Topological Data Structure Research Articles

Metric learning for semi-supervised clustering using pairwise constraints and the geometrical structure of data

Metric learning is a powerful approach for semi-supervised clustering. In this paper, a metric learning method considering both pairwise constraints and the geometrical structure of data is introduced for semi-supervised clustering. At first, a smooth metric is found (based on an optimization problem) using positive constraints as supervisory information. Then, an extension of this method employing both positive and negative constraints is introduced. As opposed to the existing methods, the extended method has the capability of considering both positive and negative constraints while considering the topological structure of data. The proposed metric learning method can improve performance of semi-supervised clustering algorithms. Experimental results on real-world data sets show the effectiveness of this method.

ParTopS: compact topological framework for parallel fragmentation simulations

Cohesive models are used for simulation of fracture, branching and fragmentation phenomena at various scales. Those models require high levels of mesh refinement at the crack tip region so that nonlinear behavior can be captured and physical results obtained. This imposes the use of large meshes that usually result in computational and memory costs prohibitively expensive for a single traditional workstation. If an extrinsic cohesive model is to be used, support for dynamic insertion of cohesive elements is also required. This paper proposes a topological framework for supporting parallel adaptive fragmentation simulations that provides operations for dynamic insertion of cohesive elements, in a uniform way, for both two- and three-dimensional unstructured meshes. Cohesive elements are truly represented and are treated like any other regular element. The framework is built as an extension of a compact adjacency-based serial topological data structure, which can natively handle the representation of cohesive elements. Symmetrical modifications of duplicated entities are used to reduce the communication of topological changes among mesh partitions and also to avoid the use of locks. The correctness and efficiency of the proposed framework are demonstrated by a series of arbitrary insertions of cohesive elements into some sample meshes.

TOPOLOGY-GUIDED TESSELLATION OF QUADRATIC ElEMENTS

Topology-based methods have been successfully used for the analysis and visualization of piecewise-linear functions defined on triangle meshes. This paper describes a mechanism for extending these methods to piecewise-quadratic functions defined on triangulations of surfaces. Each triangular patch is tessellated into monotone regions, so that existing algorithms for computing topological representations of piecewise-linear functions may be applied directly to the piecewise-quadratic function. In particular, the tessellation is used for computing the Reeb graph, a topological data structure that provides a succinct representation of level sets of the function.

RGB Subdivision

We introduce the RGB Subdivision: an adaptive subdivision scheme for triangle meshes, which is based on the iterative application of local refinement and coarsening operators, and generates the same limit surface of the Loop subdivision, independently on the order of application of local operators. Our scheme supports dynamic selective refinement, as in Continuous Level Of Detail models, and it generates conforming meshes at all intermediate steps. The RGB subdivision is encoded in a standard topological data structure, extended with few attributes, which can be used directly for further processing. We present an interactive tool that permits to start from a base mesh and use RGB subdivision to dynamically adjust its level of detail.

A simplicial complex‐based DBMS approach to 3D topographic data modelling

This paper introduces a new compact topological 3D data structure. The proposed method models the real world as a complete decomposition of space and this subdivision is represented by a constrained tetrahedral network (TEN). Operators and definitions from the mathematical field of simplicial homology are used to define and handle this TEN structure. Only tetrahedrons need to be stored explicitly in a (single column) database table, while all simplexes of lower dimensions, constraints and topological relationships can be derived in views. As a result the data structure is relatively compact and easy to update, while it still offers favourable characteristics from a computational point of view as well as presence of topological relationships.

A new approach to discover interlacing data structures in high-dimensional space

The discovery of structures hidden in high-dimensional data space is of great significance for understanding and further processing of the data. Real world datasets are often composed of multiple low dimensional patterns, the interlacement of which may impede our ability to understand the distribution rule of the data. Few of the existing methods focus on the detection and extraction of the manifolds representing distinct patterns. Inspired by the nonlinear dimensionality reduction method ISOmap, in this paper we present a novel approach called Multi-Manifold Partition to identify the interlacing low dimensional patterns. The algorithm has three steps: first a neighborhood graph is built to capture the intrinsic topological structure of the input data, then the dimensional uniformity of neighboring nodes is analyzed to discover the segments of patterns, finally the segments which are possibly from the same low-dimensional structure are combined to obtain a global representation of distribution rules. Experiments on synthetic data as well as real problems are reported. The results show that this new approach to exploratory data analysis is effective and may enhance our understanding of the data distribution.

A general topology-based framework for adaptive insertion of cohesive elements in finite element meshes

Large-scale simulation of separation phenomena in solids such as fracture, branching, and fragmentation requires a scalable data structure representation of the evolving model. Modeling of such phenomena can be successfully accomplished by means of cohesive models of fracture, which are versatile and effective tools for computational analysis. A common approach to insert cohesive elements in finite element meshes consists of adding discrete special interfaces (cohesive elements) between bulk elements. The insertion of cohesive elements along bulk element interfaces for fragmentation simulation imposes changes in the topology of the mesh. This paper presents a unified topology-based framework for supporting adaptive fragmentation simulations, being able to handle two- and three-dimensional models, with finite elements of any order. We represent the finite element model using a compact and “complete” topological data structure, which is capable of retrieving all adjacency relationships needed for the simulation. Moreover, we introduce a new topology-based algorithm that systematically classifies fractured facets (i.e., facets along which fracture has occurred). The algorithm follows a set of procedures that consistently perform all the topological changes needed to update the model. The proposed topology-based framework is general and ensures that the model representation remains always valid during fragmentation, even when very complex crack patterns are involved. The framework correctness and efficiency are illustrated by arbitrary insertion of cohesive elements in various finite element meshes of self-similar geometries, including both two- and three-dimensional models. These computational tests clearly show linear scaling in time, which is a key feature of the present data-structure representation. The effectiveness of the proposed approach is also demonstrated by dynamic fracture analysis through finite element simulations of actual engineering problems.

A Flexible Kernel for Adaptive Mesh Refinement on GPU

We present a flexible GPU kernel for adaptive on-the-fly refinement of meshes with arbitrary topology. By simply reserving a small amount of GPU memory to store a set of adaptive refinement patterns, on-the-fly refinement is performed by the GPU, without any preprocessing nor additional topology data structure. The level of adaptive refinement can be controlled by specifying a per-vertex depth-tag, in addition to usual position, normal, color and texture coordinates. This depth-tag is used by the kernel to instanciate the correct refinement pattern, which will map a refined connectivity on the input coarse polygon. Finally, the refined patch produced for each triangle can be displaced by the vertex shader, using any kind of geometric refinement, such as Bezier patch smoothing, scalar valued displacement, procedural geometry synthesis or subdivision surfaces. This refinement engine does neither require multipass rendering nor any use of fragment processing nor special preprocess of the input mesh structure. It can be implemented on any GPU with vertex shading capabilities.

Extrinsic cohesive modelling of dynamic fracture and microbranching instability in brittle materials

AbstractDynamic crack microbranching processes in brittle materials are investigated by means of a computational fracture mechanics approach using the finite element method with special interface elements and a topological data structure representation. Experiments indicate presence of a limiting crack speed for dynamic crack in brittle materials as well as increasing fracture resistance with crack speed. These phenomena are numerically investigated by means of a cohesive zone model (CZM) to characterize the fracture process. A critical evaluation of intrinsic versus extrinsic CZMs is briefly presented, which highlights the necessity of adopting an extrinsic approach in the current analysis. A novel topology‐based data structure is employed to enable fast and robust manipulation of evolving mesh information when extrinsic cohesive elements are inserted adaptively. Compared to intrinsic CZMs, which include an initial hardening segment in the traction–separation curve, extrinsic CZMs involve additional issues both in implementing the procedure and in interpreting simulation results. These include time discontinuity in stress history, fracture pattern dependence on time step control, and numerical energy balance. These issues are investigated in detail through a ‘quasi‐steady‐state’ crack propagation problem in polymethylmethacrylate. The simulation results compare reasonably well with experimental observations both globally and locally, and demonstrate certain advantageous features of the extrinsic CZM with respect to the intrinsic CZM. Copyright © 2007 John Wiley & Sons, Ltd.

Developmental Word Grounding Through a Growing Neural Network With a Humanoid Robot

This paper presents an unsupervised approach of integrating speech and visual information without using any prepared data. The approach enables a humanoid robot, Incremental Knowledge Robot 1 (IKR1), to learn word meanings. The approach is different from most existing approaches in that the robot learns online from audio-visual input, rather than from stationary data provided in advance. In addition, the robot is capable of learning incrementally, which is considered to be indispensable to lifelong learning. A noise-robust self-organized growing neural network is developed to represent the topological structure of unsupervised online data. We are also developing an active-learning mechanism, called "desire for knowledge," to let the robot select the object for which it possesses the least information for subsequent learning. Experimental results show that the approach raises the efficiency of the learning process. Based on audio and visual data, they construct a mental model for the robot, which forms a basis for constructing IKRI's inner world and builds a bridge connecting the learned concepts with current and past scenes.

A study on two‐stage self‐organizing map and its application to clustering problems

AbstractThis paper presents a two‐stage self‐organizing map algorithm that we call two‐stage SOM which combines Kohonen's basic SOM (BSOM) and Aoki's SOM with threshold operation (THSOM). In the first stage of two‐stage SOM, we use BSOM algorithm in order to acquire topological structure of input data, and then we apply THSOM algorithm so that inactivated code vectors move to appropriate region reflecting the distribution of the input data. Furthermore, we show that two‐stage SOM can be applied to clustering problems. Some experimental results reveal that two‐stage SOM is effective for clustering problems in comparison with conventional methods. © 2007 Wiley Periodicals, Inc. Electr Eng Jpn, 159(1): 46–53, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/eej.20268

自己増殖型ニューラルネットワークを用いたヒューマノイドロボットの発達的言語獲得

This paper presents an unsupervised approach of integrating speech and visual information without using any prepared data(training data). The approach enables a humanoid robot, Incremental Knowledge Robot 1 (IKR1), to learn words' meanings. The approach is different from most existing approaches in that the robot learns online from audio-visual input, rather than from stationary data provided in advance. In addition, the robot is capable of incremental learning, which is considered to be indispensable to lifelong learning. A noise-robust self-organized incremental neural network(SOINN) is developed to represent the topological structure of unsupervised online data. We are also developing an active learning mechanism, called ``desire for knowledge'', to let the robot select the object for which it possesses the least information for subsequent learning. Experimental results show that the approach raises the efficiency of the learning process. Based on audio and visual data, we construct a mental model for the robot, which forms a basis for constructing IKR1's inner world and builds a bridge connecting the learned concepts with current and past scenes.

Model Support Algorithm in Powder Based Three Dimensional Printer

Generally, rapid prototype system, it bases on the operation principle to process layer by layer, and has several kinds of hardware already. The input of STL file format has become the de facto standard in rapid prototyping. With the application of complex material, the variety of product-type using RP technology will be dramatically extended and, no more, be limited in model production. However under the structure of 3D printing, large item production will be possible only if specific support were applied. This study proposes the hardware and software integrated method with the subject of the three-dimensional printing. The hardware utilizes ink-jet of general two-dimensional printer and combines with the traversing mechanism and the process controls and model material, and piles up in accordance with default layer thickness. As the supporting structure in STL-file format is generated through the concepts of the 3D-printing support generating method which is carried out by performing projection and displacement of selected triangular surfaces and then the generated 3D model is further constructed by Open GL and saved in STL-file format. As STL file is read, and then builds adjacency to store into the topology data structure. After the operation of direct slicing rule, hundreds of contours in each layer are obtained by using Visual C ++ programming language and OpenGL library to implement the algorithm above. It can confirm exactness that the software output and systematic feasibility of integrating the hardware through the result of system integration test on the 3D printer.

An incremental network for on-line unsupervised classification and topology learning

This paper presents an on-line unsupervised learning mechanism for unlabeled data that are polluted by noise. Using a similarity threshold-based and a local error-based insertion criterion, the system is able to grow incrementally and to accommodate input patterns of on-line non-stationary data distribution. A definition of a utility parameter, the error-radius, allows this system to learn the number of nodes needed to solve a task. The use of a new technique for removing nodes in low probability density regions can separate clusters with low-density overlaps and dynamically eliminate noise in the input data. The design of two-layer neural network enables this system to represent the topological structure of unsupervised on-line data, report the reasonable number of clusters, and give typical prototype patterns of every cluster without prior conditions such as a suitable number of nodes or a good initial codebook.

A Study on Two-stage Self-Organizing Map and Its Application to Clustering Problems

This paper presents a two-stage self-organizing map algorithm what we call Two-stage SOM which combines Kohonen's basic SOM (BSOM) and Aoki's SOM with threshold operation (THSOM). In the first stage of Two-stage SOM, we use BSOM algorithm in order to acquire topological structure of input data, and then we apply THSOM algorithm so that inactivated code-vectors move to appropriate region reflecting the distribution of the input data. Furthermore, we show that Two-stage SOM can be applied to clustering problems. Some experimental results reveal that Two-stage SOM is effective for clustering problems in comparison with conventional methods.

A compact adjacency-based topological data structure for finite element mesh representation

SUMMARY This paper presents a novel compact adjacency-based topological data structure for finite element mesh representation. The proposed data structure is designed to support, under the same framework, both two- and three-dimensional meshes, with any type of elements defined by templates of ordered nodes. When compared to other proposals, our data structure reduces the required storage space while being ‘complete’, in the sense that it preserves the ability to retrieve all topological adjacency relationships in constant time or in time proportional to the number of retrieved entities. Element and node are the only entities explicitly represented. Other topological entities, which include facet, edge, and vertex, are implicitly represented. In order to simplify accessing topological adjacency relationships, we also define and implicitly represent oriented entities, associated to the use of facets, edges, and vertices by an element. All implicit entities are represented by concrete types, being handled as values, which avoid usual problems encountered in other reduced data structures when performing operations such as entity enumeration and attribute attachment. We also extend the data structure with the use of ‘reverse indices’, which improves performance for extracting adjacency relationships while maintaining storage space within reasonable limits. The data structure effectiveness is demonstrated by two different applications: for supporting fragmentation simulation and for supporting volume rendering algorithms. Copyright 2005 John Wiley & Sons, Ltd.

Variable-scale Topological Data Structures Suitable for Progressive Data Transfer: The GAP-face Tree and GAP-edge Forest

This paper presents the first data structure for a variable scale representation of an area partitioning without redundancy of geometry. At the highest level of detail, the areas are represented using a topological structure based on faces and edges; there is no redundancy of geometry in this structure as the shared boundaries (edges) between neighbor areas are stored only once. Each edge is represented by a Binary Line Generalization (BLG)-tree, which enables selection of the proper representation for a given scale. Further, there is also no geometry redundancy between the different levels of detail. An edge at a higher importance level (less detail) does not contain copies of the lower-level edges or coordinates (more detail), but it is represented by efficiently combining their corresponding BLG trees. Which edges have to be combined follows from the generalization computation, and this is stored in a data structure. This data structure turns out to be a set of trees, which will be called the (Generalized Area Partitioning) GAP-edge forest. With regard to faces, the generalization result can be captured in a single tree structure for the parent-child relationships—the GAP face-tree. At the client side there are no geometric computations necessary to compute the polygon representations of the faces, merely following the topological references is sufficient. Finally, the presented data structure is also suitable for progressive transfer of vector maps, assuming that the client maintains a local copy of the GAP-face tree and the GAP-edge forest.

A procedural approach of inspecting CAD model errors

Neutral format translators such as STEP and IGES have been used to exchange CAD model between the various CAD systems in the product development process. But imperfect interoperability imposes costs on the industry due to higher costs of design and production and slower implementation of design changes. In this paper, we propose a procedural approach to inspect the CAD model's errors that occur in a neutral format. We employed two separate inspection processes for checking CAD model errors without wasting excessive resources: the topological data structure and the geometrical data structure. We developed the 3D CAD model inspection system to check the topological and geometrical errors and applied it to case study. An inspection record can be edited in HTML, and thus can be shared by any designer at any time.

The use of topological data-structures in the distributed solution of non-conventional finite element formulations

In this paper we present the advantages of using topological data-structures in the distributed solution of non-conventional finite element models. These data-structures may also be used with other FE formulations. The implementation, derived from Baumgart's winged-edge data-structure, allows to efficiently process all the information related to the finite element mesh and to each problem and formulation. The parallel performance on an heterogeneous and on an homogeneous distributed platform is presented for a series of applications in order to evaluate the efficiency of the approach.

A Growing Neural Network for Online Unsupervised Learning

New online learning is proposed for unsupervised classification and topology representation. The combination of similarity threshold and local accumulated error suits the algorithm for nonstationary data distribution. A novel online criterion for removal of nodes is proposed to classify the data set well and eliminate noise. The use of a utility parameter, error-radius, is able to judge if insertion is successful and control the increase of nodes. As shown in experiment results, the system can represent the topological structure of unsupervised online data, report the reasonable number of clusters, and give typical prototype patterns of every cluster without priori conditions such as a suitable number of nodes or a good initial codebook.