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

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

8

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)

9

Example

5

1

0,00,0

3,02,01,05,04,0

j

jaa

aaaaa

0 1

3 2

4 5

7 6

3

3

2

2

1

1

5

5

4

4

0

0

3

33,0

2

22,0

1

11,0

5

55,0

4

44,0

0

00,0

0,0

3,0

0,0

2,0

0,0

1,0

0,0

5,0

0,0

4,0

0

v

u

v

u

v

u

v

u

v

u

v

u

v

ua

v

ua

v

ua

v

ua

v

ua

v

ua

a

a

a

a

a

a

a

a

a

a

10

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

uauauauauauauaua

uauauauauauauaua

uauauauauauauaua

uauauauauauauaua

77,344,3

77,266,255,2

55,1

55,044,0

7

6

5

4

7,34,3

7,26,25,2

5,1

5,04,0

3

2

1

0

3,32,30,3

3,22,21,20,2

2,11,10,1

3,02,01,00,0

00

0

000

00

0

0

uaua

uauaua

ua

uaua

u

u

u

u

aa

aaa

a

aa

u

u

u

u

aaa

aaaa

aaa

aaaa

ij

jiii aa ,, 7,6,5,4:Boundary3,2,1,0:Internal

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"). 

16

[Differential Geometry Primer]

• Gaussian curvature

17

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

18

DETAILS

19

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

23

2

1

3

0

1

3

0

2

3

2

0

3

2

0

2

2

1

0

2

1

0

u

u

u

RM

v

v

v

MA

u

u

u

RM

v

v

v

MA

u

u

u

RM

v

v

v

MAE cccbbbaaaLSCM

0

3

2

1

0

3

2

1

0

v

v

v

v

u

u

u

u

-AaRMa AaMa

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

0

2

1

2

1

3

0

3

0

v

v

u

u

v

v

u

u

Variable change columns swap

0

1

3

2

24

2

0

1

a

0

2 3

b

0

3 1

c

0

2

1

2

1

3

0

3

0

v

v

u

u

v

v

u

u

Af Al

2

1

2

1

3

0

3

0

v

v

u

u

v

v

u

u

Af Al

25

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.

26

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)

28

Wheel Consistency

2

1

3

1

2

3

1sin

sin

sin

sin

sin

sin

sinsinsinsinsinsin

0sinsinsinsinsinsin

3

3

2

2

1

1

321321

321321

a b

c

1sin

sin

sin

sin

sin

sin

sin

sin

sinsin:law Sine

3

3

2

2

1

1

1

1

11

c

a

b

c

a

b

a

b

ba

29

1963

1995

2003

2002

30

Other Issues

Segmentation and atlas

32

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)

33

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

35

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

1

36

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

37

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

38

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

uauauauauauauaua

uauauauauauauaua

uauauauauauauaua

uauauauauauauaua

0

7

6

5

4

3

2

1

0

u

u

u

u

u

u

u

u

Ai Ab

7

6

5

4

3

2

1

0

u

u

u

u

u

u

u

u

Ai Ab

39BACK

[From Siggraph Course 2007]

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

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

40

kkjjii

kkjjii

vvvYXv

uuuYXu

),(

),(

MT solely depends on the geometry of the triangle T

u=

41

k

j

i

kji

kji

k

j

i

kjikkjjii

u

u

u

TTuuu

X

u

Y

X

TuuuuuuuuuYXu

2

1

2

1

12

1),(

Similarly, we can get Y

u

BACK