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Skinning Cubic Bzier Splines and Catmull-Clark Subdivision Surfaces Songrun Liu George Mason University Alec Jacobson Columbia University Yotam Gingold George Mason University SPONSORED BY SA2014.SIGGRAPH.ORG Raster Deformation

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SPONSORED BY SA2014.SIGGRAPH.ORG

Skinning Cubic Bézier Splines and Catmull-Clark Subdivision Surfaces

Songrun Liu Alec Jacobson Yotam Gingold George Mason University Columbia University George Mason University

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SA2014.SIGGRAPH.ORG SPONSORED BY

Raster Deformation

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Raster Deformation

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Vector Graphics Deformation

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Vector Graphics Deformation

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Vector Graphics Deformation

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Our Approach

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Our Approach

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Our Approach

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Our Approach

Deforming All Control Points

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Our Approach

Deforming All Control Points Deforming Joint Control Points

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Related Work

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2D Vector Graphics

  • Splines in 2D — Cubic Bézier splines.

BC(t) =

n

X

i=0

bi,n(t)Ci, t ∈ [0, 1]

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SA2014.SIGGRAPH.ORG SPONSORED BY

2D Vector Graphics

  • Splines in 2D — Cubic Bézier splines.

BC(t) =

n

X

i=0

bi,n(t)Ci, t ∈ [0, 1]

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2D Vector Graphics

  • Splines in 2D — Cubic Bézier splines.

Siggraph

BC(t) =

n

X

i=0

bi,n(t)Ci, t ∈ [0, 1]

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SA2014.SIGGRAPH.ORG SPONSORED BY

2D Vector Graphics

  • Splines in 2D — Cubic Bézier splines.

Siggraph

BC(t) =

n

X

i=0

bi,n(t)Ci, t ∈ [0, 1]

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3D Vector Graphics

  • Subdivision Surfaces in 3D — Catmull-Clark Subdivision Surfaces.
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3D Vector Graphics

  • Subdivision Surfaces in 3D — Catmull-Clark Subdivision Surfaces.
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3D Vector Graphics

  • Inherently C2 everywhere except extraordinary vertices (C1)

normal control point

  • Subdivision Surfaces in 3D — Catmull-Clark Subdivision Surfaces.
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3D Vector Graphics

  • Inherently C2 everywhere except extraordinary vertices (C1)
  • Sharp creases can also be specified
  • Subdivision Surfaces in 3D — Catmull-Clark Subdivision Surfaces.

sharp crease

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Linear Blend Skinning(LBS)

v0 = X

j2H

wj(v)Tj ✓ v 1 ◆

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Linear Blend Skinning(LBS)

v0 = X

j2H

wj(v)Tj ✓ v 1 ◆ H

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Linear Blend Skinning(LBS)

v0 = X

j2H

wj(v)Tj ✓ v 1 ◆ wj H

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Linear Blend Skinning(LBS)

v0 = X

j2H

wj(v)Tj ✓ v 1 ◆ wj Tj H

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Linear Blend Skinning(LBS)

v0 = X

j2H

wj(v)Tj ✓ v 1 ◆ wj Tj H

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Linear Blend Skinning(LBS)

v0 = X

j2H

wj(v)Tj ✓ v 1 ◆

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Difficulty Skinning Cubic Bézier Curve

v0 = X

j2H

wj(v)Tj ✓ v 1 ◆

v = BC(t)

Apply skinning to

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Difficulty in Deforming A Cubic Bézier Curve

v0 = X

j2H

wj(v)Tj ✓ v 1 ◆

v = BC(t)

Apply skinning to

v

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Difficulty in Deforming A Cubic Bézier Curve

v = BC(t)

Apply skinning to

v0 = X

j2H

wj(v)Tj ✓ v 1 ◆

v

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Difficulty in Deforming A Cubic Bézier Curve

v = BC(t)

Apply skinning to

v0 = X

j2H

wj(v)Tj ✓ v 1 ◆

v

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Difficulty in Deforming A Cubic Bézier Curve

v = BC(t)

Apply skinning to

v0 = X

j2H

wj(v)Tj ✓ v 1 ◆

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Difficulty in Deforming A Cubic Bézier Curve

v = BC(t)

Apply skinning to

v0 = X

j2H

wj(v)Tj ✓ v 1 ◆

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Difficulty in Deforming A Cubic Bézier Curve

No longer a cubic Bézier curve!

v = BC(t)

Apply skinning to

v0 = X

j2H

wj(v)Tj ✓ v 1 ◆

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Difficulty in Deforming A Cubic Bézier Curve

Target(t)

No longer a cubic Bézier curve!

v = BC(t)

Apply skinning to

v0 = X

j2H

wj(v)Tj ✓ v 1 ◆

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How it works

  • Minimizing the L2 norm

Target(t)

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How it works

  • Minimizing the L2 norm

BC0(t)− Target(t)

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E(C0) = Z

L

kBC0(t) Target(t)k2 dt

How it works

  • Minimizing the L2 norm
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E(C0) = Z

L

kBC0(t) Target(t)k2 dt

How it works

  • Minimizing the L2 norm
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E(C0) = Z

L

kBC0(t) Target(t)k2 dt

How it works

  • Minimizing the L2 norm
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SA2014.SIGGRAPH.ORG SPONSORED BY

How it works

  • Minimizing the L2 norm

E(C0) = Z

L

kBC0(t) Target(t)k2 dt

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How it works

  • Minimizing the L2 norm

E(C0) = Z

L

kBC0(t) Target(t)k2 dt

C0 = X

j2H

Tj ˆ Wj ˆ A1

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How it works

  • Minimizing the L2 norm

Pre-computed

E(C0) = Z

L

kBC0(t) Target(t)k2 dt

C0 = X

j2H

Tj ˆ Wj ˆ A1

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SA2014.SIGGRAPH.ORG SPONSORED BY

How it works

  • Minimizing the L2 norm

Pre-computed

LBS: v0 =

X

j2H

wj(v)Tj ✓ v 1 ◆

E(C0) = Z

L

kBC0(t) Target(t)k2 dt

C0 = X

j2H

Tj ˆ Wj ˆ A1

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SA2014.SIGGRAPH.ORG SPONSORED BY

How it works

  • Minimizing the L2 norm

Pre-computed

LBS: v0 = X

j2H

Tjwj(v) ✓ v 1 ◆

E(C0) = Z

L

kBC0(t) Target(t)k2 dt

C0 = X

j2H

Tj ˆ Wj ˆ A1

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Our approach extends to 3D

E(C0) = Z

L

kBC0(t) Target(t)k2 dt

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Our approach extends to 3D

E(C0) = Z

D

kGC0(u, v) Target(u, v)k2 dudv

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3D Live Demo

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3D Results

Original Original Original Original

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Why 2D is more complicated?

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Why 2D is more complicated? In practice:

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Why 2D is more complicated? In practice:

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Why 2D is more complicated? In practice:

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Bézier Spline Junctions

Angle C0 C1 Original Deformed G1 flexible

(optional)

default

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Bézier Spline Junctions

Angle C0 C1 Original Deformed G1 flexible

(optional)

default

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Bézier Spline Junctions

Angle C0 C1 Original Deformed G1 flexible

(optional)

default

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Bézier Spline Junctions

Angle C0 C1 Original Deformed G1 flexible

(optional)

default

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2D Live Demo

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Applied on different weights

Harmonic Coordinates [Joshi et al. 2007] Shepard’s Weights [Shepard 1968] Bounded Biharmonic Weights [Jacobson et al. 2011]

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Performance

# faces pre-compute time(secs) seconds per update 4150 9 0.0002

in C++

# curves pre-compute time(secs) seconds per update 1324 3.2 0.03

in Python

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Limitation and Future Work

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Limitation and Future Work

  • Other primitives in vector graphics
  • circular arcs, NURBS, clothoid splines.
  • linear or radial gradients, diffusion curves, texture.
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Limitation and Future Work

  • Other primitives in vector graphics
  • circular arcs, NURBS, clothoid splines.
  • linear or radial gradients, diffusion curves, texture.

Additional Control Points Deformed Original

no control points

  • Adaptivity
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Limitation and Future Work

Additional Control Points Deformed Original

no control points

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Limitation and Future Work

[Schneider 1990]

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Limitation and Future Work

[Schneider 1990] [Our approach]

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Limitation and Future Work

Additional Control Points Deformed Original

no control points

  • Adaptivity

[Schneider 1990] [Our approach]

  • Other primitives in vector graphics
  • circular arcs, NURBS, clothoid splines.
  • linear or radial gradients, diffusion curves, texture.
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Limitation and Future Work

Additional Control Points Deformed Original

no control points

  • Adaptivity
  • Other applications
  • motion path planning

[Schneider 1990] [Our approach]

  • Other primitives in vector graphics
  • circular arcs, NURBS, clothoid splines.
  • linear or radial gradients, diffusion curves, texture.
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SA2014.SIGGRAPH.ORG SPONSORED BY

Thank You.

Songrun Liu, sliu11@gmu.edu Alec Jacobson, jacobson@cs.columbia.edu Yotam Gingold, ygingold@gmu.edu project webpage: http://cs.gmu.edu/~ygingold/splineskin/, code coming soon.

Acknowledgements: We thank support from NSF, Google, Intel, The Walt Disney Company, and Autodesk. Special thanks to Michelle Lee for her 2D artwork and Blender Foundation for 3D models from “Big Buck Bunny”.

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Junction between Curves (smooth) C1 continuity G1 flexibility

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Junction between Curves (smooth) C1 continuity G1 flexibility

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Junction between Curves (smooth) C1 continuity G1 flexibility

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Junction between Curves (sharp) No Angle Preservation Angle Preservation

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Junction between Curves (sharp) No Angle Preservation Angle Preservation

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2D Vector Graphics

BC(t) = CM¯ t = C     −1 3 −3 1 3 −6 3 −3 3 1         t3 t2 t 1     = Cmt,

  • Splines in 2D — Cubic Bézier splines.
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2D Vector Graphics

BC(t) = CM¯ t = C     −1 3 −3 1 3 −6 3 −3 3 1         t3 t2 t 1     = Cmt,

  • Splines in 2D — Cubic Bézier splines.
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2D Vector Graphics

BC(t) = CM¯ t = C     −1 3 −3 1 3 −6 3 −3 3 1         t3 t2 t 1     = Cmt,

  • Splines in 2D — Cubic Bézier splines.
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SA2014.SIGGRAPH.ORG SPONSORED BY

2D Vector Graphics

BC(t) = CM¯ t = C     −1 3 −3 1 3 −6 3 −3 3 1         t3 t2 t 1     = Cmt,

  • Splines in 2D — Cubic Bézier splines.
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How it works

E(C0) = Z

D

  • GC0(p) −

h

X

i=1

wi(GC(p))TiGC(p)

  • 2

dp

A(p) = mpmT

p

ˆ A = Z

D

A(p) dp C0 =

h

X

i=1

Ti ˆ Wi ˆ A1 ˆ Wi = C Z

D

wi(Cmp)A(p) dp

Pre-computed

  • Minimizing the L2 norm
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How it works

E(C0) = Z

D

  • GC0(p) −

h

X

i=1

wi(GC(p))TiGC(p)

  • 2

dp

A(p) = mpmT

p

ˆ A = Z

D

A(p) dp C0 =

h

X

i=1

Ti ˆ Wi ˆ A1 ˆ Wi = C Z

D

wi(Cmp)A(p) dp

Pre-computed

v0 = X

j2H

wj(v)Tj ✓ v 1 ◆

LBS:

  • Minimizing the L2 norm
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How it works

E(C0) = Z

D

  • GC0(p) −

h

X

i=1

wi(GC(p))TiGC(p)

  • 2

dp

C0 =

h

X

i=1

Ti ˆ Wi ˆ A1 ˆ Wi = C Z

D

wi(Cmp)A(p) dp ˆ A−1 =     16 −24 16 −4. −24 69 1

3

−57 1

3

16 16 −57 1

3

69 1

3

−24 −4 16 −24 16    

Pre-computed

  • Play the video of minimizing the L2 norm
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# control vertices # control faces precompute time (secs) seconds per update example pyramid 5 5 1e-3 6e-6 torus 32 32 8e-3 6e-6 butterfly 1216 1222 0.3 5e-5 squirrel 4307 4278 4 2e-4 rabbit 4139 4150 9 2e-4

3D Performance Dual-core, 2.4 GHz Intel Core i5, in C++.

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# control vertices # control faces precompute time (secs) seconds per update example pyramid 5 5 1e-3 6e-6 torus 32 32 8e-3 6e-6 butterfly 1216 1222 0.3 5e-5 squirrel 4307 4278 4 2e-4 rabbit 4139 4150 9 2e-4

3D Performance Dual-core, 2.4 GHz Intel Core i5, in C++.

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# G1 & angle constraints precomp. secs

seconds per update example

# curves # handles

regular flexible Boxes (Fig. 6) 4 2 4 0.03 2e-4 0.008 Spoon (Fig. 3) 7 2 7 0.08 2e-4 0.02 Clam (Fig. 1) 56 3 50 0.1 8e-4 0.1 Boy 226 5 118 0.7 5e-3 0.4 Zapfino 379 3 316 1.4 4e-3 0.65 Worm 395 2 261 8.1 3e-2 2.75 Penguin on Lion 400 9 168 2.5 1e-2 0.64 Man 516 4 240 1.3 1e-2 0.78 Seven Turtles 1324 5 914 3.2 3e-2 2.5 Octopus 1706 8 1181 11 2e-2 4.2 Coat of Arms 9496 4 8159 42 1e-1 16.1

2D Performance Dual-core, 2 GHz Intel Core i7, in Python.

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# G1 & angle constraints precomp. secs

seconds per update example

# curves # handles

regular flexible Boxes (Fig. 6) 4 2 4 0.03 2e-4 0.008 Spoon (Fig. 3) 7 2 7 0.08 2e-4 0.02 Clam (Fig. 1) 56 3 50 0.1 8e-4 0.1 Boy 226 5 118 0.7 5e-3 0.4 Zapfino 379 3 316 1.4 4e-3 0.65 Worm 395 2 261 8.1 3e-2 2.75 Penguin on Lion 400 9 168 2.5 1e-2 0.64 Man 516 4 240 1.3 1e-2 0.78 Seven Turtles 1324 5 914 3.2 3e-2 2.5 Octopus 1706 8 1181 11 2e-2 4.2 Coat of Arms 9496 4 8159 42 1e-1 16.1

2D Performance Dual-core, 2 GHz Intel Core i7, in Python.

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# G1 & angle constraints precomp. secs

seconds per update example

# curves # handles

regular flexible Boxes (Fig. 6) 4 2 4 0.03 2e-4 0.008 Spoon (Fig. 3) 7 2 7 0.08 2e-4 0.02 Clam (Fig. 1) 56 3 50 0.1 8e-4 0.1 Boy 226 5 118 0.7 5e-3 0.4 Zapfino 379 3 316 1.4 4e-3 0.65 Worm 395 2 261 8.1 3e-2 2.75 Penguin on Lion 400 9 168 2.5 1e-2 0.64 Man 516 4 240 1.3 1e-2 0.78 Seven Turtles 1324 5 914 3.2 3e-2 2.5 Octopus 1706 8 1181 11 2e-2 4.2 Coat of Arms 9496 4 8159 42 1e-1 16.1

2D Performance Dual-core, 2 GHz Intel Core i7, in Python.