Single view geometry

1
??????2???_at_?????

• ???
• ?? ?
• ??????????? -??????-?
• ????

2
Single view geometry
Camera model Single view geom.
3
???????
(x,y,z)??(x,y,z)??? (????????)
z
z
x
O
x
?? ?????????? Z???????
(f Z)
??????Z?????????
??????????????????f1???
4
??? (orthographic projection)

• ??????????????????????
• ?Z??????????, X??Y????????????????
• ?????Z??????????
• ????????????
• ????????????????
• ????????????,????
• ????CG??????????
• ????CV??????????
• ??????

5
??????
(x, y, z)
O
?????Z????????
Projection from (x,y,z) to (x,y,z)
or
6
?????(weak perspective projection)

• ??????????????????????????????????????????
• ????????????????????????????
• ?????????????????????????????????????????????

7
?????

• ???????????????????????
• 1) ???????p??????P???????
• 2) ????????????p?????
• ???????????????????
• ?????????
• x X/Z X/Z
• y Y/Z Y/Z
• Z???

8
????????????

• ?????????????????
• ?????????????????
• ????????????????????Z????Z????????????????
• ?????Z??????D??????????????Z??????????????
• ?DZ?110??????????????????????

9
??????(paraperspective projection)

• ??????????????????p??????P?????????
• ?????????????????????????????????P????????
• ???????????????L??????????????L???????P???????

L
L
?????
??????
10
??????

• ???????? ??????????
• ??????????????????????????????????????????????????
  ?????????????

11
????????

• Step 1 ???????????(?X, ?Y, 1)??
• ???????????
• X X ( Z- Z)?X
• Y Y ( Z- Z)?Y
• Z Z
  
• Step 2 ??????
• ????????????????
• x X/Z 1/ZX - ?X/Z Z ?X
• y Y/Z 1/ZY - ?Y/ ZZ ?Y

12
??????

• ????X, Y, Z????1/ Z, ?X/ Z, ?Y/ Z, ?X,
  ?Y????????? gt X, Y, Z??????
• ?????????????????
• ????????????????????????????????????

13
????????????

• ?????????????????p??????P???????????????
• ?????????P??????????
• ????????????????????????????
• ???????????????????

14
E.g., Shape Reconstruction
Right
Left
???
????
??????????????????????3D??????
15
Affine cameras
????f ???
????????????
16
2??????
17
2??????
18
2????(????)
19
2????????
????
????
???!
??
???
20
???????????????????
(homogeneous coordinates)
1???? ????
???!

• ?????????????1??????
• ?????????????????????

21
???????2D??

• Basic 2D transformations as 3x3 matrices

????Translate
????Scale
??Rotate
???Shear
22
?????

• ?????????????????????????

p T(tx,ty) R(Q)
S(sx,sy) p
23
???? (3??)

• 3??????,????????4?????????
• (x/f, y/f, z/f) ??????????
• f1????????(x, y, z)????

( x, y, z )
24
3????????
P(x, y, z) ??P(x, y, z) ????????(?????????)
(A ????????)
25
?????????

• ???????????
• ??????????
• ??????????

• ???????????

H1,H2 2D ? 3 x 3 ????????? H1,H2 3D ? 4 x 4
?????????
26
??????????
fw1
?????????????????????(??????????)
27
?????

28
Two-view geometry
3D reconstruction Epipolar geometry E-matrix
comp. F-matrix comp. H-matrix comp. Structure
comp.
29
???Stereo Vision (Stereopsis)

• Main problem recover 3D depth from two (or
  more) image views

??????????????????????3D???????????????????? ?2???
??????????????????????????????????????????? ?????
???????????????????????2??????OK
30
??????
??????????? ??????????????
??????
?(disparity)
??Z?????f????B???x???
31
??????

• Algorithm
• Rotate both left and right camera so that they
  share the same X axis Or-Ol T
• Define a rotation matrix Rrect for the left
  camera
• Rotation Matrix for the right camera is RrectRT
• Rotation can be implemented by image
  transformation

???????????
32
??????

• Algorithm
• Rotate both left and right camera so that they
  share the same X axis Or-Ol T
• Define a rotation matrix Rrect for the left
  camera
• Rotation Matrix for the right camera is RrectRT
• Rotation can be implemented by image
  transformation

??????????? ??(R,t???)
33
Correlation Approach
LEFT IMAGE

• For each point (xl, yl) in the left image, define
  a window centered at the point

34
Correlation Approach
RIGHT IMAGE
(xl, yl)

• search its corresponding point within a search
  region in the right image

35
Correlation Approach
RIGHT IMAGE
(xl, yl)
dx
(xr, yr)

• the disparity (dx, dy) is the displacement when
  the correlation is maximum

36
??????????

• ????
• ????????????
• ?????????
• (???????)
• ??(disparty)??????
• ??????????

corner
line
structure
37
?????????????

• SSD(Sum of Squared Differences)
• ??????????
• ?????????????
• ??????????????
• ????????????
• ?????????????

???????
???????????
???????????
??
38
?????????????

• SAD(Sum of Absolute Differeces)
• ????????????
• SSD????????????????
• ???????????

39
?????????????

• CC(Cross Correlation) ???????
• ????????(??)
• ???????

40
??????????
2?????? 3???????
41
Examples
Left Image
Right Image
Depth Map
42
??????

• 1.?????????????????????????????????