S bdi i i S h fSubdivision Schemes for Geometric ModellingGeometric Modelling

Kai HormannKa o aUniversity of Lugano

Outline

Sep 5 – Subdivision as a linear processbasic concepts, notation, subdivision matrixbasic concepts, notation, subdivision matrix

Sep 6 – The Laurent polynomial formalismalgebraic approach, polynomial reproduction

Sep 7 – Smoothness analysisSep 7 Smoothness analysisHölder regularity of limit by spectral radius method

Sep 8 – Subdivision surfacesoverview of most important schemes & properties

1 Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei

p p p

Lane–Riesenfeld algorithm

multiplying the symbol by (1+z)/2 increases the smoothness of the limit curves by 1y

geometrically, this averages the control points

Lane–Riesenfeld algorithmeach refinement step first inserts edge midpointsp g p

then applies m–1 averaging steps

symbol for these schemes:symbol for these schemes:

regularity of the limit curves: r=m+1–log2(2)=m

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limit curves are uniform B-splines of degree m

Tensor-product schemes

extend this idea to quadrilateral meshesfirst insert edge and face midpointsfirst insert edge and face midpoints- splits the quadrilateral into four new quadrilaterals

then apply averaging stepsthen apply averaging steps- each averaging step averages in two directions

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Doo–Sabin subdivision

Example1 averaging step1 averaging step

dual scheme4 i t f di d ld i t- 4 new points per face, discard old points

4 stencils for the new points

tensor products of the stencils [3,1]/4 and [1,3]/4 from Chaikin’s scheme, e.g.

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, g

Doo–Sabin subdivision

Examplefor a regular quad mesh,for a regular quad mesh,this gives tensor-productquadratic B-splines- C 1 limit surfaces

a general quad mesh has extraordinary verticesa general quad mesh has extraordinary vertices- where not 4, but 3, 5, or even more faces meet- non-quadrilateral facesq

after one refinement step

special rules and

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analysis needed

Doo–Sabin subdivisionα2

Examplegeneral stencil for a new point

α1

α2

general stencil for a new pointin a face with n vertices α0

αn 2

, i = 1,…,n–1

αn–1

αn–2

- note: stencil coefficients sum to 1

- reduces to the regular stencil above for n=4g

limit surfaces are C1-continuous

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Doo–Sabin subdivision

Exampleduality of the schemeduality of the scheme- valence-n vertex → n-gon, edge → 4-gon, n-gon → n-gon

all vertices are regular (valence 4) after first refinement- all vertices are regular (valence 4) after first refinement

- number of irregular faces remains the same

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Catmull–Clark subdivision

Example2 averaging steps2 averaging steps

primal scheme- new vertex ( ) edge ( ) and face ( ) points- new vertex ( ), edge ( ), and face ( ) points

stencilsregular setting: tensor product of cubic B spline stencils- regular setting: tensor product of cubic B-spline stencils

ββ

γ

α

β

ββ

ββ

γ γ

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βγ γ

Comparison

9 Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei

Triangle meshes

extend Lane–Riesenfeld to triangle meshesfirst insert edge midpointsfirst insert edge midpoints- splits the triangles into four new triangles

then apply averaging stepsthen apply averaging steps- each averaging step averages in three directions

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Loop subdivision

Example1 averaging step1 averaging step

primal schemet ( ) d d ( ) i t- new vertex ( ) and edge ( ) points

stencilsβ

α

β

ββ

αβ

ββ

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ββ

Loop subdivision

12 Subdivision Schemes for Geometric Modelling – DRWA 2011 – Alba di Canazei

Butterfly subdivision

extend idea of the 4-point scheme to mesheskeep old points, insert a new point for each edgekeep old points, insert a new point for each edge

ExampleButterfly scheme

stencil for the new point- derived from fitting a local

interpolating bivariate cubic polynomial (only 8 insteadpolynomial (only 8 instead of 10 degrees of freedom because of symmetry)

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Butterfly scheme

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Summary

“refine-and-smooth” algorithm by Lane and Riesenfeld gives the B-spline curve schemesg p

idea can be extended to surface schemesDoo–Sabin → C1 surfacesDoo–Sabin → C surfaces

Catmull–Clark→ C1 surfaces (C2 in regular region)1 2Loop → C1 surfaces (C2 in regular region)

interpolatory schemes, based on local p y ,polynomial interpolation and evaluation

Butterfly → C1 surfaces (after minor modification)

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Butterfly → C surfaces (after minor modification)

Literature

N. Dyn, D. LevinSubdivision Schemes in Geometric ModellingActa Numerica, Vol. 11, 2002, pp. 73–144

A. Iske, E. Quak, M.S. FloaterTutorials on Multiresolution in Geometric ModellingChapters 1–4 (N. Dyn, M. Sabin), Springer 2002Chapters 1 4 (N. Dyn, M. Sabin), Springer 2002

M. SabinAnalysis and Design of Univariate Subdivision SchemesS i 2010Springer 2010

J. Peters, U. ReifSubdivision SurfacesSpringer 2008

L.-E. Andersson, N. F. StewartMathematics of Subdivision Surfaces

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Mathematics of Subdivision SurfacesSIAM 2010