Post on 17-Apr-2020
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
Mesh Parameterization: Theory and PracticeNon‐Linear Methods
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
Mesh Parameterization: Theory and PracticeNon‐Linear Methods
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
Mesh Parameterization: Theory and PracticeNon‐Linear Methods
Hierarchical SolutionHierarchical Solution
When adding vertices “improve” local
neighbourhood only
Mesh Parameterization: Theory and PracticeNon‐Linear Methods
MIPS [Hormann & Greiner 2000]MIPS [Hormann & Greiner 2000]
• Measure conformality ( ):
• Scale independent
– in contrast to linear conformal formulations
Linear (LSCM) [Levy’02]
Mesh Parameterization: Theory and PracticeNon‐Linear Methods
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
Mesh Parameterization: Theory and PracticeNon‐Linear Methods
• 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)
Mesh Parameterization: Theory and PracticeNon‐Linear Methods
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]
Mesh Parameterization: Theory and PracticeNon‐Linear Methods
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
Mesh Parameterization: Theory and PracticeNon‐Linear Methods
ABF :Formulation ABF :Formulation
• Distortion:– 2D/3D angle difference
( ) 22
3..1,
1,tj
tj
jTt
tj
tj
tj ww
ββα =−∑
=∈
Mesh Parameterization: Theory and PracticeNon‐Linear Methods
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
Mesh Parameterization: Theory and PracticeNon‐Linear Methods
ComparisonComparison
Linear (LSCM) stretch minimization
MIPS ABF ++
Mesh Parameterization: Theory and PracticeNon‐Linear Methods
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
Mesh Parameterization: Theory and PracticeNon‐Linear Methods
ConvergenceConvergence
1 Iteration 2 iterations 10 iterations
Mesh Parameterization: Theory and PracticeNon‐Linear Methods
Speedup: ABF vs ABF++ Speedup: ABF vs ABF++
Mesh Parameterization: Theory and PracticeNon‐Linear Methods
Circle Patterns [Kharevych:06]Circle Patterns [Kharevych:06]
• Three Points make a Triangle…or a Circle
• Local geometry
Edge angles
Mesh Parameterization: Theory and PracticeNon‐Linear Methods
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
Mesh Parameterization: Theory and PracticeNon‐Linear Methods
Feasible anglesFeasible angles
• Minimize
• Subject to
– Compare to ABF: replace reconstruction constraint
• Solve with quadratic programming
• Convert:
β
Mesh Parameterization: Theory and PracticeNon‐Linear Methods
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
Mesh Parameterization: Theory and PracticeNon‐Linear Methods
Intrinsic DelaunayIntrinsic Delaunay
• Enforce:
• Large distortion if 3D mesh not Delaunay
• Solution: Intrinsic Delaunay triangulation
– perform implicit (local) edge flips in 3D
Mesh Parameterization: Theory and PracticeNon‐Linear Methods
Planar, Non‐Linear MethodsPlanar, Non‐Linear Methods
circle patterns stretch minimization
MIPS ABF ++
Mesh Parameterization: Theory and PracticeNon‐Linear Methods
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
Mesh Parameterization: Theory and PracticeNon‐Linear Methods
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
Mesh Parameterization: Theory and PracticeNon‐Linear Methods
Circle Patterns + Cone Singularities Circle Patterns + Cone Singularities
Mesh Parameterization: Theory and PracticeNon‐Linear Methods
ABF + Cone SingularitiesABF + Cone Singularities
Mesh Parameterization: Theory and PracticeNon‐Linear Methods
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...)
Mesh Parameterization: Theory and PracticeNon‐Linear Methods
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.