Topological Data Structure Research Articles

3D TOPOLOGICAL SUPPORT IN SPATIAL DATABASES: AN OVERVIEW

Abstract. The storage of spatial data that consists of spatial and non-spatial properties requires a database management system that possesses spatial functions that can cater to the spatial characteristics of data. These characteristics include the geometrical shape, topological and positional information. Parallel to how geometries describe the shape of an object, topological information is also an important spatial property which describes how the geometries in a space are related to each other. This information describes the connectivity, containment and adjacencies of spatial objects which are the foundation for more complex analysis such as navigation, data reconstruction, spatial queries and others. However, the topological support provided by spatial databases varies. This paper provided an overview on the current implementations of topological support in spatial databases such as ArcGIS, QGIS, PostgreSQL and others. The native topology in most spatial databases was found to be 2D topology maintained by 2D topology rules with limited representation of 3D topological relationships. Consequently, 3D objects represented by 2D topology had to be decomposed into objects of lower dimensions. Approaches to implement additional topological support for spatial databases included the use of topological data models, data structures, operators, and rules. 3D applications such as 3D cadastre required more detailed representations of topological information which required a more comprehensive 3D topological data model. Nonetheless, comprehensive preservation of topological information also mandates voluminous storage and higher computational efficiency. Thus, the appropriate 3D topological support should be provided in spatial databases to accurately represent 3D objects and meet 3D analysis requirements.

MarkovHC: Markov hierarchical clustering for the topological structure of high-dimensional single-cell omics data with transition pathway and critical point detection.

Clustering cells and depicting the lineage relationship among cell subpopulations are fundamental tasks in single-cell omics studies. However, existing analytical methods face challenges in stratifying cells, tracking cellular trajectories, and identifying critical points of cell transitions. To overcome these, we proposed a novel Markov hierarchical clustering algorithm (MarkovHC), a topological clustering method that leverages the metastability of exponentially perturbed Markov chains for systematically reconstructing the cellular landscape. Briefly, MarkovHC starts with local connectivity and density derived from the input and outputs a hierarchical structure for the data. We firstly benchmarked MarkovHC on five simulated datasets and ten public single-cell datasets with known labels. Then, we used MarkovHC to investigate the multi-level architectures and transition processes during human embryo preimplantation development and gastric cancer procession. MarkovHC found heterogeneous cell states and sub-cell types in lineage-specific progenitor cells and revealed the most possible transition paths and critical points in the cellular processes. These results demonstrated MarkovHC’s effectiveness in facilitating the stratification of cells, identification of cell populations, and characterization of cellular trajectories and critical points.

Knowledge Graph Completion Based on the Joint Embedding of Entities

As the last step in the process of knowledge graph construction, knowledge graph completion predicts and inferences new factual triples according to the existing knowledge, which is of great significance to the downstream application of the knowledge graph. The knowledge graph completion technology learns representations of entities and relations in the knowledge graph and reconstructs triples factual relations in the representation space to measure their effectiveness, so as to predict and supplement entities and relations. At present, Graph Convolutional Network (GCN) is widely used in entity modeling, which combines the graph structure information to learn the vector representation of entities. However, most algorithms only focus on static knowledge graph completion, ignoring the important role of time in the knowledge graph, and the modeling of entities is not comprehensive enough. In this paper, a knowledge graph completion based on the joint embedding of entities is proposed, which combines structural feature embedding of the graph and dynamic embedding of entities to carry out multi-level learning of entities to improve the effect of entity prediction. Structural feature embedding of the graph takes advantage of the powerful feature aggregation capability of GCN to integrate the knowledge graph as the topological structure and node features of graph data into the representation learning of entities. Dynamic embedding of entities takes advantage of the characteristics that entities will behave in different states at different times and divides the entity features into two parts for learning. One part of the features changes dynamically over time and the other part retains its static properties. In this paper, the probability model ConvTransE is used to construct the model of relations and triples, ConvTransE uses the convolutional neural network to extract the semantic information of the triples and predict the missing elements. We have conducted experiments on public datasets and achieved better results on Hit@K and MRR.

Surface network extraction from high resolution digital terrain models

A surface network is a topological data structure formed by a set of thalwegs and ridges on a digital terrain model. Its computation relies on the detection of saddles on the terrain. Hence, computation methods must guarantee enough saddles are detected but also that no improper conflicts between ridges and thalwegs are created, leading to an inconsistent network. This paper presents a new approach that maximizes the number of saddles and ensures this topological consistency for high-resolution terrain models represented by a raster grid. The grid is triangulated in order to preserve saddles and to facilitate thalweg and ridge computation. It does not require any user parameter and lines remain aligned with triangulation edges, avoiding numerical errors. The method also includes a coherent partitioning of the terrain into hills and dales. A case study shows that the surface network computation can be achieved in reasonable time and hence can be applied to the analysis of large terrain models.

The Stellar decomposition: A compact representation for simplicial complexes and beyond

We introduce the Stellar decomposition, a model for efficient topological data structures over a broad range of simplicial and cell complexes. A Stellar decomposition of a complex is a collection of regions indexing the complex’s vertices and cells such that each region has sufficient information to locally reconstruct the star of its vertices, i.e., the cells incident in the region’s vertices. Stellar decompositions are general in that they can compactly represent and efficiently traverse arbitrary complexes with a manifold or non-manifold domain. They are scalable to complexes in high dimension and of large size, and they enable users to easily construct tailored application-dependent data structures using a fraction of the memory required by a corresponding global topological data structure on the complex.As a concrete realization of this model for spatially embedded complexes, we introduce the Stellar tree, which combines a nested spatial tree with a simple tuning parameter to control the number of vertices in a region. Stellar trees exploit the complex’s spatial locality by reordering vertex and cell indices according to the spatial decomposition and by compressing sequential ranges of indices. Stellar trees are competitive with state-of-the-art topological data structures for manifold simplicial complexes and offer significant improvements for cell complexes and non-manifold simplicial complexes. We conclude with a high-level description of several mesh processing and analysis applications that utilize Stellar trees to process large datasets.

Graph embedding clustering: Graph attention auto-encoder with cluster-specificity distribution

Towards exploring the topological structure of data, numerous graph embedding clustering methods have been developed in recent years, none of them takes into account the cluster-specificity distribution of the nodes representations, resulting in suboptimal clustering performance. Moreover, most existing graph embedding clustering methods execute the nodes representations learning and clustering in two separated steps, which increases the instability of its original performance. Additionally, rare of them simultaneously takes node attributes reconstruction and graph structure reconstruction into account, resulting in degrading the capability of graph learning. In this work, we integrate the nodes representations learning and clustering into a unified framework, and propose a new deep graph attention auto-encoder for nodes clustering that attempts to learn more favorable nodes representations by leveraging self-attention mechanism and node attributes reconstruction. Meanwhile, a cluster-specificity distribution constraint, which is measured by ℓ1,2-norm, is employed to make the nodes representations within the same cluster end up with a common distribution in the dimension space while representations with different clusters have different distributions in the intrinsic dimensions. Extensive experiment results reveal that our proposed method is superior to several state-of-the-art methods in terms of performance.

Faster repetition-aware compressed suffix trees based on Block Trees

The suffix tree is a fundamental data structure in stringology, but its space usage, though linear, is an important problem in applications like Bioinformatics. We design and implement a new compressed suffix tree (CST) targeted to highly repetitive texts, such as large genomic collections of the same species. Our first contribution is to enhance the Block Tree, a data structure that captures the repetitiveness of its input sequence, to represent the topology of trees with large repeated subtrees. Our so-called Block-Tree Compressed Topology (BT-CT) data structure augments the Block Tree nodes with data that speeds up tree navigation. Our Block-Tree CST (BT-CST), in turn, uses the BT-CT to compress the topology of the suffix tree, and also replaces the sampling of the suffix array and its inverse with grammar- and/or Block-Tree-based representations of those arrays.Our experimental results show that BT-CTs reach navigation speeds comparable to compact tree representations that are insensitive to repetitiveness, while using 2–10 times less space on the topologies of the suffix trees of repetitive collections. Our BT-CST is slightly larger than previous repetition-aware suffix trees based on grammar-compressed topologies, but outperforms them in time, often by orders of magnitude.

Persistent Homology: Application To Monitoring Hydraulic Fracturing

Persistent homology is a topological data analysis tool which is reflecting changes in topological structure of data along its scale. Application of persistent homology to monitoring hydraulic fracturing which is allowing researchers to consider prior information in a natural way is given in the article

SFCM: A Fuzzy Clustering Algorithm of Extracting the Shape Information of Data

Topological data analysis is a new theoretical trend using topological techniques to mine data. This approach helps determine topological data structures. It focuses on investigating the global shape of data rather than on local information of high-dimensional data. The Mapper algorithm is considered as a sound representative approach in this area. It is used to cluster and identify concise and meaningful global topological data structures that are out of reach for many other clustering methods. In this article, we propose a new method called the Shape Fuzzy C-Means (SFCM) algorithm, which is constructed based on the Fuzzy C-Means algorithm with particular features of the Mapper algorithm. The SFCM algorithm can not only exhibit the same clustering ability as the Fuzzy C-Means but also reveal some relationships through visualizing the global shape of data supplied by the Mapper. We present a formal proof and include experiments to confirm our claims. The performance of the enhanced algorithm is demonstrated through a comparative analysis involving the original algorithm, Mapper, and the other fuzzy set based improved algorithm, F-Mapper, for synthetic and real-world data. The comparison is conducted with respect to output visualization in the topological sense and clustering stability.

Tetrahedral Trees

We address the problem of performing efficient spatial and topological queries on large tetrahedral meshes with arbitrary topology and complex boundaries. Such meshes arise in several application domains, such as 3D Geographic Information Systems (GISs), scientific visualization, and finite element analysis. To this aim, we propose Tetrahedral trees , a family of spatial indexes based on a nested space subdivision (an octree or a kD-tree) and defined by several different subdivision criteria. We provide efficient algorithms for spatial and topological queries on Tetrahedral trees and compare to state-of-the-art approaches. Our results indicate that Tetrahedral trees are an improvement over R * -trees for querying tetrahedral meshes; they are more compact, faster in many queries, and stable at variations of construction thresholds. They also support spatial queries on more general domains than topological data structures, which explicitly encode adjacency information for efficient navigation but have difficulties with domains with a non-trivial geometric or topological shape.

문장형 데이터 구조를 사용한 토폴로지컬 데이터의 일반화와 처리 방법

문장형 데이터 구조를 사용한 토폴로지컬 데이터의 일반화와 처리 방법

3D Topological Validation of Compact Abstract Cell Complexes (CACC) Data Structure for Buildings in CityGML

3D models without the preservation of 3D topological information hinders the ability of 3D models to serve its full potential in terms of 3D analyses. The support of 3D topology is crucial for analyses that requires information regarding adjacencies and connectivity. One of the ways to maintain topological information is by implementing a topological data structure such as the Compact Abstract Cell Complexes (CACC) topological data structure. This paper demonstrates the topological validation for the implementation of the CACC topological data structure implemented for buildings in LoD2 CityGML. Directed graphs and adjacency matrices were constructed for the test datasets of buildings in CityGML. The in-degree and out-degree for all vertices were calculated based on the adjacency matrices. Based on the “Hand-shaking” theorem, the number of α₀-cycles of the CACC topological data structure which connects points to form 1D topological links was compared to the number of directed edges of the constructed directed graphs. Therefore, the implementation of the CACC topological data structure for buildings in LoD2 CityGML was found to be topologically sound.

Block-Constraint Laplacian-Regularized Low-Rank Representation and Its Application for Cancer Sample Clustering Based on Integrated TCGA Data

Low-Rank Representation (LRR) is a powerful subspace clustering method because of its successful learning of low-dimensional subspace of data. With the breakthrough of “OMics” technology, many LRR-based methods have been proposed and used to cancer clustering based on gene expression data. Moreover, studies have shown that besides gene expression data, some other genomic data in TCGA also contain important information for cancer research. Therefore, these genomic data can be integrated as a comprehensive feature source for cancer clustering. How to establish an effective clustering model for comprehensive analysis of integrated TCGA data has become a key issue. In this paper, we develop the traditional LRR method and propose a novel method named Block-constraint Laplacian-Regularized Low-Rank Representation (BLLRR) to model multigenome data for cancer sample clustering. The proposed method is dedicated to extracting more abundant subspace structure information from multiple genomic data to improve the accuracy of cancer sample clustering. Considering the heterogeneity of different genome data, we introduce the block-constraint idea into our method. In BLLRR decomposition, we treat each genome data as a data block and impose different constraints on different data blocks. In addition, graph Laplacian is also introduced into our method to better learn the topological structure of data by preserving the local geometric information. The experiments demonstrate that the BLLRR method can effectively analyze integrated TCGA data and extract more subspace structure information from multigenome data. It is a reliable and efficient clustering algorithm for cancer sample clustering.

Multi-level uncorrelated discriminative shared Gaussian process for multi-view facial expression recognition

In multi-view facial expression recognition, discriminative shared Gaussian process latent variable model (DS-GPLVM) gives better performance than that of linear and nonlinear multi-view learning-based methods. However, Laplacian-based prior used in DS-GPLVM only captures topological structure of data space without considering the inter-class separability of the data, and hence the obtained latent space is suboptimal. So, we propose a multi-level uncorrelated DS-GPLVM (ML-UDSGPLVM) model which searches a common uncorrelated discriminative latent space learned from multiple observable spaces. A novel prior is proposed, which not only depends on the topological structure of the intra-class data, but also on the local-between-class-scatter-matrix of the data onto the latent manifold. The proposed approach employs an hierarchical framework, in which, expressions are first divided into three sub-categories. Subsequently, each of the sub-categories are further classified to identify the constituent basic expressions. Experimental results show that the proposed method outperforms state-of-the-art methods in many instances.

Distributions of Matching Distances in Topological Data Analysis

In topological data analysis, we want to discern topological and geometric structure of data, and to understand whether or not certain features of data are significant as opposed to simply random noise. While progress has been made on statistical techniques for single-parameter persistence, the case of two-parameter persistence, which is highly desirable for real-world applications, has been less studied. This paper provides an accessible introduction to two-parameter persistent homology and presents results about matching distance between 2-D persistence modules obtained from families of point clouds. Results include observations of how differences in geometric structure of point clouds affect the matching distance between persistence modules. We offer these results as a starting point for the investigation of more complex data.

ROS-based robot offline planning simulation system

In order to realize the rapid and automatic generation of robot motion trajectory, a robot offline planning simulation system was developed based on ROS platform. This paper analyzes the model STL file, reconstructs the topology data structure, generates the model minimum bounding box, and obtains the model surface path through the slicing algorithm to generate the processing path. The system loads the Moveit motion planner, applies different motion planning methods, generates a collision-free, continuous motion trajectory, and visualizes the offline planning process. The simulation verification system can realize the functions of rapid path generation on the workpiece surface and obstacle avoidance motion planning of the robot arm. It can realize offline programming of robots and improve the efficiency of robot teaching.

Method for 3D City Building Continuous Transformation Based on an Improved LOD Topological Data Structure

A 3D city model is an intuitive tool that is used to describe cities. Currently, level-of-detail (LOD) technology is used to meet different visual demands for 3D city models by weighting the rendering efficiency against the details of the model. However, when the visual demands change, the “popping” phenomenon appears when making transformations between different LOD models. We optimized this popping phenomenon by improving the data structure that focuses on 3D city building models and combined it with the facet shift algorithm based on minimal features. Unlike generating finite LOD models in advance, the proposed continuous LOD topology data structure is able to store the changes between different LOD models. By reasonably using the change information, continuous LOD transformation becomes possible. The experimental results showed that the continuous LOD transformation based on the proposed data structure worked well, and the improved data structure also performed well in memory occupation.

SPATIAL ADJACENCY ANALYSIS OF CITYGML BUILDINGS VIA 3D TOPOLOGICAL DATA STRUCTURE

Abstract. Adjacencies between objects provides the most basic connectivity information of objects. This connectivity information provides support for more complex 3D spatial analysis such as 3D navigation, nearest neighbour and others. In 3D models, the connectivity information is maintained by building a comprehensive 3D topology. As the international standard for 3D city models, CityGML employs a simple XML links mechanism that references related entities to each other as a means of maintaining topological information. This method fulfils the purpose of relating connected entities but, it does not describe how the entities are related or in other words its adjacencies. In this study, a 3D topological data structure was utilised to preserve topological primitives and maintain connectivity information for CityGML datasets of buildings in LoD2. The adjacencies tested in this study were based on the topological links maintained by the Compact Abstract Cell Complexes 3D topological data structure. Four types of adjacencies were tested which are Point-to-Line, Line-to-Surface, Surface-to-Surface and Volume-to-Volume adjacency. As a result, all adjacencies were able to be executed for both datasets which consisted of two connected buildings and disjointed buildings. It was found that the ability of the 3D topological data structure to preserve topological primitives and build topological links supported the maintenance of connectivity information between buildings. The maintenance of connectivity information was also not limited to objects of the same dimension and could extend to connectivity between building elements in different dimensions.

Hydromechanical modeling of unrestricted crack propagation in fractured formations using intrinsic cohesive zone model

Production from unconventional reservoirs has revolutionized the oil and gas industry strongly through the development of innovative applications, such as horizontal drilling and shale stimulation by hydraulic fracturing. However, the presence of geological discontinuities such as natural fractures (NFs) complicates the hydraulic fracturing treatment and affects the general geometry of the hydraulic fracture (HF). For that reason, it is essential to develop and implement robust numerical methods capable of simulating the phenomena present during hydraulic fracturing in a naturally fractured formation. This paper presents a novel mesh fragmentation technique to simulate unrestricted hydraulic fracture propagation in fractured media. This method is based on the hydromechanical zero thickness interface element combined with the cohesive zone model (CZM). An algorithm for inserting the especial triple-noded interface elements into conventional 2D finite element meshes is described. We introduce a simple but effective topological data structure that considers each finite element independent of others. The proposed topological structure ensures the generation of new nodes and faces which are necessary for the insertion of interface elements. The implementation is validated by simulating some cases of fracturing experiments in laboratory. The numerical procedure provides very good agreement with the laboratory tests and shows the robustness of the proposed computational approach. Some advantages and limitations of the proposed methodology are discussed.

Unsupervised spectral clustering for shield tunneling machine monitoring data with complex network theory

Abstract Extraction of underlying knowledge from monitoring data is beneficial to on-site management during shield tunneling construction. However, it remains a challenge to unsupervisedly learn from these monitoring data due to the nature of high-dimensional, miscellaneous, and highly nonlinear. This study proposes a new systematic approach to classify shield machine monitoring data by integrating spectral clustering (SC) and complex network (CN) theory. In this approach, CN theory is introduced to obtain the topological relations and network structure of machine monitoring data directly. Based on the network topology, SC is employed to classify these data with unbiased similarity measurement. A river-crossing shield tunnel is used in this study to validate the effectiveness and feasibility of the proposed approach. It's demonstrated that the proposed approach outperforms the other SC methods for unsupervised classification of the machine monitoring data. The classification of machine monitoring data with the proposed approach has a potential value in machine performance and geological risk assessment during shield tunneling construction.