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

Surface Parameterization A Tutorial and Survey Michael Floater and Kai Hormann Presented by Afra Zomorodian CS 468 – 10/19/5 1

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

History • Ptolemy (100-168) • Preserve: – angles – area • Loxodrome: constant bearing Orthographic Stereographic Mercator Lambert (1512-1594) (1728-1777) Afra Zomorodian 3

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 Afra Zomorodian 4

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

What is I? • I is: – symmetric (3 DOF) 2 > 0 – g = det I = g 11 g 22 – g 12 – positive definite Afra Zomorodian 6

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

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

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

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

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

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

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

Equiareal Maps • Conformal maps are almost unique scalar • 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) Afra Zomorodian 14

Outline � Continuous • Discrete – Harmonic – Conformal – Equiareal Afra Zomorodian 15

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

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

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

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

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

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) = 2 π • Optimal angles β i = φ i s(v) • Minimize • Non-linear constraints Afra Zomorodian 21

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

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