## 3 Algorithm Overview
It can be shown as a three-dimensional array of size, \(W\times H\times N\) where \(W\) and \(H\) are the height and width, respectively, whereas \(N\) is the amount of given channels (in the case when \(f\) is the result of a convolutional layer, \(N\) is the amount of filters in this convolution layer) [54]. Then, \(p\)-norm subsampling (or pooling) having a pooling dimension \(k\) (or half-length \(k/2\)) and stride of \(r\) is implemented on the given feature map \(f\) with a three-dimensional array \(s(f)\) represented by the following entries:

\[s_{i,j,u}(f)=\bigg{(}\sum_{h=-\lfloor k/2\rfloor}^{\lfloor k/2\rfloor}\sum_{w =-\lfloor k/2\rfloor}^{\lfloor k/2\rfloor}\|f_{g}(h,w,i,j,u)\|^{p}\bigg{)}^{1/ p}, \tag{1}\]

where \(g(h,w,i,j,u)=(r.i+h,r.j+w,u)\) is a function mapping from positions in \(s\) to positions in \(f\) with stride rate \(r\) and \(p\) denotes the order of \(p\)-norm (for \(p\rightarrow\infty\), it turns out to be the commonly-employed max pooling). If \(r>k\) and pooling regions do not overlap, the present CNN network normally consists of an overlapping pooling by \(k=3\) and \(r=2\). The pooling operation stated in Equation (1) could be compared with the standard definition of a convolutional layer \(c\) applied to feature map \(f\) given as in Equation (3):

\[c_{i,j,o}(f)=\sigma\bigg{(}\sum_{h=-\lfloor k/2\rfloor}^{\lfloor k/2\rfloor}\sum _{w=-\lfloor k/2\rfloor}^{\lfloor k/2\rfloor}\sum_{u=1}^{N}\theta_{h,w,u,o}.f_{ g}(h,w,i,j,u)\bigg{)}, \tag{2}\]

where \(\theta\) is the convolutional layer weight (or the kernel weight, or the filter), \(\sigma(. )\) is the activation function, \(\sigma(x)=\max(x,0)\) is an activation function and \(o\in[1,M]\) is the amount of output features (or channels) of the convolutional layer. Through this analysis, it can be seen clearly that both operations rely on the similar elements of the previous network layer's feature map [55]. The pooling layer could also be seen as a feature-wise convolution (where \(\theta_{h,w,u,o}=1\) if \(u\) equals \(o\) and zero otherwise) in which we can replace the activation function with the \(p\)-norm. It is a perplexing question of whether and why such kinds of layers could be introduced. According to [PERSON] et al. [54], the pooling layers contribute to the CNN architecture as follows: (1) the \(p\)-norm facilitates the network representation in a more invariant way; (2) the pooling layer's spatial dimensionality reduction feature offers covering for large parts of the input image in higher layers; (3) the pooling layer's feature-wise operation nature could help in making optimization easier as opposed to the convolutional layer operation where features get mixed. By taking the second point into consideration, the pooling layer dimensionality reduction is crucial for achieving good performance. Thus, this is a key hypothesis that we are going to test in this experiment. Test Model 5: After the deep and detailed analysis of different BTCMs, final tests will be conducted over the final version of our proposed network that will incorporate all best-performing components of the BTCM tests. Later, this model will be trained, and final assessments will be presented in the Results section. The illustration of the proposed CSSA architecture in shown in Figure 2. Apart from some differences, the proposed final model shares a similar architecture to a deconvolution network [15] and SegNet [40]. The deconvolution network by [PERSON] et al. is based on 13 complete layers of VGG-16, which introduces much larger parameterization that leads to more computational resources. It is also difficult to train it end-to-end because of keeping the \(fc6\) and \(fc7\) layers, which leads to a huge number of trainable parameters to the decoder of the network. The CSSA proposed encoder excludes the \(fc6\) and \(fc7\) layers, which leads to less trainable parameters for the decoder. In this way, the proposed decoder needs less space and computational resources. SegNet is based on the ReLU activation function with no dropout units.