Gill Barequet

Efficiently approximating the minimum-volume bounding box of a point set in three dimensions

We present an efficient O(n+1/?4.5-time algorithm for computing a (1+?)-approximation of the minimum-volume bounding box of n points in R3. We also present a simpler algorithm whose running time is O(nlogn+n/?3). We give some experimental results with implementations of various variants of the second algorithm.

Filling gaps in the boundary of a polyhedron

Abstract In this paper we present an algorithm for detecting and repairing defects in the boundary of a polyhedron. These defects, usually caused by problems in CAD software, consist of small gaps bounded by edges that are incident to only one polyhedron face. The algorithm uses a partial curve matching technique for matching parts of the defects, and an optimal triangulation of 3-D polygons for resolving the unmatched parts. It is also shown that finding a consistent set of partial curve matches with maximum score, a subproblem which is related to our repairing process, is NP-hard. Experimental results on several polyhedra are presented.

Piecewise-Linear Interpolation between Polygonal Slices

In this paper we present a new technique for piecewise-linear surface reconstruction from a series of parallel polygonal cross sections. This is an important problem in medical imaging, surface reconstruction from topographic data, and other applications. We reduce the problem, as in most previous works, to a series of problems of piecewise-linear interpolation between each pair of successive slices. Our algorithm uses a partial curve matching technique for matching parts of the contours, an optimal triangulation of 3-D polygons for resolving the unmatched parts, and a minimum spanning tree heuristic for interpolating between nonsimply connected regions. Unlike previous attempts at solving this problem, our algorithm seems to handle successfully in practice any kind of data. It allows multiple contours in each slice, with any hierarchy of contour nesting, and avoids the introduction of counterintuitive bridges between contours, proposed in some earlier papers to handle interpolation between multiply connected regions. Experimental results on various complex examples, involving actual medical imaging data, are presented and show the good and robust performance of our algorithm.

Repairing CAD models

We describe an algorithm for repairing polyhedral CAD models that have errors in their B-REP. Errors like cracks, degeneracies, duplication, holes and overlaps are usually introduced in solid models due to imprecise arithmetic, model transformations, designer errors, programming bugs, etc. Such errors often hamper further processing such as finite element analysis, radiosity computation and rapid prototyping. Our fault-repair algorithm converts an unordered collection of polygons to a shared-vertex representation to help eliminate errors. This is done by choosing, for each polygon edge, the most appropriate edge to unify it with. The two edges are then geometrically merged into one, by moving vertices. At the end of this process, each polygon edge is either coincident with another or is a boundary edge for a polygonal hole or a dangling wall and may be appropriately repaired. Finally, in order to allow user-inspection of the automatic corrections, we produce a visualization of the repair and let the user mark the corrections that conflict with the original design intent. A second iteration of the correction algorithm then produces a repair that is commensurate with the intent. This, by involving the users in a feedback loop, we are able to refine the correction to their satisfaction.

BOXTREE: A Hierarchical Representation for Surfaces in 3D

We introduce the boxtree, a versatile data structure for representing triangulated or meshed surfaces in 3D. A boxtree is a hierarchical structure of nested boxes that supports efficient ray tracing and collision detection. It is simple and robust, and requires minimal space. In situations where storage is at a premium, boxtrees are effective alternatives to octrees and BSP trees. They are also more flexible and efficient than R‐trees, and nearly as simple to implement.

Piecewise-linear interpolation between polygonal slices

In this paper we present a new technique for piecewise-linear surface reconstruction from a series of parallel polygonal cross-sections. This is an important problem in medical imaging, surface reconstruction from topographic data, and other applications. We reduce the problem, as in most previous works, to a series of problems of piecewise-linear interpolation between each pair of successive slices. Our algorithm uses a partial curve matching technique for matching parts of the contours, an optimal triangulation of 3-D polygons for resolving the unmatched parts, and a minimum spanning tree heuristic for interpolating between non simply connected regions. Unlike previous attempts at solving this problem, our algorithm seems to handle successfully any kind of data. It allows multiple contours in each slice, with any hierarchy of contour nesting, and avoids the introduction of counter-intuitive bridges between contours, proposed in some earlier papers to handle interpolation between multiply connected regions. Experimental results on various complex examples, involving actual medical imaging data, are presented, and show the good and robust performance of our algorithm.

Multilevel sensitive reconstruction of polyhedral surfaces from parallel slices

1Dept. of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel; The Center for Geometric Computing, Dept. of Computer Science, Johns Hopkins University, Baltimore, MD 21218, USA 2Dept. of Computer Science, Princeton University, Princeton, NJ 08540, USA 3Dept. of Applied Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel e-mail: barequet@cs.technion.ac.il, dans@cs.princeton.edu, ayellet@ee.technion.ac.il

RSVP: a geometric toolkit for controlled repair of solid models

The paper presents a system and the associated algorithms for repairing the boundary representation of CAD models. Two types of errors are considered: topological errors, i.e., aggregate errors, like zero volume parts, duplicate or missing parts, inconsistent surface orientation, etc., and geometric errors, i.e., numerical imprecision errors, like cracks or overlaps of geometry. The output of our system describes a set of clean and consistent two-manifolds (possibly with boundaries) with derived adjacencies. Such solid representation enables the application of a variety of rendering and analysis algorithms, e.g., finite element analysis, radiosity computation, model simplification, and solid free form fabrication. The algorithms described were originally designed to correct errors in polygonal B-Reps. We also present an extension for spline surfaces. Central to our system is a procedure for inferring local adjacencies of edges. The geometric representation of topologically adjacent edges are merged to evolve a set of two-manifolds. Aggregate errors are discovered during the merging step. Unfortunately, there are many ambiguous situations where errors admit more than one valid solution. Our system proposes an object repairing process based on a set of user tunable heuristics. The system also allows the user to override the algorithms decisions in a repair visualization step. In essence, this visualization step presents an organized and intuitive way for the user to explore the space of valid solutions and to select the correct one.

A bijection between permutations and floorplans, and its applications

A floorplan represents the relative relations between modules on an integrated circuit. Floorplans are commonly classified as slicing, mosaic, or general. Separable and Baxter permutations are classes of permutations that can be defined in terms of forbidden subsequences. It is known that the number of slicing floorplans equals the number of separable permutations and that the number of mosaic floorplans equals the number of Baxter permutations [B. Yao, H. Chen, C.K. Cheng, R.L. Graham, Floorplan representations: complexity and connections, ACM Trans. Design Automation Electron. Systems 8(1) (2003) 55-80]. We present a simple and efficient bijection between Baxter permutations and mosaic floorplans with applications to integrated circuits design. Moreover, this bijection has two additional merits: (1) It also maps between separable permutations and slicing floorplans; and (2) it suggests enumerations of mosaic floorplans according to various structural parameters.

Using geometric hashing to repair CAD objects

Problems in CAD software sometimes cause defects in the boundaries of polyhedral objects-small gaps bounded by edges incident to one polyhedron face. Using geometric hashing, algorithms can resolve this problem, which has vexed layered manufacturing. Similar techniques hold promise for solving problems in such areas as computer vision, medical imaging, and molecular biology.