Parameterization

Introduction

• The goal of parameterization is to attach a coordinate system to the object– In particular, assign (2D) texture coordinates to the

3D vertices

• One application of mesh parameterization is texture mapping

2

UVMapper in Blender

3

Introduction (cont)

• Another class of application concerns remeshing algorithm– converting from a mesh representation into an

alternative one

4

Introduction (cont)

• In summary, constructing a parameterization of a triangulated surface means finding a set of coordinates (ui,vi) associated to each vertex i.

• Moreover, the parameter space does not self-intersect.

5Self-intersect

Mappings in the PMP Book

• 5.3 Barycentric mappingTutte-Floater

Discrete Laplacian (Laplace-Beltrami)

• 5.4 Conformal mapping5.4.2 Least square conformal maps

5.4.4 (Geometric-based) ABF (angle-based flattening)

• [5.5 Distortion analysis based methods]

6

Barycentric Mapping

• One of the most widely used

• Based on Tutte’s barycentric mapping theorem from graph theory [Tutte60]

7

WikipediaConvex combination

Barycentric coordinate system

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Barycentric Mapping (cont)

• Fixing the vertices of the boundary on a convex polygon. The coordinates at the internal vertices are found by solving the equation.

One simple way:(without considering mesh geometry)

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Example

5

1

0,00,0

3,02,01,05,04,0

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77,366,355,344,333,322,311,300,3

77,266,255,244,233,222,211,200,2

77,166,155,144,133,122,111,100,1

77,066,055,044,033,022,011,000,0

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Ax = b, Solved by Gauss Seidel

11DETAILS

Discrete harmonic coordinates [Eck et al 1995]

Mean value coordinates [Floater 2003]

Other Alternatives

12

Mean Value Coordinate

13

For some surfaces, fixing the boundary on a convex polygon may be problematic

“Free” boundary 14

Conformal Mapping

• Iso-u and iso-v lines are orthogonal• Minimize mesh distorsion

A conformal parameterization transforms a small circle into a small circle. It is locally a similarity transform.

vuXzyx T ,,,

15

Types of Distortion

• L stretch, L shear, AngD– Isometric (length-preserving)– Conformal (angle-preserving)– Equi-areal (area-preserving)

• Global isometric paramterization only exists for developable surfaces, with vanishing Gaussian curvature K(p) = 0 at all surface points

a developable surface is a surface with zero Gaussian curvature. That is, it is a "surface" that can be flattened onto a plane without distortion (i.e. "stretching" or "compressing"). 

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[Differential Geometry Primer]

• Gaussian curvature

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Gradient in a Triangle

Study the inverse of parameterization: maps (X,Y) of the triangle to a point (u,v)

u intersects the iso-u lines; v intersects the iso-v lines

X, Y: orthonormal basis of the trianglexi, xj, xk: vertex coordinates in the XY basis

Conformality condition

vu

iso-u lines iso-v lines

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DETAILS

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Least Square Conformal Map

Only developable surfaces admit a conformal paramterization.For general (non-developable) surface, LSCM minimizes an energy ELSCM that corresponds to the non-conformality of the application

ELSCM is invariant to translation and rotation in the parametric space. To have a unique minimizer, it is required to fixed at least two vertices.

Mimizing a quadratic form

From [Levy et al.2002], if the pinned vertices are chosen on the boundary, all the triangles are consistently oriented (no flips). 20

Quadratic Optimization

Quadratic form: a polynomial function that the degree is not larger than two.

G is symmetric

Minimizer occurs at its stationary point:

21

Least Square

(m > n)

Minimize the sum of residuals:

F is a quadratic form

Minimizer found at stationary point

22

Least Square with Reduced D.O.F.

nff xxx 1Free parameters

nnfl xxx 1Lock parameters

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Fixed vertices 1 & 2(u1,v1) & (u2,v2) locked

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Variable change columns swap

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ELSCM is invariant to translation and rotation in the parametric space. To have a unique minimizer, it is required to fixed at least two vertices.

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27

Angle-Based Flattening (ABF)[Sheffer & de Sturler 2000]

Finding (ui,vi) coordinates, in terms of angles, Stable (ui,vi) to i conversion

Constraints:Constrained quadratic optimization with equality constraintsNonlinear optimization

(wheel consistency)

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Wheel Consistency

2

1

3

1

2

3

1sin

sin

sin

sin

sin

sin

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3

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1963

1995

2003

2002

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Other Issues

Segmentation and atlas

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Model Segmentation

• Planar parameterization is only applicable to surfaces with disk topology

• Closed surfaces and surfaces with genus greater than zero have to be cut prior to planar parameterization

• Cut to reduce complexity (to reduce distortion)• Cut introduce cross-cut discontinuities• Segmentation technique (partition the surface into

multiple charts) and seam generaton technique (introduce cuts into the surface but keep it as a single chart)

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Numerical Optimization

From “Mesh Parameterization,

Theory and Practice”,

Siggraph 2007 Coursenote

Sparse Linear System

SOR (successive over-relaxation)• Simplest, both from the conceptual and the

implementation points of view

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SOR (cont)New Update Scheme: Successive Over-Relaxation

1

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Other Methods

• Conjugate gradient method• Sparse direct solvers (LU)

Summary• SOR-like methods are easy to understand and

implement, but do not perform well for more than 10K variables

• Direct methods are most efficient, but consume considerable amounts of memory

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References

• “Polygon Mesh Processing”, Mario Botsch, Leif Kobbelt, Mark Pauly, Pierre Alliez and Bruno Levy, AK Peters, 2010

• “Mesh Parameterization: Theory and Practice”, Kai Hormann, Bruno Lévy and Alla Sheffer, ACM SIGGRAPH Course Notes, 2007

• “Least Squares Conformal Maps for Automatic Texture Atlas Generation”, Bruno Lévy, Sylvain Petitjean, Nicolas Ray and Jérome Maillot, ACM SIGGRAPH conference proceedings, 2002

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0

0

0

0

77,366,355,344,333,322,311,300,3

77,266,255,244,233,222,211,200,2

77,166,155,144,133,122,111,100,1

77,066,055,044,033,022,011,000,0

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39BACK

[From Siggraph Course 2007]

Study inverse of parameterization (X,Y) (u,v)

(i,j,k) barycentric coordinates, computed as:

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kkjjii

kkjjii

vvvYXv

uuuYXu

),(

),(

MT solely depends on the geometry of the triangle T

u=

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k

j

i

kji

kji

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u

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X

u

Y

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1

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1),(

Similarly, we can get Y

u

BACK