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Advanced Spectral Processing Techniques for Point-Sampled Geometry

This paper presents an innovative spectral processing pipeline designed for the effective management of point-sampled geometry. The approach extends Fourier transform methods to 2-manifold surfaces, facilitating powerful digital geometry processing applications such as noise removal, microstructure analysis, and adaptive resampling. Key components include patch layout generation, clustering, spectral analysis, and reconstruction. The results demonstrate significant improvements in filtering and resampling, providing versatility in handling large point-based models. Future work will focus on compression and enhanced feature extraction techniques.

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Advanced Spectral Processing Techniques for Point-Sampled Geometry

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  1. Spectral Processing of Point-sampled Geometry Mark Pauly Markus Gross ETH Zürich

  2. Outline • Introduction • Spectral processing pipeline • Results • Conclusions

  3. Introduction Point-based Geometry Processing Spectral Methods Introduction • Model • Acquisition • Range scans • Depth images • … • Point • Rendering • QSplat • Surfels • …

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

  5. Introduction Applications • Spectral filtering: • Noise removal • Microstructure analysis • Enhancement • Adaptive resampling: • Complexity reduction • Continuous LOD

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

  7. Spectral Processing Pipeline Overview

  8. Spectral Processing Pipeline Patch Layout Generation Clustering  Optimization Samples  Clusters  Patches

  9. Spectral Processing Pipeline Patch Merging Optimization • Iterative, local optimization method • Quality metric: patch Size  curvature  patch boundary  spring energy regularization

  10. Spectral Processing Pipeline Hierarchical Push-Pull Filter: Scattered Data Approximation

  11. 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

  12. Spectral Processing Pipeline Spectral Analysis Original Gaussian low-pass Ideal low-pass Band-stop Enhancement

  13. Spectral Processing Pipeline Resampling • Low-pass filtering • Band-limitation • Regular Resampling • Optimal sampling rate (Sampling Theorem) • Error control (Parseval’s Theorem) Power Spectrum

  14. 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

  15. Spectral Processing Pipeline

  16. Results Surface Restoration Original Gaussian Wiener Patch noise+blur Filter Filter Layout

  17. Results Interactive Filtering

  18. Results Adaptive Subsampling 4,128,614 pts. = 100% 287,163 pts. = 6.9%

  19. Results 9% 38% 23% 4% 26% Timings Clustering Time Patch Merging SDA Analysis Reconstruction

  20. Results Timings

  21. Conclusions Summary • Versatile spectral decomposition of point-based models • Effective filtering • Adaptive resampling • Efficient processing of large point-sampled models

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

  23. Acknowledgements Our Thanks to: Marc Levoy and the Stanford Digital Michelangelo Project, Szymon Rusinkiewicz, Bernd Gärtner

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