ON THE METHOD FOR SEMI-REGULAR REMESHING PROPOSED BY IGOR GUSKOV - - PowerPoint PPT Presentation

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on the method for semi regular remeshing proposed by igor

ON THE METHOD FOR SEMI-REGULAR REMESHING PROPOSED BY IGOR GUSKOV - - PowerPoint PPT Presentation

Jul 30, 2023 243 likes •795 views

ON THE METHOD FOR SEMI-REGULAR REMESHING PROPOSED BY IGOR GUSKOV Javier Lezama *, Mattia Natali ** *Facultad de Matem atica, Astronom a y F sica, Universidad Nacional de C ordoba **University of Bergen November 22nd, 2011 J.

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SLIDE 1

ON THE METHOD FOR SEMI-REGULAR REMESHING PROPOSED BY IGOR GUSKOV

Javier Lezama *, Mattia Natali **

*Facultad de Matem´ atica, Astronom´ ıa y F´ ısica, Universidad Nacional de C´

rdoba

**University of Bergen

November 22nd, 2011

J. Lezama, M. Natali ()

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SLIDE 2

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 ()

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SLIDE 3

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 ()

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SLIDE 4

Introduction

Aim of the presentation Imput mesh Semi-regular remeshing

J. Lezama, M. Natali ()

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SLIDE 5

Introduction

Semi-Regular mesh

J. Lezama, M. Natali ()

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SLIDE 6

Introduction

Semi-Regular mesh easy level-of-detail management.

J. Lezama, M. Natali ()

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SLIDE 7

Introduction

Semi-Regular mesh easy level-of-detail management. efficient data structures.

J. Lezama, M. Natali ()

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SLIDE 8

Introduction

Semi-Regular mesh easy level-of-detail management. efficient data structures. efficient processing algorithms.

J. Lezama, M. Natali ()

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SLIDE 9

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 ()

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SLIDE 10

Chartification

Tile

J. Lezama, M. Natali ()

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SLIDE 11

Chartification

Chart

J. Lezama, M. Natali ()

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SLIDE 12

Chartification

Patch

J. Lezama, M. Natali ()

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SLIDE 13

Chartification

J. Lezama, M. Natali ()

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SLIDE 14

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 ()

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SLIDE 15

Parameterization

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

J. Lezama, M. Natali ()

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SLIDE 16

Parameterization

From base mesh to p-domain.

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SLIDE 17

Parameterization

ρb(v) = Rb(u(v))

J. Lezama, M. Natali ()

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SLIDE 18

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 ()

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SLIDE 19

Parameterization

How can we compute the u map?

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SLIDE 20

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 ()

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SLIDE 21

Optimization

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

J. Lezama, M. Natali ()

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SLIDE 22

Optimization

u(0) obtained from the parameterization process ( ρ = Rb ◦ u(0)). Using u(0) for the parameterization, we would get distortion between adjacent patches

J. Lezama, M. Natali ()

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SLIDE 23

Optimization

u(0) obtained from the parameterization process ( ρ = Rb ◦ u(0)). Using u(0) for the parameterization, we would get distortion between adjacent patches

J. Lezama, M. Natali ()

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SLIDE 24

Optimization

u(0) obtained from the parameterization process ( ρ = Rb ◦ u(0)). Using u(0) for the parameterization, we would get distortion between adjacent patches Courtesy of Andr´ e M´ aximo

J. Lezama, M. Natali ()

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SLIDE 25

Optimization

For a single base vertex b, we can consider the mapping Rb ◦ u : u−1(Ωb) → R2. Rb(u(v)) =

vi∈ω1(v)

avviRb(u(vi)),

J. Lezama, M. Natali ()

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SLIDE 26

Optimization

For a single base vertex b, we can consider the mapping Rb ◦ 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 of neighbors of v.

J. Lezama, M. Natali ()

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SLIDE 27

Optimization

For a single base vertex b, we can consider the mapping Rb ◦ 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 of neighbors of v.

J. Lezama, M. Natali ()

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SLIDE 28

Optimization

For a single base vertex b, we can consider the mapping Rb ◦ 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 of neighbors of v.

J. Lezama, M. Natali ()

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SLIDE 29

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 ()

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SLIDE 30

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 ()

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SLIDE 31

Optimization

J. Lezama, M. Natali ()

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SLIDE 32

Optimization

The parametric energy functional J(u) is then re-expressed in terms

f these 2D values. At this stage, a standard optimization procedure

is invoked to produce the locally optimal values for each vertex.

J. Lezama, M. Natali ()

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SLIDE 33

Optimization

The parametric energy functional J(u) is then re-expressed in terms

f these 2D values. At this stage, a standard optimization procedure

is invoked to produce the locally optimal values for each vertex.

J. Lezama, M. Natali ()

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SLIDE 34

Optimization

The parametric energy functional J(u) is then re-expressed in terms

f these 2D values. At this stage, a standard optimization procedure

is invoked to produce the locally optimal values for each vertex. F(y) =

v∈ ˇ

Λe 4

k=1

σ(v)wbk(ξ−1(y(v))) × (ζk(y(v)) −

v′∈ω1(v)

avv′ζk(y(v′)))2 ζk(y) = K h

6 val(bk ) (Ae

k(y1 + iy2) + Be k), k = 1, 2, 3, 4.

A1 = 1, B1 = 1/2; A2 = −1, B2 = 1

2; A3 = i, B3 = √ 3 2 ; A4 = −i, B4 = √ 3 2

J. Lezama, M. Natali ()

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SLIDE 35

Optimization

J. Lezama, M. Natali ()

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SLIDE 36

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 ()

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SLIDE 37

Resampling step

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

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SLIDE 38

Resampling step

The sampling stage uniformly refines the base mesh triangles to the desired level. Then we need to invert the mapping to construct the output remeshes at different levels.

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SLIDE 39

Resampling step

J. Lezama, M. Natali ()

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SLIDE 40

Resampling step

Example of resampled mesh.

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SLIDE 41

Resampling step

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

J. Lezama, M. Natali ()

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SLIDE 42

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 ()

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SLIDE 43

Conclusions

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

J. Lezama, M. Natali ()

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SLIDE 44

Conclusions

We showed a manifold-based method for semi-regular remeshing. The method is easy to implement and require almost no user intervention except for the choice of the base domain complexity.

J. Lezama, M. Natali ()

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SLIDE 45

Conclusions

We showed a manifold-based method for semi-regular remeshing. The method is easy to implement and require almost no user intervention except for the choice of the base domain complexity. The resampling does not require a meta-mesh construction, like other methods and is simple to implement.

J. Lezama, M. Natali ()

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SLIDE 46

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 ()

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SLIDE 47

Restrictions

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

J. Lezama, M. Natali ()

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SLIDE 48

Restrictions

This method is specifically targeted at the semi-regular mesh construction, and will not work for morphing applications. This method assumes that the input mesh is a valid triangular mesh with topological noise removed.

J. Lezama, M. Natali ()

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SLIDE 49

Restrictions

This method is specifically targeted at the semi-regular mesh construction, and will not work for morphing applications. This method assumes that the input mesh is a valid triangular mesh with topological noise removed. While this method can handle meshes with boundaries, sharp creases are not preserved in the current implementation.

J. Lezama, M. Natali ()

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SLIDE 50

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 ()

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SLIDE 51

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 ()

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SLIDE 52

References

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

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SLIDE 53

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 ()

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SLIDE 54

References

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

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