The Experts below are selected from a list of 10410 Experts worldwide ranked by ideXlab platform
Benjamin B Kimia - One of the best experts on this subject based on the ideXlab platform.
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a formal classification of 3d medial axis points and their local geometry
IEEE Transactions on Pattern Analysis and Machine Intelligence, 2004Co-Authors: Peter Giblin, Benjamin B KimiaAbstract:This paper proposes a novel hypergraph skeletal representation for 3D shape based on a formal derivation of the generic structure of its medial axis. By classifying each skeletal point by its order of contact, we show that, genetically, the medial axis consists of five types of points, which are then organized into sheets, Curves, and points: 1) sheets (manifolds with boundary) which are the locus of bitangent spheres with regular tangency A/sub 1//sup 2/ (A/sub k//sup n/ notation means n distinct k-fold tangencies of the sphere of contact, as explained in the text); two types of Curves, 2) the Intersection Curve of three sheets and the locus of centers of tritangent spheres, A/sub 1//sup 3/, and 3) the boundary of sheets, which are the locus of centers of spheres whose radius equals the larger principal curvature, i.e., higher order contact A/sub 3/ points; and two types of points, 4) centers of quad-tangent spheres, A/sub 1//sup 4/, and 5) centers of spheres with one regular tangency and one higher order tangency, A/sub 1/A/sub 3/. The geometry of the 3D medial axis thus consists of sheets (A/sub 1//sup 2/) bounded by one type of Curve (A/sub 3/) on their free end, which corresponds to ridges on the surface, and attached to two other sheets at another type of Curve (A/sub 1//sup 3/), which support a generalized cylinder description. The A/sub 3/ Curves can only end in A/sub 1/ A/sub 3/ points where they must meet an A/sub 1//sup 3/ Curve. The A/sub 1//sup 3/ Curves meet together in fours at an A/sub 1//sup 4/ point. This formal result leads to a compact representation for 3D shape, referred to as the medial axis hypergraph representation consisting of nodes (A/sub 1//sup 4/ and A/sub 1/ A/sub 3/ points), links between pairs of nodes (A/sub 1//sup 3/ and A/sub 3/ Curves) and hyperlinks between groups of links (A/sub 1//sup 2/ sheets). The description of the local geometry at nodes by itself is sufficient to capture qualitative aspects of shapes, in analogy to 2D. We derive a pointwise reconstruction formula to reconstruct a surface from this medial axis hypergraph together with the radius function. Thus, this information completely characterizes 3D shape and lays the theoretical foundation for its use in recognition, morphing, design, and manipulation of shapes.
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a formal classification of 3d medial axis points and their local geometry
Computer Vision and Pattern Recognition, 2000Co-Authors: Peter Giblin, Benjamin B KimiaAbstract:This paper proposes a novel hypergraph skeletal representation for 3D shape based on a formal derivation of the generic structure of its medial axis. By classifying each skeletal point by its order of contact, we shout that generically the medial axis consists of five types of points which are then organized into sheets, Curves, and points: (i) sheets (manifolds with boundary) which are the locus of bitangent spheres with regular tangency/sup 1/ A/sub 1//sup 2/. Two types of Curves (ii) the Intersection Curve of three sheets and the locus of centers of tritangent spheres, A/sub 1//sup 3/, and (iii) the boundary of sheets which are the locus of centers of spheres whose radius equals the larger principle curvature, i.e., higher order contact A/sub 3/ points; and two types of points (iv) centers of quad-tangent spheres, A/sub 1//sup 4/, and, (v) centers of spheres with one regular tangency and one higher order tangency, A/sub 1/A/sub 3/ The geometry of the 3D medial axis thus consists of sheets (A/sub 1//sup 2/) bounded by one type of Curve (A/sub 3/) on their free end, which corresponds to ridges on the surface, and attached to two other sheets at another type of Curves (A/sub 1//sup 3/), which support a generalized cylinder description. The A/sub 3/ Curves can only end in A/sub 1/A/sub 3/ points where they must meet an A/sub 1//sup 3/ Curve. The A/sub 1//sup 3/ Curves can either meet one A/sub 3/ Curve or meet three other A/sub 1//sup 3/ Curve at an A/sub 1//sup 4/ point. This formal result leads to a compact representation for 3D shape, referred to as the medial axis hypergraph representation consisting of nodes (A/sub 1//sup 4/ and A/sub 1/A/sub 3/ points), links between pairs of nodes (A/sub 1//sup 3/ and A/sub 3/ Curves) and hyperlinks between groups of links (A/sub 1//sup 2/ sheets). The description of the local geometry at nodes by itself is sufficient to capture qualitative aspects of shapes, in analogy to 2D. We derive a pointwise reconstruction formula to reconstruct a surface from this medial axis hypergraph. Thus, the hypergraph completely characterizes 3D shape and lays the theoretical foundation for its use in recognition, morphing, design and manipulation of shapes.
Peter Giblin - One of the best experts on this subject based on the ideXlab platform.
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a formal classification of 3d medial axis points and their local geometry
IEEE Transactions on Pattern Analysis and Machine Intelligence, 2004Co-Authors: Peter Giblin, Benjamin B KimiaAbstract:This paper proposes a novel hypergraph skeletal representation for 3D shape based on a formal derivation of the generic structure of its medial axis. By classifying each skeletal point by its order of contact, we show that, genetically, the medial axis consists of five types of points, which are then organized into sheets, Curves, and points: 1) sheets (manifolds with boundary) which are the locus of bitangent spheres with regular tangency A/sub 1//sup 2/ (A/sub k//sup n/ notation means n distinct k-fold tangencies of the sphere of contact, as explained in the text); two types of Curves, 2) the Intersection Curve of three sheets and the locus of centers of tritangent spheres, A/sub 1//sup 3/, and 3) the boundary of sheets, which are the locus of centers of spheres whose radius equals the larger principal curvature, i.e., higher order contact A/sub 3/ points; and two types of points, 4) centers of quad-tangent spheres, A/sub 1//sup 4/, and 5) centers of spheres with one regular tangency and one higher order tangency, A/sub 1/A/sub 3/. The geometry of the 3D medial axis thus consists of sheets (A/sub 1//sup 2/) bounded by one type of Curve (A/sub 3/) on their free end, which corresponds to ridges on the surface, and attached to two other sheets at another type of Curve (A/sub 1//sup 3/), which support a generalized cylinder description. The A/sub 3/ Curves can only end in A/sub 1/ A/sub 3/ points where they must meet an A/sub 1//sup 3/ Curve. The A/sub 1//sup 3/ Curves meet together in fours at an A/sub 1//sup 4/ point. This formal result leads to a compact representation for 3D shape, referred to as the medial axis hypergraph representation consisting of nodes (A/sub 1//sup 4/ and A/sub 1/ A/sub 3/ points), links between pairs of nodes (A/sub 1//sup 3/ and A/sub 3/ Curves) and hyperlinks between groups of links (A/sub 1//sup 2/ sheets). The description of the local geometry at nodes by itself is sufficient to capture qualitative aspects of shapes, in analogy to 2D. We derive a pointwise reconstruction formula to reconstruct a surface from this medial axis hypergraph together with the radius function. Thus, this information completely characterizes 3D shape and lays the theoretical foundation for its use in recognition, morphing, design, and manipulation of shapes.
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a formal classification of 3d medial axis points and their local geometry
Computer Vision and Pattern Recognition, 2000Co-Authors: Peter Giblin, Benjamin B KimiaAbstract:This paper proposes a novel hypergraph skeletal representation for 3D shape based on a formal derivation of the generic structure of its medial axis. By classifying each skeletal point by its order of contact, we shout that generically the medial axis consists of five types of points which are then organized into sheets, Curves, and points: (i) sheets (manifolds with boundary) which are the locus of bitangent spheres with regular tangency/sup 1/ A/sub 1//sup 2/. Two types of Curves (ii) the Intersection Curve of three sheets and the locus of centers of tritangent spheres, A/sub 1//sup 3/, and (iii) the boundary of sheets which are the locus of centers of spheres whose radius equals the larger principle curvature, i.e., higher order contact A/sub 3/ points; and two types of points (iv) centers of quad-tangent spheres, A/sub 1//sup 4/, and, (v) centers of spheres with one regular tangency and one higher order tangency, A/sub 1/A/sub 3/ The geometry of the 3D medial axis thus consists of sheets (A/sub 1//sup 2/) bounded by one type of Curve (A/sub 3/) on their free end, which corresponds to ridges on the surface, and attached to two other sheets at another type of Curves (A/sub 1//sup 3/), which support a generalized cylinder description. The A/sub 3/ Curves can only end in A/sub 1/A/sub 3/ points where they must meet an A/sub 1//sup 3/ Curve. The A/sub 1//sup 3/ Curves can either meet one A/sub 3/ Curve or meet three other A/sub 1//sup 3/ Curve at an A/sub 1//sup 4/ point. This formal result leads to a compact representation for 3D shape, referred to as the medial axis hypergraph representation consisting of nodes (A/sub 1//sup 4/ and A/sub 1/A/sub 3/ points), links between pairs of nodes (A/sub 1//sup 3/ and A/sub 3/ Curves) and hyperlinks between groups of links (A/sub 1//sup 2/ sheets). The description of the local geometry at nodes by itself is sufficient to capture qualitative aspects of shapes, in analogy to 2D. We derive a pointwise reconstruction formula to reconstruct a surface from this medial axis hypergraph. Thus, the hypergraph completely characterizes 3D shape and lays the theoretical foundation for its use in recognition, morphing, design and manipulation of shapes.
Dinesh Manocha - One of the best experts on this subject based on the ideXlab platform.
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an efficient surface Intersection algorithm based on lower dimensional formulation
ACM Transactions on Graphics, 1997Co-Authors: Shankar Krishnan, Dinesh ManochaAbstract:We present an efficient algorithm to compute the Intersection of algebraic and NURBS surfaces. Our approach is based on combining the marching methods with the algbraic formulation. In particular, we propose and matrix computations. We present algorithms to compute a start point on each component of the Intersection Curve (both open and closed components), detect the presence of singularities, and find all the Curve branches near the singularity. We also suggest methods to compute the step size during tracing to prevent component jumping. The algorithm runs an order of magnitude faster than previously published robust algorithms. The complexity of the algorithm is output sensitive.
Shankar Krishnan - One of the best experts on this subject based on the ideXlab platform.
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an efficient surface Intersection algorithm based on lower dimensional formulation
ACM Transactions on Graphics, 1997Co-Authors: Shankar Krishnan, Dinesh ManochaAbstract:We present an efficient algorithm to compute the Intersection of algebraic and NURBS surfaces. Our approach is based on combining the marching methods with the algbraic formulation. In particular, we propose and matrix computations. We present algorithms to compute a start point on each component of the Intersection Curve (both open and closed components), detect the presence of singularities, and find all the Curve branches near the singularity. We also suggest methods to compute the step size during tracing to prevent component jumping. The algorithm runs an order of magnitude faster than previously published robust algorithms. The complexity of the algorithm is output sensitive.
Gershon Elber - One of the best experts on this subject based on the ideXlab platform.
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surface surface Intersection computation using a bounding volume hierarchy with osculating toroidal patches in the leaf nodes
Computer-aided Design, 2020Co-Authors: Youngjin Park, Sanghyun Son, Myungsoo Kim, Gershon ElberAbstract:Abstract We present an efficient and robust algorithm for computing the Intersection Curve of two freeform surfaces using a Bounding Volume Hierarchy (BVH), where the leaf nodes contain osculating toroidal patches. The covering of each surface by a union of tightly fitting toroidal patches greatly simplifies the geometric operations involved in the surface–surface-Intersection computation, i.e., the bounding of surface normals, the detection of surface binormals, the point projection from one surface to the other surface, and the Intersection of local surface patches. Moreover, the hierarchy of simple bounding volumes (such as rectangle-swept spheres) accelerates the geometric search for the potential pairs of surface patches that may generate some Curve segments in the surface–surface-Intersection. We demonstrate the effectiveness of our approach by using test examples of intersecting two freeform surfaces, including some highly non-trivial examples with tangential Intersections. In particular, we test the Intersection of two almost identical surfaces, where one surface is obtained from the same surface, using a rotation around a normal line by a smaller and smaller angle θ = 1 0 − k degree, k = 0 , … , 5 . The Intersection results are often given as surface subpatches in some highly tangential areas, and even as the whole surface itself, when θ = 0 . 0000 1 ∘ .
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topologically guaranteed univariate solutions of underconstrained polynomial systems via no loop and single component tests
Computer-aided Design, 2011Co-Authors: Michael Bartoň, Gershon Elber, Iddo HannielAbstract:Abstract We present an algorithm which robustly computes the Intersection Curve(s) of an underconstrained piecewise polynomial system consisting of n equations with n + 1 unknowns. The solution of such a system is typically a Curve in R n + 1 . This work extends the single solution test of Hanniel and Elber (2007) [6] for a set of algebraic constraints from zero-dimensional solutions to univariate solutions, in R n + 1 . Our method exploits two tests: a no-loop test (NLT) and a single-component test (SCT) that together isolate and separate domains D where the solution Curve consists of just one single component. For such domains, a numerical Curve tracing is applied. If one of those tests fails, D is subdivided. Finally, the single components are merged together and, consequently, the topological configuration of the resulting Curve is guaranteed. Several possible applications of the solver, namely solving the surface–surface Intersection problem, computing 3D trisector Curves, flecnodal Curves or kinematic simulations in 3D are also discussed.
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topologically guaranteed univariate solutions of underconstrained polynomial systems via no loop and single component tests
Solid and Physical Modeling, 2010Co-Authors: Michael Bartoň, Gershon Elber, Iddo HannielAbstract:We present an algorithm which robustly computes the Intersection Curve(s) of an under-constrained piecewise polynomial system consisting of n equations with n + 1 unknowns. The solution of such a system is typically a Curve in Rn+1. This work extends the single solution test of [6] for a set of algebraic constraints from zero dimensional solutions to univariate solutions, in Rn+1. Our method exploits two tests: a no loop test (NLT) and a single component test (SCT) that together isolate and separate domains D where the solution Curve consists of just one single component. For such domains, a numerical Curve tracing is applied. If one of those tests fails, D is subdivided. Finally, the single components are merged together and, consequently, the topological configuration of the resulting Curve is guaranteed. Several possible application of the solver, like 3D trisector Curves or kinematic simulations in 3D are discussed.
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intersecting a freeform surface with a ruled or a ringed surface
Geometric Modeling and Processing, 2004Co-Authors: Joonkyung Seong, Myungsoo Kim, Kujin Kim, Gershon ElberAbstract:We present efficient and robust algorithms for intersecting a freeform surface with a ringed surface or a ruled surface. A ringed surface is given as a one-parameter family of circles. By computing the Intersection between a freeform surface and each circle in the family, we can solve the Intersection problem. We propose two approaches which are closely related to each other. The first approach detects certain critical points; and the Intersection Curve is constructed by connecting them in a correct topology. The second approach converts the Intersection problem to that of finding the zero set of two polynomial equations in the parameter space. The Intersection between a freeform surface and a ruled surface can be computed in a similar way.
