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

• 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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M O T D : p = 1 M O T D : 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)
• 50

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)
• 57

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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