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Wavelets for Surface Reconstruction Josiah Manson Guergana Petrova - - PowerPoint PPT Presentation
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
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Data Acquisition
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Properties of Wavelets
Fourier Series Wavelets Represents all functions Locality Smoothness Depends
- n wavelet
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Wavelet Bases
Haar D4
) (x ) (x ) (x ) (x
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Example of Function using Wavelets
k j j k j k k
k x c k x c x f
, ,
) 2 ( ) ( ) (
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Example of Function using Wavelets
k k
k x c x f ) ( ) (
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Example of Function using Wavelets
k j j k j k k
k x c k x c x f
, ,
) 2 ( ) ( ) (
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Example of Function using Wavelets
k j j k j k k
k x c k x c x f
}, 1 , { ,
) 2 ( ) ( ) (
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Example of Function using Wavelets
k j j k j k k
k x c k x c x f
}, 2 , 1 , { ,
) 2 ( ) ( ) (
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Strategy
- Estimate wavelet coefficients of indicator
function
- Use only local combination of samples to
find coefficients
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Computing the Indicator Function
k j j k j k k
k x c k x c x
, ,
) 2 ( ) ( ) (
) (x
[Kazhdan 2005]
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Computing the Indicator Function
R j k j
dx k x x c ) 2 ( ) (
,
k j j k j k k
k x c k x c x
, ,
) 2 ( ) ( ) (
) (x
[Kazhdan 2005]
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Computing the Indicator Function
M j R j k j
dx k x dx k x x c ) 2 ( ) 2 ( ) (
,
k j j k j k k
k x c k x c x
, ,
) 2 ( ) ( ) (
) (x
[Kazhdan 2005]
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Computing the Indicator Function
M p M
d p n p F dx x F ) ( ) ( ) (
Divergence Theorem
k j j k j k k
k x c k x c x
, ,
) 2 ( ) ( ) (
) (x
M j R j k j
dx k x dx k x x c ) 2 ( ) 2 ( ) (
,
[Kazhdan 2005]
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Computing the Indicator Function
M p M
d p n p F dx x F ) ( ) ( ) (
Divergence Theorem
k j j k j k k
k x c k x c x
, ,
) 2 ( ) ( ) (
) (x
M j R j k j
dx k x dx k x x c ) 2 ( ) 2 ( ) (
,
[Kazhdan 2005]
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Computing the Indicator Function
M p k j
d p n p F ) ( ) (
,
k j j k j k k
k x c k x c x
, ,
) 2 ( ) ( ) (
) (x
M j R j k j
dx k x dx k x x c ) 2 ( ) 2 ( ) (
,
[Kazhdan 2005]
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Computing the Indicator Function
i i i i k j
d n p F ) (
,
k j j k j k k
k x c k x c x
, ,
) 2 ( ) ( ) (
) (x
M p k j
d p n p F ) ( ) (
,
M j R j k j
dx k x dx k x x c ) 2 ( ) 2 ( ) (
,
[Kazhdan 2005]
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Finding
) 2 ( ) ( k x x F
j
) (x F
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Finding
) (x F
) 2 ( ) ( ) ( k x x F dx d x F
j
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Finding
) (x F
) 2 ( ) ( ) ( k x x F dx d x F
j
x j
ds k s x F ) 2 ( ) (
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Extracting the surface
k j j k j k k
k x c k x c x
, ,
) 2 ( ) ( ) (
Coefficients Indicator function Surface Dual marching cubes
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Haar unsmoothed
Smoothing the Indicator Function
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Smoothing the Indicator Function
Haar unsmoothed Haar smoothed
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Comparison of Wavelet Bases
Haar D4
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Advantages of Wavelets
- Coefficients calculated only near surface
– Fast – Low memory
- Multi-resolution representation
- Out of core calculation is possible
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Streaming Pipeline
Input Output
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Results
Michelangelo’s Barbuto 329 million points (7.4 GB of data), 329MB memory, 112 minutes
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Results
Michelangelo’s Awakening 381 million points (8.5 GB), 573MB memory, 81 minutes Produced 590 million polygons
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Results
Michelangelo’s Atlas 410 million points (9.15 GB), 1188MB memory, 98 minutes Produced 642 million polygons
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Results
Michelangelo’s Atlas 410 million points (9.15 GB), 1188MB memory, 98 minutes Produced 642 million polygons
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Robustness to Noise in Normals
0° 30° 60° 90°
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Comparison of Methods
MPU 551 sec 750 MB Poisson 289 sec 57 MB Haar 17 sec 13 MB D4 82 sec 43 MB
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Relative Hausdorf Errors
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 armadilloman happy buddha dragon elephant2 hand malaysia tall teeth venus2
Scaled Error
MPU Poisson Haar Haar Smooth D4 D4 Smooth
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Conclusions
- Wavelets provide trade-off
between speed/quality
- Works with all orthogonal
wavelets
- Guarantees closed, manifold
surface
- Out of core