Oil and gas industry uses different types of ceramic proppants in millions of kilograms per year. X-ray microtomography (microCT) imaging can be applied for investigation of quality of the material. For analysis, it is necessary to segment spherical contacting particles of proppant. We apply a marker-controlled watershed for segmentation. The method of markers detection has several parameters which have crucial influence on segmentation outcome. To optimize segmentation quality, we propose unsupervised (nonreference) measure based on a compactness of 3D connected regions, where compactness is calculated via central geometric moments of second order. In addition, we demonstrate advantages of our technique for compactness estimation over the method based on ratio of surface area and volume of a region.
fracturing technology. Annually in the world hundreds of thousands of tons of ceramic proppant are produced, the particles of which are granules of spherical shape with a size of about 1 mm. Mechanical strength and conductivity are the most important attributes of proppant pack for optimal fracturing job design. Crush test is one of conventional procedures for measurement of proppant characteristics. Proppant grains crush test should be performed at axial stress up to 100 MPa. Some particles are crushed during the test. The fraction of the crushed particles depends on stress and quality of the proppant. For some proppants the crush-rate is only a few percent. X-ray micro-tomography (microCT) makes capable measurement of morphometric characteristics of each particle as well as their fragments in initial state and after stress.
Figure 1 shows example of real reconstructed 3D microCT image of proppant pack before stress.
image. Also fragments of crushed particles can be found in images scanned after loading. The usual image of proppant pack contains several hundreds of particles. Intensities of particle regions are much lighter than dark background. However, there is many contacting regions of granules and their fragments. Separation of the contacting regions is a challenging task.
regions. An algorithm for detection of markers has several parameters, which influence to segmentation quality considerably. Since the fraction of broken particles can be equal to several percent, even single segmentation errors leads to bias of an assessment of quality of the material.
preliminary experiments have shown the optimal segmentation parameters vary from image to image; it is impossible to set parameters once based on a previously processed sample. Therefore, it is important to develop an unsupervised (non-reference) quality metrics that allow selection of the parameters automatically with aim to minimize the number of errors of segmentation of proppant particles.
Figure 2 demonstrates flow-chart of segmentation algorithm. Reconstructed grayscale 3D image is
downsampled to image G having size 1000x1000x500 voxels and 8-bits depth. It is necessary to
reduce requirements to memory space and decrease processing time. Lighter voxels of ceramic
particles differ from dark voxels of background and holder. Histogram of intensities of the image is
bimodal. For images with such histograms thresholding by Otsu algorithm [
Conventional way for separation of overlapping or contacting convex regions without holes is an
application of watershed algorithm to the inverted outcome of the geodesic distance transform [
pores are open, they connect with background voids. That is why filling of holes in 3D keeps these pores unchanged. It is required to perform filling of holes for 2D slices. Theoretically, the filling of holes should be done for slices in all three mutually perpendicular directions. Such approach ensures that the open pores, which are penetrate a particle and are parallel to image axes, are filled. However, in practice open pores are tortuous, so, it is enough to fill the holes for 2D slices in only one direction. Binary image Tf is outcome of slice-by-slice filling of holes for T image. It is worth to note, filling of holes in 2D can lead to filling of space between touching particles. Fortunately, it happens seldom and it has no negative impact on the next processing steps.
voxel of Tf image Euclidean distance to the nearest voxel equal to zero. Inverted D plays the role of
relief for watershed algorithm. Local minima on (-D) are basins origins. Frequently there are several
local minima inside a connected region because even survived particles have non-ideal convex shape,
a lot of fragments of crushed particles are concave. It leads to over-segmentation. To avoid
oversegmentation due to the huge number of local minima the marker-controlled watershed is used, where
markers play role of water sources [
T f
R f Markers detection: R = = Rf G T D R M L
L w
max_filter_size delta_h (b) delta_h max_filter_size delta_h
K
Segmentation outcome depends on parameters delta_h and max_filter_size. Decrease of both leads to
over-segmentation, because regions of granules split in fragments. Increase of both leads to
undersegmentation, because neighbouring particles and fragments
merge. How to set the parameters
properly? Subjective choice based on visual analysis of segmentation result is a troublesome due to 3D
nature of data. It is hard to find optimal parameters visually. Paper [
We know a-priori that studied particles have a rounded shape. We use the fact for formulation of criterion of unsupervised segmentation. A sphere is the most compact body in three-dimensional space. In the next section, we describe a compactness for characterizing the closeness of a region shape to a sphere. Maximal average compactness of segmented regions corresponds to the best segmentation: 1 = ∑ , where N is the number of segmented regions having volume greater then Vt; Ci is compactness of i-th region. The threshold Vt is introduced to exclude from consideration too small fragments. In addition, it is hard issue: what is compactness for regions consisting of only a few voxels?
note, proposed segmentation criterion is correct in assumption of existing of at least several tens of survived spherical particles in image after loading. If almost all particles are crushed then the metrics is ineffective.
generally understood as the degree to which a given shape differs from a region bounded by a circle.
There are plenty publications, which describe compactness as ratio of area of region to squared
perimeter. Analogous definition of compactness for 3D regions is ratio of squared volume to area of
surface of a region in cubic power [
area for calculation of compactness for 2D and 3D images correspondingly. First, a perimeter and a surface area are difficult to calculate invariantly to rotation and quite precisely due to the digital nature of images. Second, the surface of the regions is not ideal, there are various types of noises that arise during the registration and segmentation of regions. Third, the evaluation of compactness is seriously affected by holes in regions.
Bribiesca [
marching cubes algorithm [
∑ ( , , )( − 100) ( − 010) ( − 001) , where geometric moments are:
= ∑ ∑
∑ ( , , ) ,
where ( , , ) is indicator function of particle region, I equals one for voxels belonging the region, 000 = . First order geometric moments are centroids. Compactness lies in the range from 0 to 1.
0005/3
Paper [
calculate the compactness coefficients for spheres of different radii with different random noises such as protuberances and cavities on a surface. Figure 4 shows a slice of the sphere with typical noise on the surface. In the next experiment, we determine which method for compactness calculation allows us to distinguish sphere from regions formed by cutting off segments of different sizes from the sphere better.
Plots in figure 5 show various compactnesses depending on radius of sphere: ; , where surface area is calculated by marching cubes algorithm; , where surface area is calculated by means of erosion, Bribiesca’s compactness; theoretical ideal case, where compactness equals 1. for both considered approaches are very different from 1 and have significant fluctuations. is close to 1 except for spheres having radius less than 10. Bribiesca’s compactness is almost 1 for all range of considered radiuses.
Plots in figure 6 show compactnesses for bodies from hemisphere (L=60) to sphere (L=0). Figure 7 illustrates meaning of L parameter. and allow to distinguish sphere and sphere with clipped segment. Bribiesca’s compactness is too similar even for sphere and hemisphere.
We conclude, application of compactness calculated via second order central geometrical moments has obvious advantages in comparison with well-known approaches based on surface area and volume of 3D region. In addition, in contrast to а, number of open and close pores. has robustness to presence of the modest
images with fixed parameters delta_h and max_filter_size as well as with parameters obtained by maximization of metrics. It is worth to note, those fixed parameters were optimal according to criterion for two images segmented previously. We apply gradient descent algorithm for looking for maximal . Figure 8 shows metrics depending on parameters delta_h and max_filter_size.
Table 1 contains for each tested image the number of erroneously segmented particles by segmentation with fixed parameters and by proposed unsupervised segmentation. Total number of particles in each image is 615. Segmentation with fixed parameters leads to 30-50 errors. Figure 9 shows slice of segmented and labeled image by segmentation with fixed parameters. One can see, one ellipsoidal particle was segmented as two regions, ten pairs of neighboring particles were pairwise combined into one region. Unsupervised segmentation had no errors, or just one or two errors in the worst case. Figure 10 demonstrates the slice of 3D image, where unsupervised segmentation was applied. Segmentation outcome is fully correct.