QuickCSG: Fast Arbitrary Boolean Combinations of N Solids

QuickCSG computes the result for general N-polyhedron boolean expressions without an intermediate tree of solids. We propose a vertex-centric view of the problem, which simplifies the identification of final geometric contributions, and facilitates its spatial decomposition. The problem is then cast in a single KD-tree exploration, geared toward the result by early pruning of any region of space not contributing to the final surface. We assume strong regularity properties on the input meshes and that they are in general position. This simplifying assumption, in combination with our vertex-centric approach, improves the speed of the approach. Complemented with a task-stealing parallelization, the algorithm achieves breakthrough performance, one to two orders of magnitude speedups with respect to state-of-the-art CPU algorithms, on boolean operations over two to dozens of polyhedra. The algorithm also outperforms GPU implementations with approximate discretizations, while producing an output without redundant facets. Despite the restrictive assumptions on the input, we show the usefulness of QuickCSG for applications with large CSG problems and strong temporal constraints, e.g. modeling for 3D printers, reconstruction from visual hulls and collision detection

QuickCSG: Fast Arbitrary Boolean Combinations of N Solids

QuickCSG computes the result for general N-polyhedron boolean expressions without an intermediate tree of solids. We propose a vertex-centric view of the problem, which simplifies the identification of final geometric contributions, and facilitates its spatial decomposition. The problem is then cast in a single KD-tree exploration, geared toward the result by early pruning of any region of space not contributing to the final surface. We assume strong regularity properties on the input meshes and that they are in general position. This simplifying assumption, in combination with our vertex-centric approach, improves the speed of the approach. Complemented with a task-stealing parallelization, the algorithm achieves breakthrough performance, one to two orders of magnitude speedups with respect to state-of-the-art CPU algorithms, on boolean operations over two to dozens of polyhedra. The algorithm also outperforms GPU implementations with approximate discretizations, while producing an output without redundant facets. Despite the restrictive assumptions on the input, we show the usefulness of QuickCSG for applications with large CSG problems and strong temporal constraints, e.g. modeling for 3D printers, reconstruction from visual hulls and collision detection

Fast Exact Booleans for Iterated CSG using Octree-Embedded BSPs

We present octree-embedded BSPs, a volumetric mesh data structure suited for performing a sequence of Boolean operations (iterated CSG) efficiently. At its core, our data structure leverages a plane-based geometry representation and integer arithmetics to guarantee unconditionally robust operations. These typically present considerable performance challenges which we overcome by using custom-tailored fixed-precision operations and an efficient algorithm for cutting a convex mesh against a plane. Consequently, BSP Booleans and mesh extraction are formulated in terms of mesh cutting. The octree is used as a global acceleration structure to keep modifications local and bound the BSP complexity. With our optimizations, we can perform up to 2.5 million mesh-plane cuts per second on a single core, which creates roughly 40-50 million output BSP nodes for CSG. We demonstrate our system in two iterated CSG settings: sweep volumes and a milling simulation

Generating Property-Directed Potential Invariants By Backward Analysis

This paper addresses the issue of lemma generation in a k-induction-based formal analysis of transition systems, in the linear real/integer arithmetic fragment. A backward analysis, powered by quantifier elimination, is used to output preimages of the negation of the proof objective, viewed as unauthorized states, or gray states. Two heuristics are proposed to take advantage of this source of information. First, a thorough exploration of the possible partitionings of the gray state space discovers new relations between state variables, representing potential invariants. Second, an inexact exploration regroups and over-approximates disjoint areas of the gray state space, also to discover new relations between state variables. k-induction is used to isolate the invariants and check if they strengthen the proof objective. These heuristics can be used on the first preimage of the backward exploration, and each time a new one is output, refining the information on the gray states. In our context of critical avionics embedded systems, we show that our approach is able to outperform other academic or commercial tools on examples of interest in our application field. The method is introduced and motivated through two main examples, one of which was provided by Rockwell Collins, in a collaborative formal verification framework.Comment: In Proceedings FTSCS 2012, arXiv:1212.657

QuickCSG: Arbitrary and Faster Boolean Combinations of N Solids

While studied over several decades, the computation of boolean operations on polyhedra is almost always addressed by focusing on the case of two polyhedra. For multiple input polyhedra and an arbitrary boolean operation to be applied, the operation is decomposed over a binary CSG tree, each node being processed separately in quasilinear time. For large trees, this is both error prone due to intermediate geometry and error accumulation, and inefficient because each node yields a specific overhead. We introduce a fundamentally new approach to polyhedral CSG evaluation, addressing the general N-polyhedron case. We propose a new vertex-centric view of the problem, which both simplifies the algorithm computing resulting geometric contributions, and vastly facilitates its spatial decomposition. We then embed the entire problem in a single KD-tree, specifically geared toward the final result by early pruning of any region of space not contributing to the final surface. This not only improves the robustness of the approach, it also gives it a fundamental speed advantage, with an output complexity depending on the output mesh size instead of the input size as with usual approaches. Complemented with a task-stealing parallelization, the algorithm achieves breakthrough performance, one to two orders of magnitude speedups with respect to state-of-the-art CPU algorithms, on boolean operations over two to several dozen polyhedra. The algorithm is also shown to outperform recent GPU implementations and approximate discretizations, while producing an exact output without redundant facets.Quoique étudié depuis des décennies, le calcul d'opérations booléennes sur des polyèdres est quasiment toujours fait sur deux opérandes. Pour un plus grand nombre de polyèdres et une opération booléenne arbitraire à effectuer, l'opération est décomposée sur un arbre binaire CSG (géométrie constructive), dans lequel chaque nœud est traité séparément en temps quasi-linéaire. Pour de grands arbres, ceci est à la fois source d'erreurs, à cause des calculs géométriques intermédiaires, et inefficace à cause des traitements superflus au niveau des nœuds. Nous introduisons une approche fondamentalement nouvelle qui traite le cas général de N polyèdres. Nous proposons une vue du problème centrée sur les sommets, ce qui simplifie l'algorithme et facilite sa décomposition spatiale. Nous traitons le problème dans un seul KD-tree, qui est dirigé vers le résultat final, en élaguant les régions de l'espace qui ne contribuent pas à la surface finale. Non seulement ceci améliore la robustesse de l'approche mais ça lui donne un avantage en vitesse, car la complexité dépend plus de la taille de la sortie que celle d'entrée. En la combinant avec une parallélisation basée sur du vol de tâche, l'algorithme a des performances inouïes, d'un ou deux ordres de grandeur plus rapide que les algorithmes de l'état de l'art sur CPU et GPU. De plus il produit un résultat exact, sans aucune primitive géométrique superflue

QuickCSG: Arbitrary and Faster Boolean Combinations of N Solids

While studied over several decades, the computation of boolean operations on polyhedra is almost always addressed by focusing on the case of two polyhedra. For multiple input polyhedra and an arbitrary boolean operation to be applied, the operation is decomposed over a binary CSG tree, each node being processed separately in quasilinear time. For large trees, this is both error prone due to intermediate geometry and error accumulation, and inefficient because each node yields a specific overhead. We introduce a fundamentally new approach to polyhedral CSG evaluation, addressing the general N-polyhedron case. We propose a new vertex-centric view of the problem, which both simplifies the algorithm computing resulting geometric contributions, and vastly facilitates its spatial decomposition. We then embed the entire problem in a single KD-tree, specifically geared toward the final result by early pruning of any region of space not contributing to the final surface. This not only improves the robustness of the approach, it also gives it a fundamental speed advantage, with an output complexity depending on the output mesh size instead of the input size as with usual approaches. Complemented with a task-stealing parallelization, the algorithm achieves breakthrough performance, one to two orders of magnitude speedups with respect to state-of-the-art CPU algorithms, on boolean operations over two to several dozen polyhedra. The algorithm is also shown to outperform recent GPU implementations and approximate discretizations, while producing an exact output without redundant facets.Quoique étudié depuis des décennies, le calcul d'opérations booléennes sur des polyèdres est quasiment toujours fait sur deux opérandes. Pour un plus grand nombre de polyèdres et une opération booléenne arbitraire à effectuer, l'opération est décomposée sur un arbre binaire CSG (géométrie constructive), dans lequel chaque nœud est traité séparément en temps quasi-linéaire. Pour de grands arbres, ceci est à la fois source d'erreurs, à cause des calculs géométriques intermédiaires, et inefficace à cause des traitements superflus au niveau des nœuds. Nous introduisons une approche fondamentalement nouvelle qui traite le cas général de N polyèdres. Nous proposons une vue du problème centrée sur les sommets, ce qui simplifie l'algorithme et facilite sa décomposition spatiale. Nous traitons le problème dans un seul KD-tree, qui est dirigé vers le résultat final, en élaguant les régions de l'espace qui ne contribuent pas à la surface finale. Non seulement ceci améliore la robustesse de l'approche mais ça lui donne un avantage en vitesse, car la complexité dépend plus de la taille de la sortie que celle d'entrée. En la combinant avec une parallélisation basée sur du vol de tâche, l'algorithme a des performances inouïes, d'un ou deux ordres de grandeur plus rapide que les algorithmes de l'état de l'art sur CPU et GPU. De plus il produit un résultat exact, sans aucune primitive géométrique superflue

BSP-fields: An Exact Representation of Polygonal Objects by Differentiable Scalar Fields Based on Binary Space Partitioning

The problem considered in this work is to find a dimension independent algorithm for the generation of signed scalar fields exactly representing polygonal objects and satisfying the following requirements: the defining real function takes zero value exactly at the polygonal object boundary; no extra zero-value isosurfaces should be generated; C1 continuity of the function in the entire domain. The proposed algorithms are based on the binary space partitioning (BSP) of the object by the planes passing through the polygonal faces and are independent of the object genus, the number of disjoint components, and holes in the initial polygonal mesh. Several extensions to the basic algorithm are proposed to satisfy the selected optimization criteria. The generated BSP-fields allow for applying techniques of the function-based modeling to already existing legacy objects from CAD and computer animation areas, which is illustrated by several examples