Wavelets for Surface Reconstruction Josiah Manson Guergana Petrova - PowerPoint PPT Presentation

Wavelets for Surface Reconstruction Josiah Manson Guergana Petrova Scott Schaefer

Convert Points to an Indicator Function

Data Acquisition

Properties of Wavelets Fourier Series Wavelets Represents all functions Locality Depends Smoothness on wavelet

Wavelet Bases Haar D4   ( x ) ( x )   ( x ) ( x )

Example of Function using Wavelets         j f ( x ) c ( x k ) c ( 2 x k ) k j , k k j , k

Example of Function using Wavelets     f ( x ) c ( x k ) k k

Example of Function using Wavelets         j f ( x ) c ( x k ) c ( 2 x k ) k j , k  k j 0 , k

Example of Function using Wavelets         j f ( x ) c ( x k ) c ( 2 x k ) k j , k  k j { 0 , 1 }, k

Example of Function using Wavelets         j f ( x ) c ( x k ) c ( 2 x k ) k j , k  k j { 0 , 1 , 2 }, k

Strategy • Estimate wavelet coefficients of indicator function • Use only local combination of samples to find coefficients

Computing the Indicator Function  [Kazhdan 2005] ( x )          j ( x ) c ( x k ) c ( 2 x k ) k j , k k j , k

Computing the Indicator Function  [Kazhdan 2005] ( x )          j ( x ) c ( x k ) c ( 2 x k ) k j , k k j , k      j c ( x ) ( 2 x k ) dx j , k R

Computing the Indicator Function  [Kazhdan 2005] ( x )          j ( x ) c ( x k ) c ( 2 x k ) k j , k k j , k          j j c ( x ) ( 2 x k ) dx ( 2 x k ) dx j , k R M

Computing the Indicator Function  [Kazhdan 2005] ( x )          j ( x ) c ( x k ) c ( 2 x k ) k j , k k j , k          j j c ( x ) ( 2 x k ) dx ( 2 x k ) dx j , k R M Divergence Theorem           F ( x ) dx F ( p ) n ( p ) d   M p M

Computing the Indicator Function  [Kazhdan 2005] ( x )          j ( x ) c ( x k ) c ( 2 x k ) k j , k k j , k          j j c ( x ) ( 2 x k ) dx ( 2 x k ) dx j , k R M Divergence Theorem           F ( x ) dx F ( p ) n ( p ) d   M p M

Computing the Indicator Function  [Kazhdan 2005] ( x )          j ( x ) c ( x k ) c ( 2 x k ) k j , k k j , k          j j c ( x ) ( 2 x k ) dx ( 2 x k ) dx j , k R M       F ( p ) n ( p ) d j , k   p M

Computing the Indicator Function  [Kazhdan 2005] ( x )          j ( x ) c ( x k ) c ( 2 x k ) k j , k k j , k          j j c ( x ) ( 2 x k ) dx ( 2 x k ) dx j , k R M       F ( p ) n ( p ) d j , k   p M       ( ) F p n d j , k i i i i

 Finding F ( x )       j F ( x ) ( 2 x k )

 Finding F ( x ) d       j F ( x ) F ( x ) ( 2 x k ) dx

 Finding F ( x ) d       j F ( x ) F ( x ) ( 2 x k ) dx x     j F ( x ) ( 2 s k ) ds  

Extracting the surface Coefficients          j ( x ) c ( x k ) c ( 2 x k ) k j , k k j , k Indicator function Dual marching cubes Surface

Smoothing the Indicator Function Haar unsmoothed

Smoothing the Indicator Function Haar unsmoothed Haar smoothed

Comparison of Wavelet Bases Haar D4

Advantages of Wavelets • Coefficients calculated only near surface – Fast – Low memory • Multi-resolution representation • Out of core calculation is possible

Streaming Pipeline Output Input

Results Michelangelo’s Barbuto 329 million points (7.4 GB of data), 329MB memory, 112 minutes

Results Michelangelo’s Awakening 381 million points (8.5 GB), 573MB memory, 81 minutes Produced 590 million polygons

Results Michelangelo’s Atlas 410 million points (9.15 GB), 1188MB memory, 98 minutes Produced 642 million polygons

Results Michelangelo’s Atlas 410 million points (9.15 GB), 1188MB memory, 98 minutes Produced 642 million polygons

Robustness to Noise in Normals 0 ° 30 ° 60 ° 90 °

Comparison of Methods Poisson Haar MPU D4 289 sec 17 sec 551 sec 82 sec 57 MB 13 MB 750 MB 43 MB

Relative Hausdorf Errors 1 0.9 0.8 0.7 Scaled Error MPU 0.6 Poisson Haar 0.5 Haar Smooth D4 0.4 D4 Smooth 0.3 0.2 0.1 0 armadilloman happy buddha dragon elephant2 hand malaysia tall teeth venus2

Conclusions • Wavelets provide trade-off between speed/quality • Works with all orthogonal wavelets • Guarantees closed, manifold surface • Out of core