## 3 Proposed Method
### Basics of DeepCA Feature Extraction
In this regard, the class of an input image is represented as the score matrix of the model \((S_{model})\), which can be determined from a minimum error that can be calculated by a decision function as follows:_  
\[S_{model}=min\left(f_{err}(S_{r_{i}},I_{class})\right) \tag{9}\]  
_where \(f_{err}\) denotes an error estimation function, such as a mean squared error (MSE) function._  
**Definition 6**. : _(Rule vector). For feature extraction task, the rule vector is the most significant aspect for all layers of DeepCA in terms of tuning the score matrix values. However, several types of CA rules with different rule space sizes are available [59]. In this work, we address the general and totalistic rule spaces; for the general rule space, the rule members consist of entire rules according to their neighborhood configurations. For example, the general rule space of a Moore neighborhood can be formed from rule \(0\) to \(2^{29}\) (or \(1.34\times 10^{154}\)) [59]. We formalize all these rules as the rule vector \(R\) in the \(n^{th}\) layer as follows:_  
\[R^{n}=\ \{r_{1},r_{2},\ldots,r_{m}\} \tag{10}\]  
_where \(r_{i}|i=1\) to \(m\) denote the rule members in the rule vector. \(m\) denotes the size of the rule space of the CA. However, this paper uses \(m=2^{9}\) (or \(512\)) based on the totalistic rule type._  
Since the general rule space of a Moore neighborhood has a massive size, reducing the rule space is necessary to reduce the time consumption required during the training process and to enable the determination of the rule vector. In order to reduce the number of general rules in the space, we employ the totalistic rule type proposed by [59; 60] that limits the rules from \(2^{29}\) to only \(2^{9}\) (or \(512\)) rules [61] (see Figures 2 and 3). Figure 4a shows the particular setting of the totalistic rule type for a Moore neighborhood with approximately \(2^{m}\) neighbor according to [59; 60; 61; 62]. Figure 3: Examples of totalistic rules. Figure 2: Examples of general rules. **Definition 7**. : _(Rule type equivalence). The neighborhood type and the number of possible states are relative to the size of the rule space in general. Considering a Moore neighborhood of size \(3\times 3\) with two possible states, the range of the general rule space is between 0 and \(2^{512}-1\) (see Figure 2 (Top)), whereas the range of the totalistic rule space is between 0 (with neighborhood code 000000000) and \(2^{9}-1\) (rule 511 with neighborhood code \"111111111\") (see Figure 3 (Top)). In addition, the totalistic rule space is also a part of the general rule space. Therefore, these spaces also have a significant relationship. For instance, to find the neighborhood codes of any rule based on the numbering of rule conventions with neighbors that shown in Figure 4a, the evolution result obtained for input data with rule 50 (or rule code \"0... 110010\" (see. the yellow box in Figure 2)) in the general rule space depends on three neighborhood codes: \"000000001\", \"000000100\", and \"000000101\" (see Figure 2 (Bottom)), whereas the evolution result obtained under the totalistic rule depends on only one neighborhood code \"000110010\" (see Figure 3 (Bottom)). In this regard, the following formalization demonstrates how the general rule and totalistic rule spaces are equivalent._  
\[r_{totalistic}(x)\equiv r_{general}(2^{(x)}) \tag{11}\]  
_For instance, rule 50 in the totalistic rule type (represented as \(r_{totalistic}(50)=\) \"000110010\") is equivalent to rule \(r_{general}(2^{50})=\) \"000110010\" in the general type. On the other hand, the rule \(r_{general}(2^{1}+2^{4}+2^{5})\) consists of neighborhood codes \"000000001\", \"000000100\", and \"000000101\", which are equivalent to \(r_{totalistic}(1)\), \(r_{totalistic}(4)\), and \(r_{totalistic}(5)\), respectively._  
**Definition 8**. : _(Rule-0). Even though rule \(-0\) denotes no operation, the rule \(-0\) s of the general rule and totalistic rule spaces still exhibit some differences.