Multiscale Representations for Point Cloud Data

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 Replace hand with terrain point cloud!

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 Trading off

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 Add rectangular here if we decide to use it! Firm up smoothness understanding before talk

Error Metric From L2 error To Hausdorff error Computationally simple Suppress thin structures To Hausdorff error Measures maximum deviation Expected in urban terrain.

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

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 : Combine into one slide – give the gyst and move on! Scale Smoothness Order

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

Experiment: Smooth Function 16,400 points Planar Surflets 0.03 bpp “3200:1” Compression 22

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 L2 distance Surflets offer excellent encoding for piecewise smooth surfaces Multiscale surface polynomial tiling Multiscale quantization Predictive Encoding Open Question: Asymptotic optimality for Hausdorff metric