Multiscale Representations for Point Cloud Data Andrew Waters Manjari Narayan Richard Baraniuk Luke Owens Daniel Freeman Matt Hielsberg Guergana Petrova.
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Multiscale Representations for Point Cloud Data
Andrew WatersManjari NarayanRichard Baraniuk
Luke OwensDaniel FreemanMatt Hielsberg
Guergana PetrovaRon 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 ? 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 Innovation ?
Our Approach
1. Fit piecewise polynomial surface to point cloud
– Octree polynomial representation
2. 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
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
SmoothnessOrder
Scale
Optimal -- [Chandrasekaran, Wakin, Baron, Baraniuk 2006]
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”
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 surfletsoct-tree: 120 nodes1100 bits (“1400:1” compression)
Experiments: Mountain
263,000 pointspiecewise planar surflets2000 Nodes21000 Bits (“1500:1” Compression)
Summary• Multiscale, lossy compression for large point clouds
– Error metric: Hausdorff distance, not L2 distance
– Surflets offer excellent encoding for 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
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