## 2 Proposed Network
### Spatial Attention Module  
We produce spatial attention weights using the inter-spatial relationships of channels to obtain rich and global spatial information that helps to create a robust global feature map. Figure 4 depicts the overview of our proposed spatial attention module. A feature map \(A\in R^{C\times H\times W}\) is provided as an input to the spatial attention module where \(C\) denotes channel size and \(H\times W\) are height and width of the feature map, which are \(256\times 128\times 128\) in this paper. This input \(A\) is then fed into a \(1\times 1\) linear transformation layer and a normalized feature map \(S_{signal}\) is created for all channels using the sigmoid activation function.  
\[S^{\prime}_{ij}=\frac{1}{1+exp(-S_{ij})} \tag{2}\]  
where \(S_{ij}\) is the scalar value at \(i\)th and \(j\)th position and \(S^{{}^{\prime}}_{ij}\) denotes corresponding activated scalar value at \(i\)th and \(j\)th pixel position. The output of this operation is a sigmoid activated map, i.e., \(S_{signal}\in R^{C\times H\times W}\).  
Additionally, the input \(A\in R^{C\times H\times W}\) is fed into a convolutional block, which generates a feature map \((F_{CONV3})\). This convolutional block consists of three convolutional layers with \(1\times 1,3\times 3,1\times 1\) kernel size, respectively. To keep channel-wise details, the number of channels \((C)\) for each convolutional layer remains unchanged which is 256. ReLU activation is followed by the first two convolutional operations while batch normalization has been  
Figure 3: (**Left**) CenterNet framework. (**Middle**) LogoNet with spatial attention module and added final output feature maps. (**Right**) Dual-Attention LogoNet.  
performed for all the convolutional layers. Softmax normalization strategy is applied across the channels over the output feature space of the convolutional block (\(F_{CONV3}\)). During softmax normalization, all positional scalar values in the same pixel-position across all feature channels are considered. New scalar value is synthesized for each pixel across the channels using the value of other pixels at the same index. In Equation (3), if \(P_{i,j,k}\) is a scalar value at \(i\)th and \(j\)th pixel position in \(k\)th channel, a normalized scalar value \(P^{{}^{\prime}}_{i,j,k}\) can be obtained as:  
\[P^{{}^{\prime}}_{i,j,k}=\frac{exp(P_{i,j,k})}{\sum_{k=1}^{C}exp(P_{i,j,k})} \tag{3}\]  
where \(C\) denotes the number of channels in feature map \(F_{CONV3}\). A softmax normalized feature maps \(P_{softmax}\in R^{C\times H\times W}\) has been produced using these normalized scalar values (\(P^{{}^{\prime}}_{i,j,k}\)).  
We perform element-wise product of both generated normalized feature map, i.e., \(S_{sigmoid}\) and \(P_{softmax}\). The input feature map (\(A\)) is added as a skip connection to this product to obtain final attention-weighted feature map.  
\[A_{attention}=A\quad+\quad(S_{sigmoid}\odot P_{softmax}) \tag{4}\]  
where \(\odot\) is the element-wise product.  
As we mentioned in our previous conference paper [15], our convolutional layers block follows the module structure proposed in [23], but our method is totally different from their approach. For multi-label image classification, they employed a regularization module to generate attention weights. These attention weights were provided to the classification layer of the feature extractor network which was ResNet [30]. Whereas, we generate a weighted feature map and perform an element-wise addition with the input to obtain a robust representation of input image. Our proposed technique uses both sigmoid and softmax functions as activation to learn important spatial weights.