## 2 Background
### Cellular Automata (CA)  
CA were first proposed by [44; 45] and are mathematical models used to simulate a complex system in terms of a dynamical model in which discrete time and space consist of a regular lattice of cells in any dimension. The operations of CA usually depend on time; the state of a cell depends on the imposed evolution rule relative to the present state and the neighborhood configuration. The CA structure is defined as a set of cells (also known as a neighborhood) relative to the central cell. Several types of CA are available depending on their dimensions and neighborhoods; the basic class contains elementary CA (ECA) or the one-dimensional CA proposed by [46]. However, the most generally used CA are the two-dimensional CA involving Moore and Von Neumann neighborhoods. In this paper, we simulate the proposed model only on a Moore neighborhood that aims to directly connect to the 8-connected pixels of a digital image structure [47]. We adopt this configuration not only to enhance the image contrast but also to extract the features of breast cancer images. More specifically, the neighborhood \(N(x,y)\) in the range \(r=1\) can be represented as follows:  
\[N(x_{0},y_{0})^{E}=\{(x_{-1},y_{0}),(x_{0},y_{0}),(x_{1},y_{0})\} \tag{1}\]  
where \(N\) represents a neighborhood function and \(E\) denotes the ECA. \[N(x_{0},y_{0})^{V}=\{(x_{0},y_{-1}),(x_{-1},y_{0}),(x_{0},y_{0}),(x_{1},y_{0}), (x_{0},y_{1})\} \tag{2}\]  
where \(V\) represents a Von Neumann neighborhood, \(x\) and \(y\) denote the coordinates of the neighborhood, and \(x_{0}\) and \(y_{0}\) denote the center points of the neighborhood. \[N(x_{0},y_{0})^{M}=\{(x,y):|x-x_{0}|\leq r,|y-y_{0}|\leq r\} \tag{3}\]  
where \(M\) represents the Moore neighborhood, \(x\) and \(y\) denote the position of the neighbor in the neighborhood, \(x_{0}\) and \(y_{0}\) are the center points, and \(r\) denotes the range of the neighborhood. CA models are the dynamical models that have been successfully used in several tasks, especially in image processing scenarios, such as edge detection [48; 49; 50; 51], noise filtering [47; 52; 53; 54], saliency detection [16; 17; 55], and image segmentation [15; 56; 57]. For instance, an edge detection algorithm was proposed by [48], who ran a CA model on grayscale images by using a uniform CA rule in a Von Neumann neighborhood to effectively detect image edges. The spatiotemporal CA-based filtering approach (st-CAF) was proposed by [54] to cope with the problem of denoising image sequences; a spatiotemporal neighborhood was adopted to identify a type of noise and tune the algorithm. The saliency detection method proposed by [17] combines global and local information by using multilayer CA. In this model, the skip links and edge penalty terms are added to an encoder-decoder network to transfer information from high-level layers to lower-level layers; the network relies on a CNN structure to determine the global saliency map and generate the foreground and background codebooks. Then, the local saliency map is obtained by utilizing these codebooks. Finally, a multilayer CA framework is applied to produce the final saliency map from the global and local saliency maps. Among the learning methods that rely on CA approaches, CA learning and prediction (CALP) is an ensemble learning model proposed by [58] to address the handwritten pattern problem in classification tasks. First, the handwritten patterns evolve from various parameters controlled by the rules of CA. Then, various evolutions of the handwritten pattern are used to build classifiers in an aggregate fashion. Furthermore, ref. [21] proposed a CA-based reservoir system based on a deep learning approach to address reservoir computing. The model has two main processes, encoding and decoding, which rely on CA theory. The ECA (or 1-D neighborhood) was adopted as a reservoir to correctly learn the binary pattern from the input layer and map it to the binary output via a readout layer. In addition, in our previous work, ref.