Single view geometry

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??????2???_at_?????

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

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

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

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

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?????

• Step 1) ???????p??????P????
• ???????? ? ??
• Step 2) ????????????p?????
• ???????P???????p????
• ? ???????????????
• ? ??
• ?????????

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???????(Step 1)

• ????????p??????P??????????????????X???Y?????????Z?
  ???????
• ?????P?Z?????Z???????????
• ??????????????
• XX
• YY
• Z' Z

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???????(Step 2)

• ????????????????????
• ? ??????????????
• x X/Z X/Z
• y Y/Z Y/Z
• Z???
• ???????X?Y?????????
• ? ?????????

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????????????

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

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??????(paraperspective projection)

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

L
L
?????
??????
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??????

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

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????????

• 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

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??????

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

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????????????

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

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E.g., Shape Reconstruction
Right
Left
??????????????????????3D??????
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Affine cameras
????f ???
????????????
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??????

• ?????????????????????????????(????)???????????????
  ????????????????????
• ???????????????????????
• ??????????????????????????????????????????(???????
  )???
• ???????????????3D????2D??????

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??????

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

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???????????
??????????? ????????
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2??????
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2??????
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2????(????)
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2????????
????
????
???!
??
???
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???????????????????
(homogeneous coordinates)
1???? ????
???!

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

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???????2D??

• Basic 2D transformations as 3x3 matrices

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

• ?????????????????????????

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

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

( x, y, z )
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3????????
P(x, y, z) ??P(x, y, z) ????????(?????????)
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?????????

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

• ???????????

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

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

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

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

• 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

???????????
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??????

• 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???)
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Correlation Approach
LEFT IMAGE

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

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Correlation Approach
RIGHT IMAGE
(xl, yl)

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

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

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

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??????????

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

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

• ????????????????????
• ??????

• ?????????????????

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Examples
Left Image
Right Image
Depth Map
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• 1.?????????????????????????????????