Single view geometry

1
??????1???_at_?????

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

2
Single view geometry
Camera model Single view geom.
3
??????(Camera model)

• ???????3???????????
• ????????????
• ? ?????????????????

?
4
??( Projections )

• ??m????n?????? (mgtn)
• CV 3D to 2D

????
???
???
???????
????
5
???????

• ?3????????????????????????????????
• ????????????????????????????
• ???????????????????????????????????

??????????????????,??????????????
6
?????????(pinhole camera)

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

7
?????????
f
???????????????(image plane) ????????????(pinhol
e)?????(optical center) ????????????????????f(fo
cal length)
8
?????????

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

9
?????????
????????(Virtual image plane)???????(????)????????
????????????????? ???????????????????????????????
? ??????????????????? ???????????????????????
??????????? ??????????
10
Pinhole camera image
Amsterdam what do you see in this picture?

• straight line
• size
• parallelism/angle
• shape
• shape of planes
• depth

• Photo by Robert Kosara, robert_at_kosara.net
• http//www.kosara.net/gallery/pinholeamsterdam/pic
  01.html

11
Pinhole camera image
Amsterdam

• straight line
• size
• parallelism/angle
• shape
• shape of planes
• depth

• Photo by Robert Kosara, robert_at_kosara.net
• http//www.kosara.net/gallery/pinholeamsterdam/pic
  01.html

12
Pinhole camera image
Amsterdam

• straight line
• size
• parallelism/angle
• shape
• shape of planes
• depth

• Photo by Robert Kosara, robert_at_kosara.net
• http//www.kosara.net/gallery/pinholeamsterdam/pic
  01.html

13
Pinhole camera image
Amsterdam

• straight line
• size
• parallelism/angle
• shape
• shape of planes
• depth

• Photo by Robert Kosara, robert_at_kosara.net
• http//www.kosara.net/gallery/pinholeamsterdam/pic
  01.html

14
Pinhole camera image
Amsterdam

• straight line
• size
• parallelism/angle
• shape
• shape of planes
• depth

• Photo by Robert Kosara, robert_at_kosara.net
• http//www.kosara.net/gallery/pinholeamsterdam/pic
  01.html

15
Pinhole camera image
Amsterdam

• straight line
• size
• parallelism/angle
• shape
• shape of planes
• parallel to image
• depth

• Photo by Robert Kosara, robert_at_kosara.net
• http//www.kosara.net/gallery/pinholeamsterdam/pic
  01.html

16
Pinhole camera image
Amsterdam what do you see?

• straight line
• size
• parallelism/angle
• shape
• shape of planes
• parallel to image
• Depth ?
• stereo
• motion
• size
• structure

• - We see spatial shapes rather than individual
  pixels
• - Knowledge top-down vision belongs to human
• - Stereo Motion most successful in 3D CV
  application
• - You can see it but you don't know how

17
???? (Perspective Projection)
????????
O
18
???? (Perspective Projection)

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

???
19
???????
(x,y,z)??(x,y,z)??? (????????)
z
z
x
O
x
?? ?????????? Z???????
(f Z)
??????Z?????????
??????????????????f1???
20
?????Homogenous Coordinates
21
?????????Camera Parameters

• ?????Image coordinates (ximage, yimage)
• ????Image center (ox, oy)
• ??????Camera coordinates (xcamera, ycamera)
• ???????Real world coordinates (X, Y, Z)
• ????Focal length f
• ????????Effective size of pixel in millimeter
  
• (kx, ky)

22
?????????
23
?????????
24
???????????Intrinsic Camera Parameters
ox
(0,0)
xim

• fx
• fy
• ox
• oy
• ????????????????????????????????

oy
Pixel (xim,yim)
yim
Size (Sx,Sy)
25
???????????II
(xd, yd)
(x, y)

• ???????Lens Distortions

k1 , k2
26
???????????II
(xd, yd)
(x, y)
k1 , k2

• ???????Lens Distortions

• Modeled as simple radial distortions
• r2 xd2yd2
• (xd , yd) distorted points
• k1 , k2 distortion coefficients
• A model with k2 0 is still accurate for a CCD
  sensor of 500x500 with 5 pixels distortion on
  the outer boundary

27
???????????Extrinsic Camera Parameters

• ???????????????????????????????????
• ????Translation (3x1????)
• ??Rotation (3x3??)

R
t
O
28
????(Translation)

• (tx, ty, tz) Translation vector

29
????(Translation)

• Inverse translation

30
??(Rotation)

• Z-???

31
??(Rotation)

• X-???
• Y-???
• Z-???
• ????

32
??(Rotation)

• Inverse rotation

• ?????????!!

33
?????Euler?

• ???? 3x3
• ????Orthogonal
• 9 elements gt 33 constraints (orthogonal) gt 22
  constraints (unit vectors) gt 3 DOF ??? (degrees
  of freedom)

• ?, ?, ? ?X, Y, Z???????
• ??
• ???????????????
• ????????

34
?????Euler?

• If angle ? is small, then cos? 1 and sin? ?
  ?? ? ? ? ?

35
?????????

• ????????????????
• tx, ty, tz ? r1,1r3,3 ???????????

36
?????????

• ????????????????

37
???????????
38
???????
(x,y,z)??(x,y,z)??? (????????)
z
z
x
O
x
?? ?????????? Z???????
(f Z)
??????Z?????????
??????????????????f1???
39
??? (orthographic projection)

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

40
??????
(x, y, z)
O
Projection from (x,y,z) to (x,y,z)
or
41
??????

• 1.????????(???????)???????????????????????????