1 Free-Form Deformations Dr. Scott Schaefer. 2/28 Deformation

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Free-Form Deformations

Dr. Scott Schaefer

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Deformation

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Deformation

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Deformation Applications

Toy Story © Disney / Pixar

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Challenges in Deformation

Large meshes – millions of polygons Need efficient techniques for computing and

specifying the deformation

Digital Michelangelo Project

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Deformation Handles

Low-resolution auxiliary shape controls deformation of high-resolution model

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Deformation Handles

Low-resolution auxiliary shape controls deformation of high-resolution model

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Deformation Handles

Low-resolution auxiliary shape controls deformation of high-resolution model

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Deformation Handles

Low-resolution auxiliary shape controls deformation of high-resolution model

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FFD Contributions

Smooth deformations of arbitrary shapes Local control of deformation Performing deformation is fast

Widely usedGame/Movie industryPart of nearly every 3D modeler

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Free-Form Deformations

Embed object in uniform grid Represent each point in space as a weighted

combination of grid vertices

0x 1x 2x 3x

v

i

iixwv

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Free-Form Deformations

Embed object in uniform grid Represent each point in space as a weighted

combination of grid vertices Assume xi are equally spaced and use

Bernstein basis functions

0x 1x 2x 3x

v

ii

iid

iii xtt

i

dxwv )1(

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Free-Form Deformations

Embed object in uniform grid Represent each point in space as a weighted

combination of grid vertices Assume xi are equally spaced and use

Bernstein basis functions

0x 1x 2x 3x

v0 3

132 1

txtti

dxwv

ii

iid

iii

)1(

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Free-Form Deformations

Embed object in uniform grid Represent each point in space as a weighted

combination of grid vertices Assume xi are equally spaced and use

Bernstein basis functions

0x 1x 2x 3x

v0 3

132 1

iidi vv

i

dw

)1(

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2D Example

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2D Example

0 1

0

1

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2D Example

0 1

0

1

32

54 ,

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2D Example

0 1

0

1

32

54 , 3

2

54

v

u

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2D Example

32

54 , 3

2

54

v

u

22 )1()1( vu 2)1()1(2 vuu 22 )1( vu

22vu22)1( vu 2)1(2 uvu

vvu )1()1(2 2 vvu )1(2 2 vvuu )1()1(4

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2D Example

32

54 , 3

2

54

v

u

2251

2258

22516

2254

22532 225

64

2254

22532

22564

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Applying the Deformation

jip ,

v

ji

jiji pwv,

,,

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Applying the Deformation

jip ,

v

ji

jiji pwv,

,,

jip ,ˆ

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Applying the Deformation

jip ,

v

ji

jiji pwv,

,,

jip ,ˆ

ji

jiji pwv,

,, ˆˆ

v̂

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Smoothness of Deformation

Constraining Bezier control points controls smoothness

1C

0C

1C

2C

Image taken from “Free-form Deformations of Solid Geometric Models”

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Volume Preservation

Must ensure that the jacobian of the deformation is 1 everywhere

),,(),,,(),,,(ˆ,ˆ,ˆ zyxHzyxGzyxFzyx

1

zH

yH

xH

zG

yG

xG

zF

yF

xF

Images taken from “Free-form Deformations of Solid Geometric Models”

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Advantages

Smooth Deformation of arbitrary shapes Local control of deformations Computing the deformation

is easy Deformations are very fast

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Disadvantages

Must use cubical cells for deformation Restricted to uniform grid Deformation warps space… not surface

Does not take into account geometry/topology of surface

May need many FFD’s to achieve a simple deformation

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Summary

Widely used deformation technique Fast, easy to compute Some control over volume

preservation/smoothness

Uniform grids are restrictive