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### Progressive Encoding of Complex Isosurfaces

Haeyoung Lee Mathieu Desbrun Peter Schröder

USC USC Caltech

Motivation

- Largest meshes come from volume data
- MRI, CT, Laser Scan
- Scientific simulation
- Challenging to store and/or transmit

Background on Compression

- Mesh Encoding vs. Geometry Encoding
- Connectivity + Geometry, or Geometry only
- Single-rate vs. Progressive Compression
- Progressivity is preferred for huge meshes

Progressive

Single-rate

Our Context

- High genus and many components
- Remeshing impractical
- best known coders unusable!
- Extracted from volume data
- Very special mesh structure

V: 280039CC:183Genus: 425

Skull, extracted from 257x257x257 MRI volume data

Outline

- Definitions
- Previous Work
- Our progressive compression
- Connectivity
- Geometry
- Our results
- Conclusion and Future work

Definitions

- Volume data
- Binary Sign
- Isosurface
- Piercing edge
- Homogeneous
- Inhomogeneous

Previous Work (1)

- Single-rate Isosurface Compression
- Connectivity: locate piercing edges
- Saupe & Kuska ’01,’02: Octree
- Zhang et al ’01: Binary sign and cell map
- Yang & Wu ’02: 3D chessboard
- Taubin ’02 (BLIC): Binary Sign map
- Geometry: displacements along piercing edges

Much lower rates than general mesh encoders

Previous Work (2)

- Progressive Isosurface Compression
- Laney et al. 2002
- Distance transformation & wavelet decomposition
- Samet and Kochut 2002
- Octree encoding, without explicit geometry

Problems:

- Very limited test sets
- Bitrates much worse than single-rate encoders

Our Contributions

- Progressive Isosurface Codec
- Connectivity Encoding
- Novel octree encoding of binary bitmaps
- Geometry Encoding
- Dual contouring for crack-free visualization
- Best bitrates so far
- even better than any single-rate isosurface encoders

Our Design Choices (1)

- Adaptive Octree for Connectivity Encoding
- Enable progressive localization
- Provide contexts for entropy coding
- Avoid redundancy

Horse: 9*9*9 (level 3) 17*17*17 (level 4) 33*33*33 (level 5)

Our Design Choices (2)

- Dual Contouring [Ju et al 02, SW02]
- Watertight meshes
- Sharp features for hermite data
- Vertices in cells, not on edges

Our Encoder At A Glance

- Read in & Process volume data
- Build Octree
- Create Isosurface by DC

- Encode Connectivity

during a breadth-first traversal

- Encode Geometry

Connectivity Encoding

- Sign bits (Inside/Outside)
- Encode binary signs at grid vertices
- Cells with children: encode necessary signs
- Cells without children: deduce sign from the parent
- Leaf bits (Leaf/Non-leaf)
- Encode the presence of children
- Identify non-empty cells

Context Modeling

- Compression ratios depend on context choice
- Sign bitstream:
- 15-bit context (best bit rates): 7 neighbors + 8 of parent
- Differs from JBIG
- Leaf bitstream:
- 1-bit context: previous bit (best bit rates)

w/o geo

w/ geo

w/ geo

Geometry Encoding?- Sometimes, octree bits enough!
- Octree provides coarse geometry during decoding
- Barycenters of midpoints of the piercing edges

P

Center

Center

Geometry Encoding- Local Coordinate System
- Least-square fitted plane
- through midpoints of piercing edges
- Two passes
- normal(z) & tangential(x,y)
- Context : 8 signs of the cell

Implementation

- Beware of Memory Footprint!
- Octree data structure can be overkill
- 2573 grids use up more than 1Gb
- We use a “linearized” data structure
- Unfolds the octree in a bitmap
- No pointers, no recursive calls
- Allows 10253 grids (or bigger) on your PC

Our Results (1)

- Total: 6.10b/v on average out of 10 models
- Connectivity:
- 0.65 b/v on average
- 24% better than Taubin’s single-rate BLIC
- Geometry:
- 5.45 b/v on average
- For a distortion similar to 12-bit quantization

Our Results (2)

8+ 100% geo.

145,708

0.47

5

622

303.47

Oct. level

Bytes

Distort (10-4)

8

20,324

3.66

7

8,411

32.72

507 bytes

166.18

7

8,605 bytes

22.02

8 + 33% geo.

92,156 bytes

4.06

8 + 100% geo.

226,554 bytes

0.65

Our Results (3)Octree level

Bytes passed

Distortion(10-4)

Results (5)

- Encoding a raw mesh often requires > 15b/v

3.45 b/v

(0.09 + 3.39)

3.95 b/v

(0.58 + 3.37)

3.21 b/v

(0.51 + 2.70)

Conclusion

- Progressive isosurface compression
- Progressive coding of binary octree
- Encoding of dual contouring mesh vertices
- Context modeling with arithmetic coding
- Competitive compression ratios
- 24% better than the leading single-rate on connectivity alone

Future Work

- Reducing bit rate further
- Sophisticated binary valued wavelet?
- View-dependent compression
- View-dependent encoding
- View-dependent decoding
- Volume compression
- Neighboring isosurfaces

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