Performance of scanline algorithms for direct display of CSG

Direct display algorithms display a CSG model without first converting the model into a boundary representation. Three such algorithms are described. All three are based on the scanline display algorithm, and are able to handle both polygonal and quadratic faces.The first algorithm is based on Atherton's recursive subdivision scanline algorithm, the second is a combination of a scanline and a ray casting algorithm, and the third is a scanline version of the Trickle algorithm. A multiprocessor system in which these algorithms can be incorporated is also described.The performances of the algorithms are compared. It turns out that the algorithms efficiently display CSG models on general-purpose architectures. A comparison is also made between the performances for polygon-approximated models and exact models for objects bounded by quadratic faces, such as spherical, cylindrical and conical faces, to get an indication of how many polygons can at most be used to approximate quadratic faces and still have better performance.     

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.     

Techniques for reducing Boolean evaluation time in CSG scan-line algorithms

Atherton's scan-line hidden surface algorithm for Boolean combinations of plane-faced primitives is intended for fast image generation of complex models.

An important step in the algorithm is the Boolean evaluation at the required positions on each scan-line. This step, however, can be very time consuming if performed in a straightforward way.

Some techniques are presented here that considerably reduce the time needed for the Boolean evaluation. These include a fast bottom-up evaluation technique and partial backface elimination.     

限定TIN与CSG集成仿真模型生成算法研究

为了有效地组织和管理三维GIS空间数据,实现对三维空间实体高效、完整地表达,提出了限定不规则三角网（TIN）与构造实体几何（CSG）集成的仿真建模算法。采用TIN模型描述地形,CSG模型描述建筑物,通过抽取建筑物地面轮廓线作为地形三角剖分时的限定约束条件,将两种模型有机集成在一起。同时,实现了两类模型可视化渲染操作的同步进行。仿真试验结果表明,算法在保证模拟精度的前提下,能有效地简化建模过程,在一定程度上降低了可视化渲染计算的复杂度。     

明暗效果快速多变的CSG实本显示技术

CSG(Constructive Solid Geometry)是实体造型中一种重要表示方法.传统的CSG实体显示方法不仅费时,而且难以控制明暗效果.本文提出的方法以一种新的CSG体素表面光强生成模型为基础,结台深度缓冲技术,能快速有效地显示CSG实体,方便灵活地控制明暗效果.     

Visualization algorithm for CSG polyhedral solids

An algorithm is presented here to visualize here to visualize CSG solids in wireframe with hidden faces eliminated. The approach taken is to construct the image of the CSG solid directly from the CSG tree. This algorithm takes into account the face coherence property and the depth of the faces to minimize the number of rays fired during the process. It mixes a two-dimensional polygonal clipping and a ray-casting algorithm.     

大规模网格模型间的快速视觉布尔运算

为了解决产品设计阶段中大规模网格模型间的布尔运算无法实现立等可得的速度瓶颈,提出了一种新算法。该算法利用离散化采样获得射线段点云模型,将三角面片间的3D布尔运算转换为射线段间的1D布尔运算,对相交处的交点进行高精度的求解和插值处理,使得布尔运算速度大为提高,从而大大提升复杂拓扑结构的产品设计效率。通过该算法所获得射线段点云模型可获得等同于基于三角网格的渲染效果,该方法可进行工程应用。     

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.     

Experimental CSG environment for modelling solid

The purpose of this research is to develop an experimental environment for solid modelling which can provide a wide range of graphics primitives and operations toward making the modelling process more flexible. The system can handle different types of representations and combine them with an extended-CSG structure. A new geometric editor to enter and modify the data is also reported. To handle different types of representation, the system has an object-oriented structure and the idea of a common function is introduced to give the interface between them. Ray tracing is used as the rendering method. New techniques for generating images for complex objects are shown with several modelling examples.     

Construction and optimization of CSG representations

Boundary representations (B-reps) and constructive solid geometry (CSG) are widely used representation schemes for solids. While the problem of computing a B-rep from a CSG representation is relatively well understood, the inverse problem of B-rep to CSG conversion has not been addressed in general. The ability to perform B-rep to CSG conversion has important implications for the architecture of solid modelling systems and, in addition, is of considerable theoretical interest.

The paper presents a general approach to B-rep to CSG conversion based on a partition of Euclidean space by surfaces induced from a B-rep, and on the well known fact that closed regular sets and regularized set operations form a Boolean algebra. It is shown that the conversion problem is well defined, and that the solution results in a CSG representation that is unique for a fixed set of halfspaces that serve as a ‘basis’ for the representation. The ‘basis’ set contains halfspaces induced from a B-rep plus additional non-unique separating halfspaces.

An important characteristic of B-rep to CSG conversion is the size of a resulting CSG representation. We consider minimization of CSG representations in some detail and suggest new minimization techniques.

While many important geometric and combinatorial issues remain open, a companion paper shows that the proposed approach to B-rep to CSG conversion and minimization is effective in E², In E³, an experimental system currently converts natural-quadric B-reps in PARASOLID to efficient CSG representations in PADL-2.     

Constructive generation of the medial axis for solid models

The medial axis (MA) is a simplified representation of complicated models which is widely used in current research. However, the efficient generation of the MA for complicated solid models continues to pose a challenge. In this study, a constructive approach for the generation of the MA is proposed for solid models after they are voxelized. With this method, the MA of the model constructed from two operand models via a Boolean operation is efficiently generated by merging the MAs of the operand models in a certain way, instead of regenerating them from scratch. To support the proposed method, the affected region of the resultant model is computed first using a Boolean operation. Second, only the MA in the affected region is regenerated by distance dilation. Third, the complete MA of the resultant model is constructed by combining the newly generated MA with the unchanged MAs of the operand models. In this study, the accuracy and complexity are analyzed for the final MA and some examples are given to illustrate the performance of the proposed method.     

Applications of the 4×4 determinant method and the POLYGON ENGINE

In the first half of the paper, various types of processing pertaining to a polygon, using the 4×4 determinant theories are explained along with a new containment test algorithm of a point in a polygon. In the latter half of the paper, a general-purpose geometric processor, the POLYGON ENGINE, is presented which can deal with various types of interference problems, such as Boolean operations in solid modelling, hidden line and surface eliminations, ray tracing and so on. It is, a successor of the TRIANGLE PROCESSOR and is also based upon the 4×4 determinant theories [4–6]. While the TRIANGLE PROCESSOR processes a triangulated polygon on a triangle-by-triangle basis, the POLYGON ENGINE can treat a polygon without triangulation. The latter is expected to be more functional, more efficient and easier to use.     

An optimal parallel algorithm for triangulating a set of points in the plane

This paper presents an optimal parallel algorithm for triangulating an arbitrary set ofn points in the plane. The algorithm runs inO(logn) time usingO(n) space andO(_n) processors on a Concurrent-Read, Exclusive-Write Parallel RAM model (CREW PRAM). The parallel lower bound on triangulation is (logn) time so the best possible linear speedup has been achieved. A parallel divide-and-conquer technique of subdividing a problem into subproblems is employed.     

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.     

An enhanced linear Kalman filter (EnLKF) algorithm for parameter estimation of nonlinear rational models

In this study, an enhanced Kalman Filter formulation for linear in the parameters models with inherent correlated errors is proposed to build up a new framework for nonlinear rational model parameter estimation. The mechanism of linear Kalman filter (LKF) with point data processing is adopted to develop a new recursive algorithm. The novelty of the enhanced linear Kalman filter (EnLKF in short and distinguished from extended Kalman filter (EKF)) is that it is not formulated from the routes of extended Kalman Filters (to approximate nonlinear models by linear approximation around operating points through Taylor expansion) and also it includes LKF as its subset while linear models have no correlated errors in regressor terms. No matter linear or nonlinear models in representing a system from measured data, it is very common to have correlated errors between measurement noise and regression terms, the EnLKF provides a general solution for unbiased model parameter estimation without extra cost to convert model structure. The associated convergence is analysed to provide a quantitative indicator for applications and reference for further research. Three simulated examples are selected to bench-test the performance of the algorithm. In addition, the style of conducting numerical simulation studies provides a user-friendly step by step procedure for the readers/users with interest in their ad hoc applications. It should be noted that this approach is fundamentally different from those using linearisation to approximate nonlinear models and then conduct state/parameter estimate.     

A linear algorithm for eliminating hidden-lines from a polygonal cylinder

A variety of applications have motivated interest in the hidden-line and hidden-surface problem. This has resulted in a number of fundamentally different solutions. However no algorithm has been shown to be optimal. A common trait among algorithms for hidden-line elimination is a worst case complexity ofO(n ²). It is the interent here to introduce an algorithm that exhibits a linear worst case complexity. The use of a restricted class of input, has been employed to achieve asymptotic improvement in complexity as well as simplifying the problem enough to permit theoretic analysis of the algorithm. The class of input is still general enough to conform to the requirements of a number of applications.     

An efficient algorithm for ray casting of CSG animation frames

This paper presents a new algorithm to generate ray-cast CSG animation frames. We consider sequences of frames where only the objects can move; in this way, we take advantage of the high screen area coherence of this kind of animation. A new definition of bounding box allows us to reduce the number of pixels to be computed for the frames after the first. We associate a CSG subtree and two new flags, denoting if the box has changed in the current frame and if it will change in the next frame, with each box. We show with three examples the advantages of our technique when compared with an algorithm which entirely renders each frame of an animation. Intersections with CSG objects may be reduced to about one-fifth, while the rendering may be computed up to four times faster for the test sequences. © 1998 John Wiley & Sons, Ltd.     

A PDP constructive algorithm for system modelling

A new constructive algorithm is presented for building neural networks that learn to reproduce output temporal sequences based on one or several input sequences. This algorithm builds a network for the task of system modelling, dealing with continuous variables in the discrete time domain. The constructive scheme makes it user independent. The network's structure consists of an ordinary set and a classification set, so it is a hybrid network like that of Stokbro et al. [6], but with a binary classification. The networks can easily be interpreted, so the learned representation can be transferred to a human engineer, unlike many other network models. This allows for a better understanding of the system structure than just its simulation. This constructive algorithm limits the network complexity automatically, hence preserving extrapolation capabilities. Examples with real data from three totally different sources show good performance and allow for a promising line of research.     

An Algorithm to Evaluate Quantified Boolean Formulae and Its Experimental Evaluation

The high computational complexity of advanced reasoning tasks such as reasoning about knowledge and planning calls for efficient and reliable algorithms for reasoning problems harder than NP. In this paper we propose Evaluate, an algorithm for evaluating quantified Boolean formulae (QBFs). Algorithms for evaluation of QBFs are suitable for experimental analysis of problems that belong to a wide range of complexity classes, a property not easily found in other formalisms. Evaluate is a generalization of the Davis–Putnam procedure for SAT and is guaranteed to work in polynomial space. Before presenting the algorithm, we discuss several abstract properties of QBFs that we singled out to make it more efficient. We also discuss various options that were investigated about heuristics and data structures and report the main results of the experimental analysis. In particular, Evaluate is orders of magnitude more efficient than a nested backtracking procedure that resorts to a Davis–Putnam algorithm for handling the innermost set of quantifiers. Moreover, experiments show that randomly generated QBFs exhibit regular patterns such as phase transition and easy-hard-easy distribution.     

Reconstructing head models from photographs for individualized 3D‐audio processing

Visual fidelity and interactivity are the main goals in Computer Graphics research, but recently also audio is assuming an important role. Binaural rendering can provide extremely pleasing and realistic three‐dimensional sound, but to achieve best results it's necessary either to measure or to estimate individual Head Related Transfer Function (HRTF). This function is strictly related to the peculiar features of ears and face of the listener. Recent sound scattering simulation techniques can calculate HRTF starting from an accurate 3D model of a human head. Hence, the use of binaural rendering on large scale (i.e. video games, entertainment) could depend on the possibility to produce a sufficiently accurate 3D model of a human head, starting from the smallest possible input. In this paper we present a completely automatic system, which produces a 3D model of a head starting from simple input data (five photos and some key‐points indicated by user). The geometry is generated by extracting information from images and accordingly deforming a 3D dummy to reproduce user head features. The system proves to be fast, automatic, robust and reliable: geometric validation and preliminary assessments show that it can be accurate enough for HRTF calculation.