Multiscale Representations for Point Cloud Data

1. Multiscale Representations for Point Cloud Data Andrew Waters Manjari Narayan Richard Baraniuk Luke Owens Ron DeVore

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

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

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

6. Error Metric • From L2 error • Computationally simple • Suppress thin structures • To Hausdorff error • Measures maximum deviation

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 : Scale Smoothness Order

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

17. Experiment: Building 22,000 points piecewise planar surflets Octree: 150 nodes 1100 bits “1400:1” compression 0.05 bpp

18. Experiment: Mountain 263,000 points piecewise planar surflets Octree: 2000 Nodes 21000 Bits “1500:1” Compression 0.08 bpp

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