zl1 a sharpness dependent filter for mesh smoothing chun-yen chen kuo-young cheng available in cagd...
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A sharpness dependent filter for mesh smoothing
Chun-Yen Chen
Kuo-Young Cheng
available in CAGD Vol.22. 5(2005) 376-391
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Outline
Introduction about authorsIntroduction about worksSharpness dependent filterSharpness dependent weighting
functionComparisonConclusion
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Introduction about authors
Chun-Yen Chen, Kuo-Young Cheng Institute of Information Science,
Academia Sinica, Nankang, TaipeiDepartment of Computer Science and Information Engineering, National Taiwan University
Computer Graphics, Chinese Processing
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Outline
Introduction about authorsIntroduction about works Sharpness dependent filterSharpness dependent weighting
functionComparisonConclusion
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Introduction about works
Mesh straight-line graph embedded in R³
a pair (K, V), where K is a simplicial complex representing the connectivity of vertices, edges, and faces and V=( ) describes the geo-metric positions of the vertices in R³
… …
1 2, ,..., nv v v
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Introduction about works
Mesh
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Introduction about works
Mesh smoothing problem arising
creating high-fidelity computer graphics objects using imperfectly-measured data from the real world
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Introduction about works
Mesh smoothing main task
adjusting the position of mesh vertex the to remove undesirable noise and uneven edges while retaining desirable geometric features
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Introduction about works
Mesh smoothing regarded as a filter design problem, to
remove high-frequency tiny part on surfaces
filter: a function or a procedure which remove unwanted parts of a signal
Taubin, G., 1995. A signal processing approach to fair surface design. Siggraph’95.
Taubin, G., 1996. Optimal surface smoothing as filter design. Research Report RC-20404. IBM Thomas J.Watson Research Center.
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Introduction about works
Mesh smoothing dilemma
how can one get rid of the noise by smoothing the surface, while preserving sharp edge to keep the underlying geometry intact or feature?
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Introduction about works
Related works Notations mesh S={V, F}, where V and F are the sets of
vertices and faces, respectively vertex element, face element collection of neighboring vertices of vertex Laplacian operator
iv V if FviNv iv
( ) uu vvL V V V
iv
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Introduction about works
Laplacian smoothing (Taubin, 1995, 2000)
'
( )
( )
m
m m
V V L V
V V L V
1( ) ( )
vv ii
i ij j ij Nvijj Nv
L v w v vw
adjust vertex for smoothingCompensate shrinkage
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Introduction about works
Laplacian smoothing (Taubin, 1995, 2000) iterative process anti-shrinkage good overall smoothing, bad feature
preserving 200 smoothing steps100 smoothing steps
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Introduction about works
MCF (Mean Curvature Flow) isotropic filter design (Desbrun et al.,
1999)
mean curvature,
discrete mean curvature operator
( ) ( )iH i ik
t
v
v n v
1 2( ) ( )( )
2i i
H i
k kk
v vv( )H ik v
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Introduction about works
MCF (Mean Curvature Flow) isotropic filter design (Desbrun et al.,
1999)
new vertex position
'I dtK V V
'V
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Introduction about works
MCF (Mean Curvature Flow) anisotropic filter design (Meyer et al.,
2002) 1 2( ) , ( ) ( )iH i iAM w k k k
t
i
vv v n v
1 2
1 2 1 2
1 2 1 1 1 2
2 2 1 2
1 2
1 ,
0 , , 0
, / min , ,
1
/ min , ,
min , ,
H H
H H
H
k T k T
k T k T k k
w k k k k k k k k
k k k k k k
k k k k
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Introduction about works
MCF (Mean Curvature Flow)isotropic filter
feature non-preservinganisotropic filter
feature preserving
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Introduction about works
Bilateral Filter (Fleishman, et al., 2003; Jones et al., 2003)
'
, ,
,
j i
j i
s j i c j i i j i iN
i i i
s j i c j i iN
G G
G G
q v
q v
q v q v n q v n
v v nq v q v n
2 2/ 2( ) sxsG x e
2 2/ 2( ) cxcG x e
in
iv
jq
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Introduction about works
Mean-filter design (Ohtake et al., 2001)
surface normal based① compute weighted average normal
' 1,
fjj i
i
fjj i
ij ff N
f ij fiij ff N
wand w
Nw
nn
n
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Introduction about works
Mean-filter design (Ohtake et al., 2001)
surface normal based② update each vertex
' ' ' ''
1j j
vv j ij i
i i ij i j f ff Nijf N
w v cw
JJJJJJJJJJJJJJ
v v n n
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Introduction about works
Mean-filter design (Ohtake et al., 2001)feature non-preserving
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Introduction about works
Median-filter design (Yagou et al., 2002)
surface normal based① compute weighted average normal
fjj i
fjj i
ij UU N
T
ij UU N
wm
w
n
n
( )
( )fj i
j
ij
jU N
A Uw
A U
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Introduction about works
Median-filter design (Yagou et al., 2002)
surface normal based② adjust normal choose as media angle in N(T) replace m(T) by m( )
iU
( ), ( )i im T m U
i
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Introduction about works
Median-filter design (Yagou et al., 2002)
surface normal based③ update each vertex
' ' ' ''
1j
vv j ij i
i i ij i j U jU NijU N
w m mw
JJJJJJJJJJJJJJ
P P PC
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Introduction about works
Median-filter design (Yagou et al., 2002) feature preserving
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Introduction about works
Remarkmean-filter flat region
median-filter edgemin-filter corner (Gonzalez, Woods, 2002)
to smooth mesh appropriately, combine filters above together
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Introduction about works
This paperpropose a sharpness dependent filter design based on the fairing of surface normal, selecting a mean-filter for flat region and a min-filter for sharp region automatically
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Introduction about works
This paper
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Outline
Introduction about authorsIntroduction about works Sharpness dependent filterSharpness dependent weighting
functionComparisonConclusion
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Sharpness dependent filter
Basic concepts sharpness
a measure of the distribution of the included angles between polygon face normals
1n 2n
2n 1n
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Sharpness dependent filter
Basic concepts sharpness dependent weighting
functiondefined as the distribution of sharpness
cutoff value of sharpcriteria for sharp and non-sharp, derived by Bayesian classification (Chen, et al., 2004)
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Sharpness dependent filter
Algorithm1. compute mean normal
for each polygon face
is the No. of neighboring faces of
imif
' 1,
fjj i
i
fjj i
ij ff N
f ij fiij ff N
w nn and w
Nw n
fiN if
( )im
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Sharpness dependent filter
Algorithm2. determine the closet face
normal, , for each as follows① calculate the angle between
normals
normalized in a range [0,1]
ifi
,i jij f fn n
( , ) , |i j i j
ff f f f j in n n n f N
,i jf fn n
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Sharpness dependent filter
Algorithm2. determine the closet face
normal, , for each as follows
② find the minimum value of
ifmini ij
min min | fi ij j if N
min,if i in
ii
iifn
mini
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Sharpness dependent filter
Algorithm3. Calculate the local sharpness
21f
j i
mi ij if
f Ni
sN
is1
fj i
mi ijf
f NiN
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Sharpness dependent filter
Algorithm4. compute a new face normal
user-defined sharpness dependent weighting functon
'
1 ( )
/i
i i i i
f
n W s m W s
n n n
'
ifn
W s
( )im
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Sharpness dependent filter
Algorithm5. update each vertex
position
area weight contributed by
' ' ' ''
1j
vv j ij i
i i ij i j f jf Nijf N
v v w v c n nw
JJJJJJJJJJJJJJ
'ijw
'
jfn
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Sharpness dependent filter
Algorithm6. proceed to next iteration step
until a steady state, i.e.,
is a preset tolerance
'1i if fn n
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Sharpness dependent filter
Remarkwe’ve got a filter design for mesh smoothing based on the weighting function defined
by sharpness
mean-filter
min-filer
( )W s
( ) 1W s ( ) 0W s
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Sharpness dependent filter
Remark
how to select weighing function ?( )W s
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Outline
Introduction about authorsIntroduction about worksSharpness dependent filterSharpness dependent weighting
functionComparisonConclusion
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Sharpness dependent weighting function
Selection principle experiment to compare sharpness
distribution of most noisy models Fandisk
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Sharpness dependent weighting function
Selection principle experiment to compare sharpness
distribution of most noisy models Two-hole structure
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Sharpness dependent weighting function
Selection principle experiment to compare sharpness
distribution of most noisy models Golf driver head
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Sharpness dependent weighting function
Selection principle monotonic decreasing function,
vanishing beyond the cutoff of sharpness
Gaussian function
Laplacian function
El Fallah Ford function
2
2( ) exp
2
sW s
2( ) exp
sW s
2
1( )
1 /W s
s
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Sharpness dependent weighting function
Selection principle monotonic decreasing function,
vanishing beyond the cutoff of sharpness
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Sharpness dependent weighting function
selection user-defined, chosen such that for large sharpness and for
small sharpness
Remember cutoff value of sharpness for Remember cutoff value of sharpness for sharp and non-sharp obtained by sharp and non-sharp obtained by applying Bayesian classification?applying Bayesian classification?
( ) 1W s
( ) 0W s
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Sharpness dependent weighting function
selection user-defined, chosen such that for large sharpness and for
small sharpness obtain the best cutoff, , should
be small when Gaussian weighting function
( ) 1W s
( ) 0W s
th ( )W ss th
( ) exp( 5)10
thW th
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Sharpness dependent weighting function
selectionsharpness factor
to control degree of sharpness for feature preserving
1/ 2sf
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Sharpness dependent weighting function
Remarksharpness factor controls the degress of sharpness for feature preserving
, non-feature preservingthe larger , the stronger the feature preserving
sf
0sf sf
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Outline
Introduction about authorsIntroduction about worksSharpness dependent filterSharpness dependent weighting
functionComparisonConclusion
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Comparison
Different sharpness factor1/ 2sf
2( ) exp
sW s
sf=0, 10, 15, 20
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Comparison
With other feature preserving filter Like anisotropic MCF
bilateral filter median filter
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Comparison
With other feature preserving filterFandisk model A MCF Bilateral Median
Sharpness
Gaussian, sh=23.4, 16 steps
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Comparison
With other feature preserving filterTwo hole structure
Laplacian, sh=32, 97 steps
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Comparison
With other feature preserving filterGolf driver head
Bilateral
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Comparison
With other feature preserving filterGolf driver head
A MCF
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Comparison
With other feature preserving filterGolf driver head
Sharpness
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Comparison
With other feature preserving filterGuardian lion
Bilateral
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Comparison
With other feature preserving filterGuardian lion
A MFC
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Comparison
With other feature preserving filterGuardian lion
Sharpness
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Comparison
How about shrinkage? Little volume shrinkage, nearly intact
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Comparison
How about execution time? 2.8 GHz Pentium 4 processor with 1 GB
RAM
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Outline
Introduction about authorsIntroduction about worksSharpness dependent filterSharpness dependent weighting
functionComparisonConclusion
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Conclusion
Highlights Define sharpness to measure feature
areas of models Use sharpness dependent weighting
function to automatically select filter to smooth for different feature
Experiments to evaluate weighting function
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Thank U