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Controlled-Distortion Constrained Global Parametrization

Ashish Myles

Denis Zorin

New York University

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Global parametrizationMap surface to plane

Quadrangulation, remeshing Geometry images Smooth surface fitting Texture/bump/displacement

mapping

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Goal: singularity placement

singularities(valence ≠ 4)

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Seamless global parametrizationRequirements

Feature alignment Map feature curves

to axis-aligned curves

Seamlessness All curvatures

multiples of π/2

Parametrize Quadrangulate

cone singularities(quad valence 3)

π/2 π/2

Integrated curvature(angle deficit) π/2

Seam-constraints

Feature linegrid-aligned

0 curvature (flat)

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Cones for features and distortion

Distortion induced by feature-alignment

Cones required toalign to features

π/2

0 – π/2

π/2

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Distortion vs # conesProposition 5. For a closed smooth surface with bounded total curvature, and any ϵ > 0, there is a seamless parametrization which has total metric distortion EARAP < ϵ.

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Goal: automatic cone placementConstraints: (1) seamless (2) feature-aligned

Automatically determine cone locations + curvatures (mixed integer problem)

Trade off parametric distortion with number of cones.

No conesHigh distortion

Bad

2 conesLow distortion

Good

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Cross-field smoothing

Easy feature alignmentNo direct distortion control

Conformal maps

Easy to measure distortionOver-constrained for multiple features

Contribution: Unification of techniques

scale by eϕ

cross-field rotation ∗ω = dϕ + ∗h closest conformal map

ϕ2

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Smooth cross-field not distortion-aware

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Smooth cross-field not distortion-aware

parametricdomain collapsed region

To avoid collapse

23 cones

31 cones

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Seamless + feature-alignment constraints

1. Internal curvature –∇2ϕ = Knew – Korig

2. Feature line curvature ∇nϕ = κnew – κorig

3. Relative alignment constraints

Conformal maps cannot handle condition 3.

But cross-fields can.

Conformal maps: over-constrained

π/2

0

π/2 boundary curvature

0 boundarycurvature

Surface alreadysatisfies 1 and 2.

Completelydetermine ϕ

π/2

– π/2

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Relation of conformal to cross-fields

Simple 2D case∇θ⊥ = ∇ϕ

(∇θ computes field rotation)

Conformal map

scale by eϕ

rotate by eiθ

3D surfaces (exterior calculus) ∗ω = dϕ

(Connection form ω ∼ ∇θ) [Crane et al. 2010]

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Conformal + Cross-field

Interpretation of harmonic part: Not seamless! Distortion ||ϕ||2 only

accurate if ||h||2 low

Need to minimize ||h||2

cross-fieldrotation

closestconformal map

harmonic part

Constant scale across

∗ω = dϕ + ∗h

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System setup and solveCompute ∗ω = dϕ + ∗h (determines cross-field)

Minimize energy ||ϕ||2 + α ||ω||2

1. Satisfy linear constraints for seamlessness and alignment

2. Minimize ||h||2 (gives linear equations)

Mixed Integer problem Integer variables (curvature, relative-alignment) Solve via iterative rounding (similar to [Bommes 2009])

Linearconstraints

Quadratic energy(cross-field smoothness)(distortion)

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α = 0.001

90 conesα = 0.01

66 cones

α = 0.1

47 cones

α = 1.0

44 cones

α = 100

32 conesno harmonic minimization

Field smoothing[Bommes 2009]

32 cones

0.0

0.6

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Comparison to cross-field smoothing

Field smoothing[Bommes 2009]

Ours 0.0

0.6

50 cones

72 cones

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Comparison to cross-field smoothing

0.0

0.6Field smoothing[Bommes 2009]

Ours

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Comparison to cross-field smoothing

0.0

0.6

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Field smoothing[Bommes 2009]

Ours

Comparison to cross-field smoothing

To avoid collapse

0.0

0.623 cones

31 cones

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Comparison to cross-field smoothing

0.0

0.8

30 cones

42 cones

Field smoothing[Bommes 2009]

Ours

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SummaryUnification via ∗ω = dϕ + ∗h of

1) cross-field smoothing and2) conformal approach

Support both1) feature alignment and2) distortion control

Trade-off between1) distortion and2) # cones

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Questions?