Leonardo vindicated: Pythagorean trees for minimal reconstruction of the natural branching structures

Trees continue to fascinate with their natural beauty and as engineering masterpieces optimal with respect to several independent criteria. Pythagorean tree is a well-known fractal design that realistically mimics the natural tree branching structures. We study various types of Pythagorean-like fractal trees with different shapes of the base, branching angles and relaxed scales in an attempt to identify and explain which variants are the closest match to the branching structures commonly observed in the natural world. Pursuing simultaneously the realism and minimalism of the fractal tree model, we have developed a flexibly parameterised and fast algorithm to grow and visually examine deep Pythagorean-inspired fractal trees with the capability to orderly over- or underestimate the Leonardo da Vinci's tree branching rule as well as control various imbalances and branching angles. We tested the realism of the generated fractal tree images by means of the classification accuracy of detecting natural tree with the transfer-trained deep Convolutional Neural Networks (CNNs). Having empirically established the parameters of the fractal trees that maximize the CNN's natural tree class classification accuracy we have translated them back to the scales and angles of branches and came to the interesting conclusions that support the da Vinci branching rule and golden ratio based scaling for both the shape of the branch and imbalance between the child branches, and claim the flexibly parameterized fractal trees can be used to generate artificial examples to train robust detectors of different species of trees.

Hyperbox based machine learning algorithms: A comprehensive survey

With the rapid development of digital information, the data volume generated by humans and machines is growing exponentially. Along with this trend, machine learning algorithms have been formed and evolved continuously to discover new information and knowledge from different data sources. Learning algorithms using hyperboxes as fundamental representational and building blocks are a branch of machine learning methods. These algorithms have enormous potential for high scalability and online adaptation of predictors built using hyperbox data representations to the dynamically changing environments and streaming data. This paper aims to give a comprehensive survey of literature on hyperbox-based machine learning models. In general, according to the architecture and characteristic features of the resulting models, the existing hyperbox-based learning algorithms may be grouped into three major categories: fuzzy min-max neural networks, hyperbox-based hybrid models, and other algorithms based on hyperbox representations. Within each of these groups, this paper shows a brief description of the structure of models, associated learning algorithms, and an analysis of their advantages and drawbacks. Main applications of these hyperbox-based models to the real-world problems are also described in this paper. Finally, we discuss some open problems and identify potential future research directions in this field.