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Spectral Processing of Point-sampled Geometry. Mark Pauly Markus Gross ETH Zürich. Outline. Introduction Spectral processing pipeline Results Conclusions. Introduction. Point-based Geometry Processing. Spectral Methods. Introduction. Model Acquisition Range scans

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spectral processing of point sampled geometry
Spectral Processing of Point-sampled Geometry

Mark Pauly Markus Gross

ETH Zürich

outline
Outline
  • Introduction
  • Spectral processing pipeline
  • Results
  • Conclusions
introduction

Introduction

Point-based

Geometry Processing

Spectral Methods

Introduction
  • Model
  • Acquisition
  • Range scans
  • Depth images
  • Point
  • Rendering
  • QSplat
  • Surfels
spectral transform

Introduction

Spectral Transform
  • Extend Fourier transform to 2-manifold surfaces
    • Spectral representation of point-based objects
    • Powerful methods for digital geometry processing
applications

Introduction

Applications
  • Spectral filtering:
    • Noise removal
    • Microstructure analysis
    • Enhancement
  • Adaptive resampling:
    • Complexity reduction
    • Continuous LOD
fourier transform

Introduction

Fourier Transform
  • Benefits:
    • Sound concept of frequency
    • Extensive theory
    • Fast algorithms
  • Limitations:
    • Euclidean domain, global parameterization
    • Regular sampling
    • Lack of local control
patch layout generation

Spectral Processing Pipeline

Patch Layout Generation

Clustering  Optimization

Samples  Clusters  Patches

patch merging optimization

Spectral Processing Pipeline

Patch Merging Optimization
  • Iterative, local optimization method
  • Quality metric:

patch Size

 curvature

 patch boundary

 spring energy regularization

spectral analysis

Spectral Processing Pipeline

Spectral Analysis
  • 2D Discrete Fourier Transform (DFT)
    • Direct manipulation of spectral coefficients
  • Filtering as convolution:
    • Convolution: O(N2)  Multiplication: O(N)
  • Inverse Fourier Transform
    • Filtered patch surface
spectral analysis12

Spectral Processing Pipeline

Spectral Analysis

Original

Gaussian low-pass

Ideal low-pass

Band-stop

Enhancement

resampling

Spectral Processing Pipeline

Resampling
  • Low-pass filtering
    • Band-limitation
  • Regular Resampling
    • Optimal sampling rate (Sampling Theorem)
    • Error control (Parseval’s Theorem)

Power Spectrum

reconstruction

Spectral Processing Pipeline

Reconstruction
  • Filtering can lead to discontinuities at patch boundaries
      • Create patch overlap, blend adjacent patches

region of overlap

Sampling rates

Point positions

Normals

surface restoration

Results

Surface Restoration

Original Gaussian Wiener Patch

noise+blur Filter Filter Layout

adaptive subsampling

Results

Adaptive Subsampling

4,128,614 pts.

= 100%

287,163 pts.

= 6.9%

timings

Results

9%

38%

23%

4%

26%

Timings

Clustering

Time

Patch

Merging

SDA

Analysis

Reconstruction

summary

Conclusions

Summary
  • Versatile spectral decomposition of point-based models
  • Effective filtering
  • Adaptive resampling
  • Efficient processing of large point-sampled models
future work

Conclusions

Future Work
  • Compression
    • Scalar Representation + Spectral Compression
  • Hierarchical Representation
    • Modeling and Animation
  • Feature Detection & Extraction
    • Robust Computation of Laplacian
acknowledgements
Acknowledgements

Our Thanks to:

Marc Levoy and the Stanford Digital Michelangelo Project, Szymon Rusinkiewicz, Bernd Gärtner

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