The following content has been provided by the University of Erlangen-Nürnberg.

And we started out to look at 2D registration using point correspondencies.

And given two sets of points, given the assignment of the points, that means we have matched points,

how can we compute rotation and translation but not deformations?

So we looked at the problem of rigid registration where no deformation was allowed.

And if we have a set of point features, correspondencies in 2D,

it turns out that the registration problem basically can be reduced to a system of linear equations.

Using complex numbers that basically represent two-dimensional points in the space of the real and the imaginary part,

we were able to show by just looking at the definition of complex numbers

that multiplication of complex numbers is nothing else but rotation and scaling.

And if we multiply a complex number that represents a two-dimensional point in the complex plane

with a complex number that has unit length, we basically can characterize rotations.

So going beyond what we have learned in standard engineering mathematics,

we can say complex numbers are not only nice because we can compute the square root of negative numbers.

That's a very weird idea actually.

No, from a geometrical point of view, complex numbers look very easygoing.

They allow us to characterize two-dimensional rotations.

And using also summation, we can also incorporate translations.

And having this in mind, it's straightforward to write down the equations and to do all the computations.

Basically, this is only a few lines in MATLAB or just a few lines in C if you use a linear algebra library.

So very nice to see that.

And then we started to look at 3D registration problems using point correspondences.

And the rigid registration should also be limited to rigid transformations.

So only rotations and translations are considered.

3D, 3D rigid point-based registration.

And in this context, we have to consider the transformations that we can basically apply to do 3D, 3D registration.

So again, the scenario is as follows.

We have two point sets in 3D.

X, Y, Z.

Here we have some points in 3D.

And we have a second coordinate system, X, Y, Z, with another set of points.

And we know the matching.

That means the assignment, which feature belongs to which feature.

This is just a sketchy illustration here.

And given these point correspondences, how can we compute rotation and translation?

And just by heart, you could say, oh, that's not so difficult.

I can do the following.

I can set up a least square estimator as usual.

Let's call these points here P i in R3 and these points here Q i in R3.

Then basically I can say Q i is rotation matrix multiplied with P i plus T.

That's the equation that holds for all points that are corresponding to each other.

And now we can say, OK, this is basically leading to Q i minus R P i minus T,

the L2 norm of it, sum over all i.

And we compute R and T such that this thing here is minimized.

So we look for a rotation and a translation such that this error basically turns out to be as close as possible to zero.

That's a standard engineering idea.

We have a parametric mapping.

We have the correspondencies.

We just say the difference has to be zero in the ideal case, close to zero in the numerical situation

where we have points corrupted by noise, and then we minimize things.