Whereas conventional state-of-the-art image processing systems of recording and output devices almost exclusively utilize square arranged methods, biological models, however, suggest an alternative, evolutionarily-based structure. Inspired by the human visual perception system, hexagonal image processing in the context of machine learning offers a number of key advantages that can benefit both researchers and users alike. The hexagonal deep learning framework Hexnet leveraged in this contribution serves therefore the generation of hexagonal images by utilizing hexagonal deep neural networks (H-DNN). As the results of our created test environment show, the proposed models can surpass current approaches of conventional image generation. While resulting in a reduction of the models' complexity in the form of trainable parameters, they furthermore allow an increase of test rates in comparison to their square counterparts.

Abstract-- We study the sample placement and shortest tour problem for robots tasked with mapping environmental phenomena modeled as stationary random fields. The objective is to minimize the resources used (samples or tour length) while guaranteeing estimation accuracy. We give approximation algorithms for both problems in convex environments. These improve previously known results, both in terms of theoretical guarantees and in simulations. In addition, we disprove an existing claim in the literature on a lower bound for a solution to the sample placement problem.

Super-resolution (SR) aims to increase the resolution of imagery. Applications include security, medical imaging, and object recognition. We propose a deep learning-based SR system that takes a hexagonally sampled low-resolution image as an input and generates a rectangularly sampled SR image as an output. For training and testing, we use a realistic observation model that includes optical degradation from diffraction and sensor degradation from detector integration. Our SR approach first uses non-uniform interpolation to partially upsample the observed hexagonal imagery and convert it to a rectangular grid. We then leverage a state-of-the-art convolutional neural network (CNN) architecture designed for SR known as Residual Channel Attention Network (RCAN). In particular, we use RCAN to further upsample and restore the imagery to produce the final SR image estimate. We demonstrate that this system is superior to applying RCAN directly to rectangularly sampled LR imagery with equivalent sample density. The theoretical advantages of hexagonal sampling are well known. However, to the best of our knowledge, the practical benefit of hexagonal sampling in light of modern processing techniques such as RCAN SR is heretofore untested. Our SR system demonstrates a notable advantage of hexagonally sampled imagery when employing a modified RCAN for hexagonal SR.