parameterization
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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
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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]
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WikipediaConvex combination
Barycentric coordinate system
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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)
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Example
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77,366,355,344,333,322,311,300,3
77,266,255,244,233,222,211,200,2
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77,066,055,044,033,022,011,000,0
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Ax = b, Solved by Gauss Seidel
11DETAILS
Discrete harmonic coordinates [Eck et al 1995]
Mean value coordinates [Floater 2003]
Other Alternatives
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Mean Value Coordinate
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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 ,,,
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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").
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[Differential Geometry Primer]
• Gaussian curvature
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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
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DETAILS
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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
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Least Square with Reduced D.O.F.
nff xxx 1Free parameters
nnfl xxx 1Lock parameters
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Fixed vertices 1 & 2(u1,v1) & (u2,v2) locked
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Variable change columns swap
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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.
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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)
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Wheel Consistency
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1963
1995
2003
2002
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Other Issues
Segmentation and atlas
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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)
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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
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SOR (cont)New Update Scheme: Successive Over-Relaxation
1
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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
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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
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0
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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
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39BACK
[From Siggraph Course 2007]
Study inverse of parameterization (X,Y) (u,v)
(i,j,k) barycentric coordinates, computed as:
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kkjjii
kkjjii
vvvYXv
uuuYXu
),(
),(
MT solely depends on the geometry of the triangle T
u=
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k
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Similarly, we can get Y
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BACK
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