Before we continue in the text of this lecture, let me just summarize where we currently are

and which topics we have considered so far.

There are always different ways to look at things.

Today I want to look at it from a more mathematical point of view, but before I talk about math

or algorithms, let me just say a word about modalities.

We have seen in one of the first lectures that in medical imaging we have tons of different

modalities.

Each modality has its own advantage and the images look quite differently.

And there are different principles borrowed from physics that can be applied to visualize

the inner of the human body.

It ranges from endoscopy using standard CCD chips over ultrasound, spect-PAT, MR, X-ray,

CT imaging.

So there is a bunch of modalities that is used today in hospitals and each modality

has its own advantages and disadvantages.

They all have in common, they are usually expensive.

It's the field where we are in.

They generate huge amounts of data and they are expensive.

And what we are currently considering is an important chapter on pre-processing.

And we talk about acquisition specific pre-processing.

We look at the acquisition devices and look at the principles that are applied to generate

the signal.

And time is no longer important for us today, how long theyeah, that's the difference between

Okay, good.

We talked about image intensifiers, so we have the electrons flying through the earth

magnetic field and dependent on the position of the image intensifier the relationship between the magnetic field and the moving electrons, the flying electrons changes and so we have different deviations and these deviations cause distortions in the image.

And we have to correct these.

Okay.

And how can we correct these?

We have always these pair of images where we have here our undistorted and our distorted image and we compute a mapping using a calibration pattern.

Calibration pattern is a highly or a precisely manufactured piece of metal that can be put on the image intensifier and there are points and these points can be detected automatically by software, by algorithms in the image.

And then people compute the correspondences and based on these correspondences you can estimate the mapping between the undistorted and the distorted image.

Rule of thumb is always sample in the space where you expect the result.

That was something that you should remember as well.

And we looked at many things here in terms of math.

We have seen that the mapping can be characterized by a parametric function, by two bivariate parametric functions.

You put in the x and y value of the current position where you want to have the intensity value of and then there is the first function that tells you which is the x coordinate in the distorted image and you have a second function that tells you what is the y coordinate in the distorted image.

Okay.

And we started out to look at polynomials, very simple mappings and they do a pretty good job actually and in current implementations and in products still they use fifth order polynomials to undistort the images.

Then besides the parametric function we have seen that we can define a linear estimator to get the coefficients of the polynomials and we have seen that we have applied a least squares or least square estimator.

So basically we have seen that we have a measurement matrix, the parameter matrix and we subtract this and minimize this least square error.

And we are pretty happy with that.

So far we are pretty happy.

For those of you who attend pattern analysis or pattern recognition it is way more applied.

There you will learn that you can do way better by looking at different norms besides the L2 norm that we are considering here.

But for us it is more than sufficient.

Okay.

So we minimize the distance between Mx and b and that we cannot solve this Mx equals b equation is due to the fact that b is not in the range of the matrix M.

We have discussed this in more detail yesterday again.

So we minimize the error here.

And this is one way to do it and we can use SVD to solve it and SVD is something that we have discussed in a little more detail in one lecture right at the beginning of this summer, of this winter semester.