Octree generation from object silhouettes in perspective views

Octrees are useful for object representation when fast access to coarse spatial occupancy information is necessary. This paper presents an efficient algorithm for generating octrees from multiple perspective views of an object. The algorithm first obtains a polygonal approximation of the object silhouette. This polygon is then decomposed into convex components. For each convex component, a pyramid is formed treating the view point as its apex and the convex components as a cross section. The octree representation of each of these pyramids is obtained by performing intersection detection of the object with the cubes corresponding to octree nodes. The intersection detection step is made efficient by decomposing it into a coarse-to-fine sequence of intersection tests. The octree for one silhouette is obtained by taking the union of octrees obtained for each component. An intersection of octrees corresponding to different viewing directions gives the final octree of the object. An implementation of the algorithm is given. The accuracy of the octree representation of the objects is evaluated. The ratio of the actual volume of the object to the volume of the object reconstructed from the octree representation is used as a performance index of the algorithm.     

Boolean operations with implicit and parametric representation of primitives using R-functions

We present a new and efficient algorithm to accurately polygonize an implicit surface generated by multiple Boolean operations with globally deformed primitives. Our algorithm is special in the sense that it can be applied to objects with both an implicit and a parametric representation, such as superquadrics, supershapes, and Dupin cyclides. The input is a constructive solid geometry tree (CSG tree) that contains the Boolean operations, the parameters of the primitives, and the global deformations. At each node of the CSG tree, the implicit formulations of the subtrees are used to quickly determine the parts to be transmitted to the parent node, while the primitives' parametric definition are used to refine an intermediary mesh around the intersection curves. The output is both an implicit equation and a mesh representing its solution. For the resulting object, an implicit equation with guaranteed differential properties is obtained by simple combinations of the primitives' implicit equations using R-functions. Depending on the chosen R-function, this equation is continuous and can be differentiable everywhere. The primitives' parametric representations are used to directly polygonize the resulting surface by generating vertices that belong exactly to the zero-set of the resulting implicit equation. The proposed approach has many potential applications, ranging from mechanical engineering to shape recognition and data compression. Examples of complex objects are presented and commented on to show the potential of our approach for shape modeling.     

Manufacturing applications of octrees

It is well understood that effective and comprehensive CAD representation methods are necessary for modelling all the design and manufacturing functions in an automated environment. Recent research work has concentrated on using Constructive Solid Geometry (CSG) methods and Boundary Representation (B-Rep) schemes for representing design and manufacturing functions. This paper presents several novel ideas on the application of octree and quadtree based representation schemes in several manufacturing areas such as robotic task planning, tolerance representation and inspection, feature-based design, and assembly modelling and verification. Suggestions on the application of extended octrees for modelling metal forming operations are also presented.     

3-D object decomposition with extended octree model and its application in geometric simulation of NC machining

The paper discusses the development of a prototype solid modeling system based on the extended octree modeling approach and its applications in 3-D NC machining simulation and automatic verification. Along with a simple hierarchical data structure, the extended octree model uses the face boundary information (i) to represent complex objects, (ii) to improve object representational accuracy, and (iii) to accelerate the model updating procedures in a graphic simulation process. The improved representational accuracy makes it possible to carry out automatic NC program verification by generating the machined model of the part through the simulation and comparing it with the designed model of the part. The paper also addresses the issues of model conversions from CSG and B-Rep schemes to the corresponding extended octree models and the issues of carrying out Boolean operations on those extended octree models. A prototype system has been implemented and is integrated with AutoCAD AME solid modeler for object modeling and for NC simulation purpose.     

An algorithm for converting the boundary representation of a CAD model to its octree representation

Present CAD systems store the solid model of an object using a convenient representation. Boundary models and CSG (Constructive Solid Geometry) models are the most frequently used representations. Based on recent research findings, octree representation of an object presents a promising approach in solving problems in the areas of Computer Graphics, Manufacturing and Robotics. The most notable use of octree representations is in CAD-based robotic path planning problems. Octree models have also been used in fast rendering of 3-D solid models using ray tracing methods. This paper presents an algorithm for converting the boundary representation of polyhedral models to its octree representation. Such an algorithm would provide the link between an object generated using a solid modelling system and the application involving an octree representation of an object. The algorithm is demonstrated by converting a polyhedral boundary model of a sample object to its octree representation.     

A strategy for repetitive neighbor finding in octree representations

An effective strategy for repetitive neighbor finding in octrees is presented. It is based on a new method which uses a matrix representation for octrees and some auxiliary locational codes for neighbors’ generation. This approach is universal and generates all neighbors (larger, smaller, equal) for any octree node using only simple algebraic operations.     

Deferred boundary evaluation of complex CSG models

In this paper we present a deferred method for evaluating a complete CSG tree based on triangulated solids. It allows the exact evaluation of the surface of the entire model in a single step, using regularized Boolean classifications. The overall performance with this approach is better than with the classical method, which incrementally evaluates a CSG tree with single Boolean operations. The deferred algorithm does not use any intermediate result for the nodes of the CSG tree. It uses a very simple data structure and an octree that speeds up spatial queries for the entire CSG tree. The algorithm intensively uses multitasking and is ready for working with very complex CSG expressions, including the application of an out-of-core based approach.     

Viewing Transformations of Voxel-Based Objects Via Linear Octrees

This article gives three viewing-transformation algorithms for displaying on a screen 3D pictures represented by linear octrees. All the procedures take advantage of the recursive labeling used to identify the successive decomposition of an object into octants. The first algorithm performs transformations directly on the linear octree, while the second and third algorithms determine the 3D border of the given object first and then project onto the screen the surface voxels thus found. All the algorithms perform the viewing transformations in O(RN) time, where R is the resolution of the picture and N is the number of elements in the linear octree. One of the algorithms provides views of the object at different layers of gray level, while another allows internal views.     

Algorithms for the physical rendering and assembly of octree models

Hierarchical decomposition techniques are well established for the representation of 2D images, the calculation of distance maps, and the modelling of volume data. However, recent work has suggested that their use can be extended to the manufacture of physical objects for low cost prototyping and visualization. This paper details various decomposition and assembly planning routines created to support this process. Specifically the decomposition methods are described to generate octants appropriate for the physical assembly process. Having established methods for generating suitable octrees, three different algorithms for planning the assembly of octrees are presented. The comparative performance of these different approaches is discussed.     

Etree: a database-oriented method for generating large octree meshes

This paper presents the design, implementation, and evaluation of the etree, a database-oriented method for large out-of-core octree mesh generation. The main idea is to map an octree to a database structure and perform all octree operations by querying and updating the database. We apply two standard database techniques, the linear octree and the B-tree, to index and store the octants on disk. Then we introduce two new techniques, auto-navigation and local balancing, to address the special needs of mesh generation. Preliminary evaluation suggests that the etree method is an effective way of generating very large octree meshes on desktop machines.     

基于自适应延迟切割的三角网格布尔运算优化

规则化的布尔运算被广泛应用在三维建模系统中.近年来,随着图形硬件的发展,基于三角网格的规则化布尔算法由于输出结果能直接被图形硬件处理,表现出了明显的优势.但是传统的算法由于采用CSG树局部评估策略,使得面片在相交测试中反复被切割,并且由于面片分类在切割后的模型之间直接进行,导致算法无法在保证鲁棒性的同时实现高性能.为了避免这些问题,本文呈现了一种CSG树全局评估算法来统一执行单次和连续布尔运算.算法由两部分组成：自适应的延迟切割和全局化面片分类.在自适应的延迟切割阶段,算法通过仔细处理多个三角面片相交导致的各种情况使得延迟切割被扩展到整个CSG树来避免由于面片的反复切割带来的数值误差累积并利用自适应的八叉树使得相交测试能在线性时间内完成.在全局化面片分类阶段,算法通过分治法使得分类始终在切割后的面片和原始输入模型之间进行来保证分类的精度;通过结合组分类策略和自适应的八叉树来进一步优化了分类性能。实验结果表明,本文提出的算法无论是在执行单次或连续布尔运算时都能在保证鲁棒性同时性能优于其他的算法,因此本文算法可广泛应用于交互式建模系统中,如数字雕刻、计算机辅助设计和制造（CAD/CAM）等.     

Octree Representation and Its Applications in CAD

In this paper,a survey of octree representation and its applications in CAD is presented.The octree representation may be categorized as pure octree representation and polytree(or extended octree),and the latter is actually a boundary representation decomposed by octree.Linear octree which is a variant of regular octree representation has the advantage of saving memory space.The mapping between Cartesian coordinates and node addresses in linear octree is discussed.Then,algorithms for converting a boundary representation of 3D object into an octree are investiged and major approaches for transforming an octree encoded object are presented.After that,some of the applications of octree representation in CAD are listed,in particular,the applications in solid modeling,in accelerating ray tracing and in generating meshes for FEM.     

Creation and boundary evaluation of CSG-models

This paper deals with one particular type of solid model representation. Constructive solid geometry (CSG) describes solids in terms of primitive shapes combined by Boolean operators. The goal of this paper is to present an algorithm suitable for the visualization of a given CSG-object for verification purposes. To accomplish this, the CSG representation is to be converted into a polygonal boundary representation (B-Rep), which allows the easy display of the solid's visible edges and surfaces. An algorithm performing this conversion is called a boundary evaluator. This paper describes the fundamental concepts leading to this new algorithm and presents some examples.The boundary evaluator described here follows the following steps: (1) A polygonal mesh is created that covers the primitive shapes. (2) Interfering polygons are repertedly subdivided until the resulting mesh contains the intersection curves between the primitives. (3) The location and validity of all polygons must be determined since only valid polygons reflect the solid's final shape. (4) Adjoining coplanar polygons are collected and merged by deleting their common edges. The final result of this procedure is a consistent, nonredundant polygon mesh describing the given solid model. Now the object may be displayed by simple perspective plots, by isometric projections, by hidden line drawings, or through the use of hidden surface removal techniques as color-shaded pictures. The algorithm presented here has been successfully used to create a variety of renderings of various CSG-models. The interpretation, display, and validation of computer-generated CSG models is essential for the successful use of such models in the engineering design process.     

Rapid Jewelry Manufacture using Spatial Subdivision

The paper presents a theory for making rapid prototypes of jewelry from octree subdivision. Traditionally, the input of jewelry for making jewel rapid prototypes is the CAD models’ representation. In this paper, a new representation for jewelry is introduced. Aesthetic jewelry is first bounded by a minimum bounding cube. The bounding cube is then subdivided into octrees. The cubes obtained are in three types: cubes fully filled the jewel model; cubes partially filled the jewel model and cubes out of the jewel model. Cubes out of the jewel model; are expelled, while cubes fully filled the jewel model are retained. For cubes partially filled in the jewel model, additional octree subdivision is performed. The number of octree subdivision levels depends upon the end-user requirement. After the octree subdivision process is completed, cubes partially filled the jewel model from different subdivision levels are replaced by boundary voxels. The final voxel model undergoes direct slicing to build the rapid prototype.     

Hierarchical data structures and algorithms for computer graphics.I. Fundamentals

The fundamentals of hierarchical data structures are reviewed and it is shown how they are used in the implementation of some basic operations in computer graphics. The properties of hierarchical structures are discussed, focusing on quadtrees and octrees. The latter are defined, some of the more common ways in which they are implemented are examined, and an explanation of the quadtree/octree complexity theorem is provided. Vector quadtrees and vector octrees are discussed. The performance of basic operations using quadtrees is considered     

An octree-based method for shape from inconsistent silhouettes

Shape-from-Silhouette (SfS) is the widely known problem of obtaining the 3D structure of an object from its silhouettes. Two main approaches can be employed: those based on voxel sets, which perform an exhaustive search of the working space, and those based on octrees, which perform a top-down analysis that speeds up the computation. The main problem of both approaches is the need for perfect silhouettes to obtain accurate results. Perfect background subtraction hardly ever happens in realistic scenarios, so these techniques are restricted to controlled environments where the consistency hypothesis can be assumed. Recently, some approaches (all of them based on voxel sets) have been proposed to solve the problem of inconsistency. Their main drawback is the high computational cost required to perform an exhaustive analysis of the working space. This paper proposes a novel approach to solve SfS with inconsistent silhouettes from an octree based perspective. The inconsistencies are dealt by means of the Dempster–Shafer (DS) theory and we employ a Butterworth function for adapting threshold values in each resolution level of the octree. The results obtained show that our proposal provides higher reconstruction quality than the standard octree based methods in realistic environments. When compared to voxel set approaches that manage inconsistency, our method obtains similar results with a reduction in the computing time of an order of magnitude.     

基于任意骨架的隐式曲面造型技术

给出了一个新的基于任意多面体网格骨架的构造性自由曲面造型算法.算法首先由每个给定骨架构造出一个距离场,然后利用隐函数光滑过渡技术和CSG(constructive solid geometry)表示技术将所构造的隐式曲面自由地两两粘合成一张光滑曲面.隐式曲面的多边形化算法则用来生成最终曲面网格.以任意骨架作为基本体素,突破了传统隐式曲面以点为基本骨架的限制.而且,距离曲面很好地逼近了原骨架形状,使用户可直观地对复杂曲面进行交互设计.而形变函数的引入,则极大地丰富了此方法的造型能力.实验结果表明,基于该算法的原型系统能够方便、直观地构造复杂的自由曲面.