### Linear estimation of parameters

• Ignoring radial distortion (for the time being) and setting (measuring in pixels), we have and .
• Then, combining equations we have

and

• Assuming to be known (at the center of the image) and setting and , we have

and

• Eliminating we have

• Rearranging, we have

or

which is a linear homogeneous equation in the eight unknowns , , , , , , and .
• The unknown scale factor can be fixed by setting . Image correspondences of seven points in general position are sufficient to solve for the remaining unknowns. Let the solution be , , , , , , and
• We can estimate the correct scale factor by noting that the two rows of the rotation matrix are supposed to be normal, i.e.,

• The scale factor for the solution can then be determined from

and

This also allows recovery of .
• In the above procedure we didn't enforce orthogonality of the first two rows of . Given vectors and , we can find two orthogonal vectors and close to the originals as follows:

and

which gives

The solution of this quadratic in is numerically ill behaved because will be quite small. We can use the approximate solution

since and are both near 1.
• can then be recovered as .
• Once we have we can estimate and from the basic equations above. This will require one more correspondence to be given.
• The above procedure may be problematic if is close to 0. In such a case the entire experimental data may have to be first translated by a fixed amount.
Subhashis Banerjee 2008-01-20