Filtered BackProjection (Radiologic Technologists : Illustrated guide to FBP)

The Filtered BackProjection (FBP) algorithm is the basis for image reconstruction (converting from the measured data to the image) on modern CT scanners. FBP is a fast and direct method to generate CT images. As the name suggests it is composed of two main steps filtering the data (along the row direction in the detector) and performing the backprojection operation where the data is painted back in the image along the direction which it was measured.

chapters:
00:00 Intro
00:58 CT Image Matrix
02:25 Forward Projection
06:15 Backprojection
08:13 Projection reconstruction
13:48 Sharpening filter
14:43 Filtered Backprojection
15:29 Outro




For more details please see: https://howradiologyworks.com/FBP
In this video we will go over the details of the projection and backprojection operations first, and then build upon those concepts to discuss the FBP algorithm.
What is Forward Projection in image reconstruction?
The forward projection algorithm is used in iterative reconstruction and gives insight into how projection data is acquired in CT imaging. The forward projection process is an addition operation; where we are adding up values in each pixel along the ray direction as seen in this Figure. This Figure demonstrates parallel beam projections from the left to the right (green arrows) through the image pixels.

In this case you can see that from this single view of parallel beam projections that each of the detector values is the same. So if we have just one view of projection data it will not be
sufficient to reconstruct the image. This is the power of CT imaging compared with x-ray radiography in that more views allows for better differentiation of structures.
In parallel beam acquisitions projection data from 180 degrees is required to be able to accurately reconstruct the object. As an example of another forward projection we demonstrate here a projection which is 90 degrees rotated from the first forward projection.

In this case we can see that now different information is available from this projection. In the first forward projection all of the detector data had the same value, and in this case each detector has a different value. It is these unique measurements from each view which come together to enable us to reconstruct a CT image.

The backprojection operation is essentially trying to undo the forward projection operation. Since the forward projection operation mapped from the image into the detector space the backprojection operation maps from the detector back to the image.

What does the data look like in CT and why is it called a sinogram?

In the demonstration above we talked about the mathematical operation of forward projection through the object. When we turn on our x-ray tube and acquire one view on the detector the x-ray system performs the forward projection for us.

For each view you get one projection which corresponds to one line or one row in the sinogram. In the sinogram the x direction usually is the detector channel direction and the y direction is the view direction.

As in this case the sinogram is typically displayed for one detector row. In this toy example we assume that we just have one detector row in our parallel beam system, so this is all the projection data.

As you look at the sinogram you can see that there are several waves or sinusoids which overlap one another in the sinogram. This is where the name sinogram comes from. In fact for each point in the image space there is a well defined cosine wave that it will trace out.

The next step that we want to discuss is the conversion from the sinogram to the image. That process is what we call image reconstruction.

Why not just use Back Projection for Image Reconstruction?

Now we want to make an image from the sinogram. We know that the sinogram is generated from essentially a forward projection through the object (by the x-rays themselves).

The obvious first step would be to use the backprojection operation to make the image since we know that is doing a reverse type operation to the forward projection. So lets see what happens if we use just backprojection to make an image.

As we showed above the backprojection operation will spread back the information into the image for each view. One analogy you could use is that the backprojection operation is like painting the image information back from each view, one view at a time.

So if we perform the backprojection operation for just one view the image will look like a smeared painting where the artist could only pull the brush across one time. As we discussed above each backprojection provides information for that view, but only for that view. That is why the image looks so blurred with information from just one projection.

The power of computed tomography is that we can combine the information from each of the views in order to reconstruct an image.

Watch the short video a few times to see this information populated.