Biological evidence shows that there are neural networks specialized for recognition of signals and patterns acting as associative memories. The spiking neural networks are another kind which receive input from a broad range of other brain areas to produce output that selects particular cognitive or motor actions to perform. An important contribution of this work is to consider the geometric processing in the modeling of feed-forward neural networks. Since quaternions are well suited to represent 3D rotations, it is then well justified to extend real-valued neural networks to quaternion-valued neural networks for task of perception and control of robot manipulators. This work presents the quaternion spiking neural networks which are able to control robots, where the examples confirm that these artificial neurons have the capacity to adapt on-line the robot to reach the desired position. Also, we present the quaternionic quantum neural networks for pattern recognition using just one quaternion neuron. In the experimental analysis, we show the excellent performance of both quaternion neural networks.

This article presents a revisited proposal for the formulation of objects and geometric relations and constraints in the conformal space. For modeling, graphics engineering, kinematics, and dynamics, the solution of problems using only points and lines; or the formulation of rigid motion (SE(3) using vectors calculus, matrix algebra, or calculus is indeed very awkward. In contrast, we use incidence algebra and conformal geometric algebra to effectively represent geometric objects and compute relations and constraints between geometric entities. In conformal geometric algebra, one can compute efficiently the linear transformations SO(3) and SE(3) of these geometric entities using rotors, translators, and motors. Since these operators and geometric entities have no redundant coefficients, they can be computed very fast. The authors present a new and complete set of equations using incidence algebra and conformal geometric algebra. The use of the proposed equations depends upon the applications. You can enclose certain objects with geometric shapes in your setting using points, lines, planes, circles, spheres, hyperplanes, and hyperspheres. Then, quadratic programming for optimization can be applied to find the minimal directed distance or a minimal path to be followed among many geometric objects. These methods and equations can be used to tackle a variety of problems in graphics, augmented virtual reality, GIS, Robotics, and Human-Machine Interaction. For real-time applications, the procedures and equations presented in this work can be used to develop efficient algorithms, which can be sped up using FPGA or CUDA (Nvidia).

Since their first applications, Convolutional Neural Networks (CNNs) have solved problems that have advanced the state-of-the-art in several domains. CNNs represent information using real numbers. Despite encouraging results, theoretical analysis shows that representations such as hyper-complex numbers can achieve richer representational capacities than real numbers, and that Hamilton products can capture intrinsic interchannel relationships. Moreover, in the last few years, experimental research has shown that Quaternion-Valued CNNs (QCNNs) can achieve similar performance with fewer parameters than their real-valued counterparts. This paper condenses research in the development of QCNNs from its very beginnings. We propose a conceptual organization of current trends and analyze the main building blocks used in the design of QCNN models. Based on this conceptual organization, we propose future directions of research.