David Levin Tel-Aviv University Afrigraph 2009 Shape Preserving Deformation David Levin Tel-Aviv University Afrigraph 2009 Based on joint works with Yaron.

Shape Preserving Deformation

David LevinDavid Levin Tel-Aviv University Tel-Aviv University

Afrigraph 2009Afrigraph 2009

Based on joint works with Based on joint works with

Yaron Lipman andYaron Lipman and Daniel Cohen-OrDaniel Cohen-Or

Tel-Aviv UniversityTel-Aviv University

Table of Contents

Representation and manipulation of discrete surfaces. Rigid motion invariant discrete surface

representation. Moving Frames for Surface Deformation. Volume representations and Green coordinates.

Representation and manipulation of discrete surfaces.

In this part we address the deformationdeformation problem:

Approach: use special surface representations.surface representations. Rotation invariant surface representation. Moving frames. Volume representation.

Linear Rotation-invariant Coordinates for Meshes SIGGRAPH 2005, special issue of ACM Transactions on Graphics.

Shape is invariant to rigid-motions:

We wish to find a rigid-motion invariant shape representation shape representation for discrete surfaces (meshes)for discrete surfaces (meshes):

Local support. Easy to transform from other representations. Easy to reconstruct.

i i

Moving Frames on surfaces

Moving frames on surfaces, generalization of Frenet-frame: Now we have two parameters.

1,2 1,3

2,1 2,3

3,1 3,2

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

d

d

d

1 2 3

2 1 3

3 1 2

e e e

e e e

e e e

,i k k ik

d , , ,i j i k k jk

d

v n

n v b

b n

( ) ( )C t tv1 2( ) ( ) ( )d 1 2g e e

, ,i j j i

Rotation invariant Surface representation

(i)3e

(i)1e

(i)2e

i(1)g

i(2)g

i(n)g

( ) ( ) ( )1 2 3

1,2,...,

i i i

k n

(i) (i) (i)i(k) i 1 2 3g g e e e

On Discrete Surfaces (Meshes): place a frame at each vertex(In a rigid-motion invariant manner.)

ig

1 2( ) ( ) ( )d 1 2g e e Tangential form:

Rotation invariant Surface representation

(i)3e

(i)1e

(i)2e

i(k)3e

i(k)1e

i(k)2e

( ) ( ) ( )1,1 1,2 1,3

( ) ( ) ( )2,1 2,2 2,3

( ) ( ) ( )3,1 3,2 3,3

i i i

i i i

i i i

i(k) (i) (i) (i) (i)1 1 1 2 3

i(k) (i) (i) (i) (i)2 2 1 2 3

i(k) (i) (i) (i) (i)3 3 1 2 3

e - e e e e

e - e e e e

e - e e e e

1,2 1,3

2,1 2,3

3,1 3,2

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

d

d

d

1 2 3

2 1 3

3 1 2

e e e

e e e

e e e

On Discrete Surfaces (Meshes)

Rotation invariant Surface representation

( ) ( ) ( )1,1 1,2 1,3

( ) ( ) ( )2,1 2,2 2,3

( ) ( ) ( )3,1 3,2 3,3

i i i

i i i

i i i

i(k) (i) (i) (i) (i)1 1 1 2 3

i(k) (i) (i) (i) (i)2 2 1 2 3

i(k) (i) (i) (i) (i)3 3 1 2 3

e - e e e e

e - e e e e

e - e e e e

Discrete surfaceDiscrete surface ((linearlinear )! )!equationsequations

(i)3e

(i)1e

(i)2e

( ) ( ) ( )1 2 3i i i (i) (i) (i)

i(k) i 1 2 3g g e e e

Rotation invariant Surface representation

11 . .Solve for the Solve for the framesframes((Linear Least SquaresLinear Least Squares))

( ) ( ), , ,

,i ij j k i j k

Rigid motionRigid motion invariantinvariant

representationrepresentation..

( ) ( ) ( )1,1 1,2 1,3

( ) ( ) ( )2,1 2,2 2,3

( ) ( ) ( )3,1 3,2 3,3

i i i

i i i

i i i

(j) (i) (i) (i) (i)1 1 1 2 3

(j) (i) (i) (i) (i)2 2 1 2 3

(j) (i) (i) (i) (i)3 3 1 2 3

e - e e e e

e - e e e e

e - e e e e

1 1

2 2 2

( ) ( ) ( )

(2) (2) (2)

, ,

, ,

1(k ) (k ) (k ) 1 1 11 2 3 1 2 3

(k ) (k ) (k )1 2 3 1 2 3

e e e v , v , v

e e e v , v , v

( ) ( ) ( )1 2 3i i i (i) (i) (i)

i(k) i 1 2 3g g e e e

1 1 1

1

( ) ( ) ( )1 2 3n n n

n (i) (i) (i)1 2 3g e e e

22 . .Solve for the Solve for the vertices positionvertices position..((Integration – Linear Least SquaresIntegration – Linear Least Squares))

Theorem 1Theorem 1: The set of coefficients define a unique discrete surface up to rigid motion. ( ) ( ),,i i

j j k

Applications

DeformationDeformation (prescribing frames on the surface):

Applications

MorphingMorphing (linear blending of the coordinates) – can handle large rotations.

Linear blending ofrotation-invariantCoords.

Linear blending ofworld Coords.

(1 )

Applications

MorphingMorphing (linear blending of the coordinates) – can handle large rotations.

can handle large rotations

Applications

Moving Frames for Surface DeformationACM Transactions on Graphics (2007).

In previous method we look for new orientations of local frames. The deformation energy can be quantified in a more precise manner. We define: Geometric distance Geometric distance which reflects resemblance between

curves and as such should ignore rigid motions.

Use a classical rigid motion invariant representation: CurvatureCurvature and torsiontorsion to define:( )t

( )t

2 2

0

( , )L

Dist C C dl

3, :[0, ]C C L R

Geometric Deformation of Curves

Reduction: Instead for looking for f, we search for R:

Theorem:2 2 21

2 FR

Geometric Deformation of Curves

Reduction: Instead for looking for f, we search for R:

22 2

0 0

( , )L L

FDist C C dl R dl

Dirichlet energy

Geometric Deformation of Curves

Critical solutions of Dirichlet integral are called harmonic harmonic mappingsmappings.

We have (still for curves): Shape is preserved if:

Frenet Frames where rotated by harmonic quantityharmonic quantity. The Frenet Frames rotation map is geodesic curvegeodesic curve on SO(3).

But we know what are geodesics on SO(3): one parameter subgroup of SO(3)

2

0

( , )L

FDist C C R dl

1log( )( ) , [0,1]R tR t e t

Geometric Deformation of Curves

Algorithm:

(3)SO

1R

10 1log( )

0( ) [0,1]R R tR t R e t

0R0R

0

( ) ( ) ( )t

C t R s C s ds

C

C

2R1R

2R

Geometric Distance and Optimal Rotation Field [Lipman et al. 2007] Let us seek optimal deformation .

Maintain the first fundamental form (isometries) and minimize the second fundamental form. Define a geometric distancegeometric distance between isometric surfaces:

2( , )

FM

Dist M M H H d

H the matrix of (Gauss map) in local frame,3de

2 2( , )

C

Dist C C d

( , ( )) minDist C f C

( , ) 0 , congruentDist C C C C

( , ( )) minDist M f M

( , ) 0 , congruentDist M M M M

Geometric Distance and Optimal Rotation Field [Lipman et al. 2007] Similarly to the curve case we look for mapping R.

Analogous to the curve case we have and therefore

21( , )

2 FM

Dist M M dR d

221

2 F FdR H H

The shapeshape is preserved if the rotation map is harmonicharmonic

Algorithm To Minimize we choose a parameterization

of SO(3): OrthogonalOrthogonal parameterization

ConformalConformal parameterization:

M

FddRMMDist 2

)~

,(

1 2 3 3, , : (3)R SO

12 2 22 1 2 2 2 2 32 4sin sin ( )

2FM M

dR d d d d d

Rotation angle Rotation axis

And when one rotation axis is involved we can assume 2 3, 0 0

1 2 3 3, , : (3)R SO

2 2

2

164

(4 )FM M

dR d d

Discretization Choose your favorite Laplace-Beltrami discretization (E.g. cot weight). Solve for the rotationsSolve for the rotations (with constraints). Two cases:

One rotation axis: exist linear solution– the angle of rotation should be harmonic function of the mesh.

More than one rotation angle: A non-linear problem. However linear approximation is enough.

IntegrateIntegrate: use the first discrete surface equation.

3

1

2

)1()1()(

2cot

jjijiji

TF

i

ddR

3

1

2

)1()1()()3()2()1(

2cot),,(

jjijijiiii

TF

WddRi

(1) (2) (3) (1) (2) (3)

2 2

( , , ) ( ) ( ) ( )

( ) 1/(4 )

i i i i i iW w w w

w

Results Discrete surface

equationsRotations solved

In SO(3)

Results

Green CoordinatesYaron Lipman, David Levin, Daniel Cohen-Or

Tel Aviv University

( ) ( )C

G uu u G d

n n

Related work

Free-form )space( deformations.Free-form )space( deformations. Lattice-based

(Sederberg & Parry 86, Coquillart 90, …) Curve-/handle-based

(Singh & Fiume 98, Botsch et al. 05, …) Cage-based (Mean Value Coordinates)

(Floater 03, Ju et al. 05, Langer et al. 06, Joshi et al. 07, Lipman et al. 07, Langer et al.08,… ) Vector Field constructions

(Angelidis 04, Von Funck 06,…)

Generalized Barycentric Coordinates Recent generalization to barycentric coordinatesbarycentric coordinates allow defining

space deformations using a flexible control polyhedra [Floater 03, Ju et al 05, Joshi et al. 07]:

Advantages:Advantages: Fast, simple, robust, not limited to surfaces. Disadvantage Disadvantage (in our context): Do not preserve shape.

( )ii

ip p q

piq C

1iq

( ; ) ( )ii

F C ip p p q

iq C

Joshi et.al. 2007

1iq

Shape Preserving Free-Form deformation

What is the class of space mappings we looklook for? (preserve shape) Conformal mappings Conformal mappings are shape preserving:

Angle preserving. Conformal mappings induce local similarity transformations :

: d df R R

Shape Preserving Free-Form deformation Two problems:

Conformal mappingsConformal mappings are hard to compute. Schwarz-Christoffel mapping – computationally hard [ T.A.Driscoll L.N.Trefethen 02].

Conformal mappingsConformal mappings exist only in 2D.

Shape Preserving Free-Form deformation What class of space mappings can be producedcan be produced using the

standard Free-from deformation formulation?

Each axis is treated independently. Affine invariance.

: ( )ii

F ip p q

Scalar affine weights (coordinates)

Constant points)w.r.t. p(

No hope for conformal mappings in this formulation of free-form deformations

Result break

Conformal space deformation Harmonic coordinates

Shape Preserving Free-Form deformation IdeaIdea: use the normals to blend between the coordinates…

( )ii

ip p q ( )( )jj

j+1 jp q -q

p

iq

i+1q

i-1q

( )i+1 iq -q

Shape Preserving Free-Form deformation Then the deformation is defined by:

p

iq

i+1q

i-1q

( )i+1 iq -q

( ) ( )( )i ji j

i j+1 jp p q p q -q

p

iq

i+1q

i-1q

( )i+1 iq -q

( ; )F Cp

( ; ) ( ) ( )( )i ji j

F C i j+1 jp p p q p q -q

Green Coordinates

QuestionQuestion: what are and ?

Green’s third identityGreen’s third identity encodes a similar relation: A harmonic function can be written as

( )i p ( )j p

( ; ) ( ) ( )( )i ji j

F C i j+1 jp p p q p q -q

( ) ( )C

G uu u G d

n n

η

2

13

(2 )( , )

1log 2

2

d

d

dd

G

d

ξ - ηξ η

ξ - η

η

C

Green Coordinates

Use the coordinate functions for u:

iq

i+1q

i-1q

C( ) ( )C

G uu u G d

n n

η

j jj T t t

Gd G d

n

jη ξ n

( , ,...)x y

ξ

nn

1

( )k k

d

j jk

ξ q ξξ

1jq

2jq

3jq

( )j i+1 it q -q

in 2D

in 3D

face nDjt

edge

triangle

simplicial

Green Coordinates

Use the coordinate functions for u:

1i i

J j

dj

j jj T i T tj

Gd Gd

n

t

η qt

{ }

1i

j i j j

i j ji V t N q j Tt tj

Gd Gd

n

η q tt

( , )x y

( )C

G uu u G d

n n

η

j jj T t t

Gd G d

n

jη ξ niq

i+1q

i-1q

C ( )j i+1 it q -q

( )i ( )j

Green Coordinates Theorem 1:Theorem 1:

The mapping for d=2 is conformal for all .

( ; ) ( ) ( )( )i ji j

F C i j+1 jp p p q p q -q

( ; )F Cp p C

HarmonicCoordinates

GreenCoordinates

Might go out of the cage

Green Coordinates

Theorem 2:Theorem 2: The coordinate functions have closed form expressionsclosed form expressions

for d=2,3.( ), ( )i j

2D Green Coordinates 3D Green Coordinates

Results

Green Coordinates Harmonic coordinates

Results

Green Coordinates

Mean-Value Coordinates

Harmonic Coordinates

Distortion histogram

Observation (Conjecture):Observation (Conjecture): The mapping is quasi-conformal for d=3.

( ; )F Cp p

Results

Green Coordinates

Mean ValueCoordinates

Results

Green Coordinates

Mean Valuecoordinates

Results

Green Coordinates

Mean Valuecoordinates

Results

Green Coordinates

Mean ValueCoordinates

Green Coordinates Theorem 3:Theorem 3:

The extension through every simplicial face is uniqueunique in the sense of analytic continuationanalytic continuation in 2D and real-analytic continuationreal-analytic continuation in 3D.

The formula for the unique extension is simply

where is an affine mapaffine map.

( ; )F Cp

( ; )F Cp

( ; ) ( ; ) ( ; ', )F C F C L P l p p p

( ; ', )L P lp

Employment of partial cages

Green Coordinates

Mean-Value Coordinates

Thanks for listening…