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

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´ ordoba **University of Bergen November 22nd, 2011 J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 1 / 35

Introduction 1 Chartification 2 Parameterization 3 Optimization 4 Resampling step 5 Conclusions 6 Restrictions 7 How it works 8 References 9 J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 2 / 35

Introduction Table of contents Introduction 1 Chartification 2 Parameterization 3 Optimization 4 Resampling step 5 Conclusions 6 Restrictions 7 How it works 8 References 9 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 J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 5 / 35

Introduction Semi-Regular mesh easy level-of-detail management. J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 5 / 35

Introduction Semi-Regular mesh easy level-of-detail management. efficient data structures. 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 Introduction 1 Chartification 2 Parameterization 3 Optimization 4 Resampling step 5 Conclusions 6 Restrictions 7 How it works 8 References 9 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 Introduction 1 Chartification 2 Parameterization 3 Optimization 4 Resampling step 5 Conclusions 6 Restrictions 7 How it works 8 References 9 J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 11 / 35

Parameterization Original mesh: M = ( V M , E M , F M ) Base mesh: B = ( V B , E B , F B ). u : V M → | B | u : | M | → | B | ¯ J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 12 / 35

Parameterization From base mesh to p-domain. J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 13 / 35

Parameterization ρ b ( v ) = R b ( u ( v )) J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 14 / 35

Parameterization ρ b ( v ) = R b ( u ( v )) D b = { t ∈ F M : t = ( v 1 , v 2 , v 3 ) , v k ∈ u − 1 (Ω b ) , k = 1 , 2 , 3 } . J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 14 / 35

Parameterization How can we compute the u map? J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 15 / 35

Optimization Table of contents Introduction 1 Chartification 2 Parameterization 3 Optimization 4 Resampling step 5 Conclusions 6 Restrictions 7 How it works 8 References 9 J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 16 / 35

Optimization u (0) obtained from the parameterization process ( ρ = R b ◦ u (0) ) . J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 17 / 35

Optimization u (0) obtained from the parameterization process ( ρ = R b ◦ u (0) ) . Using u (0) for the parameterization, we would get distortion between adjacent patches J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 17 / 35

Optimization u (0) obtained from the parameterization process ( ρ = R b ◦ u (0) ) . Using u (0) for the parameterization, we would get distortion between adjacent patches J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 17 / 35

Optimization u (0) obtained from the parameterization process ( ρ = R b ◦ 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 () SEMI-REGULAR REMESHING November 22nd, 2011 17 / 35

Optimization For a single base vertex b , we can consider the mapping R b ◦ u : u − 1 (Ω b ) → R 2 . � R b ( u ( v )) = a vv i R b ( u ( v i )) , v i ∈ ω 1 ( v ) J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 18 / 35

Optimization For a single base vertex b , we can consider the mapping R b ◦ u : u − 1 (Ω b ) → R 2 . � R b ( u ( v )) = a vv i R b ( u ( v i )) , v i ∈ ω 1 ( v ) where a vv i are MVP coefficients and ω 1 ( v ) is the one-ring of neighbors 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 mapping R b ◦ u : u − 1 (Ω b ) → R 2 . � R b ( u ( v )) = a vv i R b ( u ( v i )) , v i ∈ ω 1 ( v ) where a vv i are MVP coefficients and ω 1 ( v ) is the one-ring of neighbors 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 mapping R b ◦ u : u − 1 (Ω b ) → R 2 . � R b ( u ( v )) = a vv i R b ( u ( v i )) , v i ∈ ω 1 ( v ) where a vv i are MVP coefficients and ω 1 ( v ) is the one-ring of neighbors of v . J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 18 / 35

Optimization J ( u ) = def � � σ ( v ) w b ( u ( v )) v ∈ V m b : ω 1 ( v ) ∪{ v }⊂ u − 1 (Ω b ) 2   � ×  R b ( u ( v )) − a vv i R b ( u ( v i ))  v ∈ ω 1 ( v ) 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 � � σ ( v ) w b ( u ( v )) v ∈ V m b : ω 1 ( v ) ∪{ v }⊂ u − 1 (Ω b ) 2   � ×  R b ( u ( v )) − a vv i R b ( u ( v i ))  v ∈ ω 1 ( v ) 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 terms of 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 () SEMI-REGULAR REMESHING November 22nd, 2011 21 / 35

Optimization The parametric energy functional J ( u ) is then re-expressed in terms of 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 () SEMI-REGULAR REMESHING November 22nd, 2011 21 / 35

Optimization The parametric energy functional J ( u ) is then re-expressed in terms of these 2D values. At this stage, a standard optimization procedure is invoked to produce the locally optimal values for each vertex. 4 � � σ ( v ) w b k ( ξ − 1 ( y ( v ))) F ( y ) = v ∈ ˇ k =1 Λ e � a vv ′ ζ k ( y ( v ′ ))) 2 × ( ζ k ( y ( v )) − v ′ ∈ ω 1 ( v ) ζ k ( y ) = K h val ( bk ) ( A e k ( y 1 + iy 2 ) + B e k ) , k = 1 , 2 , 3 , 4 . 6 √ √ A 1 = 1 , B 1 = 1 / 2; A 2 = − 1 , B 2 = 1 3 3 2 ; A 3 = i , B 3 = 2 ; A 4 = − i , B 4 = 2 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 Introduction 1 Chartification 2 Parameterization 3 Optimization 4 Resampling step 5 Conclusions 6 Restrictions 7 How it works 8 References 9 J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 23 / 35

Resampling step The sampling stage uniformly refines the base mesh triangles to the desired level. J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 24 / 35

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. 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 sharp features of the Fandisk model. J. Lezama, M. Natali () SEMI-REGULAR REMESHING November 22nd, 2011 27 / 35