Surface Parameterization

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1 Surface Parameterization A Tutorial and Survey Michael Floater and Kai Hormann Presented by Afra Zomorodian CS /19/5 1

2 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 Afra Zomorodian 2

3 History Ptolemy ( ) Preserve: angles area Loxodrome: constant bearing Orthographic Stereographic Mercator Lambert ( ) ( ) Afra Zomorodian 3

4 Theory Gauß ( ) differential calculus differential geometry Riemann ( ) 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 Afra Zomorodian 4

5 Maps Parameterized surface S 3 Regular: x i are smooth (C ) partials are linearly independent First fundamental form Afra Zomorodian 5

6 What is I? I is: symmetric (3 DOF) g = det I = g 11 g 22 g 122 > 0 positive definite Afra Zomorodian 6

7 Types of Mappings Isometric: preserves lengths developable surfaces Conformal: preserves angles Stereographic and Mercator projections Equiareal: preserves area Lambert projection scalar (Theorem) isometric conformal + equiareal Afra Zomorodian 7

8 Planar Mappings f : 2 2 f(x, y) = (u(x,y), v(x,y)) I = J T J, J is Jacobian of f Singular values of J σ 1, σ 2 are square roots of eigenvalues λ 1, λ 2 of I Afra Zomorodian 8

9 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 Afra Zomorodian 9

10 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) Afra Zomorodian 10

11 A Complex View 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 Afra Zomorodian 11

12 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 Afra Zomorodian 12

13 Minimizing Distortion No guarantee on angles Harmonic maps minimize Dirichlet Energy: For surface S 3, Generalized Laplace Laplace-Beltrami operator Afra Zomorodian 13

14 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) scalar Lots and lots of equiareal maps (badly behaved, too) Afra Zomorodian 14

15 Outline Continuous Discrete Harmonic Conformal Equiareal Afra Zomorodian 15

16 Discrete Surfaces Surface S 3 PL surface S T = {T 1,, T M } Polygonal domain S * 2 PL map f : S T S *, f linear on each T i Uniquely determined by image of vertices Afra Zomorodian 16

17 Discrete Harmonic Maps Finite Element Method fix boundary somehow minimize Dirichlet energy for internal vertices Quadratic minimization (Nice) Linear System Afra Zomorodian 17

18 Convex Combination Maps Discrete harmonic maps tend to harmonic maps 1-1: non-degenerate triangles that are not flipped Normalized weights Linear system If w ij positive, so are λ ij Convex Combination Maps (Theorems) Discrete harmonic maps are 1-1: Barycentric Maps (1/d i ) [Tutte, 3-connected graphs] Any weights such that j Ni λ ij = 1 If opposite angles sum < π (eg Delaunay) Afra Zomorodian 18

19 Discrete Conformal Maps 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) Afra Zomorodian 19

20 Computing MIPS Relationship between E M and E D Non-linear minimization MIPS energy on degenerate triangles on (K i ) (star-shaped neighborhood of v i ) Local functional is convex, so use Newton s method Afra Zomorodian 20

21 Angle-Based Flattening φ i : angles in S T α i : angles in S * φ(v): sum of φ i around vertex v For interior vertices, α(v) = 2π Optimal angles β i = φ i s(v) Minimize Non-linear constraints Afra Zomorodian 21

22 Discrete Equiareal Maps Minimize energy functional? Badly behaved Multiple minima with E(f) = 0 Minimize mixture of energies Afra Zomorodian 22

23 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! Afra Zomorodian 23

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