Camera Calibration

1
Camera Calibration
March 6, 2007
2

• Finding cameras internal parameters that effect
  the imaging process.
• Position of the image center on the image
  (usually not just (width/2, height/2).
• Focal length
• Scaling factors for row and column pixels.
• Lens distortion.

3

• Camera pixels are not necessarily squares
• Analog output and digitalizing.

4
Each camera can be considered as a function a
function that takes each 3D point to a point in
2D image plane. (x, y, z) ------ gt (X,
Y) Camera calibration is about finding (or
approximating) this function. Difficulty of the
problem depends on the assumed form of this
function, e.g., perspective model, radial
distortion of the lense.
5

• Good camera calibration is needed when we want to
  reconstruct the geometry from images
• Robotics, human-robot interaction
• Robotics, robot navigation.
• Location of objects in 3D

6
Suppose the camera is calibrated and the 3D
positions of the feature points are known for the
first frame
7
(No Transcript)
8

• Literature on this subject is tremendous.
  Hundreds of published papers.
• Linear algebraic methods present here
• Can be used as initialization for iterative
  non-linear method.
• Vanishing points and projective geometry

9
Calibration finding the function (defined by
the camera) that maps 3D points to 2D image
plane. First thing required is to obtain pairs
of corresponding 3D and 2D points. (X1, Y1, Z1)
(x1, y1), (X2, Y2, Z2) (x2, y2), .
10

• Calibration Target Two perpendicular planes
  with chessboard pattern.
• We know the 3D positions of the corners with
  respect to a coordinates system defined on the
  target.
• Place a camera in front of the target and we can
  locate the corresponding corners on the image.
  This gives us the correspondences.
• Recover the equation that describes imaging
  projection and cameras internal parameters. At
  the same time, also recover the relative
  orientation between the camera and the target
  (pose).

11

• Corner detector
• Canny Edge detector plus fitting lines to the
  detected edges. Find the intersections of the
  lines.
• Manual input.

Matching 3D and 2D points (we know the number of
corners) by counting. This gives corresponding
pairs ( world point) lt --- gt (image point
) (X1, Y1, Z1) (x1, y1),
12
(No Transcript)
13
Rotation matrix
14
Let C - Rt T
15
Use Homogeneous coordinates, the perspective
projection becomes linear.
16
(No Transcript)
17
(No Transcript)
18
(No Transcript)
19

• Estimate matrix P using scene points and their
  images
• Estimate the intrinsic and extrinsic parameters
• Left 3x3 submatrix is the product of an upper
  triangular matrix and an orthogonal matrix.

20
(No Transcript)
21
(No Transcript)
22
(No Transcript)
23
(No Transcript)
24
(No Transcript)
25
We know P and we can get the translation C.
Then we get M KR. Find the QR decomposition of
M to get K and R !!
Further Improvement
26
Parallel lines in 3D has a vanishing point on
the 2D image plane. Point of intersections of
these lines. Take three bundles of mutually
perpendicular lines in 3D and compute the three
vanishing points. The image center is the
orthocenter of the triangle formed by the 3
vanishing points!!
Homework 3