[PPT] - Surface Representations Leif Kobbelt RWTH Aachen University 1 PowerPoint Presentation

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Surface Representations

Leif Kobbelt RWTH Aachen University

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Leif Kobbelt RWTH Aachen University

Outline

(mathematical) geometry representations

– parametric vs. implicit

approximation properties
types of operations

– distance queries – evaluation – modification / deformation

data structures

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Leif Kobbelt RWTH Aachen University

(mathematical) geometry representations

– parametric vs. implicit

approximation properties
types of operations

– distance queries – evaluation – modification / deformation

data structures

Outline

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Leif Kobbelt RWTH Aachen University

Mathematical Representations

parametric

– range of a function – surface patch

implicit

– kernel of a function – level set

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f : R2 → R3, SΩ = f(Ω) F : R3 → R, Sc = {p : F(p) = c}

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Leif Kobbelt RWTH Aachen University

2D-Example: Circle

parametric
implicit

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f : t → r cos(t) r sin(t)

,

S = f([0, 2π]) F(x, y) = x2 + y2 − r2 S = {(x, y) : F(x, y) = 0}

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Leif Kobbelt RWTH Aachen University

parametric
implicit

2D-Example: Island

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f : t → r cos(t) r sin(t)

,

S = f([0, 2π]) F(x, y) = x2 + y2 − r2 S = {(x, y) : F(x, y) = 0} ??? ??? ???

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Leif Kobbelt RWTH Aachen University

piecewise parametric
piecewise implicit

Approximation Quality

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f : t → r cos(t) r sin(t)

,

S = f([0, 2π]) F(x, y) = x2 + y2 − r2 S = {(x, y) : F(x, y) = 0} ??? ??? ???

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Leif Kobbelt RWTH Aachen University

piecewise parametric
piecewise implicit

Approximation Quality

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f : t → r cos(t) r sin(t)

,

S = f([0, 2π]) F(x, y) = x2 + y2 − r2 S = {(x, y) : F(x, y) = 0} ??? ??? ???

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Leif Kobbelt RWTH Aachen University

Requirements / Properties

continuity

– interpolation / approximation

topological consistency

– manifold-ness

smoothness

– C0, C1, C2, ... Ck

fairness

– curvature distribution

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f(ui, vi) ≈ pi

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Leif Kobbelt RWTH Aachen University

continuity

– interpolation / approximation

topological consistency

– manifold-ness

smoothness

– C0, C1, C2, ... Ck

fairness

– curvature distribution

Requirements / Properties

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f(ui, vi) ≈ pi

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Leif Kobbelt RWTH Aachen University

Requirements / Properties

continuity

– interpolation / approximation

topological consistency

– manifold-ness

smoothness

– C0, C1, C2, ... Ck

fairness

– curvature distribution

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f(ui, vi) ≈ pi

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Leif Kobbelt RWTH Aachen University

continuity

– interpolation / approximation

topological consistency

– manifold-ness

smoothness

– C0, C1, C2, ... Ck

fairness

– curvature distribution

Requirements / Properties

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f(ui, vi) ≈ pi

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Leif Kobbelt RWTH Aachen University

continuity

– interpolation / approximation

topological consistency

– manifold-ness

smoothness

– C0, C1, C2, ... Ck

fairness

– curvature distribution

Requirements / Properties

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f(ui, vi) ≈ pi

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Leif Kobbelt RWTH Aachen University

Topological Consistency

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Leif Kobbelt RWTH Aachen University

Topological Consistency

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Leif Kobbelt RWTH Aachen University

Topological Consistency

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Mesh Repair ...

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Leif Kobbelt RWTH Aachen University

Closed 2-Manifolds

parametric

– disk-shaped neighborhoods – + injectivity

implicit

– surface of a “physical” solid –

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f(Dε[u, v]) = Dδ[f(u, v)] F(x, y, z) = c, ∇F(x, y, z) = 0

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Leif Kobbelt RWTH Aachen University

Closed 2-Manifolds

parametric

– disk-shaped neighborhoods – + injectivity

implicit

– surface of a “physical” solid –

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f(Dε[u, v]) = Dδ[f(u, v)] F(x, y, z) = c, ∇F(x, y, z) = 0

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Leif Kobbelt RWTH Aachen University

Closed 2-Manifolds

parametric

– disk-shaped neighborhoods – + injectivity

implicit

– surface of a “physical” solid –

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f(Dε[u, v]) = Dδ[f(u, v)] F(x, y, z) = c, ∇F(x, y, z) = 0

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Leif Kobbelt RWTH Aachen University

Closed 2-Manifolds

parametric

– disk-shaped neighborhoods –

implicit

– surface of a “physical” solid –

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f(Dε[u, v]) = Dδ[f(u, v)] F(x, y, z) = c, ∇F(x, y, z) = 0

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Leif Kobbelt RWTH Aachen University

Closed 2-Manifolds

parametric

– disk-shaped neighborhoods –

implicit

– surface of a “physical” solid –

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f(Dε[u, v]) = Dδ[f(u, v)] F(x, y, z) = c, ∇F(x, y, z) = 0

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Leif Kobbelt RWTH Aachen University

Smoothness

position continuity : C0
tangent continuity : C1
curvature continuity : C2

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Leif Kobbelt RWTH Aachen University

Smoothness

position continuity : C0
tangent continuity : C1
curvature continuity : C2

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Leif Kobbelt RWTH Aachen University

Smoothness

position continuity : C0
tangent continuity : C1
curvature continuity : C2

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Leif Kobbelt RWTH Aachen University

Fairness

minimum surface area
minimum curvature
minimum curvature variation

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Leif Kobbelt RWTH Aachen University

(mathematical) geometry representations

– parametric vs. implicit

approximation properties
types of operations

– distance queries – evaluation – modification / deformation

data structures

Outline

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Leif Kobbelt RWTH Aachen University

Polynomials

computable functions
Taylor expansion
interpolation error (mean value theorem)

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p(t) =

p

i=0

ci ti =

p

i=0

c

i Φi(t)

p(ti) = f(ti), ti ∈ [0, h]

f(h) =

p

i=0

1 i! f (i)(0) hi + O(hp+1) f(t) − p(t) = 1 (p + 1)! f (p+1)(t∗)

p

i=0

(t − ti) = O(h(p+1))

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Leif Kobbelt RWTH Aachen University

computable functions
Taylor expansion
interpolation error (mean value theorem)

Polynomials

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p(ti) = f(ti), ti ∈ [0, h]

p(t) =

p

i=0

ci ti =

p

i=0

c

i Φi(t)

f(t) − p(t) = 1 (p + 1)! f (p+1)(t∗)

p

i=0

(t − ti) = O(h(p+1))

f(h) =

p

i=0

1 i! f (i)(0) hi + O(hp+1)

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Leif Kobbelt RWTH Aachen University

Polynomials

computable functions
Taylor expansion
interpolation error (mean value theorem)

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p(ti) = f(ti), ti ∈ [0, h]

p(t) =

p

i=0

ci ti =

p

i=0

c

i Φi(t)

f(h) =

p

i=0

1 i! f (i)(0) hi + O(hp+1)

f(t) − p(t) = 1 (p + 1)! f (p+1)(t∗)

p

i=0

(t − ti) = O(h(p+1))

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Leif Kobbelt RWTH Aachen University

Implicit Polynomials

interpolation error of the function values
approximation error of the contour

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F(p + p) − F(p) p ≈ ∇F(p) p = λ ∇F(p) F(x, y, z) − P(x, y, z) = O(h(p+1))

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Leif Kobbelt RWTH Aachen University

Implicit Polynomials

interpolation error of the function values
approximation error of the contour

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F(p + p) − F(p) p ≈ ∇F(p) p = λ ∇F(p) F(x, y, z) − P(x, y, z) = O(h(p+1))

p p+Δp

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Leif Kobbelt RWTH Aachen University

Implicit Polynomials

interpolation error of the function values
approximation error of the contour

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p ≈ F(p + p) − F(p) ∇F(p) p = λ ∇F(p) (gradient bounded from below) F(x, y, z) − P(x, y, z) = O(h(p+1))

p p+Δp

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Leif Kobbelt RWTH Aachen University

Implicit Polynomials

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large gradient small gradient F(x,y,z) F(x,y,z) x,y,z x,y,z

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Leif Kobbelt RWTH Aachen University

Polynomial Approximation

approximation error is O(hp+1)
improve approximation quality by

– increasing p ... higher order polynomials – decreasing h ... smaller / more segments

issues

– smoothness of the target data ( maxt f(p+1)(t) ) – handling higher order patches (e.g. boundary conditions)

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Leif Kobbelt RWTH Aachen University

Polynomial Approximation

approximation error is O(hp+1)
improve approximation quality by

– increasing p ... higher order polynomials – decreasing h ... smaller / more segments

issues

– smoothness of the target data ( maxt f(p+1)(t) ) – handling higher order patches (e.g. boundary conditions)

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MOTD: p = 1 MOTD: p = 1

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Leif Kobbelt RWTH Aachen University

Piecewise Definition

parametric

– Euler formula: V - E + F = 2 (1-g) – regular quad meshes

F ≈ V
E ≈ 2V
average valence = 4

– quasi-regular – semi-regular

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Leif Kobbelt RWTH Aachen University

Piecewise Definition

parametric

– Euler formula: V - E + F = 2 (1-g) – regular triangle meshes

F ≈ 2V
E ≈ 3V
average valence = 6

– quasi-regular – semi-regular

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Leif Kobbelt RWTH Aachen University

Piecewise Definition

quasi regular (→ remeshing, Pierre)

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Leif Kobbelt RWTH Aachen University

Piecewise Definition

semi regular

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Leif Kobbelt RWTH Aachen University

Piecewise Definition

semi regular

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Leif Kobbelt RWTH Aachen University

Piecewise Definition

implicit

– regular voxel grids O(h-3) – three color octrees

surface-adaptive refinement O(h-2)
feature-adaptive refinement O(h-1)

– irregular hierarchies

binary space partition O(h-1)

(BSP)

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Leif Kobbelt RWTH Aachen University

3-Color Octree

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12040 cells 1048576 cells

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Leif Kobbelt RWTH Aachen University

Adaptively Sampled Distance Fields

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12040 cells 895 cells

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Leif Kobbelt RWTH Aachen University

Binary Space Partitions

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254 cells 895 cells

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Leif Kobbelt RWTH Aachen University

Message of the Day ...

polygonal meshes are a good compromise

– approximation O(h2) ... error * #faces = const. – arbitrary topology – flexibility for piecewise smooth surfaces – flexibility for adaptive refinement

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Leif Kobbelt RWTH Aachen University

Message of the Day ...

polygonal meshes are a good compromise

– approximation O(h2) ... error * #faces = const. – arbitrary topology – flexibility for piecewise smooth surfaces – flexibility for adaptive refinement

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Leif Kobbelt RWTH Aachen University

Message of the Day ...

polygonal meshes are a good compromise

– approximation O(h2) ... error * #faces = const. – arbitrary topology – flexibility for piecewise smooth surfaces – flexibility for adaptive refinement

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Leif Kobbelt RWTH Aachen University

Message of the Day ...

polygonal meshes are a good compromise

– approximation O(h2) ... error * #faces = const. – arbitrary topology – flexibility for piecewise smooth surfaces – flexibility for adaptive refinement

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Leif Kobbelt RWTH Aachen University

Message of the Day ...

polygonal meshes are a good compromise

– approximation O(h2) ... error * #faces = const. – arbitrary topology – flexibility for piecewise smooth surfaces – flexibility for adaptive refinement – efficient rendering

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Leif Kobbelt RWTH Aachen University

Message of the Day ...

polygonal meshes are a good compromise

– approximation O(h2) ... error * #faces = const. – arbitrary topology – flexibility for piecewise smooth surfaces – flexibility for adaptive refinement – efficient rendering

implicit representation can support efficient

access to vertices, faces, ....

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Leif Kobbelt RWTH Aachen University

(mathematical) geometry representations

– parametric vs. implicit

approximation properties
types of operations

– evaluation – distance queries – modification / deformation

data structures

Outline

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Leif Kobbelt RWTH Aachen University

Evaluation

smooth parametric surfaces

– positions – normals – curvatures

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f(u, v) n(u, v) = fu(u, v) × fv(u, v) c(u, v) = C

fuu(u, v), fuv(u, v), fvv(u, v)
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Leif Kobbelt RWTH Aachen University

Evaluation

smooth parametric surfaces

– positions – normals – curvatures

generalization to triangle meshes

– positions (barycentric coordinates)

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(α, β) → α P1 + β P2 + (1−α−β) P3

f(u, v) n(u, v) = fu(u, v) × fv(u, v) c(u, v) = C

fuu(u, v), fuv(u, v), fvv(u, v)
0 ≤ α,

0 ≤ β, α + β ≤ 1

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Leif Kobbelt RWTH Aachen University

Evaluation

smooth parametric surfaces

– positions – normals – curvatures

generalization to triangle meshes

– positions (barycentric coordinates)

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(α, β, γ) → α P1 + β P2 + γ P3

f(u, v) n(u, v) = fu(u, v) × fv(u, v) c(u, v) = C

fuu(u, v), fuv(u, v), fvv(u, v)
α + β + γ = 1

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Leif Kobbelt RWTH Aachen University

Evaluation

smooth parametric surfaces

– positions – normals – curvatures

generalization to triangle meshes

– positions (barycentric coordinates)

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α u + β v + γ w → α P1 + β P2 + γ P3

f(u, v) n(u, v) = fu(u, v) × fv(u, v) c(u, v) = C

fuu(u, v), fuv(u, v), fvv(u, v)
α + β + γ = 0

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Leif Kobbelt RWTH Aachen University

Evaluation

smooth parametric surfaces

– positions – normals – curvatures

generalization to triangle meshes

– positions (barycentric coordinates)

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α u + β v + γ w → α P1 + β P2 + γ P3

f(u, v) n(u, v) = fu(u, v) × fv(u, v) c(u, v) = C

fuu(u, v), fuv(u, v), fvv(u, v)
α + β + γ = 0

→ Parametrization Bruno

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Leif Kobbelt RWTH Aachen University

Evaluation

smooth parametric surfaces

– positions – normals – curvatures

generalization to triangle meshes

– positions (barycentric coordinates) – normals (per face, Phong)

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f(u, v) n(u, v) = fu(u, v) × fv(u, v) c(u, v) = C

fuu(u, v), fuv(u, v), fvv(u, v)
N = (P2 − P1) × (P3 − P1)

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Leif Kobbelt RWTH Aachen University

Evaluation

smooth parametric surfaces

– positions – normals – curvatures

generalization to triangle meshes

– positions (barycentric coordinates) – normals (per face, Phong)

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f(u, v) n(u, v) = fu(u, v) × fv(u, v) c(u, v) = C

fuu(u, v), fuv(u, v), fvv(u, v)
α u + β v + γ w → α N1 + β N2 + γ N3

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Leif Kobbelt RWTH Aachen University

Evaluation

smooth parametric surfaces

– positions – normals – curvatures

generalization to triangle meshes

– positions (barycentric coordinates) – normals (per face, Phong) – curvatures ... (→ smoothing, Mark)

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f(u, v) n(u, v) = fu(u, v) × fv(u, v) c(u, v) = C

fuu(u, v), fuv(u, v), fvv(u, v)
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Leif Kobbelt RWTH Aachen University

Distance Queries

parametric

– for smooth surfaces: find orthogonal base point – for triangle meshes

use kd-tree or BSP to find closest triangle
find base point by Newton iteration

(use Phong normal field)

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[p − f(u, v)] × n(u, v) = 0

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Leif Kobbelt RWTH Aachen University

Modifications

parameteric

– control vertices – free-form deformation – boundary constraint modeling

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f (u, v) =

n

i=0

m

j=0

cijN n

i (u) N m j (v)

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Leif Kobbelt RWTH Aachen University

Modifications

parameteric

– control vertices – free-form deformation – boundary constraint modeling

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f (u, v) =

n

i=0

m

j=0

cijN n

i (u) N m j (v)

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Leif Kobbelt RWTH Aachen University

parameteric

– control vertices – free-form deformation – boundary constraint modeling

Modifications

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→ Remeshing Pierre

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Leif Kobbelt RWTH Aachen University

parameteric

– control vertices – free-form deformation – boundary constraint modeling

Modifications

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Leif Kobbelt RWTH Aachen University

parameteric

– control vertices – free-form deformation – boundary constraint modeling

Modifications

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Leif Kobbelt RWTH Aachen University

parameteric

– control vertices – free-form deformation – boundary constraint modeling

Modifications

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Leif Kobbelt RWTH Aachen University

parameteric

– control vertices – free-form deformation – boundary constraint modeling

Modifications

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→ Mesh Editing Mario

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Leif Kobbelt RWTH Aachen University

(mathematical) geometry representations

– parametric vs. implicit

approximation properties
types of operations

– distance queries – evaluation – modification / deformation

data structures

Outline

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Leif Kobbelt RWTH Aachen University

Mesh Data Structures

how to store geometry & connectivity?
compact storage

– file formats

efficient algorithms on meshes

– identify time-critical operations – all vertices/edges of a face – all incident vertices/edges/faces of a vertex

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Leif Kobbelt RWTH Aachen University

Face Set (STL)

face:

– 3 positions

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Triangles x11 y11 z11 x12 y12 z12 x13 y13 z13 x21 y21 z21 x22 y22 z22 x23 y23 z23 ... ... ... xF1 yF1 zF1 xF2 yF2 zF2 xF3 yF3 zF3

36 B/f = 72 B/v no connectivity!

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Leif Kobbelt RWTH Aachen University

Shared Vertex (OBJ, OFF)

vertex:

– position

face:

– vertex indices

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Vertices x1 y1 z1 ... xV yV zV Triangles v11 v12 v13 ... ... ... ... vF1 vF2 vF3

12 B/v + 12 B/f = 36 B/v no neighborhood info

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Leif Kobbelt RWTH Aachen University

Face-Based Connectivity

vertex:

– position – 1 face

face:

– 3 vertices – 3 face neighbors

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64 B/v no edges!

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Leif Kobbelt RWTH Aachen University

Edge-Based Connectivity

vertex

– position – 1 edge

edge

– 2 vertices – 2 faces – 4 edges

face

– 1 edge

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120 B/v edge orientation?

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Leif Kobbelt RWTH Aachen University

Halfedge-Based Connectivity

vertex

– position – 1 halfedge

halfedge

– 1 vertex – 1 face – 1, 2, or 3 halfedges

face

– 1 halfedge

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96 to 144 B/v no case distinctions during traversal

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Leif Kobbelt RWTH Aachen University 72

One-Ring Traversal

1. Start at vertex

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Leif Kobbelt RWTH Aachen University 73

One-Ring Traversal

1. Start at vertex
2. Outgoing halfedge

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Leif Kobbelt RWTH Aachen University 74

One-Ring Traversal

1. Start at vertex
2. Outgoing halfedge
3. Opposite halfedge

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Leif Kobbelt RWTH Aachen University 75

One-Ring Traversal

1. Start at vertex
2. Outgoing halfedge
3. Opposite halfedge
4. Next halfedge

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Leif Kobbelt RWTH Aachen University 76

One-Ring Traversal

1. Start at vertex
2. Outgoing halfedge
3. Opposite halfedge
4. Next halfedge
5. Opposite

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Leif Kobbelt RWTH Aachen University 77

One-Ring Traversal

1. Start at vertex
2. Outgoing halfedge
3. Opposite halfedge
4. Next halfedge
5. Opposite
6. Next
7. ...

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Leif Kobbelt RWTH Aachen University

Halfedge-Based Libraries

CGAL

– www.cgal.org – computational geometry – free for non-commercial use

OpenMesh

– www.openmesh.org – mesh processing – free, LGPL licence

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Leif Kobbelt RWTH Aachen University

Literature

Kettner, Using generic programming for designing a

data structure for polyhedral surfaces, Symp. on

Comp. Geom., 1998
Campagna et al, Directed Edges - A Scalable

Representation for Triangle Meshes, Journal of Graphics Tools 4(3), 1998

Botsch et al, OpenMesh - A generic and efficient

polygon mesh data structure, OpenSG Symp. 2002

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Leif Kobbelt RWTH Aachen University

Outline

(mathematical) geometry representations

– parametric vs. implicit

approximation properties
types of operations

– distance queries – evaluation – modification / deformation

data structures

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