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Complex-Valued Neural Networks (CNNs) are the most successful extension of the real-valued neural networks. They play a significant role in deep learning. Following this great fact, CNNs are utilized in computer vision and its critical field Image analysis (IA). Symmetry vision is a theory closely related to assembly and uniformity. More accurately, symmetrical rallies can be categorized as covering self-similarities. In 2D images, the self-similarity comes from rigid transformations (such as operators, polynomials and differential and integral equations) that map one part onto another (in calculus is called an injective map and in the conformal analysis is called a univalent map). In this paper, we define a new style of CNNs called Conformal Neural Networks (CoNNs) based on the concept of conformal theory in the open unit disk. Moreover, we introduce a general symmetric 2D-parameter Beltrami equation (SBE) in a complex domain. The dominant idea is to map the external and inner of the domain conformally to unit disks, using a symmetric differential operator (which is suggested to be the solution of SBE). As an application, we present a new algorithm to obtain the arches of the foot from their marks by training CoNNs. The results prove that the effectiveness of our suggested procedure is a stable form arrangement. [ABSTRACT FROM AUTHOR]

A biological dynamic system carries engineering properties such as control systems and signal processing (or image processing) addicted to molecular biology at the level of bio-molecular communication networks. Dynamical system features and signal reply functions of cellular signaling pathways are some of the main topics in biological dynamic systems (for example the biological segmentation). In the present paper, we introduce new generalized hybrid time-space dynamical systems of growing bacteria. We impose the approximate analytic solution for the system. The generalization adapted the concepts of the Riemann–Liouville fractional operators for time and the Srivastava–Owa fractional operators for space. Moreover, we introduce a numerical perturbation method of two operators to obtain the approximate solutions. We establish the existence and uniqueness results and impose some applications in the sequel. Moreover, we study the Ulam stability and apply these stable solutions to improve the segmentation of a class of growing bacteria. [ABSTRACT FROM AUTHOR]