computer graphics • subdivision surfaces © 2006 fabio pellacini • 1

subdivision surfaces

computer graphics • subdivision surfaces © 2006 fabio pellacini • 2

subdivision curves

• start with a piecewise linear curve• apply recursively subdivision rule

[Zorin and Schröder,2000]

computer graphics • subdivision surfaces © 2006 fabio pellacini • 3

subdivision surfaces

• start with polygon mesh• apply recursively subdivision rule

[Zor

inan

d Sc

hröd

er,2

000]

computer graphics • subdivision surfaces © 2006 fabio pellacini • 4

subdivision surfaces

• subdiv.: limit surface of subdivision process– provably smooth surface! (i.e. continuous function)– subdiv. process: input is mesh, output is refined mesh

– limit surface)(1 ii f MM =+

iiMS

∞→= lim

[DeRose et al.,1998]

computer graphics • subdivision surfaces © 2006 fabio pellacini • 5

subdivision schemes

• two stages• mesh subdivision new topology

– create new vertices and faces

• vertex placement new geometry– compute vertex position

computer graphics • subdivision surfaces © 2006 fabio pellacini • 6

subdivision schemes

• different schemes available– depend on input/output topology and geometry

• quadrilaterals input: – Catmull-Clark

• triangles input: – Loop

computer graphics • subdivision surfaces © 2006 fabio pellacini • 7

loop subdivision scheme

• new topology: new faces– split each triangle in 4 triangles

[Zor

inan

d Sc

hröd

er,2

000]

computer graphics • subdivision surfaces © 2006 fabio pellacini • 8

loop subdivision scheme

• new geometry: new vertex positions– need rules for even/odd (white/black) vertices

[Zor

inan

d Sc

hröd

er,2

000]

computer graphics • subdivision surfaces © 2006 fabio pellacini • 9

loop subdivision rules

• new vertex pos: weighted average of neighbors

[Zor

inan

d Sc

hröd

er,2

000]

computer graphics • subdivision surfaces © 2006 fabio pellacini • 10

boundaries and creases

• rules on boundaries– only consider vertices along the boundary

• creases: mark edges sharp– only consider vertices along a crease

computer graphics • subdivision surfaces © 2006 fabio pellacini • 11

extraordinary vertices

• vertices with non-standard num. of neighbors– unavoidable for complex topology– make subdivision ideal for this case

[Zor

inan

d Sc

hröd

er,2

000]

computer graphics • subdivision surfaces © 2006 fabio pellacini • 12

loop subdivision rules

[Zor

inan

d Sc

hröd

er,2

000]

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ +−=

22cos41

83

851

nnπβ

computer graphics • subdivision surfaces © 2006 fabio pellacini • 13

Catmull-Clark subdivision rules

[Zor

inan

d Sc

hröd

er,2

000]

kk 4/1;2/3 == γβ

computer graphics • subdivision surfaces © 2006 fabio pellacini • 14

determining subdivision rules

• how to determine subdivision rule?– analyze property of limit surface– goal: smoothness

• limit surface has provable smoothness!– C1 at extraordinary points– C2 everywhere else

computer graphics • subdivision surfaces © 2006 fabio pellacini • 15

subdivision schemes comparison

[Zorin and Schröder,2000]

Loop Catmull-Clark

computer graphics • subdivision surfaces © 2006 fabio pellacini • 16

subdivision schemes comparison

[Zorin and Schröder,2000]

Loop Catmull-Clark

computer graphics • subdivision surfaces © 2006 fabio pellacini • 17

subdivision schemes comparison

[Zorin and Schröder,2000]

Loop Catmull-ClarkControl Mesh

computer graphics • subdivision surfaces © 2006 fabio pellacini • 18

subdivision schemes comparison

[Zorin and Schröder,2000]

Loop Catmull-Clark

computer graphics • subdivision surfaces © 2006 fabio pellacini • 19

subdivision surface manipulation

• manipulate original mesh, while displaying subdiv

[Zor

inan

d Sc

hröd

er,2

000]

computer graphics • subdivision surfaces © 2006 fabio pellacini • 20

subdivision surface manipulation

• manipulate original mesh, while displaying subdiv

computer graphics • subdivision surfaces © 2006 fabio pellacini • 21

subdivision surface manipulation

• add crease for sharp features– on edges

computer graphics • subdivision surfaces © 2006 fabio pellacini • 22

subdivision surface manipulation

• add crease for sharp features– on vertices

computer graphics • subdivision surfaces © 2006 fabio pellacini • 23

subdivision surface manipulation

• add crease for sharp features– on edges and vertices

computer graphics • subdivision surfaces © 2006 fabio pellacini • 24

subdivision surfaces

• heavily used in Computer Graphics– old algorithms but practicalities introduced recently– works well with animation

• arbitrary topology, no stitching

– Pixar’s Geri’s Game 1998• Catmull-Clark subdiv• complex control mesh• heavy use of creases

computer graphics • subdivision surfaces © 2006 fabio pellacini • 25

surface representation comparison

computer graphics • subdivision surfaces © 2006 fabio pellacini • 26

representation comparison

polygonmeshes

implicitsurfaces

parametricsurfaces

subdivisionsurfaces

accurate no yes yes yes

concise no yes yes yes

intuitive specification no no yes no

local support yes no yes yes

affine invariant yes yes yes yes

arbitrary topology yes no no yes

guaranteed continuity no yes yes yes

parameterization no no yes no

efficient display yes no yes yes

efficient intersection no yes no no

[based on Finkelstein 2004]