Collaborators Mark Foskey Fast Discretized Geometric Young Kim Algorithms for Union and Shankar Krishnan Envelope Computations Ming C. Lin Avneesh Sud Gokul Varadhan Dinesh Manocha UNC-Chapel Hill http://gamma.cs.unc.edu/ The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Basic Computational Arrangement Problems Pipeline • Enumerate a set S of primitives that Arrangements contribute to the final surface Decomposition of space into connected open cells Fundamental problem in computational geometry • Compute the arrangement A(S) by and related areas performing intersection and trimming Underlying structure in many computations geometric applications Swept Volumes • Traverse the arrangement and extract a Minkowski Sums substructure δ A(S) CSG or Boolean operations Many more……. The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 1
Example: CSG Union Operation CSG Operations Boundary = outer envelope in the arrangement of the primitives Design of complex parts UNION The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Boundary Evaluation of Minkowski Sum Complex CSG Models A + B = { a+b | a ∈ A, b ∈ B } OFFSET Bradley Fighting Vehicle + 1200+ solids 8,000+ CSG operations + + The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 2
Offset Computation Minkowski Sums: Motivation Offset : Minkowski Configuration space computation sum with a Offsets sphere Morphing Packing and layout Friction model Input : 2982 triangles The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Minkowski Computation Swept Volume (SV) Decompose A and B into convex pieces Volume generated by sweeping an object in space along a trajectory Compute pairwise convex Minkowski sums Compute their union Goal: Compute a boundary representation • Issues: of SV –High combinatorial complexity = O(n 6 ) –Exact computation almost impractical The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 3
Sweep Equation Swept Volume: Applications Γ (t) = Ψ (t) + R(t) Γ , 0 ≤ t ≤ 1 Numerically Controlled Machine Verification Γ : Generator (polyhedron) Ψ (t) : Smooth vector in R 3 (sweeping path) R (t) : Local orientation Swept Volume of Γ := ∪Γ (t) Γ (t) Γ t ) R ( Ψ (t) Tool and workpiece Material removal • No scaling, shearing, and deformation The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Swept Volume Computation Swept Volume: Applications Collision detection between discrete instances Enumerate ruled and developable surfaces Boundary of SV = outer envelope of the arrangement The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 4
Swept Volume Computation Computation of Swept Volumes • Generate ruled and developable surfaces X-Wing Model 2496 triangles • Compute their arrangement 3931 ruled and developable surfaces Intersection curves of degree as high as nine • Traverse the arrangement and extract the outermost boundary (outer envelope computation) Sweep Trajectory Arrangement Boundary of SV The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Approximation Pipeline Complexity of Arrangements High computational and combinatorial Enumerate surface primitives complexity Compute distance fields on a voxel grid Perform filtering operations on distance fields Super-quadratic in number of surfaces Use improved reconstruction algorithms Accuracy and robustness problems No good practical implementations are available The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 5
Approximation Pipeline Organization Enumerate surface primitives Fast distance field computation Compute distance fields on a voxel grid Max-norm based voxelization Perform filtering operations on distance fields Boundary reconstruction Use improved reconstruction algorithms Analysis • Max-norm computations for reliable voxelization Applications • Recover all connected components Boundary evaluation • Faithfully reconstruct sharp features Swept volume computation Medial axis computation Minkowski sums The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Distance Fields Organization Fast distance field computation Distance Function For a site a scalar function f:R n -> R representing Max-norm based voxelization the distance from a point P ε R n to the site Boundary reconstruction Distance Field Analysis For a set of sites, the minima of all distance Applications functions representing the distance from a Boundary evaluation point P ε R n to closest site Swept volume computation Medial axis computation Minkowski sums The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 6
Voronoi Diagrams Ordinary Generalized Point sites Higher-order site geometry Given a collection of geometric primitives, it Varying distance metrics Nearest Euclidean is a subdivision of space into cells such that distance all points in a cell are closer to one primitive Higher-order than to any other Sites Voronoi Site 2.0 Voronoi Region 0.5 Weighted Distances The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Voronoi Diagram & Distance Distance Fields: Applications Fields Minimization diagram of distance functions generates a Voronoi Diagram Collision Detection Projection of lower envelope of Surface Reconstruction distance functions Robot Motion Planning Non-Photorealistic Rendering Surface Simplification Mesh Generation Shape Analysis The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 7
Approximating the Distance GPU Based Computation Function Avoid per-pixel distance evaluation HAVOC2D, HAVOC3D [Hoff et al. 99,01] Point-sample the distance function Evaluate distance at each pixel for all sites Reconstruct by rendering polygonal mesh Evaluate the distance function using graphics hardware Point Line Triangle Point Line Triangle The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL GPU Based Computation Meshing the Distance Function Triangular mesh approximation of distance functions Render distance meshes using graphics hardware Shape of distance function for Need a bounded-error a 2D point is a cone tessellation of the cone The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 8
Graphics Hardware Results in the Frame Buffer Acceleration Rasterization to reconstruct distance values Color Buffer Depth Buffer Depth test to perform minimum operator Voronoi Regions Distance Field Perspective, 3/4 view Parallel, top view The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Shape of 3D Distance 3D Voronoi Diagrams Functions Graphics hardware can generate one 2D slice at a time Sweep along 3 rd dimension (Z-axis) computing 1 slice at a time Slices of the distance Distance meshes used to Distance Field of the Teapot Model function for a 3D point site approximate slices The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 9
Shape of 3D Distance Bottlenecks in HAVOC3D Functions Point Line segment Triangle Rasterization: Distance mesh can fill entire slice Complexity for n sites and k slices = O( kn ) Lot of Fill ! Readback: Stalls the graphics pipeline Not suitable for interactive applications 1 sheet of a Elliptical cone Plane hyperboloid The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Improved Distance Field Improved Distance Field Computation (DiFi) Computation (DiFi) Use graphics hardware Reduce fill : Cull using estimated Exploit spatial coherence between slices voronoi region bounds Use the programmable hardware to perform computations Along Z: Cull sites whose voronoi regions don’t intersect with current slice [Sud and Manocha 2003] In XY plane: Restrict fill per site using planar bounds of the voronoi region The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 10
Voronoi Diagram Properties Voronoi Diagram Properties Within a bounded Within a bounded region, all voronoi regions have region, all voronoi a bounded volume regions have a bounded volume As site density increases, average spatial bounds decrease 9 Sites, 2D 27 Sites, 2D The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Voronoi Diagram Properties Voronoi Diagram Properties Voronoi regions are Voronoi regions connected are connected Valid for l 2 , l inf etc. Valid for l 2 , l inf norms norms Special cases: Overlapping features The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL 11
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