ON THE METHOD FOR SEMI-REGULARREMESHING PROPOSED BY IGOR GUSKOV

Javier Lezama *, Mattia Natali **

*Facultad de Matematica, Astronomıa y Fısica, Universidad Nacional de Cordoba**University of Bergen

November 22nd, 2011

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 1 / 35

1 Introduction

2 Chartification

3 Parameterization

4 Optimization

5 Resampling step

6 Conclusions

7 Restrictions

8 How it works

9 References

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 2 / 35

Introduction

Table of contents

1 Introduction

2 Chartification

3 Parameterization

4 Optimization

5 Resampling step

6 Conclusions

7 Restrictions

8 How it works

9 References

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 3 / 35

Introduction

Aim of the presentation

Imput mesh Semi-regular remeshing

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Introduction

Semi-Regular mesh

easy level-of-detail management.

efficient data structures.

efficient processing algorithms.

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 5 / 35

Introduction

Semi-Regular mesh

easy level-of-detail management.

efficient data structures.

efficient processing algorithms.

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 5 / 35

Introduction

Semi-Regular mesh

easy level-of-detail management.

efficient data structures.

efficient processing algorithms.

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 5 / 35

Introduction

Semi-Regular mesh

easy level-of-detail management.

efficient data structures.

efficient processing algorithms.

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 5 / 35

Chartification

Table of contents

1 Introduction

2 Chartification

3 Parameterization

4 Optimization

5 Resampling step

6 Conclusions

7 Restrictions

8 How it works

9 References

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 6 / 35

Chartification

Tile

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Chartification

Chart

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Chartification

Patch

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Chartification

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 10 / 35

Parameterization

Table of contents

1 Introduction

2 Chartification

3 Parameterization

4 Optimization

5 Resampling step

6 Conclusions

7 Restrictions

8 How it works

9 References

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 11 / 35

Parameterization

Original mesh: M = (VM ,EM ,FM) Base mesh: B = (VB ,EB ,FB).

u : VM → |B|u : |M| → |B|

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Parameterization

From base mesh to p-domain.

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Parameterization

ρb(v) = Rb(u(v))

Db = t ∈ FM : t = (v1, v2, v3), vk ∈ u−1(Ωb), k = 1, 2, 3.

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 14 / 35

Parameterization

ρb(v) = Rb(u(v))

Db = t ∈ FM : t = (v1, v2, v3), vk ∈ u−1(Ωb), k = 1, 2, 3.

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 14 / 35

Parameterization

How can we compute the u map?

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Optimization

Table of contents

1 Introduction

2 Chartification

3 Parameterization

4 Optimization

5 Resampling step

6 Conclusions

7 Restrictions

8 How it works

9 References

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 16 / 35

Optimization

u(0) obtained from the parameterization process ( ρ = Rb u(0)).

Using u(0) for the parameterization, we would get distortion betweenadjacent patches

Courtesy of Andre Maximo

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 17 / 35

Optimization

u(0) obtained from the parameterization process ( ρ = Rb u(0)).

Using u(0) for the parameterization, we would get distortion betweenadjacent patches

Courtesy of Andre Maximo

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 17 / 35

Optimization

u(0) obtained from the parameterization process ( ρ = Rb u(0)).

Using u(0) for the parameterization, we would get distortion betweenadjacent patches

Courtesy of Andre Maximo

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 17 / 35

Optimization

u(0) obtained from the parameterization process ( ρ = Rb u(0)).

Using u(0) for the parameterization, we would get distortion betweenadjacent patches

Courtesy of Andre Maximo

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 17 / 35

Optimization

For a single base vertex b, we can consider the mappingRb u : u−1(Ωb)→ R2.

Rb(u(v)) =∑

vi∈ω1(v)

avviRb(u(vi )),

where avvi are MVP coefficients and ω1(v) is the one-ring ofneighbors of v .

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 18 / 35

Optimization

For a single base vertex b, we can consider the mappingRb u : u−1(Ωb)→ R2.

Rb(u(v)) =∑

vi∈ω1(v)

avviRb(u(vi )),

where avvi are MVP coefficients and ω1(v) is the one-ring ofneighbors of v .

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 18 / 35

Optimization

For a single base vertex b, we can consider the mappingRb u : u−1(Ωb)→ R2.

Rb(u(v)) =∑

vi∈ω1(v)

avviRb(u(vi )),

where avvi are MVP coefficients and ω1(v) is the one-ring ofneighbors of v .

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 18 / 35

Optimization

For a single base vertex b, we can consider the mappingRb u : u−1(Ωb)→ R2.

Rb(u(v)) =∑

vi∈ω1(v)

avviRb(u(vi )),

where avvi are MVP coefficients and ω1(v) is the one-ring ofneighbors of v .

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 18 / 35

Optimization

J(u) =def∑v∈Vm

∑b:ω1(v)∪v⊂u−1(Ωb)

σ(v)wb(u(v))

×

Rb(u(v))−∑

v∈ω1(v)

avviRb(u(vi ))

2

where σ(v) is the area associated with the vertex v .

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 19 / 35

Optimization

J(u) =def∑v∈Vm

∑b:ω1(v)∪v⊂u−1(Ωb)

σ(v)wb(u(v))

×

Rb(u(v))−∑

v∈ω1(v)

avviRb(u(vi ))

2

where σ(v) is the area associated with the vertex v .

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 19 / 35

Optimization

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 20 / 35

Optimization

The parametric energy functional J(u) is then re-expressed in termsof these 2D values. At this stage, a standard optimization procedureis invoked to produce the locally optimal values for each vertex.

F (y) =∑v∈Λe

4∑k=1

σ(v)wbk (ξ−1(y(v)))

× (ζk(y(v))−∑

v ′∈ω1(v)

avv ′ζk(y(v ′)))2

ζk(y) = Kh6

val(bk )

(Aek(y1 + iy2) + Be

k ), k = 1, 2, 3, 4.

A1 = 1,B1 = 1/2; A2 = −1,B2 = 12 ; A3 = i ,B3 =

√3

2 ; A4 = −i ,B4 =√

32

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 21 / 35

Optimization

The parametric energy functional J(u) is then re-expressed in termsof these 2D values. At this stage, a standard optimization procedureis invoked to produce the locally optimal values for each vertex.

F (y) =∑v∈Λe

4∑k=1

σ(v)wbk (ξ−1(y(v)))

× (ζk(y(v))−∑

v ′∈ω1(v)

avv ′ζk(y(v ′)))2

ζk(y) = Kh6

val(bk )

(Aek(y1 + iy2) + Be

k ), k = 1, 2, 3, 4.

A1 = 1,B1 = 1/2; A2 = −1,B2 = 12 ; A3 = i ,B3 =

√3

2 ; A4 = −i ,B4 =√

32

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 21 / 35

Optimization

The parametric energy functional J(u) is then re-expressed in termsof these 2D values. At this stage, a standard optimization procedureis invoked to produce the locally optimal values for each vertex.

F (y) =∑v∈Λe

4∑k=1

σ(v)wbk (ξ−1(y(v)))

× (ζk(y(v))−∑

v ′∈ω1(v)

avv ′ζk(y(v ′)))2

ζk(y) = Kh6

val(bk )

(Aek(y1 + iy2) + Be

k ), k = 1, 2, 3, 4.

A1 = 1,B1 = 1/2; A2 = −1,B2 = 12 ; A3 = i ,B3 =

√3

2 ; A4 = −i ,B4 =√

32

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 21 / 35

Optimization

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 22 / 35

Resampling step

Table of contents

1 Introduction

2 Chartification

3 Parameterization

4 Optimization

5 Resampling step

6 Conclusions

7 Restrictions

8 How it works

9 References

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 23 / 35

Resampling step

The sampling stage uniformly refines the base mesh triangles to thedesired level.

Then we need to invert the mapping to construct the outputremeshes at different levels.

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 24 / 35

Resampling step

The sampling stage uniformly refines the base mesh triangles to thedesired level.

Then we need to invert the mapping to construct the outputremeshes at different levels.

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 24 / 35

Resampling step

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 25 / 35

Resampling step

Example of resampled mesh.

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Resampling step

Next figure shows a failure of the algorithm to fully capture the sharpfeatures of the Fandisk model.

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Conclusions

Table of contents

1 Introduction

2 Chartification

3 Parameterization

4 Optimization

5 Resampling step

6 Conclusions

7 Restrictions

8 How it works

9 References

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 28 / 35

Conclusions

We showed a manifold-based method for semi-regular remeshing.

The method is easy to implement and require almost no userintervention except for the choice of the base domain complexity.

The resampling does not require a meta-mesh construction, like othermethods and is simple to implement.

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 29 / 35

Conclusions

We showed a manifold-based method for semi-regular remeshing.

The method is easy to implement and require almost no userintervention except for the choice of the base domain complexity.

The resampling does not require a meta-mesh construction, like othermethods and is simple to implement.

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 29 / 35

Conclusions

We showed a manifold-based method for semi-regular remeshing.

The method is easy to implement and require almost no userintervention except for the choice of the base domain complexity.

The resampling does not require a meta-mesh construction, like othermethods and is simple to implement.

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 29 / 35

Restrictions

Table of contents

1 Introduction

2 Chartification

3 Parameterization

4 Optimization

5 Resampling step

6 Conclusions

7 Restrictions

8 How it works

9 References

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 30 / 35

Restrictions

This method is specifically targeted at the semi-regular meshconstruction, and will not work for morphing applications.

This method assumes that the input mesh is a valid triangular meshwith topological noise removed.

While this method can handle meshes with boundaries, sharp creasesare not preserved in the current implementation.

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 31 / 35

Restrictions

This method is specifically targeted at the semi-regular meshconstruction, and will not work for morphing applications.

This method assumes that the input mesh is a valid triangular meshwith topological noise removed.

While this method can handle meshes with boundaries, sharp creasesare not preserved in the current implementation.

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 31 / 35

Restrictions

This method is specifically targeted at the semi-regular meshconstruction, and will not work for morphing applications.

This method assumes that the input mesh is a valid triangular meshwith topological noise removed.

While this method can handle meshes with boundaries, sharp creasesare not preserved in the current implementation.

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 31 / 35

How it works

Table of contents

1 Introduction

2 Chartification

3 Parameterization

4 Optimization

5 Resampling step

6 Conclusions

7 Restrictions

8 How it works

9 References

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 32 / 35

References

Table of contents

1 Introduction

2 Chartification

3 Parameterization

4 Optimization

5 Resampling step

6 Conclusions

7 Restrictions

8 How it works

9 References

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 33 / 35

References

Igor Guskov, Manifold-based approach to semi-regular remeshing.Graphical Models 69 (2007) 1-18.http://www.sciencedirect.com/science/article/pii/S1524070306000385

Sofware available at: http://www.guskov.org/trireme

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 34 / 35

References

Igor Guskov, Manifold-based approach to semi-regular remeshing.Graphical Models 69 (2007) 1-18.http://www.sciencedirect.com/science/article/pii/S1524070306000385

Sofware available at: http://www.guskov.org/trireme

J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 34 / 35

References

Thanks, Obrigado, Merci, Takk, Grazie, Gracias =)

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