Parameterization II - Computer Graphics Laboratory

Parameterization II
Some slides from the Mesh Parameterization Course from Siggraph Asia 2008
1

Non-Convex Non Convex Boundary
• Convex boundary creates
significant distortion
• “Free” boundary is better
2

Fixed vs Free Boundary
3

Fixed vs Free Boundary
4

Cut graph
7

Free Boundary Methods - Tactics
• Solve for the (u,v) coordinates using the discrete first
fundamental form
– MIPS [Hormann et al., 2000]
– Stretch optimization [Sander et al., 2001]
– LSCM (conformal, linear) [Levy et al., 2002]
– DCP ( (conformal, f l li linear) ) [Desbrun et al., 2002]
• Solve for the angles of the map (conformal)
– ABF [Sheffer et al., 2001], ABF++ [Sheffer et al., 2004]
– LinABF (linear) [Zayer et al., al 2007]
9

Free Boundary Methods - Tactics
• Solve for the edge g lengths g of the map pby yp prescribing g
curvature
– Circle patterns [Kharevych et al., 2006]
– CPMS (linear) [ [Ben-Chen h et al., l 2008] ]
– CETM [Springborn et al., 2008]
• Balance area/conformality
– ARAP [Liu et al., 2008]
• More…
10

Back to the First Fundamental Form
11

Linear Map Surgery
• Singular g Value Decomposition p (SVD) ( ) of
with rotations and
and scale factors ( (singular values) )
12

Notion of Distortion
• isometric or length-preserving
length preserving
• conformal or angle-preserving
• equiareal q or area-preserving p g
• everything defined pointwise on
13

Computing the Stretch Factors
• first fundamental form
• eigenvalues of
• singular values of
and
14

Measuring Distortion
• local distortion measure
• has minimum at
– isometric measure
– conformal measure
• overall distortion
15

Piecewise Linear Parameterizations
• piecewise linear atomic maps
• distortion constant per triangle
• overall distortion
16

Distortion Based Methods
• Define energy functional F as a function of J Jp, I Ip, σ1,σ2 • Expand their expression in F in function of the
unknown u uii, v i
• Design an algorithm to find the u uii,vv i's i s that
minimizes F
17

Linear Methods
• the terms and are quadratic
in the parameter points
• Dirichlet energy
• Conformal energy
• minimization yields linear problem
[Pinkall & Polthier 1993]
[Eck et al. 1995]
[Lévy et al. 2002]
[Desbrun [ et al. 2002] ]
18

Linear Methods
• both result in barycentric mappings with
discrete harmonic weights for interior vertices
• Dirichlet maps require to fix all boundary vertices
• Conformal maps only two
– result depends on this choice
– best choice → [Mullen et al. 2008]
• both maps not necessarily bijective
19

Non-linear Non linear Methods
• MIPS energy [Hormann &Greiner2000]
& Greiner 2000]
• Area-preserving MIPS [Degener et al. 2003]
20

Non-linear Non linear Methods
• Green-Lagrange Green Lagrange deformation tensor
[Maillot et al al. 1993]
• Stretch energies g ( , , and symmetric y stretch) )
[Sander et al. 2001]
[Sorkine et al. 2002]
21

Conformal Parameterization - LSCM
Cauchy Riemann equations:
No Piecewise Linear solution in general
v
u
⎧∂ ⎪ v ∂u
⎪
=−
⎪ ∂ x ∂ y
⎨
⎪∂ ⎪ v ∂u
⎪ =
⎪∂ ⎪⎩
∂y ∂ ∂x
22

Conformal Parameterization – LSCM
Minimize
∑
T
[Levy et al., 2002]
⎛ ∂ ∂v ⎞ ⎛ ∂u
⎞
⎟
∂
⎟
⎜ ⎟ ⎜−⎟
⎜ x⎟ ⎜
y⎟
⎜
∂ ⎟ ⎜ ∂ ⎟
⎜ ⎟−⎜ ⎟
⎜∂v ⎟
⎜ ⎟
⎜ ⎟ ⎜ ∂ u ⎟
⎜ ⎜
⎜
⎟ ⎟
y ⎟ ⎜
⎝∂⎠ ⎝⎜ ∂x
⎠⎟
2
Fix two vertices to
determine rot rot,transl,scaling
transl scaling
23

Conformal Parameterization - LSCM
Equivalent formulation formulation, uses only edge lengths
ratios and angles
For each triangle: p1 p2 p − p = R
α
p −p
2
1 3 2 3
l l1
( ) ( ) l
l 2 α
p 3
l 1
24

x 1
i
Conformal Parameterization – DCP
γi
γ
βi
0
x 0
α
[D [Desbrun b et t al., l 2002]
β
x 2
For one triangle:
90
γ ( x − x ) + β(
x − x ) = R ( x −x
)
cot cot
2 0 1 0 1 2
For complete one-ring of triangles
(interior vertex):
n
(cot γ + cot β ) x − x = 0
∑
i=
1
i i i
( )
For incomplete one-ring of triangles
(boundary vertex):
n
90
i
∑ γγi + ββi
( x xi − x 0 ) = R x xi − x
1 0
∑
1 i2
i=
1
i2
(cot cot ) ( )
0
25

Isotropic Parameterizations
Conformal = Harmonic
26

Angle Based Flattening (ABF)
• Fact: Triangular 2D mesh is defined
by its angles (up to similarity)
• Define problem in angle space
• Angle based formulation:
– Di Distortion i as ffunction i of f angles l
– Validity - set of angle constraints
27

Constrained Constrained Minimization
• Notation: β are (given) 3D angles angles. α
i
i are (unknown)
2D angles.
• Objective: minimize (relative) deviation of angles:
3T
∑
1
D ( α ) = ( α − β )
i i i
i =
2
28

Constraints
• All angles are positive (linear inequalities)
inequalities).
• Sum of angles in each triangle is π (linear
equalities).
• Sum of angles around each vertex is 2π
(linear equalities).
• All one-rings close properly (non-linear
equalities).
29

Solving
• Use non-linear solver (Lagrange multipliers,
NNewton t method) th d) tto solve l ffor
αi • Use LSCM to embed in plane based on α i
30

Examples
LSCM
ABF
31

Examples
LSCM
ABF
32

Linear Methods
uniform harmonic mean value conformal
33

Linear Methods
mean value conformal
34

Non-Linear Non Linear Methods
ABF++ circle patterns p
MIPS
stretch
35

Non-Linear Non Linear Methods
ABF++ circle patterns MIPS
stretch
36

LSCM [2002]
ABF++ [2005]
LinABF [2007]
(and many more…)
Circle Patterns [2006]
Ricci Flow [2007]
CPMS, CETM [2008]
Curvature Prescription
11. Cut C t mesh h to t disk di k
2. Compute new angles/lengths
3. Embed in plane
� Discontinuities in scale

LSCM [2002]
ABF++ [2005]
LinABF [2007]
(and many more…)
Circle Patterns [2006]
Ricci Flow [2007]
CPMS, CETM [2008]
Curvature Prescription
11. Compute new angles/lengths
2. Cut Mesh to disk
3. Embed in plane
� No discontinuities in scale

Continuous definition
Scale dependent p
Gaussian Curvature
Gauss-Bonnet theorem: ∫ κ = 2πχ
Discrete definition: κ(v) = 2π – Σα i
Scale independent !
Gauss-Bonnet theorem: ∑
κ = 2πχ
v
α i
κ=π/2
κ ≡1/r 2

v
α i
Discrete Metric
Triangle edge lengths (a,b,c)
are the discrete metric
The metric defines the angles by the cosine law
The angles define the Gaussian curvature

CPMS [2008]
Original edge lengths (metric)
Conformal factor (per vertex)
Extend φ to edge
φ () t = φ t +φ
(1 − t )
2 3
Integrate e φ on edge
l l e φ �
Scale to get new edge lengths = ∫
1 1
( t )

What Happens to the Curvature?
• Continuous conformal map
Curvature
after map
2 φ
2
e κ� = κ − ∇ φ
Curvature
before map
Conformal
factor

What Happens to the Curvature?
• Continuous conformal mapp
2φ2 e κ� = κ −∇ φ
• Discrete conformal map (for small changes)
κ� ≈ κ −∇
• Where did the exponent go?
– Discrete curvature doesn’t scale!
2 φ

The Flattening Algorithm for a Topological Disk
• Set target g curvature to 0 on interior vertices
• Set φ to 0 on the boundary
• Find φ on interior vertices by solving
• Scale edge lengths to get conformal metric
• Embed new edge lengths using LSCM
∇
2
φ = κ� − κ

Some Results

Comparison with Non-Linear Non Linear Methods
ABF++ circle patterns MIPS
stretch CPMS
46

Reducing Area Distortion
• Gather Gaussian curvature into “cone cone points” points
– Target curvature not 0 everywhere
• First compute new edge lengths, then cut through
cone points
• No discontinuities in scale
47

Conformal Equivalence of Triangle Meshes [2008]
52

Conformal Equivalence of Triangle Meshes [2008]
• Given input mesh and target curvatures curvatures, find u st s.t.
– New mesh is conformally equivalent to source mesh
– Target curvatures are achieved
• Mi Minimize i i convex energy �� global l b l minimum i i
• First optimization step equivalent to CPMS
53

Conformal Equivalence of Triangle Meshes [2008]
54

Parameterization - Conclusions
• Many a yMANY methods et ods out tthere e e
– Fixed / free boundary
– Bijective / non bijective
– Conformal / area preserving
– Linear / non linear
– First cut then embed / cone points
• Best method depends on mesh
– Close to disk � linear methods can work well
– Require large distortion �� prefer non-linear methods
58