A 3D Aggregate Reconstruction and Random Generation Method Based on Spherical DOG Wavelet

Abstract

The present invention discloses a method for three-dimensional reconstruction and random generation of aggregates based on spherical DOG wavelets, comprising: 1. using a three-dimensional scanner to acquire three-dimensional point cloud coordinates of the aggregate surface; 2. performing data simplification on the original point cloud; 3. converting the simplified point cloud data from a Cartesian coordinate system to a spherical coordinate system; 4. discretizing the position and scale of the spherical DOG wavelet based on spherical subdivision to construct a spherical DOG wavelet framework; 5. using a regularization method to solve the spherical DOG wavelet coefficients; 6. using the spherical DOG wavelet series to obtain the aggregate surface function; and 7. randomly varying the amplitude of the spherical DOG wavelet coefficients to randomly generate aggregate particles of different morphologies. The present invention converts a large number of discrete coordinate points on the aggregate surface into a small number of spherical DOG wavelet coefficients to reduce storage space. Based on the spherical DOG wavelet coefficients, a continuous surface function of the aggregate can be obtained, facilitating the calculation of the three-dimensional morphological parameters of the aggregate and interference determination for integration into a geometric model.

Description

Technical Field

[0001] The invention belongs to the technical field of three-dimensional aggregate shape reconstruction, and in particular relates to a three-dimensional aggregate reconstruction and random generation method based on spherical DOG wavelet. Background Art

[0002] Aggregates comprise approximately 50% to 70% of concrete and significantly influence its rheological properties, crack development, and mechanical properties. For a long time, researchers have focused primarily on the impact of aggregate size and gradation on concrete performance, with insufficient attention paid to the influence of aggregate shape. With the widespread application of high-strength, high-performance concrete in recent years, the impact of aggregate shape on various concrete properties has gradually gained attention.

[0003] Early studies on aggregate shape were often limited to two-dimensional (2D) analysis. Digital cameras and microscopes were used to capture aggregate projections. Aggregate metrics such as angularity, roundness, and roughness were calculated based on digital imaging and Fourier reconstruction methods. These metrics were then used to qualitatively explain changes in concrete properties. However, because the 2D aggregate profile depends on the orientation of the image and exhibits significant randomness, it cannot accurately characterize the 3D aggregate morphology. Recent advances in CT and 3D scanning technologies have made it possible to directly obtain 3D point cloud coordinate information from aggregate surfaces. Given the enormous volume of raw surface measurement data, direct calculation of morphological parameters is difficult. Spherical harmonic basis functions, derived from the extension of Fourier functions onto a sphere, have been widely used to process raw 3D aggregate measurement data. Results show that spherical harmonic reconstruction methods can accurately reconstruct the original aggregate morphology using only a few hundred spherical harmonic coefficients, with an error of less than 1%, and a storage compression rate of up to 1%.

[0004] However, due to the global support of spherical harmonic basis functions, when reconstructing the shape of multi-angular particles using a finite-term spherical harmonic series, ringing effects can introduce severe truncation errors as the order of the spherical harmonic reconstruction increases. Therefore, given the recent shortage of natural aggregates and the widespread use of angular and sharp machine-made aggregates, it is essential to find a compactly supported basis function to achieve multi-angular particle reconstruction without truncation errors. To date, no method for reconstructing the shape of three-dimensional aggregate particles based on compactly supported wavelet functions exists. Summary of the Invention

[0005] In order to overcome the shortcomings of the existing technology, a spherical DOG wavelet-based three-dimensional reconstruction and random generation method for aggregates is proposed, in order to more accurately reconstruct the three-dimensional morphology of multi-angular aggregate particles using fewer coefficients, thereby calculating the morphological parameters and determining the interference of the geometric model.

[0006] In order to achieve the above-mentioned object, the present invention adopts the following technical solutions:

[0007] The method for reconstructing and randomly generating the three-dimensional morphology of aggregates based on spherical DOG wavelet of the present invention is characterized in that it comprises the following steps:

[0008] Step 1: Use a 3D scanner to obtain the original 3D point cloud data of the aggregate surface;

[0009] Step 2: Using the volume error VE and the surface area error SE as indicators, the original 3D point cloud data is simplified to obtain the simplified 3D point cloud data;

[0010] Step 3: Convert the simplified 3D point cloud data defined in the 3D Cartesian coordinate system into the spherical coordinate system to obtain simplified 3D point cloud data defined in the spherical coordinate system;

[0011] Step 4: Discretize the position and scale of the continuous spherical DOG wavelet function based on the spherical decomposition method of the regular icosahedron, and construct the spherical DOG wavelet frame F;

[0012] Step 5: Use the framework function in the spherical DOG wavelet framework F to decompose the simplified 3D point cloud data defined in the spherical coordinate system, and use the Tikhonov regularization method to solve the spherical DOG wavelet coefficients {a k |k=1,2,…,M}; where a k represents the kth spherical DOG wavelet expansion coefficient to be determined; M is the number of frame functions in the spherical DOG wavelet frame F;

[0013] Step 6: According to the spherical DOG wavelet coefficient {a k |k=1,2,…,M} and framework function Obtaining approximate surface function of aggregate 3D morphology According to the aggregate approximate surface function The radius of the aggregate at each angle is calculated to obtain the coordinate data of a large number of points on the aggregate surface in the spherical coordinate system. By converting it into the coordinate data in the three-dimensional Cartesian coordinate system, the Delaunay triangulation method is further used to achieve visualization; represents the kth frame function in the spherical DOG wavelet frame F; θ represents the polar angle in the spherical coordinate system, which represents the angle between the line connecting any surface point and the origin and the positive direction of the z axis. It represents the azimuth in the spherical coordinate system, which represents the angle between the line connecting any surface point and the origin and the positive direction of the x-axis;

[0014] Step 7: By randomly changing the amplitude of the DOG wavelet coefficient of some spherical surfaces, aggregate particles with different morphologies are randomly generated according to step 6.

[0015] The method for reconstructing and randomly generating the three-dimensional morphology of aggregates based on spherical DOG wavelet according to the present invention is also characterized in that the step 2 specifically includes:

[0016] Step 2.1: Based on the Delaunay triangulation principle, triangulate the original 3D point cloud data to convert the discrete point cloud in the original 3D point cloud data into triangular facets. The sum of the areas of each triangular facet is calculated and used as the true surface area of the aggregate. The three vertices of each triangular facet are connected to the center of the aggregate to form tetrahedrons. The sum of the volumes of all tetrahedrons is calculated and used as the true volume of the aggregate.

[0017] Step 2.2: Randomly delete some points in the original 3D point cloud data according to the ratio, thereby simplifying the original 3D point cloud data to obtain the 3D point cloud data at the current level of simplification;

[0018] Step 2.3: Perform Delaunay triangulation on the currently simplified 3D point cloud data. Calculate the sum of the areas of the triangular facets obtained as the aggregate surface area at the current level of simplification. Connect the three vertices of each triangular facet with the center of the aggregate to form tetrahedrons. Calculate the sum of the volumes of all tetrahedrons and use this as the aggregate volume at the current level of simplification.

[0019] Step 2.4: Compare the aggregate volume and surface area at the current level of simplification with the true values obtained in step 2.1 to obtain the volume error VE and surface area error SE at the current level of simplification;

[0020] Step 2.5: If the current aggregate volume error VE and surface area error SE both meet the limit requirements, increase the ratio and return to step 2.2. Otherwise, the 3D point cloud data obtained at the previous level of simplification is used as the final 3D point cloud data after simplification.

[0021] The step 4 specifically includes:

[0022] Step 4.1: After moving the center of the icosahedron to the center of the unit sphere, move the vertices of the icosahedron {n j |j=1,2,3,…,12} is projected onto the unit sphere, and the resulting spherical grid is denoted as G0; where n j represents the j-th vertex of the regular icosahedron;

[0023] Step 4.2: Find the midpoints of the sides of the triangles in the regular icosahedron, connect the midpoints of the sides of the triangles to divide the triangle into four small triangles, and replace the vertices {m c |c=1,2,3,…,40} is projected onto the unit sphere, and the resulting spherical grid is denoted as G1; where m cRepresents the vertices of each small triangle;

[0024] Step 4.3: Repeat the process of step 4.2 to obtain spherical grids of different subdivision levels {G q |q=0,1,2,…,K}; where K is the maximum number of spherical subdivisions; G q represents the spherical grid obtained at the qth subdivision level;

[0025] Step 4.4: Separately divide the spherical grids {G q The grid points in |q=0,1,2,…,K} are used as the central poles of the spherical DOG wavelet at the corresponding scale. The spherical DOG wavelet with the central pole at X is obtained according to formula (1):

[0026]

[0027] In formula (1), is the central pole position of the spherical DOG wavelet, where θ0 represents the polar angle of the central pole, represents the azimuth of the central pole; γ is the angle between the central pole and any point X′ on the sphere, where θ′ represents the polar angle of any point on the sphere, Represents the azimuth of any point on the sphere; a = 2 -q , q is the scale, and is consistent with the spherical subdivision level; α is the constant value of the adjustment function shape, and α>1; λ a (γ) is a function related to γ and a, and is obtained from formula (2):

[0028]

[0029] Step 4.5: Select the maximum spherical subdivision level used for the three-dimensional morphology reconstruction of the aggregate as q max , and thus the spherical DOG wavelet frame F for the three-dimensional morphology reconstruction of aggregates is obtained using formula (3):

[0030]

[0031] In formula (3), X (q,j) is the spherical grid G q The jth grid point in a = 2 -q .

[0032] The step 5 specifically includes:

[0033] Step 5.1: Rewrite the frame function in the spherical DOG frame F into the form of formula (4) in a certain order:

[0034]

[0035] In formula (4), M is the total number of functions in the spherical DOG wavelet frame F; is the kth spherical DOG wavelet frame function;

[0036] Step 5.2: Convert the simplified 3D point cloud data defined in the spherical coordinate system obtained in step 3 into a linear combination of the frame function, thereby obtaining the observation equation shown in formula (5):

[0037]

[0038] In formula (5), N is the number of points in the final simplified 3D point cloud data; is the measured value at the nth point cloud data; It is the calculated value of the k-th frame function in the spherical DOG wavelet frame F at the n-th point cloud data;

[0039] Step 5.3: Rewrite the observation equation of Equation (5) into the matrix form of Equation (6):

[0040] r=Gm (6)

[0041] In formula (6), G is the equation coefficient matrix, and r is the measured data matrix, and m is the spherical DOG wavelet coefficient matrix, m=[a1,a2,…,a M ] T ;

[0042] Step 5.4: Use formula (7) to obtain the coefficient matrix m composed of the spherical DOG wavelet coefficients:

[0043] m=(G T C D -1 G+ρ 2 R) -1 G T C D -1 r (7)

[0044] In formula (7), C D is the covariance matrix of the three-dimensional point cloud data after simplification of the aggregate surface; R is the regularization matrix with dimension M×M, and the element R in the kth row and k'th column of R is obtained by formula (8): kk′ ;ρ is the regularization parameter;

[0045]

[0046] In formula (8), S represents the sphere; Ω represents the integral differential element.

[0047] The step 7 specifically includes:

[0048] Step 7.1: Randomly change the amplitudes of some coefficients in the coefficient matrix m to obtain a randomly generated coefficient matrix m1;

[0049] Step 7.2: Calculate the corresponding approximate surface function according to the randomly generated coefficient matrix m1, thereby obtaining the randomly generated aggregate according to the corresponding approximate surface function.

[0050] Compared with the prior art, the present invention has the following beneficial effects:

[0051] 1. The present invention uses a 3D scanner to obtain the original 3D point cloud data of the aggregate surface, and reconstructs the discrete 3D point cloud data based on the compactly supported spherical DOG wavelet framework to obtain a continuous approximate surface function of the aggregate. This greatly facilitates the calculation of aggregate morphological parameters such as volume, surface area, and curvature, and helps to determine particle interference when establishing a concrete microscopic model.

[0052] 2. The present invention converts the storage mode of the original three-dimensional point cloud data into the storage mode of spherical DOG wavelet coefficients. Since the spherical DOG wavelet coefficients have the characteristic of energy concentration towards low frequency, the main morphological information of the original aggregate can be retained by using fewer wavelet coefficients, which is beneficial to reducing the memory usage of aggregate information.

[0053] 3. Compared with the existing spherical harmonic series reconstruction method, the present invention can effectively avoid the truncation error introduced when reconstructing multi-angular particles using globally supported spherical harmonic basis functions due to the compact support characteristics of the spherical DOG wavelet function, and can realize shape modeling with only local variations. Considering the characteristic that residual mortar only exists in local positions of recycled aggregate, it can be used for random modeling of recycled aggregate particle shapes. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] Figure 1 This is a flow chart of the aggregate three-dimensional reconstruction and random generation method based on spherical DOG wavelet of the present invention;

[0055] Figure 2 A schematic diagram of the original surface of the aggregate obtained by the three-dimensional scanner of the present invention;

[0056] Figure 3 Schematic diagram of calculating aggregate volume and surface area based on point cloud data in the present invention;

[0057] Figure 4 Schematic diagram of the aggregate surface when the point cloud data of the present invention is simplified to different numbers;

[0058] Figure 5 Schematic diagram of spherical subdivision for discretizing the scale and position of the spherical DOG wavelet according to the present invention;

[0059] Figure 6 This is a diagram showing the calculation results of the OCV regularization parameter when solving the spherical DOG wavelet expansion coefficients of the present invention;

[0060] Figure 7 Schematic diagram of the machine-made sand aggregate of the present invention after reconstruction by different spherical DOG wavelet frameworks;

[0061] Figure 8 This is a schematic diagram of a new aggregate surface randomly generated by randomly changing the spherical wavelet coefficients of the present invention. DETAILED DESCRIPTION

[0062] In this embodiment, Figure 1 As shown in the figure, a method for reconstructing and randomly generating the three-dimensional shape of aggregates based on spherical DOG wavelets converts a large number of discrete coordinate points on the aggregate surface into a small number of spherical DOG wavelet coefficients to reduce storage space. Based on the spherical DOG wavelet coefficients, a continuous surface function of the aggregate is obtained, facilitating the calculation of the three-dimensional shape parameters of the aggregate and the interference determination of the geometric model. Specifically, the method includes the following steps:

[0063] Step 1: Use a 3D scanner to obtain the original 3D point cloud data of the aggregate surface;

[0064] Step 2: Using the volume error VE and the surface area error SE as indicators, the original 3D point cloud data is simplified to obtain the simplified 3D point cloud data;

[0065] Step 2.1: Based on the Delaunay triangulation principle, the original 3D point cloud data is triangulated to convert the discrete point cloud in the original 3D point cloud data into triangular facets. The sum of the areas of the triangular facets is calculated and used as the true surface area of the aggregate. The three vertices of each triangular facet are connected to the center of the aggregate to form tetrahedrons. The sum of the volumes of all tetrahedrons is calculated and used as the true volume of the aggregate. The aggregate surface area S is calculated based on the 3D point cloud data. r and volume V r The method is shown in formula (1) and formula (2).

[0066]

[0067] In formula (1), S is the surface area of sand particles; S i is the area of the i-th triangle; p1, p2 and p3 represent the three vertices of each triangle respectively.

[0068]

[0069] In formula (2), V is the aggregate volume; V i is the volume of the i-th tetrahedron; O represents the center of the sand grain.

[0070] Step 2.2: Randomly delete some points in the original 3D point cloud data according to the ratio, thereby simplifying the original 3D point cloud data to obtain the 3D point cloud data at the current level of simplification;

[0071] Step 2.3: Perform Delaunay triangulation on the currently simplified 3D point cloud data. Calculate the sum of the areas of the triangular facets obtained as the aggregate surface area at the current level of simplification. Connect the three vertices of each triangular facet with the center of the aggregate to form tetrahedrons. Calculate the sum of the volumes of all tetrahedrons and use this as the aggregate volume at the current level of simplification.

[0072] Step 2.4: Compare the aggregate volume and surface area at the current level of simplification with the true values obtained in step 2.1 to obtain the volume error VE and surface area error SE at the current level of simplification;

[0073] Step 2.5: If the current aggregate volume error VE and surface area error SE both meet the limit requirements, increase the ratio and return to step 2.2. Otherwise, the 3D point cloud data obtained at the previous level of simplification is used as the final 3D point cloud data after simplification.

[0074] Step 3: According to equations (3)-(4), the simplified three-dimensional point cloud data defined in the three-dimensional Cartesian coordinate system is converted into the spherical coordinate system to obtain the simplified three-dimensional point cloud data defined in the spherical coordinate system;

[0075]

[0076]

[0077]

[0078] In formula (3), x i ,y i and z i Represents the 3D point cloud data defined in the 3D Cartesian coordinate system after ith simplification; x c ,y c and z c represents the coordinate of the aggregate center point in the three-dimensional Cartesian coordinate system, which is obtained according to formula (4); θ i is the angle between the line connecting the i-th point and the center point in the aggregate surface point cloud and the positive direction of the z-axis, that is, the zenith angle, is the angle between the line connecting the i-th point and the center point in the aggregate surface point cloud and the positive direction of the x-axis, that is, the azimuth, where counterclockwise rotation is positive and clockwise rotation is negative, r i is the radial distance between the i-th point and the center point in the aggregate surface point cloud.

[0079]

[0080] In formula (1), H is the total number of aggregate surface point clouds.

[0081] Step 4: Considering that the continuous spherical DOG wavelet function is difficult to apply, the position and scale of the continuous spherical DOG wavelet function are discretized based on the spherical decomposition method of the regular icosahedron, and the spherical DOG wavelet framework F is constructed;

[0082] Step 4.1: After moving the center of the icosahedron to the center of the unit sphere, move the vertices of the icosahedron {n j |j=1,2,3,…,12} is projected onto the unit sphere, and the resulting spherical grid is denoted as G0; where n j represents the j-th vertex of the regular icosahedron;

[0083] Step 4.2: Find the midpoints of the sides of the triangles in the regular icosahedron, connect the midpoints of the sides of the triangles to divide the triangle into four small triangles, and replace the vertices {m c |c=1,2,3,…,40} is projected onto the unit sphere, and the resulting spherical grid is denoted as G1; where m c Represents the vertices of each small triangle;

[0084] Step 4.3: Repeat the process of step 4.2 to obtain spherical grids of different subdivision levels {G q |q=0,1,2,…,K}; where K is the maximum number of spherical subdivisions; G q represents the spherical grid obtained at the qth subdivision level;

[0085] Step 4.4: Separately divide the spherical grids {G q The grid points in |q=0,1,2,…,K} are used as the central poles of the spherical DOG wavelet at the corresponding scale. The spherical DOG wavelet with the central pole at X is obtained according to formula (5):

[0086]

[0087] In formula (5), is the central pole position of the spherical DOG wavelet, where θ0 represents the polar angle of the central pole, represents the azimuth of the central pole; γ is the angle between the central pole and point X' on the sphere, where θ′ represents the polar angle of any point on the sphere, Represents the azimuth of any point on the sphere; a = 2 -q , q is the scale, and is consistent with the spherical subdivision level; α is the constant value of the adjustment function shape, and α>1; λ a(γ) is a function related to γ and a, and is obtained from formula (6):

[0088]

[0089] Step 4.5: Select the maximum spherical subdivision level used for the three-dimensional morphology reconstruction of the aggregate as q max , and thus the spherical DOG wavelet frame F for the three-dimensional morphology reconstruction of aggregates is obtained using formula (7):

[0090]

[0091] In formula (7), X (q,j) is the spherical grid G q The jth grid point in a = 2 -q ;q max Indicates the maximum spherical subdivision level used for reconstructing the three-dimensional morphology of the aggregate.

[0092] Step 5: Use the first M frame functions in the spherical DOG wavelet frame F to decompose the simplified 3D point cloud data defined in the spherical coordinate system, and use the Tikhonov regularization method to solve the spherical DOG wavelet coefficients {a k |k=1,2,…,M}; where a k represents the kth spherical DOG wavelet expansion coefficient to be determined; M is the number of frame functions in the spherical DOG wavelet frame F;

[0093] Step 5.1: Rewrite the frame function in the spherical DOG frame F into the form of formula (8) in a certain order:

[0094]

[0095] In formula (8), M is the total number of functions in the spherical DOG wavelet frame F; is the kth spherical DOG wavelet frame function;

[0096] Step 5.2: Convert the simplified 3D point cloud data defined in the spherical coordinate system obtained in step 3 into a linear combination of the frame function, thereby obtaining the observation equation shown in formula (9):

[0097]

[0098] In formula (9), N is the number of points in the final simplified 3D point cloud data; is the measured value at the nth point cloud data; is the calculated value of the k-th frame function in the spherical DOG wavelet frame F at the n-th point cloud data;

[0099] Step 5.3: Rewrite the observation equation of Equation (9) into the matrix form of Equation (10):

[0100] r=Gm (10)

[0101] In formula (10), G is the equation coefficient matrix: r is the measured data matrix: m is the spherical DOG wavelet coefficient matrix, m=[a1,a2,…,a M ] T ;

[0102] Step 5.4: Since the spherical DOG wavelet obtained by position and scale discretization has a large redundancy, solve the spherical DOG wavelet coefficient a k (k=1,2,3,…) will lead to non-unique solutions, that is, ill-posed problems. Therefore, it is necessary to use the Tikhonov regularization method to constrain the solution by introducing the regularization matrix and regularization parameters. Further, using formula (11), the coefficient matrix m composed of the spherical DOG wavelet coefficients is obtained:

[0103] m=(G T C D -1 G+ρ 2 R) -1 G T C D -1 r (11)

[0104] In formula (11), C D is the covariance matrix of the 3D point cloud data after simplification of the aggregate surface; R is an M×M regularization matrix, and the element R in the kth row and k'th column of R is obtained by formula (12): kk′ ρ is the regularization parameter, which can be calculated using methods such as OCV. The basic idea is to first remove a known surface point, calculate the parameter to be determined, and then calculate the difference e between the actual value of the point and the model estimate. i , when H(ρ) in formula (13) reaches the minimum value, the corresponding ρ is the optimal regularization parameter selected.

[0105]

[0106] In formula (12), S represents the sphere; Ω represents the integral differential element;

[0107]

[0108] Step 6: According to the spherical DOG wavelet coefficient {a k |k=1,2,…,M} and framework function Obtaining approximate surface function of aggregate 3D morphology and perform visualization;

[0109] Step 6.1: After obtaining the spherical DOG wavelet coefficient matrix m, the approximate surface function of the aggregate can be obtained by using the spherical DOG wavelet series shown in formula (14):

[0110]

[0111] Step 6.2: Approximating the function based on the aggregate surface The radius of the aggregate at each angle is calculated to obtain the coordinate data of a large number of points on the aggregate surface in the spherical coordinate system. By converting it into the coordinate data in the three-dimensional Cartesian coordinate system, the Delaunay triangulation method is further used to achieve visualization:

[0112]

[0113] Step 7: By randomly changing the amplitude of the DOG wavelet coefficient of some spherical surfaces, aggregate particles with different morphologies are randomly generated according to step 6.

[0114] Step 7.1: Randomly change the amplitudes of some coefficients in the coefficient matrix m to obtain a randomly generated coefficient matrix m1;

[0115] Step 7.2: Calculate the corresponding approximate surface function according to the randomly generated coefficient matrix m1, thereby obtaining the randomly generated aggregate according to the corresponding approximate surface function.

[0116] Example: Combination Figure 1 As shown, a method for reconstructing and randomly generating the three-dimensional morphology of aggregates based on spherical DOG wavelet includes the following steps:

[0117] Step 1: Use a 3D scanner to obtain aggregate particle surface data:

[0118] Limestone machine-made sand particles with a particle size range of 4.75mm-9.50mm were selected, and a high-precision 3D scanner was used to obtain the 3D point cloud data of their surface, which contained a total of 53,000 surface points. The schematic diagram of the original 3D point cloud data after Delaunay triangulation is shown below. Figure 2 As shown;

[0119] Step 2: Reduction of original 3D point cloud data:

[0120] Perform Delaunay triangulation on the original 3D point cloud data of the aggregate and calculate the real volume V and real surface area S of the machine-made sand aggregate (the calculation diagram is shown in the figure). Figure 3 shown), are 177.09mm 3and 205.98mm 2 .

[0121] The original 3D point cloud data is reduced to different quantities according to different ratios (the results are as follows Figure 4 As shown in Figure 3, the volume error and surface area error at each level of simplification should not exceed 1%, and the number of simplified point clouds is finally selected as 6k.

[0122] Step 3: Coordinate system conversion of simplified point cloud data:

[0123] According to formulas (3)-(4), the simplified three-dimensional point cloud data of the machine-made sand aggregate surface is converted from the Cartesian coordinate system to the spherical coordinate system;

[0124] Step 4: Discretize the position and scale of the spherical DOG wavelet function and establish the spherical DOG wavelet framework:

[0125] Select q respectively max 1, 2, 3, 4, where the spherical grid diagram is as follows Figure 5 As shown, four spherical DOG wavelet frames F are established.

[0126] Step 5: Use Tikhonov regularization method to solve the spherical DOG wavelet coefficients:

[0127] Further, the spherical DOG wavelet expansion coefficient matrix m of the aggregate under different spherical DOG wavelet frames is calculated according to formula (10); Figure 6 Given the max The result of calculating the regularization parameter using the OCV method when it is 4.

[0128] Step 6: Reconstruct the aggregate surface function for visualization:

[0129] Using the spherical DOG wavelet series, we can get the approximate surface function of aggregate under different spherical DOG wavelet frames.

[0130] Calculate the radius value of each grid point in the spherical grid G5, and further use formula (15) to convert the coordinate value of each point in the spherical coordinate system into the coordinate value in the three-dimensional Cartesian coordinate system;

[0131] Delaunay triangulation is used to connect the discrete points of the aggregate surface into triangular patches for visualization. Figure 7 This is the three-dimensional shape of the machine-made sand aggregate reconstructed by different spherical DOG wavelet frames.

[0132] Step 7: Randomly change the spherical DOG wavelet coefficients to randomly generate aggregates of arbitrary shapes:

[0133] Randomly change the amplitudes of the 101st to 103th coefficients in the spherical DOG wavelet expansion coefficient matrix m to generate a new coefficient matrix m1;

[0134] The corresponding approximate surface function is calculated according to the randomly generated coefficient matrix m1, and the randomly generated aggregate is obtained according to the corresponding approximate surface function, such as Figure 8 As shown in the figure, the difference in radius value of aggregate at each angle before and after the coefficient is changed is represented by color.

Claims

1. A method for reconstructing and randomly generating three-dimensional aggregate morphology based on spherical DOG wavelet, characterized in that: The following steps are involved: Step 1: Use a 3D scanner to obtain the original 3D point cloud data of the aggregate surface; Step 2: Using the volume error VE and the surface area error SE as indicators, the original 3D point cloud data is simplified to obtain the simplified 3D point cloud data; Step 3: Convert the simplified 3D point cloud data defined in the 3D Cartesian coordinate system into the spherical coordinate system to obtain simplified 3D point cloud data defined in the spherical coordinate system; Step 4: Discretize the position and scale of the continuous spherical DOG wavelet function based on the spherical decomposition method of the regular icosahedron, and construct the spherical DOG wavelet frame F; Step 5: Use the framework function in the spherical DOG wavelet framework F to decompose the simplified 3D point cloud data defined in the spherical coordinate system, and use the Tikhonov regularization method to solve the spherical DOG wavelet coefficients {a k |k=1,2,…,M}; Among them, a k represents the kth spherical DOG wavelet expansion coefficient to be determined; M is the number of frame functions in the spherical DOG wavelet frame F; Step 6: According to the spherical DOG wavelet coefficient {a k |k=1,2,…,M} and framework function Obtaining approximate surface function of aggregate 3D morphology According to the aggregate approximate surface function The radius of the aggregate at each angle is calculated to obtain the coordinate data of a large number of points on the aggregate surface in the spherical coordinate system. By converting it into the coordinate data in the three-dimensional Cartesian coordinate system, the Delaunay triangulation method is further used to achieve visualization; represents the kth frame function in the spherical DOG wavelet frame F; θ represents the polar angle in the spherical coordinate system, which represents the angle between the line connecting any surface point and the origin and the positive direction of the z axis. It represents the azimuth in the spherical coordinate system, which represents the angle between the line connecting any surface point and the origin and the positive direction of the x-axis; Step 7: By randomly changing the amplitude of the DOG wavelet coefficient of some spherical surfaces, aggregate particles with different morphologies are randomly generated according to step 6.

2. The method for reconstructing and randomly generating aggregate three-dimensional morphology based on spherical DOG wavelet according to claim 1 is characterized in that: The step 2 specifically includes: Step 2.1: Based on the Delaunay triangulation principle, triangulate the original 3D point cloud data to convert the discrete point cloud in the original 3D point cloud data into triangular facets. The sum of the areas of each triangular facet is calculated and used as the true surface area of the aggregate. The three vertices of each triangular facet are connected to the center of the aggregate to form tetrahedrons. The sum of the volumes of all tetrahedrons is calculated and used as the true volume of the aggregate. Step 2.2: Randomly delete some points in the original 3D point cloud data according to the ratio, thereby simplifying the original 3D point cloud data to obtain the 3D point cloud data at the current level of simplification; Step 2.3: Perform Delaunay triangulation on the currently simplified 3D point cloud data. Calculate the sum of the areas of the triangular facets obtained as the aggregate surface area at the current level of simplification. Connect the three vertices of each triangular facet with the center of the aggregate to form tetrahedrons. Calculate the sum of the volumes of all tetrahedrons and use this as the aggregate volume at the current level of simplification. Step 2.4: Compare the aggregate volume and surface area at the current level of simplification with the true values obtained in step 2.1 to obtain the volume error VE and surface area error SE at the current level of simplification; Step 2.5: If the current aggregate volume error VE and surface area error SE both meet the limit requirements, increase the ratio and return to step 2.

2. Otherwise, the 3D point cloud data obtained at the previous level of simplification is used as the final 3D point cloud data after simplification.

3. The method for reconstructing and randomly generating aggregate three-dimensional morphology based on spherical DOG wavelet according to claim 1 is characterized in that: The step 4 specifically includes: Step 4.1: After moving the center of the icosahedron to the center of the unit sphere, move the vertices of the icosahedron {n j |j=1,2,3,…,12} is projected onto the unit sphere, and the resulting spherical grid is denoted as G0; where n j represents the j-th vertex of the regular icosahedron; Step 4.2: Find the midpoints of the sides of the triangles in the regular icosahedron, connect the midpoints of the sides of the triangles to divide the triangle into four small triangles, and replace the vertices {m c |c=1,2,3,…,40} is projected onto the unit sphere, and the resulting spherical grid is denoted as G1; where m c Represents the vertices of each small triangle; Step 4.3: Repeat the process of step 4.2 to obtain spherical grids of different subdivision levels {G q |q=0,1,2,…,K}; where K is the maximum number of spherical subdivisions; G q represents the spherical grid obtained at the qth subdivision level; Step 4.4: Separately divide the spherical grids {G q The grid points in |q=0,1,2,…,K} are used as the central poles of the spherical DOG wavelet at the corresponding scale. The spherical DOG wavelet with the central pole at X is obtained according to formula (1): In formula (1), is the central pole position of the spherical DOG wavelet, where θ0 represents the polar angle of the central pole, represents the azimuth of the central pole; γ is the angle between the central pole and any point X′ on the sphere, where θ′ represents the polar angle of any point on the sphere, Represents the azimuth of any point on the sphere; a = 2 -q , q is the scale, and is consistent with the spherical subdivision level; α is the constant value of the adjustment function shape, and α>1; λ a (γ) is a function related to γ and a, and is obtained from formula (2): Step 4.5: Select the maximum spherical subdivision level used for the three-dimensional morphology reconstruction of the aggregate as q max , and thus the spherical DOG wavelet frame F for the three-dimensional morphology reconstruction of aggregates is obtained using formula (3): In formula (3), X (q,j) is the spherical grid G q The jth grid point in a = 2 -q .

4. The method for reconstructing and randomly generating three-dimensional aggregate morphology based on spherical DOG wavelet according to claim 3 is characterized in that: The step 5 specifically includes: Step 5.1: Rewrite the frame function in the spherical DOG frame F into the form of formula (4) in a certain order: In formula (4), M is the total number of functions in the spherical DOG wavelet frame F; is the kth spherical DOG wavelet frame function; Step 5.2: Convert the simplified 3D point cloud data defined in the spherical coordinate system obtained in step 3 into a linear combination of the frame function, thereby obtaining the observation equation shown in formula (5): In formula (5), N is the number of points in the final simplified 3D point cloud data; is the measured value at the nth point cloud data; It is the calculated value of the k-th frame function in the spherical DOG wavelet frame F at the n-th point cloud data; Step 5.3: Rewrite the observation equation of Equation (5) into the matrix form of Equation (6): r=Gm (6) In formula (6), G is the equation coefficient matrix, and r is the measured data matrix, and m is the spherical DOG wavelet coefficient matrix, m=[a1,a2,…,a M ] T ; Step 5.4: Use formula (7) to obtain the coefficient matrix m composed of the spherical DOG wavelet coefficients: m=(G T C D -1 G+ρ 2 R) -1 G T C D -1 r (7) In formula (7), C D is the covariance matrix of the three-dimensional point cloud data after simplification of the aggregate surface; R is the regularization matrix with dimension M×M, and the element R in the kth row and k'th column of R is obtained by formula (8): kk′ ;ρ is the regularization parameter; In formula (8), S represents the sphere; Ω represents the integral differential element.

5. The method for reconstructing and randomly generating three-dimensional aggregate morphology based on spherical DOG wavelet according to claim 4 is characterized in that: The step 7 specifically includes: Step 7.1: Randomly change the amplitudes of some coefficients in the coefficient matrix m to obtain a randomly generated coefficient matrix m1; Step 7.2: Calculate the corresponding approximate surface function according to the randomly generated coefficient matrix m1, thereby obtaining the randomly generated aggregate according to the corresponding approximate surface function.

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