Boolean Set Operations Research Articles

A simple algorithm for medial axis transform computation

Although medial axis transform is introduced as a shape description for many engineering applications, the computational algorithm is still challenging. This is especially true for the shape with free form boundary. This paper presents an algorithm for medial axis transform computation from a perspective of minimum distance between the points in a two-dimensional shape and its boundary. The minimum distance is given by a resultant distance function which is a superposition of the individual distance function between a point within a shape and each boundary point. By elaborating the resultant distance function, the medial axis transform will be obtained naturally. The distance function is modeled as a solid cone and the superposition is equivalent to the union Boolean set operation. The implementation of the approach is simplified using a solid modeling kernel. Several examples of two-dimensional shapes with free form boundaries are raised to illustrate the concept and algorithm.

Contour Offset Approach to Generate Tool Path for Polyhedral Surfaces Machining

Polyhedral surfaces are used as representation model for CAM and process planning purposes because of its simplicity for data exchange and geometric computation. However, there is few tool path planning strategies for such surfaces but the iso-plane method. In this paper, contour parallel path are generated for three-axis ball-end milling. This tool path is based on a novel algorithm for offsetting curves on polyhedral surfaces presented in this paper. It reduces the task of removing complex interfering of offset curve from 3D surface to 2D plane by flattening mesh surface, and avoids costly 3D Boolean set operations and expensive distance calculation. This results in an efficient tool-path generation. Empirical examples illustrate the feasibility the proposed method.

Boolean operations on multi-region solids for mesh generation

An algorithm for Boolean operations on non-manifold models is proposed to allow the treatment of solids with multiple regions (internal interfaces) and degenerate portions (shells and wires), in the context of mesh generation. In a solid modeler, one of the most powerful tools to create three-dimensional objects with any level of geometric complexity is the Boolean set operators. They are intuitive and popular ways to combine solids, based on the operations applied to point sets. To assure that the resulting objects have the same dimension as the original objects, without loose or dangling parts, a regularization process is usually applied after a Boolean operation. In practice, the regularization is performed classifying the topological elements and removing internal or lower-dimensional structures. However, in many engineering applications, the adopted geometric model may contain idealized internal parts, as in the case of multi-region models, or lower-dimensional parts, as in the case of solids that contain dangling slabs that are represented as zero-thickness surfaces or wireframes in the model. Therefore, the aim of this work is the development of a generic algorithm that allows the application of the Boolean set operations in a geometric modeling environment applied to finite and boundary element mesh generation. This environment adopts a non-manifold boundary representation that considers an undefined number of topological entities (group concept), and works with objects of different dimensions and with objects not necessarily plane or polyhedral (parametric curved surfaces). Numerical examples are presented to illustrate the proposed methodology.

Parallel and efficient Boolean on polygonal solids

We present a novel framework which can efficiently evaluate approximate Boolean set operations for B-rep models by highly parallel algorithms. This is achieved by taking axis-aligned surfels of Layered Depth Images (LDI) as a bridge and performing Boolean operations on the structured points. As compared with prior surfel-based approaches, this paper has much improvement. Firstly, we adopt key-data pairs to store LDI more compactly. Secondly, robust depth peeling is investigated to overcome the bottleneck of layer-complexity. Thirdly, an out-of-core tiling technique is presented to overcome the limitation of memory. Real-time feedback is provided by streaming the proposed pipeline on the many-core graphics hardware.

Improved Binary Space Partition merging

This paper presents a new method for evaluating boolean set operations between Binary Space Partition (BSP) trees. Our algorithm has many desirable features, including both numerical robustness and O ( n ) output sensitive time complexity, while simultaneously admitting a straightforward implementation. To achieve these properties, we present two key algorithmic improvements. The first is a method for eliminating null regions within a BSP tree using linear programming. This replaces previous techniques based on polygon cutting and tree splitting. The second is an improved method for compressing BSP trees based on a similar approach within binary decision diagrams. The performance of the new method is analyzed both theoretically and experimentally. Given the importance of boolean set operations, our algorithms can be directly applied to many problems in graphics, CAD and computational geometry.

Shape sensitivity of constructively represented geometric models

Most solid models are archived using boundary representations, but they are created, edited, and optimized using high level constructive methods that rely on parameterized Boolean set operations and feature-based techniques. Downstream applications often require optimization of integral-valued performance measures over such models that include volume, mass, and energy properties, as well as more general distributed fields (stress, temperature, etc.). A key computational utility in all such applications is the computation of the sensitivity of the performance measure with respect to the parameters in the solid's construction history. We show that for a class of performance measures defined as domain integrals, the sensitivity with respect to a parameter requires integration over a subset of the solid's boundaries that is affected by that parameter. In contrast to earlier methods, the proposed approach for computing sensitivities does not require solid's boundary to remain homeomorphic, and may be used with most types of constructive representations, including CSG and feature-based representations, where the defining Boolean expression may not be known. The simplicity and effectiveness of the proposed technique are illustrated on several common shape optimization problems.

Boolean Set Operations with Cubic Algebraic Patches

In this paper we discuss a symbolic-numeric algorithm for Boolean operations, closed in the algebra of curved polyhedra whose boundary is triangulated with algebraic patches (A-patches). This approach uses a linear polyhedron as a first approximation of both the arguments and the result. On each triangle of a boundary representation of such linear approximation, a piecewise cubic algebraic interpolant is built, using a C(1)-continuous prism algebraic patch (prism A-patch) that interpolates the three triangle vertices, with given normal vectors. The boundary representation only stores the vertices of the initial triangulation and their external vertex normals. In order to represent also flat and/or sharp local features, the corresponding normal-per-face and/or normal-per-edge may be also given, respectively. The topology is described by storing, for each curved triangle, the two triples of pointers to incident vertices and to adjacent triangles. For each triangle, a scaffolding prism is built, produced by its extreme vertices and normals, which provides a containment volume for the curved interpolating A-patch. When looking for the result of a regularized Boolean operation, the 0-set of a tri-variate polynomial within each such prism is generated, and intersected with the analogous 0-sets of the other curved polyhedron, when two prisms have non-empty intersection. The intersection curves of the boundaries are traced and used to decompose each boundary into the 3 standard classes of subpatches, denoted in, out and on. While tracing the intersection curves, the locally refined triangulation of intersecting patches is produced, and added to the boundary representation.

Boolean operations on 3D selective Nef complexes: Data structure, algorithms, optimized implementation and experiments

Nef polyhedra in d-dimensional space are the closure of half-spaces under boolean set operations. In consequence, they can represent non-manifold situations, open and closed sets, mixed-dimensional complexes, and they are closed under all boolean and topological operations, such as complement and boundary. They were introduced by W. Nef in his seminal 1978 book on polyhedra. The generality of Nef complexes is essential for some applications. In this paper, we present a new data structure for the boundary representation of three-dimensional Nef polyhedra and efficient algorithms for boolean operations. We use exact arithmetic to avoid well-known problems with floating-point arithmetic and handle all degeneracies. Furthermore, we present important optimizations for the algorithms, and evaluate this optimized implementation with extensive experiments. The experiments supplement the theoretical runtime analysis and illustrate the effectiveness of our optimizations. We compare our implementation with the Acis CAD kernel. Acis is mostly faster, by a factor up to six. There are examples on which Acis fails. The implementation was released as Open Source in the Computational Geometry Algorithm Library ( Cgal) release 3.1 in December 2004.

Exact and approximate construction of offset polygons

The Minkowski sum of two sets A , B ∈ R 2 , denoted A ⊕ B , is defined as { a + b ∣ a ∈ A , b ∈ B } . We describe an efficient and robust implementation of the construction of the Minkowski sum of a polygon in R 2 with a disc, an operation known as offsetting the polygon. Our software package includes a procedure for computing the exact offset of a straight-edge polygon, based on the arrangement of conic arcs computed using exact algebraic number-types. We also present a conservative approximation algorithm for offset computation that uses only rational arithmetic and decreases the running times by an order of magnitude in some cases, while having a guarantee on the quality of the result. The package will be included in the next public release of the Computational Geometry Algorithms Library, Cgal Version 3.3. It also integrates well with other Cgal packages; in particular, it is possible to perform regularized Boolean set-operations on the polygons the offset procedures generate.

Progressive Conversion from B-rep to BSP for Streaming Geometric Modeling

We introduce a novel progressive approach to generate a Binary Space Partition (BSP) tree and a convex cell decomposition for any input triangles boundary representation (B-rep), by utilizing a fast calculation of the surface inertia. We also generate a solid model at progressive levels of detail. This approach relies on a variation of standard BSP tree generation, allowing for labeling cells as in, out and fuzzy, and which permits a comprehensive representation of a solid as the Hasse diagram of a cell complex. Our new algorithm is embedded in a streaming computational framework, using four types of dataflow processes that continuously produce, transform, combine or consume subsets of cells depending on their number or input/output stream. A varied collection of geometric modeling techniques are integrated in this streaming framework, including polygonal, spline, solid and heterogeneous modeling with boundary and decompositive representations, Boolean set operations, Cartesian products and adaptive refinement. The real-time B-rep to BSP streaming results we report in this paper are a large step forward in the ultimate unification of rapid conceptual and detailed shape design methodologies.

Semiautomated Building Extraction Based on CSG Model-Image Fitting

mostly implemented in a semi-automatic manner, solving the Building extraction based on pre-established models has been model-image fitting problem based on some high-level inforrecognized as a promising idea for acquiring 3D data for build- mation given by the operator. The spatial data of a building ings from aerial images. This paper proposes a novel building object are determined, when model-image fitting is achieved. extraction method developed from the concept of fitting CSG In contrast to the traditional point-by-point mapping proce(Constructive Solid Geometry) primitives to aerial images. To dure, model-based building extraction features object-based be practicable, this method adopts a semiautomatic proce- data acquisition. Although the idea and benefits of modeldure, carrying out high-level tasks (building detection, model based building extraction have been acknowledged, the workselection, and attribution) interactively by the operator and ing principle is not well established. Therefore, the focus of performing optimal model-image fitting automatically with a this study is to establish a practical theory for model-based least-squares fitting algorithm. Buildings, represented by CSG building extraction. models, can be reconstructed part by part after fitting each Building modeling and model-image fitting are the key parameterized CSG primitive to the edge pixels of aerial issues in model-based building extraction. The issue on buildimages. Reconstructed building parts can then be combined ing modeling is how to establish a set of representative and using CSG Boolean set operators. Consequently, a building is complete building models. This paper reviews some building represented by a CSG tree in which each node links two model schemes known in the field of digital photogrammetry branches of combined parts. This paper demonstrates ten and discusses how CSG modeling is employed in the proposed examples of building extraction from aerial photos taken at a method. The issue in model-image fitting is how to develop a scale of 1:5,000 and scanned at a pixel size of 25 m. All of computer algorithm that is able to determine the pose and the tests were performed in the prototypal system implemented shape parameters of an object model such that the edge lines of in a CAD-based environment cooperated with a number of the wire frame, as projected into the images, are optimally specially designed programs. The process time for each prim- coincident with the corresponding edge pixels. It is assumed itive is about 20 seconds and the successful rate of model- that the image orientations are known and that the pose and image fitting was about 90 percent. Evaluated with some check shape parameters are approximately determined through an points, the fitting accuracy was about 0.3 m horizontally and interactive manual process. To deal with this problem, this 1m vertically. The test results are encouraging and promote paper proposes a tailored least-squares model-image fitting the theory of model-based building extraction. algorithm as the key component of the building extraction framework.

同次処理に基づく整数演算を用いた多面体ソリッドモデラ

In solid modeling systems, the stability of Boolean set operations is an important issue. Solid modeling systems that employ floating-point arithmetic tend to be unstable because of inconsistent decisions caused by numerical errors. The use of exact integer arithmetic solves this problem. By using exact integer arithmetic based on totally homogeneous processing, error-free arithmetic is implemented. In this paper, we propose a robust polyhedral solid modeling system. The system employs exact integer arithmetic based on totally homogeneous processing. All of the numerical data of solid models for Boolean set operations are represented in terms of integer representations. Boolean set operations and transformations of solid models are performed in the integer domain. Several examples of Boolean set operations, which are very difficult in floating point arithmetic, are presented to show that our system does not cause failure in such situations. Methods that improve the efficiency of exact integer arithmetic are also presented to avoid the increase of computation time caused by the increase of the data lengths of integers.

Toward a Meta-Model for Computational Engineering

Geometric modeling and finite element analysis have matured in recent decades. Both methods are used extensively in engineering design. However, the link between geometric modeling, physical modeling and finite element analysis is currently cumbersome, error-prone, and ad-hoc. Topological domain modeling provides the missing link. In this paper, we propose a combined topological modeling and finite element modeling method that allows not only topological modeling, but also promotes geometric and physical modeling, by providing a topological base space for the definition of finite element meshes, fields, and the definition and solution of boundary value problems. We call the method the Constructive Topological Domain Method (CTDM). In this method, Primitive Topological Domains (PTDs), each possessing a natural coordinate space, are combined in multiple n-dimensional Cartesian coordinate spaces, called charts, using generalizations of Boolean set operations, to create Constructed Topological Domains (CTDs) capable of acting as the base spaces of fiber bundles. The charts are glued together to create an atlas, within which the CTD is defined. The fiber of the bundle may describe, in addition to geometry, physical fields like density, stress, and temperature. Finite element meshes may be defined upon each of the PTDs from which the CTD is constructed, enabling the definition and solution of boundary value problems, thus avoiding the difficult and messy problem of creating a single finite element mesh to represent the entire CTD. A modified finite element method, to handle the individually meshed PTDs, is described. The boundary conditions may be specified as analytical or as finite element-based fields upon each of the PTDs. The CTDM appears to be a promising approach to robust mathematical and computational modeling of physical objects. Simple examples are presented.

Model-Based 3D Reconstruction of Buildings from Multiple Aerial Images

This paper proposes a model-based building reconstruction method. Modeled by Constructive Solid Geometry (CSG), buildings are reconstructed part by part by fitting each parameterized CSG primitive to the corresponding edge pixels of aerial images. The shape and pose parameters associated to a primitive provide a link between perception (images) and prior knowledge (primitive) of a building part, so that the fitting method proceeds to determine the shape and pose parameters so as to fit a primitive with the corresponding images. Having all of the building parts been uniquely represented by parametric primitives, a building can be reconstructed by using CSG Boolean set operators to combine the building parts. Consequently, a building is represented by a CSG-tree in which each node links two branches of combined parts. This paper demonstrates 10 examples of extracting various buildings. The process time for each primitive is about 20 sec and the successful rate of model-image fitting is about 90%.

BOOLE: A BOUNDARY EVALUATION SYSTEM FOR BOOLEAN COMBINATIONS OF SCULPTURED SOLIDS

In this paper we describe a system, BOOLE, that generates the boundary representations (B-reps) of solids given as a CSG expression in the form of trimmed Bézier patches. The system makes use of techniques from computational geometry, numerical linear algebra and symbolic computation to generate the B-reps. Given two solids, the system first computes the intersection curve between the two solids using our surface intersection algorithm. Using the topological information of each solid, it computes various components within each solid generated by the intersection curve and their connectivity. The component classification step is performed by ray-shooting. Depending on the Boolean operation performed, appropriate components are put together to obtain the final solid. We also present techniques to parallelize this system on shared memory multiprocessor machines. The system has been successfully used to generate B-reps for a number of large industrial models including parts of a notional submarine storage and handling room (courtesy - Electric Boat Inc.) and Bradley fighting vehicle (courtesy - Army Research Labs). Each of these models is composed of over 8000 Boolean operations and is represented using over 50,000 trimmed Bézier patches. Our exact representation of the intersection curve and use of stable numerical algorithms facilitate an accurate boundary evaluation at every Boolean set operation and generation of topologically consistent solids.

多変数Ｓｔｕｒｍ列を利用した曲線・曲面の交点の存在判定

The interference processing of curves and surfaces is an important technique in CAD/CAM applications. The lack of the stability in the interference processing is due to numerical errors caused by floating point arithmetic and the approximate computations. In the interference processing, the geometric Newton-Raphson method is generally used for approximations, because the solutions of high degree polynomial equations cannot be exactly computed. Most geometric algorithms, such as Boolean set operation algorithms, modify the topological data structure according to the existence of the solutions of polynomial equations. The number of solutions of algebraic equations can be exactly counted with Sturm's theorem and exact arithmetic. In this paper, a stable interference method for curves and surfaces using Sturm's theorem and exact arithmetic is proposed. The proposed approach can exactly evaluate the existence of intersection points in curve/surface intersection.

Voxel-Based Virtual Multi-Axis Machining

Although multi-axis CNC machines are well known for complicated part machining involving few time-consuming set-ups, difficulties arise in generating collision-free tool paths and determining cutting parameters. Machining simulation is used to validate the generated information for multi-axis CNC machines before it is physically downloaded and executed. The objective of this research is to develop a voxel-based simulator for multi-axis CNC machining. The simulator displays the machining process in which the initial workpiece is incrementally converted into the finished part. The voxel representation is used to model efficiently the state of the in-process workpiece, which is generated by successively subtracting tool swept volumes from the workpiece. The voxel representation also simplifies the computation of regularised Boolean set operations and of material removal volumes. By using the material removal rate measured by the number of removed voxels, the feedrate can be adjusted adaptively to increase the machining productivity.

浮動小数点演算ユニットを利用した４×４行列式の適応的符号判定処理

In solid modeling systems, the stability of Boolean set operation algorithms is an important issue. The stability can be achieved by using the homogeneous processing, which is proposed by one of the authors, and the exact integer arithmetic. The variable integer data type can represent very large integers by changing the size of the date length. However, the increase of the data length of integers leads to the increase of the computation time. In geometric algorithms, the signs of determinants are necessary to modify the topological data structure. The previous method adaptively and repeatedly extracts 8 bits from variable integer data to detect the signs of 4×4 determinants. The previously proposed method detects the signs of determinants using an FPU. In this paper, a more efficient method is proposed for detecting the signs of 4×4 determinants by extracting 16 bits. The algorithm can detect the signs of 4×4 determinants faster than the previous algorithm.

The dual of sweep

As an infinite union operation, sweep of a moving object through space is a powerful and natural addition to the Boolean set operations that incorporates motion-related information for the purposes of shaping, collision detection, and simulation of moving objects. Use of sweep has been hindered by limited computational support and by the fact that it is a ‘material growing’ operation, whereas many applications, such as interference elimination and mechanism design, appear to be better modeled by a ‘material removal’ operation. This article formally defines a new geometric modeling operation of unsweep. Given an arbitrary subset E of Euclidean space and a general motion M, unsweep( E, M) returns the largest subset of E that remains inside the original E under M. When M is a translation, unsweep ( E, M) naturally reduces to the standard Minkowski difference of E and the trajectory generated by the inverted motion M̂. In this sense, the operation of unsweep is a generalization of Minkowski difference that corresponds to a ‘material removal’ operation, it can be also defined as an infinite intersection operation, and is the dual of sweep in a precise set-theoretic sense. We show that unsweep has attractive theoretical and computational properties, including a practical point membership test for arbitrary general motions. Using duality, the established properties of unsweep is extended to the general sweep operation, and can be used to improve the computational support for general sweeps.

An undecidable fragment of the theory of set constraints

Set constraints are inclusions between expressions denoting sets of trees. In atomic set constraints, the syntax of set expressions is restricted not to contain any Boolean set operators. Using a reduction from Hilbert's Tenth Problem we prove the undecidability of the ∃ ∗∀ ∗-fragment of the first-order theory of atomic set constraints. This improves recent results by Seynhaeve et al.