backbone complexity is:<\/p>\n $$\\begin{aligned} \\text {FLOPs}_{backbone} = \\sum _{s=1}^{S} d_s \\cdot \\left( 2 H_s W_s w_{s-1} w_s + \\frac{9 H_s W_s w_s^2}{g} \\right) \\end{aligned}$$<\/p>\n \n (22)\n <\/p>\n where \\(d_s\\) is the number of blocks in stage s, \\(H_s\\) and \\(W_s\\) are the feature map spatial dimensions, \\(w_s\\) is the channel width, and the two terms correspond to \\(1 \\times 1\\) convolutions and \\(3 \\times 3\\) group convolution respectively. The group convolution with \\(g = 8\\) reduces computational cost by factor 8 compared to standard convolution. The feature pyramid network complexity is:<\/p>\n $$\\begin{aligned} \\text {FLOPs}_{FPN} = \\sum _{l=1}^{5} H_l W_l \\cdot (w_l \\cdot C_{fpn} + C_{fpn}^2) \\end{aligned}$$<\/p>\n \n (23)\n <\/p>\n where \\(C_{fpn} = 256\\) is the FPN channel number. The detection head complexity is:<\/p>\n $$\\begin{aligned} \\text {FLOPs}_{head} = \\sum _{l=1}^{5} H_l W_l \\cdot C_{fpn} \\cdot \\left( 4 \\times 9 \\times C_{fpn} + (K + 4 + 1) \\right) \\end{aligned}$$<\/p>\n \n (24)\n <\/p>\n where the factor 4 accounts for the four convolutional layers in each branch, and \\((K + 4 + 1)\\) represents the output dimensions for classification, regression, and centerness. With the adopted configuration and \\(640 \\times 640\\) input resolution, the total computational cost is approximately 2.4 GFLOPs, enabling inference time of 8.7 ms on RTX 3070 GPU, which satisfies the industrial real-time constraint \\(T_{process} < T_{max} = 10\\) ms.<\/p>\n The space complexity during inference is dominated by feature map storage. The RegNet-X design with group convolution and optimized width progression, combined with the lightweight FPN structure, constrains peak memory usage to approximately 150 MB for \\(640 \\times 640\\) input images, enabling deployment on resource-constrained edge devices.<\/p>\n","protected":false},"excerpt":{"rendered":"Problem statement Steel surface defect detection can be modeled as a real-time object detection problem under industrial constraints.…\n","protected":false},"author":2,"featured_media":659296,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[114,8763,4068,85,46,19179,4069,141],"class_list":["post-659295","post","type-post","status-publish","format-standard","has-post-thumbnail","category-business","tag-business","tag-engineering","tag-humanities-and-social-sciences","tag-il","tag-israel","tag-mathematics-and-computing","tag-multidisciplinary","tag-science"],"_links":{"self":[{"href":"https:\/\/www.newsbeep.com\/il\/wp-json\/wp\/v2\/posts\/659295","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.newsbeep.com\/il\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.newsbeep.com\/il\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/il\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/il\/wp-json\/wp\/v2\/comments?post=659295"}],"version-history":[{"count":0,"href":"https:\/\/www.newsbeep.com\/il\/wp-json\/wp\/v2\/posts\/659295\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/il\/wp-json\/wp\/v2\/media\/659296"}],"wp:attachment":[{"href":"https:\/\/www.newsbeep.com\/il\/wp-json\/wp\/v2\/media?parent=659295"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.newsbeep.com\/il\/wp-json\/wp\/v2\/categories?post=659295"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.newsbeep.com\/il\/wp-json\/wp\/v2\/tags?post=659295"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"Fig.](\"https:\/\/www.newsbeep.com\/il\/wp-content\/uploads\/2026\/10\/41598_2026_54738_Fig2_HTML.png\")
ATSS algorithm for adaptive sample selection<\/p>\n![\"Fig.](\"https:\/\/www.newsbeep.com\/il\/wp-content\/uploads\/2026\/10\/41598_2026_54738_Fig3_HTML.png\")
<\/p>\n![\"Algorithm](\"https:\/\/www.newsbeep.com\/il\/wp-content\/uploads\/2026\/10\/41598_2026_54738_Figa_HTML.png\")
<\/p>\n