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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