So, good morning everybody. Can we start? Can we start? Good morning. That's not good.

Let's check again.

Danke schön.

So, we are getting closer to the end of the semester. We still have a quite tough program.

Maybe I lost too much time by telling stories about industry and hospitals. But we will look into a few important subtypes of reconstruction algorithms today.

And then by Thursday we have to start with image fusion. So, I skip the big picture today for the sake of new topics that we can squeeze in for the final lecture.

I hope that with the registration of the oral exam everything is going fine now. Everything is set up. I receive quite a few emails regarding how should I apply and how should I register.

I usually forward these things because I end up with the same questions like you because I have never registered for this.

Okay, so far we have seen two different types of image reconstruction techniques.

Good morning. One class of reconstruction algorithms is using the analytical approach.

The other part is using the algebraic approach.

So, these are algorithms that are based on algebra. And the second part, and this is something we are going to learn also today, is using statistical reconstruction algorithms.

Just for us, for our understanding, here we do all the computations with the integral equations to the end and discretize at the end of the day the method we have found.

Filtered back projection for instance is based on the Fourier slice theorem that was proven in the continuous using analysis methods.

So, we push through the consideration of the problem in the continuous until we end up with a reconstruction method, an analytical reconstruction method that we discretize at the end of the day.

And in the algebraic approach we use the numeric approximation of the integral. That means we replace the integral by a sum right away at the beginning when we start to consider Baer's law.

So, we discretize in a very early stage of our algorithms. And here we transform everything into a probabilistic context and then we do a maximum likelihood estimation saying,

compute the parameters that we are looking for. That means the object to be reconstructed in a way that some probabilities are maximized based on the observations.

How that looks in detail we are going to see today. So, we are working with integrals, we are working with sums and we are working with probabilities and maximum likelihood estimators.

And with respect to algebraic reconstruction we have seen that the reconstruction of a volume using x-ray projections can be rewritten in terms of a system of linear equations.

And each pixel that we measure contributes one equation for the system of equations we have. Each volume element, each voxel that we have to compute is one unknown.

That means the number of unknowns of our system of equations is driven by the volume size that we want to reconstruct.

And basically we have Ax is P, that's the system of linear equations we have to solve. And we won't be able to compute the inverse of this matrix right away because most or quite often we have scenarios where we have more unknowns than equations.

So we have ambiguities in the solutions and we have to deal with that. And instead of solving this type of equation for x, we replace P minus Ax is zero by a numerical term saying that the squared Euclidean difference

of these two vectors should be minimum. So x is the volume we want to reconstruct and we want to compute x in a way that this difference here is minimized.

And if we have this linear setup, we can say the solution can even be written in closed form using this zero inverse that we also have seen within the context of the singular value decomposition.

And this here requires the inversion of this matrix and this is also something that we cannot do right away by using this Gram matrix, that's how it is named, using the Gram matrix and computing is inverse.

So we have to use numerical approximations and we have seen last Thursday one iteration scheme with a geometric association of the system of linear equations, how we can iteratively get closer and closer to the point of intersection of these linear manifolds.

And the intersection of linear manifolds means that the solution for the system of linear equation. In a more abstract way we can say we have here a function of f minus, of fP minus Ax that has to be minimized and in this case f is just the squared Euclidean distance.

And the reconstruction itself is thus an optimization problem.

Let me just move this here a little bit up. Can I make this dark?

Maybe this way.

So reconstruction means we compute the minimum f of P minus Ax.

Once again, what is P Stefan? What is P?

Come on, you have to ask if you don't remember that.

These are the measured intensity values just packed into a vector. What is A?

That's the system matrix that is characterized and given by the projection geometry and we have to optimize this.

If this is a quadratic function that has to be minimized, we know quadratic functions that can be minimized look like.

So this is a convex function and we look for this minimum and we can compute the minimum value. That means, for instance, if it sits here, we can compute the x value for the minimum by a closed form solution.

Of course, quadratic function means compute the derivative. The derivative of a quadratic function is linear and the linear problem can be solved in closed form like this here.

So we can compute a closed form solution for it.

Quite often we use numerical methods to compute the minimum of this function.

One way to do that is to do a gradient descent method.

That's one approach that is also applied quite often in algebraic reconstruction techniques.

Instead of computing closed form the point for the minimum, we set up an iterative scheme that does gradient descent.

If I compute the gradient of the function, I get a vector. Where does this vector point to?

Ascent.

The gradient always tells you in which direction does the function increase the most.

Keep in mind the nice picture you are in the Alps with your girlfriend or boyfriend. Your girlfriend or boyfriend asks you which direction should we go to reach the top of the mountain.

Then you just sit there and say no problem, I'm studying computer science, medical engineering at the University of Jalang. I know how to do that.