Javier Lezama. Matia Natali

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

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

Introduction

Aim of the presentation

Imput mesh Semi-regular remeshing

Introduction

Semi-Regular mesh

easy level-of-detail management.

efficient data structures.

efficient processing algorithms.

Introduction

Semi-Regular mesh

Introduction

Semi-Regular mesh

Introduction

Semi-Regular mesh

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

Chartification

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

Parameterization

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

u : VM → |B|u : |M| → |B|

Parameterization

From base mesh to p-domain.

Parameterization

ρb(v) = Rb(u(v))

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

Parameterization

ρb(v) = Rb(u(v))

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

Parameterization

How can we compute the u map?

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

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

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 .

Optimization

Rb(u(v)) =∑

vi∈ω1(v)

avviRb(u(vi )),

Optimization

Rb(u(v)) =∑

vi∈ω1(v)

avviRb(u(vi )),

Optimization

Rb(u(v)) =∑

vi∈ω1(v)

avviRb(u(vi )),

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 ))

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

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 ))

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

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 =

2 ; A4 = −i ,B4 =√

Optimization

F (y) =∑v∈Λe

4∑k=1

× (ζk(y(v))−∑

v ′∈ω1(v)

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 =

2 ; A4 = −i ,B4 =√

Optimization

F (y) =∑v∈Λe

4∑k=1

× (ζk(y(v))−∑

v ′∈ω1(v)

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 =

2 ; A4 = −i ,B4 =√

Optimization

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

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.

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.

Resampling step

Example of resampled mesh.

Resampling step

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

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

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.

Conclusions

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

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.

Restrictions

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

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

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

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

References

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

Javier Lezama. Matia Natali

Documents

Transcript of Javier Lezama. Matia Natali