Hierarchical Triangulation Research Articles

Development of Algorithm and Data Structure for 2D Regions Discrete Model Elements Accelerated Search

This paper is devoted to improving the search request processing productivity for planar discrete models used in engineering software. A data structure has been developed to accelerate the search for discretization elements based on a hierarchical triangular mesh. The developed indexing structure is built by a downward iterative algorithm, which constructs each new level of the hierarchy based on the previous level or indexed triangulation by simplifying it, which ensures that the morphology of the triangulation mesh is preserved throughout all levels of the hierarchical indexing structure. The developed building algorithm ensures the presence of tree-like connections between the levels of the hierarchical triangulation mesh, which allows downward navigation between geometrically close triangles. Search acceleration is achieved by performing a directed search in the top level of the indexing structure and then navigating between levels using downward links until the indexed triangle is found. The program implementation was carried out using C++17, and visualization of triangulation grids and isolines was carried out using the ObjectARX library. Based on the software implementation, an executive library was created.

Compression of Multi-Resolution Terrain Data Based on Binary Tree

In order to overcome the incontinuity and redundancy of data storage caused by the grids in representing the multi-resolution terrain, an adaptive hierarchical triangulation based on binary tree is used to represent the multi-resolution terrain model. It realizes the representation of the terrain data at different levels of detail and enables data compression by local omission of data values. The structure of binary tree is stored using a bit code of the underlying binary tree, while the height data are stored using an array, and relative pointers which allow a selective tree traversal. This method realizes the continuity and reduces the data volume in data storage of multi-resolution digital elevation model (DEM), and it is possible to work directly on the compressed data. We show that significant compression rates can be obtained already for small threshold values, and in a visualization application, it is possible to extracted and drawn triangulations at interactive rates.

Quasi-hierarchical Powell–Sabin B-splines

Hierarchical Powell–Sabin splines are C 1 -continuous piecewise quadratic polynomials defined on a hierarchical triangulation. The mesh is obtained by partitioning an initial conforming triangulation locally with a triadic split, so that it is no longer conforming. We propose a normalized quasi-hierarchical basis for this spline space. The basis functions have a local support, they form a convex partition of unity, and they admit local subdivision. We show that the basis is strongly stable on uniform hierarchical triangulations. We consider two applications: data fitting and surface modelling.

A web-mapping system for real-time visualization of the global terrain

In this paper, we mainly present a web-based 3D global terrain visualization application that provides more powerful transmission and visualization of global multiresolution data sets across networks. A client/server architecture is put forward. The paper also reports various relevant research work, such as efficient data compression methods to reduce the physical size of these data sets and accelerate network delivery, streaming transmission for progressively downloading data, and real-time multiresolution terrain surface visualization with a high visual quality by M-band wavelet transforms and a hierarchical triangulation technique. Finally, an experiment is performed using different levels of detailed data to verify that the system works appropriately.

A minimum entropy approach to adaptive image polygonization

This paper introduces a novel adaptive image segmentation algorithm which represents images by polygonal segments. The algorithm is based on an intuitive generative model for pixel intensities and its associated cost function which can be effectively optimized by a hierarchical triangulation algorithm. A triangular mesh is iteratively refined and reorganized to extract a compact description of the essential image structure. After analyzing fundamental convexity properties of our cost function, we adapt an information-theoretic bound to assess the statistical significance of a given triangulation step. The bound effectively defines a stopping criterion to limit the number of triangles in the mesh, thereby avoiding undesirable overfitting phenomena. It also facilitates the development of a multiscale variant of the triangulation algorithm, which substantially improves its computational demands. The algorithm has various applications in contextual classification, remote sensing, and visual object recognition. It is particularly suitable for the segmentation of noisy imagery.

Multiresolution Compression and Visualization of Global Topographic Data

We present a multiresolution model for terrain surfaces which is able to handle large-scale global topographic data. It is based on a hierarchical decomposition of the sphere by a recursive bisection triangulation in geographic coordinates. Error indicators allow the representation of the data at various levels of detail and enable data compression by local omission of data values. The resulting adaptive hierarchical triangulation is stored using a bit code of the underlying binary tree and additionally, relative pointers which allow a selective tree traversal. This way, it is possible to work directly on the compressed data. We show that significant compression rates can be obtained already for small threshold values. In a visualization application, adaptive triangulations which consist of hundreds of thousands of shaded triangles are extracted and drawn at interactive rates.

Hierarchical representation and coding of surfaces using 3-D polygon meshes

This paper presents a novel procedure for the representation and coding of three-dimensional (3-D) surfaces using hierarchical adaptive triangulation. The proposed procedure is based on pyramidal analysis using the quincunx sampling minimum variance interpolation (QMVINT) filters. These are reduced pyramids with quincunx sampling applied to the parametric representation of the surface, chosen so as to minimize the variance of the interpolation error, and thus, when combined with the appropriate encoding of the coefficients, optimize the compression of the mesh information transmitted. At the same time, it produces a hierarchy of meshes based on quincunx sampling where coarse meshes are as similar to their finer versions as possible. This is very much desirable in progressive transmission. Depending on its interpolation error and the available bitrate, each filtered sample is a candidate for becoming a vertex of the mesh. The result is a progressive sequence of meshes consisting of more triangles wherever large variations exist and fewer in uniform regions. Complete correspondence between triangles at each level is identified, resulting in an efficient hierarchical representation of the mesh. The algorithm can be also used for the triangulation of a specific region of interest. Experimental results demonstrate that the proposed scheme provides an improvement in quality (MSE) by a factor of two when compared with other well known adaptive triangulation schemes.

Representation of 3-D elevation in terrain databases using hierarchical triangulated irregular networks: a comparative analysis

3-D terrain representation plays an important role in a number of terrain database applications. Hierarchical Triangulated Irregular Networks (TINs) provide a variable-resolution terrain representation that is based on a nested triangulation of the terrain. This paper compares and analyzes existing hierarchical triangulation techniques. The comparative analysis takes into account how aesthetically appealing and accurate the resulting terrain representation is. Parameters, such as adjacency, slivers, and streaks, are used to provide a measure on how aesthetically appealing the terrain representation is. Slivers occur when the triangulation produces thin and slivery triangles. Streaks appear when there are too many triangulations done at a given vertex. Simple mathematical expressions are derived for these parameters, thereby providing a fairer and a more easily duplicated comparison. In addition to meeting the adjacency requirement, an aesthetically pleasant hierarchical TINs generation algorithm is expected to reduce both slivers and streaks while maintaining accuracy. A comparative analysis of a number of existing approaches shows that a variant of a method originally proposed by Scarlatos exhibits better overall performance.

Multiresolution models for topographic surface description

Multiresolution terrain models describe a topographic surface at various levels of resolution. Besides providing a data compression mechanism for dense topographic data, such models enable us to analyze and visualize surfaces at a variable resolution. This paper provides a critical survey of multiresolution terrain models. Formal definitions of hierarchical and pyramidal models are presented. Multiresolution models proposed in the literature (namely, surface quadtree, restricted quadtree, quaternary triangulation, ternary triangulation, adaptive hierarchical triangulation, hierarchical Delaunay triangulation, and Delaunay pyramid) are described and discussed within such frameworks. Construction algorithms for all such models are given, together with an analysis of their time and space complexities.

Multiresolution Surface Modeling Based on Hierarchical Triangulation

This paper describes a sequential multiresolution surface modeling technique. The initial model consists in a surface triangulation resulting from the integration of a set of range views. Thereafter, a hierarchical triangulation algorithm iteratively removes vertices of the triangulation, always minimizing the retriangulation error. The equiangularity of the surface triangulation is optimized in 3-D space throughout the optimization process. Besides compressing surface information, the proposed technique preserves the topology of the triangulation and surface orientation discontinuities. Experimental models built from complex multipart objects with holes are presented.

Hierarchical triangulation for multiresolution surface description

A new hierarchical triangle-based model for representing surfaces over sampled data is proposed, which is based on the subdivision of the surface domain into nested triangulations, called a hierarchical triangulation (HT) . The model allows compression of spatial data and representation of a surface at successively finer degrees of resolution. An HT is a collection of triangulations organized in a tree, where each node, except for the root, is a triangulation refining a face belonging to its parent in the hierarchy. We present a topological model for representing an HT, and algorithms for its construction and for the extraction of a triangulation at a given degree of resolution. The surface model, called a hierarchical triangulated surface (HTS) is obtained by associating data values with the vertices of triangles, and by defining suitable functions that describe the surface over each triangular patch. We consider an application of a piecewise-linear version of the HTS to interpolate topographical data, and we describe a specialized version of the construction algorithm that builds an HTS for a terrain starting from a high-resolution rectangular grid of sampled data. Finally, we present an algorithm for extracting representations of terrain at variable resolution over the domain.

Accuracy-Based Sampling and Reconstruction with Adaptive Meshes for Parallel Hierarchical Triangulation

Recent advances in range finding techniques have made the task of acquiring surface data for 3-D objects easier and more accurate. With most advanced techniques, range and color data are acquired simultaneously. Since the number of such acquired data is generally very large, a surface model capable of compressing data while maintaining a specified accuracy is required. The objective of this work is to construct a polyhedral representation of input data for surfaces. This representation adapts to local intrinsic surface properties while preserving their discontinuities. In this paper, we present an accuracy-based adaptive sampling and reconstruction technique for hierarchical triangulation of 3D objects. We have developed a parallel algorithm for adaptive mesh generation that recursively bisects mesh elements by increasing the number of mesh nodes according to local surface properties, such as surface orientation, curvature, and color. The recursive subdivision based on such viewpoint-invariant properties yields a hierarchical surface triangulation that is intrinsic to the surface. This approach also satisfies the absolute accuracy criteria, since nodes are generated as required until the entire surface has been approximated within a specified level of accuracy. We have also developed a parallel algorithm that detects and preserves both depth (C0) and orientation (C1) discontinuities, without the formation of cracks which normally occur during independent subdivision. The algorithm has been successfully applied to adaptive sampling and reconstruction of both range and color images of human faces and Japanese antique dolls with fine grained color-texture.

Adaptive mesh generation for surface reconstruction—parallel hierarchical triangulation without cracks

This paper describes a method of precisely and effectively visualizing a complex 3-dimensionally curved surface from a set of range data by using meshes (parallel hierarchical triangulation) which are adaptive to the degree of its complexity. The method can reconstruct the curved surface without cracks by approximating its intrinsic shape and discontinuity with a polygon. The algorithm used can be processed by fact parallel local calculations with an upper limit in time and space. This paper proposes a parallel recursive algorithm for generating an adaptive mesh using a 3D-curvature based on the intrinsic shape of an object, and another parallel recursive algorithm to avoid the formation of a crack which occurs between two subdivisions. Experimental results applied to range images of human faces are shown. The proposed method can be applied to general images (e.g., gray-level image and color images) other than range images so that the compression or reconstruction of an image can be carried out by adaptively using its geometrical features (e.g., the pixel value, its first-order differential, and second order differential).

Structured graph representation of a hierarchical triangulation

The problem of representing a 2 1 2 -dimensional surface at variable degrees of accuracy is considered. A hierarchical model, called a hierarchical triangulation, is described which approximates a surface by a network of triangular, planar facets whose projections in the plane are nested. In the paper, a dual representation of such a triangulation in the form of a structured graph is defined and the fundamental properties of this representation are discussed. An algorithm for converting a hierarchical triangulation into its dual form is given. As an example of application, an algorithm is described for the conversion of the dual graph into the standard triangle-oriented structure used for encoding triangulated irregular networks.

Structured graph representation of a hierarchical triangulation

Structured graph representation of a hierarchical triangulation

A hierarchical structure for surface approximation

Hierarchical triangulation is a method for point selection and surface representation where the surface is approximated at successively finer levels of detail by triangular patches whose projections in the horizontal plane are nested. A tree data structure for this representation can be constructed in O( n 2) worst case and O( n log n) average case time, where n is the number of data points considered. Efficient algorithms for approximation of the elevation of an arbitrary point, contour extraction, and conversion of the hierarchical structure into an ordinary triangulated irregular network, are demonstrated. The convergence and the optimality of the approximation and the relationship of the hierarchical triangulation to a structured graph representation are examined.