Theory and algorithms for swept manifold intersections

Recent developments in such fields as computer aided geometric design, geometric modeling, and computational topology have generated a spate of interest towards geometric objects called swept volumes. Besides their great applicability in various practical areas, the mere geometry and topology of these entities make them a perfect testbed for novel approaches aimed at analyzing and representing geometric objects. The structure of swept volumes reveals that it is also important to focus on a little simpler, although a very similar type of objects - swept manifolds. In particular, effective computability of swept manifold intersections is of major concern. The main goal of this dissertation is to conduct a study of swept manifolds and, based on the findings, develop methods for computing swept surface intersections. The twofold nature of this goal prompted a division of the work into two distinct parts. At first, a theoretical analysis of swept manifolds is performed, providing a better insight into the topological structure of swept manifolds and unveiling several important properties. In the course of the investigation, several subclasses of swept manifolds are introduced; in particular, attention is focused on regular and critical swept manifolds. Because of the high applicability, additional effort is put into analysis of two-dimensional swept manifolds - swept surfaces. Some of the valuable properties exhibited by such surfaces are generalized to higher dimensions by introducing yet another class of swept manifolds - recursive swept manifolds. In the second part of this work, algorithms for finding swept surface intersections are developed. The need for such algorithms is necessitated by a specific structure of swept surfaces that precludes direct employment of existing intersection methods. The new algorithms are designed by utilizing the underlying ideas of existing intersection techniques and making necessary technical modifications. Such modifications are achieved by employing properties of swept surfaces obtained in the course of the theoretical study. The intersection problems is also considered from a little different prospective. A novel, homology based approach to local characterization of intersections of submanifolds and s-subvarieties of a Euclidean space is presented. It provides a method for distinguishing between transverse and tangential intersection points and determining, in some cases, whether the intersection point belongs to a boundary. At the end, several possible applications of the obtained results are described, including virtual sculpting and modeling of heterogeneous materials

Hierarchical ordering of reticular networks

The structure of hierarchical networks in biological and physical systems has long been characterized using the Horton-Strahler ordering scheme. The scheme assigns an integer order to each edge in the network based on the topology of branching such that the order increases from distal parts of the network (e.g., mountain streams or capillaries) to the "root" of the network (e.g., the river outlet or the aorta). However, Horton-Strahler ordering cannot be applied to networks with loops because they they create a contradiction in the edge ordering in terms of which edge precedes another in the hierarchy. Here, we present a generalization of the Horton-Strahler order to weighted planar reticular networks, where weights are assumed to correlate with the importance of network edges, e.g., weights estimated from edge widths may correlate to flow capacity. Our method assigns hierarchical levels not only to edges of the network, but also to its loops, and classifies the edges into reticular edges, which are responsible for loop formation, and tree edges. In addition, we perform a detailed and rigorous theoretical analysis of the sensitivity of the hierarchical levels to weight perturbations. We discuss applications of this generalized Horton-Strahler ordering to the study of leaf venation and other biological networks.Comment: 9 pages, 5 figures, During preparation of this manuscript the authors became aware of a related work by Katifori and Magnasco, concurrently submitted for publicatio

COVID-19 Heterogeneity in Islands Chain Environment

As 2021 dawns, the COVID-19 pandemic is still raging strongly as vaccines finally appear and hopes for a return to normalcy start to materialize. There is much to be learned from the pandemic's first year data that will likely remain applicable to future epidemics and possible pandemics. With only minor variants in virus strain, countries across the globe have suffered roughly the same pandemic by first glance, yet few locations exhibit the same patterns of viral spread, growth, and control as the state of Hawai'i. In this paper, we examine the data and compare the COVID-19 spread statistics between the counties of Hawai'i as well as examine several locations with similar properties to Hawai'i

GiA Roots: Software for the high throughput analysis of plant root system architecture

Background: Characterizing root system architecture (RSA) is essential to understanding the development and function of vascular plants. Identifying RSA-associated genes also represents an underexplored opportunity for crop improvement. Software tools are needed to accelerate the pace at which quantitative traits of RSA are estimated from images of root networks.Results: We have developed GiA Roots (General Image Analysis of Roots), a semi-automated software tool designed specifically for the high-throughput analysis of root system images. GiA Roots includes user-assisted algorithms to distinguish root from background and a fully automated pipeline that extracts dozens of root system phenotypes. Quantitative information on each phenotype, along with intermediate steps for full reproducibility, is returned to the end-user for downstream analysis. GiA Roots has a GUI front end and a command-line interface for interweaving the software into large-scale workflows. GiA Roots can also be extended to estimate novel phenotypes specified by the end-user.Conclusions: We demonstrate the use of GiA Roots on a set of 2393 images of rice roots representing 12 genotypes from the species Oryza sativa. We validate trait measurements against prior analyses of this image set that demonstrated that RSA traits are likely heritable and associated with genotypic differences. Moreover, we demonstrate that GiA Roots is extensible and an end-user can add functionality so that GiA Roots can estimate novel RSA traits. In summary, we show that the software can function as an efficient tool as part of a workflow to move from large numbers of root images to downstream analysis

Comparison of Pattern Detection Methods in Microarray Time Series of the Segmentation Clock

While genome-wide gene expression data are generated at an increasing rate, the repertoire of approaches for pattern discovery in these data is still limited. Identifying subtle patterns of interest in large amounts of data (tens of thousands of profiles) associated with a certain level of noise remains a challenge. A microarray time series was recently generated to study the transcriptional program of the mouse segmentation clock, a biological oscillator associated with the periodic formation of the segments of the body axis. A method related to Fourier analysis, the Lomb-Scargle periodogram, was used to detect periodic profiles in the dataset, leading to the identification of a novel set of cyclic genes associated with the segmentation clock. Here, we applied to the same microarray time series dataset four distinct mathematical methods to identify significant patterns in gene expression profiles. These methods are called: Phase consistency, Address reduction, Cyclohedron test and Stable persistence, and are based on different conceptual frameworks that are either hypothesis- or data-driven. Some of the methods, unlike Fourier transforms, are not dependent on the assumption of periodicity of the pattern of interest. Remarkably, these methods identified blindly the expression profiles of known cyclic genes as the most significant patterns in the dataset. Many candidate genes predicted by more than one approach appeared to be true positive cyclic genes and will be of particular interest for future research. In addition, these methods predicted novel candidate cyclic genes that were consistent with previous biological knowledge and experimental validation in mouse embryos. Our results demonstrate the utility of these novel pattern detection strategies, notably for detection of periodic profiles, and suggest that combining several distinct mathematical approaches to analyze microarray datasets is a valuable strategy for identifying genes that exhibit novel, interesting transcriptional patterns

Computational Differential Topology

Abstract. Some of the more differential aspects of the nascent field of computational topology are introduced and treated in considerable depth. Relevant categories based upon stratified geometric objects are proposed, and fundamental problems are identified and discussed in the context of both differential topology and computer science. New results on the triangulation of objects in the computational differential categories are proven, and evaluated from the perspective of effective computability (algorithmic solvability). In addition, the elements of innovative, effectively computable approaches for analyzing and obtaining computer generated representations of geometric objects based upon singularity/stratification theory and obstruction theory are formulated. New methods for characterizing complicated intersection sets are proven using differential analysis and homology theory. Also included are brief descriptions of several implementation aspects of some of the approaches described, as well as applications of the results in such areas as virtual sculpting, virtual surgery, modeling of heterogeneous biomaterials, and high speed visualizations

Weak Witnesses for Delaunay triangulations of Submanifold

International audienceThe main result of this paper is an extension of de Silva's Weak Delaunay Theorem to smoothly embedded curves and surfaces in Euclidean space. Assuming a sufficiently fine sampling, we prove that i + 1 points in the sample span an i-simplex in the restricted Delaunay triangulation iff every subset of i+1 points has a weak witness