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 sub cube – Refine until reaching target metric • Question: What’s the right error metric?

Error Metric •

L

2 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 Smoothness Scale 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 based on distribution, error

Predictive Encoding Par Child 4) Code with bits Fewer bits More bits

Optimality Properties • Surflet encoding for L 2 functions error metric for smooth [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 L 2 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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