Spectral processing of point-sampled geometry Mark Pauly and Markus Gross Presented by Benjamin Haas Seminar on graphical data processing June 13, 2001

Spectral processing of point-sampled geometry Introduction Motivation Algorithm overview Algorithm details Results Conclusion

Spectral processing of point-sampled geometry Introduction Research area General description Motivation Algorithm overview Algorithm details Results Conclusion

Point-sampled geometry Introduction Point-sampled geometry Typically from laser range scanner Model sizes ~ 106 points No connectivity information Contains noise Fast rendering possible

Introduction Surface patches

Surface (distance map) as a 2-d signal Introduction Surface (distance map) as a 2-d signal

Spectral processing of point-sampled geometry Introduction Motivation Point-sampled geometry Fourier methods for geometry Algorithm overview Algorithm details Results Conclusion

Why using point samples? Motivation Point-sampled geometry Why using point samples? Increasing model size Decreasing triangle/quadrilateral size Smaller memory requirements Less computation time Hardware rendering to be expected soon

Why using Fourier methods? Motivation Fourier methods for geometry Why using Fourier methods? Frequency for curvature / LOD Filters expressed in frequency domain Convolution theorem Parseval's theorem (error estimation) O(n2) O(n)

Spectral processing of point-sampled geometry Introduction Motivation Algorithm overview The spectral processing pipeline Algorithm details Results Conclusion

The spectral processing pipeline Algorithm overview The spectral processing pipeline

Spectral processing of point-sampled geometry Introduction Motivation Algorithm overview Patch layout creation Scattered data approximation (Spectral analysis) Resampling Patch blending Algorithm details Results Conclusion

Two-step patch layout creation Algorithm details Patch layout creation Two-step patch layout creation Point samples Clusters Patches

Neighborhood detection using BSP tree Algorithm details Patch layout creation Clustering Neighborhood detection using BSP tree

Score function for optimization Algorithm details Patch layout creation Patch merging Score function for optimization

Algorithm details Patch layout creation Normal cone condition

Example: Patch layout for human face Algorithm details Patch layout creation Example: Patch layout for human face

Scattered data approximation Algorithm details Scattered data approximation Scattered data approximation

Scattered data approximation Algorithm details Scattered data approximation Scattered data approximation

The spectral processing pipeline Algorithm overview The spectral processing pipeline

Low-pass filtered signal  Band limitation Sampling theorem (Nyquist) Algorithm details Resampling Resampling Low-pass filtered signal  Band limitation Sampling theorem (Nyquist)  Optimal sampling rate Parseval’s theorem  Error control

Patch blending (reconstruction) Algorithm details Patch blending Patch blending (reconstruction) overlapping patch boundaries Blending of sampling rates Blending of points Blending of normals

Spectral processing of point-sampled geometry Introduction Motivation Algorithm overview Algorithm details Noise removal Enhancement Restoration Subsampling Results Conclusion

Smoothing and enhancement Results Smoothing and enhancement

Flexible spectral filtering Results Flexible spectral filtering

Noise removal Gaussian filter Wiener filter Original Results filtered with Gaussian filter filtered with Wiener filter Original (with noise)

Results Subsampling Original resolution Adapted resolution

Spectral processing of point-sampled geometry Introduction Motivation Algorithm overview Algorithm details Results Pros and cons Discussion Conclusion

+ Sound concept of frequency + Flexible, elegant and fast filtering Conclusion Pros and cons + Sound concept of frequency + Flexible, elegant and fast filtering - Patch merging, SDA and reconstruction are costly - Inconsistencies at patch boundaries (no mathematical foundation at present) Well suitable for interactive surface processing