Quantifying phenotypic variation through local persistent homology and imaging

Understanding the genetic basis of phenotypic variation in organisms is a central problem in developmental and evolutionary biology. In plant science, to gain insights on such problems as how plants respond to environmental changes and how to breed the next generation of crops, a sound quantification of the variation in complex plant phenotypes is crucial. For example, the shape of leaves, the architecture of root systems, and the morphology of pollen grains are all important and interesting phenotypic traits that require mathematical informed methods to model their variation comprehensively.;In this dissertation, we develop topological methods and algorithms based on persistent homology, which let us construct informative summaries of the shape of data. We propose a localized form of persistent homology represented by a continuous persistence diagram field. We prove that such fields are stable and robust to noise and outliers. This technique lets us produce compact, and yet rich summaries of global and local morphology useful for modeling and quantifying variation in complex shapes. This enables statistical approaches such as quantitative trait loci (QTL) analysis, time series analysis of dynamical traits, and the investigation of correlations between morphological traits to study their evolution and developmental constraints.;We apply the methods to: (i) QTL analysis of multiple tomato introgression lines through a study of leaf shape and root architecture; (ii) time series analysis of dynamic growing maize root systems; (iii) quantitative analysis of morphology of grass pollen grains; and (iv) an analysis of the complexity of dryland spatial vegetation patterns.