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Multiscale Representations for Point Cloud Data. Andrew Waters Manjari Narayan Richard Baraniuk. Luke Owens Daniel Freeman Matt Hielsberg Guergana Petrova Ron DeVore. 3D Surface Scanning. Explosion in data and applications. Terrain visualization Mobile robot navigation.
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Multiscale Representations for Point Cloud Data Andrew Waters Manjari Narayan Richard Baraniuk Luke Owens Daniel Freeman Matt Hielsberg Guergana Petrova 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 • Mesh Compression [Khodakovsky, Schröder, Sweldens 2000] • Geometric Mesh Compression [Huang, Peng, Kuo, Gopi 2006] • Point Cloud Compression [Schnabel, Klein 2006]
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 ?
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 ?
Our Approach • Fit piecewise polynomial surface to point cloud • Octree polynomialrepresentation • Encode polynomial coefficients • Rate-distortion coder • multiscale quantization • predictive encoding
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?
Error Metric • L2 error • Computationally simple • Suppress thin structures • Hausdorff error • Measures maximum deviation
Tree Decomposition -- data in square i Assume surflet dictionary with finite elements
Tree Decomposition root
Tree Decomposition root
Tree Decomposition root
Tree Decomposition root Cease refining a branch once node falls below threshold
Surflet Hallmarks • Multiscale representation • Allow for transmission of incremental detail • Prune tree for coarser representation • Extend 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 wisely as we increase scale, : • Intuition: • Coarse scale: poor fits (fewer bits) • Fine scale: good fits (more bits)
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
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”
Predictive Encoding Par Child
Predictive Encoding Par 1) Project parent into child domain Child
Predictive Encoding Par 2) Compute Hausdorff Error Child
Predictive Encoding Par 3) Determine probability based on distribution, error Child
Predictive Encoding Par 4) Code with bits Child Fewer bits More bits
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?
Experiments: Building 22,000 points piecewise planar surflets oct-tree: 120 nodes 1100 bits (“1400:1” compression)
Experiments: Mountain 263,000 points piecewise planar surflets 2000 Nodes 21000 Bits (“1500:1” Compression)
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