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Closest Point Transform: The Characteristics/Scan Conversion Algorithm Sean Mauch Caltech April, 2003

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2 Problem: Convert a Surface to a Level Set Solid mechanics. –Explicit representation of the solid boundary, b-rep. –Triangle mesh surface. Fluid mechanics. –Implicit representation, level set. –Position and velocity of the closest point on the surface computed at each grid point. Triangle Mesh.Slice of Distance.Closest Point.

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3 Desiderata for CPT Algorithm Speed. –The CPT is computed at every time step of the solver. –Fluid mechanics solver has computational complexity O(N), N = grid size. –If the CPT algorithm is not fast enough, it will dominate the computation. Locality. –The CPT is only needed in a narrow layer around the surface. –Algorithm must compute CPT to arbitrary distance from surface. Concurrent computing. –Algorithm must scale with the size of the b-rep and the distribution of the fluid grid. Multi-resolution grids. –Algorithm must accommodate AMR, adaptive mesh refinement.

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4 Previous Work: Brute Force Iterate over both triangles in the mesh and grid points. Computational complexity is O(N M) where N is the number of grid points and M is the number of triangles in the mesh. For each triangle t: For each grid point p: Compute the distance et c. from p to t.

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5 Previous Work: Finite Difference Methods Sethian’s Fast Marching Method uses upwind finite differencing to compute approximate distance by solving the eikonal equation. Solves a related, but different problem. Cannot convert a surface to a level set, but can calculate distance given a level set as an initial condition. Computational complexity is O(N log N).

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6 Previous Work: Tree Data Structures A lower-upper-bound tree (LUB-tree) stores the mesh in a tree data structure that can give upper and lower bounds on the distance at each level. Computational complexity O(N log M).

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7 Motivation: Eikonal Equation, Characteristics The distance u satisfies the eikonal equation: | grad u |=1. The eikonal equation can be solved with the method of characteristics. A grid point x is connected to its closest point on the surface ξ by the characteristic line. Characteristic regions for the edges of a polygonal curve.

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8 Motivation: Voronoi Diagrams A Voronoi cell is a convex polygon/polyhedron which contains exactly the closest points. Unbounded and bounded Voronoi diagrams.

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9 Motivation: Method of Characteristics The closest points to faces/edges/vertices lie in prisms/wedges/pyramids. These polyhedra contain at least all the closest points. (a) shows the face polyhedra. (b) shows the edge polyhedra. (c) shows a single edge polyhedron. (d) shows the vertex polyhedra.

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10 Motivation: Scan Conversion 2-D polygon scan conversion. –Computer graphics rendering algorithm. –Determine pixels inside the polygon. Linear complexity. 3-D polyhedron scan conversion. –Slice to obtain polygons. 2-D polygon scan conversion. Slicing a polyhedron to obtain polygons.

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11 The CSC Algorithm The characteristics/scan conversion (CSC) algorithm. –For each face/edge/vertex. Scan convert the polyhedron. Find the distance, closest point, et c. to that primitive for the scan converted points. Computational complexity. –O(M) to build the b-rep and the polyhedra. –O(N) to scan convert the polyhedra and compute the distance, etc.

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12 Computational Complexity for CSC Verification of linear complexity. Log-log plots of scaled execution time versus grid size and mesh size. The super-linear speed-up is due to coarser inner loops and decreased polyhedron volume.

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13 Concurrency for CSC Algorithm Embarrassingly concurrent. Data distribution. –Each processor needs its portion of the fluid grid and its portion of the solid boundary. –Each processor runs the sequential CSC algorithm on its portion of the fluid grid using its portion of the solid boundary. Multi-threaded concurrency. –Each polyhedron represents an independent task. –Tasks can be executed concurrently.

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14 Summary of Results for CSC Algorithm Advantages. –Optimal computation complexity. –CPT computed to arbitrary distance around the surface. –No significant additional memory required. –Embarrassingly concurrent. –Works for multi-resolution grids. Disadvantages. –More difficult to implement than other algorithms.

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