1. Multiscale Representations for Point Cloud Data

2. 3D Surface Scanning Explosion in data and applications
Terrain visualization
Mobile robot navigation

3. Data Deluge The Challenge: Massive data sets
Millions of points
Costly to store/transmit/manipulate
Goal: Find efficient algorithms for representation and compression
Replace hand with terrain point cloud!

4. Selected Related Work Point Cloud Compression [Schnabel, Klein 2006]
Geometric Mesh Compression [Huang, Peng, Kuo, Gopi 2006]
Surflets [Chandrasekaran, Wakin, Baron, Baraniuk 2004]
Multiscale tiling of piecewise surface polynomials
Trading off

5. Optimality Properties
Surflet encoding for L2 error metric for piecewise constant/smooth functions
Polynomial order determined by smoothness of the image
Optimal asymptotic approximation rate for this function class
Optimal rate-distortion performance for this function class
Our innovation:
More physically relevant error metric
Extension to point cloud data
Smoothness
Dimension
Rate
Add rectangular here if we decide to use it!
Firm up smoothness understanding before talk

6. Error Metric From L2 error To Hausdorff error Computationally simple
Suppress thin structures
To Hausdorff error
Measures maximum deviation
Expected in urban terrain.

7. Our Approach Octree decomposition of point cloud
Fit a surflet at each node
Polynomial order determined by the image smoothness
Encode polynomial coefficients
Rate-distortion coder
multiscale quantization
predictive encoding

8. Step 1: Tree Decomposition (2D)
-- data in square i
Assume surflet dictionary
with finite elements
Stop refining a branch once node falls below threshold

9. Step 1: Tree Decomposition (2D)
root

10. Step 1: Tree Decomposition (2D)
root

11. Step 1: Tree Decomposition (2D)
root

12. Step 1: Tree Decomposition (2D)
root

13. Octree Hallmarks Multiscale representation
Enable transmission of incremental details
Prune tree for coarser representation
Grow tree for finer representation

14. Step 2: Encode Polynomial Coeffs
Must encode polynomial coefficients and configuration of tree
Uniform quantization
suboptimal
Key: Allocate bits nonuniformly
multiscale quantization adapted to octree scale
variable quantization according to polynomial order

15. Multiscale Quantization
Allocate more bits at finer scales:
Allocate more bits to lower order coefficients
Taylor series :
Combine into one slide – give the gyst and move on!
Scale
Smoothness
Order

16. Step 3: Predictive Encoding
“Likely”
“Less likely”
Insight: Smooth images small innovation at finer scale
Coding Model: Favor small innovations over large ones
Encode according to distribution:
Encode with –log(p) bits:
Fewer bits
More bits

17. Experiment: Smooth Function
16,400 points
Planar Surflets
0.03 bpp
“3200:1” Compression 22

18. Experiment: Building 22,000 points Planar Surflets 0.4 bpp
“300:1” Compression

19. Experiment: Mountain 263,000 points Planar Surflets .08 bpp
“1200:1” Compression

20. Comparison: Binary and Octree

21. Summary Multiscale, lossy compression for large point clouds
Error metric: Hausdorff distance, not L2 distance
Surflets offer excellent encoding for piecewise smooth surfaces
Multiscale surface polynomial tiling
Multiscale quantization
Predictive Encoding
Open Question: Asymptotic optimality for Hausdorff metric