1. The document discusses computer vision techniques for augmented reality. It covers topics like camera calibration, 3D reconstruction, and tracking algorithms. 2. Key aspects covered include using camera intrinsics and extrinsics to relate the camera coordinate system to the world coordinate system. It also discusses estimating rotation and translation parameters from 2D point correspondences. 3. Techniques for tracking objects over multiple frames are summarized, including minimizing errors between current and previous frame projections and refining estimates using bundle adjustment.

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