ComputerVision Technology for AR

1. 第4章
拡張現実感のための
コンピュータビジョン技術
!1

2. !2

3. 3
↪

4. ↪
↪
↪
4

5. 5

6. Xworld
Yworld
Zworld
u
v
Xcamera
Ycamera
Zcamera
u
Xworld
6

7. [
u
v
1]
fu 0 cu
0 fv cv
0 0 1
r11 r12 r13 t1
r21 r22 r23 t2
r31 r32 r33 t3
Xworld
Yworld
Zworld
1
s =
s ˜u = K [r1 r2 r3 T] ˜Xworld
s ˜u = K [R T] ˜Xworld
Xworld u
7

8. Xworld u
8

9. ˜Xcamera =
[
R T
0T
1]
˜Xworld
Xcamera = RXworld+T
9

10. Xworld =
[
0
0
0] Xcamera = T
T =
[
0
0
0]
Xworld =
[
1
0
0]
Xcamera = r1
Xcamera = RXworld+T
˜Xcamera =
[
R T
0T
1]
˜Xworld
r1
Xworld
10

11. s ˜u = K [E 0] ˜Xcamera
Xcamera
u
s ˜u = K [R T] ˜Xworld
˜Xcamera =
[
R T
0T
1]
˜Xworld
Xworld
11

12. Zworld = 0
s ˜u = K [r1 r2 r3 T]
Xw
Yw
0
1
= K [r1 r2 T]
Xworld
Yworld
1
= H ˜XXY
12

13. u, Xworld
H Xworld, Yworld
13

14. [b1 b2 b3] = ± K−1
H
λ =
∥b1∥ + ∥b2∥
2
[r1 r2 T] =
1
λ
[b1 b2 b3]
r3 = r1 × r2
Xworld Yworld Zworld
14

15. s ˜u = K [R T] ˜Xworld
s ˜u = H ˜XXY
15

16. 16

17. Xworld
uXworld
[R T]
17

18. Yworld
Xworld
Zworld
Yworld
Xworld
Zworld
Yworld
Xworld
Zworld
Yworld
Xworld
Zworld
18

19. s ˜u = K [R T] ˜Xworld
[R T]
19

20. uXworld
20

21. [
u
v
1]
fu 0 cu
0 fv cv
0 0 1
r11 r12 r13 t1
r21 r22 r23 t2
r31 r32 r33 t3
Xworld
Yworld
Zworld
1
s =
s ˜u = K [r1 r2 r3 T] ˜Xworld
s ˜u = K [R T] ˜Xworld
Xworld u
21

22. Xworld u
22

23. 23

24. 24

25. 25

26. 26

27. u, Xworld
H Xworld, Yworld
27

28. 28

29. 29

30. 30

31. 31

32. 32

33. H
[R T]
33

34. [Rt−1 Tt−1]
[Rt Tt]
E =
∑
i
∥u′i − proj(K, R, T, Xwi)∥2
s ˜u = K [R T] ˜Xworldproj
i
Xwi
34

35. ⇢
35

36. 36

37. 37

38. min
H
∥It−1 − It(H)∥ It
It HIt(H)
s ˜u = H ˜XXY
38

39. 39

40. P = K [R T]
s ˜u = P ˜Xworld
[R T]
P
Xworld u
40

41. s ˜u = P ˜Xworld
s′˜u′ = P′ ˜Xworld
P, P′
u, u′
Xworld
(u, Xworld) P, P′
u, u′
Xworld
u, u′, P, P′ Xworld
Ycamera
Xcamera
Zcamera
Xcamera′
Zcamera′
Ycamera′
u
u′
Yworld
Zworld
Xworld
P
P′
41

42. 42

43. P, P′
u, u′
Xworld
(u, Xworld)
u, u′
Xworld
P, P′
43

44. E =
∑
i
∑
j∈Fi
∥u′j − proj(K, Ri, Ti, Xwj)∥2
i
j
Fi i
[R T]
Xwj
E =
∑
i
∥u′i − proj(K, R, T, Xwi)∥2
[R T]
<cf.>
44

45. xEx′ = 0 E R tx, x′
45

46. 46

47. Xcamera = RworldXworld + Tworld
Xcamera = s(RatamXatam + Tatam)
Xcamera
XcameraXatam s T
Xatam = R
Xworld
s
+ T
= R′Xworld + T
R
[
Rworld Tworld
0T
1 ]
=
[
sRatam sTatam
0T
1 ] [
R′ T
0T
1]
Yatam
Zatam
Xatam
Zcamerai
Ycamerai
Xcamerai
Zcameraj
Xcameraj
Ycameraj
47

48. Xworld =
[
0
0
0] Xcamera = T
T =
[
0
0
0]
Xworld =
[
1
0
0]
Xcamera = r1
Xcamera = RXworld+T
˜Xcamera =
[
R T
0T
1]
˜Xworld
r1
Xworld
48

49. [
Rworld Tworld
0T
1 ]
=
[
sRatam sTatam
0T
1 ] [
R′ T
0T
1]
s
i, j
Tworldi = sRatamiT + sTatami
Tworldj = sRatamjT + sTatamj
s =
∥Tworldi − RatamiR−1
atamjTworldj∥
∥Tatami − RatamiR−1
atamjTatamj∥
[R T]s
[Rworldi Tworldi] [Rworldj Tworldj]
[Ratami Tatami] [Ratamj Tatamj]
49

50. 50

51. M = [R T]
min
M
∥I − Ir(M)∥
M
M
51

52. 52

53. 第4章
拡張現実感のための
コンピュータビジョン技術
Fin.
53