computer vision geometric camera models and camera calibration
TRANSCRIPT
Bahadir K. Gunturk 2
Coordinate Systems
Let O be the origin of a 3D coordinate system spanned by the unit vectors i, j, and k orthogonal to each other.
i
j
kO
Px OP i
��������������
y OP j��������������
z OP k��������������
OP x y z i j k��������������
x
y
z
PCoordinate vector
Bahadir K. Gunturk 3
Homogeneous Coordinatesn
a
b
c
HH
O
P
x
y
z
P
0HP OH ����������������������������
2 2 2( ) 0ax by cz a b c
0ax by cz d
0
1
x
ya b c d
z
Homogeneous coordinates 0 H P
P
TH
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Coordinate System Changes
Rotation
where
Exercise: Write the rotation matrix for a 2D coordinate system.
ˆ
ˆ
ˆ
AX B
AY B
AZ B
B i P
B j P
B k P
Bahadir K. Gunturk 6
Coordinate System Changes
Rotation + Translation
In homogeneous coordinates
Rigid transformation matrix
Bahadir K. Gunturk 9
Intrinsic Camera Parameters
We need take into account the dimensions of the pixels.
CCD sensor array
Bahadir K. Gunturk 10
Intrinsic Camera Parameters
The center of the sensor chip may not coincide with the pinhole center.
Bahadir K. Gunturk 11
Intrinsic Camera Parameters
The camera coordinate system may be skewed due to some manufacturing error.
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Intrinsic Camera Parameters
In homogeneous coordinates
These five parameters are known as intrinsic parameters
Bahadir K. Gunturk 13
Intrinsic Camera Parameters
In a simpler notation: 1K
zp 0 P
With respect to the camera coordinate system
Bahadir K. Gunturk 14
Extrinsic Camera Parameters
Translation and rotation of the camera frame with respect to the world frame
In homogeneous coordinates
1K
zp 0 PUsing , we get 1
11
1
X
C CYW W
TZ
Wu
WR Ov K
Wz
00
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Combine Intrinsic & Extrinsic Parameters
We can further simplify to
1
11
1
X
C CYW W
TZ
Wu
WR Ov K
Wz
00
1
11
X
YC CW W
Z
Wu
Wv K R O
Wz
3x4 matrix with 11 degrees of freedom: 5 intrinsic, 3 rotation, and 3 translation parameters.
Bahadir K. Gunturk 16
Camera Calibration
Camera’s intrinsic and extrinsic parameters are found using a setup with known positions in some fixed world coordinate system.
Bahadir K. Gunturk 18
Camera Calibration
Mathematically, we are given n points
We want to find M
i
i
u
v
wiwiwi
x
y
z
1,...,i nand where
Bahadir K. Gunturk 20
Camera Calibration
Scale and subtract last row from first and second rows
to get
Bahadir K. Gunturk 21
Camera Calibration
Write in matrix form for n points
to get
Let m34=1; that is, scale the projection matrix by m34.
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Camera Calibration
The least square solution of is
From the matrix M, we can find the intrinsic and extrinsic parameters.
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Camera Calibration
Consider the case where skew angle is 90. Since we set m34=1, we need to take that into account at the end.
Notice that
Since R is a rotation matrix,
Therefore,
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Camera Calibration
We get
See Forsyth & Ponce for details and skew-angle case.
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Parameters of a Stereo System Intrinsic Parameters
Characterize the transformation from camera to pixel coordinate systems of each camera
Focal length, image center, aspect ratio
Extrinsic parameters Describe the relative
position and orientation of the two cameras
Rotation matrix R and translation vector T
pl
pr
P
Ol Or
Xl
Xr
Pl Pr
fl fr
Zl
Yl
Zr
Yr
R, T
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Calibrated Camera
( , ,1)[ ( ')] 0 with
' ( ', ',1)
T
T
p u vp t Rp
p u v
' 0Tp Ep
' 0 with Tp Ep E t R Essential matrix