Octree Approach Research Articles

VOLUMINATOR 2.0 – SPEEDING UP THE APPROXIMATION OF THE VOLUME OF DEFECTIVE 3D BUILDING MODELS

Abstract. Semantic 3D city models are increasingly used as a data source in planning and analyzing processes of cities. They represent a virtual copy of the reality and are a common information base and source of information for examining urban questions. A significant advantage of virtual city models is that important indicators such as the volume of buildings, topological relationships between objects and other geometric as well as thematic information can be derived. Knowledge about the exact building volume is an essential base for estimating the building energy demand. In order to determine the volume of buildings with conventional algorithms and tools, the buildings may not contain any topological and geometrical errors. The reality, however, shows that city models very often contain errors such as missing surfaces, duplicated faces and misclosures. To overcome these errors (Steuer et al., 2015) have presented a robust method for approximating the volume of building models. For this purpose, a bounding box of the building is divided into a regular grid of voxels and it is determined which voxels are inside the building. The regular arrangement of the voxels leads to a high number of topological tests and prevents the application of this method using very high resolutions. In this paper we present an extension of the algorithm using an octree approach limiting the subdivision of space to regions around surfaces of the building models and to regions where, in the case of defective models, the topological tests are inconclusive. We show that the computation time can be significantly reduced, while preserving the robustness against geometrical and topological errors.

VOLUMINATOR 2.0 – SPEEDING UP THE APPROXIMATION OF THE VOLUME OF DEFECTIVE 3D BUILDING MODELS

Semantic 3D city models are increasingly used as a data source in planning and analyzing processes of cities. They represent a virtual copy of the reality and are a common information base and source of information for examining urban questions. A significant advantage of virtual city models is that important indicators such as the volume of buildings, topological relationships between objects and other geometric as well as thematic information can be derived. Knowledge about the exact building volume is an essential base for estimating the building energy demand. In order to determine the volume of buildings with conventional algorithms and tools, the buildings may not contain any topological and geometrical errors. The reality, however, shows that city models very often contain errors such as missing surfaces, duplicated faces and misclosures. To overcome these errors (Steuer et al., 2015) have presented a robust method for approximating the volume of building models. For this purpose, a bounding box of the building is divided into a regular grid of voxels and it is determined which voxels are inside the building. The regular arrangement of the voxels leads to a high number of topological tests and prevents the application of this method using very high resolutions. In this paper we present an extension of the algorithm using an octree approach limiting the subdivision of space to regions around surfaces of the building models and to regions where, in the case of defective models, the topological tests are inconclusive. We show that the computation time can be significantly reduced, while preserving the robustness against geometrical and topological errors.

Grid-Free Octree Approach for Modeling Heat Transfer to Complex Geometries

The grid-free octree direct simulation Monte–Carlo algorithm has been explored to simulate flows through and heat transfer to highly irregular geometries for finite Knudsen numbers. The approach was extended to accurately compute the volume of irregular octree clusters as well as a line-of-sight test before performing collisions in octree clusters that are intersected by a solid boundary. The modified grid-free octree direct simulation Monte–Carlo algorithm was tested for the classic Fourier and micro-Poiseuille flows. A single sphere as well as an aggregate of spheres were modeled in a 500 K heat bath and supersonic flow. For the heat bath case, it was observed that a two-level Cartesian grid direct simulation Monte–Carlo code was able to achieve the same heat flux to a single sphere as the octree approach if 10 times more computational particles were used. Furthermore, even with increased computational load, the two-level Cartesian approach encountered a difference of 15% from the octree calculation for heat flux for interior spheres due to shielding by outer spheres. It was found that, for supersonic flows surrounding a single sphere and an aggregate, the octree approach was able to resolve the detached bow shock as well as the wake region, and in the aggregate case, the spatial resolution was sufficient to capture distinct multiple shocks in terms of temperature jump and pressure contours. Computational efficiency in terms of central processing unit–graphical processing unit computing is also discussed.

A real-time virtual sculpting application by using an optimized hash-based octree

sculpting system by using a hash-based octree data structure with an optimized approach to reduce the huge memory and computation cost and also the tree traversal time. The system first reads the selected 3D virtual object model, which must be in a triangular watertight polygon mesh format. Then both the surface and the interior volume of the virtual object is voxelized in order to generate its volumetric dataset. Afterwards, a hash-based data structure with a novel optimized approach is applied to the dataset. After each sculpting process applied to the virtual object, its surface is locally reconstructed and a feedback force is calculated synchronously. Finally, to give the user a realistic sense of interaction, the feedback force is sent back via a haptic device. Creation durations of a traditional octree, a hash-based octree, and our optimized hash-based octree are compared. Tree traversal and surface reconstruction times of our optimized hash-based octree approach are also compared with the pointer-based one in off-line and real-time while sculpting a 3D model. It is shown that the memory requirement and durations of our optimized approach are the lowest.

Generation of Polyhedral Delaunay Meshes

A polyhedral mesh fulfills the Delaunay condition if the vertices of each polyhedron are co-spherical and each polyhedron circum- sphere is point-free. If Delaunay tessellations are used together with the finite volume method, it is not necessary to partition each polyhedron into tetrahedra; co-spherical elements can be used as final elements. This paper presents a mixed-element mesh gen- erator based on the modified octree approach that has been adapted to generate polyhedral Delaunay meshes. The main difference with its predecessor is to include a new algorithm to compute Delaunay tessellations for each 1-irregular cuboids (cuboids with at most one Steiner point on their edges) that minimize the number of mesh elements. In particular, we show that when Steiner points are located at edge midpoints, 24 different co-spherical elements can appear while tessellating 1-irregular cubes. By inserting internal faces and edges to these new elements, this number can be reduced to 13. When 1-irregular cuboids with aspect ratio equal to √2 are tessellated, 10 co-spherical elements are required. If 1-irregular cuboids have aspect ratio between 1 and √2, all the tessellations are adequate for the finite volume method. The proposed algorithm can be applied to any point set to compute the Delaunay tessellation inside the convex hull of the point set. Simple polyhedral Delaunay meshes generated by using the adapted mesh generator are shown.

Automatic off-body overset adaptive Cartesian mesh method based on an octree approach

This paper describes a method for generating adaptive structured Cartesian grids within a near-body/off-body mesh partitioning framework for the flow simulation around complex geometries. The off-body Cartesian mesh generation derives from an octree structure, assuming each octree leaf node defines a structured Cartesian block. This enables one to take into account the large scale discrepancies in terms of resolution between the different bodies involved in the simulation, with minimum memory requirements. Two different conversions from the octree to Cartesian grids are proposed: the first one generates Adaptive Mesh Refinement (AMR) type grid systems, and the second one generates abutting or minimally overlapping Cartesian grid set. We also introduce an algorithm to control the number of points at each adaptation, that automatically determines relevant values of the refinement indicator driving the grid refinement and coarsening. An application to a wing tip vortex computation assesses the capability of the method to capture accurately the flow features.

Finite element simulation of skeletal muscular structures obtained from images of histological serial sections

In this study, we present a method for the three-dimensional reconstruction of objects obtained from histological serial sections (exemplified by those of a pennate striated skeletal muscle) and its application to the finite element method. A hyperelastic material model is used for modeling biological soft tissue. The reconstruction process relies on the direct construction of a volumetric mesh using an octree approach which leads to a stable finite element method. Stability can be expressed in the spectral matrix condition number. To visualize stress patterns within the underlying anatomy the simulation results are projected onto images of the histological scenario.

Octree-based animated geometry compression

Geometry compression is the coding of 3D geometric data in a form that requires less space to store and less bandwidth to transmit. This 3D geometric data provides a computer graphics system with the scene description necessary to render images. Geometric data is quite large and, therefore, needs effective compression methods to decrease the transmission and storage requirements. A large amount of research has been focused on static geometry compression, but only limited research has addressed animated geometry compression, the compression of temporal sequences of geometric data. The goal of this paper is to represent 3D animation sequences with a reduced set of motion vectors that take advantage of the large data coherence in space and time. We proposed an octree-based motion representation method in which a small set of motion vectors are generated for each frame by analyzing the motion between consecutive frames. The vectors represent the differential motion from a previous frame to the current frame and are used to predict the vertex positions for the new frame. The octree motion representation process generates a hierarchical octree structure for each frame in the sequence. The octree approach can represent 3D animated sequences with high compression factors while maintaining reasonable quality. This approach is easy to implement and have a low cost encoding process and a fast decoding process, which make it very suitable for real time application.

Generation of 3D mixed element meshes using a flexible refinement approach

This paper describes and discusses the main characteristics and implementation issues of a 3D mixed element mesh generator based on a generalization of the modified octree approach. This mesh generator uses primitive elements of different type as internal nodes, a flexible refinement approach as refinement strategy (primitive elements are not always bisected), and bricks, pyramids, prisms and tetrahedra as final elements. The mesh generation process is divided in several steps: the generation of the initial mesh composed of primitive elements, the refinement of primitive elements until the point density requirements are fulfilled, the generation of a graded mesh between dense and coarse regions, and finally, the recognition of the final elements. The main algorithms and data structures are described in detail for each step of the mesh generation process. As result, examples of meshes that satisfy the Delaunay condition and that can be used with the control volume method are shown.

Interactive volume rendering of large sparse data sets using adaptive mesh refinement hierarchies

In this paper, we present an algorithm that accelerates 3D texture-based volume rendering of large, sparse data sets, i.e., data sets where only a traction of the voxels contain relevant information. In texture-based approaches, the rendering performance is affected by the fill-rate, the size of texture memory, and the texture I/O bandwidth. For sparse data, these limitations can be circumvented by restricting most of the rendering work to the relevant parts of the volume. In order to efficiently enclose the corresponding regions with axis-aligned boxes, we employ a hierarchical data structure, known as an AMR (adaptive mesh refinement) tree. The hierarchy is generated utilizing a clustering algorithm. A good balance is thereby achieved between the size of the enclosed volume, i.e., the amount to render in graphics hardware and the number of axis-aligned regions, i.e., the number of texture coordinates to compute in software. The waste of texture memory by the power-of-two restriction is minimized by a 3D packing algorithm which arranges texture bricks economically in memory. Compared to an octree approach, the rendering performance is significantly increased and less parameter tuning is necessary.

Octree reinforced thin shell objects rapid prototyping by fused deposition modelling

Rapid prototyping has gained wide industrial acceptance in recent years. Its main advantage is that it improves the “time to market” and the design quality of a product. Most rapid prototyping processes have to build the solid volume layer by layer. The tracing of the cross-sectional area in each layer is the most time-consuming step. The process can be speeded up if the material volume to be traced can be reduced by extracting some empty volumes in the original solid. However, the hollow solid generated by employing the negative solid offset technique is subject to the problem of no interior support. In this paper, a sub-boundary octree approach of generating thin shell solids with reinforced interior structuring is proposed. The procedures to generate a reinforced thin shell solid are studied. Various testing models are produced by a fused deposition modelling rapid prototyper and the results are supportive.

Octree reinforced thin-shell rapid prototyping

Rapid prototyping has been accepted widely as an essential element in speeding up industrial practice. By using rapid prototyping methods, prototypes can be completed rapidly in comparison with the use of conventional methods. This paper investigates a ‘sub-boundary’ octree approach to produce a thin-shell-based rapid prototyping model. A ‘sub-boundary’ octree means an octree that is wholly in the interior of the original solid. The operation to differenciate the evaluated octree from the solid is studied, the operation being valid for solids that are represented by constructive solid geometry or boundary representation. This is achieved by unioning the octree solid with the reduced volume solid from a negative offset. The proposed method can speed up the rapid prototyping process by enabling the employment of solids of reduced volume with reinforced interior structuring. Possible future work is also discussed. The areas including the development of a sufficient and efficient data structure for the storing and editing of the ‘sub-boundary’ octree skeleton, the implementation of efficient algorithms to derive the ‘sub-boundary’ octree approximations from solids, and the verification of the constructability of the theoretical results

Graded tetrahedral finite element meshes

AbstractVertices in the body centred cubic (bcc) lattice are used to create a tetrahedral spatial decomposition. With this spatial decomposition an octree approach is combined with Delaunay triangulations to decompose solids into tetrahedral finite element meshes. Solids must have their surfaces triangulated and the vertices in the triangulation are finite element nodes. Local densities of interior tetrahedra are controlled by the densities of surface triangles. Accuracy of the decomposition into finite elements depends on the accuracy of the surface triangulation which can be constructed with state of the art computer aided design systems.

An Octree Approach to Rough Machining

This paper presents an octree subdivision based algorithm of automatic cutter path generation for machining on a numerical control machining centre. Both the stock and the part are represented by solid models. The machining strategy is based on a layer by layer approach taking care of the cutter strength and the variation of the part geometry. This approach allows the problem to be reduced from a three-dimensional to a two-dimensional cutter path generation problem which is solved by a quadtree representation of the plane. The method can be easily extended to include the jigs and fixtures and any other obstacles which would be represented by solid models. Algorithms are described in Pascal-like structure.

ARTS: Accelerated Ray-Tracing System

In this article we propose algorithms that address the two basic problems encountered in generating continuous-tone images by ray tracing: speed and aliasing. We examine previous approaches to the problem and then propose a scheme based on the coherency of an auxiliary data structure imposed on the original object domain. After investigating both simple spatial enumeration and a hybrid octree approach, we developed 3DDDA, a 3D line generator for efficient traversing of both structures. 3DDDA provides an order of magnitude improvement in processing speed compared to other known ray-tracing methods. Processing time is found to be virtually independent of the number of objects involved in the scene. For large numbers of objects, this method actully becomes faster than scan-line methods. To remove jags from edges, a scheme for identifying edge orientation and distance from pixel center to true edge has been implemented. The time required for antialiasing depends on the total length of the edges encountered, but it is normally only a fractional addition to the time needed to produce the scene without antialiasing.