Multiscale Representations for Point Cloud Data 3 D

Multiscale Representations for Point Cloud Data

3 D 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 Smoothness Rate • Our innovation: – More physically relevant error metric – Extension to point cloud data 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 (2 D) -- data in square i Assume surflet dictionary with finite elements Stop refining a branch once node falls below threshold

Step 1: Tree Decomposition (2 D) root

Step 1: Tree Decomposition (2 D) root

Step 1: Tree Decomposition (2 D) root

Step 1: Tree Decomposition (2 D) 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” “Less likely” • Insight: Encode with Smooth –log(p) images bits: small innovation Fewer bits at finer scale More bits • Coding Model: Favor small innovations over large ones • Encode according to distribution:

Experiment: Smooth Function 16, 400 points Planar Surflets 0. 03 bpp “ 3200: 1” Compression

Experiment: Building 22, 000 points Planar Surflets 0. 4 bpp “ 300: 1” Compression

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

Comparison: Binary and Octree

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