How to obtain the 8th degree polynomial from the linear PnP Algorithm

In the previous blog I have introduced an algorithm to solve the PnP problem by Long Quan and Zhongdan Lan, but there is still one problem during the calculation, that is with the equation system

how to get the 8th polynomial:

and here I will derive the whole process.

Still like before with this illustration let's face the problem more intuitively. With this tetrahedron we achieve the all angles and lengths we need, and what should be obtained is the length of three legs, S1, S2 and S3.

And our new equation system looks like this:

Besides we also define another two helpful tools to help us eliminate the invariable.

So combine (5) with (2) to eliminate the invariable S3 and we can get:

The same to combine (4) with (3) will come the following:

And do the same thing with (4) and (5) to take place of (1), the S2 and S3 will be totally removed.

And next we will do a serial of supersessions to really solve the problem, but you guys should be patient!

So with this replacement, (6), (7) and (8) can be rewritten as

And square the equation (11) with the help of (9) and (10) to eliminate the u^2 and v^2, we can acquire the following equation:

So the following job is another replacement, still that word "BE PATIENT!"

And in this way, expand the equation and replace the u^2 and v^2 with above equations we get another relevant equation:

Now we are already close to the 8th degree polynomial, this is the good news! But bad one is we need another replacement,hehe. Here we go: Let's square the equation (12) and enjoy the last replacement, as follows:

Finally the 8th degree polynomial is available as follows:

With this equation man can apply it on lots of algorithm calculation for the perspective n points pose estimation. And the main idea of this solution is from Linnainmaa, Harwood and Davis.

Posted in: tracking technology,Virtual Reality