Permission Grids

1
Permission Grids

• Practical, Error-bounded Simplification

2
In a Nutshell
3
Overview

• What good are error bounds?
• Desirable properties of error-bounded
  simplification
• Previous approaches
• Permission grid in detail
• Results
• Current and future work

4
Why Error Bounds?

• Surface reconstruction accuracy
• Progressive refinement
• Finite element methods
• View dependent refinement
• Medical applications
• Collision detection

5
Measuring Error

• D(p, S) minimum distance to S
• D(S1, S2) max over p in S1 d(p, S2)
• One-sided Hausdorff distance S1 g S2
• max( D(S1, S2), D(S2, S1) )
• Two-sided Hausdorffdistance

D(T, C)
D(C, T)
6
Desirable Properties

• No assumptions about input mesh
• Allow topological changes
• Level of detail control
• Efficient

7
Previous Approaches

• Vertex clustering
• Rossignac and Borrel 93
• Low and Tan 97
• Lindstrom 00
• Drawbacks
• Hard to control
• Low-quality LOD
• Error bound questionable

8
Previous Approaches

• Planar Approximations
• Kalvin and Taylor 96
• Drawbacks
• No topology change
• Manifold only ?
• Slow
• High face counts

9
Previous Approaches

• Local re-projection
• Ciampalini, et al, 97
• Klein, et al, 96
• Kobbelt, et al, 98
• Ronfard and Rossignac, 96
• Drawbacks
• Manifold only
• No topology change
• Input-sensitive

10
Previous Approaches

• Local re-mapping
• Bajaj and Schikore 96
• Cohen, et al, 96

• Drawbacks
• Manifold only
• No topology change
• Slow

11
Previous Approaches

• Tolerance Volumes
• Cohen, et al, 96
• Input-sensitive
• Manifold only
• No topology change
• Guéziec, 96
• No topology change
• Very complex
• Slow

12
Permission Grids

• Another Tolerance Volume
• Shape is union of a-prisms
• Satisfies all desired properties

13
Algorithm Overview

• Choose grid resolution
• Voxelize a-prisms of the input mesh
• While there are candidate simplification
  operations
• Test the triangles generated by the next
  operation
• If any fail, reject the operation
• Otherwise, perform it

14
Resolution and r

• Effective e dependent on grid resolution
• Define precision r e / voxel length
• E.g., r 4

15
Effect of Changing r
1.25
2
4
8
16
Voxelizing a-prisms

• Anti-aliased volume graphics
• Dachille and Kaufman, 00

17
Testing Triangles

• 3D generalization of 2D rasterization
• Kaufman and Shimony 86
• Project into 2D
• Rasterize
• Reproject into 3D

18
Advantages

• Simple
• Examine only one triangle at a time
• No assumptions on input
• Parallelizable
• Topological change possible
• Output-sensitive
• Can incorporate additional sources of information
  into simplification

19
Advantages

• Incorporate additional sources of information
  into simplification

20
Level of Detail Control

• Expand grid by one voxel when stuck
• Increases e by the voxel length

21
Disadvantages
FromOriginal

• Conservative grid expansion
• Limit on e
• Only one-sided error bound
• Boundaries need special treatment

Expanded4 times
22
Boundaries

• Use supplementary grid with voxelized boundary
  edges

23
Results

• Venus at 0.5

Original Permission Grid
QSlim
Simplification Envelopes
24
Results

• Venus at 1

QSlim
Original
Permission Grid
SimplificationEnvelopes
25
Results

• Bottom of Stanford Bunny at 1

Original
Permission Grid
QSlim
Simplification Envelopes
26
Results

• Buddha at 1

Original Permission Grid
QSlim
27
Results Summary

• Generally an order of magnitude faster than
  Simplification Envelopes
• Overhead of approx 2-5x QSlim
• More simplification for given e
• Better max/average errors for given e

28
Future Directions

• Compression
• Adaptively-sampled distance fields
• Spatial data structure, allows distance to mesh
  reconstruction anywhere
• Compact
• No modification required to change e
• Can optimize placement instead of reject