Detecting Carbon Nanotube Orientation with Topological Analysis of Scanning Electron Micrographs.

As the aerospace industry is increasingly demanding stronger, lightweight materials, ultra-strong carbon nanotube (CNT) composites with highly aligned CNT network structures could be the answer. In this work, a novel methodology applying topological data analysis (TDA) to scanning electron microscope (SEM) images was developed to detect CNT orientation. The CNT bundle extensions in certain directions were summarized algebraically and expressed as visible barcodes. The barcodes were then calculated and converted into the total spread function, V(X, θ), from which the alignment fraction and the preferred direction could be determined. For validation purposes, the random CNT sheets were mechanically stretched at various strain ratios ranging from 0 to 40%, and quantitative TDA was conducted based on the SEM images taken at random positions. The results showed high consistency (R2 = 0.972) compared to Herman's orientation factors derived from polarized Raman spectroscopy and wide-angle X-ray scattering analysis. Additionally, the TDA method presented great robustness with varying SEM acceleration voltages and magnifications, which might alter the scope of alignment detection. With potential applications in nanofiber systems, this study offers a rapid and simple way to quantify CNT alignment, which plays a crucial role in transferring the CNT properties into engineering products.

Geometric and topological approaches to shape variation in Ginkgo leaves.

Leaf shape is a key plant trait that varies enormously. The range of applications for data on this trait requires frequent methodological development so that researchers have an up-to-date toolkit with which to quantify leaf shape. We generated a dataset of 468 leaves produced by Ginkgo biloba, and 24 fossil leaves produced by evolutionary relatives of extant Ginkgo. We quantified the shape of each leaf by developing a geometric method based on elastic curves and a topological method based on persistent homology. Our geometric method indicates that shape variation in modern leaves is dominated by leaf size, furrow depth and the angle of the two lobes at the leaf base that is also related to leaf width. Our topological method indicates that shape variation in modern leaves is dominated by leaf size and furrow depth. We have applied both methods to modern and fossil material: the methods are complementary, identifying similar primary patterns of variation, but also revealing different aspects of morphological variation. Our topological approach distinguishes long-shoot leaves from short-shoot leaves, both methods indicate that leaf shape influences or is at least related to leaf area, and both could be applied in palaeoclimatic and evolutionary studies of leaf shape.

The Persistent Homology Mathematical Framework Provides Enhanced Genotype-to-Phenotype Associations for Plant Morphology.

Efforts to understand the genetic and environmental conditioning of plant morphology are hindered by the lack of flexible and effective tools for quantifying morphology. Here, we demonstrate that persistent-homology-based topological methods can improve measurement of variation in leaf shape, serrations, and root architecture. We apply these methods to 2D images of leaves and root systems in field-grown plants of a domesticated introgression line population of tomato (Solanum pennellii). We find that compared with some commonly used conventional traits, (1) persistent-homology-based methods can more comprehensively capture morphological variation; (2) these techniques discriminate between genotypes with a larger normalized effect size and detect a greater number of unique quantitative trait loci (QTLs); (3) multivariate traits, whether statistically derived from univariate or persistent-homology-based traits, improve our ability to understand the genetic basis of phenotype; and (4) persistent-homology-based techniques detect unique QTLs compared to conventional traits or their multivariate derivatives, indicating that previously unmeasured aspects of morphology are now detectable. The QTL results further imply that genetic contributions to morphology can affect both the shoot and root, revealing a pleiotropic basis to natural variation in tomato. Persistent homology is a versatile framework to quantify plant morphology and developmental processes that complements and extends existing methods.

Morphological Plant Modeling: Unleashing Geometric and Topological Potential within the Plant Sciences.

The geometries and topologies of leaves, flowers, roots, shoots, and their arrangements have fascinated plant biologists and mathematicians alike. As such, plant morphology is inherently mathematical in that it describes plant form and architecture with geometrical and topological techniques. Gaining an understanding of how to modify plant morphology, through molecular biology and breeding, aided by a mathematical perspective, is critical to improving agriculture, and the monitoring of ecosystems is vital to modeling a future with fewer natural resources. In this white paper, we begin with an overview in quantifying the form of plants and mathematical models of patterning in plants. We then explore the fundamental challenges that remain unanswered concerning plant morphology, from the barriers preventing the prediction of phenotype from genotype to modeling the movement of leaves in air streams. We end with a discussion concerning the education of plant morphology synthesizing biological and mathematical approaches and ways to facilitate research advances through outreach, cross-disciplinary training, and open science. Unleashing the potential of geometric and topological approaches in the plant sciences promises to transform our understanding of both plants and mathematics.

A morphometric analysis of vegetation patterns in dryland ecosystems.

Vegetation in dryland ecosystems often forms remarkable spatial patterns. These range from regular bands of vegetation alternating with bare ground, to vegetated spots and labyrinths, to regular gaps of bare ground within an otherwise continuous expanse of vegetation. It has been suggested that spotted vegetation patterns could indicate that collapse into a bare ground state is imminent, and the morphology of spatial vegetation patterns, therefore, represents a potentially valuable source of information on the proximity of regime shifts in dryland ecosystems. In this paper, we have developed quantitative methods to characterize the morphology of spatial patterns in dryland vegetation. Our approach is based on algorithmic techniques that have been used to classify pollen grains on the basis of textural patterning, and involves constructing feature vectors to quantify the shapes formed by vegetation patterns. We have analysed images of patterned vegetation produced by a computational model and a small set of satellite images from South Kordofan (South Sudan), which illustrates that our methods are applicable to both simulated and real-world data. Our approach provides a means of quantifying patterns that are frequently described using qualitative terminology, and could be used to classify vegetation patterns in large-scale satellite surveys of dryland ecosystems.

Rapid automated landmarking for morphometric analysis of three-dimensional facial scans.

Automated phenotyping is essential for the creation of large, highly standardized datasets from anatomical imaging data. Such datasets can support large-scale studies of complex traits or clinical studies related to precision medicine or clinical trials. We have developed a method that generates three-dimensional landmark data that meet the requirements of standard geometric morphometric analyses. The method is robust and can be implemented without high-performance computing resources. We validated the method using both direct comparison to manual landmarking on the same individuals and also analyses of the variation patterns and outlier patterns in a large dataset of automated and manual landmark data. Direct comparison of manual and automated landmarks reveals that automated landmark data are less variable, but more highly integrated and reproducible. Automated data produce covariation structure that closely resembles that of manual landmarks. We further find that while our method does produce some landmarking errors, they tend to be readily detectable and can be fixed by adjusting parameters used in the registration and control-point steps. Data generated using the method described here have been successfully used to study the genomic architecture of facial shape in two different genome-wide association studies of facial shape.

Human Facial Shape and Size Heritability and Genetic Correlations.

The human face is an array of variable physical features that together make each of us unique and distinguishable. Striking familial facial similarities underscore a genetic component, but little is known of the genes that underlie facial shape differences. Numerous studies have estimated facial shape heritability using various methods. Here, we used advanced three-dimensional imaging technology and quantitative human genetics analysis to estimate narrow-sense heritability, heritability explained by common genetic variation, and pairwise genetic correlations of 38 measures of facial shape and size in normal African Bantu children from Tanzania. Specifically, we fit a linear mixed model of genetic relatedness between close and distant relatives to jointly estimate variance components that correspond to heritability explained by genome-wide common genetic variation and variance explained by uncaptured genetic variation, the sum representing total narrow-sense heritability. Our significant estimates for narrow-sense heritability of specific facial traits range from 28 to 67%, with horizontal measures being slightly more heritable than vertical or depth measures. Furthermore, for over half of facial traits, >90% of narrow-sense heritability can be explained by common genetic variation. We also find high absolute genetic correlation between most traits, indicating large overlap in underlying genetic loci. Not surprisingly, traits measured in the same physical orientation (i.e., both horizontal or both vertical) have high positive genetic correlations, whereas traits in opposite orientations have high negative correlations. The complex genetic architecture of facial shape informs our understanding of the intricate relationships among different facial features as well as overall facial development.

Genome-Wide Association Study Reveals Multiple Loci Influencing Normal Human Facial Morphology.

Numerous lines of evidence point to a genetic basis for facial morphology in humans, yet little is known about how specific genetic variants relate to the phenotypic expression of many common facial features. We conducted genome-wide association meta-analyses of 20 quantitative facial measurements derived from the 3D surface images of 3118 healthy individuals of European ancestry belonging to two US cohorts. Analyses were performed on just under one million genotyped SNPs (Illumina OmniExpress+Exome v1.2 array) imputed to the 1000 Genomes reference panel (Phase 3). We observed genome-wide significant associations (p < 5 x 10-8) for cranial base width at 14q21.1 and 20q12, intercanthal width at 1p13.3 and Xq13.2, nasal width at 20p11.22, nasal ala length at 14q11.2, and upper facial depth at 11q22.1. Several genes in the associated regions are known to play roles in craniofacial development or in syndromes affecting the face: MAFB, PAX9, MIPOL1, ALX3, HDAC8, and PAX1. We also tested genotype-phenotype associations reported in two previous genome-wide studies and found evidence of replication for nasal ala length and SNPs in CACNA2D3 and PRDM16. These results provide further evidence that common variants in regions harboring genes of known craniofacial function contribute to normal variation in human facial features. Improved understanding of the genes associated with facial morphology in healthy individuals can provide insights into the pathways and mechanisms controlling normal and abnormal facial morphogenesis.

Genomewide Association Study of African Children Identifies Association of SCHIP1 and PDE8A with Facial Size and Shape.

The human face is a complex assemblage of highly variable yet clearly heritable anatomic structures that together make each of us unique, distinguishable, and recognizable. Relatively little is known about the genetic underpinnings of normal human facial variation. To address this, we carried out a large genomewide association study and two independent replication studies of Bantu African children and adolescents from Mwanza, Tanzania, a region that is both genetically and environmentally relatively homogeneous. We tested for genetic association of facial shape and size phenotypes derived from 3D imaging and automated landmarking of standard facial morphometric points. SNPs within genes SCHIP1 and PDE8A were associated with measures of facial size in both the GWAS and replication cohorts and passed a stringent genomewide significance threshold adjusted for multiple testing of 34 correlated traits. For both SCHIP1 and PDE8A, we demonstrated clear expression in the developing mouse face by both whole-mount in situ hybridization and RNA-seq, supporting their involvement in facial morphogenesis. Ten additional loci demonstrated suggestive association with various measures of facial shape. Our findings, which differ from those in previous studies of European-derived whites, augment understanding of the genetic basis of normal facial development, and provide insights relevant to both human disease and forensics.

Automatic stage identification of Drosophila egg chamber based on DAPI images.

The Drosophila egg chamber, whose development is divided into 14 stages, is a well-established model for developmental biology. However, visual stage determination can be a tedious, subjective and time-consuming task prone to errors. Our study presents an objective, reliable and repeatable automated method for quantifying cell features and classifying egg chamber stages based on DAPI images. The proposed approach is composed of two steps: 1) a feature extraction step and 2) a statistical modeling step. The egg chamber features used are egg chamber size, oocyte size, egg chamber ratio and distribution of follicle cells. Methods for determining the on-site of the polytene stage and centripetal migration are also discussed. The statistical model uses linear and ordinal regression to explore the stage-feature relationships and classify egg chamber stages. Combined with machine learning, our method has great potential to enable discovery of hidden developmental mechanisms.

Morphometrics, 3D Imaging, and Craniofacial Development.

Recent studies have shown how volumetric imaging and morphometrics can add significantly to our understanding of morphogenesis, the developmental basis for variation, and the etiology of structural birth defects. On the other hand, the complex questions and diverse imaging data in developmental biology present morphometrics with more complex challenges than applications in virtually any other field. Meeting these challenges is necessary in order to understand the mechanistic basis for variation in complex morphologies. This chapter reviews the methods and theory that enable the application of modern landmark-based morphometrics to developmental biology and craniofacial development, in particular. We discuss the theoretical foundations of morphometrics as applied to development and review the basic approaches to the quantification of morphology. Focusing on geometric morphometrics, we discuss the principal statistical methods for quantifying and comparing morphological variation and covariation structure within and among groups. Finally, we discuss the future directions for morphometrics in developmental biology that will be required for approaches that enable quantitative integration across the genotype-phenotype map.

Correlations Between the Morphology of Sonic Hedgehog Expression Domains and Embryonic Craniofacial Shape.

Quantitative analysis of gene expression domains and investigation of relationships between gene expression and developmental and phenotypic outcomes are central to advancing our understanding of the genotype-phenotype map. Gene expression domains typically have smooth but irregular shapes lacking homologous landmarks, making it difficult to analyze shape variation with the tools of landmark-based geometric morphometrics. In addition, 3D image acquisition and processing introduce many artifacts that further exacerbate the problem. To overcome these difficulties, this paper presents a method that combines optical projection tomography scanning, a shape regularization technique and a landmark-free approach to quantify variation in the morphology of Sonic hedgehog expression domains in the frontonasal ectodermal zone (FEZ) of avians and investigate relationships with embryonic craniofacial shape. The model reveals axes in FEZ and embryonic-head morphospaces along which variation exhibits a sharp linear relationship at high statistical significance. The technique should be applicable to analyses of other 3D biological structures that can be modeled as smooth surfaces and have ill-defined shape.

Signals from the brain induce variation in avian facial shape.

BACKGROUND: How developmental mechanisms generate the phenotypic variation that is the raw material for evolution is largely unknown. Here, we explore whether variation in a conserved signaling axis between the brain and face contributes to differences in morphogenesis of the avian upper jaw. In amniotes, including both mice and avians, signals from the brain establish a signaling center in the ectoderm (the Frontonasal ectodermal zone or "FEZ") that directs outgrowth of the facial primordia. RESULTS: Here we show that the spatial organization of this signaling center differs among avians, and these correspond to Sonic hedgehog (Shh) expression in the basal forebrain and embryonic facial shape. In ducks this basal forebrain domain is present almost the entire width, while in chickens it is restricted to the midline. When the duck forebrain is unilaterally transplanted into stage matched chicken embryos the face on the treated side resembles that of the donor. CONCLUSIONS: Combined with previous findings, these results demonstrate that variation in a highly conserved developmental pathway has the potential to contribute to evolutionary differences in avian upper jaw morphology. Developmental Dynamics 244:1133-1143, 2015. © 2015 Wiley Periodicals, Inc.

Classification of grass pollen through the quantitative analysis of surface ornamentation and texture.

Taxonomic identification of pollen and spores uses inherently qualitative descriptions of morphology. Consequently, identifications are restricted to categories that can be reliably classified by multiple analysts, resulting in the coarse taxonomic resolution of the pollen and spore record. Grass pollen represents an archetypal example; it is not routinely identified below family level. To address this issue, we developed quantitative morphometric methods to characterize surface ornamentation and classify grass pollen grains. This produces a means of quantifying morphological features that are traditionally described qualitatively. We used scanning electron microscopy to image 240 specimens of pollen from 12 species within the grass family (Poaceae). We classified these species by developing algorithmic features that quantify the size and density of sculptural elements on the pollen surface, and measure the complexity of the ornamentation they form. These features yielded a classification accuracy of 77.5%. In comparison, a texture descriptor based on modelling the statistical distribution of brightness values in image patches yielded a classification accuracy of 85.8%, and seven human subjects achieved accuracies between 68.33 and 81.67%. The algorithmic features we developed directly relate to biologically meaningful features of grass pollen morphology, and could facilitate direct interpretation of unsupervised classification results from fossil material.

A Computational Model of Multidimensional Shape.

We develop a computational model of shape that extends existing Riemannian models of curves to multidimensional objects of general topological type. We construct shape spaces equipped with geodesic metrics that measure how costly it is to interpolate two shapes through elastic deformations. The model employs a representation of shape based on the discrete exterior derivative of parametrizations over a finite simplicial complex. We develop algorithms to calculate geodesics and geodesic distances, as well as tools to quantify local shape similarities and contrasts, thus obtaining a formulation that accounts for regional differences and integrates them into a global measure of dissimilarity. The Riemannian shape spaces provide a common framework to treat numerous problems such as the statistical modeling of shapes, the comparison of shapes associated with different individuals or groups, and modeling and simulation of shape dynamics. We give multiple examples of geodesic interpolations and illustrations of the use of the models in brain mapping, particularly, the analysis of anatomical variation based on neuroimaging data.

Models of normal variation and local contrasts in hippocampal anatomy.

We develop a model of continuous spherical shapes and use it to analyze the anatomy of the hippocampus. To account for the geometry of bends and folds, the model relies on a geodesic metric that is sensitive to first-order deformations. We construct an atlas of the hippocampus as a mean shape and develop statistical models to characterize quantitative and qualitative normal shape variation. We also develop a localization tool to identify local contrasts in the anatomy of different populations. The tool is applied to the detection, characterization and visualization of anatomical differences such as local enlargement and gains in volume on the right hippocampus of blind subjects.

Learning representations for object classification using multi-stage optimal component analysis.

Learning data representations is a fundamental challenge in modeling neural processes and plays an important role in applications such as object recognition. Optimal component analysis (OCA) formulates the problem in the framework of optimization on a Grassmann manifold and a stochastic gradient method is used to estimate the optimal basis. OCA has been successfully applied to image classification problems arising in a variety of contexts. However, as the search space is typically very high dimensional, OCA optimization often requires expensive computational cost. In multi-stage OCA, we first hierarchically project the data onto several low-dimensional subspaces using standard techniques, then OCA learning is performed hierarchically from the lowest to the highest levels to learn about a subspace that is optimal for data discrimination based on the K-nearest neighbor classifier. One of the main advantages of multi-stage OCA lies in the fact that it greatly improves the computational efficiency of the OCA learning algorithm without sacrificing the recognition performance, thus enhancing its applicability to practical problems. In addition to the nearest neighbor classifier, we illustrate the effectiveness of the learned representations on object classification used in conjunction with classifiers such as neural networks and support vector machines.

Elastic shape models for interpolations of curves in image sequences.

Many applications in image analysis are concerned with the temporal evolution of shapes in video sequences. In situations involving low-contrast, low-quality images, human aid is often needed to extract shapes from images. An interesting approach is to use expert help to extract shapes in certain well-separated frames, and to use automated methods to extract shapes from intermediate frames. We present a technique to interpolate between expert generated shapes. This technique preserves salient features in the interpolated shapes, and allows analysts to model a continuous evolution of shapes, instead of a coarse sampling generated by the expert. The basic idea is to establish a correspondence between points on the two end shapes, and to construct a geodesic flow on a shape space maintaining that correspondence. This technique is demonstrated using echocardiagraphic images and infrared human gait sequences.

Analysis of planar shapes using geodesic paths on shape spaces.

For analyzing shapes of planar, closed curves, we propose differential geometric representations of curves using their direction functions and curvature functions. Shapes are represented as elements of infinite-dimensional spaces and their pairwise differences are quantified using the lengths of geodesics connecting them on these spaces. We use a Fourier basis to represent tangents to the shape spaces and then use a gradient-based shooting method to solve for the tangent that connects any two shapes via a geodesic. Using the Surrey fish database, we demonstrate some applications of this approach: 1) interpolation and extrapolations of shape changes, 2) clustering of objects according to their shapes, 3) statistics on shape spaces, and 4) Bayesian extraction of shapes in low-quality images.