Thank you for the very nice introduction. And if you have any questions, you can stop

me and ask directly. So today, as you see, I will talk about deep learning for CT image

reconstruction from insufficient data. And here is the outline. So first, I will talk

about how CT works and introduce what limited angle tomography is. And the same, I will

talk about the robustness of deep learning. Then we will propose our own hybrid reconstruction

method which combines deep learning with conventional reconstruction method. So we call it data

consistent reconstruction. And we apply it in two scenarios, as you mentioned. First,

yeah, limited angle tomography. And then later, yeah, another application. So for image reconstruction

from truncated data. So first CT. So for CT image reconstruction, we need a sufficient

number of X-ray projection images. So this is one example. So yeah, here. So this multiple

views of a patient is the region of the torso. So we can see the X-resource and the detector

are rotating. Then we can see the different views of this patient. And actually, for one

view, so if we draw one horizontal line like this, then actually this line is the projection

of one horizontal slice. Then on this line, for each pixel, actually it is the integral

of one X-ray. So different pixels corresponds to different X-rays. So each integral, yeah,

in the discrete space. So actually, you only need to sum up all the pixel values. Then

now I will talk about how CT reconstruction works. So here we use a very simple example.

So here, we have an image with four pixels only. In this projection view, so in the vertical

direction, this projection has two pixels only, so seven and two. Then for the first

X-ray, it passes these two pixels, X1 and X3. Then we get an equation X1 plus X3 equals

seven. And here, for the second pixel, so the second ray, so it passes through X2 and

X4. Then we get another equation X2 plus X4 equals two. Then when we rotate the X-resource,

then we have another view. Then again, we have two pixels, five and four. Then for five,

it passes through X1 and X2. Then for four, it passes through X3 and X4. Then here, then

we have four linear equations. Its solution is very simple. We can easily get X1 equals

three, X2 equals two, X3 equals four, X4 equals zero. Then with these solutions, then we know

the distribution of the anatomical structures inside of the body. So this is a basic idea

for CT image reconstruction. Here, we can also formulate this in matrix notation. For

the projections, actually, it is a vector. Here, it has four elements, seven, two, five,

four. And eight is the system matrix. Here, each row corresponds to one X-ray. The elements

in this row, actually, mean stands for the weight for each image pixel inside this X-ray.

And the number of X-ray rows, the number of the rows equals the number of X-rays. And

here, X is the vector of the object or the image we want to reconstruct. Then here, so

P equals AX. So we want to reconstruct X from what we obtained, P, and with the known system

matrix A. So it is a typical inverse problem. Then here, but in practice, we cannot simply

use compute the inverse of A or the pursuit inverse of A because the system matrix A is

very large. We need about 64 PB data in real applications, for example. That's why we can

never store the system matrix to compute its inverse. So in the CT community, we have a

spatial algorithm called filtered back projection for CT image reconstruction. So it consists

of two steps. First, filtering the projection data. Then afterwards, we do the back project.

So this back projection operation serves as the inverse of the system matrix A. So this

is the standard algorithm for CT image reconstruction. And now, yeah, talk about my PhD project.

So I use combing computed tomography. So combing computed tomography is widely used for planning,

and also monitoring of many interventional therapies. And in order to get a 3D reconstruction,

the source and the detector need to rotate at least pi plus a comb angle. So in practice,

typically around 200 degrees, then we can see the different views of the imaged object.

And however, in some situations, the secondary rotation might be limited by some external

of a circle or some other system parts. For example, this is a Simmons RTC multipurpose

system that because of its short C arm, it can rotate only 150 degrees. Then if we want

to use such a system for image reconstruction, then some data are missing. So image reconstruction