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Yesterday we have introduced the core ideas. What is the key message of yesterday's lecture?

It's basically if we search for functions instead of parameters,

we know that we have to compute the Euler-Lagrosse partial differential equation

and we have to solve the Euler-Lagrosse partial differential equation

to find a necessary condition for an optimum of the function we want to optimize.

And you also should remember that the optimization problem we are actually considering

is an integral over a function that depends on a variable x

and that depends on the function and its first derivative.

And we have seen yesterday a very simple example

how the Euler-Lagrosse partial differential equation can be applied.

We have considered two points in the plane

and we were looking for the function that minimizes the distance

or that has the minimum path length basically connecting these

or the minimum length connecting these two points.

Maybe that's the right term.

So we have used Pythagoras to compute the approximation of a small distance

and then we integrated over the delta s for the whole function

and we set up the Euler-Lagrosse partial differential equation

and found out that basically we end up with a straight line.

What is actually intuitively clear to everybody in the audience

that this is the right answer without knowing mathematics.

So this example is in the range of the example

computing the mean value of numbers using a maximum likelihood estimator.

At the end of the day we just get the intuitive result,

just sum up the numbers and divide by the number.

So we like the example and I want to point out that sometimes I ask

in particular this example, even it's so simple,

I ask students can you write it down and explain to me

the core idea of variational calculus

and sometimes I hear an answer like this is not image processing,

it was such a simple example and blah, blah, blah

and they say yes it was simple and it was not image processing

but I want to see it and then I can't do it.

So that's not positive if you do not know it.

But on the other hand all these details, I mean I will not get,

I will not get or go into all the details

but the rough idea at least you should be able to explain.

Okay, good. So nice example. Let's continue.

Sometimes in medical image processing we do not have an

optimization problem as described last time

but we do have a more extended version in a sense

that higher order derivatives show up.

For instance we have seen yesterday the problem of image smoothing

where we said we have given an image F, we have to up,

oh, was the picture missing all the time?

And nobody told me?

So you missed my smiley here.

You still miss it here.

Once again.