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

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

Data Deluge The Challenge: Massive data sets – Millions of points – Costly to store/transmit/manipulate Goal: Find efficient algorithms for representation and compression

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

Optimality Properties Surflet encoding for L 2 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 Rate Dimension

Error Metric From L 2 error – Computationally simple – Suppress thin structures To Hausdorff error – Measures maximum deviation

Our Approach 1.Octree decomposition of point cloud – Fit a surflet at each node – Polynomial order determined by the image smoothness 2.Encode polynomial coefficients – Rate-distortion coder multiscale quantization predictive encoding

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

Step 1: Tree Decomposition (2D) root

Step 1: Tree Decomposition (2D) root

Step 1: Tree Decomposition (2D) root

Step 1: Tree Decomposition (2D) root

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

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

Multiscale Quantization Allocate more bits at finer scales: Allocate more bits to lower order coefficients – Taylor series : Smoothness Order Scale

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

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

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

Summary Multiscale, lossy compression for large point clouds – Error metric: Hausdorff distance, not L 2 distance – Surflets offer excellent encoding for piecewise smooth surfaces Multiscale surface polynomial tiling Multiscale quantization Predictive Encoding Open Question: Asymptotic optimality for Hausdorff metric