Mesh Parameterization: Theory and Practice Mesh Parameterization ...

Mesh Parameterization: 

Theory and Practice 

Non‐Linear Methods 

Alla Sheffer

Direct 

Planar, Non‐Linear Methods 

MIPS 

stretch minimization 

circle patterns 

ABF ++ 

Indirect 

Mesh Parameterization: Theory and Practice 

Non‐Linear Methods

Direct Methods 

• Define (non‐linear) distortion metric to minimize 

– Function of & 

– On mesh constant per triangle (more later) 

– Examples 

• Conformal –MIPS[Hormann & Greiner 2000] 

• Stretch [Sander et al. 2001, Sorkine et al. 2002] 

– ( , , and symmetric stretch) 

• Area preserving [Degener et al. 2003] 

• Solve resulting optimization problem 



Direct Methods: Solution Mechanism 

• Challenge ‐ Metric complexity 

– Convergence 

– Speed 

• Solution Mechanism 

– Start from linear solution as initial guess 

– “Gauss‐Seidel” solver 

• Move one vertex at a time 

– Explicitly check/prevent for flips 

– Use hierarchical solution 



Hierarchical Solution 

When adding vertices 

“improve” local 

neighbourhood only 



MIPS [Hormann & Greiner 2000] 

• Measure conformality ( ): 

• Scale independent 

– in contrast to linear conformal formulations 


Non‐Linear Methods 

Linear (LSCM) [Levy’02]

Stretch Minimization 

[Sander et al,2001] 

• Measure “averaged” per triangle stretch 

• Balances stretch 

– Increase one σ to decrease another 

MIPS 

Stretch Minimizing 



So what is f ? 

• Barycentric coordinates per triangle 

2D 

p 

p2 

p3 

f(p) 

= 

p 1 

q 1 

f 

q 2 

f(p) 

( p , p 

) 

2 

, p3 

q1 

+ p, 

p3, 

p1 

q2 

+ p, 

p1 

, p2 

q3 

p1 

, p2, 

p3 

3D 

q 3 

• Obtain J f by explicit derivation 



Alternative Variables 

• Alternative: 

– Use parameters which define 2D mesh uniquely 

– Search in alternative parameter space & convert to UV 

– Enforce constraints defining 2D mesh in parameter space 

• Examples 

– 2D mesh angles [Sheffer & de Sturler:00; Kharevych:06] 

– Gradients [Gu & Yau:03; Ray:06] 

– Angle deficit [Gotsman:07] 



Angle Space: ABF [Sheffer & de Sturler:00] 

• Triangular 2D mesh is defined by its angles 

• Formulate parameterization as problem in angle space 

[Sheffer & de Sturler,00] 

• Angle based flattening (ABF): 

– Distortion as function of angles (conformality) 

– Validity: set of angle constraints 

• No flips 

– Convert solution to UV 



ABF :Formulation 

• Distortion: 

– 2D/3D angle difference 

∑ 

( 

t t 

) 

2 t 

α − β = 1 

2 

t 

w j j j 

, w 

j 

t 

t∈T 

, j= 

1..3 

β 

j 



ABF Formulation 

• Constraints: 

– Triangle validity: 

– Planarity: 

– Reconstruction (sine rule) 

• Distortion: 

∑ 

( 

t t 

) 

2 t 

α − β = 1 

2 

t 

w j j j 

, w 

j 

t 

t∈T 

, j= 

1..3 

β 

j 

• Solve: constrained 

optimization (Lagrange 

multipliers) 

– Positivity 

t 

α j 

> 0 



Comparison 

MIPS ABF ++ 

Linear (LSCM) 




ABF++: Up to x100 speedup 

[Sheffer et al:05] 

• Solver: 

∇ 

2 

– Newton 

ABF 

• At each step solve 

Fδ = −∇F, 

∇ 

• Conversion 

⎛ 

= ⎜ 

⎝ 

– Triangle unfolding 

2 

F 

A 

T 

B 

• accumulates error 

B⎞ 

⎟ 

0 ⎠ 

• Solver: 

ABF++ 

– Gauss‐Newton 

• Allows drastic system simplification 

Diagonal 

• Conversion: 

t 

– LSCM ( as target angles) 

α j 

∇ 

2 

F 

= 

⎛ Λ 

⎜ 

T 

⎝ B 

B⎞ 

⎟ 

0 ⎠ 

• allow less accurate solution 



Convergence 

1 Iteration 2 iterations 10 iterations 



Speedup: ABF vs ABF++ 



Circle Patterns [Kharevych:06] 

• Three Points make a Triangle… 

or a Circle 

• Local geometry 

Edge angles 



Geometry Preserving 

Edge Angles 

• Edge angle constraints 

– positivity 

– planarity 

• Extract from 3D geometry? 

• Idea: extract “feasible” triangle 

angles & convert to edge angles 

– feasible angles close to 3D 

angles 



Feasible angles 

• Minimize 

• Subject to 

β 

– Compare to ABF: replace reconstruction constraint 

• Solve with quadratic programming 

• Convert: 



2D Geometry From Edge Angles 

• To get radii from edge angles solve global minimization 

problem 

– Convex energy ‐ Unique minimum 

• Given radii and edge angles get UV by unfolding 



Intrinsic Delaunay 

• Enforce: 

• Large distortion if 3D mesh not Delaunay 

• Solution: Intrinsic Delaunay triangulation 

– perform implicit (local) edge flips in 3D 



Planar, Non‐Linear Methods 

MIPS ABF ++ 

circle patterns 




Cone Singularities [Kharevych:06] 

• What separates boundary from interior in angle space? 

• Answer: Sum of angles at vertex 

• Formulation specific 

– Circle patterns 

• Planarity 

– ABF/ABF++ 

• Planarity & Reconstruction 



Cone Singularities 

• Idea: Reduce boundary to small set of vertices 

• Implementation: 

– Enforce “interior” constraints at all other vertices 

• To unfold choose any sequence of edges connecting 

“boundary” vertices 



Circle Patterns + Cone Singularities 



ABF + Cone Singularities 



Recent Advances 

• [Zayer:07] Reformulate ABF to increase convergence (1 iter + 

LSCM) 

• [Gotsman:07]: Formulate in terms of angle deficit at 

interior/boundary vertices 

– Single linear system (almost...) 



Main References 

• Hormann, K. Greiner, G.,MIPS: An efficient global 

parametrization method. In Curve and Surface Design, 1999. 

• P. Sander, J. Snyder, S. Gortler, H. Hoppe. Texture mapping 

progressive meshes, ACM SIGGRAPH 2001, 409‐416 

• L. Kharevych, B. Springborn, and P. Schröder. Discrete conformal 

mappings via circle patterns. ACM Transactions on Graphics, 

25(2):412‐438, 2006. 

• A. Sheffer and E. de Sturler. Surface parameterization for 

meshing by triangulation flattening. In Proc. 9th International 

Meshing Roundtable (IMR 2000), 161‐172, 2000.

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