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Convolutional neural networks (CNNs) have gained growing interest in recent years for their advantages in detecting cracks on concrete bridge components. Class imbalance is a fundamental problem in crack segmentation, resulting in unsatisfactory segmentation for tiny cracks. Besides, limited by the local receptive field, CNNs often cannot integrate local features with global dependencies, thus significantly affecting the detection accuracy of tiny cracks across the entire image. To solve those problems in segmenting tiny cracks, a multiscale feature fusion network with attention mechanisms named “Tiny‐Crack‐Net” (TCN) is proposed. The modified residual network was used to capture the local features of tiny cracks. The dual attention module was then incorporated into the architecture to better separate the tiny cracks from the background. Also, a multiscale fusion operation was implemented to preserve the edge details of tiny cracks. Finally, a joint learning loss of the cross‐entropy and similarity was proposed to alleviate the poor convergence induced by the severe class imbalance of the pixels representing tiny cracks. The capability of the network in segmenting tiny cracks was remarkably enhanced by the aforementioned arrangements, and the “Tiny‐Crack‐Net” achieved a Dice similarity coefficient of 87.96% on an open‐source data set, which was at least 5.84% higher than those of the six cutting‐edge networks. The effectiveness and robustness of the “Tiny‐Crack‐Net” were validated with field test results, which showed that the intersection over union (IOU) for cracks with a width of 0.05 mm or wider reaches 91.44%.

In the article, we present the best possible parameters α1, β1, α2, β2 ∈ ℝ and α3, β3 ∈ [1/2, 1] such that the double inequalities$$\begin{array}{} \begin{split} \displaystyle \alpha_{1}C(a, b)+(1-\alpha_{1})A(a, b) & \lt T_{3}(a, b) \lt \beta_{1}C(a, b)+(1-\beta_{1})A(a, b), \\ \alpha_{2}C(a, b)+(1-\alpha_{2})Q(a, b) & \lt T_{3}(a, b) \lt \beta_{2}C(a, b)+(1-\beta_{2})Q(a, b), \\ C(\alpha_{3}; a, b) & \lt T_{3}(a, b) \lt C(\beta_{3}; a, b) \end{split} \end{array}$$hold for a, b > 0 with a ≠ b, and provide new bounds for the complete elliptic integral of the second kind, where A(a, b) = (a + b)/2 is the arithmetic mean, $\begin{array}{} \displaystyle Q(a, b)=\sqrt{\left(a^{2}+b^{2}\right)/2} \end{array}$ is the quadratic mean, C(a, b) = (a2 + b2)/(a + b) is the contra-harmonic mean, C(p; a, b) = C[pa + (1 – p)b, pb + (1 – p)a] is the one-parameter contra-harmonic mean and $\begin{array}{} T_{3}(a,b)=\Big(\frac{2}{\pi}\int\limits_{0}^{\pi/2}\sqrt{a^{3}\cos^{2}\theta+b^{3}\sin^{2}\theta}\text{d}\theta\Big)^{2/3} \end{array}$ is the Toader mean of order 3.

Cracks are often the most intuitive indicators for assessing the condition of in-service structures. Intelligent detection methods based on regular convolutional neural networks (CNNs) have been widely applied to the field of crack detection in recently years; however, these methods exhibit unsatisfying performance on the detection of out-of-plane cracks. To overcome this drawback, a new type of region-based CNN (R-CNN) crack detector with deformable modules is proposed in the present study. The core idea of the method is to replace the traditional regular convolution and pooling operation with a deformable convolution operation and a deformable pooling operation. The idea is implemented on three different regular detectors, namely the Faster R-CNN, region-based fully convolutional networks (R-FCN), and feature pyramid network (FPN)-based Faster R-CNN. To examine the advantages of the proposed method, the results obtained from the proposed detector and corresponding regular detectors are compared. The results show that the addition of deformable modules improves the mean average precisions (mAPs) achieved by the Faster R-CNN, R-FCN, and FPN-based Faster R-CNN for crack detection. More importantly, adding deformable modules enables these detectors to detect the out-of-plane cracks that are difficult for regular detectors to detect.

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