So, good morning everybody. Today we will continue with the problem domain of rigid

image registration. Before we continue to look into the representation of 3D rotations

and how to compute rotations from point correspondences, let us do a brief overview and a brief summary

of what we have considered so far. So, we are talking this semester about diagnostic

medical image processing and we have basically four key chapters. The first one is a very

short chapter where I have briefly introduced different modalities in medical imaging. The

second chapter is on pre-processing. The third chapter is on 3D or general on reconstruction

from projections. And now we are currently considering the chapter on image fusion. So,

the story is we have different modalities and we use different chemical and physical

principles to acquire images of the human body and of the inner of the human body. Then

associated with the image acquisition devices, there are particular artifacts that come into

the game that are due to the design of the acquisition device and we have discussed various

pre-processing methods that take the knowledge about the acquisition device, about the artifacts

that we expect and eliminates the artifacts as much as possible in the final images. So,

we can acquire images of quite good quality. Then we consider the questions what can we

do if I have multiple images with one modality, can I do for instance reconstruction. We talked

about computer tomography and the reconstruction of volumes from x-ray projections. And now

we talk about fusion and we have different modalities, multiple images of a single modality,

higher dimensional information and now we have multiple images from multiple modalities,

how can we bring them all together into a single coordinate system. So, if you use your

tongue and you have to answer the question is there a storyline in the lecture, you have

to say yes and there is a range from one to five and five is the best and I mean can I

do anything better like that. No, so choose the five and support me. And again it's as

usual highly confidential and we just record your IP address and track you down. Okay,

that's the big picture. So, if I ask you about the big picture that's what you should draw.

Then we talked about many mathematical principles, that's also a potential question. Which mathematical

concepts did we repeat from engineering math that you should know and that are interesting

for us. What mathematical principles did we use to solve particular problems in image

processing and we have seen various mathematical concepts that are important for solving image

processing problems and let me briefly summarize these. So, we talked at the beginning about

the singular value decomposition. Decomposition that is also abbreviated by SVD. I personally

like it very much, it's a very powerful tool and I also put some bias on your understanding

of linear algebra. So, SVD is the tool for everything. So, I even read in PhD thesis

in preliminary versions before submission of my PhD students. Yes, we use the SVD because

this is a method to compute, direct method to compute the inverse of the SVD is a normal

form for matrix. So, you can factorize matrices and with this factorization you can solve

particular or various problems and it's not a method to compute the inverse. What can

we do for instance, we can compute the rank by counting the non-zero singular values.

We have a very good understanding whether a matrix is close to be singular, the smallest

singular value. We know how we can estimate the pseudo inverse. Then what else did we

discuss using SVD? You can force a certain rank of the matrix by setting certain singular

values to zero. The singular value decomposition tells you where the unit ball is mapped to

by a matrix mapping down to the edge with the principal components that's related to

the PCA for the factor analysis. So, it has a lot of nice features and powerful tool and

the nice thing is that you basically are not required to know how it's computed. It's

available in libraries like Linberg, it's available in Meta, there is a book so we attend

lectures on numeric. For us it's basically a backup for all the numerical problems that

we have in terms of linear algebra. So, we can use it and if we have solved a certain

problem using SVD and we have to perform for it, it's not much better numerical methods

or certain situations. So, that's something that we have to consider. So, the key message