Analysis techniques for subdivision schemes

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Analysis techniques for subdivision schemes
Joe Warren Rice University
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A teaser for subdivision gurus

• Consider the basis function n(t) for the
  standard, four point interpolatory scheme.
• Note this function is not piecewise polynomial!
• What is the exact value of n(?) as a rational
  number?

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Choosing a modeling technology

• Many possibilities points, polygons,
  algebraics, implicits, NURBS, subdivision
  surfaces
• Choice based on various factors
• Ease of use
• Expressiveness
• Computational efficiency
• Analysis techniques
• Subdivision surfaces strong on first three
  points, perceived as weak on last

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Typical analysis problems

• Determine the smoothness of a given curve/surface
  scheme.
• Perform exact evaluation of positions, tangents
  and inner products for a given curve/surface
  scheme.
• Interpolate a networks of curves using a given
  surface scheme.

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Analysis of subdivision schemes using linear
algebra

• Convergence/smoothness analysis for subdivision
  schemes
• D S T D
• Exact computation of values, derivatives and
  inner products for subdivision schemes
• N S N
• ST E S E
• Interpolation of curves using subdivision schemes
• M S C M

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Example Cubic subdivision
Add new vertices between pairs of consecutive
vertices, repositioned via rules
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Example Catmull-Clark subdivision
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Subdivision as linear algebra

• Express positions of vertices of coarse and fine
  mesh as column vectors pk and pk1
• Construct subdivision matrix S whose rows
  correspond to rules for scheme
• Relate vectors via the subdivision relation
• pk1 S pk
• Apply linear algebra to S to analyze and
  manipulate the scheme

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Subdivision matrix for cubics
Rows of subdivision matrix S alternate between
rules and Consider only 5x5
sub-matrix centered at fixed mesh vertex
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Part I Smoothness analysis

• Given a scheme, are the limit meshes produced by
  the scheme Cn continuous?
• Observe that pk is related to p0 via
• pk Sk p0
• Compute diagonal matrix ? of eigenvalues and
  matrix Z of right eigenvectors satisfying
• SZ? Z-1

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Right eigenvectors polynomials

• Theorem If scheme is Cn continuous, let Zp
  denoted those right eigenvectors of Z with
  eigenvalues (½)j where 0 j n.
• Then the right eigenvectors of Zp converge to
  polynomials of degree at most n.
• C2 cubic example

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Smoothness condition

• Build difference matrix D that annihilates Zp
• D Zp 0
• Construct matrix T that satisfies relation
• (2n D) S T D
• T is subdivision matrix for differences D pk
• (2n D) pk1 T D pk
• Theorem If there exists kgt0 withTk8lt 1, the
  limit meshes for the scheme are Cn

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Smoothness for cubic splines

• Form difference matrix D as nullspace of Zp
• Compute subdivision matrix T for differences
• Observe that T8½ so scheme is C2

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Non-uniform subdivision schemes

• Tri/quad subdivision - mesh of triangles/quads
• Methods - Loop/Stam, Levin2, Schaefer/Warren

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C2 analysis for tri/quad schemes

• Construct matrix S for tri/quad interface
• Compute Zp, D, and T from S
• Attempt to find kgt0 such that Tk8lt1
• Problems
• D and T are not uniquely determined
• Bounding Tk8 is difficult since number of
  distinct rows in Tk grows exponentially
• For non-uniform schemes, assembling these rows is
  a tricky computation

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Simpler smoothness test (Levin2)

• Construct finite sub-matrices S0 and S1 of matrix
  S centered along tri/quad interface

S0
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Simpler smoothness test (Levin2)

• Construct finite sub-matrices S0 and S1 of matrix
  S centered along tri/quad interface

S1
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Joint spectral radius condition

• Compute finite subdivision matrices T0 and T1
• 4D S0 T0 D
• 4D S1 T1 D
• Theorem If there exists a kgt0 such that
• Te1Te2Tek 8lt1
• for all ei 0,1, the scheme is C2

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Part II Exact evaluation

• Problem Given a scheme, compute exact values
  for positions and tangents of limit mesh at
  arbitrary locations.
• Problem Given a scheme, compute the exact value
  for inner products of functions defined via the
  scheme.

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Exact position on limit mesh

• Treat mesh as being parameterized by t
• Compute limit position of mesh at arbitrary t
• For piecewise polynomial schemes, blossom right
  eigenvectors (Stam)
• For non-polynomial schemes, develop method to
  evaluate mesh at rational pts ti/j
• Get tangents by evaluating difference scheme

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Two-scale relation

• Let n(t) be the basis function associated with
  subdividing the vector p0(, 0, 1, 0, )T
• n(t) satisfies a two-scale relation of the form
• Coefficients si form columns of S

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Exact evaluation at rational pts

• Key To compute n(i/j), treat value of n(t) on
  the integer grid as being unknowns
• Use two-scale relation to setup system of
  equations relating these unknowns
• Solve equations to generate exact values
• Solution is dominant left eigenvector of
  upsampled S generated using power method

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Exact values of n(i/3) for cubics

• Examples of equations generated by two-scale
• Exact values from 2 to 2

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Values of n(i/3) for 4 pt scheme

• Four point scheme non-polynomial scheme
• Exact values for 3 to 0

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Exact valuation of inner products

• Given p0 and q0, compute inner product
  
• where p(t)p8 and q(t)q8
• Applications include fitting, fairing, mass
  properties
• Previous work compute using polynomial rep with
  infinite sums at extraordinary vertices (Peters,
  Reif)

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A relation for inner products

• Define inner product matrix E satisfying
• Express p(t) and q(t) as combinations of n(t-i)
• Express as sum of
  inner products of translated n(2t) via two-scale
  eqs
• ST E S 2 E

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Enclosed area as an inner product

• Input is closed polygon whose x and y coordinates
  are p0 and q0, respectively
• Defines smooth curve with param (p(t),q(t))
• Enclosed area is inner product
• Construct scheme T for derivatives, solve for
  inner product matrix E from
• ST E T E

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Enclosed area for 4 point curves

• Rows of inner product matrix E are shifts of
• Inscribe base polygon in unit circle - graph area

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Enclosed area for 4 point curves
Rows of inner product matrix E are shifts
of Inscribe base polygon in unit circle - graph
area
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Enclosed area for 4 point curves
Rows of inner product matrix E are shifts
of Inscribe base polygon in unit circle - graph
area
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Enclosed area for 4 point curves
Rows of inner product matrix E are shifts
of Inscribe base polygon in unit circle - graph
area
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Enclosed area for 4 point curves
Rows of inner product matrix E are shifts
of Inscribe base polygon in unit circle - graph
area
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Enclosed area for 4 point curves
Rows of inner product matrix E are shifts
of Inscribe base polygon in unit circle - graph
area
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Part III - Lofting

• Problem Build a subdivision scheme C for curve
  nets lying on a Catmull-Clark surface.
• Previous work polygonal complexes (Nasri)

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A scheme for curve nets

• Ordinary parts of Catmull-Clark surfaces are
  bicubic splines.
• Subdivision rules for curve nets at
• Valence two vertices normal cubic rules
• Valence four vertices special interpolating
  rule for two intersecting cubics
• Other valences next talk

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An interpolating rule for cubics

• Modify subdivision rules at vertex v of cubic
  such that v always lies at its limit position

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The commutative relation

• Given subdivision matrix Cu for uniform cubics
  and matrix N that positions v at limit, solve for
  non-uniform rules in C using
• C N N Cu

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Solving for C using the relation

• Solve for interpolating rules by inverting N.
• Yields special rules on two-ring of vertex v
• Multiple cubics can interpolate v

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Understanding the new rules

• Construct two or more polygons passing through
  common vertex v
• Apply rules to each polygon independently

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Understanding the new rules

• Construct two or more polygons passing through
  common vertex v
• Apply rules to each polygon independently

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Understanding the new rules

• Construct two or more polygons passing through
  common vertex v
• Apply rules to each polygon independently

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Understanding the new rules

• Construct two or more polygons passing through
  common vertex v
• Apply rules to each polygon independently

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Understanding the new rules

• Construct two or more polygons passing through
  common vertex v
• Apply rules to each polygon independently

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Lofting the curve nets

• Given curve scheme, find change of basis M where
  C commutes with S for Catmull-Clark
• C M M S
• Observe that commutative relation implies
• Ck M M Sk
• Therefore, S8p0 interpolates C8q0 where
• q0M p0

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Change of basis at valence two

• Row of matrix M has form

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Change of basis at valence four

• Row of matrix M has form

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Extraordinary vertices

• For standard Catmull-Clark, curves passing
  through an extraordinary verts are not cubics
• Modify the subdivision rules in the neighborhood
  of extraordinary vertices to allow interpolation
• Idea Fix curve net rules, generate modified
  surface rules at extraordinary vertices

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Generating modified surface rules

• Expand commutative relation C MM S into block
  form
• Solve locally for S via block decomposition
• Dont form Sc explicitly, use filters!

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Conclusions

• Many important types of analysis are possible for
  subdivision schemes
• Linear algebra, especially commutative relations,
  is the key to these analysis methods
• Better theory supporting these analysis methods
  is needed
• Many thanks to Scott Schaefer!