This text is initiated by the dialogue that arose after our last publication on tomography. The comment was reproached that rings were visible on the reconstructed image. Indeed, such ring distortions (ring artifacts) often arise in tomographic reconstructions around the center of rotation of the source-object-detector system. In this article we will talk about the reasons for the appearance of such rings and how we fight with them.
In tomographic installations, there is often a highlighted point relative to which something rotates: either the object mounted in the holder on the goniometer rotates, and the source and detector are stationary; or a source-detector system rotates around a selected point. These are two fundamentally different approaches to organizing the procedure for collecting tomographic projections; there are problems in both cases. So where do the ring-type artifacts come from in the reconstructed image and how to reduce their severity? The result of the reconstruction (horizontal section of a porous object with artifacts in the form of concentric circles) is shown in Fig. 1.
Fig. 1 Result of reconstruction without ring suppression 
In the method of X-ray tomography, a set of projections measured at different angles is used to restore the spatial distribution of the coefficient or the “effective” attenuation coefficient of the probe radiation. A tomographic projection is an image in each pixel of which contains the result of measuring the radiation intensity by one detector cell. For clarity, we will consider the reconstruction of not a whole object, but only one of its sections (see Fig. 1). For such a reconstruction, we do not need to use the registered projection of the image as a whole, but rather take the same line of the detector from each angular projection (see Fig. 2)
Fig. 2 Tomographic projection. The horizontal section involved in the construction of the synogram is highlighted in red
We will build a new image – a synogram, collecting the corresponding lines of all angular projections (Fig. 3). The i-th line of the received image corresponds to the i-th projection angle. Those. each column contains measurements of the same cell corresponding to different projection angles. A synogram of such an image is not called randomly. It is easy to see that in the central region it consists of sinusoids.
Fig. 3 Section of a synogram collected from lines of tomographic projections
In the image, especially on the bright left and right edges, where there is no shadow of the object, vertical stripes are visible. The presence of vertical stripes in the synogram is the reason for the appearance of concentric circles in the reconstructed image. There may be several reasons for the appearance of vertical stripes. The different response of the detector cells to the same incoming signal is one of them. Detector manufacturers are trying to compensate for this effect when the detector is launched on the market. Periodically updated, the so-called pixel map can compensate for the degradation that occurs during the life cycle of the device. Its creation is a costly procedure, since it requires a calibrated source. Those. either the user must have their own such source, or forced to contact companies providing such services. An alternative is to use vertical strip suppression algorithms. The second possible source of the appearance of bands on the synogram is the stitching of image areas. The fact is that the imaged object does not always fit entirely in the field of view of the detector. Mankind is inexorably moving towards increasing the spatial resolution of the tomography method. I would like to tomograph large objects, for example, a human head (a vertical size of several tens of centimeters), with a NANOMETER resolution. It is easy to calculate how many pixels a matrix must have in order to register the desired projection. Now they are trying to solve the problem by stitching together the registered sections of the parts of the object, shot with overlap. When stitching, similar artifacts occur. Another source of bands is the instability of the beam itself, i.e. change in beam intensity from projection to projection. Whatever the reason for the appearance of vertical stripes, during reconstruction they generate ring artifacts, which are usually removed by post-processing of the reconstructed images. We will fight with rings by filtering vertical stripes.
Description of the algorithm and experiments
Since the reconstruction does not receive the result from the detector, but is normalized to an empty beam and the logarithmized image (Fig. 4), it is this algorithm that is input to the algorithm described below.
Fig. 4 The result of the logarithm normalized to an empty bundle of the synogram
In the method of suppressing vertical stripes, the Guided Filtering algorithm (Fig. 5 ) is used as the main tool.
Fig. 5 Schematic diagram of filtration 
Guided Filtering is based on master and slave images. We want to build a leading image on which the sinusoids will be shown well, and the severity of the vertical stripes will weaken. The first step is to calculate the derivative in the horizontal direction (Fig. 6), i.e. in the direction perpendicular to the direction of the stripes.
Fig. 6 Horizontal derivative of the prologarithmic synogram
In the enlarged portion of the image (Fig. 7), the noise caused by the instability of the beam during the measurement appears as discontinuities in the vertical bands.
Fig. 7 Enlarged section of the image fig. 6
For each column, we perform the one-dimensional convolution operation to reduce the contribution of the high-frequency noise component (Fig. 8).
Fig. 8 Convolution Result
The enlarged section is shown in Fig. 9.
Fig. 9 Enlarged image area
However, we are still on the path to building the lead image. We apply the cumulative summation operation line by line to the image presented in Fig. 8. The result led us out of the space of derivatives, preserving the low-frequency dependences (Fig. 10).
Fig. 10 The result of the line-by-line application of the cumulative summation operation
Subtract the result from the logarithmic synogram, completing the process of constructing the leading image (Fig. 11). It remains for us to perform the filtering operation (Fig. 5).
Fig. 11 Leading Image
The result of the operation with the window (9.1) and E = 0.00001 is shown in Fig. 12.
Fig. 12 Result of a performed filtering operation
Fig. 13 Difference between input image and filter result
In fig. Figure 14 shows the results of tomographic reconstruction using unfiltered (left) and filtered (right) projections.
Fig. 13 Results of tomographic reconstruction
We described the suppression algorithm on the synograms of vertical bands, the presence of which leads to the appearance of concentric circles in the reconstructed images. Everyone who has to work with tomographic images knows about this pain. An analysis of the dynamics of the severity of the circles in the reconstructed image helps to choose the optimal values of the algorithm parameters. In conclusion, we want to note that the algorithm will be useful to everyone who suffers from the presence of stripes on the images. The direction of the bands does not play a decisive role, because it is enough to rotate the image and apply our algorithm to obtain an image with a significantly reduced degree of expression of the bands. Thank you for the attention.