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

Tree Decomposition

Assume surflet dictionary with finite elements

-- data in square i

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

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”

Predictive Encoding

Par

Child

Predictive Encoding

1) Project parent into child domain

Par

Child

Predictive Encoding

2) Compute Hausdorff ErrorPar

Child

Predictive Encoding

3) Determine probability based on distribution, error

Par

Child

Predictive Encoding

4) Code with bits

Fewer bits

More bits

Par

Child

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

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