fourier-based and poisson surface reconstruction
DESCRIPTION
Fourier-based and Poisson surface reconstruction. Reporter: Lincong Fang 24th Dec, 2008. Implicit surfaces. F ( x , y ) =0. F ( x , y )>0. F ( x , y )TRANSCRIPT
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Fourier-based and Poisson surface reconstruction
Reporter: Lincong Fang24th Dec, 2008
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Implicit surfaces
F F ((xx,,yy)>0)>0
F F ((xx,,yy)<0)<0
F F ((xx,,yy) =0) =0
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Papers Michael Kazhdan. Reconstruction of solid models from oriented
point set. SGP2005. Oliver Schall, Alexander Belyaev, Hans-Peter Seidel. Error-
guided adaptive Fourier-based surface reconstruction. CAD 2007.
Michael Kazhdan, Matthew Bolitho, Hugues Hoppe. Poisson surface reconstruction. SGP 2006.
Kun Zhou, Minmin Gong, Xin Huang, Baining Guo. Highly parallel surface reconstruction. Technique report 2008.
![Page 6: Fourier-based and Poisson surface reconstruction](https://reader035.vdocuments.us/reader035/viewer/2022062410/568159a3550346895dc6faaf/html5/thumbnails/6.jpg)
Papers Michael Kazhdan. Reconstruction of solid models from oriented
point set. SGP2005. Oliver Schall, Alexander Belyaev, Hans-Peter Seidel. Error-
guided adaptive Fourier-based surface reconstruction. CAD 2007.
Michael Kazhdan, Matthew Bolitho, Hugues Hoppe. Poisson surface reconstruction. SGP 2006.
Kun Zhou, Minmin Gong, Xin Huang, Baining Guo. Highly parallel surface reconstruction. Technique report 2008.
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Characteristic Function
0M
1M
MM
1 if ( , , )( , , )
0 otherwiseM
x y z Mx y z
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Reconstruction
Mp
MppM if0
if1
M
01
0
00
0
1
1
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Fourier Coefficients
The Fourier coefficients of the characteristic function give an expression of M as a sum of complex exponentials:
( )
, ,
ˆ( , , ) ( , , ) i lx my nzM M
l m n
x y z l m n e
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Divergence Theorem
3 (three-dimensional solid)M R
3 3( , , ) : (vector-valued function)x y zF F F F R R ??????????????
( ) ( ), ( )M M
F p dp F p n p dp
??????????????????????????????????????????
( ) yx zFF F
F px y z
??????????????
=
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Monte-Carlo integration
1
| |( ) ( ),
N
iiMi
MF p dp F p n
N
????????????????????????????????????????????????????????
{ , }iip n M????????????????????????????
3 (three-dimensional solid)M R
3 3( , , ) : (vector-valued function)x y zF F F F R R ??????????????
( ) ( ), ( )M M
F p dp F p n p dp
??????????????????????????????????????????
( ) yx zFF F
F px y z
??????????????
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, ,( )( , , )l m nM
F x y z dxdydz ??????????????
Fourier Coefficients
3
( )( , , ) ( , , ) i lx my nzM MR
l m n x y z e dxdydz
( )x y zi lp mp np
p Me dxdydz
( ), ( )M
F p n p dp
????????????????????????????
, ,
1
1( ),
N
l m n jjj
F p nN
??????????????????????????????????????????
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( )
( ), ,
( )
( , , )
i lx my nz
i lx my nzl m n
i lx my nz
ie
l m ni
F x y z el m n
ie
l m n
??????????????
( )2 2 2
( ), , 2 2 2
( )2 2 2
( , , )
i lx my nz
i lx my nzl m n
i lx my nz
ile
l m nim
F x y z el m n
ine
l m n
??????????????
Anisotropic
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, ,
1
1( , , ) ( ),
N
l m n jM jj
l m n F p nN
??????????????????????????????????????????
1
1( ) ( )
j
N
jpj
N p p nN
??????????????????????????????????????????????????????????????????????
,
1
1( ) j
Nl p
j
j
N l e nN
????????????? ??????????????????????????????????????????? ,
21
( ) ,|| ||
j
Ni l p
jMj
il e n l
N l
????????????? ?????????????????????????????????????????????????????????
2
( ) ( ),|| ||
jM
il N l l
l ??????????????????????????????????????????????????????????????????????
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Iso-surface
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Non-Uniform Samples
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Non-Uniform Samples
i=1/2
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Non-Uniform Samples
i=1/1
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Non-Uniform Samples
i=1/3
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Non-Uniform Samples
i=1/4
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Results (Resolution)
res=1283
tris=49,008time=0:01
res=2563
tris=199,796time=0:07
100,000 Points100,000 Points 100,000 Points
res=643
tris=11,672time<0:01
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Results (Sample Count)
res=2563
tris=200,704time=0:07
res=2563
tris=206,216time=0:07
100,000 Points1000 Points
res=2563
tris=199,796time=0:07
10,000 Points
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Results (Non-Uniform Sampling)
res=2563
tris=111,680time=0:09
res=2563
tris=220,324time=0:09
100,000 Points100,000 Points100,000 Points
res=2563
tris=199,712time=0:09
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Results (Positional Noise)
disp=radius/128res=2563
tris=139,468time=0:07points=100,000
disp=0res=2563
tris=141,808time=0:07points=100,000
disp=radius/32res=2563
tris=124,300time=0:07points=100,000
disp=radius/64res=2563
tris=134,848time=0:07points=100,000
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Results (Normal Noise)
angle=0o
res=2563
tris=141,808time=0:07points=100,000
angle=30o
res=2563
tris=141,876time=0:07points=100,000
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Results (Normal Noise)
angle=15o
res=2563
tris=141,776time=0:07points=100,000
angle=45o
res=2563
tris=142,048time=0:07points=100,000
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Papers Michael Kazhdan. Reconstruction of solid models from oriented
point set. SGP2005. Oliver Schall, Alexander Belyaev, Hans-Peter Seidel. Error-
guided adaptive Fourier-based surface reconstruction. CAD 2007.
Michael Kazhdan, Matthew Bolitho, Hugues Hoppe. Poisson surface reconstruction. SGP 2006.
Kun Zhou, Minmin Gong, Xin Huang, Baining Guo. Highly parallel surface reconstruction. Technique report 2008.
![Page 31: Fourier-based and Poisson surface reconstruction](https://reader035.vdocuments.us/reader035/viewer/2022062410/568159a3550346895dc6faaf/html5/thumbnails/31.jpg)
Error-guided adaptive
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Error-guided adaptive
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Error-guided adaptive
0
0
N
i ii
g N
ii
w ff
w
2(|| || )i i iw G c p
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Results
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FFT time: 1min15s, peak memory: 1.4GB
AdFFT time: 21min10s, peak memory: 1.6GB
Comparison
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Papers Michael Kazhdan. Reconstruction of solid models from oriented
point set. SGP2005. Oliver Schall, Alexander Belyaev, Hans-Peter Seidel. Error-
guided adaptive Fourier-based surface reconstruction. CAD 2007.
Michael Kazhdan, Matthew Bolitho, Hugues Hoppe. Poisson surface reconstruction. SGP 2006.
Kun Zhou, Minmin Gong, Xin Huang, Baining Guo. Highly parallel surface reconstruction. Technique report 2008.
![Page 37: Fourier-based and Poisson surface reconstruction](https://reader035.vdocuments.us/reader035/viewer/2022062410/568159a3550346895dc6faaf/html5/thumbnails/37.jpg)
Indicator function
M
Indicator functionOriented points
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Poisson surface reconstruction
M
Indicator gradient
0 0
0
0
0
0
V
min
VV
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Octree
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Reconstruction
O
iii opfp
0
V
| |
2min || ||x R
Lx v
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Function Space
O
iii opfp
0
3
. 1( ) ( )
. .o
q o cf q f
o w o w
*( , , ) ( ( ) ( ) ( )) nf x y z B x B y B z1 | |<0.5
( )0 otherwise
tB t
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Function Space
3
. 1( ) ( )
. .o
q o cf q f
o w o w
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Function Space
3
. 1( ) ( )
. .o
q o cf q f
o w o w
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Function Space
3
. 1( ) ( )
. .o
q o cf q f
o w o w
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Function Space
3
. 1( ) ( )
. .o
q o cf q f
o w o w
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Splat the samples
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Splat the samples
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Splat the samples
,( )
( ) .D
q o s os S o Ngbr s
V F q s N
????????????????????????????
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Vector Field
O
iii opfp
0
V
| |
2min || ||x R
Lx v
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Indicator Function
O
iii opfp
0
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Surface Extraction
3{ | ( ) }M q R q
1( . )
| | s S
s pS
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Resolution
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Results (Resolution)
Octree depths 6 Octree depths 8 Octree depths 10
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Comparison to FFT
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Papers Michael Kazhdan. Reconstruction of solid models from oriented
point set. SGP2005. Oliver Schall, Alexander Belyaev, Hans-Peter Seidel. Error-
guided adaptive Fourier-based surface reconstruction. CAD 2007.
Michael Kazhdan, Matthew Bolitho, Hugues Hoppe. Poisson surface reconstruction. SGP 2006.
Kun Zhou, Minmin Gong, Xin Huang, Baining Guo. Highly parallel surface reconstruction. Technique report 2008.
![Page 56: Fourier-based and Poisson surface reconstruction](https://reader035.vdocuments.us/reader035/viewer/2022062410/568159a3550346895dc6faaf/html5/thumbnails/56.jpg)
Papers Michael Kazhdan. Reconstruction of solid models from oriented
point set. SGP2005. Oliver Schall, Alexander Belyaev, Hans-Peter Seidel. Error-
guided adaptive Fourier-based surface reconstruction. CAD 2007.
Michael Kazhdan, Matthew Bolitho, Hugues Hoppe. Poisson surface reconstruction. SGP 2006.
Kun Zhou, Minmin Gong, Xin Huang, Baining Guo. Highly parallel surface reconstruction. Technique report 2008.
![Page 57: Fourier-based and Poisson surface reconstruction](https://reader035.vdocuments.us/reader035/viewer/2022062410/568159a3550346895dc6faaf/html5/thumbnails/57.jpg)
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