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.     

An efficient algorithm for finding the CSG representation of a simple polygon

Modeling two-dimensional and three-dimensional objects is an important theme in computer graphics. Two main types of models are used in both cases: boundary representations, which represent the surface of an object explicitly but represent its interior only implicitly, and constructive solid geometry representations, which model a complex object, surface and interior together, as a boolean combination of simpler objects. Because neither representation is good for all applications, conversion between the two is often necessary.We consider the problem of converting boundary representations of polyhedral objects into constructive solid geometry (CSG) representations. The CSG representations for a polyhedronP are based on the half-spaces supporting the faces ofP. For certain kinds of polyhedra this problem is equivalent to the corresponding problem for simple polygons in the plane. We give a new proof that the interior of each simple polygon can be represented by a monotone boolean formula based on the half-planes supporting the sides of the polygon and using each such half-plane only once. Our main contribution is an efficient and practicalO(n logn) algorithm for doing this boundary-to-CSG conversion for a simple polygon ofn sides. We also prove that such nice formulae do not always exist for general polyhedra in three dimensions.The first author would like to acknowledge the support of the National Science Foundation under Grants CCR87-00917 and CCR90-02352. The fourth author was supported in part by a National Science Foundation Graduate Fellowship. This work was begun while the first author was visiting the DEC Systems Research Center.     

Hierarchical structure to winged-edge structure: a conversion algorithm

The winged-edge data structure is advantageous for traversing the topological graph of the boundary representation of a solid object. This paper presents an algorithm for converting hierarchical boundary representations into representations in the winged-edge data structure. As a result of the conversion, the adjacency relationships of geometric entities embedded in hierarchical boundary representations,-which may be evaluated through boundary evaluation on solid objects defined via Boolean set-operations, can be easily and efficiently accessed.     

Automatic Generation of Multiresolution Boundary Representations

The paper focuses on automatic simplification algorithms for the generation of a multiresolution family of solid models from an initial boundary representation of a polyhedral solid. An algorithm for general polyhedra based on an intermediate octree representation is proposed. Simplified elements of the multiresolution family approximate the initial solid within increasing tolerances. A discussion among different octree-based simplification methods and the standard marching cubes algorithm is presented.     

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.     

Representing dimensions, tolerances, and features in MCAE systems

A method is presented for explicitly representing dimensions, tolerances, and geometric features in solid models. The method combines CSG and boundary representations in a graph structure called an object graph. Dimensions are represented by a relative position operator. The method can automatically translate changes in dimensional values into corresponding changes in geometry and topology. The representation provides an important foundation for higher-level application programs to automate tolerance analysis and synthesis. The implementation of a prototype interactive polyhedral modeler based on this representation is presented     

Efficient representation and extraction of 2-manifold isosurfaces using kd-trees

In this paper, we propose the utilization of a kd-tree based hierarchy as an implicit object representation. Compared to an octree, the kd-tree based hierarchy is superior in terms of adaptation to the object surface. In consequence, we obtain considerably more compact implicit representations especially in case of thin object structures. We describe a new isosurface extraction algorithm for this kind of implicit representation. In contrast to related algorithms for octrees, it generates 2-manifold meshes even for kd-trees with cells containing multiple surface components. The algorithm retains all the good properties of the Dual Contouring approach by Ju et al. [ACM Trans. Graphics 21 (2002) 339–346] like feature preservation, computational efficiency, etc. In addition, we present a simplification framework for the surfaces represented by the kd-tree based on quadric error metrics. We adapt this framework to quantify the influence of topological changes, thereby allowing controlled topological simplification of the object. The advantages of the new algorithm are demonstrated by several examples.     

Adaptive physics based tetrahedral mesh generation using level sets

We present a tetrahedral mesh generation algorithm designed for the Lagrangian simulation of deformable bodies. The algorithm’s input is a level set (i.e., a signed distance function on a Cartesian grid or octree). First a bounding box of the object is covered with a uniform lattice of subdivision-invariant tetrahedra. The level set is then used to guide a red green adaptive subdivision procedure that is based on both the local curvature and the proximity to the object boundary. The final topology is carefully chosen so that the connectivity is suitable for large deformation and the mesh approximates the desired shape. Finally, this candidate mesh is compressed to match the object boundary. To maintain element quality during this compression phase we relax the positions of the nodes using finite elements, masses and springs, or an optimization procedure. The resulting mesh is well suited for simulation since it is highly structured, has topology chosen specifically for large deformations, and is readily refined if required during subsequent simulation. We then use this algorithm to generate meshes for the simulation of skeletal muscle from level set representations of the anatomy. The geometric complexity of biological materials makes it very difficult to generate these models procedurally and as a result we obtain most if not all data from an actual human subject. Our current method involves using voxelized data from the Visible Male [1] to create level set representations of muscle and bone geometries. Given this representation, we use simple level set operations to rebuild and repair errors in the segmented data as well as to smooth aliasing inherent in the voxelized data.     

高度场八叉树的体特征表达算法

体特征表达对用户理解和认知虚拟环境有着至关重要的作用。当前的体特征表达算法由于存储量大且不易于在GPU中加速等问题,渲染效率低下,难以满足场景可视化的实时性需求。针对这一问题,提出了一种高效的高度场八叉树体特征表达算法,不仅解决了传统高度场仅能表达2.5维模型,无法表达真三维模型的问题,而且为体特征表达提供了一种新的可行途径。算法使用八叉树结构生成三维模型的高度场表示,将传统的z向高度场扩展到x,y,z三个方向的高度场。首先,提出了三角面片预处理方法,保证模型精度和数据的完整性;其次,提出了基于投影变换的高度场表示判断及栅格化方法,将几何图元转换成二维空间的高度场数据;最后,提出了基于高度场八叉树的光线投射算法。实验结果表明,算法能极大地减少存储量,具有较高的光线投射效率,表达三维模型时取得较好效果。     

An approach to geometric modeling of solids bounded by sculptured surfaces

In this paper we present the design and implementation of a geometric modeling system for modeling solids bounded by sculptured surfaces. The three most important solid representation schemes—constructive solid geometry, boundary representation, and octree representation—are combined together in our system in such a way that the resulting scheme enjoys many of the advantages of its individual schemes. We also developed algorithms for conversion of objects from their boundary representation to octree representation, and for the boundary evaluation of octree encoded objects. The system was implemented on a DEC 1090 computer in PASCAL, and we have presented a simple illustrative example to show the use of the system to create solid objects bounded by sculptured surfaces.     

Free-form deformation of constructive shell models

Free-form deformation (FFD) is known to be a powerful technique for deforming an object independent of its representation. A point on a solid is deformed by specifying the point relative to a coordinate system defined with a lattice of control points. Adjusting the control points of the lattice deforms the object. The deformed object can be visualized by sampling points on the object surfaces, or by approximating the object with a polyhedral model. However, a conversion of the polyhedral model to the underlining solid representation is required if further solid operations (e.g. Boolean operation) is to be applied. This paper presents a technique for applying FFD on constructive shell models (CSR). A CSR object is constructed by subtracting a set of depression (negative) trunctets from the union of a set of outer (positive) trunctets and a polyhedron core. FFD is applied to the polyhedron core and the trunctets of the model. A trunctet is deformed by applying FFD to a set of selected surface points on the trunctet. The deformed trunctet is obtained by interpolating a surface through the deformed surface points. In the deformation of a trunctet, an outer trunctet may become a depression trunctet (or vice versa). By using the concept of mating trunctet, and an approach for classifying the shape of a trunctet, shape changes of a trunctet in a deformation can be determined. Deformation of a trunctet may also result in an invalid trunctet that bulges out of its enclosing tetrahedron. Besides, the degree of the algebraic surface used determines the size of a trunctet relative to the distance between adjacent lattice control points. A subdivision of a trunctet may have to be performed to maintain the validity of a deformation.     

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.     

Enabling cuts on multiresolution representation

Multiresolution representations are widely used in many visualization contexts and applications, since they provide optimal management of the data representation by using at each time instant a level of detail most appropriate for the application requirements. Unfortunately, current solutions do not allow the topology of the object to be changed, and this tends to prevent its adoption in applications where topological changes are needed, such as virtual surgery applications. By extending a known multiresolution model based on simplicial complexes, we develop a new approach which supports dynamic topological modifications of the represented object without greatly increasing the representation complexity.     

Toward a model of representation changes

This paper presents the first steps in the development of a computer model of the process of changing representations in problem solving. The task of discovering representations that yield efficient solution strategies for problems is viewed as heuristic search in the space of representations. Two dimensions of this representation space are information structure and information quantity. Changes of representation are characterized as isomorphisms and homomorphisms, corresponding to changes of information structure and information quantity, respectively. A language for expressing representations is given. Also, a language for describing representation transformations and an interpreter for applying the transformations to representations has been developed. In addition, transformations can be automatically inverted and composed to generate new transformations. Among the example problems used to illustrate and support this model are tic-tac-toe, integer arithmetic, the Tower of Hanoi problem, the arrow puzzle, the five puzzle, the mutilated checkerboard problem, and floor plan design. The system has also been used to generate some new NP-complete problems.     

A haptic-rendering technique based on hybrid surface representation

A novel haptic rendering technique using a hybrid surface representation addresses conventional limitations in haptic displays. A haptic interface lets the user touch, explore, paint, and manipulate virtual 3D models in a natural way using a haptic display device. A haptic rendering algorithm must generate a force field to simulate the presence of these virtual objects and their surface properties (such as friction and texture), or to guide the user along a specific trajectory. We can roughly classify haptic rendering algorithms according to the surface representation they use: geometric haptic algorithms for surface data, and volumetric haptic algorithms based on volumetric data including implicit surface representation. Our algorithm is based on a hybrid surface representation - a combination of geometric (B-rep) and implicit (V-rep) surface representations for a given 3D object, which takes advantage of both surface representations.     

Accurate solid modeling using polyhedral approximations

Although curved-surface solid modeling systems achieve a higher level of accuracy than faceted systems, they also introduce a host of topological, geometric, and numerical complications. A method for calculating accurate boundary representations of solid models is introduced that reduces the impact of these complications. The method uses a pair of bounding polyhedral approximations to enclose the boundary of each object. A structural analysis automatically determines where to make adaptive refinements to the polyhedrons to assure the topological validity of the results. Potential singularities are localized. The implementation is an experimental extension to the Geometric Design Processor (GDP) solid modeling system     

Octree creation via C.S.G. definition

The common method for generating the octrees of complex objects, is based upon generating the octrees of several pre-defined primitives and applying Boolean operations on them. Regardless how the octrees representing the primitives are generated (top-down or bottom-up) the octree of a desired object is obtained by performing Boolean operations among the primitives comprising the object according to the object's CSG (constructive solid Geometry) representation. When carrying out this procedure, most of the computing and memory resources are used for generating and storing the octants comprising the primitives. However, the majority of those octants are not required for the representation of the final object. In this paper the extention of the top-down approach to the CSG level (i.e., generating the octree of an object directly from its CSG representation) is proposed. With this method there is no need to generate the octrees of the primitives comprising the object nor to perform Boolean operations on them. Moreover, only these octants which belong to the final object are generated.