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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

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Spectral processing of point sampled geometry l.jpg

Spectral Processing of Point-sampled Geometry

Mark Pauly Markus Gross

ETH Zürich


Outline l.jpg

Outline

  • Introduction

  • Spectral processing pipeline

  • Results

  • Conclusions


Introduction l.jpg

Introduction

Point-based

Geometry Processing

Spectral Methods

Introduction

  • Model

  • Acquisition

  • Range scans

  • Depth images

  • Point

  • Rendering

  • QSplat

  • Surfels


Spectral transform l.jpg

Introduction

Spectral Transform

  • Extend Fourier transform to 2-manifold surfaces

    • Spectral representation of point-based objects

    • Powerful methods for digital geometry processing


Applications l.jpg

Introduction

Applications

  • Spectral filtering:

    • Noise removal

    • Microstructure analysis

    • Enhancement

  • Adaptive resampling:

    • Complexity reduction

    • Continuous LOD


Fourier transform l.jpg

Introduction

Fourier Transform

  • Benefits:

    • Sound concept of frequency

    • Extensive theory

    • Fast algorithms

  • Limitations:

    • Euclidean domain, global parameterization

    • Regular sampling

    • Lack of local control


Overview l.jpg

Spectral Processing Pipeline

Overview


Patch layout generation l.jpg

Spectral Processing Pipeline

Patch Layout Generation

Clustering  Optimization

Samples  Clusters  Patches


Patch merging optimization l.jpg

Spectral Processing Pipeline

Patch Merging Optimization

  • Iterative, local optimization method

  • Quality metric:

patch Size

 curvature

 patch boundary

 spring energy regularization


Scattered data approximation l.jpg

Spectral Processing Pipeline

Hierarchical Push-Pull Filter:

Scattered Data Approximation


Spectral analysis l.jpg

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 l.jpg

Spectral Processing Pipeline

Spectral Analysis

Original

Gaussian low-pass

Ideal low-pass

Band-stop

Enhancement


Resampling l.jpg

Spectral Processing Pipeline

Resampling

  • Low-pass filtering

    • Band-limitation

  • Regular Resampling

    • Optimal sampling rate (Sampling Theorem)

    • Error control (Parseval’s Theorem)

Power Spectrum


Reconstruction l.jpg

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


Slide15 l.jpg

Spectral Processing Pipeline


Surface restoration l.jpg

Results

Surface Restoration

Original Gaussian Wiener Patch

noise+blur Filter Filter Layout


Interactive filtering l.jpg

Results

Interactive Filtering


Adaptive subsampling l.jpg

Results

Adaptive Subsampling

4,128,614 pts.

= 100%

287,163 pts.

= 6.9%


Timings l.jpg

Results

9%

38%

23%

4%

26%

Timings

Clustering

Time

Patch

Merging

SDA

Analysis

Reconstruction


Timings20 l.jpg

Results

Timings


Summary l.jpg

Conclusions

Summary

  • Versatile spectral decomposition of point-based models

  • Effective filtering

  • Adaptive resampling

  • Efficient processing of large point-sampled models


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Conclusions

Future Work

  • Compression

    • Scalar Representation + Spectral Compression

  • Hierarchical Representation

    • Modeling and Animation

  • Feature Detection & Extraction

    • Robust Computation of Laplacian


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Acknowledgements

Our Thanks to:

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


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