Closest Point Transform: The Characteristics/Scan Conversion

Algorithm

Sean Mauch

Caltech

April, 2003

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.

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.

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.

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).

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).

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.

8

Motivation: Voronoi Diagrams

• A Voronoi cell is a convex polygon/polyhedron which contains exactly the closest points.

Unbounded and bounded Voronoi diagrams.

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.

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 polyhedronto obtain polygons.

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.

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.

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.

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.