javier lezama. matia natali
DESCRIPTION
Whorkshop Impa Noviembre 2011TRANSCRIPT
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
J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 4 / 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
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
J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 7 / 35
Chartification
Chart
J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 8 / 35
Chartification
Patch
J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 9 / 35
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.
J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 26 / 35
Resampling step
Next figure shows a failure of the algorithm to fully capture the sharpfeatures of the Fandisk model.
J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 27 / 35
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 =)
J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 35 / 35