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

TL;DR

This paper investigates Pythagorean-inspired fractal trees to identify variants that closely mimic natural tree structures, developing a fast algorithm and validating realism through CNN classification accuracy, supporting da Vinci's branching rule and the golden ratio.

Contribution

It introduces a flexible, parameterized fractal tree model that matches natural branching patterns and can generate training data for tree species detection.

Findings

Parameters maximizing CNN accuracy support da Vinci's rule.

Golden ratio-based scaling aligns with natural tree structures.

Fractal trees can generate realistic artificial training data.

Abstract

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.