## 3 Methods
### Scale-Invariant Calibrated Convolution  
A structural diagram of the proposed method is shown in Figure 2. As with grouped convolution, CCGD divides the standard convolutional filters into multiple parts \([\mathcal{F}_{1},\mathcal{F}_{2},\mathcal{F}_{3},\mathcal{F}_{S}]\). Here, however, each filter is not positioned in parallel, but rather made responsible for executing a particular function.  
An input, \(X\), is evenly divided into two subsets, \(X_{1}\) and \(X_{2}\), each of which then enters a designated pathway for learning feature representations at a different level. The scale-invariant attention pathway is further subdivided into two sub-paths. The first component of the top path is a \(3\times 3\) convolution. Meanwhile, the \(X_{2}\) features are sent along a bottom path to execute a Scale-Invariant Calibration (SIC) operation. Motivated by the DoG, this module contains continuous down-sampling and up-sampling operations, \(S_{r}(\cdot)\), with different rates, to mimic the Gaussian filtering in Equation (1). This minimizes the extra computational load and parameters. A Gaussian blur approximation, \(G\), is then obtained, which has the same resolution as the input, \(X_{2}\). This can be written as follows:  
\[G=\{G_{i}\}_{i=1}^{n}=S_{r}(X_{2}) \tag{2}\]  
where \(n\) denotes the number of iterations for \(S_{r}(\cdot)\). The difference operation on the bottom path is performed upon \(G\) as follows:  
\[D=\sum_{i=1}^{n-1}(G_{i+1}-G_{i})^{2} \tag{3}\]  
where \(D\) is a set of difference maps. To avoid any possible differentiation problems in the back-propagation and to enhance the effect of the difference maps, we use a square function to replace the absolute value operation. The calibrated operation can therefore be defined as follows:  
\[X_{2}^{\prime}=\mathcal{F}_{2}(X_{2})\odot\delta\big{(}D+\mathcal{F}_{g}(X_{2 })\big{)} \tag{4}\]  
where \(\delta\) represents the sigmoid activation, and \(\odot\) denotes element-wise multiplication. A \(3\times 3\) group convolution, \(\mathcal{F}_{g}(\cdot)\), is used to introduce a learnable feature region as an extra replenishment.  
For the attention mechanism in CNNs, the mathematical description can be expressed as follows:  
\[X^{\prime}=\mathcal{F}(X)\odot AM(X) \tag{5}\]  
where \(AM(\cdot)\) represents the attention map, which is used to assist in the calibration of the output features of a convolution. Correspondingly, \(\delta\big{(}D+\mathcal{F}_{g}(X_{2})\big{)}\) is the attention map that calibrates the output features of \(\mathcal{F}_{2}(\cdot)\). Where, the DoG is a key technology for generating robust attention maps. Then, the final output after CCGD can be formulated as follows:  
\[Y=Y_{2}\cup Y_{1}=\mathcal{F}_{3}\big{(}X_{2}^{\prime}\big{)}\cup\mathcal{F}_ {1}(X_{1}) \tag{6}\]  
The advantages of CCGD are three-fold. First, by utilizing the calibrated steps defined in Equation (4), each feature is allowed to adaptively build powerful pixel-level dependencies on the basis of its scale-invariant information, enabling recalibration of the intermediate feature representation. Second, as \(\mathcal{F}_{2}(\cdot)\) and \(\mathcal{F}_{3}(\cdot)\) are combined into series form, the receptive field of the scale-invariant attention pathway can be expanded explicitly, thus providing a feature map that is different from the original pathway, \(\mathcal{F}_{1}(\cdot)\), in the scale space. This gives CCGD a strong multi-scale representation ability. Third, instead of introducing added parameters, the scale-invariant calibration operation only employs lightweight continuous down-sampling, up-sampling, and \(3\times 3\) grouped convolutions. The deeper the network, the more significant this becomes.