Computer%20Vision

1
Computer Vision

• Geometric Camera Models and
• Camera Calibration

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
P
O
k
j
Coordinate vector
3
Homogeneous Coordinates
n
H
P
O
Homogeneous coordinates
4
Coordinate System Changes

• Translation

5
Coordinate System Changes

• Rotation

where
Exercise Write the rotation matrix for a 2D
coordinate system.
6
Coordinate System Changes

• Rotation Translation

In homogeneous coordinates
Rigid transformation matrix
7
Perspective Projection

• Perspective projection equations

8
Intrinsic Camera Parameters
Perspective projection
9
Intrinsic Camera Parameters
We need take into account the dimensions of the
pixels.
CCD sensor array
10
Intrinsic Camera Parameters
The center of the sensor chip may not coincide
with the pinhole center.
11
Intrinsic Camera Parameters
The camera coordinate system may be skewed due to
some manufacturing error.
12
Intrinsic Camera Parameters
In homogeneous coordinates
These five parameters are known as intrinsic
parameters
13
Intrinsic Camera Parameters
In a simpler notation
With respect to the camera coordinate system
14
Extrinsic Camera Parameters

• Translation and rotation of the camera frame with
  respect to the world frame

In homogeneous coordinates
Using , we get
15
Combine Intrinsic Extrinsic Parameters
We can further simplify to
3x4 matrix with 11 degrees of freedom 5
intrinsic, 3 rotation, and 3 translation
parameters.
16
Camera Calibration

• Cameras intrinsic and extrinsic parameters are
  found using a setup with known positions in some
  fixed world coordinate system.

17
Camera Calibration
Y
X
Z
courtesy of B. Wilburn
18
Camera Calibration

• Mathematically, we are given n points
• We want to find M

and
where
19
Camera Calibration

• We can write

20
Camera Calibration

• Scale and subtract last row from first and second
  rows

to get
21
Camera Calibration

• Write in matrix form for n points

to get
Let m341 that is, scale the projection matrix
by m34.
22
Camera Calibration

• The least square solution of is
• From the matrix M, we can find the intrinsic and
  extrinsic parameters.

23
Camera Calibration

• Consider the case where skew angle is 90. Since
  we set m341, we need to take that into account
  at the end.

Notice that
Since R is a rotation matrix,
Therefore,
24
Camera Calibration

• We get

See Forsyth Ponce for details and skew-angle
case.
25
Applications
First-down line
courtesy of Sportvision
26
Applications
Virtual advertising
courtesy of Princeton Video Image
27
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

28
Calibrated Camera
Essential matrix
29
Uncalibrated Camera
Fundamental matrix