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1

2 i Simultanious Registration

3 ii

4 iii Cyrax Cyrax CCD Cyrax ( ) ( ) ( ) ( ) ( ) ( )

5 iv Cyrax

6 1 1

7 CAD

8 (alignment) (merging). 1.1 ( ) ( ) (x; y; z x; y; z )

9 : ( ) ( )

10 :

11

12

13 Cyrax Cyrax

14 9 2

15 [1] ( ) 2 Shum[2]

16 [3] Besl[4] ICP(Iterative Closest Point) [5-10] ICP ( ) 2 2 ICP 1 Zhang[5] ICP

17 k-d tree Chen[6] Dorai [11] Chen Fledmar[12] Bergervin[7] Chen 2 1 Masda[9] Least Median Squares 2 [10] [13-17]

18 13 3 Simultanious Registration Simultanious Registration

19 3 Simultanious Registration : Cyrax 6mm ( )( ) ( )( ) ( ) ( ) ( ) ( ) 100m(50cm - 50m ) 2mm(@50m) 2mm(@50m) 6mm (0-50m) Cyrax ( 3.2 ) Cyrax 3.. Cyrax ( ) ( ).. Cyrax

20 3 Simultanious Registration : Cyrax

21 3 Simultanious Registration : Cyrax.... CyraX CyraX CCD ( ).. ( ). depth. depth depth depth. 1

22 3 Simultanious Registration 17. CyraX ( ) n S 0...S n ~x x ~ 0 ( ) T ( ) i. T ( ) i ~x = ~ x 0 (3.1) ( ) ~ i S i. ~ i R () i t () i ~ i. ~x 2 S i T () i. T () i ~x = R () i ~x + t () i (3.2) T ( ) i T () i T () i = T () i T ( ) i (3.3)

23 3 Simultanious Registration 18. S i S j D(S i ;S j ) ". " 2 =min ~ =min =min =min X D(T () i S i ;T () j S j ) i6=j X D(T () i T ( ) i ~ i6=j X ~ i6=j;k X ~ i6=j;k S i ;T () j T ( ) j S j ) (R () i ~n ( ) ik ((R() j ~y ( ) + ~ ijk t () j ) 0 (R () i ~x ( ) ik + ~ t () i ))) 2 ka ijk ~ 0 sijk k 2 (3.4). A ijk = 0 :::0 {z } 6i21 C {z} ijk 621 s ijk = ~n ( ) ik 1 (~x( ) ik 0 ~y( ) ijk ) (3.5) 0 :::0 {z } 6(l0i01)21 {z } 6j :::0 C ijk = ~n ( ) ik 2 ~y( ) ijk 0~n ( ) ik 0C {z ijk } 0 :::0 {z } 6(l0j01)21 (3.6) 621! T (3.7) 5.

24 3 Simultanious Registration : CCD ( ) Cyrax ( )

25 20 4

26 Model Model Scene Scene Model 6 (x; y; z x; y; z ) 2..( 4.1 )

27 :

28 Model Scene " " 1. 3 ( )

29 4 24.( 4.1 ) 4.1: 4.1 n S 1 ;S 2 ;:::;S n 1 S 1 T w i S i i 0 1 S i01 T i;i01 S 0 1 ;S0 2 :::;S0 n S 1 S n T w ;T 2 ;:::;T n.( 4.2 ) 8 S 0 1 = T w S 1 >< >: S 0 2 = T 2 S 2 = T 2;1 T w S 1 S 0 3 = T 3 S 3 = T 3;2 T 2;1 T w S 3. S 0 n01 = T n01 S n01 = T n01;n02 T n02;n03 :::T 2;1 T w S n01 S 0 n = T n S n = T n;n01 T n01;n02 :::T 2;1 T w S n (4.1).

30 4 25 i S i i 0 1 S i01 1T i;i01 T i;i01 1 1T w T w i 1T i 1T i =1T i;i01 1T i01;i02 :::1T 2;1 1T w (4.2).. 4.2: S S 0 i ;S j ( S 0 i S j ) 3. 2 S 0 i ;S j

31 S j % 40% Model 3 M 1 ;M 2 ;M 3 3 Scene 3 S 1 ;S 2 ;S 3. 4S 1 S 2 S 3 4M 1 M 2 M 3 T Model. M 1 M 2 u 12 M 1 M 3 u 13 2 u 12 u 13 m. u 12 u 13 n 12,n 13

32 :. n 12 = u 12 k u 12 k n 13 = u 13 k u 13 k (4.3) (4.4) 2 u 12 u 13 2 m 1 ; m 2. 1 m 1 = n 12 m 2 m 2 = n 13 0 n 12 2 cos m k n 13 0 n 12 2 cos m k (4.5). 2 m 1 ; m 2 m 3 m 3 = m 1 2 m 2 (4.6) 3 m 1 =(m 1x ;m 1y ;m 1z ) T ; m 2 =(m 2x ;m 2y ;m 2z ) T ; m 3 = (m 3x ;m 3y ;m 3z ) T. Scene s 1 =(s 1x ;s 1y ;s 1z ) T ; s 2 =(s 2x ;s 2y ;s 2z ) T ; s 3 =(s 3x ;s 3y ;s 3z ) T.( 4.4 )

33 : s 1 ; s 2 ; s 3 m 1 ; m 2 ; m 3 R 0. (s 1 s 2 s 2 )=R 0 (m 1 m 2 m 2 ) (4.7) R 0 =(s 1 s 2 s 2 )(m 1 m 2 m 2 ) 01 (4.8) M = (m 1 m 2 m 2 ) M 01 = M T (4.9) R 0 =(s 1 s 2 s 2 )(m 1 m 2 m 2 ) T (4.10) T t R. Scene P S T Model P M P M = RP S + t (4.11)

34 4 29. R = R 0 (4.12) R Model Scene 1 t. P M = RP S + t (4.13) t = P M 0 RP S (4.14) 1 t 3. G M = M 1 + M 2 + M 3 3 G S = S 1 + S 2 + S 3 3 4M 1 M 2 M 3 4S 1 S 2 S 3 (4.15) (4.16) t = G M 0 RG S (4.17) t R t T

35 ( ) 30 ( ) Model Scene : (12 )

36 :

37 : 12.

38 Simultanious Registration

39 n S 0...S n (12 )

40 ( )

41 5 36 ~x ~ x 0 ( ) T ( ) i. T ( ) i ~x = ~ x 0 (5.1) T i R i ~ t i R ( ) i ~x + ~ t i ( ) = ~ x 0 (5.2). k-d tree 2 z ( ) ~ i S i. ~ i R () i t () ~ i. ~x 2 S i T () i. i 1 2

42 k-d tree Model Scene Model Model Scene ( 5.1) 2 ( ) 5.4 T () i ~x = R () i ~x + t () i (5.3)

43 : T ( ) i T () i T () i = T () i T ( ) i (5.4). S i S j D(S i ;S j ) ". " 2 =min ~ =min =min X D(T () i S i ;T () j S j ) i6=j X D(T () i T ( ) i ~ i6=j X ~ i6=j;k S i ;T () (R () i ~n ( ) ik ((R() j ~y ( ) j T ( ) j S j ) + ~ ijk t () j ) 0 (R () i ~x ( ) ik + ~ t () i ))) 2

44 : X =min ka ~ ijk 0 sijk k 2 (5.5) ~ i6=j;k. A ijk = 0 :::0 {z } 6i21 C {z} ijk 621 s ijk = ~n ( ) ik 1 (~x( ) ik 0 ~y( ) ijk ) (5.6) 0 :::0 {z } 6(l0i01)21 {z } 6j :::0 C ijk = ~n ( ) ik 2 ~y( ) ijk 0~n ( ) ik 0C {z ijk } 0 :::0 {z } 6(l0j01)21 (5.7) 621! T (5.8) (4).Newton-Taylor Levenberg-Marqurdt ~ (8)

45 5 40. ~ =( X i6=j;k A T ijk A ijk) 01 X i6=j;k A T ijk s ijk (5.9) 5.6 ". (9). R = 0 1 0c 3 c 2 c 3 1 0c 1 0c 2 c C A (5.10) " = R m n 1 ((R s y + t s ) 0 (R m x + t m ))) = R m n 1 ((R s y + R m y 0 R m y + t s ) 0 (R m x + t m ))) = R m n 1 ((R s 0 R m )y +(t s 0 t m ) 0 R m (x 0 y)) = R m n 1 ((R s 0 R m )y +(t s 0 t m )) 0 R m n 1 (R m n(x 0 y)) = R m n 1 ((R s 0 R m )y +(t s 0 t m )) 0 n 1 (x 0 y)) (5.11) 2 s. 1 1 = R m n 1 ((R s 0 R m )y +(t s 0 t m )) = n 1 ((R s 0 R m )y +(t s 0 t m )) = n 1 (( = 0 n x n y 0 1 C A 1 0 0c 3s c 2s c 3s 0 0c 1s 0 0c 2s c 1s C A 0 0 0c 3m c 2m c 3m 0 0c 1m 0c 2m c 1m 0 1 C (0c 3s y y + c 2s y z ) 0 (0c 3m y y + c 2m y z )+t sx 0 t mx (0c 1s y z + c 3s y x ) 0 (0c 1m y z + c 3m y x )+t sy 0 t my A )y +(t s + t m )) 1 C A n z (0c 2s y x + c 1s y y ) 0 (0c 2m y x + c 1m y y )+t sz 0 t mz

46 5 41 = = n x n y 0 n z 1 C A 1 n x n y n z C A 1 n y y z 0 n z y y n z y x 0 n x y z 0 n x y y 0 n y y x n x n y n z 1 0 C A 1 0c 3s y y + c 2s y z 0c 1s y z + c 3s y x 0c 2s y x + c 1s y y t sx t sy t sz 1 C A C A 1 t sx t sy t sz 1 C A + 0 c 1s c 2s 0 c 3s n x n y 1 0 C A C A 1 n 1 z 0 C A + n x n y n z =(n 2 y T 0n 0n 2 y T n ) 1 0 C A 1 0 n x n y n z t mx t my t mz 1 C A 1 1 C A 0 n y y z 0 n z y y n z y x 0 n x y z n x y y 0 n y y x t mx t my t mz c 1m c 1m c 1m t mx t my t mz c 1s c 1s c 1s t sx t sy t sz 1 C A 1 C A 0c 3m y y + c 2m y z 0c 1m y z + c 3m y x 0c 2m y x + c 1m y y 1 0 C A 1 c 1m c 2m c 3m 1 C A 1 C A (5.12) (Model

47 : ( ) Scene ) 5.8.

48 : ( )

49 : ( )

50 : yz

51 : zx

52 :

53 48 6

54 \ " cm

55 Newton-Taylor.Newton- Taylor Newton-Taylor 2...

56 51 [1],,,. 3., Vol.J75-D, No.4, pp , [2] H.Shum, K.Ikeuchi and R.Reddy. Principal Component Analysis with missing Data and Its Application to Polyhedral Object modeling. IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 17, No. 9, pp , [3], P.Boulanger. ( 3D )., Vol.34,No.6,pp , [4] P.J.Besl, N.D.McKay. A Method of Registration of 3-D Shapes. IEEE Trans. Pattern Analysis and Machine Intelligence Vol. 14,No.2 pp , [5] Z.Zhang. Iterative Point Matching for Registration of Free-Form Curves and Surfaces. International Journal of Computer Vision, Vol. 13, No. 2, pp , [6] Yang Chen, Gerard Medioni. Object modelling by registaration of Maltiple range Images. Image and Vision Computing Vol. 10, No.3 pp , [7] R.Bergevin, M.Soucy, H.Gagnon and D.Laurendeau. Toward a General Multi-View Registration Technique. IEEE Trans. Pattern Analysis and Machine Intelligence, Vol.18, No. 5, pp , [8] G.Turk, M.Levoy. Zippered Polygon Meshes from Range Images. InACM SIGGRAPH Computer Graphics, pp , 1994.

57 52 [9] T.Masuda and N.Yokoya. A Robust Method for registration and Segmentation of Multiple Range Images. Computer Vision and Image Understanding, Vol. 61, No. 3, pp , [10],, pp , [11] C.Dorai, G.Wang, A.K.Jain and C.Mercer. Registration and Integration of Multiple Object Views for 3D Model Construction. IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. 20, No. 1, pp , [12] J.Feldmar, N.Ayache and F.Berring. 3D-2D Projective Registration of Free-Form Curves and Surfaces. Computer Vision and Image Understanding, Vol. 65, No. 3, pp ,1997. [13] M.Rutishauser, M.Stricker and M.Trobina. Merging Range Images of Arbitrary. In Proceedings of IEEE Computer Vision and Pattern recognition, pp , [14] A.Hilton, A.J.Stoddart, J.Illingworth and T.Windeatt. Reliable Surface Reconstruction from Multiple Range Images. In Proceedings of IEEE Computer Vision and Pattern recognition, pp , [15] Y.Chen and G.Medioni. Description of Complex Objects from Multiple Range Images Using an inating Ballon Model. Computer Vision and Image Understanding, Vol. 61, No. 3, pp , [16] H.Hoppe, T.DeRose and T.Duchamp. Surface Reconstruction from Unorganized points. In ACM SIGGRAPH Computer Graphics, Vol. 26-2, pp , [17] P.Hebert, D.Laurendeau and D.Poussart. Surface Prole Reconstruction: Reliable Geometric Primitive Extraction. In Proceedings of IAPR International Conference on Pattern recognition, pp. A , 1994.

58 53 [18] Peter J. Neugebauter. Reconstruction of Real-World Objects Via Simultaneous Registration and Robust Combination of Multiple Range Images. International Journal of Shape Modeling, Vol. 3 No.1 & 2, pp , June [19] Mark D. Wheeler, Yoichi Sato, Katsushi Ikeuchi. Consensus Surfaces for Modeling 3D Objects from Multiple Range Images. Proceedingd of DARPA Image Understanding Workshop '97. [20] Berthold K.P.Horn. Closed-form Solution of Absolute Orientation Using Unit Quaternions. Optical Society of America, Vol. 4 No.4/April [21] O.D.Faugeras, M.Hebert. The Representation, Recognition, and Locating og 3-D Objects. The International Journal of Robotics Research, pp , May 1986 [22] R.Bergevin, D.Laurendeau and D.Poussart. Registrating Range Views of Multipart Objects. Computer Vision and Image Understanding 61 pp. 1-16, [23] K.Higuchi, M.Hebert and K.Ikeuchi Building 3-d models from Unregistrated Range Images. Graphical Models and Image Processing 57 pp , [24] B.Curless and M.Levoy. A Volumetric Method for Building Complex Models from Range Images. Proc. of SIGGRAPH '96 New Orleans, Louisiana pp , [25] G.Borgefors. Chamfering: A Fast Method for Obtaining Approximations of the Euclidean Distance in N Dimensions. 3rd Scand. Conf. on Image Analysis, Copenhagen, Denmark [26] W.E.Lorensen and H.E. Cline. Marching Cubes: A High Resolution 3d Surface Construction Algorithm. Proc. of SIGGRAPH '87 Anaheim, California pp , 1987.

59 54 1.,. 3 (Constructing a 3D Modeling Using a High Resolution Range Sensor) 4 pp , ,,, pp , 1999

60 . CAD

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