Quantitative Determination of Partial Voxel Compositions with X-ray CT Image-Based Data-Constrained Modelling

Abstract

X-ray CT imaging is an important three-dimensional non-destructive testing technique, which has been widely applied in various fields. However, segmenting image voxels as discrete material compositions may lose information below the voxel size. In this study, six samples with known volume fractions of compositions were imaged using laboratory micro-CT. Optical microscopic images of the samples reveal numerous small particles of compositions smaller than the CT voxel size within the samples. By employing the equivalent energy method to determine the X-ray beam energy for sample imaging experiments, data-constrained modelling (DCM) was used to obtain the volume fractions of different compositions in the samples for each voxel. The results demonstrated that DCM effectively captured information about compositions occupying CT voxels partially. The computed volume fractions of compositions using DCM closely matched the known values. The results of DCM and four automatic threshold segmentation algorithms were compared and analyzed. The results showed that DCM has obvious advantages in processing those samples containing a large number of particles smaller than the CT voxel size. This work is the first quantitative evaluation of DCM for laboratory CT image processing, which provides a new idea for multi-scale structure characterization of materials based on laboratory CT.

1. Introduction

As an important non-destructive testing technique, X-ray computed tomography (CT) imaging has evolved beyond medical applications in the past two decades. Based on CT imaging, recent applications include virtual unwrapping of ancient scrolls to decipher text [1], reconstruction of ancient artifact structures [2] and biological morphology [3], characterization of polymer porosity in 3D-printed materials [4], and studying corrosion kinetics in metal additive manufacturing [5]. Advances in micro-CT and nano-CT technologies enabled imaging of three-dimensional material compositional distributions at voxel sizes ranging from micrometers to tens of nanometers, presenting new opportunities for quantitative characterization of the microscopic structures of materials. Particularly in the characterization of energy reservoir samples, CT imaging has been employed for in situ dynamic studies of methane hydrate formation and dissociation in sandstone [6], quantifying pore morphology and structure in oil and gas reservoir rocks [7,8,9], and establishing physical property numerical simulations such as fluid flow [10], thermal conduction [11], and diffusion [12].

X-rays attenuate as they pass through material samples due to interactions with the material, resulting in intensity attenuation recorded on the detector, forming a projection image. By imaging a material sample from various angles and employing sophisticated mathematical inversion algorithms, the cross-section images of the sample can be reconstructed. The intensity values of each pixel in the reconstructed cross-section images are proportional to the sum of the X-ray linear attenuation coefficients of the compositions contained in those pixels. The X-ray linear attenuation coefficient primarily depends on the X-ray beam energy, material density, and elemental composition. Establishing the correlation between the X-ray linear attenuation coefficients of the CT slice and the physical compositions of the material is crucial for image recognition and quantitative analysis. The image threshold segmentation algorithm is currently the mainstream method for processing CT images. It determines the image segmentation thresholds for different compositions based on the pixel intensity distribution of the CT image. This method assigns voxels to specific compositions based on predefined intensity thresholds, facilitating image recognition and quantitative analysis. However, practical CT applications often encounter reconstructed voxels containing multiple physical compositions smaller than the voxel size, leading to a weighted average of X-ray linear attenuation coefficients within each voxel. This is the so-called partial volume effect. When using image threshold segmentation algorithms for image recognition, the presence of voxels occupied partially by multiple compositions introduces biases regardless of the classification of these voxels into specific compositions. Studies by Coutsiers Morell et al. [13] and Mahesh et al. [14] have highlighted that the partial volume effect significantly affects the evaluation of tooth morphology volumes in dental CT imaging.

In recent years, various methods have been proposed to avoid the partial volume effect. Arabi et al. [15] proposed a dual-projection scanning method where slight detector movements were employed at each angle during scanning to collect two projection images. This method decreased the effective imaging pixel size and reduced the averaging effect of the partial volume effect on material composition densities. Jomaa et al. [16] developed a new partial volume effect correction method based on the imaging system’s point spread function. To address the issue of blurred edges of image compositions due to the partial volume effect, a smoothing-based approach [17] is proposed to improve edge detection accuracy. Machine learning and deep learning methods have also been applied by researchers for the precise segmentation of medical CT images [18,19,20]. Goo et al. [21] used partial voxel interpolation to reduce errors in cardiac CT ventricular volume measurements. Bull et al. [22] estimated the length of partial cracks in epoxy materials based on the ratio of grayscale values between pixels containing partial cracks to those of pure epoxy materials. Godinho et al. [23] integrated decision tree classification algorithms and particle histogram parameters to achieve the quantitative assessment of mineral compositions inside particles, considering the inter-phase partial volume effect. Some researchers have combined imaging techniques of different resolutions, such as nano-CT with micro-CT or electron tomography [24,25,26,27], to obtain multi-scale structural information of samples.

In many fields, it is often necessary to establish the three-dimensional digital structure of materials for numerical simulations of their physical and chemical properties. While X-ray CT allows non-destructive imaging of sample three-dimensional structures, the trade-off between imaging field of view and resolution remains challenging. High imaging resolution enables capturing detailed structural features of the samples, but limits the imaging field of view, restricting the statistical representativeness of numerical simulation results. For instance, in unconventional energy reservoir structure characterization, natural rock samples contain pores of different scales. Due to the influence of the partial volume effect and other factors, pore structure information will be lost when extracting rock pores, resulting in the deviation of permeability calculation. Establishing a three-dimensional physical field of the samples based on high-resolution results can lead to impractical data volumes. Jackson et al. used deep learning to enhance low-resolution images on a large-sample scale, creating multi-scale models capable of accurately simulating fluid dynamics experiments from pore (micrometer) to continuum (centimeter) scales [28]. Yang et al. [29] proposed a flexible ternary segmentation method to accurately express the contribution of partial pores to rock permeability.

Despite advancements in methods to mitigate the partial volume effect on CT image analysis and physical modelling, accurately obtaining quantitative information on partially occupied CT voxel compositions to characterize material multi-scale structures remains challenging. Yang et al. [30] proposed data-constrained modelling, which assumed that the theoretical X-ray linear attenuation coefficient of each voxel is the volume-weighted average of the X-ray linear attenuation coefficients of all compositions contained within it. By minimizing the difference between theoretical and reconstructed X-ray linear attenuation coefficients and maximizing the Boltzmann statistical distribution probability of the materials of all voxels, the volume fractions of different compositions within each voxel can be calculated. This method has been successfully applied in synchrotron X-ray CT data analysis. The excellent monochromaticity of synchrotron X-rays greatly facilitates determining the theoretical X-ray linear attenuation coefficients of sample CT voxels. Combining this method with synchrotron CT has achieved successful applications in some fields [31,32,33].

Despite achieving a certain degree of success in synchrotron CT image analysis, the practical application of DCM is limited due to the high cost of the beamline time of synchrotron CT. Laboratory micro-CT is more common, but it uses an X-ray tube as the light source, producing continuous spectrum (polychromatic) X-rays. Compared to the quasi-monochromatic X-rays used in synchrotron CT, the continuous spectrum X-rays pose challenges in establishing the correlation between material elemental compositions and CT slice X-ray linear attenuation coefficients, which is energy-dependent. This study is a quantitative investigation of six artificial samples with known overall compositions and volume fractions. They were imaged using laboratory micro-CT equipment, and employed DCM to calculate multi-scale composition distributions of these samples, quantitatively analyzing the accuracy of DCM calculation results.

2. Materials and Methods

2.1. Preparation of Samples

Six cylindrical samples with a diameter of approximately 8 mm were prepared for CT imaging in this study. The samples were fabricated by uniformly mixing varying masses of silica and iron powder. The powder of different qualities was packed into a plastic tube with an outer diameter of 10 mm and inner diameter of 8 mm and pressed tightly. The particle sizes of silica and iron powder were both 1 μm, with densities of 2.32 g/cm³ and 7.86 g/cm³, respectively. A pure aluminum cylinder with a diameter of 6 mm was adhered to the top of each sample for calibrating the X-ray CT imaging equivalent energy. Figure 1 shows the prepared pure silica sample, with other samples having a similar structure to this sample. The region c in the samples was designated for characterization, and the masses of its compositions are listed in

Table 1. Information about the samples.

Sample Number	Mass (g)
	SiO2	Fe
1	0.3192	0.0000
2	0.2844	0.0175
3	0.3067	0.0443
4	0.2572	0.0970
5	0.2654	0.2246
6	0.2170	0.3149

. The regions b and d in the samples were prepared with the same proportion of compositions as the region c to mitigate imaging artifacts due to X-ray diffraction or insufficient radiation at the top and bottom of the region c. Regions b and d were separated from region c using cardboard spacers. Region a is the pure aluminum standard sample. Region e consisted of vacuum putty used to secure the plastic tube to the CT imaging sample holder.

2.2. X-ray CT Imaging

The X-ray CT imaging was performed on nano-voxel 3000H X-ray micro-CT equipment manufactured by Tianjin Sanying Precision Instruments Co., Ltd. (Tianjin, China). During imaging, region c of the samples, as shown in Figure 1, was positioned at the center of the detector, with a part of the standard aluminum sample also within the field of view. The source-to-object distance (SOD) was 47.36 mm, the source-to-detector distance (SDD) was 816.86 mm, and the detector pixel size was 127 μm. The effective projection image pixel size was 7.363 μm. Prior to the experiment, 10 flat fields were collected to facilitate background subtraction before the reconstruction of CT slices. For samples 2–6, containing iron powder, the X-ray tube operated at 130 kV and 80 μA. Two projections were collected for every 0.25 degree of rotation of each sample, and the exposure time of each projection was 2 s. Each pair of two projections captured at the same projection angle were merged into one frame to reduce random noise. Each sample was rotated 360 degrees, and a total of 1440 projection frames were collected. A 0.5 mm thick Cu filter was placed directly at the X-ray exit to reduce beam hardening effects. For sample 1, composed of pure silica, an X-ray tube voltage of 80 kV and a 0.5 mm thick aluminum filter were used to ensure optimal imaging contrast. Three projections were collected for every 0.25 degree of rotation of sample 1 and merged into one frame to minimize the random noise. Other CT imaging parameters of sample 1 were consistent with samples 2–6. After acquiring the projections of the samples, X-ray CT slices with 32-bit floating-point format were reconstructed using VoxelStudio Recon software (version 2.5.1.25), which was supplied with the CT instrument. The pixel values of the reconstructed CT slices are the X-ray attenuation coefficients. Figure 2 is typical CT slices of the samples.

In Figure 2, the intensity value of the images is proportional to the X-ray linear absorption coefficient at each pixel. Imaging display parameters in Figure 2b–f are identical, including brightness, contrast, and dynamic range. Because the imaging parameters of sample 1, shown in Figure 2a, are different from those of other samples, the direct comparison of Figure 2a with others is not feasible. In Figure 2, the outermost ring-shaped low-absorption areas visible in the CT slices of all samples correspond to the polycarbonate tube walls. In Figure 2b–f, the average brightness of the sample CT slices increases as the iron powder content increases. Figure 2a shows that the intensity value distribution on the CT slice of the silica sample without iron is uneven, where a higher value indicates lower porosity and a lower value indicates higher porosity. This indicates varying densities across different regions of the sample due to uneven manual compression during sample preparation.

In order to avoid the influence of pixels outside the sample boundaries, six sub-regions of 800 × 800 pixels were selected from the CT slices in Figure 2 for statistical X-ray linear absorption coefficient histograms of each sample. As shown in Figure 3, the histograms exhibit clear single-peak characteristics for all six samples. For CT slices of samples 2–6 with identical imaging parameters, the peak height of the gray value distribution decreases, and the corresponding X-ray linear absorption coefficient value and the width of the peak increase with the increase in iron content in the samples. The decrease in peak height and increase in peak width indicate that the partial volume effect in the sample increases with the iron content. The histogram of sample 1 shows a wide peak, indicating that the pores and silica in the pixels in most areas of the CT slice exist in the form of co-occupying the pixel.

2.3. Sample Composition Content

The volume fractions of the compositions in region c of the samples were calculated for quantitative comparison with the DCM calculation results. A built-in sample boundary detection algorithm in the DCM software (version 2.6.3.2663) [34] was used to extract the regions corresponding to region c of the samples. Figure 4 shows the three-dimensional boundary of region c of sample 1. The quantities of CT voxels included within region c of each sample were statistically analyzed to determine the physical volume corresponding to each sample’s region c based on the sizes of individual CT voxels. Figure 4 shows that the top of sample 1’s region c has a ring-shaped protruding edge. This is due to the uneven force applied during the sample preparation process. To accurately determine the volume fractions of the compositions in the samples, these protruding edges were included in the total volumes of sample region c. Combined with the mass and density of each composition added to region c during preparation, the volume fractions of different compositions in each sample were calculated, as shown in

Table 2. The actual volume fraction of each composition of the samples.

Sample Number	Voxel Quantity of Region c	Volume Fractions of Each Composition (%)
		Pore	SiO2	Fe
1	671,853,771	48.70	51.30	0.00
2	593,476,116	47.32	51.74	0.94
3	606,861,481	43.10	54.57	2.33
4	565,165,194	45.38	49.15	5.47
5	698,846,217	48.75	41.01	10.24
6	593,903,650	43.65	39.45	16.90

.

2.4. Optical Microscope Analysis

In the process of the sample preparation, a certain amount of mixed composition powder was added and compacted, and the cross-section morphology of the sample was observed using the reflected light mode of a transreflectance-integrated polarizing microscope. The microscope was an XPF-550C manufactured by Shanghai Caikon Optical Instrument Co., Ltd. (Shanghai, China). For each sample, a total of 3 cross-sections were observed, and 3 pictures were collected from different positions of each section. A total of 9 pictures were collected for each sample. The microscope objective was 10×, and images were captured using a CK-500 camera, calibrated for magnification using a microscope standard scale. The images captured by the microscope were converted to 8-bit grayscale images using ImageJ software (version v1.51k). A median filter algorithm (radius was set to 2 pixels) was applied to eliminate salt-and-pepper noise. Phansalkar’s automatic local thresholding method [35] (radius was set to 50 pixels) was employed for threshold segmentation of the images, generating binary images of iron particles in the sample cross-sections. The area statistical analysis of iron content in the images was performed using the “Analyze Particles” function in ImageJ. Figure 5 illustrates the typical distribution of compositions captured by the microscope and the extracted iron particles from the cross-sections for samples 2–6.

From Figure 5, it can be observed that iron particles are distributed relatively evenly in the prepared samples. Numerous iron particles smaller than the voxel size of the CT slices exist in the sample cross-sections. These iron particles exist in the form of partial occupation of the CT voxels. The iron particle distribution in the 9 microscope images of each sample was statistically analyzed. The statistical results are presented in

Table 3. Numerical analysis of iron particle distribution in the microscope images. S₁ represents the area of iron particles smaller than the CT pixel size; S₂ represents the area of all iron particles; S₃ represents the area of the images.

Sample Number	S1/S3 (%)	S2/S3 (%)	S1/S2 (%)
2	1.247	1.412	88.314
3	2.328	2.674	87.061
4	3.182	3.491	91.149
5	6.166	7.220	85.402
6	8.013	9.620	83.295

. It can be seen from Table 3 that the observed areas of most iron particles are smaller than the CT pixel size of 7.363 × 7.363 μm². It should be noted that the iron particles in the samples have different geometric shapes; the optical microscope images only show one cross-section of the three-dimensional iron particles. In addition, limited optical microscope resolution and focusing issues may also result in particle sizes being larger than their actual sizes. Although the data in Table 3 include the statistical results of nine optical microscopy images taken from different cross-sections and positions of the samples, each optical microscopy image covers only 0.415% of the sample’s cross-sectional area. For non-uniform samples prepared manually, the lack of statistical representativeness of the data may also lead to significant differences between the area fractions of iron particles in Table 3 and the volume fractions of iron particles in Table 2.

3. Data-Constrained Modelling for the Samples

3.1. Data-Constrained Modelling

For each sample, the volume fractions of different compositions in each CT voxel of region c were obtained with data-constrained modelling [30] by minimizing the objective function T shown in Equation (1).

$$\left\{\begin{array}{ll}\mathrm{T}={\displaystyle \sum _{\mathrm{n}=1}^{\mathrm{N}}\left[{\displaystyle \sum _{\mathrm{m}=0}^{\mathrm{M}}{({\mathsf{\mu }}^{\left(\mathrm{m}\right)}{\mathrm{V}}_{\mathrm{n}}^{\left(\mathrm{m}\right)}-{\widehat{\mathsf{\mu }}}_{\mathrm{n}})}^2+{\displaystyle \sum _{\mathrm{m}=0}^{\mathrm{M}}{\mathrm{V}}_{\mathrm{n}}^{\left(\mathrm{m}\right)}{\mathrm{S}}^{\left(\mathrm{m}\right)}}+{\mathrm{E}}_{\mathrm{n}}}\right]}\hfill & \hfill \left(\mathrm{I}\right)\\ {\displaystyle \sum _{\mathrm{m}=0}^{\mathrm{M}}{\mathrm{V}}_{\mathrm{n}}^{\left(\mathrm{m}\right)}=1}\hfill & \hfill \left(\mathrm{II}\right)\\ 0\le {\mathrm{V}}_{\mathrm{n}}^{\left(\mathrm{m}\right)}\le 1\hfill & \hfill \left(\mathrm{III}\right)\end{array}\right.$$

(1)

where n (n = 1, 2, 3, …, N) represents the CT voxel number in region c of the sample, and N is the quantity of voxels in region c, as shown in Table 2. M denotes the total number of composition types in the sample. For a pure silica sample, M = 1, with m = 0, 1 representing pore and silica compositions, respectively. For samples 2–6 containing mixed compositions of iron powder and silica, M is 2, with m = 0, 1, 2 representing pore, silica, and iron compositions, respectively. V_n^(m) represents the volume fraction of composition m in voxel n. In (I) of Equation (1), ${\widehat{\mathsf{\mu }}}_{\mathrm{n}}$ represents the reconstructed X-ray linear absorption coefficient value at voxel n in the CT slice, and μ^(m) represents the theoretical X-ray linear absorption coefficient value of composition m. Since continuous spectrum X-rays were used in this study, an equivalent energy method was employed to determine the theoretical absorption coefficient values of each composition. In order to determine the equivalent X-ray energy of the experiments, the average linear X-ray absorption coefficients of the standard aluminum samples imaged together with the region c of each sample were calculated. For each X-ray CT experiment, we selected one CT slice of the standard aluminum sample every 25 CT slices, and calculated the average linear X-ray absorption coefficient of these slices as the average linear X-ray absorption coefficient value in this experiment. Using the built-in X-ray linear absorption coefficient database in the DCM software [34], the corresponding X-ray energy for the absorption coefficient values of the standard aluminum sample was determined. This energy was used as the equivalent X-ray energy for this experiment. The theoretical X-ray linear absorption coefficients of each composition of the samples were determined according to this equivalent energy. The equivalent energies of the samples are listed in

Table 4. Computing parameters of DCM for each sample. S⁽¹⁾ represents the self-energy parameter of the silica composition.

Sample Number	Equivalent Energy (keV)	μ(m) (1/cm)			S(1)
		Pore	SiO2	Fe
1	33.630	0.0000	1.5170	—	0.0000000
2	79.510	0.0000	0.4534	4.6860	0.0001920
3	75.600	0.0000	0.4716	5.2520	0.0009310
4	80.145	0.0000	0.4506	4.6040	0.0021175
5	82.030	0.0000	0.4429	4.3750	0.0017580
6	80.060	0.0000	0.4510	4.6150	0.0013900

. Table 4 shows that despite using the same imaging parameters and X-ray tube voltage for samples 2–6, the equivalent energies are different. Sample 3, in particular, shows a significant difference compared to the others. The reason may be related to the fact that the CT imaging experiments for these samples were conducted at different times. After system restarts, the stabilization time of the equipment or errors in the voltage control and feedback system may lead to differences in the equivalent energies even when the same settings are used before and after the restart. Additionally, varying degrees of image artifacts and differences in the relative positions of the aluminum samples during different experiments may result in slight differences in the statistical values of X-ray linear absorption coefficients at the same acceleration voltage. This may also contribute to variations in the equivalent energies across different experiments. Because the theoretical linear X-ray absorption coefficients of the compositions in the samples were obtained based on the equivalent energies from each experiment, the equivalent energy errors caused by the acceleration voltage control feedback system did not introduce additional DCM calculation errors. Equivalent energy errors due to image artifacts may also lead to a decrease in DCM accuracy, but quantifying this impact is challenging.

In Equation (1), S^(m) represents the self-energy parameter of composition m. When the volume fraction of a certain composition in the sample was known, the parameter corresponding to that composition was adjusted to ensure that the calculated volume fraction matched the known value. This improved the accuracy of the DCM calculations. For sample 1, this parameter was set to 0. For samples 2–6, due to pronounced partial volume effects and metal artifacts on their CT slices, the DCM calculation results had significant deviations from the known values when the self-energy parameters were set to 0. For these samples, the self-energy parameter of silica was adjusted to ensure that the volume fractions calculated by DCM matched the known values. E_n in Equation (1) represents the interaction parameters [30], which were set as the default values of the DCM software [34] to reduce the impact of imaging noise on the DCM calculations. Reference [30] provides a detailed introduction to the numerical calculation methods, and the interaction energy parameters and self-energy parameters of DCM. The DCM software integrates DCM calculation capabilities, an X-ray absorption coefficient database, and some commonly used image processing functions, which can be freely downloaded from the URL provided in reference [34]. The DCM calculation parameters μ^(m) and S^(m) for samples 1–6 are shown in Table 4.

3.2. Calculation Results of Data-Constrained Modelling

Table 5. Calculation results and error analysis of DCM for the samples.

Sample Number	Volume Fractions Obtained by DCM (%)			Value Differences (%)			Relative Errors (%)
	Pore	SiO2	Fe	Pore	SiO2	Fe	Pore	SiO2	Fe
1	49.30	50.70	—	0.60	−0.60	—	1.23	1.17	—
2	46.02	51.73	2.25	−1.30	−0.01	1.31	2.75	0.02	139.36
3	43.02	54.52	2.46	−0.08	−0.05	0.13	0.19	0.09	5.58
4	45.52	49.13	5.35	0.14	−0.02	−0.12	0.31	0.04	2.19
5	47.05	40.96	11.99	−1.70	−0.05	1.75	3.49	0.12	17.09
6	41.44	39.54	19.02	−2.21	0.09	2.12	5.06	0.23	12.54

shows the volume fractions obtained by DCM and error analysis results of each composition for samples 1–6. The value differences in Table 5 are the differences between the volume fractions of the compositions computed by DCM and the corresponding actual volume fractions (Table 2). The relative errors in Table 5 are the ratios between the absolute values of the difference values and the actual values, as shown in Table 2, of each composition. From Table 5, it can be seen that for sample 1, DCM accurately obtained the volume fractions of silica and pores in the sample with the default value of the self-energy parameters as 0. The value difference of the computed volume fractions was only 0.6% compared to the actual values. For samples 2–6, adjustments were made to the self-energy parameter of SiO₂ in order to align the calculated volume fractions of SiO₂ as closely as possible with the actual values. In these cases, the calculated volume fractions of pores and iron are also reasonably consistent with the actual results. Comparatively, samples 3 and 4 exhibit a higher calculation accuracy than other samples containing iron. The reason for this is the low iron content in sample 2 (only 0.9%), where even small calculation errors can result in large relative errors. For samples 5 and 6, with actual iron volume fractions of 10.24% and 16.90%, respectively, the two samples exhibit more pronounced partial volume effects and metal artifacts on their CT slices due to the significantly higher absorption coefficient of iron compared to silica. This led to increased calculation errors in DCM.

As shown in Figure 6a, most areas are mixed colors of blue and green, indicating that pores and silica mostly exist in the form of partially occupied pixels. In Figure 6b–d, the color of red corresponding to iron composition is less visible, primarily due to its lower volume fractions in these three samples, where its color is masked by others. Figure 6d shows some yellow areas, suggesting the coexistence of iron and pores in those regions. A large number of light red regions are clearly visible in Figure 6e,f, corresponding to the high volume fractions of iron in samples 5 and 6. However, areas displaying as 100% red are relatively scarce, indicating that iron in these samples also occupies voxels partially. Furthermore, an increase in iron particle volume fractions may lead to an increase in X-ray linear absorption coefficients of voxels around them due to metal artifacts and point spread functions, potentially resulting in an overestimation of iron content in the calculations. In Figure 6a–c, some compositions exhibit a ring-shaped distribution form, which is more likely caused by the ring artifacts present in the CT slices of the samples, as shown in Figure 2. Despite our efforts to improve CT imaging quality through methods such as background subtraction and the use of filters, there may still be unavoidable image artifacts present in the CT slices, such as ring artifacts, motion artifacts, and beam hardening artifacts. These image artifacts may also reduce the accuracy of the DCM calculation results, and it is difficult to obtain a quantitative assessment of their impact on the accuracy of the DCM calculations.

3.3. Comparison of DCM and Threshold Segmentation Methods

A sub-region of 796 × 796 pixels with distinct porous structures was cropped from the 1150th CT slice of sample 1. In this sub-region, the result of DCM was compared with those of four automatic image threshold segmentation algorithms, including Shanbhag [36], Triangle [37], Otsu [38], and Percentile [39]. The calculation of the image threshold segmentation was performed using ImageJ software. Figure 7 shows the original slice of the sub-region and the results of DCM and the automatic image threshold segmentation algorithms. It is seen from Figure 7 that for this type of sample image, with unimodal intensity distributions, the results obtained from threshold segmentation severely deviate from reality and are difficult to apply practically. The DCM not only accurately extracted the prominent cracks present in the CT slice but also identified numerous pixels occupied by partial pores. DCM, especially under the constraints of known information, can be effectively applied to the processing of similar images.

4. Conclusions

In this study, six samples with known volume fractions of compositions were imaged by laboratory X-ray CT equipment. Optical microscopic images of the samples showed the presence of a large number of particles smaller than the CT voxel size. Volume fractions of each composition in the samples were computed using DCM, and the results were compared with the results of image threshold segmentation algorithms. The main conclusions are as follows:

(1)

The X-ray theoretical linear absorption coefficients of different compositions in the DCM can be determined effectively by the equivalent energy method. For the sample containing a single silica composition, combining DCM with laboratory micro-CT allows accurate determination of the volume fractions of silica and pores in the sample.

(2)

For silica samples with added iron particles, the volume fractions of pore and iron particles can be obtained by constraining the self-energy parameters of silica in the DCM. The calculation accuracy of the volume fractions is related to the content of iron. The accuracy of DCM calculations decreases slightly when the iron particle contents are 10.24% and 16.90%, due to the influence of the partial volume effect and the point spread function.

(3)

A comparison between DCM and automatic image threshold segmentation algorithms reveals significant advantages of DCM when handling images containing numerous particles smaller than CT voxels. Unlike simple binary segmentation (0 or 1), DCM provides volume fractions of each composition in each voxel. This enables the quantitative assessment of the multi-scale structural information of the samples. This method facilitates the establishment of multi-scale structural three-dimensional grids for numerical simulations of the physical and chemical properties of the samples.