Subdivision and Smoothing of Surfaces

Subdivision

• Connectivity• Coordinates

Subdivision

Subdivision

Subdivision

Subdivision

Subdivision

Subdivision

Does Subdivision Change Surface Topology?

• V’=?• F’=?• E’=?• V’-E’+F’=?

Does Subdivision Change Surface Topology?

• V’=V+E• F’=4*F• E’=2*E+3*F• V’-E’+F’=?

Subdivision

• Why do we need it?– More samples inside a triangle– Smoother surfaces by using different weights:

A

B

C DP

Subdivision

• Subdivision with Loop’s masks:– New vertex

A

B

C DP

)(81)(

83 DCBAP

Subdivision

• Subdivision with Loop’s masks:– Old vertex

))2cos41

83(

85(1 2

nn

A1

A3A4

A5A

A2

A6

n

j jAAnA1

)1('

Subdivision

• Note:– The position of a new vertex is the weighted sum of the

positions of the old vertices in its neighborhood. The total weights sum to one.

– The same can be said about an old vertex.

))2cos41

83(

85(1 2

nn

n

j jAAnA1

)1('

)(81)(

83 DCBAP

More on subdivision later.

Irregular Subdivision

Irregular Subdivision

Image Blurring

original 1 time: 4 pixels

1 time: 64 pixels

1 time: 250 pixels

Surface Fairing

Image credit: Desbrun et al.

Heat Diffusion

Vector Field Visualization

Image credit: Diewald et al.

Curvature Estimation

Image credit: Alliez et al.

• They need to solve Laplace’s equations.

What Do These Application Have In Common?

Smoothing

• Of surfaces

• On surfaces

Surface Smoothing

• Solving Laplace:– Given a function:– The gradient is:

– And the Laplace is:

),,( zyxf

zf

yf

xfzyxf ),,(

2

2

2

2

2

2

),,(zf

yf

xffzyxf

Heat Diffusion

• Heat conduction equation:

– where T is temperate and t is time.– and controls the speed of diffusion– Boundary conditions:

• Heat sources:• Walls

– When is balance achieved?

),,( zyxTtT

Heat Diffusion in 1D

• 1D heat conduction equation:

– Reaches balance when – This happens

• either:• or

2

2

)(xTxT

tT

02

2

xT

CT

baxT

• Discrete Laplace:

– Reaches balance when

– Maximum and minimum can only occur on the boundaries

– Update rule:

Heat Diffusion in 1D

)22

()( 112

2nnnn TTTT

dxTdxT

tT

211

nnn

TTT nn-1 n+1

2)()( 111 nnnnmm TTTTTT

• Discrete Laplace:

• Why does it shrink?

Curve Evolution

)()(

21)(

xNjij TTiT

tT

Image credit: Delphine Nain

• Reverse shrinkage:– maintain area

Curve Evolution

Image credit: Delphine Nain

• Directions of changes:– Normal (is what we need)– Tangential (does not change shape but change

the position of points)

Curve Evolution

• Similar to curve evolution except now we have a surface:– Normal (is what we need)– Tangential (does not change shape but change

the position of points)

Surface Evolution

• Consider the weighting functions:

– such that

– rewrite

Surface Evolution

)(

111 ))()(()()(iNj

im

jm

ijim

im vTvTwtvTvT

)(

1iNj

ijw

)(iNjij

ijij e

ew

– Uniform:

• No geometry info

– Cord:

• Uses geometry info• Not directly related to normal and curvature

Weighting Functions

1ije

ijij l

e 1

– Mean curvature:• Curvature flow• In ideal situations the vertex only moves in the

normal direction• Could be negative

– Mean value:

• Always non-negative

Weighting Functions

2)cot(cot 21

ije

jv

iv

21 1 2

2))2/tan()2/(tan( 21

ije

– Explicit• Unstable• Requires small step-size

– Implicit:• Stable• Large step-size• Requires solving a system of equations

– Gauss-Seidel• Stable• Large step-size• Iterative

Update Scheme

)(

111 ))()(()()(iNj

im

jm

ijim

im vTvTwtvTvT

)(

1 ))()(()()(iNj

im

jm

ijim

im vTvTwtvTvT

)(

11 ))()(()()(iNj

im

jcurrent

ijim

im vTvTwtvTvT

Smoothing on Surfaces

Any questions?