Fourier-based and Poisson surface reconstruction Reporter: Lincong Fang 24th Dec, 2008

Fourier-based and Poisson surface reconstruction

Reporter: Lincong Fang24th Dec, 2008

Implicit surfaces

F F ((xx,,yy)>0)>0

F F ((xx,,yy)<0)<0

F F ((xx,,yy) =0) =0

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.






Characteristic Function

0M

1M

MM

1 if ( , , )( , , )

0 otherwiseM

x y z Mx y z

Reconstruction

Mp

MppM if0

if1

M

01

0

00

0

1

1

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

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

??????????????

=

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

??????????????

, ,( )( , , )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

??????????????????????????????????????????

( )

( ), ,

( )

( , , )

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

, ,

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 ??????????????????????????????????????????????????????????????????????

Iso-surface

Non-Uniform Samples

Non-Uniform Samples

i=1/2

Non-Uniform Samples

i=1/1

Non-Uniform Samples

i=1/3

Non-Uniform Samples

i=1/4

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

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

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

Results (Positional Noise)

disp=radius/128res=2563

tris=139,468time=0:07points=100,000

disp=0res=2563






Results (Normal Noise)

angle=0o

res=2563


angle=30o

res=2563


Results (Normal Noise)

angle=15o

res=2563


angle=45o

res=2563







Error-guided adaptive



0

0

N

i ii

g N

ii

w ff

w

2(|| || )i i iw G c p

Results

FFT time: 1min15s, peak memory: 1.4GB

AdFFT time: 21min10s, peak memory: 1.6GB

Comparison






Indicator function

M

Indicator functionOriented points

Poisson surface reconstruction

M

Indicator gradient

0 0

0

0

0

0

V

min

VV

Octree

Reconstruction

O

iii opfp

0

V

| |

2min || ||x R

Lx v

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

Function Space

3

. 1( ) ( )

. .o

q o cf q f

o w o w

Function Space

3

. 1( ) ( )

. .o

q o cf q f

o w o w

Function Space

3

. 1( ) ( )

. .o

q o cf q f

o w o w

Function Space

3

. 1( ) ( )

. .o

q o cf q f

o w o w

Splat the samples

Splat the samples

Splat the samples

,( )

( ) .D

q o s os S o Ngbr s

V F q s N

????????????????????????????

Vector Field

O

iii opfp

0

V

| |

2min || ||x R

Lx v

Indicator Function

O

iii opfp

0

Surface Extraction

3{ | ( ) }M q R q

1( . )

| | s S

s pS

Resolution

Results (Resolution)

Octree depths 6 Octree depths 8 Octree depths 10

Comparison to FFT











Documents

Fourier-based and Poisson surface reconstruction Reporter: Lincong Fang 24th Dec, 2008