The optical flow has been one of the common approaches for image tracking. Its advantage is that no prior knowledge for image features is required. Since movement information can be obtained based on brightness data only, this method is suitable for tracking tasks of unknown objects. Besides, insects are always masters in chasing and catching preys in the natural world due to their unique compound eye structure. If the edge of the compound eye can be applied to tracking of moving objects, it is highly expected that the tracking performance will be greatly improved. Conventional images are built on a Cartesian reference system, which is quite different from the hexagonal framework for the compound eye of insects. This thesis explores the distinction of the hexagonal image framework by incorporating the hexagonal concept into the optical flow technology. Consequently, the reason behind why the compound eye is good at tracking moving objects can be revealed. According to simulation results for test images with different features, the hexagonal optical flow method appears to be superior to the traditional optical flow method in the Cartesian reference system.

The visual processing in the real world is different from the digital world. Monkey’s and a human’s visual world is richer and more colourful affording sight of flies, regardless of whether they are immobile or airborne. The study of the evolutionary process of our visual system indicates the existence of variationally spatial arrangement; from densely hexagonal in the fovea to a sparse circular structure in the peripheral retina. Normally we use a rectangular grid for the processing of images. But as per the perspective of the human eyes, the new approach is to change the grid from rectangular to hexagonal. Applying hexagonal grid in image processing is very advantageous and easy for mimicking human visual system. The main advantages for using the hexagonal structure in image processing is its resemblance to the arrangement of photoreceptors in the human eyes. The visual processing in the real world is different from the digital world. Monkey’s and a human’s visual world is richer and more colourful affording sight of flies, regardless of whether they are immobile or airborne. The study of the evolutionary process of our visual system indicates the existence of variationally spatial arrangement; from densely hexagonal in the fovea to a sparse circular structure in the peripheral retina. Normally we use a rectangular grid for the processing of images. But as per the perspective of the human eyes, the new approach is to change the grid from rectangular to hexagonal. Applying hexagonal grid in image processing is very advantageous and easy for mimicking human visual system. The main advantages for using the hexagonal structure in image processing is its resemblance to the arrangement of photoreceptors in the human eyes.

While current approaches to digital image processing within the context of machine learning and deep learning are motivated by biological processes within the human brain, they are, however, also limited due to the current state of the art of input and output devices as well as the algorithms that are concerned with the processing of their data. In order to generate digital images from real-world scenes, the utilized digital images' underlying lattice formats are predominantly based on rectangular or square structures. Yet, the human visual perception system suggests an alternative approach that manifests itself within the sensory cells of the human eye in the form of hexagonal arrangements. As previous research demonstrates that hexagonal arrangements can provide different benefits to image processing systems in general, this contribution is concerned with the synthesis of both worlds in the form of the biologically inspired hexagonal deep learning for hexagonal image processing. This contribution is therefore concerned with the design, the implementation, and the evaluation of hexagonal solutions to currently developed approaches in the form of hexagonal deep neural networks. For this purpose, the respectively realized hexagonal functionality had to be built from the ground up as hexagonal counterparts to otherwise conventional square lattice format based image processing and deep learning based systems. Furthermore, hexagonal equivalents for artificial neural network based operations, layers, as well as models and architectures had to be realized. This also encompasses the related evaluation metrics for hexagonal lattice format based representations of digital images and their conventional counterparts in comparison. Therefore, the developed hexagonal image processing and deep learning framework Hexnet functions as a first general application-oriented open science framework for hexagonal image processing within the context of machine learning. To enable the evaluation of hexagonal approaches, a set of different application areas and use cases within conventional and hexagonal image processing – astronomical, medical, and industrial image processing – are provided that allow an assessment of hexagonal deep neural networks in terms of their classification capabilities as well as their general performance. The obtained and presented results demonstrate the possible benefits of hexagonal deep neural networks and their hexagonal representations for image processing systems. It is shown that hexagonal deep neural networks can result in increased classification capabilities given different basic geometric shapes and contours, which in turn partially translate into their real-world applications. This is indicated by a relative improvement in F1-score for the proposed hexagonal and square models, ranging from 1.00 (industrial image processing) to 1.03 (geometric primitives) with single classes even reaching a relative improvement of over 1.05. However, possible disadvantages are also given by the increased complexity of hexagonal algorithms. This is evident by the present potential in regard to runtime optimizations that have yet to be realized for certain hexagonal operations in comparison to their currently deployed square equivalents.:1 Introduction and Motivation 2 Fundamentals and Methods 3 Implementation 4 Test Results, Evaluation, and Discussion 5 Conclusion and Outlook