Multiscale Representations for Point Cloud Data

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 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

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

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

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

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 : Scale Smoothness Order

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

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 21000 Bits “1500:1” Compression 0.08 bpp

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

Multiscale Representations for Point Cloud Data