Crystal Analyzer Based Multispectral Microtomography Using CCD-Sensor

To solve the problems of spectral tomography, an X-ray optical scheme was proposed, using a crystal analyzer in Laue geometry between the sample and the detector, which allowed for the selection of predetermined pairs of wavelengths from the incident polychromatic radiation to obtain projection images. On a laboratory X-ray microtomography setup, an experiment was carried out for the first time where a mixture of micro-granules of sodium chloride NaCl, silver behenate AgC22H43O2, and lithium niobate LiNbO3 was used as a test sample to identify their spatial arrangement. The elements were chosen based on the presence of absorption edges in two of the elements in the energy range of the polychromatic spectrum of the probing radiation. The method of projection distortion correction was used to preprocess the obtained projections. To interpret the obtained reconstruction results, the segmentation method based on the analysis of joint histograms was used. This allowed us to identify each of the three substances. To compare the results obtained, additional “reference” tomographic measurements were performed: one in polychromatic and two in monochromatic (MoKα-, MoKβ-lines) modes. It took three times less time for the tomographic experiment with the crystal analyzer, while the reconstruction accuracy was comparable to that of the “reference” tomography.


Introduction
The method of computed tomography, which appeared in the early 20th century as a hardware non-destructive method of layer-by-layer visualization of the internal structure of objects, has not stopped developing for over a century [1], offering users more and more new hardware and software solutions, which, in addition to imaging, allow the identification of areas of the internal structure of tomographic objects [2,3]. Since the discovery of X-rays, it has become clear that the contrast of X-ray images of an object depends on its elemental composition and density [4]. Materials with similar characteristics, such as soft tissues of the human body, are weakly distinguishable from each other in X-ray images when objects are probed with polychromatic radiation. To "illuminate" the desired areas, contrasting is used by introducing absorbing elements, such as iodine, xenon, strontium, etc., into the corresponding tissues of the object [5]. By examining the difference in X-ray images before and after contrasting, it is possible to understand the structure of the organ of interest. However, to do this, it is necessary to take two images separated in projection images recorded on the detector are distorted due to the deviation of the projection direction from the incident radiation direction as a result of diffraction on the crystal, and, therefore, data correction at the preprocessing stage of the tomographic reconstruction algorithm is required [33].
This paper describes the first tomographic experiment using a crystal analyzer in the proposed optical scheme for nondestructive testing applications. To obtain spectral projections, the crystal analyzer was placed between the sample and the detector. A mixture of micro-granules of sodium chloride NaCl, silver behenate AgC 22 H 43 O 2 , and lithium niobate LiNbO 3 was used as a test sample. The potential integration of lithium niobate into silicon photonics [35,36] adds it to the lineup of promising materials today. As a reference experiment, classical tomographic measurements in polychromatic and monochromatic modes were performed, where a single crystal monochromator was used in the optical pathway between the source and the sample. The results of tomography in the polychromatic mode do not allow for a component-by-component separation, while tomography in two modesat the energy before the absorption edge of one of the components and after the absorption edge-allows for a component-by-component separation of the mixture under study. In the latter experiment, however, it takes approximately 3 times longer to measure. The method of correction of projection distortions arising in the scheme with the crystal analyzer, and the results of its application, are given in the section "Distortion correction algorithm". The following section presents the results of reconstruction according to the measurements with the crystal analyzer and single crystal monochromator. The segmentation method based on the analysis of joint histograms was used to interpret the obtained results. KES imaging for detecting micro-particles of silver behenate and lithium niobate in sodium chloride is described in detail in the same section. The results obtained are discussed in the Conclusion.

Description of Laboratory Set-Up
Measurements were performed on a laboratory homemade tomographic setup designed and developed in FSRC "Crystallography and photonics" RAS [37,38]. The setup consists of an X-ray source (conventional X-ray tube), multi-axis goniometer, and CCD camera. In subsequent experiments, an X-ray tube with a molybdenum anode and focus 2.0 × 14.0 mm (h × v) was used. Probing conditions: accelerating voltage-40 kV, current-40 mA. The size of the illuminated area was regulated by two mutually perpendicular slits and was 3 × 3 mm. A 330 µm thick silicon single crystal Si(111) was used as an analyzer. The use of a thicker crystal leads to a decrease in the intensities of the diffracted beams, as well as the appearance of a "double" image [39]. Initially, the crystal was mounted so that the plane (111) was perpendicular to the incident beam. Then, by rotating around the X, Y, and Z axes (Figure 1), the analyzer was adjusted to the maximum reflections K α and K β for the {111}-type crystallographic planes. The procedure of crystal analyzer alignment in Laue geometry is described in more detail in [34].
Immediately in front of the crystal on the goniometer, the sample in the study was mounted. Tomographic measurements were performed with an angular step of 1 degree in the rotation range from 0 to 200 degrees around the vertical Y axis. X-ray projection images obtained for the two characteristic lines, as well as those that passed through the "crystal-filter" system, were recorded on a two-dimensional Ximea xiRay11 CCD camera with a sensitive element (pixel) size of 9 µm. To prevent overexposure of the detector by an intense polychromatic beam, a 400 µm copper filter was used. Taking into account the low intensity of the K β -line, the exposure time of one frame was 120 s.
The use of a crystal analyzer made it possible to select two characteristic lines of the molybdenum anode-K α and K β -from the polychromatic radiation of the standard X-ray tube. As a result of one measurement, three radiographic projections (one polychromatic and two monochromatic) were recorded at once. Let us call this tomographic experiment "spectral" tomography. Immediately in front of the crystal on the goniometer, the sample in the study was mounted. Tomographic measurements were performed with an angular step of 1 degree in the rotation range from 0 to 200 degrees around the vertical Y axis. X-ray projection images obtained for the two characteristic lines, as well as those that passed through the "crystal-filter" system, were recorded on a two-dimensional Ximea xiRay11 CCD camera with a sensitive element (pixel) size of 9 μm. To prevent overexposure of the detector by an intense polychromatic beam, a 400 μm copper filter was used. Taking into account the low intensity of the Kβ-line, the exposure time of one frame was 120 s.
The use of a crystal analyzer made it possible to select two characteristic lines of the molybdenum anode-Kα and Kβ-from the polychromatic radiation of the standard X-ray tube. As a result of one measurement, three radiographic projections (one polychromatic and two monochromatic) were recorded at once. Let us call this tomographic experiment "spectral" tomography.
For comparison, a series of separate experiments were performed in the traditional scheme, when a single crystal monochromator Si(111) in Bragg geometry was placed in front of the sample to select Kα and Kβ. Furthermore, measurements were performed without the single crystal monochromator, i.e., in polychromatic radiation. The operating mode of the X-ray tube (emission spectrum) and the range of rotation angles of the studied sample were chosen the same as in the previous case. The exposure times were 3 s for polychromatic radiation, 60 s for the Kα-line case, and 120 s for the Kβ-line case. Let us call these three measurements "reference" tomography.

Description of the Sample
For the X-ray tube with a molybdenum anode, the Kα-and Kβ-lines were 17.479 and 19.608 keV, respectively. Only two chemical elements with K absorption edges are in this energy range: Zr (17.998 keV) and Nb (18.986 keV). Both metals were not investigated in their pure form in this work. As an alternative, micro-granules of lithium niobate crystal (LiNbO3), whose absorption edge is close to the absorption jump of pure niobium, were chosen.
To observe the expected effect of spectral tomography on the available equipment, the sample was prepared as follows. At the bottom of the capillary micro-granules of silver behenate (AgC22H43O2), lithium niobate (LiNbO3) was successively poured, and sodium chloride (NaCl) was poured last. At the boundaries of the layers, mixtures were formed mechanically. Since the color of the components is the same white, the layers are visually indistinguishable ( Figure 2). For comparison, a series of separate experiments were performed in the traditional scheme, when a single crystal monochromator Si(111) in Bragg geometry was placed in front of the sample to select K α and K β . Furthermore, measurements were performed without the single crystal monochromator, i.e., in polychromatic radiation. The operating mode of the X-ray tube (emission spectrum) and the range of rotation angles of the studied sample were chosen the same as in the previous case. The exposure times were 3 s for polychromatic radiation, 60 s for the K α -line case, and 120 s for the K β -line case. Let us call these three measurements "reference" tomography.

Description of the Sample
For the X-ray tube with a molybdenum anode, the K α -and K β -lines were 17.479 and 19.608 keV, respectively. Only two chemical elements with K absorption edges are in this energy range: Zr (17.998 keV) and Nb (18.986 keV). Both metals were not investigated in their pure form in this work. As an alternative, micro-granules of lithium niobate crystal (LiNbO 3 ), whose absorption edge is close to the absorption jump of pure niobium, were chosen.
To observe the expected effect of spectral tomography on the available equipment, the sample was prepared as follows. At the bottom of the capillary micro-granules of silver behenate (AgC 22 H 43 O 2 ), lithium niobate (LiNbO 3 ) was successively poured, and sodium chloride (NaCl) was poured last. At the boundaries of the layers, mixtures were formed mechanically. Since the color of the components is the same white, the layers are visually indistinguishable ( Figure 2).
It should be noted that the NaCl compound has no absorption edges in the polychromatic energy range of the probing spectrum, while for AgC 22 H 43 O 2 , the absorption edge is equal to the value of 25.514 keV of pure silver. The maximum particle size was no more than 100 µm. The mixture thus obtained was placed in a thin capillary with a diameter of 1.15 mm and a wall thickness of 10 µm (stated by the manufacturer). Figure 3 shows an example of projection at the simultaneous fixation of three X-ray beams that passed through the crystal analyzer-two deflected with energies of characteristic lines and passed through the crystal. Since the crystal analyzer was mounted behind the object, the contrast in each of the images is due to the attenuation of the corresponding energy lines by the object material.
To calculate the parameters in the method of correction of geometric distortions arising on the monochromatic projections using the measuring circuit with the crystal analyzer [33], a gold metal grid with a variable period and wall thickness of 20 µm was used as a test object ( Figure 4). It should be noted that the NaCl compound has no absorption edges in the polychromatic energy range of the probing spectrum, while for AgC22H43O2, the absorption edge is equal to the value of 25.514 keV of pure silver. The maximum particle size was no more than 100 μm. The mixture thus obtained was placed in a thin capillary with a diameter of 1.15 mm and a wall thickness of 10 μm (stated by the manufacturer). Figure 3 shows an example of projection at the simultaneous fixation of three X-ray beams that passed through the crystal analyzer-two deflected with energies of characteristic lines and passed through the crystal. Since the crystal analyzer was mounted behind the object, the contrast in each of the images is due to the attenuation of the corresponding energy lines by the object material. To calculate the parameters in the method of correction of geometric distortions arising on the monochromatic projections using the measuring circuit with the crystal analyzer [33], a gold metal grid with a variable period and wall thickness of 20 μm was used as a test object ( Figure 4).  It should be noted that the NaCl compound has no absorption edges in the polychromatic energy range of the probing spectrum, while for AgC22H43O2, the absorption edge is equal to the value of 25.514 keV of pure silver. The maximum particle size was no more than 100 μm. The mixture thus obtained was placed in a thin capillary with a diameter of 1.15 mm and a wall thickness of 10 μm (stated by the manufacturer). Figure 3 shows an example of projection at the simultaneous fixation of three X-ray beams that passed through the crystal analyzer-two deflected with energies of characteristic lines and passed through the crystal. Since the crystal analyzer was mounted behind the object, the contrast in each of the images is due to the attenuation of the corresponding energy lines by the object material. To calculate the parameters in the method of correction of geometric distortions arising on the monochromatic projections using the measuring circuit with the crystal analyzer [33], a gold metal grid with a variable period and wall thickness of 20 μm was used as a test object ( Figure 4).

Distortion Correction Algorithm
The projection formed by a polychromatic beam passing through the object was recorded without changing the direction of ray propagation and, therefore, without geometric distortions (Figure 4, bottom projection). Standard methods of tomographic image pro-

Distortion Correction Algorithm
The projection formed by a polychromatic beam passing through the object was recorded without changing the direction of ray propagation and, therefore, without geometric distortions ( Figure 4, bottom projection). Standard methods of tomographic image processing [40], including algorithms of tomographic reconstruction, can be applied to the set of such projections. The situation is different with the projections obtained for the Kβ-and Kα-lines ( Figure 4, top left and right, respectively); they are registered after the reflection on the crystal analyzer planes and, accordingly, are geometrically distorted. To perform a pixel-by-pixel comparison of the images of reconstructions obtained from three sets of projections, polychromatic and reflections, it is necessary to bring them into the same coordinate system. For this purpose, let us take the coordinate system of the polychromatic projection as the basic one, and for the projections from the reflections, we search for the parameters of the geometric transformation relative to the polychromatic projection. In our optical scheme, straight lines are transformed into straight lines without bending, so the distortion of images is described by the projective transformation [41]. The projective transformation T, in homogeneous coordinates, maps a point (x, y, w) to a point ( x , y , w ) and is given in the form: To convert from homogeneous coordinates to ordinary coordinates, we divide the first two coordinates by the third w: where the number w is the scale factor.
In the matrix of coefficients T, the parameter t 33 will be taken equal to 1 since the value of the parameter is taken out as a multiplier, which is reduced when passing from the homogeneous to the usual coordinates. Thus, 8 degrees of freedom (8 unknown parameters) remain in the matrix. It is possible to find them by considering four points, no three of which lie on the same line. Since each point has two coordinates, the system of linear equations built on 4 points will find 8 unknown parameters. For the system to be nonsingular, no three points must lie on the same line [41].
Consider the image of the polychromatic projection and the projection from the K β reflection. Since the problem is to find the transformation parameters at which the points of projections from reflections pass into the corresponding points of the undistorted polychromatic projection, it is sufficient to select 4 points on the images that pass into each other to find the distortion parameters. To simplify the search for such points, a test object was taken, a gold metal grid, which has characteristic points-in the same corners of the grid are the intersections of lines. An example of selected characteristic points at the corners of the grid is shown in Figure 5. The points that must cross into each other are highlighted with crosses of the same color (pink, orange, blue, green).
A system of linear equations of 8 equations with 8 unknowns is made from the selected points. To ensure that the system is not singular, the condition of not lying on the same line of three points must be satisfied. The system is solved using the Gaussian method (sequential elimination of unknowns). Solving the system gives a direct transformation, i.e., a transformation for the transition from a normal image to a distorted one. Applying the inverse transformation matrix to the projection images from the reflections gives the aligned images. To calculate the inverse transformation, the search for the inverse matrix is performed by the Gauss-Jordan elimination method [42].
The program "Spectral Tomo Marker" was written to find the parameters of projective transformation, the main window of which is shown in Figure 6.  Points on the projection from the Kβ reflex are points of a projectively distorted quadrilateral, for which it is necessary to find the distortion parameters.
A system of linear equations of 8 equations with 8 unknowns is made from the selected points. To ensure that the system is not singular, the condition of not lying on the same line of three points must be satisfied. The system is solved using the Gaussian method (sequential elimination of unknowns). Solving the system gives a direct transformation, i.e., a transformation for the transition from a normal image to a distorted one. Applying the inverse transformation matrix to the projection images from the reflections gives the aligned images. To calculate the inverse transformation, the search for the inverse matrix is performed by the Gauss-Jordan elimination method [42].
The program "Spectral Tomo Marker" was written to find the parameters of projective transformation, the main window of which is shown in Figure 6.  The program allows you to load a frame with a test sample, from which three areas corresponding to three projections are cut out (the top line of images in the program). Then, the user selects 4 points on each of the three projections. The parameters of two projective transformations (for the projections from the characteristic lines) are calculated on the selected points and then applied to the areas of the original test frame with projections. To check the correctness of the proposed algorithm, the method of constructing a three-channel image is used, which allows visual determination of whether the proposed approach to geometric distortion compensation is correct (bottom line of images in the program). Then, the console version of the program is run, which processes all the frames of the sample experiment with the found parameters. After all registered projections are processed, threedimensional reconstruction of the object is performed by one of the classical algorithms of tomographic reconstruction.

Results and Discussion
After applying a distortion correction algorithm to the projections obtained in the crystal analyzer scheme (Figure 1), a reconstruction was performed (Figure 7) [43,44]. The crystal analyzer installed behind the sample highlighted three energy ranges. On the left, Figure 7 shows the reconstruction from the projections in the polychromatic beam transmitted through the crystal analyzer. The center is from the projections in the MoK α radiation highlighted by the crystal analyzer, and the right is the MoK β -radiation. Horizontal (Figure 7a) and vertical (Figure 7b) cross-sections of the reconstructed images after Gaussian filtering to remove noise are depicted.
three-channel image is used, which allows visual determination of whether the proposed approach to geometric distortion compensation is correct (bottom line of images in the program). Then, the console version of the program is run, which processes all the frames of the sample experiment with the found parameters. After all registered projections are processed, three-dimensional reconstruction of the object is performed by one of the classical algorithms of tomographic reconstruction.

Results and Discussion
After applying a distortion correction algorithm to the projections obtained in the crystal analyzer scheme (Figure 1), a reconstruction was performed (Figure 7) [43,44]. The crystal analyzer installed behind the sample highlighted three energy ranges. On the left, Figure 7 shows the reconstruction from the projections in the polychromatic beam transmitted through the crystal analyzer. The center is from the projections in the MoKα radiation highlighted by the crystal analyzer, and the right is the MoKβ-radiation. Horizontal ( Figure 7a) and vertical (Figure 7b) cross-sections of the reconstructed images after Gaussian filtering to remove noise are depicted.    Figure 8 shows the results of tomographic reconstruction from the projections obtained in the tomographic scheme with a single crystal monochromator. Three tomographic measurements were performed. In the first, the sample was probed with a wide polychromatic spectrum. In the second, the sample was probed with MoK α energy radiation. In the third, MoK β -line radiation was used. The lines were cut by a single crystal monochromator placed between the source and the sample, i.e., in the second and third cases, monochromatic radiation fell on the sample. The images were reconstructed using three separate tomographic measurements: in the polychromatic spectrum and at two energies corresponding to the characteristic MoK α and MoK β -lines. tion. In the third, MoKβ-line radiation was used. The lines were cut by a single crystal monochromator placed between the source and the sample, i.e., in the second and third cases, monochromatic radiation fell on the sample. The images were reconstructed using three separate tomographic measurements: in the polychromatic spectrum and at two energies corresponding to the characteristic MoKα and MoKβ-lines. Figure 8 shows horizontal (a) and vertical (b) cross-sections of the reconstructed images of the sample. On the left is the reconstruction from the projections in the polychromatic beam (without the monochromator), in the center are the projections in the MoKα reflection of the monochromator, and on the right is the MoKβ reflection of the monochromator.  The cross sections (horizontal and vertical) of the reconstructions in Figures 7 and 8 obtained from the projections in the polychromatic mode do not provide information on the spatial distribution of the mixture components. However, analysis of the images obtained in different spectral ranges makes it possible to identify the location of lithium niobate because the niobium absorption edge lies between the MoKα and MoKβ energies.
Note that visually the results of reconstruction from the "spectral" and "reference" tomography projections are quite similar. However, the time spent on measurements in the scheme with the crystal analyzer is approximately three times less than the time spent on the "reference" tomography.
When probing with monochromatic radiation, the problem of qualitative interpretation of the results is trivial in the case when each voxel of the volume contains a single Note that visually the results of reconstruction from the "spectral" and "reference" tomography projections are quite similar. However, the time spent on measurements in the scheme with the crystal analyzer is approximately three times less than the time spent on the "reference" tomography.
When probing with monochromatic radiation, the problem of qualitative interpretation of the results is trivial in the case when each voxel of the volume contains a single material. The absorption curves for the materials are tabulated in a wide range of energies, and the value of the absorption coefficient can be related to the composition of the local volume. If, however, a mixture of several components whose density may vary is present in a local volume, the linear absorption coefficient of the mixture turns into a linear combination of components, and the approximation of the contributions of each component may be coarse since there is only one equation and several unknown coefficients describing the contribution of each component to the mixture. In such a case, tomography is performed several times at different wavelengths to obtain additional information about the composition [45].
When using a probing polychromatic spectrum, the "averaged" value of the absorption coefficient is more difficult to interpret [46]. First, the linear combination in terms of the material mixture components is supplemented by a linear combination in terms of the contributions of the energy lines of the probing spectrum. Due to the presence of the so-called "spectrum hardening effect" when passing through the object, these weight coefficients change their values. In this regard, the approximation of the final result becomes less accurate [47,48].
To interpret the obtained quantitative brightness values in voxels of the reconstructed volumes, a parallelepipedic region was chosen, measuring 199 × 80 × 80 (1,273,600 in total) voxels, lying completely inside the sample, without capturing the capillary walls and space around the sample. The mask corresponding to the voids inside the object was determined for the obtained volume. For this purpose, a threshold binarization was applied to the images reconstructed from polychromatic projections using the Otsu method [49]. The voxels of this mask were excluded from further processing and analysis but were not removed from the 3D images. Thus, the dimensions of the resulting images did not change.
Histograms of absorption coefficient distributions within selected regions of reconstructed volumes with distant voxel values belonging to the space between crystals are shown in Figure 9. The threshold cutoff does not allow us to distinguish any of the mixture components. Even the averaged absorption coefficients for silver behenate, lithium niobate, and sodium chloride cannot be determined from the histogram.
tion coefficient is more difficult to interpret [46]. First, the linear combination in terms of the material mixture components is supplemented by a linear combination in terms of the contributions of the energy lines of the probing spectrum. Due to the presence of the socalled "spectrum hardening effect" when passing through the object, these weight coefficients change their values. In this regard, the approximation of the final result becomes less accurate [47,48].
To interpret the obtained quantitative brightness values in voxels of the reconstructed volumes, a parallelepipedic region was chosen, measuring 199 × 80 × 80 (1,273,600 in total) voxels, lying completely inside the sample, without capturing the capillary walls and space around the sample. The mask corresponding to the voids inside the object was determined for the obtained volume. For this purpose, a threshold binarization was applied to the images reconstructed from polychromatic projections using the Otsu method [49]. The voxels of this mask were excluded from further processing and analysis but were not removed from the 3D images. Thus, the dimensions of the resulting images did not change.
Histograms of absorption coefficient distributions within selected regions of reconstructed volumes with distant voxel values belonging to the space between crystals are shown in Figure 9. The threshold cutoff does not allow us to distinguish any of the mixture components. Even the averaged absorption coefficients for silver behenate, lithium niobate, and sodium chloride cannot be determined from the histogram. However, the histograms (Figure 9, right column) clearly show that the distributions of reduced absorption coefficients in polychromatic spectra in the cases of "reference" tomography and tomography with the crystal analyzer are different, despite the same X-ray source used. The difference in spectra is explained by the following. In the "reference" However, the histograms (Figure 9, right column) clearly show that the distributions of reduced absorption coefficients in polychromatic spectra in the cases of "reference" tomography and tomography with the crystal analyzer are different, despite the same X-ray source used. The difference in spectra is explained by the following. In the "reference" tomography experiment, a spectrum attenuated by the sample was recorded. In the experiment with the crystal analyzer, on the other hand, the registered spectrum does not contain the brightly expressed (reflected by the crystal) MoK α and MoK β spectral lines, parts of the spectrum absorbed by the silicon crystal analyzer, 330 µm thick, or the copper filter, 400 µm thick. The filter was installed in front of the detector window to realize the simultaneous registration of transmitted X-rays, which were deflected by the crystal analyzer. Thus, the polychromatic beam in the "reference" scheme contained relatively more long-wavelength radiation, which resulted in a higher effective value of the reconstructed attenuation coefficient.
To interpret the obtained reconstruction results, the joint histograms of the absorption coefficient values distribution were plotted ( Figure 10). The histograms for the "reference" tomography case are shown in Figure 10a, and the crystal analyzer case in Figure 10b. Since to measure the projections of the "reference" tomography (when measuring the projections in monochromatic modes) the sample had to be removed during the adjustment of the MoK α and MoK β reflections of the crystals, the reconstructed 3D images of the object appeared displaced relative to each other. Their alignment, which is necessary to obtain joint histograms, was performed manually.
Three regions corresponding to AgC 22 H 43 O 2 , LiNbO 3 , and NaCl are already observed in the joint histograms. To identify the boundaries of these areas and classify the voxels according to their belonging to a particular substance, a clustering procedure was carried out using the Variational Bayesian estimation of a Gaussian mixture algorithm implemented in the Scikit-learn software library [50]. An illustration of the clustering results is shown in Figure 11, where each dot corresponds to one voxel of the sample volume, and the color of the dot identifies the substance to which the voxel belongs. To improve readability, the graphs show only 10,000 randomly selected points out of 1,273,600. periment with the crystal analyzer, on the other hand, the registered spectrum does not contain the brightly expressed (reflected by the crystal) MoKα and MoKβ spectral lines, parts of the spectrum absorbed by the silicon crystal analyzer, 330 μm thick, or the copper filter, 400 μm thick. The filter was installed in front of the detector window to realize the simultaneous registration of transmitted X-rays, which were deflected by the crystal analyzer. Thus, the polychromatic beam in the "reference" scheme contained relatively more long-wavelength radiation, which resulted in a higher effective value of the reconstructed attenuation coefficient.
To interpret the obtained reconstruction results, the joint histograms of the absorption coefficient values distribution were plotted ( Figure 10). The histograms for the "reference" tomography case are shown in Figure 10a, and the crystal analyzer case in Figure  10b. Since to measure the projections of the "reference" tomography (when measuring the projections in monochromatic modes) the sample had to be removed during the adjustment of the MoKα and MoKβ reflections of the crystals, the reconstructed 3D images of the object appeared displaced relative to each other. Their alignment, which is necessary to obtain joint histograms, was performed manually. Three regions corresponding to AgC22H43O2, LiNbO3, and NaCl are already observed in the joint histograms. To identify the boundaries of these areas and classify the voxels according to their belonging to a particular substance, a clustering procedure was carried out using the Variational Bayesian estimation of a Gaussian mixture algorithm implemented in the Scikit-learn software library [50]. An illustration of the clustering results is shown in Figure 11, where each dot corresponds to one voxel of the sample volume, and the color of the dot identifies the substance to which the voxel belongs. To improve readability, the graphs show only 10,000 randomly selected points out of 1,273,600. By analyzing the reconstruction results for the two performed tomographic measurements (with the crystal analyzer and "reference" tomography), it can be concluded that the obtained estimates of the mean values for NaCl and AgC22H43O2 are within the measurement error, and the estimate for LiNbO3 is underestimated. This can be explained by the fact that the size of some of the granules used in the mixture was smaller than the By analyzing the reconstruction results for the two performed tomographic measurements (with the crystal analyzer and "reference" tomography), it can be concluded that the obtained estimates of the mean values for NaCl and AgC 22 H 43 O 2 are within the measurement error, and the estimate for LiNbO 3 is underestimated. This can be explained by the fact that the size of some of the granules used in the mixture was smaller than the spatial resolution of the method. The local volume could simultaneously contain sodium chloride and lithium niobate, silver behenate and lithium niobate, or the niobate crystal occupied a smaller volume than the local voxel volume. Using the results of clustering, we can transfer the information about which voxel corresponds to which substance to the three-dimensional image of the object under study. Figures 12 and 13 show horizontal and vertical slices of the object with the initial images obtained in the MoK α -, MoK β -lines and in polychromatic radiation, and the corresponding map of substances distribution in the sample for the experiment with the crystal analyzer. The maps look similar to the images obtained for the "reference" tomography. One can see that the silver behenate is concentrated at the bottom of the object, there is an area containing lithium niobate above it, and the upper half is filled with sodium chloride, which corresponds to the filling of the capillary with the materials under study (see Section 2.2).

Conclusions
This work is the first experimental demonstration of the capabilities of spectral tomography with a crystal analyzer. The use of the crystal analyzer allowed for the separation of two characteristic lines of the molybdenum anode, specifically and , from the mixed radiation emitted by a standard X-ray tube. This resulted in the recording of three distinct X-ray projections or images.
The tomography does not use an energy-dispersive detector, which allows the collection of projections using a conventional CCD. The sample does not receive the additional radiation load that occurs with dual-energy or dual-source tomography technology.
Three additional "reference" tomographic measurements, one in polychromatic and two in monochromatic (MoKα-, MoKβ-lines) modes, were performed to compare the results obtained. It took three times less time to achieve reconstruction accuracy in the crystal analyzer tomography experiment comparable to that of the "reference" tomography. In such measurements, most of the operations fall on the software part of the tomographic method.
The test object was a capillary with silver behenate (AgC22H43O2), lithium niobate (LiNbO3), and sodium chloride (NaCl). After the correction of geometric distortions of the projections collected in the crystal analyzer circuit, tomographic reconstruction was performed. Joint histograms were plotted for the selected areas of the reconstructed 3D digital images. Analysis of the joint histograms allowed segmentation of the reconstructed images, which allowed identification of each of the three regions. Figure 13. Images of two mutually perpendicular vertical slices of the object obtained in the experiment with the crystal analyzer (first column-polychromatic radiation, second-MoK α , third-MoK β ) and the corresponding distribution map of substances (last column): green-NaCl, red-LiNbO 3 , blue-AgC 22 H 43 O 2 . Row I is the 50th layer on the x-axis, II is the 50th layer on the y-axis.

Conclusions
This work is the first experimental demonstration of the capabilities of spectral tomography with a crystal analyzer. The use of the crystal analyzer allowed for the separation of two characteristic lines of the molybdenum anode, specifically K α and K β , from the mixed radiation emitted by a standard X-ray tube. This resulted in the recording of three distinct X-ray projections or images.
The tomography does not use an energy-dispersive detector, which allows the collection of projections using a conventional CCD. The sample does not receive the additional radiation load that occurs with dual-energy or dual-source tomography technology.
Three additional "reference" tomographic measurements, one in polychromatic and two in monochromatic (MoK α -, MoK β -lines) modes, were performed to compare the results obtained. It took three times less time to achieve reconstruction accuracy in the crystal analyzer tomography experiment comparable to that of the "reference" tomography. In such measurements, most of the operations fall on the software part of the tomographic method.
The test object was a capillary with silver behenate (AgC 22 H 43 O 2 ), lithium niobate (LiNbO 3 ), and sodium chloride (NaCl). After the correction of geometric distortions of the projections collected in the crystal analyzer circuit, tomographic reconstruction was performed. Joint histograms were plotted for the selected areas of the reconstructed 3D digital images. Analysis of the joint histograms allowed segmentation of the reconstructed images, which allowed identification of each of the three regions. Funding: This work was performed within the State Assignment of FSRC "Crystallography and Photonics" RAS in part of X-ray studies and State Task No 075-01304-23-00 in part of tomographic data processing. Work on the optimization of reconstruction algorithms is supported by the Russian Science Foundation (Project number 23-21-00524).

Informed Consent Statement: Not applicable.
Data Availability Statement: The data presented in this study are available on request from the corresponding author. The data are not publicly available because the research results are part of a dissertation work that has not been defended. After the defense of the scientific qualification work, the data can be published in the public domain.

Conflicts of Interest:
The authors declare no conflict of interest.