Introduction to Data-Driven Animation: Programming with Motion Capture Jehee Lee

Transcription

1 Inroducion o Daa-Driven Animaion: Programming wih Moion Caure Jehee Lee Seoul Naional Universiy

2 Daa-Driven Animaion wih Moion Caure

3 Programming wih Moion Caure Why is i difficul? Encomass a lo of heerogeneous informaion Join angles Posiion/orienaion of a skeleal roo Their emoral rajecories A number of local/global coordinae sysems

4 Mahemaical Noaion Can we describe oeraions in simle euaions? Linear inerolaion beween wo moion clis Slice wo moion clis seuenially

5 Course Objecives Mahemaical framework and noaion Geomeric reasoning and inuiion A racical guide o rogramming wih moion caure

6 References Jehee Lee, Reresening Roaions and Orienaions in Geomeric Comuing, IEEE Comuer Grahics and Alicaions, 008. Yoonsang Lee, Sungeun Kim, Jehee Lee, Daa-Driven Bied Conrol, SIGGRAPH 00. Jehee Lee, Jinxiang Chai, Paul Reisma, Jessica Hodgins, and Nancy Pollard, Ineracive Conrol of Avaars Animaed wih Human Moion Daa, SIGGRAPH 00. Jehee Lee and Sung Yong Shin, A Coordinae-Invarian Aroach o Muliresoluion Moion Analysis, Grahical Models, 00. Hyun Joon Shin, Jehee Lee, Michael Gleicher, and Sung Yong Shin, Comuer Puery: An Imorance-based Aroach, ACM Transacions on Grahics, 00. Jehee Lee and Sung Yong Shin, A Hierarchical Aroach o Ineracive Moion Ediing for Human-like Characers, SIGGRAPH 99.

7 Course Overview Inroducion and Overview 5 min Coordinae-Invarian Geomeric Programming 0 min Wha is coordinae-invarian? Affine geomery Coordinae-invarian oeraions beween oins and vecors Programming wih Orienaions and Roaions 35 min Programming wih Moion Caure Daa 0 min Pracical examles 30 min

8 Geomeric Programming A way of handling geomeric eniies such as vecors, oins, and ransforms. Wrie geomeric rograms relying on geomeric reasoning raher han coordinae maniulaion Pioneered by Goldman and DeRose Geomeric rogramming : A coordinae-free aroach, SIGGRAPH 988 Course #5 Noes Coodinae-Invarian vs Coordinae-Free

9 Examle of coordinae-deendence Poin Poin Wha is he sum of hese wo osiions?

10 If you assume coordinaes, = x, y = x, y The sum is x+x, y+y Is i correc? Is i geomerically meaningful?

11 If you assume coordinaes, = x, y x+x, y+y Vecor sum Origin = x, y x, y and x, y are considered as vecors from he origin o and, resecively.

12 If you selec a differen origin, = x, y x+x, y+y = x, y Origin If you choose a differen coordinae frame, you will ge a differen resul

13 Poins and Vecors vecor - Poin Poin A oin is a osiion secified wih coordinae values. A vecor is secified as he difference beween wo oins. If an origin is secified, hen a oin can be reresened by a vecor from he origin. Bu, a oin is sill no a vecor in coordinae-free conces.

14 Vecor and Affine Saces Vecor sace Includes vecors and relaed oeraions No oins Affine sace Suerse of vecor sace Includes vecors, oins, and relaed oeraions

15 Coordinae-Invarian Geomeric Oeraions Addiion Subracion Scalar mulilicaion Linear combinaion Affine combinaion

16 Addiion + w w u + v v + w is a oin u u + v is a vecor u, v, w : vecors, : oins

17 Subracion - w u u - v - -w v u - v is a vecor - is a vecor - w is a oin u, v, w : vecors, : oins

18 Scalar Mulilicaion scalar vecor = vecor oin = oin 0 oin = vecor c oin = undefined if c 0,

19 Linear Combinaion v v v v v N N N i i i c c c c A linear sace is sanned by a se of bases Any oin in he sace can be reresened as a linear combinaion of bases

20 Affine Combinaion N i i i N i i N N N i i i c c c c c c

21 Examles + / : midoin of line + / 3 : no valid + + r / 3 : cener of graviy of r / + / r : a vecor from r o he midoin of and

22 Summary

23 Marix Reresenaion Use an exra coordinae In 3-dimensional saces Poin : x, y, z, Vecor : x, y, z, 0 For examle x, y, z, + x, y, z, = x+x, y+y, z+z, oin oin undefined x, y, z, - x, y, z, = x-x, y-y, z-z, 0 oin oin vecor x, y, z, + x, y, z, 0 = x+x, y+y, z+z, oin vecor oin

24 Projecive Saces Homogeneous coordinaes x, y, z, w = x/w, y/w, z/w, Useful for handling ersecive rojecion Bu, i is algebraically inconsisen!!,0,0,,,0,,,0,,,0,,0,0,,,0, 3,,0,3, 3,0,

25 Course Overview Inroducion and Overview 5 min Coordinae-Invarian Geomeric Programming 0 min Programming wih Orienaions and Roaions 35 min Reresening orienaions and roaions Analogy beween oins/vecors and orienaions/roaions Coordinae-invarian oeraions beween orienaions and roaions Programming wih Moion Caure Daa 0 min Pracical examles 30 min

26 Orienaion and Roaion No inuiive Formal definiions are also confusing Many differen ways o describe Roaion direcion cosine marix Euler angles Axis-angle Roaion vecor Helical angles Uni uaernions

27 Orienaion and Roaion Roaion Circular movemen Orienaion The sae of being oriened Given a coordinae sysem, he orienaion of an objec can be reresened as a roaion from a reference ose

28 Analogy oin : vecor is similar o orienaion : roaion Boh reresen a sor of sae : movemen Reference coordinae sysem

29 Analogy oin : vecor is similar o orienaion : roaion Boh reresen a sor of sae : movemen oin : he 3d locaion of he bunny vecor : ranslaional movemen Reference coordinae sysem

30 Analogy oin : vecor is similar o orienaion : roaion Boh reresen a sor of sae : movemen oin : he 3d locaion of he bunny vecor : ranslaional movemen Reference coordinae sysem orienaion : he 3d orienaion of he bunny roaion : circular movemen

31 D Orienaion or 0 Polar Coordinaes

32 D Orienaion or 0 Time Alhough he moion is coninuous, is reresenaion could be disconinuous

33 D Orienaion or 0 Time Many-o-one corresondences beween D orienaions and heir reresenaions

34 Exra Parameer Y x y x, y X

35 Exra Parameer x Roaion marix is ye anoher mehod of using exra arameers Y x y cos sin sin cos x, y X

36 Comlex Number of Uni Lengh Imaginary x iy Real

37 Comlex Exoneniaion Imaginary x iy cos i sin e i x iy Real

38 D Roaion Comlex numbers of uni lengh are good for reresening D orienaions, bu inadeuae for D roaions A comlex number canno disinguish differen roaional movemens ha resul in he same final orienaion Turn 0 degree couner-clockwise Turn -40 degree clockwise Turn 480 degree couner-clockwise Imaginary Real

39 D Roaion and Orienaion D Roaion The conseuence of any D roaional movemen can be uniuely reresened by a urning angle D Orienaion The non-singular arameerizaion of D orienaions reuires exra arameers Eg Comlex numbers, x roaion marices

40 Oeraions in D orienaion : comlex number roaion : scalar value exroaion : comlex number

41 D Roaion and Dislacemen Imaginary c x iy Real

42 D Roaion and Dislacemen Imaginary c c x iy c c c c or e e i i Real

43 D Orienaion Comosiion Imaginary c c undefined c c Real

44 D Roaion Comosiion Imaginary e i e i e i Real

45 Analogy

46 3D Orienaion and Roaion Given wo arbirary orienaions of a rigid objec, X X Z Z Y Y

47 3D Orienaion and Roaion Euler s fundamenal heorem We can always find a fixed axis of roaion and an angle abou he axis vˆ

48 Roaion Vecor vˆ vˆ : uni vecor :scalar angle Roaion vecor 3 arameers v vˆ x, y, z Axis-Angle + arameers, vˆ

49 3D Orienaions and Roaions Orienaions and roaions are differen in coordinae-invarian geomeric rogramming Use uni uaernions for orienaion reresenaion Alernaively, 3x3 orhogonal marices Use 3-vecors for roaion reresenaion

50 Uni Quaernions Real w ix jy kz S 3 w, x, y, z w, v k i j w x y z

51 Tangen Vecor Infiniesimal Roaion 3 T S

52 Tangen Vecor Infiniesimal Roaion 3 T S

53 Tangen Vecor Infiniesimal Roaion 3 T I S I, 0, 0, 0 0, x, y, z Angular Velociy

54 Ex and Log I log ex ex v ex vˆ cos, vˆ sin

55 3D Roaion and Dislacemen u u 3 S 3 R log ex ex v

56 3D Roaion and Dislacemen 3 S I u 3 R u log ex ex v

57 3D Roaion and Dislacemen I log ex ex v log v u 3 R u

58 Sherical Linear Inerolaion SLERP [Shoemake 985] Linear inerolaion beween wo orienaions sler, ex log

59 Sherical Linear Inerolaion sler, ex log

60 Analogy oin : vecor is similar o orienaion : roaion

61 Coordinae-Invarian Oeraions

62 Linear Combinaion of Roaions Marc Alexa, Linear combinaion of ransformaions, Siggrah 00.

63 Coordinae-Invarian Oeraions

64 Affine Combinaion of Orienaions

65 Course Overview Inroducion and Overview 5 min Coordinae-Invarian Geomeric Programming 0 min Programming wih Orienaions and Roaions 35 min Programming wih Moion Caure Daa 0 min Reresening moion daa and moion dislacemens Coordinae-invarian oeraions for moion daa Pracical examles 30 min

66 Moion Reresenaion Configuraion of an ariculaed figure Linear comonens: Angular comonens: 3 R 3 S i 0 n m The osiion of he roo segmen The orienaions of body segmens w.r.. heir arens join angles The orienaion of he roo segmen

67 Elemen-wise Oeraions Translaion Roaion ex ~ ' n n v v d m m ex ex ex ~ ' n n r v 0 0 v 0 d m m

68 Elemen-wise Oeraions Exercise joins ex ex ex ~ ' n i n i j v 0 v 0 0 d m m

69 Moion Dislacemen Indeenden ranslaion and roaion for roo ex ex ex ex ex ~ ex ~ ' n n n n n n v v v v v v v v v d m m

70 Moion Dislacemen Rigid ransformaion a roo ex ex ex ex ex ~ ' 0 0 n n n n v v v v v v d m m

71 Moion Dislacemen Rigid Transformaion a roo n n n n ' ' ' ' ' ' ' ex m m d

72 Oeraions Valid oeraions Invalid oeraions! Be careful ex ~ ex ~ 3 d d d d d d m m m d m?? m m m

73 Oeraions Time waring Proeries ' m s m d d m d d m m d d m

74 Coordinae-Invarian Oeraions

75 Course Overview Inroducion and Overview 5 min Coordinae-Invarian Geomeric Programming 0 min Programming wih Orienaions and Roaions 35 min Programming wih Moion Caure Daa 0 min Pracical examles 30 min Moion exaggeraion and syle ransfer Aligning and waring Inerolaion and ransiioning

76 Moion Exaggeraion Jehee Lee and Sung Yong Shin, A Coordinae-Invarian Aroach o Muliresoluion Moion Analysis and Synhesis, Grahical Models, 00.

77 Moion Exaggeraion M L M L M :inu daa L :simlified and lowass filered

78 Walk Run? Sru Run in a romous manner

79 Walk Run Sru

80 Syle Transfer A B A B B B A A

81 Walk Turn Lim Turn wih a lim

82 Transiion Grah TurnL TurnR Sar Righ foo forward Sand So Loo

83 Transiion Grah Jehee Lee, Jinxiang Chai, Paul Reisma, Jessica Hodgins, Nancy Pollard, Ineracive Conrol of Avaars Animaed wih Human Moion Daa, SIGGRAPH 00

84 Connecing Moion Segmens Firs Moion Second Moion Alignmen Roae and ranslae he second moion o align wo moions Waring War he moions a he boundary so ha hey can be conneced smoohly

85 Alignmen The end of one moion A should be aligned o he beginning of he nex moion B The roo locaion of he end of A: The roo locaion of he beginning of B: Aly o moion B, ha is, A n A n, 0 0 B B ex ex ex ~ n n B v u 0 0 v u d m B B A n A n 0 0 ex ~ v u

86 Alignmen Using In-Plane Transformaion Rigid ransformaion resriced wihin a lane Roaion abou he verical axis, followed by Translaion along wo horizonal axes Finding in-lane roaion close o given roaion Using Euler angles Discard roaion abou x- and z-axes No oimal R R x R y R z

87 Oimal In-Plane Transformaion The geodesic curve reresens a se of orienaion ha can be reached by roaing abou a fixed axis from any orienaion on he curve A n G yˆ, B 0 ex yˆ B 0 B 0 cos, yˆ sin B 0

88 Oimal In-Plane Transformaion Hyun Joon Shin, Jehee Lee, Michael Gleicher, Sung Yong Shin, Comuer Puery: An Imorance-based Aroach ACM Transacions on Grahics, 00

89 Waring Deform a moion smoohly so ha i seamlessly connecs o is revious moion Firs Moion d Second Moion d A B m m n 0 A scalar ransiion funcion s S B m s d

90 Blending for Smooh Transiion Case sudy: walk-o-sneak Transi smoohly over one cycle of locomoion Walk is faser han sneak One cycle of sneak is longer han one cycle of walk Jehee Lee and Sung Yong Shin, A Hierarchical Aroach o Ineracive Moion Ediing for Human-like Characers, SIGGRAPH 99

91 Blending for Smooh Transiion Blend over he overlaing ime inerval Walk Walk Walk Time scaling Blend Linear Inerolaion Sneak Sneak Time scaling Sneak

92 Blending for Smooh Transiion Linear inerolaion beween moions Sler for orienaion comonens A scalar ransiion funcion s Walk Blend Sneak A B s m s m s cos L L

93 Disconinuiy in velociy Walk Time Scaling Blend Uniform resamling + Linear Inerolaion Sneak Walk Blend Sneak Disconinuiy in velociy

94 Time Scaling Non-uniform resamling Walk Time elased Blend Sneak 0 0 Normalized ime T H f H f H3 T 0 0, T, T '0 f, T ' f H H H 3 0 H H H

95 Wra U Coordinae-Invariance does maer Disinguish orienaions from roaions as we do oins from vecors The algebraic srucure of moion daa is similar o he srucure of orienaion/roaion daa