1. Introduction

In synchrotron tomography, ring artefacts are caused by imperfect detector pixel elements as well as by defects or impurities on the scintillator crystals. They appear on synchrotron tomographic images as a number of dark concentric rings superimposed on the image structures. As the grey levels in the reconstructed images are influenced by these ring artefacts, a quantitative analysis of the measured data is difficult. Post processing such as noise reduction, binarization, or segmentation of image information is significantly complicated by the presence of such artefacts.

A common method to reduce ring artefacts is known as flatfield correction [1]. Thereby images of the background without sample before and after the data acquisition and in certain intervals during the data acquisition are measured.

The resulting flatfields include the non-uniform sensitivity of the CCD camera pixels, the non-uniform response of the scintillator screen, as well as the inhomogeneities in the incident X-ray beam. The measurement data are corrected by dividing the normalized radiographic images of the samples and the flatfields (I_corr=I_measurement/I_flatfield). However, ring artefacts are not removed completely by this method when different camera elements have intensity dependent, nonlinear response functions or the incident beam has time dependent non-uniformities.

 

Fig. 1. 2D slice of a tomographic image: The cross section of a superconductor strand (binary Powder-in-Tube strand of Nb3Sn and copper) superimposed by strong ring artefacts.

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Additionally, the effect of the non-uniform sensitivity of different detector pixel elements can be avoided by moving the sample or the detector system during the acquisition in defined horizontal and vertical steps [2, 3]. The accurately defined shifts are counted back after the measurement. Thus the characteristic of all detector elements is averaged, which leads to significantly reduced ring artefacts. However, this requires very precise translation motor components, especially for measurements with spatial resolutions of about 1µm or less. This is absolutely nessecary to avoid motion artefacts.

A third way to compensate ring artefacts is the post-processing of the radiographic or tomographic images. They can be reduced by different image processing algorithms, e.g. size and shape filtering after the reconstruction or sinogram processing during the reconstruction [4, 5, 6].

In this paper, a simple but efficient method is presented to eliminate ring artefacts by sinogram processing during the reconstruction. As an example for this artefact a reconstructed slice of a tomographic image superimposed by strong ring artefacts is shown in Fig. 1. The image represents a cross section of a multi-filament wire consisting of superconducting Nb3Sn filaments covered by a copper shield [7, 8]. The diameter of this wire is about 0.7mm. The reconstruction algorithm, that we used here, is the Filtered Back Projection (see e.g. [1]). It is implemented in the reconstruction program PyHST [9] which was developed at the ESRF, Grenoble.

The image in Fig. 1 contains strong concentric rings that stretch across the whole image. As described the information of such artefacts is stored in all radiographic projections that are captured during the measurement. One line in a radiographic image represents the projection of a slice of the specimen under a certain angle. Arranged among each other the projections of all angles between 0 and 180 degrees produce the so called sinogram which contains the information to reconstruct one tomographic slice.

2. Filtering

In the case of ring artefacts a sinogram shows dark, thin, vertical lines (see Fig. 2) which correspond to the ring shaped structures in the reconstructed tomographic image. The vertical lines in the sinogram are more easy to detect than the ring shaped structures in the tomographic slice. Hence, it is obvious to filter out the rings by modifying the grey values of these lines in the sinogram in such a way, that they disappear in their environment. If this is achieved, the rings in the tomographic slice will disappear equally. The use of common filtering methods, like unsharp masking and plain low pass filtering of the sinogram, do not show useable results [6]. Besides, the filtering parameters of these methods require a lot of adjustment for each ring and over all slices. Due to the automation of the artefact reduction during the reconstruction these filtering methods are out of the question.

 

Fig. 2. The background of the image shows a sinogram, which consists of all measured projections p_n(i) of a slice for angles n between 0 and 180 degrees that are arranged among each other. Here, the sinogram reveals vertical lines that are responsible for ring artefacts, the inset in the top right corner emphasises this. The diagram shows the sum y(i) of all grey values for each pixel column in the sinogram and the fitted graph y_s(i) of this function.

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In the following, we propose an algorithm which demonstrates an automatic ring artefact compensation during the reconstruction of tomographic images considering as applied example. At first the sum y(i) of all grey values of each column i of a sinogram is calculated over each projection p_n.

The graph of these sums shows local minima (peaks) at the same positions that represent the ring artefacts (see red curve in Fig. 2). The next step is to fit a function with the same shape but without the peaks. To align the grey values at the peak positions with the information of their neighbourhood usually high frequency corrections are carried out [5, 10]. This is done by one-dimensional low-pass filtering in fourier space. Figure 3 shows the result of this correction with comparatively good success. Although the artefacts are widely removed the fourier filter does not delete all rings completely. Moreover, the fourier transformation can cause overshoots at jump discontinuities, known as the Gibbs effect [11], which can lead to new ring artefacts. Instead, we concentrate on developing a compensation algorithm in real space.

 

Fig. 3. Tomographic slice after the high frequency correction [5, 10] of the sinogram. In comparison to the original image the slice contains less rings. But there are still some ring artefacts left.

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Here, a more simple but efficient method with only one adjustable variable N is presented. With the help of the Moving Average Filter [11] it is possible to delete the peaks (see blue curve in Fig. 2). The applied filtering method replaces a value by averaging the grey values of its neighbourhood with a span width of 2N+1. The result from Eq. (2), whose shape depends on the span factor N, is the smoothed function y_s of y. The larger the value of N the more smooth the resulting function will look like.

To filter out the rings in the reconstructed image it is nessecary to find the best value for N. Choosing a small value for N leads to weak filtering and is therefore closer to an unfiltered reconstruction. Otherwise, a very large value results in a blurring of the whole tomographic image. A compromise should be used for an ideal filtering. A clue for that is the number of pixels of the peak width. With both functions, the sum y(i) and the fitted graph y_s(i) (see Fig. 2) we are now able to normalize each projection in the sinogram by the ratio of both.

The values of y and y_s differ from each other strongly in the area of the peaks, but not outside of them. Thus, these large grey value deviations are smoothed out whereas the rest of the projection is almost unchanged by this normalization.

3. Results and Conclusion

Ideally, only the grey values at the peak positions should be changed. The rest of them should be unmodified and the sinogram is freed from the vertical lines corresponding to the ring artefacts. Although, in most cases the filtering will affect the whole image, the effect of smoothing for image areas which are outside the ring artefacts is very weak. Only the grey value distribution around the peaks in the function y(i) is smoothed but not each projection in the sinogram itself. The reconstruction of those modified projections shows that the rings disappeared (Fig. 4), which enables the analysis of the intrinsic sample structures, e.g. the segmentation of the image information, without artefacts.

 

Fig. 4. Tomographic slice after using the new sinogram correction algorithm without ring artefacts. Here, the algorithm was used with the span factor N=20.

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The presented algorithm shows a successful way to compensate ring artefacts from tomographic images. The strength is adjustable by the span width of the Moving Average Filter. As a further advantage, the process could be implemented independently from the reconstruction method and is applicable in automation tasks. We implemented this compensation method in the reconstruction program PyHST as a plugin. It is now possible to filter ring artefacts in a tomographic image automatically during its reconstruction.