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.
In this post we will go over the details of the projection and backprojection operations first, and then build upon those concepts to discuss the FBP algorithm.
We have a complete youtube video here if you would prefer the video format which is right here. Otherwise if you prefer the detailed written post please check the table of contents below.
What is Forward Projection in image reconstruction?
A small, ‘toy example’, image matrix will be used to demonstrate the concepts of forward and back projection. In the Figure below you can see a 3×3 image matrix where each pixel has a different intensity value. This is just like a CT image except we typically have at least 512×512 pixels in a CT image.
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.
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What is Back Projection in image reconstruction?
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.
In the figure below we can see the same toy example of a 3×3 matrix. In this case we start with values in the detector and the backprojection operation paints the values back into the image matrix along the direction of measurement.
In this case we assume parallel beam projections and thus the backprojection operation simply spreads the detector value back along the directions that the rays were measured in the first forward projection figure above.
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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.
The x-rays come out from the tube and travel in straight lines through the patient (ignoring those that scatter for now). The x-rays are then measured on the detector for each view around the patient.
As we described above there are multiple geometries for CT scanning. The simplest geometry is the first generation CT scanner where the projections are parallel to one another (as achieved by translation of the source and detector). The data looks similar to the other CT generations but is slightly different as each generation has a different geometry as we described in our CT Generations post.
In this schematic my bitmoji is the simulated patient.
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 (i.e. indicating the gantry rotation angle).
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. 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.
A couple more interesting notes for the sinogram are that it can be good in figuring out where some image artifacts arise. For instance if there is a bad detector cell in the sinogram it will show up as a vertical line, since the same detector is out for all views.
In a similar manner, if you look at the sinogram and one row is completely different than the others it tells you that something happened within that view such as a tube arc (i.e. tube spit).
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.
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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.
In this image the backprojection occured from the top of the image and the information is spread down the image. The power of computed tomography is that we can combine the information from each of the views in order to reconstruct an image.
As we add more views to the backprojection image you can see that the additional information fills into the image. For each view we populate the details that were captured along those ray paths. Watch the short video a few times to see this information populated.
After we have done the backprojection for all view angles we end up with an image which looks a lot like the image we’re expecting except it looks blurred.
This blurring is present because the backprojection operation is not the exact inverse of the forward projection operation, so we need once more step.
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Why use Filtered Back Projection (FBP) ?
We know that if we simply use backprojection that the images will be too blurred so we need a method to overcome this. There are several methods that we could envision using to remove the blurring from the images. In this table we are still considering the problem of parallel beam image reconstruction from a set of projections.
This list of options all are what we call mathematically exact, so we know that the deblurring operation (filtering) will provide a correct image. This is different than ‘photoshop’ type filters where there is not a strong mathematical basis for the processing.
This comparison table will likely not be covered on the AART exam but since I worked on this area for five years of my life so I wanted to add at least a little table here. If you are looking for some night-time reading to help you get to sleep feel free to checkout this more detailed summary we have on recent advances in image reconstruction.
Backproject first then Image Filter (De-blur)
Need to deblur in 2D sections at a time.
Slow since the filter is 2D.
Requires a reconstruction of the entire object, i.e. no targeted reconstruction.
Filter (De-blur projections) and the then backproject (FBP)
1D filter in the projection space.
Supports targeted reconstruction.
Smooth weights can help with moving objects.
Needs special attention needed for very large patients, that are bigger than the detector.
Filter in the projection space first and then more in image space.
1D filter in the projection space and 1D filter in the image space.
Handles large patients a little better.
More complicated filtering directions in the image.
More challenging to handle moving objects.
From this table you can see that there are multiple options to achieve direct and accurate image reconstruction in CT. The summary is that because performing filtering in the projections is fast and robust that Filtered BackProjection (FBP) won out as the standard image reconstruction algorithm for computed tomography (CT).
We know that the image is too blurry if we just use the backprojection operation to form an image, therefore the filter in the filtered backprojection algorithm should be a sharpening filter.
Here you can see the effect of performing the filter on a projection. We call this a one dimensional or 1D filter as it is performing a sharpening operation in the detector direction but is not filtering the data in the view direction.
Now you know the whole Filtered Backprojection (FBP) algorithm. First apply a fast 1D sharpening filter in the projection space, and then perform a backprojection operation to map the data back along the rays which over which it was acquired.
The filtered sinogram shown above can be back-projected from all views resulting in accurate image reconstruction of the object as shown in this quick video.
In this case you can see that after the first few views the image is very blurred in one direction but as more of the views are backprojected more of the image is recovered. Unlike the backprojection video above where the final image was blurred, now the final image is an accurate representation of the image object. This is the magic in the math behind the FBP algorithm.
This figure summarizes each of the steps which we have walked through above in the FBP algorithm. So we aren’t introducing any new steps here but just providing one more summary figure for your reference of the transformation from CT data (i.e. sinogram) to filtered sinogram and finally to the reconstructed CT image.
The filter which results in ‘exact’, ie. accurate reconstruction is sometimes called a ramp filter since it looks like a ramp or straight line when drawn in frequency space. As a technologist luckily you won’t need to changing the way that the image reconstruction algorithm is implemented on the scanner. Understanding FBP at this level should be sufficient for your needs.
A very important consideration for you and the radiologist as well is the reconstruction kernel. The reconstruction kernel is a filter that modifies the traditional ramp filter which provides accurate reconstruction.
The reconstruction kernel enables you to enhance details by performing more sharpening than the standard kernel, or provide more smoothing than the standard reconstruction kernel. When imaging bony anatomy it is advantageous to have a higher resolution kernel and when imaging soft tissue a high resolution kernel is not required. As higher resolution kernels are used there is an increase in the image noise.
This tradeoff between image resolution and image noise varies strongly based on the clinical task and since this is more involved we will have a separate dedicated post on this trade-off coming up.
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