Camera Calibration

The perspective projection from Euclidean 3-space to an image is represented as

is a homogeneous camera projection matrix (11 dof) with the decomposition

The algorithm for camera calibration has two parts:

1. Compute the matrix from a set of points with known 3D positions and their measured image positions.
2. Decompose into and via the decomposition.

1. Compute the matrix : Use correspondences between 3D points and their 2D images to determine the matrix .

1. Each correspondence generates two (linear) equations on the matrix elements of .

   

   multiplying out

   

2. Given () correspondences, a linear solution can be obtained for from the set of 2n linear simultaneous equations (cf computation of a projective transformation): , where is the 12-vector representation of the projection matrix , and is a matrix. The solution is the eigenvector with least eigenvalue of .

3. This linear solution is then used as the starting point for a non-linear minimisation of the difference between the measured and projected point:

   

Example - Calibration Object

Determine accurate corner positions by

1. Extract and link edges using Canny.
2. Fit lines to edges using orthogonal regression.
3. Intersect lines.

The final error between measured and projected points is typically less than 0.02 pixels.

2. Decompose into and :

The first submatrix, , of is the product () of an upper triangular and rotation matrix.

1. Factor into using the matrix decomposition. This determines and .
2. Then

   

Note, this procedure produces a matrix with an extra parameter k

with , and the angle between the image axes.