Transcript Closest Point Transform - California Institute of Technology

Closest Point Transform: The
Characteristics/Scan Conversion
Algorithm
Sean Mauch
Caltech
April, 2003
Problem: Convert a Surface to a Level Set
• Solid mechanics.
• Fluid mechanics.
– Explicit representation of
– Implicit representation, level set.
the solid boundary, b-rep.
– Position and velocity of the closest
– Triangle mesh surface.
point on the surface computed at
each grid point.
Triangle Mesh.
Slice of Distance.
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Closest Point.
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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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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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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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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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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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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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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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.
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Slicing a polyhedron
to obtain polygons.
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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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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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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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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