1. Multiscale Representations for Point Cloud Data Andrew Waters Manjari Narayan Richard Baraniuk Luke Owens Daniel Freeman Matt Hielsberg Guergana Petrova 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 • Mesh Compression [Khodakovsky, Schröder, Sweldens 2000] • Geometric Mesh Compression [Huang, Peng, Kuo, Gopi 2006] • Point Cloud Compression [Schnabel, Klein 2006]

5. Selected Related Work • Mesh Compression [Khodakovsky, Schröder, Sweldens 2000] • Geometric Mesh Compression [Huang, Peng, Kuo, Gopi 2006] • Point Cloud Compression [Schnabel, Klein 2006] Our Innovation ?

6. Selected Related Work • Mesh Compression [Khodakovsky, Schröder, Sweldens 2000] • Geometric Mesh Compression [Huang, Peng, Kuo, Gopi 2006] • Point Cloud Compression [Schnabel, Klein 2006] • More physically relevant error metric • Efficient lossy encoding Our Innovation ?

7. Our Approach • Fit piecewise polynomial surface to point cloud • Octree polynomialrepresentation • Encode polynomial coefficients • Rate-distortion coder • multiscale quantization • predictive encoding

8. Step 1 – Fit Piecewise Polynomials • Surflet representation[Chandrasekaran, Wakin, Baron, Baraniuk, 2004] • Divide domain (cube) into octree hierarchy • Fit surface polynomial to point cloud within each sub-cube • Refine until reaching target metric • Question: What’s the right error metric?

9. Error Metric • L2 error • Computationally simple • Suppress thin structures • Hausdorff error • Measures maximum deviation

10. Tree Decomposition -- data in square i Assume surflet dictionary with finite elements

11. Tree Decomposition root

12. Tree Decomposition root

13. Tree Decomposition root

14. Tree Decomposition root Cease refining a branch once node falls below threshold

15. Surflet Hallmarks • Multiscale representation • Allow for transmission of incremental detail • Prune tree for coarser representation • Extend tree for finer representation

16. 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

17. Multiscale Quantization • Allocate wisely as we increase scale, : • Intuition: • Coarse scale: poor fits (fewer bits) • Fine scale: good fits (more bits)

18. Polynomial Order-Aware Quantization • Consider Taylor-Series Expansion • Intuition: Higher order terms less significant • Increase bits for low-order terms Scale Optimal -- [Chandrasekaran, Wakin, Baron, Baraniuk 2006] Smoothness Order

19. 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”

20. Predictive Encoding Par Child

21. Predictive Encoding Par 1) Project parent into child domain Child

22. Predictive Encoding Par 2) Compute Hausdorff Error Child

23. Predictive Encoding Par 3) Determine probability based on distribution, error Child

24. Predictive Encoding Par 4) Code with bits Child Fewer bits More bits

25. Optimality Properties • Surflet encoding for L2 error metric for smooth functions[Chandrasekaran, Wakin, Baron, Baraniuk, 2004] • optimal asymptotic approximation rate for this function class • optimal rate-distortion performance for this function class • for piecewise constant surfaces of any polynomial order • Extension to Hausdorff error metric • tree encoder optimizes approximation • open question: optimal rate-distortion?

26. Experiments: Building 22,000 points piecewise planar surflets oct-tree: 120 nodes 1100 bits (“1400:1” compression)

27. Experiments: Mountain 263,000 points piecewise planar surflets 2000 Nodes 21000 Bits (“1500:1” Compression)

28. Summary • Multiscale, lossy compression for large point clouds • Error metric: Hausdorff distance, not L2 distance • Surflets offer excellent encodingfor piecewise smooth surfaces • octree based piecewise polynomial fitting • multiscale quantization • polynomial-order aware quantization • predictive encoding • Future research • Asymptotic optimality for Hausdorff metric dsp.rice.edu | math.tamu.edu