Multi-linear Systems for 3D-from-2D Interpretation. Lecture 1. Multi-view Geometry from a Stationary Scene. Amnon Shashua

Transcription

1 Mult-lnea Sytem fo 3D-fom-D Inteetaton Lectue Mult-vew Geomety fom a Statonay Scene Amnon Shahua Hebew Unvety of Jeualem Iael U. of Mlano, Lectue : multvew

2 Mateal We Wll Cove oday he tuctue of 3D->D oecton matx A me on oectve geomety of the lane Eola Geomety and Fundamental Matx Why 3 vew? Pme on Covaant-Contavaant Index conventon focal eno Quadfocal eno U. of Mlano, Lectue : multvew

3 he tuctue of 3D->D oecton matx U. of Mlano, Lectue : multvew 3

4 U. of Mlano, Lectue : multvew 4 he Stuctue of a Poecton Matx W V U P w O t n b U V W X Y Z O OP OO + Ut + Vn + Wb R t n b [ ] X Y Z R U V W +

5 he Stuctue of a Poecton Matx x f X Z + x (,, f ( x, y ) ) y Z x P X X Y Z y f Y Z + y Y U. of Mlano, Lectue : multvew 5

6 U. of Mlano, Lectue : multvew 6 he Stuctue of a Poecton Matx x f X Z + x y f Y Z + y x y f x f y X Y Z K(RP w + ) K[R;] U V W P M 4 3

7 Geneally, he Stuctue of a Poecton Matx coθ f x f x nθ f K y nθ x y θ y x ( x, y ) called the ncle ont f x coθ nθ called the ew f y f x aect ato U. of Mlano, Lectue : multvew 7

8 Why c the camea cente? Conde the otcal ay MQ (λ ) λmp he Camea Cente M 3 4 P M ha an3, thu c uch that Mc MP All ont along the lne Q (λ ) ae maed to the ame ont Q ( λ ) λp + ( λ ) c P Q (λ ) a ay though the camea cente c U. of Mlano, Lectue : multvew 8

9 he Eola Pont c e M c e Mc c U. of Mlano, Lectue : multvew 9

10 Choce of Canoncal Fame MP MWW P M P M WW P W P the new wold coodnate fame We have 5 degee of feedom (6 uto cale) Chooe W uch that MW [ I ;] U. of Mlano, Lectue : multvew

11 Choce of Canoncal Fame Let M [ M ; m ] MW [ M ; m ] M ( M m ) n ( / λ ) M m n / λ [ I mn + mn ; ( / λ )m + ( / λ ) m] I; [ ] We ae left wth 4 degee of feedom (uto cale): (n, λ ) U. of Mlano, Lectue : multvew

12 [I;]P [ H ;e ]P [I;] Choce of Canoncal Fame I I / λ λn P λ n I λn P λ x y P x y P µ µ U. of Mlano, Lectue : multvew

13 Choce of Canoncal Fame [ H ;e ] [H ;e ] µ [H ;e ] n I / λ µ n I H + e n ; ( / λ )e / λ [ ] [ λ H + e n ; e ] whee n, ) ( λ [ ] µ λh + e n ; e ae fee vaable U. of Mlano, Lectue : multvew 3

14 Famly of Homogahy Matce [ λh + e n ; e ]P H π λh + e n H π Stand fo the famly of D oectve tanfomaton between two fxed mage nduced by a lane n ace U. of Mlano, Lectue : multvew 4

15 Famly of Homogahy Matce H π H + d e n when d H π H K RK n d ft camea fame K RK the homogahy matx nduced by the lane at nfnty U. of Mlano, Lectue : multvew 5

16 Recontucton Poblem H ; e [ ] µ H + µ e We wh to olve fo the moton and tuctue fom matche Wthout addtonal nfomaton we cannot olve unquely fo H becaue H detemned u to a 4-aamete famly (oton of a efeence lane n ace). U. of Mlano, Lectue : multvew 6

17 A me on oectve geomety of the lane U. of Mlano, Lectue : multvew 7

18 ax + by + c Poectve Geomety of the Plane Equaton of a lne n the D lane he lne eeented by the vecto and l l ( a, b, c ) ( x, y,) Coeondence between lne and vecto ae not - becaue λa, λb, c ) eeent the ame lne ( λl ), λ ( λ he vecto (,,) doe not eeent any lne. wo vecto dffeng by a cale facto ae equvalent. h equvalence cla called homogenou vecto. Any vecto ( a, b, c ) a eeentaton of the equvalence cla. U. of Mlano, Lectue : multvew 8

19 Poectve Geomety of the Plane A ont ( x, y le on the lne (concdent wth) whch eeented by ff l l ( a, b, c ) But alo ( λ ) l ) ( λx, λy, λ ), λ ( 3 x, x, x ) eeent the ont he vecto eeent the ont ( x x 3, x x 3 ) ( x, y (,,) doe not eeent any ont. ) Pont and lne ae dual to each othe (only n the D cae!). U. of Mlano, Lectue : multvew 9

20 Poectve Geomety of the Plane ( ) ( ) note: ( a b ) c det( a, b, c ) Lewe, l U. of Mlano, Lectue : multvew q q l

21 U. of Mlano, Lectue : multvew Lne and Pont at Infnty Conde lne c b a ),, ( c b a ),, ( ) ( a b a b c c ac ac cb bc whch eeent the ont ), ( a b wth nfntely lage coodnate λ b a All meet at the ame ont a b ont at nfnty

22 Lne and Pont at Infnty he ont ( x x, x,), x, ( x, x,) le on a lne he lne he ont A lne l (,,) ( x x, x,), x, ( a, b, λ ) meet (whch the decton of the lne) called the lne at nfnty l (,,) ae called deal ont. a at b λ b a U. of Mlano, Lectue : multvew

23 A Model of the Poectve Plane ( x, x,) deal ont x l (,,) the lane x x x 3 x ( 3 λx, λx, λx ), λ Pont ae eeented a lne (ay) though the ogn Lne ae eeented a lane though the ogn U. of Mlano, Lectue : multvew 3

24 A Model of the Poectve Plane P n {[ x,..., x n ] [,..., ] : [ x,..., x n ] [λx,..., λx n ], λ } {lne though the ogn n n + R } {-dm ubace of n + R } U. of Mlano, Lectue : multvew 4

25 Poectve anfomaton n P he tudy of oete of the oectve lane that ae nvaant unde a gou of tanfomaton. Poectvty: h : P P that ma lne to lne (.e. eeve colneaty) Any nvetble 3x3 matx a Poectvty: Let,, 3 Colnea ont,.e. l l H H the ont H le on the lne H l heefoe H eeve colneaty. H called homogahy, colneaton H A homogahy detemned by 8 aamete. the dual. U. of Mlano, Lectue : multvew 5

26 Poectve anfomaton n P eectvty (6 d.o.f) A comoton of eectvte fom a lane π to othe lane and bac to π a oectvty. Evey oectvty can be eeented n th way. U. of Mlano, Lectue : multvew 6

27 Poectve anfomaton n P Examle, a eectvty n D: Lne adonng matchng ont ae concuent c a b a b c Lne adonng matchng ont (a,a ),(b,b ),(c,c ) ae not concuent U. of Mlano, Lectue : multvew 7

28 Poectve anfomaton n P l (,,) not nvaant unde H: Pont on l ae ( x, x,) H x x x xh + x h x x 3 x 3 not necealy Paallel lne do not eman aallel! l maed to H l U. of Mlano, Lectue : multvew 8

29 Poectve Ba A Smlex n R n + a et of n+ ont uch that no ubet Of n+ of them le on a hyelane (lnealy deendent). In P a Smlex 4 ont heoem: thee a unque colneaton between any two Smlexe U. of Mlano, Lectue : multvew 9

30 Poectve Invaant Invaant ae meauement that eman fxed unde colneaton # of ndeendent nvaant # d.o.f of confguaton - # d.o.f of tan. Ex: D cae H x H ha 3 d.o.f A ont n D eeented by aamete. 4 ont we have: 4-3 nvaant (co ato) D cae: H ha 8 d.o.f, a ont ha d.o.f thu 5 ont nduce nvaant U. of Mlano, Lectue : multvew 3

31 Poectve Invaant he co-ato of 4 ont: α ab ac cd bd a b b d 4 emutaton of the 4 ont fomng 6 gou: α,, α α α α,,, α α α U. of Mlano, Lectue : multvew 3

32 5 ont gve u d.o.f, thu -8 nvaant whch eeent D x y, z, u Poectve Invaant, ae the 4 ba ont (mlex) y α < z, u, x x x, β < z, u, y y y, > > y u y u z u x y, ae detemned unquely by α, β x y x Pont of nteecton eeved unde oectvty (exece) x, y unquely detemned U. of Mlano, Lectue : multvew 3

33 Eola Geomety and Fundamental Matx U. of Mlano, Lectue : multvew 33

34 Remnde: [ I ;] P [ H ; e ] P x y P µ [ λh + e n ; e ]P P µ H µ e π + H π λh + e n H π Stand fo the famly of D oectve tanfomaton between two fxed mage nduced by a lane n ace U. of Mlano, Lectue : multvew 34

35 H µ e π + Plane + Paallax P ( x, y,, µ ) [ I,] P π [ H π e ]P π H π e what doe µ tand fo? what would we obtan afte elmnatng µ U. of Mlano, Lectue : multvew 35

36 Plane + Paallax H π + µ e We have ued 4 ace ont fo a ba: 3 fo the efeence lane fo the efeence ont (calng) Snce 4 ont detemne an affne ba: d P d P Z µ Z Z d d µ called elatve affne tuctue Z Note: we need 5 ont fo a oectve ba. he 5 th ont the ft camea cente. U. of Mlano, Lectue : multvew 36

37 Note: A oectve nvaant H µ e π + H ˆ ˆ µ e µ ˆ µ π + dˆ dˆ d d d dˆ P d Z dˆ P Z µ Z Z d d h nvaant ( oectve deth ) ndeendent of both camea oton, theefoe oectve. 5 ba ont: 4 non-colana defne two lane, and A 5 th ont fo calng. U. of Mlano, Lectue : multvew 37

38 H µ e π + Fundamental Matx P ( x, y,, µ ) π H π e an [ H e ] π ( e H ) π ([ e] Hπ ) F U. of Mlano, Lectue : multvew 38

39 Fundamental Matx ([ e] H ) π F Defne a blnea matchng contant whoe coeffcent deend only on the camea geomety (hae wa elmnated) F doe not deend on the choce of the efeence lane [ e] λ [ e] H Hπ [ e] ( H + e n ) U. of Mlano, Lectue : multvew 39

40 Eole fom F Note: any homogahy matx ma between eole: c e H π e e e c U. of Mlano, Lectue : multvew 4

41 Eole fom F Fe [ e] H e [ e] e F e [ e] e H F the eola lne of - the oecton of the lne of ght onto the econd mage. U. of Mlano, Lectue : multvew 4

42 Etmatng F fom matchng ont F,..., 8 Lnea oluton F,..., 7 det( F) N on-lnea oluton det( F) cubc n the element of F, thu we hould exect 3 oluton. U. of Mlano, Lectue : multvew 4

43 U. of Mlano, Lectue : multvew 43 Etmatng F fom Homogahe F H π ew-ymmetc (.e. ovde 6 contant on F) + H e H H e n e H F H ] [ ] [ ) ( λ λ π + H e H n e H e H H F ] [ ) ( ] [ λ λ π π H π F F H homogahy matce ae equed fo a oluton fo F

44 F Induce a Homogahy π F δ ] F [δ a homogahy matx nduced by the lane defned by the on of the mage lne δ and the camea cente U. of Mlano, Lectue : multvew 44

45 Poectve Recontucton. Solve fo F va the ytem F (8 ont o 7 ont). Solve fo e va the ytem F e 3. Select an abtay vecto δ δ e 4. [ I ] and [ δ ] e ] F ae a a of camea matce. [ δ ] F + µ e U. of Mlano, Lectue : multvew 45

46 Why 3 vew? U. of Mlano, Lectue : multvew 46

47 focal Geomety he thee fundamental matce comletely decbe the tfocal geomety (a long a the thee camea cente ae not collnea) F e3 e e3 e 3 F e3 e e Lewe: e F e 3 3 e 3 e 3 e 3 F3 e e 3 e 3 Each contant non-lnea n the ente of the fundamental matce (becaue the eole ae the eectve null ace) U. of Mlano, Lectue : multvew 47 3

48 focal Geomety e 3 F e3 e 3 F3e e 3 F3 e 3 fundamental matce ovde aamete. Subtact 3 contant, hu we have that the tfocal geomety detemned by 8 aamete. h content wth the taght-fowad countng: 3x 5 8 (3 camea matce ovde 33 aamete, mnu the oectve ba) U. of Mlano, Lectue : multvew 48

49 What Goe Wong wth 3 vew? e3 e e e 3 e3 e 3 contant each, thu we have -65 aamete e 3 e 3 3 e e 3 e e 3 U. of Mlano, Lectue : multvew 49

50 What Goe Wong wth 3 vew? e 3 e 3 3 e e 3 e e 3 t α 3 t + t hu, to eeent t3 we need only aamete (ntead of 3). t t t3 8-6 aamete ae needed to eeent the tfocal geomety n th cae. but the awe fundamental matce can account fo only 5! U. of Mlano, Lectue : multvew 5

51 What Ele Goe Wong: Reoecton F F3 3 Gven, and the awe F-mat one can dectly detemne the oton of the matchng ont h fal when the 3 camea cente ae collnea becaue all thee lne of ght ae colana thu thee only one eola lne! F 3 3 F 3 U. of Mlano, Lectue : multvew 5

52 focal Contant U. of Mlano, Lectue : multvew 5

53 he focal Contant I [ ]P A e [ ]P B e [ ]P x y x y U. of Mlano, Lectue : multvew 53

54 he focal Contant [ A e ]P A e [ ]P A e [ ]P [ B e ]P I [ ]P B e [ ]P B e [ ]P ( x ) P ( y ) P U. of Mlano, Lectue : multvew 54

55 U. of Mlano, Lectue : multvew P e B e B e A e A y x Evey 4x4 mno mut vanh! of thoe nvolve all 3 vew, they ae aanged n 3 gou Deendng on whch vew the efeence vew. he focal Contant

56 U. of Mlano, Lectue : multvew 56 e B e B e A e A y x he efeence vew Chooe ow fom hee Chooe ow fom hee We hould exect to have 4 matchng contant ),, ( f he focal Contant

57 Exandng the detemnant: he focal Contant A + µ e A + e µ, B + µ e B + e µ, elmnate µ A B e e ( e )( A ) ( e )( B,, ) U. of Mlano, Lectue : multvew 57

58 he focal Contant [ A e ] P What gong on geometcally: ( A, e) a lane C y C P x 4 lane nteect at P! U. of Mlano, Lectue : multvew 58 C

59 Pme on Covaant-Contavaant conventon U. of Mlano, Lectue : multvew 59

60 Index Notaton Goal: eeent the oeaton of nne-oduct and oute-oduct A vecto ha ue-ct unnng ndex when t eeent a ont A vecto ha ubct unnng ndex when t eeent a hyelane Examle: (,, 3 ) Reeent a ont n the oectve lane (,, ) 3 Reeent a lne n the oectve lane U. of Mlano, Lectue : multvew 6

61 An oute-oduct: uv an obect (-valence teno) whoe ente ae u v,..., u,..., uvm,..., unv n v m Note: th the oute-oduct of two vecto: uv uv.. unv uv. u n. v m m n m (an- matx) a c c + d d x x A geneal -valence teno a um of an- -valence teno U. of Mlano, Lectue : multvew 6

62 Lewe, u v u v u v Ae oute-oduct contng of the ame element, but a a mang cay each a dffeent meanng (decbed late). hee ae alo called mxed teno, whee the ue-ct called conta-vaant ndex and the ubct called covaant ndex. U. of Mlano, Lectue : multvew 6

63 he nne-oduct (contacton): Summaton ule: ame ndex n contavaant and covaant oton ae ummed ove. h ometme called the Enten ummaton conventon. u v u v + u v u n v n U. of Mlano, Lectue : multvew 63

64 he nne-oduct (contacton): a u a u + a u a u n n v Note: th the famla matx-vecto multlcaton: Au whee the ue-ct un ove the ow of the matx v Note: the -valence teno a ma ont to ont U. of Mlano, Lectue : multvew 64

65 Lewe, a u v Ma hyelane (lne n D) to hyelane Note: th equvalent to A u We have een n the at that f hen H l l v H ma lne fom vew to vew a homogahy,, Let 3 Colnea ont,.e. l l H H the ont H le on the lne H l Wth the ndex notaton we get th oety mmedately! U. of Mlano, Lectue : multvew 65

66 he comlete lt: a u v Ma ont to ont a u v Ma hyelane (lne n D) to hyelane a u v Ma ont to hyelane a u v Ma hyelane to ont U. of Mlano, Lectue : multvew 66

67 Moe Examle: un ove the ow a b c h the matx oduct AB C u v un ove the column H u H H Mut be a ont ae a ont n ft fame, a hyelane n the econd fame and oduce a ont n the thd fame Mut be a matx (-valence teno) f u(,, ) then th a lce of the teno. H U. of Mlano, Lectue : multvew 67

68 H 5 H U. of Mlano, Lectue : multvew 68

69 U. of Mlano, Lectue : multvew det det det q q q q q q q q ε q q ε q he Co-oduct eno

70 U. of Mlano, Lectue : multvew 7 v u v u ] [ ] [ 3 3 u u u u u u u ε u he co-oduct teno defned uch that Poduce the matx ] [u.e., the ente ae,-,

71 U. of Mlano, Lectue : multvew 7 ] [ 3 3 u u u u u u u ε ε 3 ε + + ] [ 3 3 u u u u u ε ε ε ε

72 he focal eno U. of Mlano, Lectue : multvew 7

73 he focal eno ( e )( A ) ( e )( B ) New ndex notaton: -mage, -mage, -mage 3 A + µ e a + µ e a ont n mage a lne n mage e a ont n mage U. of Mlano, Lectue : multvew 73

74 l he focal eno l, ae the two lne concdent wth,.e. l m m m, ae the two lne concdent wth,.e. l a + µ l e m b + µ m e Elmnate µ l m m l ( e )( b ) ( e )( a ) U. of Mlano, Lectue : multvew 74

75 U. of Mlano, Lectue : multvew 75 he focal eno ) )( ( ) )( ( l m m l a e b e Reaange tem: ) ( m l a e b e he tfocal teno : a e b e,, m l

76 he focal eno l m l x y m x y he fou tlneate : x 3 - x x 33 + x 3 - y 3 - y x 33 + x 3 - x 3 - x y 33 + y 3 - y 3 - y y 33 + x 3 - U. of Mlano, Lectue : multvew 76

77 U. of Mlano, Lectue : multvew 77 he focal eno β α + δ γ + ) )( ( + + δ γ β α A tlneaty a contacton wth a ont-lne-lne whee the lne ae concdent wth the eectve matchng ont.

78 Slce of the focal eno Now that we have an exlct fom of the teno, what can we do wth t?? he eult mut be a contavaant vecto (a ont). h ont concdent wth fo all lne concdent wth e he ont eoecton equaton (wll wo when camea cente ae collnea a well). Note: eoecton oble afte obevng 7 matchng ont, (becaue one need 7 matchng tlet to olve fo the teno). h n contat to eoecton ung awe fundamental matce Whch eque 8 matchng ont (n ode to olve fo the F-mat). U. of Mlano, Lectue : multvew 78

79 Slce of the focal eno 3 U. of Mlano, Lectue : multvew 79

80 Slce of the focal eno? he eult mut be a lne. q Lne eoecton equaton O 3 matchng lne ae neceay fo O q olvng fo the teno (comaed to 7 matchng ont) O U. of Mlano, Lectue : multvew 8

81 Slce of the focal eno δ? he eult mut be a matx. H δ δ the eoecton equaton δ H a homogahy matx 3 H δ δ a famly of homogahy matce (fom to ) nduced by the famly of lane concdant wth the 3 d camea cente. U. of Mlano, Lectue : multvew 8

82 Slce of the focal eno δ the homogahy matx fom to 3 nduced by the lane defned by the mage lne δ and the econd camea cente. δ? δ he eult a ont on the eola lne of δ on mage 3 the eoecton equaton 3 F 3 δ U. of Mlano, Lectue : multvew 8 δ

83 Slce of the focal eno δ G G I a ont on the eola lne δ F 3 an( G) (becaue t ma the dual lane onto collnea ont) F 3 δ null(g) F δ 3 null(g ) F 3 δ δ U. of Mlano, Lectue : multvew 83

84 U. of Mlano, Lectue : multvew 84 8 Paamete fo the focal eno a e b e ) ( ) ( e n a e e n b e + + e e n e e n + Ha 4 aamete ( ) mnu fo global cale mnu fo calng e,e to be unt vecto mnu 3 fo ettng n uch that B ha a vanhng column n 8 ndeendent aamete We hould exect to fnd 9 non-lnea contant among the 7 ente of the teno (admblty contant).

85 8 Paamete fo the focal eno What haen when the 3 camea cente ae collnea? (we aw that awe F-mat account fo 5 aamete). A e B e 3 e e B e A e h ovde two addtonal (non-lnea) contant, thu 8-6. U. of Mlano, Lectue : multvew 85

86 Item not Coveed Degeneate confguaton (Lnea Lne Comlex, Quatc Cuve) he ouce of the 9 admblty contant (come fom the homogahy lce). Concatenaton of tfocal teno along a equence U. of Mlano, Lectue : multvew 86

87 Quadfocal eno U. of Mlano, Lectue : multvew 87

88 Lnea Lne Comlex P L S l l H π Whee π any lane concdent wth L ( l ) [ l ] x H [ l] G π x Fo all lne ang though and all lne ang though U. of Mlano, Lectue : multvew 88 π

89 G π unque l π L π v l H π λhπ + v l Gπ λ G ( Hπ + v l )[ l] x Hπ[ l] x π U. of Mlano, Lectue : multvew 89

90 L P t [ I ] P [ A v ] P [ B v ] P [ C v ] P U. of Mlano, Lectue : multvew 9

91 L P t [ C v] t q [ B v] We wh to contuct an LLC mang Q(,t) whoe enel L Q(, t) q q t l Q Fo all lne ang though and all lne q ang though U. of Mlano, Lectue : multvew 9 l

92 U. of Mlano, Lectue : multvew 9 P L t [ v] B [ v] C t ] [I λ λ + t v t C v B β α λ t C v B t v ) ( ) ( λ

93 [ I ] P [ A v ] P [ B v ] P [ C v ] P λ n L P σ π t A B A σ A + v n Q (, t) A [ λ] A[ λ] + v( n λ) σ ( n λ) B U. of Mlano, Lectue : multvew 93 C t

94 U. of Mlano, Lectue : multvew 94 + ] [ ) ( ] [ ) ( ) ( ), ( B A t v t A C v t C B v t Q n v A A t Q ) ( ] [ ] [ ), ( λ λ λ σ + t C B n ) ( λ t C v B t v ) ( ) ( λ ] [ ) ( ] [ ) ( ) )( ( + q B A t v q t A C v q t C B v ), ( q t Q

95 U. of Mlano, Lectue : multvew 95 P L q π B A B LLC between vew, wth enel L : [ ] B A ] [ q B A Fo all q,, though,, ] [ ) ( ] [ ) ( ) )( ( + q B A t v q t A C v q t C B v

96 U. of Mlano, Lectue : multvew 96 ] [ ) ( ] [ ) ( ) )( ( + q B A t v q t A C v q t C B v P q π B A B ] [ q B A Fo all q,, though,, A ) ( A B q [ ] B A q an

97 U. of Mlano, Lectue : multvew 97 P q π A B Dual Homogahy eno ) ( A B q ) ( n u un b a q ε H q

98 U. of Mlano, Lectue : multvew 98 ] [ ) ( ] [ ) ( ) )( ( + q B A t v q t A C v q t C B v ) ( ) ( ) ( + l l l l l l q H v t t q H v t q H v l l Q t q l l l l H v H v H v Q + Quadfocal eno a v b v he tlnea (tfocal) teno

99 Gauge Invaance [ I ], [ A + vw v ], [ B + vw v ], [ C + vw v ] q t l Q l l l Q Q Fo all choce of w U. of Mlano, Lectue : multvew 99

100 How Many matchng ont ae needed? q η µ ν t σ Q l η, µ, ν, σ, l t ont: 6 quadlnea equaton fo Q nd ont: 5 equaton 3 d ont: 4 equaton U. of Mlano, Lectue : multvew

101 Sngle Contacton δ l lq H P π δ U. of Mlano, Lectue : multvew

102 Double Contacton δ l µ l Q E LLC between vew, δ µ le Contacton t l Q l U. of Mlano, Lectue : multvew

103 Item not Coveed Quad contucted fom focal and Fundamental Matx Fundamental Matx fom Quadfocal focal fom Quadfocal Poecton Matce fom Quadfocal 5 Non-lnea Contant U. of Mlano, Lectue : multvew 3