Subdivision
S
bdi i i Schemes
S h
for
f
Geometric Modelling
Kai Hormann
Ka
o a
University of Lugano
Outline
ƒ Sep 5 – Subdivision as a linear process
ƒ basic concepts, notation, subdivision matrix
ƒ Sep 6 – The Laurent polynomial formalism
ƒ algebraic approach, polynomial reproduction
ƒ Sep 7 – Smoothness analysis
ƒ Hölder regularity of limit by spectral radius method
ƒ Sep 8 – Subdivision surfaces
ƒoverview of most important
p
schemes & p
properties
p
1
Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Lane–Riesenfeld algorithm
ƒ multiplying the symbol by (1+z)/2 increases
the smoothness of the limit curves by
y1
ƒ geometrically, this averages the control points
ƒ Lane–Riesenfeld algorithm
ƒ each refinement step
p first inserts edge
g midpoints
p
ƒ then applies m – 1 averaging steps
ƒ symbol for these schemes:
ƒ regularity of the limit curves: r = m + 1 – log2 ( 2 ) = m
ƒ limit curves are uniform B-splines of degree m
2
Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Tensor-product schemes
ƒ extend this idea to quadrilateral meshes
ƒ first insert edge and face midpoints
- splits the quadrilateral into four new quadrilaterals
ƒ then apply averaging steps
- each averaging step averages in two directions
3
Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Doo–Sabin subdivision
ƒ Example
ƒ 1 averaging step
ƒ dual scheme
- 4 new points
i t per face,
f
discard
di
d old
ld points
i t
ƒ 4 stencils for the new points
ƒ tensor products of the stencils [3,1] / 4 and [1,3] / 4
from Chaikin’s scheme,, e.g.
g
4
Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Doo–Sabin subdivision
ƒ Example
ƒ for a regular quad mesh,
this gives tensor-product
quadratic B-splines
- C 1 limit surfaces
ƒ a general quad mesh has extraordinary vertices
- where not 4, but 3, 5, or even more faces meet
q
faces
- non-quadrilateral
after one refinement step
ƒ special rules and
analysis needed
5
Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Doo–Sabin subdivision
α2
ƒ Example
α1
ƒ general stencil for a new point
in a face with n vertices
α0
αn –22
,
αn –1
i = 1, … , n –1
- note: stencil coefficients sum to 1
- reduces to the regular
g
stencil above for n = 4
ƒ limit surfaces are C 1-continuous
6
Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Doo–Sabin subdivision
ƒ Example
ƒ duality of the scheme
- valence-n vertex → n-gon, edge → 4-gon, n-gon → n-gon
- all vertices are regular (valence 4) after first refinement
- number of irregular faces remains the same
7
Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Catmull–Clark subdivision
ƒ Example
ƒ 2 averaging steps
ƒ primal scheme
- new vertex (z),
(z) edge (‹),
(‹) and face („) points
ƒ stencils
- regular setting: tensor product of cubic B
B-spline
spline stencils
γ
β
γ
β
γ
8
β
α
β
γ
β
γ
Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Comparison
9
Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Triangle meshes
ƒ extend Lane–Riesenfeld to triangle meshes
ƒ first insert edge midpoints
- splits the triangles into four new triangles
ƒ then apply averaging steps
- each averaging step averages in three directions
10
Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Loop subdivision
ƒ Example
ƒ 1 averaging step
ƒ primal scheme
- new vertex
t (z) and
d edge
d (‹) points
i t
ƒ stencils
β
β
β
α
β
11
β
β
Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Loop subdivision
12
Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Butterfly subdivision
ƒ extend idea of the 4-point scheme to meshes
ƒ keep old points, insert a new point for each edge
ƒ Example
ƒ Butterfly scheme
ƒ stencil for the new point
- derived from fitting a local
interpolating bivariate cubic
polynomial (only 8 instead
of 10 degrees of freedom
because of symmetry)
13
Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Butterfly scheme
14
Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Summary
ƒ “refine-and-smooth” algorithm by Lane and
gives the B-spline
p
curve schemes
Riesenfeld g
ƒ idea can be extended to surface schemes
ƒ Doo–Sabin
→ C 1 surfaces
ƒ Catmull–Clark → C 1 surfaces (C 2 in regular region)
ƒ Loop
→ C 1 surfaces (C 2 in regular region)
ƒ interpolatory
p
y schemes,, based on local
polynomial interpolation and evaluation
ƒ Butterfly
15
→ C 1 surfaces (after minor modification)
Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei
Literature
N. Dyn, D. Levin
ƒ Subdivision Schemes in Geometric Modelling
Acta Numerica, Vol. 11, 2002, pp. 73–144
A. Iske, E. Quak, M.S. Floater
ƒ Tutorials on Multiresolution in Geometric Modelling
Chapters 1
1–4
4 (N. Dyn, M. Sabin), Springer 2002
M. Sabin
ƒ Analysis and Design of Univariate Subdivision Schemes
S i
Springer
2010
J. Peters, U. Reif
ƒ Subdivision Surfaces
Springer 2008
L.-E. Andersson, N. F. Stewart
ƒ Mathematics of Subdivision Surfaces
SIAM 2010
16
Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei