Splines, Subdivision & Manifolds Luiz Velho IMPA Historical - - PowerPoint PPT Presentation

Splines, Subdivision & Manifolds

Luiz Velho IMPA

Historical Perspective

splines

1960 1970 1980 2000 2010 1990

subdivision surfaces manifolds

Bezier (60) de Casteljau (60) De Boor (72) Chaikin (74) Riesenfeld (80) Doo-Sabin (78) Catmull-Clark (78) Loop (87) Dyn (90) Zorin (89) Kobbelt (00) Velho (01) Schroeder (02) Grimm (95) Naveau (02) Zorin (04) Gu (06) Siqueira (09)

Modeling

• Splines (Regular Surfaces)
• Free-Form Modeling
• Subdivision (Arbitrary Surfaces)
• Efficient Algorithms
• Manifolds (Smooth Surfaces)
• Intrinsic Operations

Related Developments

• Parametrizations
• Surface Deformation
• Re-meshing
• Quad Structures
• Geometry Processing
• Discrete Differential Geometry (DDG)

Scale Spaces & Wavelets

• Constructions
• Meyer (80)
• Smoothness
• Daubechies (88)
• Multiresolution + FWT
• Mallat (89)
• Lifting
• Sweldens (95)

Road Map

Refinable Functions Scale Spaces Subdivision Splines

Scaling Function

• Localization (space / frequency)
• Normalization

φ(x)

Z φ(x)dx = 1

Scaling Family

• Two Parameters
• Change of Scale
• Change of Position

φs,t(x) = 1 |s|1/2 φ(x s − t)

φs(x) = 1 |s|1/2 φ(x s )

φt(x) = φ(x − t)

Scale Spaces

• Smoothing Operator
• Linear Scale Space
• Gaussian
• Physical Interpretation
• Heat equation (Smoothing ~ Diffusion)

φs(f(x)) = Z φs,t(x)f(x)dt

Discretization

• Dyadic Structure
• Hyperbolic Lattice

∆s0,t0 = {(sm

0 , nsm 0 t); m, n ∈ Z}

∆2,1 = {(2j, k2j); j, k ∈ Z}

V1 V2 V3 V4

Function Representation

• Representation Operator:
• Approximation Spaces

R(f) = (fj)j∈Z

R : L2(R) → l2(R)

space of functions space of sequences

Vj = {φj,k}k∈Z

scale basis

Basis and Representation

• Orthogonal Projection ~ Basis Vj

Projvj(f) = X

k

< f, φj,k >= X

k

f j

kφj,k

f = X

k

f j

kφj,k

f 7! (f j

k)

representation reconstruction

Two-Scale Relation

• Dilation Equation
• Scaling Basis:
• Reflexive Definition of Basis

φ0 ∈ V0 ⊂ V−1 φ0 = X

k

< φ0, φ−1,k > φ−1,k = X

k

hkφ−1,k

φ(x) = √ 2 X

k

hkφ(2x − k)

Refinable Function

Multiresolution Analysis

• Nested Approximation Spaces

Vj+1 Vj Vj-1

Dilations Translations

{0} ⊂ · · · ⊂ Vj+1 ⊂ Vj ⊂ Vj−1 ⊂ · · · ⊂ L2(R)

φj,k(x) = 2−2/jφ(2−jx − k)

B-Spline Basis

• Refinable Scaling Functions
• Basis of
• Uniform Piecewise Polynomials
• Properties
• Smoothness
• Compact Support
• Normalization
• Partition of Unity

P m

B-Splines

• Def: (repeated integration)
• B-Spline of order 1 (Haar)
• B-Spline of order m > 1

n1(x) = ⇢ 1 0 ≥ x ≥ 1;

• therwise.

1 1

nm(x) = Z nm−1(x − t)dt

Obs: recurrence relation

B-Splines & Gaussian

• Theorem:

lim

m→∞ nm(x) = G(x)

B-Spline Subdivision

• Refinement Relation
• Subdivision Mask

nm(x) =

m

X

k=0

Sm

k nm(2x − k)

Sm

k =

1 2m−1 ✓m k ◆

Refinable Funcions

• Example: Linear Spline

φ(x) = √ 2[1 2φ(2x + 1) + φ(2x) + 1 2φ(2x − 1)]

1 2 1 2

1

Example

• Linear B-Spline
• Rep of p(x) = {pi}

p(x) = X pinm(x − i)

n2(x) =    x if 0 < x ≥ 1; 2 − x if 1 < x ≥ 2;

• therwise

1

2

Scaling and Refinement

• Given f in Vj
• Compute Rep of f in Vj-1

f(x) = X

k

f j

k(x)

f j−1

k

=< f, φj−1,k > =< X

k

f j

kφj k, φj−1,k >

= X

k

f j

k < φj k, φj−1,k >

= X

k

f j

khk

Refinement & Reconstruction

• Decrease Scale = Increase Resolution

Subdivision

• Reconstruction of f by Refinement

(limit process)

• start with
• iterate

{f j

k}k∈Z

f j−1

k

= X f j

khk

j → −∞

subdivision operator

Elements of Subdivision

• Base Shape (control points)
• Limit Shape
• Subdivision Scheme

P0

P∞

S

P∞ = S∞P0

Subdivision Process

• Subdivision Iteration
• Limit Shape

Pk = SPk−1

P∞ = S∞P0

Graphical Example

P0 P1 P2 P3 · · ·

Subdivision Schemes

• Anatomy of Subdivision
• R : Refinement Operator
• G : Smoothing Operator
• Issues
• Representation (multiresolution basis)
• Convergence

S = (R, G)

Subdivision Algorithm

• Start with
• Repeat:
• Upsample and Change Level
• Update

P0

Pj(x) = ↑ Pj−1(x) Pj(x) = GPj(x)

Graphical View

• Topology
• Geometry

P k P k+1

R G

Vector Notation

• Knot

Vector

• Subdivision Matrix

P0 = (. . . , p0

−1, p0 0, p0 1, . . .)

        . . . p1

−1

p1 p1

1

. . .         =       . s−2 . s−1 . . . s2 s0 s−2 . . . s1 . s2 .               . . . p0

−1

p0 p0

1

. . .        

refinement smoothing

Characteristics

• Nature (level dependency)
• Stationary
• Non-Stationary
• Domain Structure (connectivity)
• Regular
• Non-Regular

Sk = S

Functional Setting

• Uniform Partition

y = p(x)

x

Parametric Setting

• Extends functional setting
• Control Points

p0

p1

p2

p3

x y t

p(t)

(x, y) = p(t) = (x(t), y(t)) pi = (xi, yi)

1D Subdivision

• Example: Cubic B-Spline

S(x) = 1 8x0 + 1 2x1 + 3 4x2 + 1 2x3 + 1 8x4

        . . . pj+1

i

. . .         =      

1 2 1 2 1 8 3 4 1 8

. . . .

1 2 1 2

. . .

1 8 3 4 1 8

.

1 2 1 2

              . . . pj

k

. . .        

2D Subdivision

• Parametric Surfaces

g : U ⊂ R2 → R3 (x, y, z) = g(u, v) = (x(u, v), y(u, v), z(u, v)) g(u, v) U M

Domain Discretization

• Regular Meshes

2 directions 3 directions

valence 4 valence 6

regular vertices

2D Refinement

• Quad-Mesh
• Tri-Mesh

Obs: preserves regularity

Arbitrary Meshes

• Semi-Regular Meshes
• Irregular Base Mesh
• Regular Refinement

extraordinary vertex

B-Splines & Subdivision

• Subdivision Surfaces
• Generalize Splines to Non-Regular Connectivity
• Examples:
• Catmull-Clark

(tensor product bi-cubic B-spline)

• Loop Subdivision Surface

(three-directional quartic box spline)

Splines & Manifolds

• Uniform Splines
• Particular case of Manifold Structure
• Characterization
• Charts: (basis functions at control vertices)
• Transition Function: (affine transformation)
• Partition of Unity

B-Spline Basis

• Recursive Definition
• Closed Form (at segment Sj)

with

• Truncated Power

Bn := Bn−1 ∗ B0

(x − xi)n

+ =

⇢ (x − xi)n if x ≥ xi; if x < xi.

bn(x) := n + 1 n

n+1

X

i=0

ωi,n(x − xi)n

+

bj,n(x) = bn(x − xj)

translation

xj xj−1

B-Spline Evaluation

• At a Segment Sj
• Example:

sj(x) =

m−1

X

j=0

cjbj,n(x)

Sj cj cj-1 bj-1 bj

2D Scheme

• Quadrilateral Structure

control mesh charts / transition function

Extraordinary Vertices

• Characteristic Map

k=5 k=13

• Breaks good properties (transition function, etc)

Topological Obstructions

• A closed 2-manifold M admits an affine atlas,

if and only if M is a torus