Today we’d like to return to the subject of tomography which is one of our main focuses here, at Smart Engines. And we’ll be talking about the algorithm for lines suppression in a tomographic image. The lines in a tomographic sinogram would be of no importance since the doctors and other tomograph users don’t have to read the sinogram images, but these lines cause concentric circles in the reconstructed images (see the left Figure). The main tool used for line reduction in this algorithm is the Guided Filtering method. We’ll describe the process of creating the master image for a sinogram, estimating the corrected sinogram and using it for the tomographic reconstruction to produce the reconstructed image without the circular artifacts (see the right Figure).
This article is a response to the dialogue sparked on a tomography-related subject. Someone in the comment section reproachfully mentioned that there were circles visible in the reconstructed image. In fact, such circular distortions (circular artifacts) are a normal occurrence on the tomographic reconstructions around the center of rotation of the “source-object-detector” system. In this article we’ll explain why it happens and how we can fight it.
As a rule, there’s a certain set point in the tomographic settings around which something is rotated: either the object fixed in the holder in the goniometer rotates, and the source and the detector are stationary; or the “source-detector” system is rotating around the selected point. They are two fundamentally different approaches to organizing the procedure of tomographic projection data collection. And both approaches have their own disadvantages. So, the question is what causes the circle-like artifacts in the reconstructed images and how we can reduce their appearance. The reconstruction result is shown in Figure 1 (the horizontal cross-section of a porous object with the artifacts in the form of concentric circles).
Figure 1. The reconstruction result without applying the circle suppression procedure 
The x-ray tomography method uses the set of projections calculated at different angles to recover the spatial distribution coefficient or the “effective” probing radiation attenuation coefficient. The tomographic projection is an image where each pixel contains the measurement result of the radiation intensity by a single detector unit. For presentation purposes, we won’t be reviewing a reconstruction of an entire object, but of just one of its cross-sections (see Figure 1). In order to perform this reconstruction, we don’t have to use an entire registered projection, we’ll only need to review the same detector line in each angle projection (see Figure 2).
Figure 2. Tomographic projection. The horizontal cross-section that helps to create the sinogram is highlighted in red.
Let’s create a new image (a sinogram) by collecting all the correct lines of all the angle projections (Figure 3). An i-th line of the produced image matches an i-th projection angle. This means that each column contains the measurements of the same unit for the different projection angles. This kind of image is called a sinogram for a reason. It’s easy to see that it consists of sine curves in its central region.
Figure 3. The part of the sinogram made from the lines of the tomographic projections.
There are some vertical lines in the image which are particularly visible on the brighter left and right edges where there is no shadow of the object. The presence of vertical lines in the sinogram is what causes concentric circles in the reconstructed image. These vertical lines might appear due to the number of reasons. One of them is a different reaction of the detector units to the same incoming signal. The detector manufacturers are trying to compensate for this effect at the time when the detector is placed on the market. To compensate for the degradation that happens during the device life-cycle, the so-called pixel map which periodically updates can be used. It proves to be quite costly to create one though since it requires a calibration source. This means that either the user needs to have their own calibration source, or they have to turn to the companies that provide calibration services. Another option would be to use the vertical line suppression algorithms. The second possible reason for the appearance of lines in a sinogram is bridging of different parts of an image. The thing is that the object being processed by the tomograph doesn’t always fit in the detector’s field of view in its entirety. The science world is inexorably moving towards a higher spatial resolution in tomography. There is a need to tomographically process larger objects, such as the human head (its size vertically is a few dozens centimeters) with the NANOSCALE spatial resolution. It’s easy to calculate how many pixels the matrix is supposed to have in order to register the necessary projection. Right now this problem is solved by bridging the registered areas of the parts of the object which can overlay. In the process of putting them together the similar artifacts appear. One more possible reason for the appearance of the lines in the image is the instability of the beam itself, i.e. changes in the beam intensity in different projections. Regardless of the reason for the vertical line occurrence, when reconstructed they cause the circular artifacts that can be removed during post-processing of reconstructed images. We’ll be getting rid of these circles by a means of filtering the vertical lines.
Since the input image for the reconstruction is not the image produced by the detector, but it’s an image that has been normalized to a hollow beam and logarithmized (Figure 4), then it would be the input image for the algorithm described below as well.
Figure 4. The result of logarithmization of the sinogram that has been normalized to a hollow beam
The Guided Filtering algorithm is used as a main tool in the vertical line suppression method.
Figure 5. The basic scheme of filtering 
The concept of the master image and the replica image is a foundation of the Guided Filtering method. We need to create a replica image where the sine curves would be clearly visible and the vertical lines become less visible. Our first step would be to calculate the horizontal derivative (Figure 6), i.e. the derivative in the direction perpendicular to the direction of the lines.
Figure 6. The horizontal derivative of the sinogram that has been logarithmized.
If we zoom in on the section of our image, we would be able to see the noise which is presented as discontinuities in the vertical lines. This noise is caused by the beam instability during the measurement process.
Figure 7. The enlarged section of the image from Figure 6.
Let’s perform an operation of one-dimensional convolution for each column in order to decrease the contribution of the high-frequency noise element (Figure 8).
Figure 8. The result of the convolution.
The enlarged part of the image is shown in Figure 9.
Figure 9. The enlarged part of the image.
We are still working on building the master image though. Let’s apply the operation of cumulative summation in the line-by-line fashion to the image demonstrated in Figure 8. The result has taken us away from the derivatives space while maintaining the low-frequency noises (Figure 10).
Figure 10. The result of line-by-line application of the cumulative summation operation.
Now let’s subtract the produced result from the logarithmized sinogram. And that will conclude the process of building the master image (Figure 11). We just need to perform the filtering operation now.
Figure 11. The master image
The result of the operation performed with a window (9,1) and E=0.00001 is shown in Figure 12.
Figure 12. The result of the filtering operation
Figure 13. The difference between the input image and the result of the filtering operation
The results of the tomographic reconstruction using unfiltered ( left) and filtered (right) projections are shown in Figure 14.
Figure 14. The tomographic reconstruction results
We described the algorithm for suppressing the vertical lines in sinograms. The problem is that these lines cause the appearance of concentric circles in the reconstructed images. Anyone who has to work with tomographic images knows about this nuisance. The analysis of trends in the intensity of circles in the reconstructed image helps us to set the optimal values for the algorithm parameters. To conclude our article, we’d like to note that this algorithm will be an asset for anyone who deals with the appearance of lines in an image.The direction of the lines doesn’t play a critical role. In order to generate an image with significantly less visible lines, all you need to do is to rotate an image and apply our algorithm. Thank you for your time!
1. Е.Е. Берловская, А.В. Бузмаков, А.С. Ингачева и др. Алгоритм подавления ортотропных артефактов регистрации изображений в рентгеновском и терагерцовом диапазонах. // Информационные процессы. 2019, т.19, ном. 2, стр. 1-9.
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