Quantitative Symmetry Profiling of Healthy and Diseased Leaves Using Logarithmic Coefficient Filters

Abstract

This study develops a $m$-fold-symmetry driven framework for quantitative analysis and filtering of leaf images inspired by a generalized logarithmic expansion   Leaves of healthy plants exhibit strong bilateral or higher--order rotational symmetries, while diseased leaves show structural distortions that break these patterns. Motivated by analytic function theory, we interpret the logarithmic coefficients $\gamma_n$ of a conformal mapping as angular harmonic amplitudes that encode local geometric balance. Starting from the polar representation of a leaf image, the log--intensity is expanded in angular Fourier series whose coefficients play the role of the $\gamma_n$. A general $3\times3$ window is then designed to project the local log--image onto the harmonics corresponding to multiples of a chosen symmetry order $m$. This window acts as a discrete filter that suppresses nonsymmetric components while preserving the dominant $m$-fold structure. Bilateral ($m=2$) and higher--order windows are implemented and applied directly to grayscale images without prior segmentation. Experiments on a curated dataset of healthy and unhealthy leaves show that healthy samples achieve bilateral symmetry scores exceeding $0.94$ with a clear dominance of the $m=2$ harmonic, whereas diseased leaves exhibit reduced bilateral scores, weakened $m=2$ energy, and elevated higher--order components. The proposed filter enhances the bilateral component, increases the separation between healthy and unhealthy score distributions, and remains robust to noise and illumination changes. The framework provides a mathematically interpretable tool for early detection of plant disease, automated health classification, and pre--processing for machine learning pipelines. Beyond plant pathology, the same $3\times3$ logarithmic window can be embedded in texture analysis, biomedical imaging, and any application requiring local symmetry enhancement or anomaly detection.