X-ray imaging resolutions are a fascinating topic in the scientific world. By utilizing various mathematical formulas and principles, physicists have been able to develop advanced methods for X-ray imaging. In the upcoming episode with Dr. Cardoso, we will be exploring the complex mathematics behind how an ideal detector is used with a real X-ray source.
This process enables us to capture detailed images with higher resolution than ever before! By understanding the science behind this technology, we can unlock even more possibilities for creative applications of X-ray imaging solutions. From medical studies to industrial analysis, X-rays provide an invaluable resource in many different industries that often require precise measurements and images.
Welcome to another episode of X-ray University. Today, we’re going to be talking about the resolution of X-ray images. Now, our producer, Tony, who told me that you guys get bored with me talking for over half an hour. So this time, what we’re going to do is we’re going to split this video into three different parts. Part one, two, and three, Part one.
We’re going to be talking about what is the resolution of an X-ray image when you have a real X-ray source in an ideal detector? Then we’re going to go and talk about in the next video how X-ray resolution works out when you have an ideal X-ray source and a real detector. And then lastly, on the third video, we’re going to talk about the real X resource and the real detector.
Now, you shouldn’t want to go through all the details. Just skip part one and two, go directly to part three and get the answer. If you want a little more detail and work the math, then watch parts one, two, and then see what the final result is. In Part three.
All right. So let’s get started with how an X-ray tube works. So we’ve covered this in previous videos, but I’m just going to give a quick refresh so we’re on the same page. So you have a filament that warms up and as you warm up a filament, it creates free electrons that accelerate towards a target. When we apply a voltage differential between the cathode and D.A. upon hitting the target, these free electrons create X-ray photons using this, bring some radiation process in those X-ray photons, exit the tube through a window that can be made of aluminum or really hard to paint or glass, depending on the energy levels you’re interested in getting out of the tube.
Now, depending on the spot size, right, depending on how small the target is, you end up with images like the one on the left where you have a blurry image caused by the large spot size on a tube. Or you can have a really nice image like you see here. You know, just focus on the edges.
You keep perfectly see the edge of the bear, die the fill it and the dye attach. Even with the micro voids there you can see everything. And when you use a micro focus, meaning it’s the focus port is micro. You know, a few microns with many folks usually refer to spot size 30 microns and up. So even if it’s 35 microns spot size, we call this mini focus.
And then we also have sources, the nano focus level where you have spot size and the nanometer level, which is even finer. Now how does an ideal X-ray tube work? We have this really fast animation here to show you this round. Gratings are the X-ray photons, so the cherry two by three electrons in the target and the X tube.
In this location, we call in the focal spot. And then the idea of this animation is to show you that every single X-ray photon on an ideal X-ray tube does not intersect the track of another X-ray photon. Right. So as you can see in this slide here, all the tracks from the spots, eyes to the detector detectors, the gray object here on the bottom.
All of these tracks do not parallel to each other. Ideally would be parallel, but we have a comb being the important thing in an ideal X-ray tube is that those tracks do not intersect. So what does it mean when you have a sample? It means that each of these tracks will cast a shadow that has a unique location on the detector.
Right. As you can see, there. So it’s a really nice, simple, straightforward connection between the track coming out of the focus part on the X-ray source all the way down to the detector. Pretty straightforward. Unfortunately, that’s not what happens in reality. Reality, we have a spot size that’s larger than zero, which means that the tracks out of the X-rays, photons out of the X-ray tube, they do interact.
And like you can see in this animation and I drew the tracks here so you can see better. And unfortunately, what happens is that now as we cross those tracks, see what happens on with the sample, that same spot on the sample is projected in two different locations on the detector. Let’s look at the tracks here so you can get a better view.
You see that what happens? So as you can see here, that edge of the sample, by the way, this is your this is our sample right there to the edge on the sample shows up in this location of the detector and at the same location, the detector. That’s what caused that blurry lines at the edge of the sample and that’s that’s exactly why the resolution of the image suffers.
The larger this focal spot size, the worse your image quality is going to be. I hope that makes sense. And we call resolution that distance that’s projected on the detector. Right. So you want to have a resolution number that is as low as possible. What is the relation for the ideal X-ray to zero? Right. Zero microns, because there’s no gap, there’s no distance between the edge of the track and the next the projection, the other side.
Right. So let’s look now side by side. On the left you have a real X-ray tube where the spot size has a physical side on the right we have an ideal tube where that’s focused, but size is zero zero microns, which means that every single X-ray photon does not interact with a neighbor. And as a result, you have a beautiful resolution of zero on your detector.
When you have a real X-ray tube. This focal spot has a dimension. And as a result, you end up with a resolution. That is a number. But what is that number? Right. You want to be able to calculate what is that number? So if you have a large spot size, as you can probably tell by now, that number is going to be larger, right?
Because that the number that’s going to be cast onto the detector is going to be larger. If you have the magnification change. As you can see here, you see the mag and just lowering the magnification, moving the sample closer to the detector. Right. So I’m lowering my magnification by lower magnification. As you can see, that resolution number gets smaller, smaller, smaller.
Right. So as you bring the sample closer to the detector, that resolution number is going to get smaller. Pretty cool. Now, what happens if you have a small spot size? They have a small spot size. And again, going towards that ideal scenario, as you can see, that projection is already smaller. And as we remove or reduce the magnification, you get smaller and smaller and smaller.
So how would you calculate? Let’s put some numbers behind it, right? I mean, you know, theoretically, I think we all understand we can we have an appreciation of the physical concepts that we’re dealing with here, but how do you actually put numbers to it? So let’s start with a sample that’s really small, right? So small that we are able to capture the edges of the sample being projected here in here, or should the detector go?
Remember part one of this trilogy, if you will, assumes that we have an ideal detector. What’s an ideal detector? Pixel size is zero. So we have an infinite number of pixels on our detector. Right? So as we look into putting numbers, the first thing we want to quantify is the size of the spot size. So the focus spot size is an area, right?
So it’s a dimension that we’re going to generate actual photos from. We’re going to call that F and the resolution of the source we call R, which is this distance here. We’re going to call this which you do over IP number, which is at the detector. So we call R. Now, it’s not going to be exact. There are some other factors that you have to be taking consideration.
For example, the flux out of the X-ray source is not constant. We’re assuming that’s constant here. So it’s R cause. And so there are fewer photons on the edges of the beam than in the center of the beams, because it’s a cone beam-shaped flux that coming off from the X-ray source. But for now, let’s consider it a constant so that we can go through this calculation.
As we saw before, the resolution or R, which is this number here, is a function of magnification. The lower the magnification, the smaller the resolution. So magnification will have to do with that. With this distance here, we know magnification is a function of one plus D two over D one. Right. So here we have it all put together. We have the spot size here on the top, the resolution on the detector or the source here on the bottom.
And how do we calculate this number as a function of the focal spot and the magnification? So I invite you to go back to your high school years right where you were dealing with pretty basic geometry. And these are what if you remember similar triangles, right? Similar triangles mean that. And here I cut them in half just to simplify that angle, all four is equal on the top triangle to as it is on the bottom triangle.
Right. Because those photons are intersecting. And a very simple calculation here now tells you that tangent of alpha is equal to F divided by two divided by D one. Similarly, tangent of alpha is also R is divided by two, divided by D two. And then if I equal those two guys tangent of alpha, tangent of alpha, what we get is that R divided by two D two is equal to F divided by two D one.
Right. So what does it mean? It means that R, our resolution on the source, is equal to F times D one divided by D two. Now we also know that magnification is one plus D two over D one. Look at the factor showing up there again. And if you isolate that factor, we know that D two divided by D one is equal to magnification minus one. Pretty straightforward. And we got a result that the resolution of the source is equal to F multiplied by magnification minus one.
So what does it mean? Right? It means that if you have a system with low mag in a large spot size, you’re going to end up so low mag, the sample was close to the detector. You get up, it’s a tiny number here and your magnification on your resolution. Similarly, if you have a low mag with a small spot size, right, it’s also a very small number resolution, a high mag in a large spot size, big resolution number.
High mag with a smaller spot size. So you can say that magnification plays a huge role in image resolution, meaning that if you lowered the mag, you’re going to improve resolution by quite a bit. And to a point, it doesn’t really matter what resource you have within limits, of course, but as you increase magnification, as you go high mag, the dominant impact on the resolution becomes the spot size.
So here we have three different sources, each with a specific focal spot, 15, five, and two micrometers. And in the horizontal axis, we have magnification and vertical axis without the resolution of the source or two on the image, right? The image resolution driven by the source and very simple line. Right. So that’s magnification goes up as this number gets bigger.
The resolution suffers right linearly. And as you can see, when magnification is one, meaning that the sample is right on top of the detector, the resolution is zero. It’s perfect. Why? Because right now we’re assuming that we have an ideal detector. Right. And that’s why all these lines start at zero. I hope that makes sense. So here’s a couple of basic The Matrix show low mag with a large spot size here, low mag with those small spot size, we can actually see our bones fill it and everything else higher mag with a large spot size where things get even worse.
And then a high mag, which is more spot size with the resolution drops, but you still get a pretty good image because we have a small spot size, small focus, spot size. So what happens? What is the impact of the pixel size on the resolution? And that’s going to be the topic for our next video. So if you want to learn what happens, what is the impact of a real detector on the image resolution, Watch Part two.
If you’re bored and just want to go off the final result, go to part three. See you next time. Thank you.
Dr. Bill Cardoso