Doctor of Philosophy

dissertationShape analysis is a well-established tool for processing surfaces. It is often a first step in performing tasks such as segmentation, symmetry detection, and finding correspondences between shapes. Shape analysis is traditionally employed on well-sampled surfaces where the geometry and topology is precisely known. When the form of the surface is that of a point cloud containing nonuniform sampling, noise, and incomplete measurements, traditional shape analysis methods perform poorly. Although one may first perform reconstruction on such a point cloud prior to performing shape analysis, if the geometry and topology is far from the true surface, then this can have an adverse impact on the subsequent analysis. Furthermore, for triangulated surfaces containing noise, thin sheets, and poorly shaped triangles, existing shape analysis methods can be highly unstable. This thesis explores methods of shape analysis applied directly to such defect-laden shapes. We first study the problem of surface reconstruction, in order to obtain a better understanding of the types of point clouds for which reconstruction methods contain difficulties. To this end, we have devised a benchmark for surface reconstruction, establishing a standard for measuring error in reconstruction. We then develop a new method for consistently orienting normals of such challenging point clouds by using a collection of harmonic functions, intrinsically defined on the point cloud. Next, we develop a new shape analysis tool which is tolerant to imperfections, by constructing distances directly on the point cloud defined as the likelihood of two points belonging to a mutually common medial ball, and apply this for segmentation and reconstruction. We extend this distance measure to define a diffusion process on the point cloud, tolerant to missing data, which is used for the purposes of matching incomplete shapes undergoing a nonrigid deformation. Lastly, we have developed an intrinsic method for multiresolution remeshing of a poor-quality triangulated surface via spectral bisection

Fast Image and LiDAR alignment based on 3D rendering in sensor topology

Mobile Mapping Systems are now commonly used in large urban acquisition campaigns. They are often equiped with LiDAR sensors and optical cameras, providing very large multimodal datasets. The fusion of both modalities serves different purposes such as point cloud colorization, geometry enhancement or object detection. However, this fusion task cannot be done directly as both modalities are only coarsely registered. This paper presents a fully automatic approach for LiDAR projection and optical image registration refinement based on LiDAR point cloud 3D renderings. First, a coarse 3D mesh is generated from the LiDAR point cloud using the sensor topology. Then, the mesh is rendered in the image domain. After that, a variational approach is used to align the rendering with the optical image. This method achieves high quality results while performing in very low computational time. Results on real data demonstrate the efficiency of the model for aligning LiDAR projections and optical images

СУЩЕСТВЕННЫЕ УСЛОВИЯ ТРУДА

Modern data science uses topological methods to find the structural features of data sets before further supervised or unsupervised analysis. Geometry and topology are very natural tools for analysing massive amounts of data since geometry can be regarded as the study of distance functions. Mathematical formalism, which has been developed for incorporating geometric and topological techniques, deals with point cloud data sets, i.e. finite sets of points. It then adapts tools from the various branches of geometry and topology for the study of point cloud data sets. The point clouds are finite samples taken from a geometric object, perhaps with noise. Topology provides a formal language for qualitative mathematics, whereas geometry is mainly quantitative. Thus, in topology, we study the relationships of proximity or nearness, without using distances. A map between topological spaces is called continuous if it preserves the nearness structures. Geometrical and topological methods are tools allowing us to analyse highly complex data. These methods create a summary or compressed representation of all of the data features to help to rapidly uncover particular patterns and relationships in data. The idea of constructing summaries of entire domains of attributes involves understanding the relationship between topological and geometric objects constructed from data using various features. A common thread in various approaches for noise removal, model reduction, feasibility reconstruction, and blind source separation, is to replace the original data with a lower dimensional approximate representation obtained via a matrix or multi-directional array factorization or decomposition. Besides those transformations, a significant challenge of feature summarization or subset selection methods for Big Data will be considered by focusing on scalable feature selection. Lower dimensional approximate representation is used for Big Data visualization. The cross-field between topology and Big Data will bring huge opportunities, as well as challenges, to Big Data communities. This survey aims at bringing together state-of-the-art research results on geometrical and topological methods for Big Data.Peer ReviewedPostprint (author's final draft

Complex scene modeling and segmentation with deformable simplex meshes

In this thesis we present a system for 3D reconstruction and segmentation of complex real world scenes. The input to the system is an unstructured cloud of 3D points. The output is a 3D model for each object in the scene. The system starts with a model that encloses the input point cloud. A deformation process is applied to the initial model so it gets close to the point cloud in terms of distance, geometry and topology. Once the deformation stops the model is analyzed to check if more than one object is present in the point cloud. If necessary a segmentation process splits the model into several parts that correspond to each object in the scene. Using this segmented model the point cloud is also segmented. Each resulting sub-cloud is treated as a new input to the system. If, after the deformation process, the model is not segmented a refinement process improves the objective and subjective quality of the model by concentrating vertices around high curvature areas. The simplex mesh reconstruction algorithm was modified and extended to suit our application. A novel segmentation algorithm was designed to be applied on the simplex mesh. We test the system with synthetic and real data obtained from single objects, simple. and complex scenes. In the case of the synthetic data different levels of noise are added to examine the performance of the system. The results show that the systems performs well for either of the three cases and also in the presence of low levels of noise

Extranoematic artifacts: neural systems in space and topology

During the past several decades, the evolution in architecture and engineering went through several stages of exploration of form. While the procedures of generating the form have varied from using physical analogous form-finding computation to engaging the form with simulated dynamic forces in digital environment, the self-generation and organization of form has always been the goal. this thesis further intend to contribute to self-organizational capacity in Architecture. The subject of investigation is the rationalizing of geometry from an unorganized point cloud by using learning neural networks. Furthermore, the focus is oriented upon aspects of efficient construction of generated topology. Neural network is connected with constraining properties, which adjust the members of the topology into predefined number of sizes while minimizing the error of deviation from the original form. The resulted algorithm is applied in several different scenarios of construction, highlighting the possibilities and versatility of this method

Topological data analysis of contagion maps for examining spreading processes on networks

Social and biological contagions are influenced by the spatial embeddedness of networks. Historically, many epidemics spread as a wave across part of the Earth's surface; however, in modern contagions long-range edges -- for example, due to airline transportation or communication media -- allow clusters of a contagion to appear in distant locations. Here we study the spread of contagions on networks through a methodology grounded in topological data analysis and nonlinear dimension reduction. We construct "contagion maps" that use multiple contagions on a network to map the nodes as a point cloud. By analyzing the topology, geometry, and dimensionality of manifold structure in such point clouds, we reveal insights to aid in the modeling, forecast, and control of spreading processes. Our approach highlights contagion maps also as a viable tool for inferring low-dimensional structure in networks.Comment: Main Text and Supplementary Informatio