A method for qualitatively and quantitatively characterizing and analyzing point sets in three-dimensional space

Abstract

The present invention proposes a method for qualitatively and quantitatively characterizing and analyzing point sets in three-dimensional space. The method comprises: Step 1, selecting a spatial region to be studied and calculating the k-order average distance between particles in the point set within the region; Step 2, calculating the expectation and standard deviation of the k-order nearest neighbor distance of any particle to the complete spatial randomness; Step 3, obtaining the three-dimensional high-order nearest neighbor index and related Z value in the spatial region through correlation calculation and plotting the NNI. k ‑k curve; Step 4, simulate the different distributions of particles in different morphological spaces, calculate their three-dimensional nearest neighbor index and summarize the general conclusions of their characterization and analysis of point sets in three-dimensional space.

Description

Technical Field

[0001] The present invention relates to a characterization and analysis theory of particle distribution, and specifically to a theoretical method for deriving a three-dimensional high-order nearest neighbor index (NNI) to qualitatively and quantitatively characterize the three-dimensional distribution of a point set in any region of interest (ROI) in three-dimensional space and analyze the curve to obtain point clustering information. Background Art

[0002] Methods for studying point distribution patterns include quadrat counting, kernel functions, and nearest neighbor distances, but these methods are limited to analyzing sets of points in a two-dimensional plane. Nearest neighbor distance analysis (NNI) analyzes the mean of the nearest neighbor distances between points. It has applications in botany, geography, and the distribution of composite material slices, and is used to study the distribution patterns of points in a two-dimensional plane. Its core concept is to compare the deviation from random distribution by calculating the ratio of the average nearest neighbor distance to the average distance in a random distribution pattern.

[0003] There is currently no method to study point distribution patterns in three-dimensional space. Most methods that fit the three-dimensional space by stacking two-dimensional slices and then perform analysis have many disadvantages, such as time-consuming and low accuracy, and the spatial shape of the ROI is limited to the cubic form. There is also no three-dimensional mathematical model of NNI, and no relevant conclusions have been drawn using it to represent and analyze point sets in arbitrary spatial regions. Summary of the Invention

[0004] In order to solve the above technical problems, the present invention proposes a method for qualitatively and quantitatively characterizing and analyzing point sets in three-dimensional space. This method derives a mathematical model of three-dimensional high-order NNI, simulates the different distributions of particles in ROI spaces of different shapes, and summarizes its relevant conclusions for characterizing and analyzing point sets in arbitrary spatial regions: by calculating the three-dimensional first-order NNI (i.e., NNI1) and the three-dimensional first-order Z value (i.e., Z1 value), the distribution of point sets in three-dimensional space can be qualitatively and quantitatively characterized, and by analyzing the three-dimensional high-order NNI curve, the clustering information in the point set can be obtained.

[0005] The technical solution of the present invention is: a method for qualitatively and quantitatively characterizing and analyzing a point set in a three-dimensional space, comprising the following steps:

[0006] Step 1: Select any three-dimensional ROI and calculate the k-th order average distance between particles in the point set within the ROI;

[0007] Step 2: Calculate the expectation and standard deviation of the kth nearest neighbor distance of any particle with respect to the complete spatial randomness;

[0008] Step 3: Obtain the three-dimensional high-order NNI and corresponding Z value within the spatial ROI through correlation calculation and draw the NNIk -k curve;

[0009] Step 4: Simulate the different distributions of particles in different morphological spaces, calculate their three-dimensional NNI, and summarize the general conclusions of the characterization and analysis of point sets in three-dimensional space.

[0010] Beneficial effects:

[0011] Compared with the traditional point set distribution analysis method, the embodiment of the present invention derives a mathematical model of three-dimensional high-order NNI, simulates the different distributions of particles in different morphological spaces, and summarizes that by calculating the NNI1 and Z1 values ​​in the studied spatial region, the distribution pattern and degree of aggregation of the point set in the spatial region can be qualitatively and quantitatively characterized. By analyzing the three-dimensional high-order NNI curve, the clustering information of the point set in the spatial region can be obtained. BRIEF DESCRIPTION OF THE DRAWINGS

[0012] Figure 1 A flowchart of a method for qualitatively and quantitatively characterizing and analyzing a point set in three-dimensional space provided by an embodiment of the present invention;

[0013] Figure 2 The distribution of four typical particles in a sphere provided by an embodiment of the present invention;

[0014] Figure 3 The distribution of four typical particles in a cube provided by an embodiment of the present invention: Figure (a) shows a uniform distribution, Figure (b) shows a random distribution, Figure (c) shows an aggregated distribution, and Figure (d) shows a random-aggregated distribution;

[0015] Figure 4 A two-sided test statistical distribution diagram for determining the degree of aggregation provided by an embodiment of the present invention;

[0016] Figure 5 The NNI obtained by performing three-dimensional high-order NNI calculations on different distributions in the cube provided by the embodiment of the present invention is k -k curve, Figure (a) is uniform distribution, Figure (b) is random distribution, Figure (c) is aggregate distribution, and Figure (d) is random-aggregate distribution;

[0017] Figure 6 The clustering information and NNI in the analysis cube provided by the embodiment of the present invention k -k curve correspondence. DETAILED DESCRIPTION

[0018] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0019] According to an embodiment of the present invention, Figure 1 As shown, the embodiment of the present invention addresses the problem that a point set in three-dimensional space needs to qualitatively and quantitatively characterize the distribution of particles within the studied spatial ROI and further mine clustering information. A method for deriving a three-dimensional high-order NNI mathematical model is proposed to analyze and characterize the distribution of point sets in three-dimensional space and obtain point set particle clustering information. The specific steps of this method are as follows:

[0020] Step S101: Taking an ultra-high temperature ceramic-based nanocomposite as an example, the ROI is any three-dimensional ROI selected in the composite material, and the toughening phase particles distributed therein are recorded as point cluster particles. The size of the particles represents the degree of aggregation of the particles, and the arbitrary k-order average distance between the toughening phase point cluster particles in the ROI is calculated;

[0021] Step S102, calculating the expectation and standard deviation of the k-th nearest neighbor distance of any particle with respect to the complete spatial randomness;

[0022] Step S103: Calculate the three-dimensional high-order NNI and the corresponding Z value and draw the NNI k -k curve;

[0023] Step S104: The distribution pattern and aggregation degree of the toughening phase particles determine the overall performance of the ultra-high temperature ceramic-based nanocomposite material. Therefore, the different distributions of the toughening phase particles in matrices with different morphological spaces are simulated, their three-dimensional NNI is calculated, and the general conclusions of the characterization and analysis of the point set in the three-dimensional space are summarized.

[0024] Furthermore, in step S101, any three-dimensional ROI in the ultra-high temperature ceramic-based nanocomposite material under study is selected, and any particle i is selected from the toughening phase particle point set within the ROI, and the distance set between i and the center of gravity of other particles is recorded as {W i}, take the particle closest to the center of gravity of particle i, calculate the center of gravity distance between the two particles and record it as the first-order nearest neighbor distance W i (1) , the second-order nearest neighbor distance is W i (2) , then the kth nearest neighbor distance of particle i can be expressed as W i (k), k=1,2,3,…≤n-1, the kth order average distance between particles in ROI is d(NN) k , as shown in formula (1):

[0025]

[0026] d(NN) represents a variable name, and n is the total number of particles contained in the point set;

[0027] Furthermore, in step S102, W i (k) The expectation of complete spatial randomness is d(ran) k (Generally d(ran) k It can be expressed as the average distance of the three-dimensional Poisson point distribution, denoted as d(ran) k )W i (k) The standard deviation of complete spatial randomness is denoted as σ, d(ran) k The expressions of and σ are shown in formula (2) and formula (3):

[0028]

[0029]

[0030] Among them, C, C1 and C2 are constant terms, d(ran) represents a variable name, λ is the spatial density, (3k-1)!!! and (3k-2)!!! are the third factorial, that is:

[0031] (3k-1)! ! ! =(3k-1)·(3k-4)! ! ! =...=(3k-1)·(3k-4)·(3k-7)...(3·1-1),

[0032] (3k-2)! ! ! =(3k-2)·(3k-5)! ! ! =...=(3k-2)·(3k-5)·(3k-8)...(3·1-2);

[0033] Furthermore, in step S103, the kth order nearest neighbor index NNI is calculated. k and the Z value of the kth order (ie Z k ), as shown in formula (4) and formula (5):

[0034]

[0035]

[0036] When k=1, we can input formula (4) and formula (5) to obtain NNI1 and Z1, as shown in formula (6) and formula (7):

[0037]

[0038]

[0039] Among them, C3, μ, ν are constant terms;

[0040] Furthermore, in step S104, the spatial form of the composite material is simulated as follows Figure 2 、 Figure 3 The point sets of the four distribution states of uniform distribution, random distribution, aggregate distribution, and random-aggregate distribution of the sphere and cube shown in the figure are calculated using the NNI1 and the corresponding Z1 formula shown in formula (6) and formula (7) to obtain the corresponding values, and summarize the conclusions of their qualitative and quantitative characterization of the distribution of point sets in three-dimensional space. Through relevant calculations, it can be concluded that NNI1 qualitatively characterizes the distribution pattern of particles in the studied spatial region, and the corresponding Z1 value can quantitatively characterize the degree of aggregation of particles. Specifically, it can be summarized as follows: Figure 3 As shown in the case of significance level α = 0.05, the particle distribution of the cube in (a) is uniform, NNI1 will be much greater than 1, and at the same time The null hypothesis that the distribution is random should be rejected; the particle distribution of the cube in (b) is random, the value of three-dimensional NNI1 is approximately equal to 1, and Z1∈(-1.96, 1.96), then the null hypothesis that the distribution is random should be accepted; the particle distribution of the cube in (c) is aggregated, NNI1 will be much less than 1, but always greater than 0, and when The null hypothesis that the distribution is random should be rejected; the particle distribution of the cube in (d) is random-aggregate distribution, NNI1 will be less than 1 but greater than the value of NNI1 of the aggregate distribution in (c), and the Z1 value is farther away from zero than the Z1 value in (c), indicating that the overall relative degree of aggregation of this distribution is lower due to the existence of random distribution. Figure 4 This is a two-sided test statistical distribution diagram for determining the degree of aggregation provided by an embodiment of the present invention.

[0041] Three-dimensional high-order NNI (i.e. NNI) was performed on the four particle distributions of the cube. k Calculate NNI for different k values ​​when k>1) k , you can get NNI k -k curve, analyzing the curve can obtain information such as the size and distance of particle clusters. By comparing the three-dimensional high-order NN curve, multi-scale information of particles can be obtained, such as Figure 5 As shown, the uniformly distributed NNI in (a) k The -k curve shows ups and downs at the beginning, and then the curve becomes smooth; the randomly distributed NNI in (b) kThe -k curve is relatively smooth overall, with a slower increase in the middle and faster increase at the beginning and end; (c) The aggregated NNI k -k curve will appear step-shaped, such as Figure 6 The k value at the bottom of the first step of the curve shown (i.e. k 11 ) is the average number of particles in a single cluster, and the d(NN) value at the bottom of the first step (i.e., d(NN) k11 ) is the equivalent size of any single cluster, and the d(NN) value at the top of the first step (i.e., d(NN) k12 ) is the mean distance between the nearest clusters in any cluster, and the d(NN) value at the top of the second step (i.e., d(NN) k22 ) represents the mean distance between any cluster and the next closest cluster; NNI of random-aggregate distribution in (d) k The -k curve shows similar results to (c), except that the curve is stretched laterally due to the presence of randomly distributed particles. The crack deflection rate within the ultrahigh-temperature ceramic-based nanocomposite is affected by the uniformity of the toughening phase particles, while the Vickers hardness and Young's modulus are affected by the aggregation process and polymer size of the toughening phase particles. Therefore, the above conclusions can quantify the distribution pattern and multi-scale information of the toughening phase particles in the studied ultrahigh-temperature ceramic-based nanocomposite, making it possible to further improve the mechanical properties and mechanical performance of the ultrahigh-temperature ceramic-based nanocomposite.

[0042] The embodiment of the present invention derives the general form of three-dimensional NNI, simulates the different distributions of particles in different morphological spaces, calculates their three-dimensional NNI and summarizes the general conclusions of characterizing and analyzing point sets in three-dimensional space, and can qualitatively and quantitatively characterize the distribution of point sets in three-dimensional space. k -k curve to obtain information such as the size and distance of particle clusters.

[0043] Although the above describes the illustrative specific embodiments of the present invention to facilitate understanding of the present invention by those skilled in the art, and it should be clear that the present invention is not limited to the scope of the specific embodiments, it is obvious to those skilled in the art that as long as various changes are within the spirit and scope of the present invention as defined and determined by the appended claims, these changes are obvious, and all inventions and creations utilizing the concepts of the present invention are protected.

Claims

1. A method for qualitatively and quantitatively characterizing and analyzing a point set in three-dimensional space, characterized in that: For ultra-high temperature ceramic-based nanocomposites, the method comprises the following steps: Step 1: Select the spatial region of interest and calculate the arbitrary k-order average distance between point-set particles within the region. Specifically, for ultra-high temperature ceramic-based nanocomposites, the ROI is any three-dimensional ROI selected in the composite material, and the toughening phase particles distributed therein are recorded as point-set particles. The size of the particles represents the degree of aggregation of the particles, and the arbitrary k-order average distance between the toughening phase point-set particles within the ROI is calculated. Step 2: Calculate the expectation and standard deviation of the kth nearest neighbor distance of any particle to the complete spatial randomness; Step 3: Obtain the three-dimensional high-order nearest neighbor index and corresponding Z value in the spatial area through correlation calculation and draw NNI k -k curve; Step 4: The distribution pattern and degree of aggregation of the toughening phase particles determine the overall performance of the ultra-high temperature ceramic-based nanocomposites. Simulate the different distributions of the toughening phase particles within matrices of different morphological spaces, calculate their three-dimensional nearest neighbor index (NNI), and summarize the conclusions of their characterization and analysis of the point set in the three-dimensional space. In step 1, the spatial region to be studied is selected and any k-order average distance between particles in the point set in the region is calculated, specifically including: Select any region of interest in the three-dimensional space under study, select any particle i from the point set in the ROI, record the set of distances between particle i and the center of gravity of other particles as {Wi}, take the particle closest to the center of gravity of particle i, calculate the distance between the centers of gravity of the two particles and record it as the first-order nearest neighbor distance Wi (1) , the second-order nearest neighbor distance is Wi (2) , then the kth order nearest neighbor distance of particle i can be expressed as Wi (k) ,k=1,2,3,…≤n-1,denoted as d(NN) k is the kth-order average distance between particles in the ROI, as shown in formula (1): （1） d(NN) represents the average distance, and n is the total number of particles contained in the point set; In step 2, the expectation and standard deviation of the k-th nearest neighbor distance of any particle to the complete spatial randomness are calculated, specifically including: kth order nearest neighbor distance Wi (k) The expectation of complete spatial randomness is d(ran) k , d(ran) k Expressed as the average distance of the three-dimensional Poisson point distribution, denoted as d(ran) k , Wi (k) The standard deviation of complete spatial randomness is denoted as σ, d(ran) k The expressions of and σ are shown in formula (2) and formula (3): （2） （3） Among them, C, C1 and C2 are constant terms, λ is the spatial density, (3k-1)!!! and (3k-2)!!! are the third factorial, that is: ， 。 2. The method for qualitatively and quantitatively characterizing and analyzing a point set in three-dimensional space according to claim 1, characterized in that: In step 3, the three-dimensional high-order nearest neighbor index and the corresponding Z value in the spatial region are obtained by correlation calculation and the NNI is plotted. k -k curve, including: The kth nearest neighbor index NNI k and the kth order Z value, namely Z k , as shown in formula (4) and formula (5): （4） （5） Let k = 1 and enter formula (4) and formula (5) to obtain the three-dimensional first-order NNI, namely NNI1, and the three-dimensional first-order Z value, namely Z1, as shown in formula (6) and formula (7): （6） （7） Among them, C3, is a constant term.

3. The method for qualitatively and quantitatively characterizing and analyzing a point set in three-dimensional space according to claim 1, characterized in that: In step 4, different distributions of particles in different morphological spaces are simulated, their three-dimensional nearest neighbor indices are calculated, and the conclusions of characterizing and analyzing the point set in the three-dimensional space are summarized, specifically including: Simulate point sets in four states: uniform distribution, random distribution, aggregate distribution, and random-aggregate distribution in different spatial forms, and calculate the corresponding three-dimensional first-order NNI and the corresponding three-dimensional first-order Z value and three-dimensional high-order NNI. k -k curve; for three-dimensional first-order NNI, it qualitatively characterizes the distribution pattern of particles in the studied spatial region, and combined with the corresponding three-dimensional first-order Z value, it can quantitatively characterize the degree of aggregation of particles; for three-dimensional high-order NNI, for NNI with different k values k , and get NNI k -k curve, analyzing the curve can obtain the size and distance information of particle clusters.

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