APPROXIMATE SYMMETRY DETECTION FOR REVERSE ENGINEERING

1
APPROXIMATE SYMMETRY DETECTION FOR REVERSE
ENGINEERING

• ??S???S ??S?????S

2
INTRODUCTION

• ??apt?sseta? ??a a?t?µat? reverse engineering
  s?st?µa ??a a?a?atas?e?? t?? s??µat?? ap???
  µ??a????? ??µµat???
• ?e?????feta? ??a? p?a?t???? a??????µ?? ??a t??
  e??es? ?????? s?µµet???? se ?-rep µ??t??a p??
  ????? ft?a?te? ap??p?peda, Sfa??e?, ?????d????,
  ??????, ??????
• ??t? ta µ??t??a de? t????? ??p??e? ?a??????t?te?
  ?a? pe?????sµ??? epe?d? ???e p?e??? t?? µ??t????
  ?a?????eta? a?e???t?ta
• ?a ?-rep µ??t??a d?µ????????ta? ap? t??
  efa?µ??? ep?fa?e??? se pe?????? s?µe??? p??
  p????pt??? ap? t? s?a????sµa t?? a?t??e?µ???? µe
  ??a 3D Laser Scanner.

3
INTRODUCTION

• REVERSE ENGINEERING
• ???sµ??? d?µ??????a e??? B-rep CAD µ??t????
  t?? s??µat?? e??? µ??a????? ??µµat???
• ??ad??as?a
• ?e?????ta? ap? t? ?d?? t? µ??a???? ??µµ?t?
  pa???e? ??a ?e?µet???? µ??t???. ??t? ? d?ad??as?a
  pe???aµß??e? t? s???s? t?? ep?f??e?a? t??
  a?t??e?µ???? 3D Laser Scanner ??a t?? pa?a????
  e??? ste?e?? µ??t????
• ?? µ??t??? ?atas?e???eta? µe f?s???? ep?f??e?e?.
  ? e?a?s??s?a t?? a??????µ?? ?µ?? st? ????ß? ?a?
  t? ?e????? ?t? ?? ep?f??e?e? t?p??et???ta?
  ?e????st? ?d??e? se µ? pa??????e? ?a? µ?
  ????????e? p?e????
• ?.? G?a ??a ??ß? d?µ?????e?ta? ??a µ??t??? µe 6
  ep?pede? ?d?e?,12 e??e?e? a?µ?? ?a? 8 ????f??

4
INTRODUCTION
5
INTRODUCTION

• ?EAUTIFICATION
• ???sµ??? d?ad??as?a µe t?? ?p??a ße?t???eta? ??a
  µ??t???
• ??ad??as?a???sa?µ??? t?? ?ste ?a ???e? a???ß??
  s?µµet???? ap? s?ed?? s?µµet???? p?? ?ta? ??
  e???
• ??????aµµ????ta? ??µµ?t?a p?? ?e?t??ta? st?? ?d?a
  d?e????s?
• ?????ta? epa?a?aµßa??µe?a ?a?a?t???st??? a???ß??
  ?µ???µ??fa
• ?? p????f???e? p?? apa?t???ta? ??a t?
  beautification ?aµß????ta? ap? t?? ?at?
  p??s????s? s?µµet??e? t?? e?d???? s?µe??? st??
  ep?f??e?a t?? a?t??e?µ????
• St???? t?? paper e??a? ? e??es? a?t?? t??
  s?µµet????

6
INTRODUCTION

• S????????S
• ??a p???ed?? e??a? p????? ???sµ??? ?ta? e??a?
  ???st?? ?? ?e?µet????? ?a? s??d?ast????
  p????f???e? ??a t?? ????f?? t??
• ? ?e?µet???? p????f???a e??a? ? t?p??es?a t??
  ???e ????f??
• ? s??d?ast??? ? t?p??????? p????f???a ?p?de????e?
  p??e? ????f?? ß??s???ta? se µ?a ????? a?µ? ?a?
  p??e? ????f?? ß??s???ta? p??? se µ?a ????? ?d?a
• ?? s?µµet??e? e??? p????d??? e??a? a???ß??
  e?e??e? ?? s?µµet??e? t?? s?µe??? p?? d?at?????
  t?? s??d?ast???? p????f???e? .?ts? ?ta? ?? ?d?e?
  ?aµp??????ta? ,s?µe?a ?p?? t? ???t?? ß????? t??
  ?d?a? ? ta ???t?a ?aµp???t?ta? t??? p??pe? ?a
  ???s?µ?p??????? ??a ?a d?at????e? ? s??s? µeta??
  t?? s?µµet???? t?? s?µe??? ?a? t?? s?µµet???? t??
  a?t??e?µ????

7
INTRODUCTION

• ??et????ta? µ??? pepe?asµ??a s????a s?µµet????
• G?a a?t? ta s????a , ?? s?µµet??e? e???
  a?t??e?µ???? e??a? ep?s?? s?µµet??e? ??p????
  s?????? s?µe??? p?? p????pt??? ap? t?
  a?t??e?µe??(????f??, ???t?a sfa????, ???t?a
  t????, ???t?a ß????? a?µ?? ?a? ???t?a ß?????
  ed???)
• G?a µ? s????aµµ???? s??????? s?µe??? ? a???µ??
  t?? s?µµet???? pe??????eta? st? d?p??s?? t??
  a???µ?? t?? s?µe??? ? st? 120
• ? s?µµet??a e??? a?t??e?µ???? de?
  a?t??at?pt???eta? p??ta st? s?µµet??a t???-rep
  µ??t????. G?a?t? ? a??????µ?? ap?te?e?ta? ap? d??
  st?d?a

8
INTRODUCTION

• St? p??t? st?d?? d??eta? µe???? ??s? st? p??ß??µa
  t?? p??sa?µ???? t?? µ??t???? ?ts? ?ste ?a
  a?t??at?pt???e? t?? s?µµet??e? t?? a?t??e?µ????.
  ??t? t? st?d?? ?a?????e? t??p??? ?µad?p???s?? ?a?
  a?a?????s?? s?µe??? ??a t? d?µ??????a ???? ?µ?d??
• St? de?te?? st?d?? ?a???????ta? ?? ?at?
  p??s????s? s?µµet??e? a?t?? t?? ?µ?d??.
• ? st???? t?? e??a? ? a??pt??? a??????µ?? p?? ?a
  d?µ??????s??? ??a ?da???? µ??t??? ??a ??a
  pa?at????µe?? a?t??e?µe??

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PREVIOUS WORK

• ?? Sugihara,Jiang ?a? Wolter ?de??a? ?t? ??a t??
  ?a????sµ? t?? s?µµet???? e??? p????d???
  apa?te?ta? ?????? O(nlogn) ?p?? n ? a???µ?? t??
  ????f?? µe t? ???s? s?µß?????? a??????µ??
  ta????µ?s??
• ? Zabrodski ??te? t? p??ß??µa ?atas?e??? e???
  µ?t??? ?at? p??s????s? s?µµet???? ??a µ???a.??
  ????t?? e??a? ? ?a????sµ?? µ?a? a???µ?t????
  p???t?ta? t?? µ????? p?? µp??e? ?a s?s?et?ste? µe
  p???µata ?p?? ? ?e?µ???as?a t???? t?? ste?e??
• ? Iwanowski ??te? t? d?s????a st?? ?a????sµ? t??
  ?pa???? µ?a? ???t???? s??????? s?µe??? µe
  s???e???µ??? s?µµet??a.??t? t? p??ß??µa µp??e? ?a
  a?t?µet?p?ste? sa? NP-hard p?? µp??e? ?a
  pe?????ste? st?? d?? d?ast?se??

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PREVIOUS WORK

• ? Alt ?a?????e? t? a? µp??e? ?a ß?e?e? ?s?µet??a
  p?? µp??e? ?a s?ed???e? ta s?µe?a µ?a? s???????
  ???t? st?? ea?t? t???.
• ? Tate pe?????fe? ??a? a??????µ? ??a ?a????sµ?
  µe????? s?µµet????. ? ?d?a e??a? t? ta???asµa
  ???e ?e????? a?µ?? ?a? e??es? ?s?µet???? p??
  s?et????? ?a? t?? d?? ?a? ?µad?p???s? a?t?? t?
  s?µµet???? µe ß?s? t?? ?µ???t?ta.

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ALGORITHM OUTLINE

• ? a??????µ?? p?? ???s?µ?p??e?ta? ap?te?e?ta? ap?
  d?? µ???
• 1? ??t??at?stas? s???e???µ???? ?µ?da? s?µe???
  ap? t? ???t?? ß????? t???
• ? a??????µ?? a?a??te? ep?peda a????? ??a ta ?p??a
  ? ?µad?p???s? t?? s?µe??? e??a? sta?e??(?e? e??a?
  apa?a?t?t? ? a???ß?? t?µ? t?? a?????)
• ?a??de??µa
• ?p?st??f??ta? t?e?? ?µad?p???se??
• ? ?at? p??s????s? ??ß??
• ??a ?µad?p???s? st?? ?p??a ???e s?µe?? e??a? µ???
  t??
• ??a ?µad?p???s? p?? pe????e? ??a ta s?µe?a

12
ALGORITHM OUTLINE
13
ALGORITHM OUTLINE
14
ALGORITHM OUTLINE

• 2o ???es? t?? ?at? p??s????s? s?µµet???? µ?a?
  s??????? s?µe??? p?? µ??e? µet? t?? ?µad?p???s?
  st? p??t? µ???? ?a? ?a????sµ?? t?? µeta??se?? p??
  ep?f????ta? ap? a?t?? t?? s?µµet??e?
• ?a??de??µa
• ? ??ß?? pe??st??feta? ?a? a?t? ep?f??e? µet??es?.
  ?ts? a?ad?at?ss??ta? ?? ????f?? t?? µe ß?s? t?
  s?µµet??a t??.
• ?p?????? 48 ?at? p??s????s? s?µµet??e?
• ?? µeta??se?? de? s?ed?????? µ??? ????f?? se
  ????f?? a??? ?a? a?µ?? se a?µ??. ???e a?µ?
  p??sa?µ??eta? st? µ?s? µ???? t?? a?µ?? st??
  ?p??e? µp??e? ?a s?ed?aste?. ?ts? ???? ? ??ß??
  µp??e? ?a a?a?atas?e?aste? µe t??????p???s? ??a
  t?? ?a????sµ? t?? t?p??es?a? ???e ????f??

15
ALGORITHM OUTLINE
4
0
6
2
5
1
7
3
16
ALGORITHM

• ?p??es???p?? ded?µ???? ?p?? s?µe?a, ??af?µata,
  s????a ,d?aµe??se?? ,????? p??te?a??t?ta? e??a?
  d?a??s?µe?
• St?? ?e?d???d??a ? ???? p??te?a??t?ta? ta????µe?
  a?t?µata ta ?e?????a ap? s?µe?a p?? t?p??et???ta?
  se a?t? µe t?t??? t??p? ?ste t? ?e???? µe t?
  µ????te?? ap?stas? d?a????sµ?? t?p??ete?ta? st??
  ????f? t?? ????? ?a? ?a ß?e? p??t?
• ??s? ? e?s?d?? ?s? ?a? ? ???d?? e??a? µ?a ??sta
  s?µe???.
• ? ???d?? e??a? ta ???t?a ß????? ?at???????
  ??p???? d?aµe??se?? t?? e?s?d?? (??.1 , G?. 19)

17
ALGORITHM

• ? p??t? d???e?? e??a? ? ?p?????sµ?? t??
  d?aµ???s??
• ?a?????eta? µ?a ap?stas? level of tolerance
• St? p??t? st?d?? ?p??????eta? ??a? a???µ?? ap?
  ??ste? e??d?? (???????µ?? 1,G?aµµ? 17)
• G?a ???e t?t??a ??sta ?a?e?ta? t? de?te?? st?d??
  ?a ?a????se? p??e? e??a? ?? s?µµet??e?(???????µ??
  1,G?aµµ? 19)

18
ALGORITHM

• ? f?s??? s?µas?a t?? ep?p?d?? a????? µp??e? ?a
  fa?e? ap? t? µe??t? µ?a? rigid ????s?? t??
  s??????? s?µe???
• ??a rigid ????s? t?? s??????? s?µe??? p??
  µeta???e? ???e s?µe?? se ??a ???? e??a? µ?a
  a???ß?? s?µµet??a
• ??a rigid ????s? t?? s??????? s?µe??? p??
  µeta???e? ???e s?µe?? µ?sa st?? a???? t??
  ???µast???? t?? st???? e??a? µ?a ?at? p??s????s?
  s?µµet??a
• ??? d?? s?µe?a ß??s???ta? t? ??a µ?sa st?? a????
  t?? ????? ?aµ?a ????s? de? apa?te?ta? ??a ?a
  f????µe t? ??a s?ed?? ???t? st? ????. ?ts? a?t?
  ta s?µe?a s?µpe??f????ta? sa? ?a e??a? st?? ?d?a
  t?p??es?a

19
ALGORITHM

• G?a t??a s?µe?a a?t? µp??e? ?a µ?? s?µßa??e?.
  G?a?t? ?p?????? d?ast?µata t?? ep?p?d?? a?????
  t?t??a ?ste ???e s?????? s?µe??? ?a??stata?
  sta?e??
• ?.? ?p??e? t? p??t? s?µe?? ?a e??a? st?? a????
  t?? de?te??? t? ?p??? µe t? se??? t?? e??a? st??
  a???? t?? t??t??, e?? µp??e? t? p??t? ?a µ??
  e??a? st?? a???? t?? t??t??. ?ts? de? µp????µe ?a
  p??µe ?t? ta s?µe?a ß??s???ta? st?? ?d?a
  t?p??es?a
• ?d?e? ?ata????se?? s?µe??? a?t??a??stata? ap? µ?a
  ?a? ?µ?de? s?µe??? p?? ß??s???ta? se ??a ep?ped?
  a????? a?t??a??stata? ap? ??a µ??? s?µe??
• St? p??t? ß?µa t?? a??????µ?? ?a???????ta? a?t?
  ta ep?peda ??a ta ?p??a ? a?t??at?stas? e??a?
  sta?e??

20
ALGORITHM
21
ALGORITHM
22
ALGORITHM-THE FIRST STAGE GENERATION OF LISTS OF
POINTS

1. ???eta? sa? e?s?d?? µ?a ??sta s?µe???
2. ??µ?????e?ta? ? ??sta µe ta ?e?????a t?? s?µe???
   (??. 1,G?. 6)
3. ?a????µe?ta? a?t? ? ??sta µe t?t??a se??? ?ste ?a
   a????eta? ? ap?stas? d?a????sµ??
4. ??????p??e?ta? µ?a d?aµ???s? t?t??a ?ste ?a
   ?at??????p??e? ???e s?µe?? sa? µ??ad??? (??.
   1,G?. 4)
5. ??????p??e?ta? ??a? ???f?? se ??a? ?de?? ???f?
   (??. 1,G?. 3). ?a s??de?µe?a ??µµ?t?a t?? ???f??
   e??a? ?? ?at?????e? t?? d?aµ???s??
6. ?p????e?eta? t?s? ? a???µ?? t?? ??µß?? ?a? t??
   a?µ?? ???e ??µµat???, ?s? ?a? ? ?????? a???µ??
   t?? ??µµat??? p?? de? e??a? p???e?? ?p????f??.
   ??????? t?µ????d?? a?µ?? ?a? ??a? ??µß?? ??a
   ???e t??? ?a? µ?d?? µ? p???e?? ?p????f??
7. ?ta? ???e s??de?µe?? ??µµ?t? st? ???f? e??a?
   p????? ?p????f?? t?te ? ???f?? a?apa?ast? µ?a
   sta?e?? ?µ?da

23
ALGORITHM-THE FIRST STAGE GENERATION OF LISTS OF
POINTS

• Level of tolerance
• ?????? t?µ? µ?d?? ,e?? de? apa?te?ta? ?
  ap????e?s? t?? t?µ?? t??
• ? a???? a????eta? pa?????ta? ap? t? ta????µ?µ???
  ??sta ??a ?e???? t? f??? (??. 1,G?. 12).
• ??? t? t????? ?e???? ??e? ?a? ta d?? s?µe?a st??
  ?d?a ?at?????a t?te a??????ta? ?? µet??t?? ??a
  t?? a?µ?? ?a? t??? ??µß??? ??a?t? t? ??µµ?t? t??
  ???f??
• ??a ?at?????a e??a? p????? ???f?? e?? ? a???µ??
  t?? a?µ?? e??a? ?s?? µe t? m(m-1)/2 , ?p?? m ?
  a???µ? t?? s?µe???
• ??? t? t????? ??e? ta d?? s?µe?a st??
  d?af??et???? ?at?????e?, t?te a?t?? ?? ?at?????e?
  s?????e???ta?, e?? ?? µet??t?? p??sa?µ????ta?
  ?at?????a

24
ALGORITHM-THE FIRST STAGE GENERATION OF LISTS OF
POINTS

• ?p? t?? a????s? p??? st? a???? p????pt??? ta
  ep?peda a????? ?a??? ?a? ta a?t?st???a sta?e???
  ?µad?p???se?? s?µe???.
• ? s?????? t?? ???t??? ßa??? a?t?? t?? ?µ?d??
  ?p??e?ta? st?? a????s? p??? st?? s?µµet??e? st?
  de?te?? µ????
• ???e s?µe?? st? ??a ?µ?da e??a? t? ???t?? ß?????
  µ?a? pe?????? s?µe???.
• ?f?s?? ? ?µad?p???s? e??a? sta?e?? ???e s?µe??
  a?t?? t?? pe?????? p?? ???t? st? ???t?? ß?????
  a?t?? t?? pe?????? ap? ?p???d?p?te ???? ???t??
  ß?????.

25
ALGORITHM-THE FIRST STAGE GENERATION OF LISTS OF
POINTS

• ???e s?µe?? st?? pe????? de? e??a? µa???te?a ap?
  t? ???t?? ß????? ap? t?? a????
• ??a a???? ?s? µe t? µ?s? t?? e????st?? ap?stas??
  d?? ???t??? ßa??? ?a d?µ??????se? t?? ?d?a ?µ?da
  ?p?? ? a??????µ??
• ?at??????p????ta? µe t?? ?d?? t??p? d?? s?µe?a
  s?µa??e? ?t? ???e s?µe?? st? ???? e??a? ???t? t?
  p??? se ??a s?µe?? t?? s???????

26
ALGORITHM-THE SECOND STAGE SYMMETRY ANALYSIS

• Se a?t? t? ß?µa ? a??????µ?? ?a?????e? t? s?????
  t?? µeta??se?? t?? s?µe??? p?? d?at????? t??
  ap?st?se?? t?? s?µe??? µ?sa st? ???? a????? p??
  ep???????e st? p??t? ß?µa
• St? p??t? tµ?µa t?? ?e?d???d??a ?atas?e???eta?
  µ?a ??sta s?µe??? p?? pe????e? t? ???t?? ß?????
  ???? t?? s?µe??? e?s?d??, t??a e?d??? s?µe?a st?
  convex hull ?a? ta ?p????pa s?µe?a. ?a t??a
  e?d??? s?µe?a ?a? t? ???t?? ß????? s??µat?????
  ??a tet??ed??. ? t?p??es?a a?t?? t?? s?µe???
  ?a?????e? t? t?p??es?a ???? t?? ????? s?µe??? st?
  s?????? ?a? af?? s?ed?ast??? a?t? µp????? µet? ?a
  s?ed?ast??? ??a ta s?µe?a (??. 2,G?. 6-13).

27
ALGORITHM-THE SECOND STAGE SYMMETRY ANALYSIS

• St? de?te?? tµ?µa t?? ?e?d???d??a (??. 2,G?. 17)
  ???eta? µ?a pe?????sµ??? a?a??t?s? se ??a d??t??
  t?? µeta??se??.????? t? ß???? ft?se? st? ep?ped?
  4,p?? s?µa??e? ?t? ??e? s?ed?aste? t? tet??ed??,?
  a?a??t?s? ???eta? µe e?a??a?t??? d?ad??as?a ,?
  ?p??a e????e? e?? ??a ta ???a s?µe?a ?????
  s?ed?aste? (??. 2,G?. 19)
• Se ???e µet??es? ???e st???e?? µp??e? ?a e??a?
  a???ß?? µ?a f??? p??t? st???e?? ?a? a???ß?? µ?a
  f??? de?te??.??ste? s?µe??? p?? p????pt??? ap?
  t?t??e? µeta??se?? ???µ????ta? partial
  injection
• ?? partial injections d?µ???????? ??a d??t?? µe
  ???a t?? ?de?a ??sta ?a? ta pa?d?? µ?a? partial
  injection p????pt??? ap? t?? p??s???? e???
  ?e????? st? ??sta.?a f???a t?? d??t??? e??a? ??
  µeta??se??

28
ALGORITHM-THE SECOND STAGE SYMMETRY ANALYSIS

• ??a partial injection e??a? approximately
  distance preserving e?? ? ap???t? d?af??? t??
  d?a????sµ?? d?? s?µe??? ?a? t?? d?a????sµ?? t??
  e?????? t?? e??a? µ????te?? t?? a????? ?a? t?te
  ???µ??eta? proto symmetry
• ?? pe??ss?te?e? partial injections de? e??a?
  approximately distance preserving
• ??e? ?? proto symmetries ft??????? ??a
  ?p?-d??t?? p??? µ????te?? ap? t? d??t?? t??
  partial injections .
• ?? approximately distance preserving
  µeta??se?? e??a? µ???st? ß???? f????? t??
  ?p?-d??t???
• G?a ?a sa??s??µe µ??? t? ?p?-d??t?? ??a ?a ß???µe
  ??a ta f???a t??, d?e???eta? µ?a s???s? st?
  µe?a??te?? d??t?? µe depth first t??p? ?a?
  ????? p?s? ??a ep?ped? e?? ??a? ??µß?? de? ß?e?e?
  proto symmetry

29
ALGORITHM-THE SECOND STAGE SYMMETRY ANALYSIS

• St?? ?e?d???d??a µe t?? e?t??? extend
  p??st??eta? ??a st???e?o st? d??t??,
  epe?te????ta? t? ß???? t?? ?a? t? backtrack
  ????? p?s? ??a ep?ped? e?? t? st???e?? de?
  e??a?proto symmetry
• ?e a?t?? t?? t??p? µe???eta? ?at? t?? e?t??es?
  t?? a??????µ?? ? a???µ?? t?? µeta??se??

30
PERFORMANCE ANALYSIS

• ? a??????µ?? ap?te?e?ta? ap? d?? ???t??e?. ?
  p??t? e??a? ??a loop p?? µe????? f???? ?a?e?
  t?? de?te?? ???t??a p?? e??a? ep?s?? ??a loop
• G?a ?a ?p?????s??µe t?? apa?t??µe?? ????? p??pe?
  ?a ?p?????s??µe t? ????? p?? apa?te?ta? ??a t?
  e??te???? loop , µe t?? p???p??es? ?t? t?
  es?te???? loop de? apa?te? ????? ??a ?a ???e?,
  t? p?se? f???? ?a?e?ta? t? es?te???? loop ,a???
  ?a? t? ????? p?? apa?te?ta? ??a t? es?te????
  loop
• ?? es?te???? loop apa?te? ??a ?????
  p???p????t?ta? t?? t???? t?? O(n2.5log4n) ?p?? n
  ? a???µ?? t?? s?µe???
• ?? es?te???? loop a?aµ??eta? ?a ????e? t? p???
  O(n) f????
• ?? e??te???? loop apa?te? ??a ????? O(n2logn)

31
PERFORMANCE ANALYSIS

• St? ?e???te?? pe??pt?s? ? ?????? p?? ?a ??e?aste?
  ?a e??a?
• O(n) x O(n2.5log4n) O(n2logn) O(n3.5log4n)
• ??a p?? p?a?µat??? ??? ???? ??a µ??a????
  a?t??e?µe?a e??a? t?
• O(n2log4n)
• ?? es?te???? loop ?e???? µe t? s?????s?
  tess???? s?µe??? µe ???a t?sse?a.?? ???t?? ß?????
  s??????eta? µe t?? ea?t? t?? ?a? apa?te? sta?e??
  ?????.G?a ta ep?µe?a d?? s?µe?a apa?te?ta? O(n2)
  .?? t??t? s?µe?? apa?te? O(n) ????? ??a ?a ß?e?
  µe p??a s?µe?a ?a s??????e?. ??a apa?te?ta?
  s???????? ?????? O(n2.5) pa?????ta? O(n1.5)
  p??a?? ta????sµata.G?a ?a ta???????µe ta ?p????pa
  s?µe?a t?? ??sta? apa?te?ta? a?????? ??????
  O(nlog3n) ??a ???e s????? tess???? s?µe??? ?a?
  O(log4n) ????? a?a??t?s?? ??a ???e s?µe??.?ts?
  apa?te?ta? ??a?t? t? d?ad??as?a O(n 2.5log3n) ?a?
  O(nlog 4n) ?????? a?? s????? tess???? s?µe??? ??a
  ?a ???e? t? ta???asµa

32
DATA CONDITIONING

• ??? ? a???? e??a? µ? µ?de???? t?te ? ?at?
  p??s????s? ap?stas? ap? ?a???a ap? ta t?sse?a
  p??ta s?µe?a de? a?a??????e? p??ta ??a s?µe??
  µ??ad???
• ?? p??ß??µa ß??s?eta? ?ta? ? ap?stas? d?? s?µe???
  e??a? p??? µ???? s???????µe?? µe t?? ap?stas?
  a?t?? t?? s?µe??? ap? t? s?µe?? p?? ?a
  a?a?????ste?
• ?.? G?a ??a ???? t?????? µe µ???? ß?s? µ?a µ????
  a??a?? st?? p?e???? a????e? t? ??s? t?? ????f??
  se µe?a??te?? ap?stas? ?a? p??a??t?ta st? ??s?
  ??p???? ????? s?µe???

33
DATA CONDITIONING

• ?? p??ß??µa a?t? a?t?µet?p??eta? µe d?? ?d?e?
• ?a p??ta t?sse?a s?µe?a ep??????ta? ?? t? ???t??
  ß????? ?a? t??a s?µe?a st? convex hull ?s? p??
  µa???? µeta?? t??? ?a? ap? t? ???t?? ß?????. ?e
  t?? t??p? a?t? ?a???a s?µe?? st?? ?µ?da de? ?a
  ap??e? pe??ss?te?? ap? ta p??ta t?sse?a ap? ?t?
  a?t? µeta?? t???
• ???e?tetaµµ??e? s??????? s?µe??? d?a?e??????ta?
  ?e????st?. S?µµet??e? ?µ?d?? s?µe??? ???????ta?
  se d?? ?at?????e?. ???ta ????µe p??sµat????
  ?µ?de? p?? af????? pe??st??f?? ???? ap? ??a?
  ????a , pa??????e? ?a? ????????e? a?ta?a???se??
  se a?t?? t?? ????a. ?pe?ta s??????? s?µe??? p??
  e??a? a??et?? f???? p?? µa???s?e?e?? ap? ap??a?t?
  µp????? ?a ????? µ??? p??sµat???? s?µµet??e?. ?
  ???s? t?? a??????µ?? a?t?µet?p??e? a?t? t?
  µa???? s?????? s?µe??? sa? ??aµµ???
  ta?t?p???s?.??te ? s?????? t??p?p??e?ta?
  µe?e?????ta? t?? ap?stas? ???e s?µe??? a?t?????
  ap? t?? ?e?t???? ??aµµ?.

34
TESTING

• ???s?µ?p??????a? s??????? s?µe??? ap? t?? ????f??
  p????d??? ?p??(pe?t?????,??ß??,e???s?ed??,??ß???-?
  ?t?ed??,e???s?d?de??ed??,??µß???s?d?de??ed??)
• ?a pa?ap??? p???ed?a ????? µe???? a???µ?
  s?µµet???? ?a? st?? p???? a?t??e?µe?a sta ?p??a
  µp??e? ?a ???e? reverse engineering de? p??pe?
  ?a ????? p??? ap? 200 ????f??
• St?? p??a?a 1 ? µ??f? (a,b,cd) de???e? ?t?
  ß?????a? a????? p?? d?µ??????sa? sta?e??? ?µ?de?
  ap? a,b,c s????a ta ?p??a ????? d s?µµet??e?. G?a
  d1 ????µe as?µµet?? a?t??e?µe?? ?a? ? µet??es?
  s?ed???e? t? s?µe?? st?? ea?t? t??
• G?a t?? ?p?????sµ? t?? ?????? µ?a s????t?s? t??
  µ??f?? (abnc)(loddn) ?e??????e ?at?????? ?a?
  µ???sta ??a c2 ?a? d3

35
TESTING
36
TESTING
37
TESTING

• ????µ?? ?de??a? ?t? p??sµata µe a?a????a
  d?ast?se?? 51 de? µp????? ?a a?a?????? µe t??
  a??????µ?
• ? pe??pt?s? p?? pa???e? t? ????te?? ????? e??a?
  t? as?µµet?? a?t??e?µe??. ? ?????? p?? apa?te?ta?
  e??a? O(nlog4n)
• ? ?e???te?? pe??pt?s? e??a? ??a p??sµa µe µe????
  a???µ? p?e????. ? a???µ?? s?µµet????
  s?µpe???aµß????ta? t?? a?ta?a???se?? e??a? 4n.?
  ?????? p?? apa?te?ta? e??a? O(n2log4n)

38
SYMMARY

• ? a???ß?? s?µµet??a e??? a?t??e?µ???? de? e??a?
  robust ?a??? ?atast??feta? ap? ????? sta
  ded?µ??a. ?ts? ???s?µ?p??e?ta? ? ???? t?? ?at?
  p??s????s? s?µµet??a?. ?? p??ß??µa e??a? ?
  ?a????sµ?? a??????µ?? p?? ?a t?? ?p??????e?, ?a
  p??te??e? ?a? ?a ep??????e? t??p?p???se?? st?
  µ??t??? ?ste t? ?a??????? µ??t??? ?a ??e? a???ß?
  s?µµet??a.
• ? a??????µ?? p?? p??te??eta? t???e? se ap?de?t?
  ?????
• ??µ?????e? ?at? p??s????s? s?µµet??e? p?? de?
  e??a? a??a??ete? ?a? e?te ?p?????? e?te de?
  ?p?????? se µ?a s?????? s?µe???
• ? ?p?????sµ?? e??a? robust ?? p??? t? ????ß?
  st?? t?p??es?a t?? s?µe???
• ? ep?µe??? st???? e??a? ? ep??tas? t?? a??????µ??
  ??a t?? e??es? µe????? as?µµet???? ?a??? ta
  µ??a???? a?t??e?µe?a st?? p?a?µat???t?ta de?
  e??a? e?te??? s?µµet????