Subdivision surfaces Construction and analysis

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Subdivision surfacesConstruction and analysis

• Martin Reimers
• CMA/IFI, University of Oslo
• September 24th 2004

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Subdivision concept

• Piecewise linear geometry with vertices P0
• Refine and reposition using subdivision rule
• P0, P1, P2?

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Subdivision Surfaces

• The same concept for surfaces
• Piecewise linear geometry refined P0, P1, P2?
• P0 P1
  P2 ? P1

Properties of P1? Continuous, smooth, fair?
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Overview

• Introduction
• Subdivision schemes
• Construction
• Analysis

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Spline curve subdivision

• Control polygon PP0
• Cut corners recursively, get P1, P2?
• Subdivision rule
• p2k (3pk-1 pk)/4
• p2k1( pk-13pk)/4
• Subdivision matrix S
• Limit P1 is a quadratic spline (smooth curve)

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Spline curve subdivision

• Generalization refine and average d-1 times
• Pj converge to spline of degree d
• Linear operations in a subdivision matrix S
  s.t.
• Pj1SPj
• Cubic splines
• Many curve subdivision methods
• Next surfaces

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Spline surface subdivision

• P0 regular control mesh (grid) in 3D
• e.g. points and quadrilaterals
• Refine P0 by a certain subdivision rule
• yields new mesh P1
• Linear operations, collect in a matrix S
• Pj1SPj
• Control mesh converge to spline bi-degree 2
• Generalize d-1 averages gives bi-degree d

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Doo Sabin Subdivision (78)

• Generalization General mesh
• N-sided faces
• Any valence (Not only 4)
• Masks

• Still have Pj1SPj (linearity)
• Generalize bi-quadratic spline to arbitrary mesh
• C1 continuous (smooth) limit surface
  PetersReif98

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Doo Sabin Subdivision

• Example

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Catmull-Clark Subdivision (78)

• Generalize bi-cubic spline to quad mesh

Face mask
Edge mask
Constants ?3/2n ?1/4n (others possible)
As before Pj1SPj
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Catmull-Clark scheme
Implementation version
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Catmull-Clark scheme
Pj converge to a C1 surface, C2 a.e
PetersReif98
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Catmull-Clark, special rules

• Crease/bdy masks
• Piecewise smooth surface

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Loops scheme (87)

• Edge
• Vertex

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Loops scheme (87)
Implementation version

• Edge
• Vertex

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Loop boundary mask

• Use cubic B-spline refinement rules

Vertex Rule
Edge Rule
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Loops scheme

• Generalize 3-direction (quartic) box-splines
• C1 everywhere, not C2 in irregular points
• Special masks for features (Hoppe et al)
• Alternative view
• Refine linearly
• Smooth all vertices by
• ( Laplacian smoothing )

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Loop example
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Spline based schemes

• All schemes up to now
• Affine invariance
• Convex hull property
• Polynomials locally
• Splines in regular regions
• Approximates the control mesh (like splines)

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Butterfly scheme (90)

• Dyn,Levin,Gregory
• Interpolating
• Triangles
• Not polynomials
• No convex hull property
• Not C1 for n3 or ngt7
• Modified scheme zorin is C1 but not C2

Edge mask
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Examples

• Famous schemes

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Subdivision rules/schemes

• Many categories, ways to refine mesh etc.
• Mesh types tris, quads, hex, combinations
• Interpolating/Approximating
• (Non)Uniform
• (Non)Stationary
• (Non)Linear
• Subdivision zoo
• Midpoint scheme PetersReif
• v3 scheme Kobbelt
• Refineaverage Stam/Warren et al
• 4-8 subdivision VelhoZorin
• Tri/Quad schemes
• many more
• Rules constructed to obtain objective
• Mimic splines, interpolation, smoothness/fairness,
  features,

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Surface subdivision
Splines Toy Story 1

• Is this a toy?
• Pros
• Many of the pros of splines
• Flexibility, wrt. topology/mesh
• Spline surfaces without gaps and seams!
• Simple to implement
• Simple/intuitive to manipulate
• Cons
• No global closed form (but locally)
• Evaluation not straight forward (but)
• Ck , kgt1 is hard
• Artefacts (ripples etc.)
• CC cannot make convex surface if ngt5
  PetersReif04

Geris Game (Pixar studios)
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NEXT ANALYSIS

• Consider primal spline schemes (e.g. Loop, C-C)
• Linear
• Affine invariant
• Convex hull property
• Stationary
• Symmetric
• Smooth in regular regions
• Have vertex correspondency
• Start with control mesh P0 with vertices in R3
• Subdivision yields finer meshes P1,P2,?
• What properties does limit surface P1 have?
• Study
• Limit points and normals / tangent planes
• C1 smoothness
• ( Curvature, C2 )

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Regular regions

• If mesh is regular, limit surface P1 is known
  (spline)
• Analyze irregular regions
• Subdivision
• Preserves valence
• Insert new regular vertices
• ) regular regions grow and
• irregular regions shrink
• Limit surface def on the green
• rings are C2 quartic splines over
• triangles of Pj
• Each subdivision adds a spline ring to the
  neighbourhood of
• an irregular point

Loop scheme
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The local subdivision matrix

• Study irregular vertex p and neighbourood P0
• Stationary scheme subdivision matrix S s.t.
• Pj1S Pj
• map neighbourhood to neighbourhhood
• S determines limit surface around p
• Assume S is non-defective SQ?Q-1
• Pj S Pj-1 Sj P0 Q?jQ-1 P0
• Spectral properties of S are fundamental!!!

p
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Spectral analysis

• S Q?Q-1
• Q (q0,q1,?qL) eigenvectors
• ? diag(?0,?1,?,?L) eigenvalues,
  ?0?1??L
• S often block symmetric with circulant blocks
• ) Can find Q and ? with Fourier techniques
• Dominant eigenpair ?01gt?1 and q0(1,?,1)
  (Affine invariance)
• Controls convergence
• Subdominant eigenpairs ??1?2?
• Controls tangent plane continuity
• Want 1gt ?1?2 gt?3
• Sub-subdominant eigenpairs ??3?4 ?
• Controls curvature
• Want ?2 ?

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Limit points

• Have Sj Q?jQ-1
• Eigen expansion P0? qi ai ai2 R3
• Pj ? ?ij qi ai
• Since ?01gt?i and q0 (1,?,1)
• Pj! (a0,a0,?)T as j ! 1
• Limit point p1 a0l0P0
• Left eigenvector l0 can be determined from a n1
  matrix
• Also Control mesh converge to continuous surface

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Spline rings
L

• Limit surface P1 is union of limit point and
  spline rings xj
• xj?n ! P1 ½ R3, P1 ? xj(?n) p1
• Where ?n 1,?,nL (quad based scheme)
• Limit surface can be parameterized over squares
• Basis functions B1,?,BN?n! R s.t. ? Bi 1 and
• xj ? Bi pijB Pj
• Define eigenfunctions ?i?n! R ?iBi qi
• xj B Pj ? ?ij ?i ai
• Note that ?01 since q0(1,1,?)

xj(?n)
Xj1(?n)
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Tangent plane

• Each spline xj ring is C1
• Assume 1gt?1?2gt?3 and q1,q2 lin.indep.
• Also that a1,a2 lin.indep
• xj a0 ?j ?1 a1 ?j ?2 a2 o(lj)
• Du xj ?j (Du ?1 a1 Du ?2 a2 o(1) )
• Dv xj ?j (Dv ?1 a1 Dv ?2 a2 o(1) )
• Normal is well defined and continuous if
• Normal(xj) Du xj Dv xj /Du xj Dv xj !
  a1a2 /a1a2
• Tangent vectors a1, a2 found using left
  eigenvectors of S

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Smoothness C1

• ) tangent plane cont. if det D (?1,?2)? 0
• Analysis only relevant if generic data det (a1
  ,a2) ? 0
• Want C1, more than tangent plane cont.
• P1 is C1 if it is C1 fcn. over the Tangent Plane
• xj a0 ?j (?1 a1 ?2 a2)o(?j)
• ring xj converge uniformly to tangent plane
• Big questionReif is limit xj injective?
• Yes if ?1 e1 ?2 e2 (?1 , ?2) is injective

a1
a2
a1
a0
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Characteristic map

• Reif95 Defined as ?(?1,?2) ?n! R2
• ?(i,u,v)B(i,u,v)(q1,q2)
• Theorem Reif95
• P1 is a C1 surface for generic initial data if
• ??1?2gt?3 with q1,q2 lin. independent
• ? is regular and injective
• Conditions means ? has a cont. Inverse
• (can be used to parameterize P1)
• Major point
• P1 is a graph over TP near p1, thus C1 regular

Not injective
?
Image ?(?n)
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Verifying C1 regularity

• PetersReif98 Showed C1 of DS CC
• Umlauf Showed C1 for Loop
• Check conditions 1?0gt?1?2 gt?3
• Check regularity and injectivity of ?
• Rotationally symmetric ) study segments
• Regular Use Bezier form to show that ?u ?v ? 0
• Injectivity show that boundary is simple
• Generalizations/simplifications
• Zorin, Umlauf,

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Continuous Curvature C2

• MUCH harder to get
• Holy grail C2, local, stationary, ? ? 0 ,low
  polynomial degree
• Large support?
• Not stationary?
• ? 0 in irregular pointsUmlaufPrautzsch
• High degree?
• Prautzsch/Reif polynomial scheme must have d
  2k2 for Ck
• Thus C2 requires degree 6 in regular regions
• Neccessary condition for bounded curvature ?2
  ?3
• Has been used to find good schemes (CC/DS)

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Scheme manipulation

• Take a scheme S
• Spectral analysis find SQ ? Q-1
• Manipulate ? Set e.g. ?3 ?2
• New eigenvalues ?2
• New scheme S2Q ?2 Q-1 with better properties
• e.g. bounded curvature

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Conclusion

• Subdivision is very usefull in some applications
• Better methods needed for high end CAD
• Construction generalize splines, tune spectrum
• Analysis of linear schemes based on linear
  algebra
• Nonlinear/Nonstationary Not much is known??,
  harder!