Provably Good Sampling and Meshing of Surfaces                              Jean-Daniel Boissonnat,                       ...
Provably Good Sampling and Meshing of Surfaces                                 Not	  a	  smooth	  surface	                ...
Provably Good Sampling and Meshing of Surfaces                                  Smooth	  surface	                         ...
Provably Good Sampling and Meshing of	  Surfaces	                                       Well	  distributed	  sample	  poin...
Provably Good Sampling and	  Meshing	  of	  Surfaces	                                               Well	  distributed	  s...
Provably Good Sampling and	  Meshing	  of	  Surfaces	                                                Well	  distributed	  ...
Provably Good Sampling and	  Meshing	  of	  Surfaces	                                                Well	  distributed	  ...
Provably	  Good	  Sampling and	  Meshing	  of	  Surfaces	                                                            Well	...
Provably	  Good	  Sampling and	  Meshing	  of	  Surfaces	                                                            Well	...
Presented	  by	  Anisimov	  Dmitry	    1.	  Take	  a	  smooth	  surface	  Compact,	  orientable,	  at	  least	  C2	  –	  c...
Presented	  by	  Anisimov	  Dmitry	        2.	  Sample	  this	  surface	  •  Medial	  axis	  	  M	  of	  the	  surface	  S...
Presented	  by	  Anisimov	  Dmitry	        2.	  Sample	  this	  surface	  •  Distance	  to	  the	  medial	  axis	  	  M	  ...
Presented	  by	  Anisimov	  Dmitry	        2.	  Sample	  this	  surface	  •  Minimum	  distance	  to	  the	  medial	  axis...
Presented	  by	  Anisimov	  Dmitry	      2.	  Sample	  this	  surface	  •  Some	  user-­‐defined	  func9on	  σ : S → R   Ø...
Presented	  by	  Anisimov	  Dmitry	          2.	  Sample	  this	  surface	  •  Ball	  	  	  	  	  of	  center	  	  c	  	  ...
Presented	  by	  Anisimov	  Dmitry	          2.	  Sample	  this	  surface	  •  Ball	  	  	  	  	  of	  center	  	  c	  	  ...
Presented	  by	  Anisimov	  Dmitry	        2.	  Sample	  this	  surface	  •  Construc9on	  of	  the	  ini9al	  point	  sam...
Presented	  by	  Anisimov	  Dmitry	         2.	  Sample	  this	  surface	  •  Construc9on	  of	  the	  ini9al	  point	  sa...
Presented	  by	  Anisimov	  Dmitry	        2.	  Sample	  this	  surface	  •  Construc9on	  of	  the	  ini9al	  point	  sam...
Presented	  by	  Anisimov	  Dmitry	        2.	  Sample	  this	  surface	  •  Construc9on	  of	  the	  ini9al	  point	  sam...
Presented	  by	  Anisimov	  Dmitry	       2.	  Sample	  this	  surface	  •  Construc9on	  of	  the	  ini9al	  point	  samp...
Presented	  by	  Anisimov	  Dmitry	       2.	  Sample	  this	  surface	  •  Construc9on	  of	  the	  ini9al	  point	  samp...
Presented	  by	  Anisimov	  Dmitry	       2.	  Sample	  this	  surface	  •  Construc9on	  of	  the	  ini9al	  point	  samp...
Presented	  by	  Anisimov	  Dmitry	       2.	  Sample	  this	  surface	  •  Construc9on	  of	  the	  ini9al	  point	  samp...
Presented	  by	  Anisimov	  Dmitry	       2.	  Sample	  this	  surface	  •  Construc9on	  of	  the	  ini9al	  point	  samp...
Presented	  by	  Anisimov	  Dmitry	       2.	  Sample	  this	  surface	  •  Construc9on	  of	  the	  ini9al	  point	  samp...
Presented	  by	  Anisimov	  Dmitry	   3.	  Triangulate	  this	  surface	  •  Compute	  the	  3-­‐dimensional	  Delaunay	  ...
Presented	  by	  Anisimov	  Dmitry	   3.	  Triangulate	  this	  surface	  •  Compute	  the	  set	  of	  all	  edges	  of	 ...
Presented	  by	  Anisimov	  Dmitry	   3.	  Triangulate	  this	  surface	  •  Compute	  Delaunay	  triangula9on	  of	  E	  ...
Presented	  by	  Anisimov	  Dmitry	   3.	  Triangulate	  this	  surface	  •  Compute	  Delaunay	  triangula9on	  of	  E	  ...
Presented	  by	  Anisimov	  Dmitry	   3.	  Triangulate	  this	  surface	  •  Surface	  Delaunay	  ball	  BD	  of	  restric...
Presented	  by	  Anisimov	  Dmitry	   3.	  Triangulate	  this	  surface	  •  Surface	  Delaunay	  ball	  BD	  of	  restric...
Presented	  by	  Anisimov	  Dmitry	   3.	  Triangulate	  this	  surface	  •  Bad	  surface	  Delaunay	  ball	  BD which	  ...
Presented	  by	  Anisimov	  Dmitry	   3.	  Triangulate	  this	  surface	  •  Surface	  Delaunay	  patch	                  ...
Presented	  by	  Anisimov	  Dmitry	   3.	  Triangulate	  this	  surface	  •  Loose	  ε-sample	        dM	                 ...
Presented	  by	  Anisimov	  Dmitry	   3.	  Triangulate	  this	  surface	  •  E	  is	  a	  loose	  ε-­‐sample	  of	  S	  if...
Presented	  by	  Anisimov	  Dmitry	   3.	  Triangulate	  this	  surface	  •  DelS(E) has	  ver9ces	  on	  all	  the	  conn...
Presented	  by	  Anisimov	  Dmitry	           3.	  Triangulate	  this	  surface	          •  Algorithm	        While	  L	 ...
Presented	  by	  Anisimov	  Dmitry	   3.	  Triangulate	  this	  surface	  •  Termina9on	  and	  output	  of	  the	  Algori...
Presented	  by	  Anisimov	  Dmitry	   3.	  Triangulate	  this	  surface	  •  Termina9on	  and	  output	  of	  the	  Algori...
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...
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	...
Presented	  by	  Anisimov	  Dmitry	                       References	  M.	  Botsch,	  L.	  Kobbelt,	  M.	  Pauly,	  P.	  A...
Presented	  by	  Anisimov	  Dmitry	     What	  steps?	  Did	  I	  forget	  something?	  
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Provably Good Sampling and Meshing of Surfaces

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My presentation of the article "Provably Good Sampling and Meshing of Surfaces" of J.-D. Boissonnat and S. Oudot. All Rights for text are Reserved by authors of this paper.

Date of presentation: May 2012

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Provably Good Sampling and Meshing of Surfaces

  1. 1. Provably Good Sampling and Meshing of Surfaces Jean-Daniel Boissonnat, Steve Oudot
  2. 2. Provably Good Sampling and Meshing of Surfaces Not  a  smooth  surface   Jean-Daniel Boissonnat, Steve Oudot
  3. 3. Provably Good Sampling and Meshing of Surfaces Smooth  surface   Jean-Daniel Boissonnat, Steve Oudot
  4. 4. Provably Good Sampling and Meshing of  Surfaces   Well  distributed  sample  points   Smooth  surface   Jean-Daniel Boissonnat, Steve Oudot
  5. 5. Provably Good Sampling and  Meshing  of  Surfaces   Well  distributed  sample  points   Good  triangula9on   Smooth  surface   Jean-Daniel Boissonnat, Steve Oudot
  6. 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. 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. 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. 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. 10. Presented  by  Anisimov  Dmitry   1.  Take  a  smooth  surface  Compact,  orientable,  at  least  C2  –  con9nuous  closed  surface.  Not  completely  suitable   Completely  suitable  
  11. 11. Presented  by  Anisimov  Dmitry   2.  Sample  this  surface  •  Medial  axis    M  of  the  surface  S       2d  medial  axis   3d  medial  axis  
  12. 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. 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. 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. 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. 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. 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. 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. 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  BxInsert  (ux, vx, wx) in  E
  20. 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. 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. 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. 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. 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. 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. 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. 27. Presented  by  Anisimov  Dmitry   3.  Triangulate  this  surface  •  Compute  the  3-­‐dimensional  Delaunay  triangula9on  of  E     Del(E)
  28. 28. Presented  by  Anisimov  Dmitry   3.  Triangulate  this  surface  •  Compute  the  set  of  all  edges  of  the  Voronoi  diagram  of  E   V(E)
  29. 29. Presented  by  Anisimov  Dmitry   3.  Triangulate  this  surface  •  Compute  Delaunay  triangula9on  of  E  restricted  to  S DelS(E)
  30. 30. Presented  by  Anisimov  Dmitry   3.  Triangulate  this  surface  •  Compute  Delaunay  triangula9on  of  E  restricted  to  S Not  constrained   DelS(E)
  31. 31. Presented  by  Anisimov  Dmitry   3.  Triangulate  this  surface  •  Surface  Delaunay  ball  BD  of  restricted  Delaunay  facet  f DelS(E)
  32. 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. 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. 34. Presented  by  Anisimov  Dmitry   3.  Triangulate  this  surface  •  Surface  Delaunay  patch The  intersec9on  of  a  surface  Delaunay   ball  with  S DelS(E)
  35. 35. Presented  by  Anisimov  Dmitry   3.  Triangulate  this  surface  •  Loose  ε-sample dM c
  36. 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. 37. Presented  by  Anisimov  Dmitry   3.  Triangulate  this  surface  •  DelS(E) has  ver9ces  on  all  the  connected  components  of  S V(E)
  38. 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. 39. Presented  by  Anisimov  Dmitry   3.  Triangulate  this  surface  •  Termina9on  and  output  of  the  Algorithm Ø  The  Algorithm  terminates  
  40. 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. 41. Presented  by  Anisimov  Dmitry   3.  Triangulate  this  surface  •  Output  of  the  Algorithm
  42. 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. 43. Presented  by  Anisimov  Dmitry   Applica9ons  Smooth  surface  
  44. 44. Presented  by  Anisimov  Dmitry   Applica9ons  Not  smooth  surface  
  45. 45. Presented  by  Anisimov  Dmitry   Applica9ons  Bad  triangula9on   Good  triangula9on  
  46. 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. 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. 48. Presented  by  Anisimov  Dmitry   What  steps?  Did  I  forget  something?  

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