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Provably Good Sampling and Meshing of Surfaces




                              Jean-Daniel Boissonnat,
                              Steve Oudot
Provably Good Sampling and Meshing of Surfaces




                                 Not	
  a	
  smooth	
  surface	
  



                              Jean-Daniel Boissonnat,
                              Steve Oudot
Provably Good Sampling and Meshing of Surfaces




                                  Smooth	
  surface	
  



                               Jean-Daniel Boissonnat,
                               Steve Oudot
Provably Good Sampling and Meshing of	
  Surfaces	
  



                                     Well	
  distributed	
  sample	
  points	
  




                                               Smooth	
  surface	
  



                                         Jean-Daniel Boissonnat,
                                         Steve Oudot
Provably Good Sampling and	
  Meshing	
  of	
  Surfaces	
  



                                             Well	
  distributed	
  sample	
  points	
  

                                                  Good	
  triangula9on	
  




                                                       Smooth	
  surface	
  



                                                 Jean-Daniel Boissonnat,
                                                 Steve Oudot
Provably Good Sampling and	
  Meshing	
  of	
  Surfaces	
  



                                              Well	
  distributed	
  sample	
  points	
  

                                                    Good	
  triangula9on:	
  

                                 25	
      All	
  angles	
  are	
  greater	
  than	
  25	
  degrees	
  




                                                           Smooth	
  surface	
  



                                                   Jean-Daniel Boissonnat,
                                                   Steve Oudot
Provably Good Sampling and	
  Meshing	
  of	
  Surfaces	
  



                                              Well	
  distributed	
  sample	
  points	
  

                                                    Good	
  triangula9on:	
  

                                 25	
      All	
  angles	
  are	
  greater	
  than	
  25	
  degrees	
  
                                              All	
  triangles	
  are	
  equilateral	
  




                                                           Smooth	
  surface	
  



                                                   Jean-Daniel Boissonnat,
                                                   Steve Oudot
Provably	
  Good	
  Sampling and	
  Meshing	
  of	
  Surfaces	
  



                                                          Well	
  distributed	
  sample	
  points	
  

                                                                Good	
  triangula9on:	
  

                                          25	
         All	
  angles	
  are	
  greater	
  than	
  25	
  degrees	
  
                                                          All	
  triangles	
  are	
  equilateral	
  

                                                                               The	
  best	
  approximates	
  

                                                                       Smooth	
  surface	
  



                                                               Jean-Daniel Boissonnat,
                                                               Steve Oudot
Provably	
  Good	
  Sampling and	
  Meshing	
  of	
  Surfaces	
  



                                                          Well	
  distributed	
  sample	
  points	
  

                                                                Good	
  triangula9on:	
  

                                          25	
         All	
  angles	
  are	
  greater	
  than	
  25	
  degrees	
  
                                                          All	
  triangles	
  are	
  equilateral	
  

                                                                               The	
  best	
  approximates	
  

                                                                       Smooth	
  surface	
  



                                                                         Jean-­‐Daniel	
  Boissonnat,	
  
                                                                         Steve	
  Oudot	
  
Presented	
  by	
  Anisimov	
  Dmitry	
  


  1.	
  Take	
  a	
  smooth	
  surface	
  
Compact,	
  orientable,	
  at	
  least	
  C2	
  –	
  con9nuous	
  closed	
  surface.	
  




Not	
  completely	
  suitable	
            Completely	
  suitable	
  
Presented	
  by	
  Anisimov	
  Dmitry	
  


      2.	
  Sample	
  this	
  surface	
  
•  Medial	
  axis	
  	
  M	
  of	
  the	
  surface	
  S
                         	
  	
  	
  




   2d	
  medial	
  axis	
  
                                                          3d	
  medial	
  axis	
  
Presented	
  by	
  Anisimov	
  Dmitry	
  


      2.	
  Sample	
  this	
  surface	
  
•  Distance	
  to	
  the	
  medial	
  axis	
  	
  M	
  	
  that	
  is	
   dM
                                                  	
  	
  	
  
                                                               dM



                   dM


    2d	
  medial	
  axis	
  
                                                          3d	
  medial	
  axis	
  
Presented	
  by	
  Anisimov	
  Dmitry	
  


      2.	
  Sample	
  this	
  surface	
  
•  Minimum	
  distance	
  to	
  the	
  medial	
  axis	
  	
  M	
  	
  that	
  is	
  dM
                                                             	
  	
  	
              inf




   inf
  dM = inf {dM (x), x ∈ S}
Presented	
  by	
  Anisimov	
  Dmitry	
  


    2.	
  Sample	
  this	
  surface	
  
•  Some	
  user-­‐defined	
  func9on	
  σ : S → R
   Ø  Posi9ve	
  that	
  is	
  σ	
  	
  > 0
                                	
  	
  

   Ø  1-­‐Lipschitz	
  that	
  is	
  	
  σ (x) − σ (y) ≤ x − y
Presented	
  by	
  Anisimov	
  Dmitry	
  


        2.	
  Sample	
  this	
  surface	
  
•  Ball	
  	
  	
  	
  	
  of	
  center	
  	
  c	
  	
  and	
  radius	
  	
  r	
  	
  that	
  is	
  B(c, r)
           B                                   	
                            	
  




                                  B(c, r)
Presented	
  by	
  Anisimov	
  Dmitry	
  


        2.	
  Sample	
  this	
  surface	
  
•  Ball	
  	
  	
  	
  	
  of	
  center	
  	
  c	
  	
  and	
  radius	
  	
  r	
  	
  that	
  is	
  B(c, r)
           B                                   	
                            	
  




                                  B(c, r)
Presented	
  by	
  Anisimov	
  Dmitry	
  


      2.	
  Sample	
  this	
  surface	
  
•  Construc9on	
  of	
  the	
  ini9al	
  point	
  sample	
  E
                                                            	
  




Pick	
  up	
  at	
  least	
  one	
  point	
  x	
  on	
  each	
  connected	
  component	
  of	
  S	

and	
  insert	
  it	
  in	
  	
   !
                             E
Presented	
  by	
  Anisimov	
  Dmitry	
  


       2.	
  Sample	
  this	
  surface	
  
•  Construc9on	
  of	
  the	
  ini9al	
  point	
  sample	
  E
                                                            	
  




Consider	
  a	
  ball	
  Bx	
  centered	
  at	
  x	
  of	
  radius	
  less	

           "1                           1     %
       min # dist(x, E !  {x}), dM (x), σ (x)&
           $6                           6     '
Presented	
  by	
  Anisimov	
  Dmitry	
  


      2.	
  Sample	
  this	
  surface	
  
•  Construc9on	
  of	
  the	
  ini9al	
  point	
  sample	
  E
                                                            	
  




Repeatedly	
  shoot	
  rays	
  inside	
  Bx	
  and	
  pick	
  up	
  	
  
three	
  points	
  (ux, vx, wx) of	

 S  Bx


Insert	
  (ux, vx, wx) in	
  E
Presented	
  by	
  Anisimov	
  Dmitry	
  


      2.	
  Sample	
  this	
  surface	
  
•  Construc9on	
  of	
  the	
  ini9al	
  point	
  sample	
  E
                                                            	
  




Connec9ng	
  these	
  points	
  we	
  get	
  a	
  persistent facet	
  

All	
  persistent facets are	
  Delaunay	
  facets	
  restricted	
  to	
  S	
  	
  
Presented	
  by	
  Anisimov	
  Dmitry	
  


     2.	
  Sample	
  this	
  surface	
  
•  Construc9on	
  of	
  the	
  ini9al	
  point	
  sample	
  E
                                                            	
  




All persistent facets remain	
  restricted	
  Delaunay	
  facets	
  throughout	
  
the	
  course	
  of	
  algorithm	
  
Presented	
  by	
  Anisimov	
  Dmitry	
  


     2.	
  Sample	
  this	
  surface	
  
•  Construc9on	
  of	
  the	
  ini9al	
  point	
  sample	
  E
                                                            	
  




All persistent facets remain	
  restricted	
  Delaunay	
  facets	
  throughout	
  
the	
  course	
  of	
  algorithm	
  
Presented	
  by	
  Anisimov	
  Dmitry	
  


     2.	
  Sample	
  this	
  surface	
  
•  Construc9on	
  of	
  the	
  ini9al	
  point	
  sample	
  E
                                                            	
  




All persistent facets remain	
  restricted	
  Delaunay	
  facets	
  throughout	
  
the	
  course	
  of	
  algorithm	
  
Presented	
  by	
  Anisimov	
  Dmitry	
  


     2.	
  Sample	
  this	
  surface	
  
•  Construc9on	
  of	
  the	
  ini9al	
  point	
  sample	
  E
                                                            	
  




All persistent facets remain	
  restricted	
  Delaunay	
  facets	
  throughout	
  
the	
  course	
  of	
  algorithm	
  
Presented	
  by	
  Anisimov	
  Dmitry	
  


     2.	
  Sample	
  this	
  surface	
  
•  Construc9on	
  of	
  the	
  ini9al	
  point	
  sample	
  E
                                                            	
  




All persistent facets remain	
  restricted	
  Delaunay	
  facets	
  throughout	
  
the	
  course	
  of	
  algorithm	
  
Presented	
  by	
  Anisimov	
  Dmitry	
  


     2.	
  Sample	
  this	
  surface	
  
•  Construc9on	
  of	
  the	
  ini9al	
  point	
  sample	
  E
                                                            	
  




All persistent facets remain	
  restricted	
  Delaunay	
  facets	
  throughout	
  
the	
  course	
  of	
  algorithm	
  
Presented	
  by	
  Anisimov	
  Dmitry	
  


 3.	
  Triangulate	
  this	
  surface	
  
•  Compute	
  the	
  3-­‐dimensional	
  Delaunay	
  triangula9on	
  of	
  E	
  	
  




        Del(E)
Presented	
  by	
  Anisimov	
  Dmitry	
  


 3.	
  Triangulate	
  this	
  surface	
  
•  Compute	
  the	
  set	
  of	
  all	
  edges	
  of	
  the	
  Voronoi	
  diagram	
  of	
  E	
  




         V(E)
Presented	
  by	
  Anisimov	
  Dmitry	
  


 3.	
  Triangulate	
  this	
  surface	
  
•  Compute	
  Delaunay	
  triangula9on	
  of	
  E	
  restricted	
  to	
  S	





            DelS(E)
Presented	
  by	
  Anisimov	
  Dmitry	
  


 3.	
  Triangulate	
  this	
  surface	
  
•  Compute	
  Delaunay	
  triangula9on	
  of	
  E	
  restricted	
  to	
  S	


                                                     Not	
  constrained	
  




            DelS(E)
Presented	
  by	
  Anisimov	
  Dmitry	
  


 3.	
  Triangulate	
  this	
  surface	
  
•  Surface	
  Delaunay	
  ball	
  BD	
  of	
  restricted	
  Delaunay	
  facet	
  f	





            DelS(E)
Presented	
  by	
  Anisimov	
  Dmitry	
  


 3.	
  Triangulate	
  this	
  surface	
  
•  Surface	
  Delaunay	
  ball	
  BD	
  of	
  restricted	
  Delaunay	
  facet	
  f	


                                             Any	
  ball	
  centered	
  at	
  some	
  point	
  of	
   S  f *
                                             where	
  f* is	
  Voronoi	
  edge	
  dual	
  to	
  f	





            DelS(E)
Presented	
  by	
  Anisimov	
  Dmitry	
  


 3.	
  Triangulate	
  this	
  surface	
  
•  Bad	
  surface	
  Delaunay	
  ball	
  BD which	
  is	
  stored	
  in	
  L	


                                              It	
  is	
  ball	
  B(c, r) such	
  that	
  r > σ(c)	



                          c	

                            r	




             DelS(E)
Presented	
  by	
  Anisimov	
  Dmitry	
  


 3.	
  Triangulate	
  this	
  surface	
  
•  Surface	
  Delaunay	
  patch	


                                     The	
  intersec9on	
  of	
  a	
  surface	
  Delaunay	
  
                                     ball	
  with	
  S	





          DelS(E)
Presented	
  by	
  Anisimov	
  Dmitry	
  


 3.	
  Triangulate	
  this	
  surface	
  
•  Loose	
  ε-sample	




        dM	

                   c
Presented	
  by	
  Anisimov	
  Dmitry	
  


 3.	
  Triangulate	
  this	
  surface	
  
•  E	
  is	
  a	
  loose	
  ε-­‐sample	
  of	
  S	
  if:	




             dM	

                                      c	





1.	
  ∀c ∈ S  V (E), E  B(c, ε d M (c)) ≠ ∅
      	
  
2.	
  DelS(E) has	
  ver9ces	
  on	
  all	
  the	
  connected	
  components	
  of	
  S
Presented	
  by	
  Anisimov	
  Dmitry	
  


 3.	
  Triangulate	
  this	
  surface	
  
•  DelS(E) has	
  ver9ces	
  on	
  all	
  the	
  connected	
  components	
  of	
  S	





        V(E)
Presented	
  by	
  Anisimov	
  Dmitry	
  


         3.	
  Triangulate	
  this	
  surface	
  
        •  Algorithm	

        While	
  L	
  is	
  not	
  empty	
  
•  Take	
  an	
  element	
  B(c,r) from	
  L	

•  Insert	
  c	
  into	
  E	
  and	
  update	
  Del(E)	

•  Update	
  DelS(E) by	
  tes9ng	
  all	
  the	
  Voronoi	
  edges	
  that	
  have	
  changed	
  or	
  appeared:	
  
     Ø  Delete	
  from	
  DelS(E) the	
  Delaunay	
  facets	
  whose	
  dual	
  Voronoi	
  edges	
  no	
  longer	
  intersect S	

     Ø  Add	
  to	
  DelS(E) the	
  new	
  Delaunay	
  facets	
  whose	
  dual	
  Voronoi	
  edges	
  intersect	
  S	

•  Update	
  L	
  by	
  
     Ø  Dele9ng	
  all	
  the	
  elements	
  of	
  L	
  which	
  are	
  no	
  longer	
  bad	
  surface	
  Delaunay	
  balls	
  
     Ø  Adding	
  all	
  the	
  new	
  surface	
  Delaunay	
  balls	
  that	
  are	
  bad	
  
Presented	
  by	
  Anisimov	
  Dmitry	
  


 3.	
  Triangulate	
  this	
  surface	
  
•  Termina9on	
  and	
  output	
  of	
  the	
  Algorithm	


Ø  The	
  Algorithm	
  terminates	
  
Presented	
  by	
  Anisimov	
  Dmitry	
  


 3.	
  Triangulate	
  this	
  surface	
  
•  Termina9on	
  and	
  output	
  of	
  the	
  Algorithm	


Ø  The	
  Algorithm	
  outputs	
  E	
  and	
  DelS(E)	


       E	
  is	
  a	
  loose	
  ε-­‐sample	
  of	
  S	


       DelS(E)	
  is	
  homeomorphic	
  to	
  the	
  input	
  surface	
  S	
  and	
  approximates	
  it	
  
       in	
  terms	
  of	
  its	
  Hausdorff	
  distance,	
  normals,	
  curvature,	
  and	
  area.	
  
Presented	
  by	
  Anisimov	
  Dmitry	
  


 3.	
  Triangulate	
  this	
  surface	
  
•  Output	
  of	
  the	
  Algorithm
Presented	
  by	
  Anisimov	
  Dmitry	
  


               Magic	
  Epsilon	
  
•  To	
  find	
  ε	
  you	
  must	
  solve	
  this	
  simple	
  inequality:	

                        2ε             ε    π
                             + arcsin     ≥
                       1− 8ε          1− ε 4
•  Or	
  just	
  take	
  this:	



                                    ε	
  =	
  0.091	
  	
  
Presented	
  by	
  Anisimov	
  Dmitry	
  


           Applica9ons	
  




Smooth	
  surface	
  
Presented	
  by	
  Anisimov	
  Dmitry	
  


              Applica9ons	
  




Not	
  smooth	
  surface	
  
Presented	
  by	
  Anisimov	
  Dmitry	
  


             Applica9ons	
  




Bad	
  triangula9on	
          Good	
  triangula9on	
  
Presented	
  by	
  Anisimov	
  Dmitry	
  


                   References	
  
J.-­‐D.	
  Boissonnat	
  and	
  S.	
  Oudot.	
  “Provably	
  Good	
  Sampling	
  and	
  Meshing	
  of	
  Surfaces.”	
  
Graphical	
  Models	
  67	
  (2005),	
  405-­‐51.	
  
Presented	
  by	
  Anisimov	
  Dmitry	
  


                     References	
  
M.	
  Botsch,	
  L.	
  Kobbelt,	
  M.	
  Pauly,	
  P.	
  Alliez,	
  and	
  B.	
  Levy.	
  “Polygon	
  Mesh	
  Processing.”	
  
Chapter	
  6,	
  Sec9on	
  6.5.1	
  (2010),	
  92-­‐96.	
  
Presented	
  by	
  Anisimov	
  Dmitry	
  


   What	
  steps?	
  




Did	
  I	
  forget	
  something?	
  

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Provably Good Sampling and Meshing of Surfaces

  • 1. Provably Good Sampling and Meshing of Surfaces Jean-Daniel Boissonnat, Steve Oudot
  • 2. Provably Good Sampling and Meshing of Surfaces Not  a  smooth  surface   Jean-Daniel Boissonnat, Steve Oudot
  • 3. Provably Good Sampling and Meshing of Surfaces Smooth  surface   Jean-Daniel Boissonnat, Steve Oudot
  • 4. Provably Good Sampling and Meshing of  Surfaces   Well  distributed  sample  points   Smooth  surface   Jean-Daniel Boissonnat, Steve Oudot
  • 5. Provably Good Sampling and  Meshing  of  Surfaces   Well  distributed  sample  points   Good  triangula9on   Smooth  surface   Jean-Daniel Boissonnat, Steve Oudot
  • 6. Provably Good Sampling and  Meshing  of  Surfaces   Well  distributed  sample  points   Good  triangula9on:   25   All  angles  are  greater  than  25  degrees   Smooth  surface   Jean-Daniel Boissonnat, Steve Oudot
  • 7. Provably Good Sampling and  Meshing  of  Surfaces   Well  distributed  sample  points   Good  triangula9on:   25   All  angles  are  greater  than  25  degrees   All  triangles  are  equilateral   Smooth  surface   Jean-Daniel Boissonnat, Steve Oudot
  • 8. Provably  Good  Sampling and  Meshing  of  Surfaces   Well  distributed  sample  points   Good  triangula9on:   25   All  angles  are  greater  than  25  degrees   All  triangles  are  equilateral   The  best  approximates   Smooth  surface   Jean-Daniel Boissonnat, Steve Oudot
  • 9. Provably  Good  Sampling and  Meshing  of  Surfaces   Well  distributed  sample  points   Good  triangula9on:   25   All  angles  are  greater  than  25  degrees   All  triangles  are  equilateral   The  best  approximates   Smooth  surface   Jean-­‐Daniel  Boissonnat,   Steve  Oudot  
  • 10. Presented  by  Anisimov  Dmitry   1.  Take  a  smooth  surface   Compact,  orientable,  at  least  C2  –  con9nuous  closed  surface.   Not  completely  suitable   Completely  suitable  
  • 11. Presented  by  Anisimov  Dmitry   2.  Sample  this  surface   •  Medial  axis    M  of  the  surface  S       2d  medial  axis   3d  medial  axis  
  • 12. Presented  by  Anisimov  Dmitry   2.  Sample  this  surface   •  Distance  to  the  medial  axis    M    that  is   dM       dM dM 2d  medial  axis   3d  medial  axis  
  • 13. Presented  by  Anisimov  Dmitry   2.  Sample  this  surface   •  Minimum  distance  to  the  medial  axis    M    that  is  dM       inf inf dM = inf {dM (x), x ∈ S}
  • 14. Presented  by  Anisimov  Dmitry   2.  Sample  this  surface   •  Some  user-­‐defined  func9on  σ : S → R Ø  Posi9ve  that  is  σ    > 0     Ø  1-­‐Lipschitz  that  is    σ (x) − σ (y) ≤ x − y
  • 15. Presented  by  Anisimov  Dmitry   2.  Sample  this  surface   •  Ball          of  center    c    and  radius    r    that  is  B(c, r) B     B(c, r)
  • 16. Presented  by  Anisimov  Dmitry   2.  Sample  this  surface   •  Ball          of  center    c    and  radius    r    that  is  B(c, r) B     B(c, r)
  • 17. Presented  by  Anisimov  Dmitry   2.  Sample  this  surface   •  Construc9on  of  the  ini9al  point  sample  E   Pick  up  at  least  one  point  x  on  each  connected  component  of  S and  insert  it  in     ! E
  • 18. Presented  by  Anisimov  Dmitry   2.  Sample  this  surface   •  Construc9on  of  the  ini9al  point  sample  E   Consider  a  ball  Bx  centered  at  x  of  radius  less "1 1 % min # dist(x, E ! {x}), dM (x), σ (x)& $6 6 '
  • 19. Presented  by  Anisimov  Dmitry   2.  Sample  this  surface   •  Construc9on  of  the  ini9al  point  sample  E   Repeatedly  shoot  rays  inside  Bx  and  pick  up     three  points  (ux, vx, wx) of S  Bx Insert  (ux, vx, wx) in  E
  • 20. Presented  by  Anisimov  Dmitry   2.  Sample  this  surface   •  Construc9on  of  the  ini9al  point  sample  E   Connec9ng  these  points  we  get  a  persistent facet   All  persistent facets are  Delaunay  facets  restricted  to  S    
  • 21. Presented  by  Anisimov  Dmitry   2.  Sample  this  surface   •  Construc9on  of  the  ini9al  point  sample  E   All persistent facets remain  restricted  Delaunay  facets  throughout   the  course  of  algorithm  
  • 22. Presented  by  Anisimov  Dmitry   2.  Sample  this  surface   •  Construc9on  of  the  ini9al  point  sample  E   All persistent facets remain  restricted  Delaunay  facets  throughout   the  course  of  algorithm  
  • 23. Presented  by  Anisimov  Dmitry   2.  Sample  this  surface   •  Construc9on  of  the  ini9al  point  sample  E   All persistent facets remain  restricted  Delaunay  facets  throughout   the  course  of  algorithm  
  • 24. Presented  by  Anisimov  Dmitry   2.  Sample  this  surface   •  Construc9on  of  the  ini9al  point  sample  E   All persistent facets remain  restricted  Delaunay  facets  throughout   the  course  of  algorithm  
  • 25. Presented  by  Anisimov  Dmitry   2.  Sample  this  surface   •  Construc9on  of  the  ini9al  point  sample  E   All persistent facets remain  restricted  Delaunay  facets  throughout   the  course  of  algorithm  
  • 26. Presented  by  Anisimov  Dmitry   2.  Sample  this  surface   •  Construc9on  of  the  ini9al  point  sample  E   All persistent facets remain  restricted  Delaunay  facets  throughout   the  course  of  algorithm  
  • 27. Presented  by  Anisimov  Dmitry   3.  Triangulate  this  surface   •  Compute  the  3-­‐dimensional  Delaunay  triangula9on  of  E     Del(E)
  • 28. Presented  by  Anisimov  Dmitry   3.  Triangulate  this  surface   •  Compute  the  set  of  all  edges  of  the  Voronoi  diagram  of  E   V(E)
  • 29. Presented  by  Anisimov  Dmitry   3.  Triangulate  this  surface   •  Compute  Delaunay  triangula9on  of  E  restricted  to  S DelS(E)
  • 30. Presented  by  Anisimov  Dmitry   3.  Triangulate  this  surface   •  Compute  Delaunay  triangula9on  of  E  restricted  to  S Not  constrained   DelS(E)
  • 31. Presented  by  Anisimov  Dmitry   3.  Triangulate  this  surface   •  Surface  Delaunay  ball  BD  of  restricted  Delaunay  facet  f DelS(E)
  • 32. Presented  by  Anisimov  Dmitry   3.  Triangulate  this  surface   •  Surface  Delaunay  ball  BD  of  restricted  Delaunay  facet  f Any  ball  centered  at  some  point  of   S  f * where  f* is  Voronoi  edge  dual  to  f DelS(E)
  • 33. Presented  by  Anisimov  Dmitry   3.  Triangulate  this  surface   •  Bad  surface  Delaunay  ball  BD which  is  stored  in  L It  is  ball  B(c, r) such  that  r > σ(c) c r DelS(E)
  • 34. Presented  by  Anisimov  Dmitry   3.  Triangulate  this  surface   •  Surface  Delaunay  patch The  intersec9on  of  a  surface  Delaunay   ball  with  S DelS(E)
  • 35. Presented  by  Anisimov  Dmitry   3.  Triangulate  this  surface   •  Loose  ε-sample dM c
  • 36. Presented  by  Anisimov  Dmitry   3.  Triangulate  this  surface   •  E  is  a  loose  ε-­‐sample  of  S  if: dM c 1.  ∀c ∈ S  V (E), E  B(c, ε d M (c)) ≠ ∅   2.  DelS(E) has  ver9ces  on  all  the  connected  components  of  S
  • 37. Presented  by  Anisimov  Dmitry   3.  Triangulate  this  surface   •  DelS(E) has  ver9ces  on  all  the  connected  components  of  S V(E)
  • 38. Presented  by  Anisimov  Dmitry   3.  Triangulate  this  surface   •  Algorithm While  L  is  not  empty   •  Take  an  element  B(c,r) from  L •  Insert  c  into  E  and  update  Del(E) •  Update  DelS(E) by  tes9ng  all  the  Voronoi  edges  that  have  changed  or  appeared:   Ø  Delete  from  DelS(E) the  Delaunay  facets  whose  dual  Voronoi  edges  no  longer  intersect S Ø  Add  to  DelS(E) the  new  Delaunay  facets  whose  dual  Voronoi  edges  intersect  S •  Update  L  by   Ø  Dele9ng  all  the  elements  of  L  which  are  no  longer  bad  surface  Delaunay  balls   Ø  Adding  all  the  new  surface  Delaunay  balls  that  are  bad  
  • 39. Presented  by  Anisimov  Dmitry   3.  Triangulate  this  surface   •  Termina9on  and  output  of  the  Algorithm Ø  The  Algorithm  terminates  
  • 40. Presented  by  Anisimov  Dmitry   3.  Triangulate  this  surface   •  Termina9on  and  output  of  the  Algorithm Ø  The  Algorithm  outputs  E  and  DelS(E) E  is  a  loose  ε-­‐sample  of  S DelS(E)  is  homeomorphic  to  the  input  surface  S  and  approximates  it   in  terms  of  its  Hausdorff  distance,  normals,  curvature,  and  area.  
  • 41. Presented  by  Anisimov  Dmitry   3.  Triangulate  this  surface   •  Output  of  the  Algorithm
  • 42. Presented  by  Anisimov  Dmitry   Magic  Epsilon   •  To  find  ε  you  must  solve  this  simple  inequality: 2ε ε π + arcsin ≥ 1− 8ε 1− ε 4 •  Or  just  take  this: ε  =  0.091    
  • 43. Presented  by  Anisimov  Dmitry   Applica9ons   Smooth  surface  
  • 44. Presented  by  Anisimov  Dmitry   Applica9ons   Not  smooth  surface  
  • 45. Presented  by  Anisimov  Dmitry   Applica9ons   Bad  triangula9on   Good  triangula9on  
  • 46. Presented  by  Anisimov  Dmitry   References   J.-­‐D.  Boissonnat  and  S.  Oudot.  “Provably  Good  Sampling  and  Meshing  of  Surfaces.”   Graphical  Models  67  (2005),  405-­‐51.  
  • 47. Presented  by  Anisimov  Dmitry   References   M.  Botsch,  L.  Kobbelt,  M.  Pauly,  P.  Alliez,  and  B.  Levy.  “Polygon  Mesh  Processing.”   Chapter  6,  Sec9on  6.5.1  (2010),  92-­‐96.  
  • 48. Presented  by  Anisimov  Dmitry   What  steps?   Did  I  forget  something?