mesh parameterization: theory and practice · mesh parameterization: theory and practice...

Mesh Parameterization: Theory and Practice

Mesh Parameterization: Theory and Practice

Non‐Linear MethodsAlla Sheffer

Non‐Linear MethodsAlla Sheffer

Mesh Parameterization: Theory and PracticeNon‐Linear Methods

Planar, Non‐Linear MethodsPlanar, Non‐Linear Methods

circle patterns

stretch minimizationMIPS

ABF ++

Dire

ctIn

dire

ct


Direct MethodsDirect 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 MechanismDirect 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 SolutionHierarchical Solution

When adding vertices “improve” local

neighbourhood only


MIPS [Hormann & Greiner 2000]MIPS [Hormann & Greiner 2000]

• Measure conformality ( ):

• Scale independent

– in contrast to linear conformal formulations

Linear (LSCM) [Levy’02]


Stretch Minimization [Sander et al,2001]Stretch Minimization [Sander et al,2001]

• Measure “averaged” per triangle stretch

• Balances stretch

– Increase one σ to decrease another

MIPS Stretch Minimizing


• Barycentric coordinates per triangle

• Obtain Jf by explicit derivation

So what is f ?So what is f ?

1p

2p 3pp

1q

2q3q

2D 3Df

f(p)

( ) 321321213132 ,,,,,,,, pppqpppqpppqppp ++=f(p)


Alternative VariablesAlternative 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]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 ABF :Formulation

• Distortion:– 2D/3D angle difference

( ) 22

3..1,

1,tj

tj

jTt

tj

tj

tj ww

ββα =−∑

=∈


ABF FormulationABF Formulation

• Distortion:• Constraints:

– Triangle validity:

– Planarity:

– Reconstruction (sine rule)

– Positivity

• Solve: constrained optimization (Lagrange multipliers)

( ) 22

3..1,

1,tj

tj

jTt

tj

tj

tj ww

ββα =−∑

=∈

0>tjα


ComparisonComparison

Linear (LSCM) stretch minimization

MIPS ABF ++


ABF++: Up to x100 speedup[Sheffer et al:05]ABF++: Up to x100 speedup[Sheffer et al:05]

ABF

• Solver: – Newton

• At each step solve

• Conversion– Triangle unfolding

• accumulates error

ABF++

• Solver:– Gauss‐Newton

• Allows drastic system simplification

• Conversion:– LSCM ( as target angles)

• allow less accurate solution

tjα

⎟⎟⎠

⎞⎜⎜⎝

⎛=∇−∇=∇

0, 22

TBBA

FFFδ ⎟⎟⎠

⎞⎜⎜⎝

⎛ Λ=∇

02

TBB

FDiagonal


ConvergenceConvergence

1 Iteration 2 iterations 10 iterations


Speedup: ABF vs ABF++ Speedup: ABF vs ABF++


Circle Patterns [Kharevych:06]Circle Patterns [Kharevych:06]

• Three Points make a Triangle…or a Circle

• Local geometry

Edge angles


Geometry Preserving Edge AnglesGeometry Preserving Edge Angles

• Edge angle constraints

– positivity

• Extract from 3D geometry?

• Idea: extract “feasible” triangle angles & convert to edge angles

– feasible angles close to 3D angles

– planarity


Feasible anglesFeasible angles

• Minimize

• Subject to

– Compare to ABF: replace reconstruction constraint

• Solve with quadratic programming

• Convert:

β


2D Geometry From Edge Angles2D 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 DelaunayIntrinsic Delaunay

• Enforce:

• Large distortion if 3D mesh not Delaunay

• Solution: Intrinsic Delaunay triangulation

– perform implicit (local) edge flips in 3D


Planar, Non‐Linear MethodsPlanar, Non‐Linear Methods

circle patterns stretch minimization

MIPS ABF ++


Cone Singularities [Kharevych:06]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 SingularitiesCone 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 Circle Patterns + Cone Singularities


ABF + Cone SingularitiesABF + Cone Singularities


Recent AdvancesRecent 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 ReferencesMain 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.

mesh parameterization: theory and practice · mesh parameterization: theory and practice...

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