The Size of the 3D Visibility Skeleton: Analysis and Application

Size: px

Start display at page:

Download "The Size of the 3D Visibility Skeleton: Analysis and Application"

Jonathan Logan
5 years ago
Views:

1 Th Siz of th 3D Visibility Sklton: Analysis and Application Ph.D. thsis proposal Linqiao Zhang School of Computr Scinc, McGill Univrsity March 20, 2008 thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 1/41

2 Ovrviw th visibility problm and its rlatd litratur xprimntal study of th siz of th visibility sklton in 3D a succinct 3D visibility sklton a nw application to motion planning summary of my thsis status thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 2/41

3 Th Visibility Problm Givn a st of input objcts, what an objct can s? application: computr graphics, robotics, computr vision two typs of problms: asy: nquir visibility information from a fixd dirction. solutions: ray shooting... hardr: nquir visibility information from all dirctions solutions: discrtization; visibility sklton data structur... thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 3/41

4 Visibility Sklton a graph that contains vrtics and arcs in 2D: vrtx: corrsponds to a maximal fr lin sgmnt that has 0-dgrs of frdom arc: corrsponds to a st of maximal fr lin sgmnts that hav 1-dgr of frdom, and form on connctd componnt A B thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 4/41

5 Visibility Sklton a graph that contains vrtics and arcs in 3D: vrtx: corrsponds to a maximal fr lin sgmnt that has 0-dgrs of frdom arc: corrsponds to a st of maximal fr lin sgmnts that hav 1-dgr of frdom, and form on connctd componnt thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 4/41

6 Visibility Sklton in 2D only on typ of sklton vrtx in 2D: A B A B thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 5/41

7 Visibility Sklton in 2D only on typ of sklton vrtx in 2D: A B A B only on typ of sklton arc in 2D: A B A B thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 5/41

8 Visibility Sklton in 3D ight typs of sklton vrtics in 3D: v f v 2 v EEEE VEE FEE VV f f v f f v v f FF FvE FE FVV thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 6/41

9 Visibility Sklton in 3D four typs of sklton arcs in 3D: v v f f v v EEE VE FE FVE thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 6/41

10 Litratur visibility sklton (complx): 2D: thory [G. Vgtr, M. Pocchiola, 93]; implmntation [P. Anglir, M. Pocchiola, 03] 3D: thory [F. Durand, G. Drttakis, C. Puch, 97]; brut forc implmntation [F. Durand, 97] thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 7/41

11 Litratur visibility sklton (complx): 2D: thory [G. Vgtr, M. Pocchiola, 93]; implmntation [P. Anglir, M. Pocchiola, 03] 3D: thory [F. Durand, G. Drttakis, C. Puch, 97]; brut forc implmntation [F. Durand, 97] siz of th visibility sklton: whn k is th numbr of input objcts (.g. polygons, polytops, discs, sphrs) and n is th total numbr of dgs 2D: worst cas: Θ(k 2 ); xprimntal vidnc: Θ(k) [F. Cho, D. Forsyth, 99] 3D: worst cas: Θ(n 4 ); xprimntal vidnc: Θ(n 2 ) [F. Durand t al. 99] thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 7/41

12 Litratur cont. mor on siz of th 3D visibility sklton: worst cas: Θ(n 2 k 2 ), whn inputs ar k polytops with total complxity n. [H. Bronnimann, O. Dvillrs, V. Dujmovic, H. Evrtt, M. Gliss, X. Goaoc, S. Lazard, H.-S. Na, and S. Whitsids, 07] worst cas: O(nk 2 nk), whn input polytops hav constant complxity. [M. Gliss, 07] xpctd siz: O(k), whn inputs ar randomly distributd unit sphrs. [O. Dvillrs, V. Dujmovic, H. Evrtt, X. Goaoc, S. Lazard, H.-S. Na, and S.Ptitjan, 03] thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 8/41

13 Goals of th Thsis provid an fficint and robust implmntation to nabl xprimntal studis (in 3D) xprimntally study th siz of th visibility sklton in practic sk a succinct visibility sklton find nw applications of th visibility sklton thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 9/41

14 3D Exprimntal Study: Softwar implmntd a swp plan algorithm: running tim complxity: O(n 2 k 2 log k) vrsus O(n 5 ) input: disjoint convx polytops in gnral position output: EEEE, VEE, FEE vrtics thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 10/41

15 3D Exprimntal Study: Softwar cont. fficincy: dsignd crtain prdicats,.g. ordring a pair of swp plans robustnss: usd filtrd_xact numbr typ softwar vrification: usd gomviw to visualiz th intrmdiat stps and th output of th softwar [socg07 vido] compard th output with a brut forc implmntation: tstd on 20 input scns thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 11/41

16 3D Exprimntal Study: th St Up Th univrs: a grat sphr with radius R Th objcts k: randomly distributd disjoint convx polytops Th scn dnsity µ: calculatd using th outr tangnt sphrs that gnratd th polytops R = 3 k/µ thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 12/41

17 3D Exprimntal Study: th St Up cont. numbr typ: doubl th obsrvd failur rat lss than 0.1% doubl is four tims fastr than filtrd_xact mdium prformanc machin: i686 with Pntium 2.8 GHz CPU and 2 GB main mmory masur: th numbr of EEEE, VEE, FEE vrtics running tim thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 13/41

18 3D Exprimntal Study: Paramtrs thr paramtrs: n/k: complxity of a polytop k: numbr of polytops µ: scn dnsity thr suits of xprimnts: suit I: fix µ, choos n/k 7.5, 40, 85, and vary k systmatically suit II: fix µ, choos k = 30, 60, 90, and vary n/k systmatically suit III: rpat suit I, and vary µ systmatically thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 14/41

19 3D Exprimntal Study: Paramtrs cont. sampl input of suit I: fix µ, choos n/k 7.5, 40, 85, and k = 50 n/k 7.5 n/k 40 n/k 85 thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 15/41

20 3D Exprimntal Study: Suit I Rsults siz of sklton vrsus total numbr of dgs n 4.5 x n / k = 7.5 n / k = 40 n / k = Numbr of All Vrtics n thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 16/41

21 3D Exprimntal Study: Suit I Rsults cont. siz of sklton vrsus k 2 n/k 4.5 x n / k = 7.5 n / k = 40 n / k = Numbr of All Vrtics sqrt(n / k) k 2 x 10 4 thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 17/41

22 3D Exprimntal Study: Intrprtation of k 2 n/k givn a univrs U of radius R, th xpctd siz of th visibility sklton is [O. Dvillrs, V. Dujmovic, H. Evrtt, X. Goaoc, S. Lazard, H.-S. Na, and S.Ptitjan]: Θ(k) O(k 2 ) whn U consists of k randomly distributd polytops that ar insid U nar th boundary of U boundd aspct ratio constant complxity thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 18/41

23 3D Exprimntal Study: Intrprtation of k 2 n/k cont. givn a univrs U of radius R that consists polytops of any givn dnsity polytops that ar insid U: Θ(R 3 ) polytops that ar nar th boundary of U: Θ(R 2 ) assum polytops hav constant complxity numbr of visibility sklton vrtics with support polytops ar insid U: Θ(R 3 ) nar th boundary of U: O(R 4 ) thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 19/41

24 3D Exprimntal Study: Intrprtation of k 2 n/k cont. xprimntal vidnc: 4.5 x n / k = 7.5 n / k = 40 n / k = Numbr of All Vrtics sqrt(n / k) k 2 x 10 4 thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 20/41

25 3D Exprimntal Study: Intrprtation of k 2 n/k cont. xprimntal vidnc: 3.5 x n / k = 7.5 n / k = 40 n / k = 85 VEE Vrsus Othr Typs of Vrtics VEE othrs sqrt(n / k) k 2 x 10 4 thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 21/41

26 3D Exprimntal Study: Intrprtation of k 2 n/k cont. thortical vidnc: thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 22/41

27 3D Exprimntal Study: Intrprtation of k 2 n/k cont. thortical vidnc: Conjctur: Th xpctd numbr of typ VEE vrtics is linarly rlatd to th xpctd silhoutt siz of th polytops. proving of this conjctur would gnraliz th rsults of [O. Dvillrs, V. Dujmovic, H. Evrtt, X. Goaoc, S. Lazard, H.-S. Na, and S.Ptitjan] thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 22/41

28 A Succinct 3D Visibility Sklton rcall th ight typs of visibility sklton vrtics: v f v 2 v EEEE VEE FEE VV f f v f f v v f FF FvE FE FVV thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 23/41

29 A Succinct 3D Visibility Sklton consists of typ EEEE, VEE, FEE and VV vrtics only impact: rducs th siz of th visibility sklton simplifis th computation procdur, and rducs th computation tim givs a simplr data structur prsrvs th visibility information thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 24/41

30 A Succinct 3D Visibility Sklton dsignd an algorithm to updat th succinct visibility sklton incrmntally can maintain th visibility sklton in dynamic scns at low cost 1 a b c d i 2 3 j k thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 25/41

31 Finding a Rlativly Shortst Path in a Scn of Polytop Obstacls litratur: computing th xact shortst path in 3D polytop obstacl spac is NP-hard [John Canny 1987] finding th supporting dg squnc of th shortst path is NP-hard [John Canny 1987] thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 26/41

32 Finding a Rlativly Shortst Path in a Scn of Polytop Obstacls litratur: computing th xact shortst path in 3D polytop obstacl spac is NP-hard [John Canny 1987] finding th supporting dg squnc of th shortst path is NP-hard [John Canny 1987] various polynomial algorithms to comput th (approximatd) shortst paths on on or two polytops xponntial algorithms to comput th shortst paths on mor than two polytops [M. Sharir and A. Schorr 1986], [K. Snvirantn and S. Earls 1993] thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 26/41

33 Finding a Rlativly Shortst Path in a Scn of Polytop Obstacls my approach: us computd visibility sklton vrtics to find a rlativly shortst path in polynomial tim thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 27/41

rlativly shortst path in polynomial tim thsis proposal: Th

34 Finding a Rlativly Shortst Path in a Scn of Polytop Obstacls my approach: us computd visibility sklton vrtics to find a rlativly shortst path in polynomial tim thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 27/41

35 Finding a Rlativly Shortst Path in a Scn of Polytop Obstacls my approach: us computd visibility sklton vrtics to find a rlativly shortst path in polynomial tim thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 27/41

36 Finding a Rlativly Shortst Path in a Scn of Polytop Obstacls algorithm: stp on: construct a sarch graph rprsnt ach polytop as a vrtx in th sarch graph rprsnt ach visibility sklton vrtx as an dg of th sarch graph dg lngth is th lngth of th sgmnt that is rprsntd by th visibility sklton vrtx thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 28/41

37 Finding a Rlativly Shortst Path in a Scn of Polytop Obstacls algorithm: stp on: construct a sarch graph rprsnt ach polytop as a vrtx in th sarch graph rprsnt ach visibility sklton vrtx as an dg of th sarch graph dg lngth is th lngth of th sgmnt that is rprsntd by th visibility sklton vrtx stp two: comput th initial shortst path from th sarch graph us Dijkstra s algorithm thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 28/41

38 Finding a Rlativly Shortst Path in a Scn of Polytop Obstacls algorithm: stp thr: optimiz th shortst path thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 29/41

39 Finding a Rlativly Shortst Path in a Scn of Polytop Obstacls algorithm: stp thr: optimiz th shortst path stp four: obtain a computd rlativly shortst path thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 29/41

40 Finding a Rlativly Shortst Path in a Scn of Polytop Obstacls running tim: O(n 2 k 2 logk) n is th total numbr of dgs, k is th total numbr of polytops quality of th approximation thortically: opn problm xprimntally: futur work thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 30/41

41 Intndd Contributions of my Thsis an fficint and robust implmntation [SoCG07 vido] xprimntal study of th siz of th 3D visibility sklton and th rlatd thortical rsults a succinct 3D visibility sklton finding a rlativly shortst path in a scn of polytop obstacls thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 31/41

42 Intndd Contributions of my Thsis an fficint and robust implmntation [SoCG07 vido] xprimntal study of th siz of th 3D visibility sklton and th rlatd thortical rsults a succinct 3D visibility sklton finding a rlativly shortst path in a scn of polytop obstacls th algbraic dgr of th prdicats [Computational Gomtry: Thory and Application, accptd] xprimntal study of th 2D visibility sklton [Intrnational Journal of Computational Gomtry and Applications, 2007] thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 31/41

43 Exprimntal Study: in 2D whn k is th total numbr of inputs worst cas: Θ(k 2 ) [V. Ggtr, M. Pocchiola] xpctd siz: thortically: O(k) (proof is similar to th 3D cas) inputs ar randomly distributd discs or polygons inputs ar with boundd aspct ratio xprimntally: us th availabl softwar [P. Anglir02] thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 32/41

44 Exprimntal Study: in 2D modl: Th univrs: a larg disc Th objcts: n randomly distributd disjoint unit discs with varying dnsity µ dnsity = 0.55 dnsity = 0.1 thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 33/41

45 Exprimntal Study: in 2D stting: vary: scn dnsity µ & numbr of discs n masur: siz of th 2D visibility sklton rport: man of th 10 xprimnts, and omit th standard drivations sinc th obsrvd valus ar small slow machin: i686 with AMD Athlon 1.73 GHz CPU and 1 GB main mmory thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 34/41

46 Exprimntal Study: in 2D rsults: scn dnsity: bitangnts mmory tim scn dnsity: 0.55 bitangnts mmory tim numbr of unit discs numbr of unit discs scn dnsity: bitangnts mmory tim scn dnsity: bitangnts mmory tim numbr of unit discs thsis proposal: Th Siz of th 3D numbr Visibility of unit discs Sklton: Analysis and Application p. 35/41

47 Exprimntal Study: in 2D rsults: µ n µ 2 µ for n > µ + 61 n o = 101 N bts = n : 300 µ : 0.55 thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 35/41

48 3D Exprimntal Study: Intrprtation of k 2 n/k givn a univrs U of radius R, th xpctd siz of th visibility sklton is [O. Dvillrs, V. Dujmovic, H. Evrtt, X. Goaoc, S. Lazard, H.-S. Na, and S.Ptitjan]: Θ(k) whn U consists of k randomly distributd unit sphrs Θ(k) whn U consists of k randomly distributd polytops that ar insid U boundd aspct ratio constant complxity thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 36/41

49 3D Exprimntal Study: Intrprtation of k 2 n/k givn a univrs U of radius R, th xpctd siz of th visibility sklton is [O. Dvillrs, V. Dujmovic, H. Evrtt, X. Goaoc, S. Lazard, H.-S. Na, and S.Ptitjan]: Θ(k) whn U consists of k randomly distributd unit sphrs Θ(k) O(k 2 ) whn U consists of k randomly distributd polytops that ar insid U nar th boundary of U boundd aspct ratio constant complxity thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 36/41

50 3D Exprimntal Study: Running Tim running tim vrsus n 1.5 k log k 3 x 104 n / k = 7.5 n / k = 40 n / k = Running Tim sqrt(n 3 ) k log(k) x 10 8 thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 37/41

51 3D Exprimntal Study: Suit II Rsults siz of sklton vrsus k 2 n/k x 105 k = 30 k = 60 k = 90 Numbr of All Vrtics sqrt(n / k) k 2 x 10 4 thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 38/41

52 3D Exprimntal Study: Suit II Rsults cont. siz of sklton vrsus n/k x 105 k = 30 k = 60 k = 90 Numbr of All Vrtics sqrt(n / k) thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 39/41

53 Rmaining Work of my Thsis xprimntal study of th siz of th 3D visibility sklton in trms of varying scn dnsity: on month writ up in dtail th succinct 3D visibility sklton data structur: on month writ up th proposal of th algorithm for computing th rlativly shortst collision fr path: two wks prpar th thsis manuscript: six months thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 40/41

54 Intndd Main Chaptrs of my Thsis Exprimntal Study of th 2D Visibility Sklton Dsign Aspcts of th Implmntation Th Algbraic Dgr of th Prdicats Exprimntal Study of th 3D Visibility Sklton A Succinct 3D Visibility Sklton Finding a Rlativly Shortst Path in a Scn of Polytop Obstacls thsis proposal: Th Siz of th 3D Visibility Sklton: Analysis and Application p. 41/41

To Do. Mesh Data Structures. Mesh Data Structures. Motivation. Outline. Advanced Computer Graphics (Fall 2010) Desirable Characteristics 1

To Do. Mesh Data Structures. Mesh Data Structures. Motivation. Outline. Advanced Computer Graphics (Fall 2010) Desirable Characteristics 1 Advancd Computr Graphics (Fall 200) CS 283, Lctur 5: Msh Data Structurs Ravi Ramamoorthi http://inst.cs.brkly.du/~cs283/fa0 To Do Assignmnt, Du Oct 7. Start rading and working on it now. Som parts you