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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:
- Compute the matrix from a set of points with known
3D positions
and their measured image positions.
- Decompose into
and via the decomposition.
1. Compute the matrix : Use correspondences between
3D points and their 2D images
to determine the matrix .
- Each correspondence
generates two (linear) equations on the matrix elements of .
multiplying out
- 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 .
- 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
- Extract and link edges using Canny.
- Fit lines to edges using orthogonal regression.
- 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.
- Factor into using
the matrix decomposition. This determines
and .
- Then
Note, this procedure produces a matrix
with an extra parameter k
with , and the angle between the image axes.
Next: Plane projective transformations
Up: No Title
Previous: Summary and Properties
Bob Fisher
Wed Apr 16 00:58:54 BST 1997