Introduction

Morphology, structure and texture of the grains of the minerals forming rocks and their relationships inside rocks is one of the important diagnostic features. Traditionally, structures and textures are described qualitatively. At the end of the twentieth century it became possible to describe them quantitatively due to the development of instrumental methods and computing technology. One of the first generalizations of these methods is presented in1. Quantitative measures describing petrographic structures can also be found in the modern literature. Classical structural-geological methods for determining the orientation of lineation and foliation of rocks are reported in2,3. Crystal size distribution is used as quantitative measure in4,5. Anisotropy of mineral grain orientation, or in other words shape preferred orientation is described in6,7,8. Anisotropy of crystallographic axes orientation, i.e. lattice preferred orientation is used in9,10,11 as a quantitative measure of rocks morphology. The probability of contacts of grains of different mineral types is calculated in12,13. Spatial covariance of minerals is applied in14. Shannon entropy of the distribution of chemical and/or mineralogical variables along a one-dimensional profile is discussed in15. Fractal dimension of patterns formed by different minerals16,17, fractal dimension of grain boundaries18, cracks19,20, pores21, etc. are also used as a quantitative measure for rocks characterization. However, all these attempts to give the morphology quantitative description have their flaws.

Reliable quantitative metrics of rock morphology appeared with the development of 3D tomography. Which is the most valuable, these metrics did not relate to individual grains, but to aggregates of the same mineral type. It became possible to measure anisotropy, surface area, volume, fractal dimension of the aggregate simultaneously22,23,24,25.

Computer-assisted 3D X-ray microtomography (X-μCT) is one of the commonly used methods to obtain information about the internal structure of a sample. This method provides high-resolution three-dimensional images of internal structures without sample destruction. Earlier we have applied integral geometry methods in order to quantitatively describe the three-dimensional structures obtained by 3D tomography26,27,28,29. This approach was applied in the quantitative analysis of various complex structures30,31,32,33,34,35,36,37,38. We have analyzed the structure of soils39,40, sedimentary rocks41, and composite materials42.

The topological approach has already been used to predict the properties of new materials, namely ion-exchange materials43,44. In the future it can be used to predict the properties of materials with ionic conductivity45, or to analyze the distribution of defects in artificially obtained crystal structures. This is critically important for the technology and properties of functional materials. This could for example, be such a popular material as lithium niobate46,47,48.

Traditional mathematical systems such as functional analysis and statistics generally do not provide quantitative estimates of the structure; they identify metric relationships or statistical connections. The only mathematical apparatus for describing the structure is, as far as we know, the methods of algebraic topology, which we propose. In addition, this is a non-destructive method of research for rare or unique samples. Our method is easily automated and will help to quickly study a number of samples.

Integral geometry has an important advantage: a detailed analysis of structures without large numbers of samples. Researchers can obtain quantitative information about the structure of a sample while relying on a limited amount of initial data.

The main tool of integral geometry is Minkowski functionals. These are invariant measures associated with the main geometric parameters. In the case of three-dimensional objects, there are four Minkowski functionals Mi: M0 is proportional to the volume, M1—to the surface area, M2—to the average integral curvature, M3—to the Euler–Poincaré characteristic. The Euler–Poincaré characteristic is an alternating sum of the Betti numbers bi. In the three-dimensional case b0 is the number of disconnected subregions in the region under consideration, b1 is the number of through “tunnels”, b2 is the number of isolated “voids”. Characteristics M0–M2 are metric and continuous, and characteristic M3 is integer.

We planned a methodologically pure experiment; we did not even have a possibility to adjust the initial data to a known result, we simply had no ready answer. First, X-μCT scientists took a sample and segmented it. Then topologists calculated the resulting 3D model in terms of integral geometry, interpreted the topology of the selected phases. After that geologists interpreted the found topological features in terms of the formation of mineral associations based exclusively on topology, without involving traditional geological and mineralogical methods. The only deviation from this “pure” scheme was this: we associated the X-ray density phases with real minerals previously described in the literature for the geological object under study. However, even this step could be skipped, as topologists could have simply named obtained phases “phase 1”, “phase 2” etc. without knowing which mineral it belongs to.

A sample of meimechite from the Kontozero volcano-plutonic complex was studied. Meimechite is an effusive rock. It consists of olivine phenocrysts. Olivine phenocrysts are uniformly distributed in a fine-grained mass, and the mass consists of crystallized volcanic glass, diopside microlites and magnetite. The rock also contains carbonate globules, which are rounded clusters of calcite grains.

Materials and methods

Initial data

Description of the initial sample

The studied sample is meimechite from the Kontozero volcano-plutonic complex (the Kola Peninsula, NW Russia). The rock structure is porphyritic. Relatively large olivine phenocrysts are found in a fine-grained groundmass. The main mass is composed of microlites of diopside, phlogopite, spinels and crystallized volcanic glass. Decrystallized volcanic glass is an extremely fine-grained aggregate of hydrobiotite and serpentine. In addition to olivine phenocrysts, the main fine-grained mass contains globules. Globules are formations of rounded (up to 2 mm in diameter), less often irregular shape, without any internal structure. Globules are distributed uniformly in the rock and are composed of calcite, crystallized volcanic glass and intergrowths of titanite and ilmenite. The ratios of the listed phases in the volume of each individual globule are different: from purely calcite to completely silicate (decrystallized glass + titanite and ilmenite). Olivine forms hypidiomorphic phenocrysts ranging in size from 0.2 to 3 mm in diameter. These phenocrysts are intensely serpentinized and chloritized along the periphery and along cracks. Moreover, the rim of secondary minerals surrounding olivine is usually zonal: its inner part is composed of serpentine, and the outer part is composed of chlorite. The content of olivine phenocrysts is about 30% of the rock volume, taking into account the serpentinized and chloritized areas of the grains. Diopside is present only in fine-grained mass and usually forms idiomorphic grains no larger than 50 µm in diameter. Spinelides, such as chromium-rich magnetite and chromite, are found as small (up to 20 µm in diameter) inclusions within olivine phenocrysts and in the central parts of the largest grains in the groundmass. Titanium-rich magnetite and ulvospinel compose the outer zones of such large grains and form small crystals in the fine-grained groundmass. Phlogopite forms small (up to 50 µm in diameter) xenomorphic grains. They are located in the interstices of diopside and magnetite in the fine-grained groundmass of the rock. The mineral has undergone secondary changes to a significant extent: loss of potassium and increase in water content. The products of such changes are most similar in chemical composition to hydrobiotite or vermiculite.

The sample is an irregular tetrahedron with sides of 2 × 2 × 3 cm. Figure 1 shows a photograph of the sample.

Fig. 1
figure 1

Photograph of the meimechite of the Kontozero volcano-plutonic complex. Sample size is 2 × 2 × 3 cm.

Full size image

X-ray computed tomography

A high-resolution microfocus computed tomography was used to obtain information about the internal structure of the studied sample. The studies were carried out on the basis of the X-ray system for computer tomography v|tome|x L240 manufactured by GE Sensing & Inspection Technologies GmbH, Germany, Berlin. The system is equipped with two X-ray tubes. The accelerating voltage is from 10 to 240 kV, the tube current is up to 2 mA, the scanning resolution is up to 2 μm, the maximum power of the X-ray tube is 320 W, the focal spot size is less than 0.001 mm, the dimensions of the detector are 400 by 400 mm, the distance from the source to the receiver is 1500 mm. A photograph of the setup is shown in Fig. 2.

Fig. 2
figure 2

Photograph of the v|tome|x L240 computed tomography system.

Full size image

Tomography of the sample was performed for the following equipment settings: image magnification was 17.76 times; voxel size was 51 μm; scanning type was spiral; number of projections was 2024; projection image size was 2024 × 2024; exposure was 500 ms; number of images of one projection during averaging was 3; accelerating voltage of the X-ray tube was 100 kV; emitter current was 185 μA.

The linear dimensions of the sample were approximately 2 × 2 × 3 cm, which corresponds to 436 tomography slices. Various commercial (Avizo Fire, GeoDict) and open source software products (ImageJ/Fiji, etc.) were used to analyze the 3D image obtained after direct scanning.

The tomographic imaging results in a three-dimensional grayscale image. Light voxels correspond to areas of higher X-ray density, and dark voxels correspond to areas of lower X-ray density in the image.

Preprocessing of initial data

Image processing by integral geometry methods requires segmentation of the original tomographic image. Segmentation implies dividing the original tomographic image into subregions. They correspond to the most predominant brightness levels (X-ray density), the so-called phases.

The simplest and most common method for segmenting such images is the Otsu method. The method is based on minimizing intra-class variances49. Segmentation of the image into phases in our case is performed by the multi-Otsu method50. This method is an extension of the Otsu method for the case when the number of phases exceeds 2. Since the presented meimechite sample contains more than 2 minerals, we can expect more than 2 phases. The choice of binarization thresholds, as in the original Otsu method, is performed by minimizing the intraphase dispersion. As a result of segmentation, we get the opportunity to extract a binary image for each individual phase.

Then, we extract individual objects using the connected component method in each of the extracted phases. The method is implemented in the cc3d library in Python51. This method extracts individual objects (connected components) within each phase according to the applied type of connectivity in the image. The extracting connected components begins with selecting a starting point. The process extends to neighboring voxels if they meet the criterion of belonging to the same phase. Voxels should have the same color and be neighbors according to the applied type of connectivity. This step is repeated until all possible pixels belonging to a given component (i.e., phase object) have been extracted. Then the algorithm moves to the next unvisited point and repeats the process until all pixels in the image have been classified into individual objects (in other words, connected components).

Geometrical description of the sample

Minkowski functionals and their properties

Minkowski functionals introduced in integral geometry are a set of invariant measures that can be used to quantitatively evaluate the structural properties of the object. Connected components belonging to one phase are chosen as the object of study in our paper.

Consider a convex subset X in the d-dimensional Euclidean space Rd bounded by a smooth surface ∂X. d + 1 Minkowski functionals for X are defined in the chosen space. Since we characterize three-dimensional objects in this paper, we will further consider only three-dimensional Minkowski functionals. In R3, there are four Minkowski functionals Mi(X), i = 0, …, 3, proportional to the volume V(X), the surface area S(X), the integral mean curvature C(X), and the Euler–Poincaré characteristic χ(X).

$$\begin{aligned} & M_{0} \left( X \right) = V\left( X \right), \\ & M_{1} \left( X \right) = \mathop \int \limits_{\delta X}^{{}} dS = S\left( X \right) \\ & M_{2} \left( X \right) = \mathop \int \limits_{\delta X}^{{}} \left( {\frac{1}{{r_{1} }} + \frac{1}{{r_{2} }}} \right)dS = C\left( X \right) \\ & M_{3} \left( X \right) = \mathop \int \limits_{\delta X}^{{}} \left( {\frac{1}{{r_{1} r_{2} }}} \right)dS = 2\pi \chi \left( {\delta X} \right) = 4\pi \chi \left( X \right) \\ \end{aligned}$$
(1)

where r1 and r2 are the principal radii of curvature of the surface dS, χ(∂X) and χ(X) are the Euler–Poincaré characteristics for the surface ∂X and the convex body X, respectively.

The Euler–Poincaré characteristic M3 for a body X in R3 is an integral estimate of its topological complexity and is defined as an alternating sum of the Betti numbers bi:

$$\chi \left( X \right) = b_{0} \left( X \right) \, - b_{1} \left( X \right) \, + b_{2} \left( X \right),$$
(2)

where the following interpretation can be used: b0 is the number of individual parts of the object, b1 is the number of conductive (through) channels in the object, and b2 is the number of isolated cavities inside the object (closed pores).

Considering that the studied sample is a digital 3D model, M0 is measured in voxels, M1—in the number of voxel faces, M2—in the number of voxel edges. M3 is a dimensionless value.

Let us list the main properties of Minkowski functionals:

  • Additivity. Let K be a class of convex sets in Rd. Suppose that a set X ∈ K is divided by a plane into 2 subsets X1 and X2, which are also convex, so that X = X1 ∪ X2 and X1, X2 ∈ K, then:

    $$M_{i} (X_{1} \cup X_{2} ) = M_{i} \left( {X_{1} } \right) + M_{i} \left( {X_{2} } \right) - M_{i} \left( {X_{1} \cap X_{2} } \right),\quad {\text{for}}\quad i = 0,1 \ldots d;$$
    (3)

  • Continuity: let G be a group of shifts in Rd. Denote the action of g ∈ G on a convex set X ∈ K as gX, then

    $$M_{i} \left( {g_{X} } \right) = M_{i} \left( X \right),\quad {\text{for}}\quad i = 0,1 \ldots d;$$
    (4)

  • Invariance under translation and rotation transformations: if a series of convex sets Xn ∈ K converges to X ∈ K, \({{X}_{n} }_{n\to \infty }\) → X, then:

    $$M_{i} \left( {X_{n} } \right)_{n \to \infty } \to M_{i} \left( X \right),\quad {\text{for}}\quad i = 0,1 \ldots d.$$
    (5)

Hadwiger’s theorem is of great importance here52: any additive, continuous, motion-invariant functional F(X) defined on sets X ⊂ Rd can be represented as a linear combination of Minkowski functionals:

$$F\left( X \right) = \mathop \sum \limits_{i = 1}^{n} \alpha_{i} M_{i} \left( X \right)$$
(6)

where αi is a coefficient for the Mi.

Therefore, we can say that the system of Minkowski functionals contains all the necessary information about the geometry and morphology of the sample. Thus one can use the entire set of Minkowski functionals for quantitative characterization and classification.

Calculating Minkowski functionals on digital images

Various works have proposed methods for calculating Minkowski functionals for binary images53,54,55,56,57 with the development of computational geometry29,58.

In this paper, the calculation of Minkowski functionals on digital images was performed using the quantimpy library55 in Python in the Jupyter environment. The method for calculating Minkowski functionals in this library is based on combinatorial algorithms. The algorithms can be found in the works53,54,59. The volume is equal to the number of voxels in the object in this approach, and such characteristics as surface area, mean integral curvature and Euler–Poincare characteristic are determined by combinatorial calculation of local contributions. They depend on the configurations of the voxels surrounding the one under consideration. For the image voxels, 26-connectivity was used in the calculation.

We perform regression analysis and construct a plane for the set of metric characteristics M0–M2; information about the topological invariant M3 is considered separately.

Results

Image segmentation

Figure 3 shows an image of one of the tomography slices and a histogram of the distribution of gray shades in the entire sample with the results of segmentation according to multi-Otsu. The full X-μCT image is available as S1 in62.

Fig. 3
figure 3

(a) Tomography slice 300 of the Kontozero sample. (b) Distribution histogram of gray shades of all voxels of the sample. Red lines delimit the areas selected during segmentation.

Full size image

3D tomography yielded a number of slices. They were assembled into a single 3D picture of the sample. An example of such a slice (number 300) is shown in Fig. 3a. All slices were obtained in shades of gray. The histogram in Fig. 3b shows their distribution throughout the entire volume of the sample, on the OX axis 0 is black, 255 is white. Figure 3b has a limitation on OX, since there are no significant digits below 40 and above 190.

The multi-Otsu method segmented four regions. Their boundaries are separated from each other by three red verticals in Fig. 3b. Thus, we conclude that there are four separate phases in the sample under study. Each phase consists of a large number of individual fragments. We will then call them phase objects. Their distribution using section number 300 as an example is shown in Fig. 4a in false colors for contrast. The green fraction (phase 2) (Fig. 4d) is rounded, weakly faceted whole grains, the yellow (phase 1) (Fig. 4c) forms shells of green grains, and the white is small individual dots scattered over a gray field (phase 4) (Fig. 4f). Gray in it is the host rock (phase 3) (Fig. 4e).

Fig. 4
figure 4

Tomography slice 300 of the Kontozero sample, (a) 4 segmented phases; (b) these phases are shown in grayscale mode, false colors; (c) phase 1 (yellow); (d) phase 2 (green); (e) phase 3 (gray, enclosing); (f) phase 4 (white).

Full size image

Based on the literature and our unpublished data, we assumed that there is a correspondence between the X-ray density phases and mineral species (groups of mineral species) characteristic of the geological object under study60. The following mineral association is characteristic of the meimechites of the Kontozero Massif: early olivine, diopside, serpentine developing after olivine, titanic magnetite and later carbonate minerals.

The least X-ray dense phase of the listed ones is serpentine. This most likely corresponds to phase 1. The dominant phase from most X-ray dense phase is magnetite, therefore, the influence of others can most likely be neglected. Thus, we take magnetite for phase 4 (Fig. 4f). It was found that this type of rocks is characterized by large hypidiomorphic grains of olivine in60,61. They correspond to the morphology of phase 2 objects (Fig. 4d). Shells of phase 2, which we have denoted as phase 1, most likely correspond to serpentine (Fig. 4c). Phase 3 apparently is a fine-grained mass of other minerals (Fig. 4e). The characteristic grain size of diposide, phlogopite is under the resolution of the X-μCT method for this sample (51 μm per voxel). Full images of phases separately in the whole sample can be found in62: S2—phase 1, S3—phase 2, S4—phase 3, S5—phase 4.

As we can see, this technique identified only some mineral species. In addition, the resolution of 51 microns per voxel makes it difficult to identify smaller inclusions. Most likely, they were dissolved in the host carbonate–silicate matrix and merged with phase 3. We demonstrated these phases on tomography slice 300 in the false gray mode in Fig. 4b: the darkest is phase 1, the lightest is phase 4. Table 1 shows the correspondence of the colors in Fig. 4a to the phases, their mineral component and morphological description. In what follows, we will use the term “phase” and number them in accordance with Table 1.

Table 1 Correspondence of phases, mineralogical types, colors in Fig. 4 and phase morphology.
Full size table

The combination of multi-Otsu and X-μCT methods provided the distribution of some mineral species in the volume without damaging the sample. This also makes it possible to manipulate the 3D volume model of these minerals and study their topological characteristics.

Sample analysis using integral geometry and topology

This section presents the results of our sample analysis using integral geometry and topological invariants. The Minkowski functionals introduced in integral geometry in 3D space can be divided into 2 parts. The first part is the functionals M0 (volume), M1 (surface area) and M2 (integral mean curvature) which depend on the linear dimensions of objects and are metric characteristics. The second part is the functional M3 (the number of objects, channels and pores of a given phase), proportional to the Euler–Poincaré characteristic. This characteristic is an alternating sum of topological invariants, Betti numbers. This functional itself is a topological invariant, i.e. it does not depend on the linear dimensions of the object.

Regression analysis was performed for the set of metric characteristics (functionals M0, M1 and M2). It helped to build linear models for the selected objects of each phase. These models describe the dependence between metric characteristics (section “Three-dimensional scatterplots for metric characteristics. Classifier”). In terms of topological invariants, the functional M3 and its constituent Betti numbers were calculated for each phase, and a detailed description of the relationships between the structural features and the calculated topological characteristics was also given (section “Topological invariants”).

Three-dimensional scatterplots for metric characteristics. Classifier

Let us consider any of the four extracted phases obtained by the multi-Otsu method. Various objects in the phase are extracted from the total sample mass using the connected component method. After that, in the phase, we calculate the Minkowski functionals on each object.

Figure 5 shows paired scatterplots in the coordinates of the metric characteristics M0, M1, M2 for each of the phases. From the diagrams we see that all the point clouds lie in a sector. We can judge the presence of interrelations between the metric characteristics of objects in each phase from these data. In other words, the points in the presented space do not lie chaotically, but obey a certain pattern, and the bulk of the cloud points for each phase is concentrated closer to the origin of the coordinates.

Fig. 5
figure 5

Scatterplots of Minkowski functionals calculated on the connected components of the (a) first, (b) second, (c) third, (d) fourth phases.

Full size image

Figure 5 shows that objects in each phase have their own spread of characteristic values of metric parameters. The largest spread is observed for the first phase, followed by the second phase. The smallest spread of values is observed for the third and fourth phases.

Let us move on to examining the three-dimensional scatterplots, Fig. 6. Volumetric models of all four phases are presented in62, S6.

Fig. 6
figure 6

Three-dimensional scatterplots of Minkowski functionals calculated on objects of (a) the first, (b) the second, (c) the third, (d) the fourth phase.

Full size image

Visual analysis of pairwise and three-dimensional scatterplots for the metric characteristics of objects in each phase showed that the points belonging to different phases lie in a plane. To prove that these are planes and to find how much the points deviate from it, we built a linear regression model for each phase.

As can be seen, the resulting linear model describes the equations of the plane in the coordinate space M0, M1 and M2. The equation can be written as follows:

$$a \cdot M_{0} + b \cdot M_{1} + \left( { - 1} \right) \cdot M_{2} + const = 0,$$
(7)

or in other order

$$M_{2} = a \cdot M_{0} + b \cdot M_{1} + const,$$
(8)

where a, b, const are constant real coefficients.

Table 2 shows the parameters of these models for phases 1–4: the values of the model coefficients with standard errors, as well as the values of the determination coefficient R2, reflecting the level of statistical significance of the constructed models.

Table 2 Result of calculating regression coefficients a, b, const and their standard errors aerr, berr, consterr, determination coefficient R2 for phases 1–4.
Full size table

As we can see from the Table 2, the values of the coefficients of determination R2 for the constructed models for each class are quite high. Therefore, we can say that they have statistical significance, and, therefore, the point cloud for each phase can be approximated by a plane with a high degree of accuracy. The objects of each phase lie in a plane in a given three-dimensional space. Note that the point clouds constructed for objects from different phases lie in different planes in space. The coordinates of this space are the Minkowski functionals M0, M1 and M2.

The set of objects with a very small volume is located in the vicinity of the origin, i.e. the point (0, 0, 0). And there are many more such objects than large ones (Fig. 7). And small objects can distort the overall picture. An object of 1 voxel size has no options for the volume-area-curvature relationship. An object of 2 voxels size already has the second degree of freedom: the second voxel can join the first through a face, an edge, or only through a vertex. When an object is 3 voxels in size, there are even more options for mutual arrangement. However, these small objects, can simply distort the angle of inclination of the general plane due to their smallness and the great number (Fig. 7). Therefore, we decided to throw out the smallest particles from the overall picture and see if the angle of inclination of the general plane changes. The result is shown in Fig. 8. Threshold on Fig. 8 is the volume, particles smaller than which are not taken into consideration while calculating the plane angle. For example, if the meaning on the axis “Threshold” is 5, then the plane angle is calculated for the phase objects with volume greater than 5, if 12—than 12, they are cut off the final result. Thus, the higher the value of “threshold”, the larger objects are taken into account in the angle of inclination.

Fig. 7
figure 7

Diagrams with the distribution of objects by volumes for: (a) the first, (b) the second, (c) the third, (d) the fourth phase. (d) Diagrams of the distribution of objects of all phases by volumes.

Full size image
Fig. 8
figure 8

Dependence of the planes inclination angles in the space of metric Minkowski functionals on the volume cutoff.

Full size image

To estimate the contribution of small particles to the plane inclination angle, we plotted the dependence of the plane angle on the volume cutoff. We described the planes using their normal vectors, which are defined as follows: N = {a, b, − 1}. Consider α the angle between the normal vector for each constructed plane and the vector {1, 1, − 1} as a plane characteristic. The angle is calculated as follows:

$$\left( {\overrightarrow {{N_{1} }} , \overrightarrow {{N_{2} }} } \right) = \left| {\overrightarrow {{N_{1} }} } \right| \left| {\overrightarrow {{N_{2} }} } \right|\cos \alpha$$
(9)
$$\alpha = \arccos \frac{{\left( {\overrightarrow {{N_{1} }} , \overrightarrow {{N_{2} }} } \right)}}{{\left| {\overrightarrow {{N_{1} }} } \right| \left| {\overrightarrow {{N_{2} }} } \right|}} \cdot \frac{180^\circ }{\pi }$$
(10)

The dependence of the plane inclination angle constructed on the basis of the scatter diagram for objects with a volume above the threshold value on the given threshold value is shown for each phase in Fig. 8. Note that when there are less than 50 objects left in the phase after the cutoff, the calculation stops.

There is an upper limit on the graph. We made a graph for the characteristics up to the cutoff of 150 voxels. Then the trend remains unchanged for all phases. Phase 3 has less than 50 objects with a volume of more than ~ 40 voxels (see Fig. 7c). This means that there are very large objects in phase 3, but there are few of them. Which is logical since this phase corresponds to the host rock. This is also shown by a comparison of the distribution of phase objects by volume (Fig. 7d): there are indeed quite a few very large objects in phase 3.

As we can see in Fig. 8, the angles corresponding to different phases remain different regardless of the cutoff. In addition, when the cutoff increases, the divergence of the plane angle occurs smoothly, without sharp jumps. This result indicates the stability of the approach and the possibility of its use for classifying objects within geological structures.

It is noteworthy that the angles phases 2 and 4 are close to each other. Within the error limits of the cutoff after objects of 40 voxels in size, they come very close together.

The obtained result (Fig. 8) also proves that the phases extracted by the multi-Otsu method actually correspond to the real X-ray density phases of the studied geological sample. If the phases extracted from the X-μCT image were the result of a random color match, then when cutting off small objects of the phases, the divergence of the tilt angle would be sufficient to assert chaos. However, we do not see this and can confirm that the extracted phases correspond to the mineral species of the sample.

Topological invariants

Description for Betti numbers for single phases separately

Table 3 shows the Betti numbers and the Euler–Poincaré characteristic (χ) for the selected phases. Recall that b0 is the number of objects, b1 is the number of channels, b2 is the number of pores, χ = b0−b1 + b2 (see Section “Minkowski functionals and their properties”). For clarity, we will further denote the phase to which the Betti number refers by a superscript, for example, b0(1) is the number of objects of phase 1.

Table 3 Result of calculating the Euler–Poincaré characteristic and Betti numbers on four selected phases.
Full size table

Table 3 shows us the following. The b0 number is the largest for phase 2. Therefore, this phase has the largest number of individual objects. Phase 1 comes next in terms of the number of objects. Therefore, not every grain from phase 2 has a separate shell: shells may be adjacent to several grains at once or individual grains do not have shells at all. And phase 3 is the enclosing mass. It contains approximately 2–4 times fewer objects than phases 1 and 2. Given that all these phases occupy the same volume of the sample, this may mean that phase 3 is more coherent and less fragmented than phases 1 and 2. The smallest number of objects belongs to phase 4. Considering that we attributed it to magnetite and it, in principle, forms scattered and small grains in small quantities, there is no contradiction with geological data.

Let’s analyze the Betti number b1. This is the number of through tunnels, i.e. those through which a thread can be pulled. Their number is greatest for phase 2, in other words, for olivine grains. There are approximately b0(2) ~ 286 k (k = thousand) grains in total, and such channels are approximately b1(2) ~ 387 k. This means that there are at least 100 k grains in which there is more than one such channel. This means that our grains are very perforated.

The next largest number of b1 belongs to phase 3, which is the host rock. This number b1(3) is approximately 2 times lower than b1(2). This means that the main host rock is also perforated. However, since this is the host phase and all other phases are located in it, this conclusion is logical. We also need to take into account the geometry of the sample. It has the form of a tetrahedron which has many corners with a narrow field of view. That is, other phases embedded in the host rock extend from one face of the tetrahedron to another and form a through hole. And there are many such narrow edges in the sample. Obviously, the higher the volume to area ratio of the figure, the greater the probability of a channel passing from surface to surface. In fact, this is an artifact caused by the shape of the sample and the presence of a sample surface in general. Therefore, the analysis of the Betti numbers is sensitive to the shape of the object under study, and the shape must be taken into account. Ideally, it is necessary to make balls or cut a ball out of the tomogram for such an analysis. Since we are examining one object, this is not so important for us, but if we need to compare the topology of a number of samples, then it is imperative to eliminate artifacts of the sample shape by analyzing strictly spherical volumes. We do not perform this here because firstly, we have only one sample, and secondly, our main goal is to show how the chosen method works on real object. However, further recommendation is to take the sample form into consideration.

The fourth phase is a scattering of small objects (Fig. 4f). They have a small number of through channels. And the number b1(4) is very small in this phase.

The Betti number b1(1) is approximately 2 times smaller than b0(1). This means that, despite the fact that these are shells of phase 2 grains, they contain few channels. This means that only half of the serpentine shells contain places in which grains of other phases can be embedded so that the shell completely wraps the grain at least in one position and forms a through channel. Taking into account the morphologies of phases 1 and 2 (phase 1 is rather a shell of phase 2), the most probable combination is precisely that the channels in phase 1 are occupied by phase 2.

Let us analyze the b2 number. These are pores completely enclosed within the volume of the phase under consideration. The largest number of such pores is observed in phase 3. Considering that the total number of individual objects of this phase is b0(3) ~ 70 k, and the number of closed pores in this phase is approximately b2(3) ~ 45 k, this means that on average, every second object of this phase contains a pore that can have a closed pore that can accommodate grains of one or more types of other phases, completely embedded in it. This is natural, according to the general morphology of phase 3: all the others are embedded in it. The next in number of b2 belongs to phase 2. That is, individual olivine grains are perforated through and contain small inclusions of other phases. The b2(1) value is quite small. That is, almost nothing is embedded in these shells compared to phases 2 and 3. And there is only 1 registered case when another phase is located inside it, b2(4) = 1 for phase 4. This means that these are dense whole grains.

However, if we analyze the phases separately, we have a poor understanding of which phases fill the through holes and closed pores, of which there are many, for example, in phase 2. To do this, we need to consider the Betti numbers for the phases separately in pairs and even in triplets. We mean that we take two or three phases and analyze their Betti numbers as of a one phase for all possible combinations.

Description for Betti numbers for phases in pairs and triplets

To analyze the relationships of topological features of different phases, we combined the phases in pairs (and then in triplets) and analyzed the change in the number of objects, channels, and pores in the combined phases. This means that we consider a pair or a triplet of phases as one and calculate Betti numbers. To understand how the Betti numbers change when phases are combined, Tables 4 and 5 show the sums for the corresponding phases separately in brackets. I.e. it looks like this: b0xy (b0x + b0y). We have developed a system of signs that show the directions of their change: ↑ denotes an increase in the Betti number when combining phases compared to the sum of the corresponding Betti numbers of individual phases, ↓ denotes a decrease, ~ denotes approximate equality to the sum. They are recognized as unchanged (i.e., ~ is put) if the sum of the Betti numbers of two\three separate phases and the Betti number of the combined phase differ by no more than 10%.

Table 4 The result of calculating the Euler–Poincaré characteristic and Betti numbers on pairwise combined phases and the sum of the Betti numbers (in parentheses) for these phases from Table 3. Designations of signs near the values are given in detail in the article text and in a footer.
Full size table
Table 5 Result of calculating the Euler–Poincaré characteristic and the Betti numbers on phases combined into triplets and the sum of the Betti numbers (in parentheses) for these phases from Table 3. Designations of signs near the values are given in detail in the article text and in a footer.
Full size table

Next, we theoretically considered pairwise combinations of sums of Betti numbers, Betti numbers of combined phases and their consequences. We obtained rules of interpretation. An illustration of these rules is given in Fig. 9.

Fig. 9
figure 9

Variants of transformation of an object (b0), a channel (b1), and a closed pore (b2) when combining two objects. Green—phase x, blue—phase y, red—combined phase xy.

Full size image

The Minkowski functional M3 contains three topological invariants: b0 is an object (solid), b1 is a through channel, and b2 is a closed pore. When two phases are combined, these invariants can transform in a certain way. A solid and a channel when combined with another phase can become a solid, a channel, or a pore. And a pore can either remain a pore or become a solid. Based on these transformation options, we went through all possible options and obtained rules for interpreting the change (dynamics) of Betti numbers when combining phases.

  1. 1.

    If b0x + b0y ≈ b0xy , then the objects of phases x and y do not contact (they contact with a low probability);

  2. 2.

    If b0x + b0y << b0xy, then most likely there is an error in segmentation or calculations;

  3. 3.

    If b0x + b0y >> b0xy, then the objects contact with a high probability;

  4. 4.

    If b1x + b1y ≈ b1xy, then the phases x and y do not enter each other’s channels;

  5. 5.

    If b1x + b1y << b1xy, then at the contacts of these two phases (x and y) there is a third phase z, in one of the dimensions not less than the shortest length of this surface;

  6. 6.

    If b1x + b1y >> b1xy, then the channels are filled by each other’s volumes;

  7. 7.

    If b2x + b2y ≈ b2xy, then the channels of phases x and y, when combined into phase xy, do not turn into pores and do not fill each other’s pores;

  8. 8.

    If b2x + b2y << b2xy, then at the contacts of these two phases (x and y) there is a third phase z, completely included in these two (less than the shortest length of this surface). Or the channels close and turn into pores;

  9. 9.

    If b2x + b2y >> b2xy, then phases x and y mutually fill each other’s pores.

The dynamics of the Betti numbers for combining more phases generally follow the same logic. Combinations of these conditions can provide more information about the structure of the object under study, this is the subject of further study. However, it can already be said that the consideration of combined phases is a very promising and fruitful method of integral geometry.

Tables 4 and 5 present the Betti numbers for the phases in pairs and triplets, respectively. That is, Table 4 presents these values for the case where phases 1 and 2 are considered as a single whole (phase 12), phases 1 and 3 are considered as phase 13, etc. And Table 5 presents these values for the case where phases 1, 2, and 3 are considered as a single whole (phase 123), etc. The union of phases to which the mentioned Betti number pertains is designated by a double and triple superscript, for example b0(12) and b0(123), respectively.

Let us analyze the obtained Betti numbers for the combined phases in accordance with a set of interpretation rules.

First, we analyze the combined phase 12. b0(1) + b0(2) >> b0(12), therefore, phases 1 and 2 are in contact with a high probability. b1(1) + b1(2) >> b1(12), which means that the channels of these phases are blocked by the volumes of the other phase. At the same time, b2(12) ≈ b2(1) + b2(2), therefore, the channels do not turn into closed pores when phases 1 and 2 are combined. This means that we can assume that phases 1 and 2 largely fill each other’s channels.

The number b0(13), as well as almost all other combined phases except 14, demonstrates that phases 1 and 3 contact with a high probability. b1(1) + b1(3) << b1(13) shows that there is a phase at the contacts that forms channels in phase 13. To determine which phase forms channels in the combined phase 13, let us turn to Table 5 which combines 3 phases. As we see, combining phase 134 also leads to an increase in the number of channels. Since only phase 2 remains unused, we conclude that the channels between phases 1 and 3 are filled with phase 2. The pores of phases 1 and 3 do not affect each other in any way.

Judging by the analysis of the numbers b0(14), b1(14), b2(14), phases 1 and 4 hardly interact: the sums of the numbers of individual phases are approximately equal to the corresponding numbers of the combined phases.

The Betti number b0(23) shows that phases 2 and 3 are in contact with a high probability. But the number b1(23) drops significantly compared to the sum. This indicates that the channels of these phases are closed by each other’s volumes. According to the analysis of the number b0(23), the number of pores does not increase during unification, i.e. the channels do not turn into pores. Therefore, if we remember the significant decrease in b1(23) compared to b1(2) + b1(3), we can conclude that the volumes (solids) of phases 2 and 3 flow into each other’s channels.

The dynamics of the Betti number b0(24) shows that phases 2 and 4 are in contact, but less than 1 and 2, 2 and 3, 3 and 4. Phase 2 is in greater contact with phases 1 and 3 (see the dynamics of the numbers b0(12) and b0(23)), while phase 4 is in almost no contact with phase 1 (see the dynamics of the numbers b0(14)). The dynamics of the numbers b1(24) and b2(24) shows that these phases do not enter each other’s channels and pores. In general, it can be said that these phases interact weakly. Only phases 1 and 4 interact more weakly.

But the interaction of phases 3 and 4 is different. Objects of phases 3 and 4 often contact each other, fill each other’s channels and pores. Considering that there are very few channels in phase 4 (380 channels vs. ~ 70 k objects) and almost no pores at all (see Table 3), it is obvious that phase 4 fills the pores and channels of phase 3. At the same time, phase 4 strongly combines objects of phase 3. Considering the distribution of the sizes of objects of phases 3 and 4 (phase 4 has only small objects, while phase 3 has a few very large objects and a lot of “dust” no larger than 5 voxels, Fig. 7), most likely, phase 4 particles “glue” small objects of phase 3. There are much fewer closed pores in other phases than small objects of phase 3: b2(1) + b2(2) << b0(3). Therefore, these pores of phases 1 and 2 are filled by aggregates of phases 3 and 4. The same is indicated by the analysis of the combined phase 234. Combining phase 34 with phase 2 (Table 5) shows a strong drop in the number of channels and pores, and this may indicate that the combined aggregates of phases 3 and 4 jointly fill the channels and pores of phase 2. Phase 4 is more likely to aggregate with phase 3, and is weakly associated with phases 1 and 2.

Since the particles of phase 4 glue together the small particles of phase 3, they can form ribbons or columns. And these columns fit well into the channels remaining in phases 1 and 2. The number b1(12) of combined phases 1 and 2 ~ 91 k. And these glued particles of phases 3 and 4 can fit well into these channels. And considering that for phases 1, 2, 3 together the number b1(123) is very small, only ~ 2.5 k, we can conclude that it is phase 3 that fills these channels.

When phases 1, 2, 4 (i.e. all phases except the enclosing phase) are combined, we obtain approximately b0(124) ~ 150 k grains versus b2(3) ~ 43 k closed pores in the enclosing phase. This means that all grains of phases 1, 2, 4 occupy all these closed pores together, but many small pores end up on the sample surface, i.e. they are a surface artifact (section “Description for Betti numbers for single phases separately”.). And since the sample is approximately a tetrahedron, the surface to volume ratio in it is quite large (1 to 12). Consequently, the surface occupies a fairly large share of the sample. It can be assumed that the “missing” 100 k objects of phase 124 are in contact with the sample surface, and, therefore, are recognized not as closed pores, but simply as the surface of phase 3. Look at Fig. 1—indeed, the surface of the sample is dotted with spots of different mineral species.

There remain b2(123) ~ 59 k closed pores and b1(123) ~ 2.5 k through channels in jointly phases 1, 2, 3. At the same time, the number of objects of the remaining phase 4 b0(4) ~ 69 k. Therefore, approximately 8 k elements of phase 4 are adjacent to the surface and are not part of a closed pore or through hole.

Discussion of the topology of each phase by Betti numbers

Phase 1 (serpentine). We see the following from the Betti numbers: these are quite a lot of objects. No more than half of them have through holes, but almost no closed pores. That is, these are such panels into which other phases are soldered, but through, in other words, other phases go beyond phase 1 at least on two sides. The characteristic shapes of such “panels” and their topology are shown in Fig. 10a. At the same time, at least half of the phase 1 objects are generally solid. For example, in the sections discussed in the work61, it is not entirely clear how serpentine behaves in volume, whether it completely wraps the olivine grains or the olivine grains are only partially overgrown with serpentine. Our studies show that phase 1 does not always completely cover the phase 2 grains, but often closes the through channels in them.

Fig. 10
figure 10

Characteristic phase shapes revealed using Betti numbers. (a) shape and topology of serpentine “canvases”; (b) shape and topology of olivine grain with a large number of channels filled with serpentine; (c) association of magnetite (phase 4) and minerals of the host fine-grained matrix (phase 3) filling channels in phase 2. The colors correspond to the phases in Fig. 4 and Table 1.

Full size image

Phase 2 (olivine, calcite). We see that these are many individual grains with a highly developed surface: many through holes and some closed pores. The characteristic shapes of large olivine grains and their topology are shown in Fig. 10b. By comparing the number of objects and channels, there are on average 1.35 channels per object and closed pores on average in every 15 grains. This information complements and expands the known data on olivine in the Kontozero lamprophyre. For example, 2D images in backscattered electrons (BSE) show that the olivine grain disintegrates into several smaller and well-faceted ones in61. Our data prove that in volume these grains rather represent a single array with a large number of through tunnels. Pairwise analysis showed that some of them are filled with phase 1, and some are filled with aggregated together small particles of phases 3 and 4 (see section “Description for Betti numbers for phases in pairs and triplets”).

Phase 3 (host mass). The framework into which the other phases are embedded. Phase 3 has significantly fewer objects than phase 1 and phase 2. According to the numbers, each phase 3 object has, on average, 1.8 channels and 0.6 closed pores. However, the statistical averaging approach is not applicable to the analysis of this phase. We see from the tomogram slice image (Fig. 3) and the object size distribution histograms (Fig. 7c) that phase 3 cannot be a large number of large objects. According to the morphology of this phase, it should rather be that all the other phases or several large-volume connected regions are embedded in one large piece of host rock, and all the channels and closed pores are embedded in them. The rest of the amount from b0(3) is small inclusions of this phase in all the others. This is clearly visible in the size distribution histogram (see Fig. 7a). All grains of phases 1, 2, 4 occupy together all closed pores of phase 3, and some number of grains (~ 10 k) come out on the surface of the sample. It is evident that, indeed, its edges are dotted with outcrops of individual grains in the photograph of the sample (Fig. 1).

Phase 4 (magnetite). This is a number of whole small grains. They are most likely embedded in other phases and fill channels and pores in them (as element b1 or b2). Magnetite grains are often associated with minerals of the host matrix and together with them fill channels and pores, for example in olivine and serpentine. Phase 4 objects rarely fill pores and channels of other phases alone. Characteristic forms of the association of Phase 3 and Phase 4 are shown in Fig. 10c.

Geological interpretation

The analysis by integral geometry utilizing Minkowski functionals, including a separate analysis of the Betti numbers, was first applied to the analysis of rocks with more than two X-ray density phases in this work. Previously, this approach was used to study objects with two X-ray density phases: oil and gas bearing rocks41, soils39,40, foams42, etc.

Topological analysis of X-μCT results of meimechite sample of the Kontozero volcano-plutonic complex revealed features of X-ray density phases morphology. Features are well correlated with rock-forming minerals and some secondary ones. These features are well amenable to geological and mineralogical meaningful interpretation.

We assume a direct connection between changes in Betti number dynamics and mineral co-occurrence. We found that olivine (corresponding well to X-ray density phase 2) is broken up by both a fracture network, visible in 2D BSE images, and a channel network. Olivine grains are naturally altered to serpentine (X-ray density phase 1) at their margins and fractures. Magnetite (phase 4) and fine-grained matrix minerals, which are not identified by X-μCT (phase 3), fill the channels in olivine and serpentine. These morphological observations and the results of topological analysis provide the basis for conclusions about the origin of the observed association: early (idiomorphic) olivine forms first, then it is replaced by serpentine. Fine-grained matrix minerals, which are not resolved by X-μCT, form slightly later than olivine. Magnetite is part of the fine-grained mass and is closely intergrown with the matrix minerals.

These conclusions are made solely on the basis of the topological features of the studied sample using minimal mineralogical information to identify X-ray density phases. At the same time, the geological-genetic interpretation is in good agreement with the conclusions made on the basis of geological-mineralogical methods of studying the substance. This was the goal of our methodological experiment: we wanted to demonstrate the possibility of moving from purely physical and mathematical approaches to geological-mineralogical conclusions.

We studied one sample to demonstrate the applicability and fundamental efficiency of the method. We assume that the application of the method to a geologically meaningful set of samples (e.g., for a profile through a deposit or section, etc.) can provide valuable additions to traditional geological methods. For example, a previously unknown property was revealed in our work: olivine is broken up by both a network of through cracks (which we can see in 2D images) and through channels. Moreover, it is typical (in about 2/3 of cases) to be filled with an association of magnetite and fine-grained minerals for these channels.

In order to analyze sets of geological samples on a large scale, it is necessary to develop a method for using topological parameters and obtaining morphological conclusions from them. We made a first approach to this in the next section.

Methodological recommendations

To obtain morphological conclusions from the analysis of Minkowski functionals for rocks with more than two X-ray density phases, we propose the following algorithm for performing the work.

  1. 1.

    Obtain a high-quality tomogram of the sample with the highest possible resolution

  2. 2.

    If the samples are not spherical, it is advisable to cut out spherical areas from the tomogram, if possible, of similar size.

  3. 3.

    Segment the image using the multi-Otsu method into individual X-ray density phases. Check the meaningfulness of the phase selection, i.e. the correspondence of the X-ray density phases to the physical ones (mineral species/group).

  4. 4.

    Select individual objects in each phase using the method of connectivity.

  5. 5.

    Obtain a set of Minkowski functionals M0–M3 for each object.

  6. 6.

    Analyze the Minkowski functionals M0–M2.

  7. 6.1.

    Place the points of each obtained object in the coordinate system M0–M2; a separate graph for each phase.

  8. 6.2.

    Approximate the point cloud with a plane using the linear regression, determine the quality of the fit R2. If R2 is high enough, this means that we have segmented the tomogram correctly (i.e. we have actually identified one physical object, i.e. the same mineral or group of minerals);

  9. 6.3.

    Check for artifacts caused by approaching the X-μCT resolution limit using the method of successive cutoffs from below, and, if necessary, eliminate them.

  10. 6.4.

    10 The corrected plane angle of inclination is used as a classifier of object shapes.

  11. 7.

    Analyze the Minkowski functional M3.

  12. 7.1.

    Find the Minkowski functionals M3 and the Betti numbers that comprise it.

  13. 7.1.1.

    For each X-ray density phase selected in step 4

  14. 7.1.2.

    For the combination of all phases in pairs, in triplets, etc., depending on the number of selected phases.

  15. 7.2.

    Find the difference in Betti numbers between the sums of individual phases and their combinations. Indicate in the table of Betti numbers of combined phases the growth, approximate equality or decrease of values.

  16. 8.

    Conduct a topological interpretation of item 7.

  17. 8.1.

    First, conduct a detailed analysis of the table from paragraph 7.1.1.

  18. 8.2.

    In accordance with the obtained results, consider the remaining tables obtained in paragraphs 7.1.2. and 7.2.

  19. 9.

    Conduct a geological and mineralogical interpretation of item 8.

Geological and mineralogical interpretation should follow from topological interpretation. In other words, when we substitute real mineral species/groups instead of abstract X-ray density phases, we add a time dimension. Time appears because mineral associations are reflections of crystallization processes, i.e. the history of geochemical development of the studied object. If we have a regular series of samples, then we need to make a comparative analysis between the samples. What conclusions and difficulties there may be are a matter for subsequent research.

Conclusions

A sample of meimechite was collected from the Kontozero volcanogenic-sedimentary complex. It characterizes the early stages of emplacement of silicate-carbonate melt. Its 3D tomography was made. Using the multi-Otsu method, 4 X-ray density phases were distinguished. The phases correlated well enough with the minerals known in this rock: fine-grained host matrix (diopside, phase 3), small inclusions of magnetite (phase 4), large faceted crystals of olivine (phase 2) and serpentine overgrowing olivine crystals (phase 1). The topological separation (i.e. the correctness of the segmentation) of these phases was confirmed by the classifier based on the Minkowski functionals M0–M2 with high reliability, R2 = 99.96–99.99. The analysis of Minkowski functionals was applied to a natural rock sample for the first time. More than 2 phases were distinguished in the rock.

The main conclusions of the work are as follows:

  1. 1.

    A classifier has been developed that evaluates the correctness of segmentation.

  2. 2.

    A new approach to studying the structure of objects using integral geometry is proposed, namely studying the behavior of Betti numbers when combining two or more phases.

  3. 3.

    Methodological recommendations have been formulated for obtaining morphological conclusions based on the analysis of Minkowski functionals for rocks with more than two X-ray density phases.

  4. 4.

    Analysis of Betti numbers (elements of the Minkowski functional M3) identified morphological features and phase relationships in the volume. Olivine is individual grains with a highly developed surface with a large number of through channels. The channels are filled with serpentine and an association of small magnetite grains and matrix minerals (presumably carbonates). Serpentine is a web embracing mainly olivine grains. Channels in serpentine are filled with olivine grains and associates of magnetite and fine-grained matrix minerals. Magnetite is a scattering of individual small grains with almost no channels or pores. The matrix contains a very large number of small individual grains (up to 5 voxels) and a small number of large volumes. All other phases are included in the volumes.

  5. 5.

    The results of the topological analysis are interpreted in geological-genetic terms. Early (idiomorphic) olivine forms first, then it is replaced by serpentine. At about the same time, some fine-grained matrix minerals are formed, which are not separated by the X-μCT method. These topological conclusions correlate quite well with known geological interpretations.