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]
Our Innovation ?
– More physically relevant error metric
– Efficient lossy encoding
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 subcube
– 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
Smoothness
Order
Optimal -- [Chandrasekaran, Wakin, Baron,
Baraniuk 2006]
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:
Predictive Encoding
Par
Child
Predictive Encoding
Par
Child
1) Project parent into child
domain
Predictive Encoding
Par
Child
2) Compute Hausdorff Error
Predictive Encoding
Par
Child
3) Determine probability
on distribution, error
based
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 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
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