For me it's important that you know in which field we are currently in and let me briefly

hook up to the discussion what we are looking at so far.

We look at the KL divergence between two densities.

No, let me, how should I do that here?

And the KL divergence between two densities is a measure for the statistical dependency

for instance as well.

And we have seen that for multi-model image registration we want to find a mapping between

two images such that the statistical dependency of the assigned densities is maximum.

Yeah, so we want to maximize a term like P of X and Y log P of X and Y.

So, this is the joint density divided by P of X times P of Y.

And this has to be maximized with respect to a transformation of one image of the source

image to the target image.

Okay, and this is easily stated so write it down.

Take P of X, take P of Y, take the joint density P of X and Y and compute the mutual information

parameterized in the transformation between the two images.

But if you sit down and start to implement things you run into problems.

And the first question is how do you represent the densities?

So, we need a representation of the densities.

For instance, if I have here to compute P of X.

I can say, okay this is the density of my random measure X.

Okay, and let's assume x is an intensity value and x is element of 0, 1 up to 255 if we have

8-bit quantization for our intensities.

So what you can do is you can do the following.

You can estimate here x and here p of x by just computing the relative frequencies.

So you take your image and you take the image and compute or you just count how often do

I observe the intensity value 0, how often do I observe the intensity value 1, how often

do I observe the intensity value 2 and so on.

I divide these values by the number of pixels in the image so I get relative frequencies.

So these numbers sum up to 1.

And I get basically here 1, 2 up to 255 and I get here something like that.

So you get some up to this.

So I can bring in here the relative frequencies.

Another problem or one substantial problem with this density function here is that this

is a discrete probability density function.

It's discrete.

The x has discrete values.

So if I want to optimize the function above using these histograms, I cannot compute for

instance a derivative, a gradient or something like that.

You can't do that.

It's all discrete.

So this is a discrete representation.

It's discrete.

Yeah?

For each and every different transformation you have to do these computations, especially

for the joint density.

So it's discrete.

You cannot compute a gradient and the optimization of this function with respect to t might turn

out to be quite hard.

So how can we bring in a way to represent these densities by continuous functions?

And one idea that we have seen last time is the Parson estimation.