We introduce the Hexagonal Convolutional Neural Network (HCNN), a modified version of CNN that is robust against rotation. HCNN utilizes a hexagonal kernel and a multi-block structure that enjoys more degrees of rotation information sharing than standard convolution layers. Our structure is easy to use and does not affect the original tissue structure of the network. We achieve the complete rotational invariance on the recognition task of simple pattern images and demonstrate better performance on the recognition task of the rotated MNIST images, synthetic biomarker images and microscopic cell images than past methods, where the robustness to rotation matters.

In recent years, applications of complex-valued neural networks have become wide spread. Quaternions are an extension of complex numbers, and neural networks with quaternions have been proposed. Because quaternion algebra is non-commutative algebra, we can consider two orders of multiplication to calculate weighted input. However, both orders provide almost the same performance. We propose hybrid quaternionic Hopfield neural networks, which have both orders of multiplication. Using computer simulations, we show that these networks outperformed conventional quaternionic Hopfield neural networks in noise tolerance. We discuss why hybrid quaternionic Hopfield neural networks improve noise tolerance from the standpoint of rotational invariance.

Recently, Gan and Luo have proposed a direction-of-arrival estimation method for uncorrelated and coherent signals in the presence of multipath propagation [3]. In their method, uncorrelated and coherent signals are distinguished by rotational invariance techniques and the property of the moduli of eigenvalues. However, due to the limitation of finite number of sensors, the pseudo-inverse matrix derived in this method is an approximate one. When the number of sensors is small, the approximation error is large, which adversely affects the property of the moduli of eigenvalues. Consequently, the method in [3] performs poorly in identifying uncorrelated signals under such circumstance. Moreover, in cases of small number of snapshots and low signal to noise ratio, the performance of their method is poor as well. Therefore, in this letter we first study the approximation in [3] and then propose an improved method that performs better in distinguishing between uncorrelated signals and coherent signals and in the aforementioned two cases. The simulation results demonstrate the effectiveness and efficiency of the proposed method.