Surface Parameterization A Tutorial and Survey Michael Floater and - - PowerPoint PPT Presentation

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Michael Floater and Kai Hormann Presented by Afra Zomorodian CS 468 – 10/19/5

Surface Parameterization

A Tutorial and Survey

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Problem

• 1-1 mapping from domain to surface
• Original application:

Texture mapping

– Images have a natural parameterization – Goal: map onto surfaces

• Geometry processing

– Approximation – Remeshing – Data fitting

• Input: Piecewise (PL) triangular meshes

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History

• Ptolemy (100-168)
• Preserve:

– angles – area

• Loxodrome: constant bearing

Orthographic Mercator (1512-1594) Stereographic Lambert (1728-1777)

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Theory

• Gauß (1777-1855)

– differential calculus – differential geometry

• Riemann (1826-1866)
• Riemann Mapping Theorem:

– Input: any simply-connected region of complex plane – Output: any other simply-connected region of complex plane – Statement: there exists a map that preserves angle

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Maps

• Parameterized surface S 3
• Regular:

– xi are smooth (C) – partials are linearly independent

• First fundamental form

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What is I?

• I is:

– symmetric (3 DOF) – g = det I = g11 g22 – g12

2 > 0

– positive definite

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Types of Mappings

• Isometric:

– preserves lengths – developable surfaces

• Conformal:

– preserves angles – Stereographic and Mercator projections

• Equiareal:

– preserves area – Lambert projection

• (Theorem) isometric conformal + equiareal

scalar

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Planar Mappings

• f : 2 2
• f(x, y) = (u(x,y), v(x,y))
• I = JTJ, J is Jacobian of f
• Singular values of J σ1, σ2 are square roots of

eigenvalues λ1, λ2 of I

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The Game

• Try to find

– isometric maps (only developable surfaces) – conformal maps: no distortion in angles – equiareal maps: no distortion in area

• Impossible?

– try to minimize distortion (maybe a mixture) – define some sort of energy function (sometimes implicit) – minimum is your answer – easy to compute

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Outline

• Continuous

– Conformal – Harmonic (Distortion minimizing) – Equiareal

• Discrete

× Not:

× Mean Value Coordinates (6.3) × Boundary mapping (6.4) × Linear methods (7.3) × Closed surfaces (9)

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• View as Conformal map function of complex variable

ω = f(z)

• Locally, analytic in neighborhood of z, f (z) ≠ 0
• z = x + iy, w = u + iv
• Conformal maps satisfy

Cauchy-Riemann equations:

• Laplace equations:
• Laplace operator:
• Harmonic Maps satisfy Laplace equations
• isometric conformal harmonic

A Complex View

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Harmonic Maps

• RKC Theorem:

– f harmonic, S* 2 (convex) – f maps ∂ S homeomorphically to ∂ S* f is 1-1

• Just map boundary!
• Approximate PDEs
• Inverse not harmonic
• harmonic conformal

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Minimizing Distortion

• No guarantee on angles
• Harmonic maps minimize Dirichlet Energy:
• For surface S 3, Generalized Laplace
• Laplace-Beltrami operator

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Equiareal Maps

• Conformal maps are almost unique
• Example: map unit disk onto self

– Choose z S, angle φ – By Riemann mapping theorem, unique f : S S, f(z) = 0 and arg f’(z) = φ – 3 degrees of freedom: complex number z (2) and φ (1)

• Lots and lots of equiareal maps (badly behaved, too)

scalar

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Outline

Continuous

• Discrete

– Harmonic – Conformal – Equiareal

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Discrete Surfaces

• Surface S 3
• PL surface ST = {T1, …, TM}
• Polygonal domain S* 2
• PL map f : ST S*, f linear on each Ti
• Uniquely determined by image of vertices

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Discrete Harmonic Maps

• Finite Element Method

– fix boundary somehow – minimize Dirichlet energy for internal vertices

• Quadratic minimization
• (Nice) Linear System

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• Discrete harmonic maps tend to harmonic maps
• 1-1: non-degenerate triangles that are not flipped
• Normalized weights
• Linear system
• If wij positive, so are λij
• Convex Combination Maps
• (Theorems) Discrete harmonic maps are 1-1:

– Barycentric Maps (1/di) [Tutte, 3-connected graphs] – Any weights such that j Ni λij = 1 – If opposite angles sum < π (eg Delaunay)

Convex Combination Maps

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• Unlike harmonic, ∂ does not to be fixed
• Condition number of Jacobean:
• Discretizing:
• Minimum is 2 number of triangles
• For PL: conformal isometric (developable)
• Most Isometric Parametrizations (MIPS)

Discrete Conformal Maps

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• Relationship between EM and ED
• Non-linear minimization
• MIPS energy

– on degenerate triangles – on ∂(Ki) (star-shaped neighborhood of vi)

• Local functional is convex, so use Newton’s method

Computing MIPS

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Angle-Based Flattening

• φi: angles in ST
• αi: angles in S*
• φ(v): sum of φi around vertex v
• For interior vertices, α(v) = 2π
• Optimal angles βi = φis(v)
• Minimize
• Non-linear constraints

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Discrete Equiareal Maps

• Minimize energy functional?
• Badly behaved
• Multiple minima with E(f) = 0
• Minimize mixture of energies

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Take Home

• Isometric: only developable
• Conformal: preserve angles
• Harmonic: minimize Dirichlet energy
• Equiareal: preserve area, but promiscuous
• Angles mean we deal with tangents (partials)
• Area means we deal with Jacobian
• In practice, find right energy to minimize
• Theory is good!