Reflexive and adiabatic boundary 2D linear cellular automata and evolution of image patterns

One can study geometrical models of image patterns generated by CA evolution. For this, the present paper has investigated theoretical and imaginary investigation of two-dimensional (2D) linear and uniform cellular automata (CA). We study 2D linear CA under special boundary (reflexive and adiabatic) conditions over the binary field <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbb{Z}_{2}$</tex> , i.e. two states case. We investigate the evolution of image patterns corresponding to the uniform linear rules of 2D CA with the reflexive and adiabatic boundary conditions over <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbb{Z}_{2}$</tex> . The linear rules of CA can be found to be some image copies of a given first image depending on the special boundary types.

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