## 3 Proposed Method
### Basics of DeepCA Feature Extraction  
In this section, the basics of the proposed DeepCA feature extraction are provided as follows. **Definition 1**. : _(\(n\)-layer DeepCA feature extraction). An \(n\)-layer deep CA feature extraction is defined as an 8-tuple as follows:_  
\[DeepCA=\left\langle Z^{d},Q,N,f^{n},R^{n},F^{n},S,C^{t}_{i,n}\right\rangle \tag{4}\]  
_where \(Z^{d}\) denotes a coordinate system with \(d\) dimensions; \(Q\) denotes a finite set of states; \(N\) denotes a finite subset of \(Z^{d}\) called a neighborhood vector; \(f^{n}\) denotes a local transition function at layer \(n^{th}\); \(R^{n}=\{r_{1},r_{2},\dots,r_{m}\}\) is a rule vector in the totalistic rule space at layer \(n^{th}\); \(F^{n}\) denotes the feature matrices of layer \(n^{th}\) generated by the current states of the input matrices, the local transition function \((f)\) and the neighborhood \((N)\); \(S\) denotes the score matrices generated from the feature matrices and the scoring function; and \(C^{t}_{i,n}\) denotes the \(i^{th}\) cell at layer \(n^{th}\) at time \(t\)._  
**Definition 2**. : _(Evolution rule). This rule is significant for the operation of CA in determining whether to evolve the current state of a cell to the next generation. Thus, we define a transition function to evolve cells as follows:_  
\[C^{t+1}_{i,n}=f(C^{t}_{i,n},N_{i,n}) \tag{5}\]  
_where \(C^{t}_{i,n}\) represents the \(i^{th}\) cell at the \(n^{th}\) layer in the current state (or at time \(t\)), \(f\) denotes the transition function, \(C^{t+1}_{i,n}\) represents the \(i^{th}\) cell at the \(n^{th}\) layer in the next state (or at time \(t+1\)), and \(N_{i,n}\) denotes the neighborhood configuration of the cell._  
_For instance, if we have a Moore neighborhood with the range \(r=1\), the results of \(C^{t+1}_{i,n}\) in the \(i^{th}\) cell at the \(n^{th}\) layer obtained by Equation (5) can be produced as follows:_  
\[ \begin{split} C^{t+1}_{i,n}=&\frac{1}{9}(a_{0}C^{t }_{(x,y)}+a_{1}C^{t}_{(x+1,y)}+a_{2}C^{t}_{(x+1,y+1)}\\ &+a_{3}C^{t}_{(x,y+1)}+a_{4}C^{t}_{(x-1,y+1)}\\ &+a_{5}C^{t}_{(x-1,y)}+a_{6}C^{t}_{(x-1,y-1)}\\ &+a_{7}C^{t}_{(x,y-1)}+a_{8}C^{t}_{(x+1,y-1)})\end{split}  \tag{6}\]  
_where \(a_{0},a_{1},\dots,a_{8}\) represent the values of the pixels in the image that correspond to the neighborhood._  
**Definition 3**. : _(Feature matrices). The multilayer features in DeepCA are defined as feature matrices obtained by the input image and the CA rule vector via the convolution function. In addition, the sizes of the feature matrices also enhance the classification efficiency. Therefore, the feature matrices of the \(n^{th}\) layer can be obtained by the convolution function as follows:_  
\[ \begin{split} F^{n}&=f_{conv}(I^{n},R^{n},s),\\ &=\{f_{r_{1}},f_{r_{2}},\dots,f_{r_{m}}\}\end{split}  \tag{7}\]  
_where \(f_{conv}\) is the convolution function, \(I^{n}\) is an input image at the \(n^{th}\) layer, \(R^{n}\) denotes the rule vector at the \(n^{th}\) layer, \(s\) denotes the number of strides, and \(f_{r_{1}},f_{r_{2}},\dots,f_{r_{m}}\) represent the feature matrix obtained by rules \(r_{1}\) to \(r_{m}\)._  
**Definition 4**. : _(Score matrix). The goal of the training process is to build a reference model consisting of a memory component. In this regard, we define a particular matrix as a memory component called a score matrix (\(S\)). This matrix can be obtained through any scoring function, e.g., \(maxPool\) or \(softMax\), enabling us to build matrices at the original size or the modified size of the feature matrices as:_  
\[S=f_{pool}(F^{n},N_{i,n},s) \tag{8}\]  
_where \(f_{pool}\) represents the scoring function, \(F^{n}\) denotes the feature matrices, and \(N_{i,n}\) denotes a neighborhood of the \(i^{th}\) cell at the \(n^{th}\) layer._  
**Definition 5**. : _(DeepCA decision rule). To adopt DeepCA in decision tasks, we consider the difference between the score matrices of an image \((S_{r_{i}})\) and its class \((I_{class})\).