CN115222914A - Aggregate three-dimensional morphology spherical harmonic reconstruction method based on three-dimensional point cloud data - Google Patents

Aggregate three-dimensional morphology spherical harmonic reconstruction method based on three-dimensional point cloud data Download PDF

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CN115222914A
CN115222914A CN202210722175.8A CN202210722175A CN115222914A CN 115222914 A CN115222914 A CN 115222914A CN 202210722175 A CN202210722175 A CN 202210722175A CN 115222914 A CN115222914 A CN 115222914A
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aggregate
point cloud
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仇培云
陈昊
韦江雄
高鹏
吴娟
王大海
崔力波
王江卡
余大鹏
王鹏
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South China University of Technology SCUT
Guangzhou Metro Group Co Ltd
China Railway Construction South China Construction Co Ltd
China Railway Guangzhou Investment and Development Co Ltd
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South China University of Technology SCUT
Guangzhou Metro Group Co Ltd
China Railway Construction South China Construction Co Ltd
China Railway Guangzhou Investment and Development Co Ltd
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Abstract

The invention discloses an aggregate three-dimensional shape spherical harmonic reconstruction method based on three-dimensional point cloud data, which comprises the following steps: extracting three-dimensional point cloud coordinates of aggregate particles; surface parameterization, namely converting the aggregate three-dimensional point cloud data into a spherical coordinate system from a Cartesian coordinate system, and obtaining the radius of a reconstruction point by applying an angle approximation algorithm according to the regular increment of a polar angle and an azimuth angle of a Gaussian orthogonal point; calculating a reconstructed point-sphere function based on a Legendre polynomial in combination with a recursive algorithm, and solving a spherical harmonic coefficient by using the sphere function; and summing the spherical harmonic coefficients and the generalized Fourier series on the spherical surface to obtain a surface function reflecting the real morphology of the aggregate, and carrying out visual processing on the reconstructed morphology. According to the invention, the three-dimensional point cloud data of the aggregate particles are converted into spherical harmonic coefficient description, so that the calculation of surface morphology parameters such as sphericity and edge angle of the aggregate particles is greatly facilitated; meanwhile, the storage mode of the three-dimensional point cloud data is converted into the storage mode of the spherical harmonic coefficient, the storage compression ratio of the three-dimensional point cloud data can reach 1%, and a large amount of storage space is saved.

Description

Aggregate three-dimensional morphology spherical harmonic reconstruction method based on three-dimensional point cloud data
Technical Field
The invention belongs to the technical field of aggregate three-dimensional shape reconstruction, and particularly relates to an aggregate three-dimensional shape spherical harmonic reconstruction method based on three-dimensional point cloud data.
Background
The aggregate mainly plays a role in skeleton and filling in concrete, can reduce volume change caused by wet expansion and dry shrinkage in the hardening process of the concrete, has a constraint effect on mortar, directly influences the strength of the concrete and the formation of micro cracks, has different shapes, has different constraint directions and causes different crack forming positions. Therefore, it is necessary to develop a research on the morphology of the aggregate.
The shape characteristics of the aggregate mainly comprise particle size, sphericity, angularity, surface texture and the like, and are one of important factors influencing the workability, strength and shrinkage deformation of concrete. In recent years, a plurality of researchers have developed researches on the influence of the shape characteristics of the aggregates on the performance of the concrete, and the main research directions can be divided into two types, namely the influence of the particle size and the grading of the aggregates on the performance of the concrete and the influence of the particle shape of the aggregates on the performance of the concrete. For a long time in the past, related researchers pay more attention to the influence of the grain size and the grading on the performance of the concrete, and the research on the influence of the grain shape on the performance of the concrete is not much; with the research of high-strength and high-performance concrete, the influence of the particle shape on the appearance of the concrete gradually draws attention of researchers. For the grain shape of the coarse aggregate particles, a relatively comprehensive and uniform evaluation method does not exist at present, the content of the needle flake particles is mostly used as an index for controlling the shape of the coarse aggregate particles in actual engineering, and the irregularity degree of the shape of the coarse aggregate particles cannot be represented substantially and accurately, so that the requirement of the current concrete material on fine and microscopic research cannot be met. Therefore, the three-dimensional morphology of the aggregate needs to be researched.
At present, three-dimensional reconstruction methods for aggregates are based on three modes of CT, laser scanning and SfM to obtain three-dimensional data of the aggregates. However, the three reconstruction methods have the problems of large generated data volume, difficulty in calculating morphological parameters and the like. The spherical harmonic reconstruction can greatly reduce the storage space of data by processing the three-dimensional data and reconstructing the three-dimensional morphology, and is convenient for subsequent calculation of related parameters such as the sphericity, the edge angle, the surface roughness and the like of the aggregate. Therefore, the spherical harmonic reconstruction has a large application potential.
In addition, the national institute of standards and technology, the university of dalf technology, the netherlands, and the like have established a spherical harmonic reconstruction algorithm based on the aggregate three-dimensional voxel data of CT scans. Meanwhile, in the implementation scheme of the prior art, calculation is directly carried out through three-dimensional scanning data or spherical harmonic reconstruction based on voxel data, the calculation is directly carried out through the three-dimensional scanning data, the data distribution is irregular, the calculation amount is large, and the calculation effect is poor; the data acquisition based on spherical harmonic reconstruction of voxel data is usually CT, which is very demanding for the apparatus.
Disclosure of Invention
In order to overcome the defects and shortcomings of the prior art, the invention provides an aggregate three-dimensional morphology spherical harmonic reconstruction method based on three-dimensional point cloud data.
In order to achieve the purpose, the invention adopts the following technical scheme:
a three-dimensional point cloud data-based aggregate three-dimensional topography spherical harmonic reconstruction calculation method comprises the following steps:
step (1): extracting three-dimensional point cloud coordinates of the aggregate from the three-dimensional data;
step (2): surface parameterization, namely converting the aggregate three-dimensional point cloud data into a spherical coordinate system from a Cartesian coordinate system, and taking a polar angle theta and an azimuth angle according to a Gaussian orthogonal point
Figure BDA0003711927450000021
The regular increment is calculated to obtain a radius value of a reconstruction point based on an angle approximation algorithm
Figure BDA0003711927450000022
And (3): method for solving spherical harmonic coefficient a based on Legendre polynomial combined with recursive algorithm nm ；
And (4): the spherical harmonic coefficient a obtained in the step (3) is applied nm Function of and sphere
Figure BDA0003711927450000023
Fourier series summation is carried out to obtain a curved surface function representing the shape of the aggregate
Figure BDA0003711927450000024
Using curved surface functions
Figure BDA0003711927450000025
And visualizing the spherical harmonic reconstruction result.
Further, the step (1) specifically comprises:
step (1.1): converting the format of the three-dimensional data into a preset format;
step (1.2): judging the position of a three-dimensional coordinate array of the stored aggregate according to a structure in a preset format;
step (1.3): presetting initial conditions and finishing conditions for extracting the three-dimensional coordinate array, and extracting three-dimensional point cloud coordinates of the aggregate.
Further, the preset format specifically adopts a wrl format in Meshlab.
Further, the initial conditions for extracting the three-dimensional coordinate array are as follows: the last element of the segmentation character string is equal to point, and the next line has an extraction starting character;
the three-dimensional coordinate array extraction end condition is as follows: and traversing from the beginning of extracting the three-dimensional coordinates to the bottom, and finishing the extraction of the three-dimensional coordinate array when the character is extracted and ended in the first time.
Further, the step (2) specifically comprises:
step (2.1): moving the origin of coordinates to the inside of the particles, selecting a unit vector in the positive direction of the z axis as an initial vector for polar angle calculation, selecting a unit vector in the positive direction of the x axis as an initial vector for azimuth angle calculation, and calculating polar angles, azimuth angles and radius values of corresponding point cloud coordinate points according to point cloud coordinates;
step (2.2): setting the polar angle theta to be in the value range of [0, pi]Azimuth angle of
Figure BDA0003711927450000031
Has a value range of [0,2 pi](ii) a Calculating Legendre polynomial zero points by combining a Gaussian-Legendre product formula and applying a Newton iterative algorithm, taking the zero points of Legendre polynomials as Gaussian orthogonal points, and calculating the weight of the Gaussian orthogonal points by combining Legendre polynomials; dividing the polar angle and the azimuth angle into g parts with unequal intervals according to the solved Gaussian orthogonal point; get reconstructed selection g 2 Polar angle θ (i) and azimuth angle of point
Figure BDA0003711927450000032
Step (2.3): using an angle approximation algorithm to reconstruct the polar angle and the azimuth angle obtained by the calculation in the step (2.1) and the polar angle theta (i) and the azimuth angle of the point selected in the step (2.2)
Figure BDA0003711927450000033
Respectively making difference, taking the radius of the minimum point of the sum of absolute values of the difference values as the radius of the reconstruction point, traversing all the reconstruction points to obtain the radius values of all the reconstruction selected points describing the surface morphology of the aggregate particles
Figure BDA0003711927450000034
Further, calculating the polar angle theta' and azimuth angle of the corresponding point cloud coordinate according to the point cloud coordinate
Figure BDA0003711927450000035
And the radius value r' specifically adopts a first coordinate conversion formula, wherein the first coordinate conversion formula is expressed as follows:
Figure BDA0003711927450000041
Figure BDA0003711927450000042
Figure BDA0003711927450000043
wherein a is a unit vector pointing to the positive direction of the x axis by taking the origin of coordinates as a starting point, c is a unit vector pointing to the positive direction of the z axis by taking the origin of coordinates as a starting point, d is a vector from the origin of coordinates to an aggregate surface point M, b is a projection of the vector d on an x0y plane, and x, y and z are coordinates of the aggregate point cloud in a Cartesian coordinate system.
Further, the step (3) comprises:
and (3).1): setting reconstruction series n, calculating reconstruction point-sphere function by using legendre function and legendre polynomial
Figure BDA0003711927450000044
Reconstructing a point-sphere function
Figure BDA0003711927450000045
The method specifically comprises the following steps:
Figure BDA0003711927450000046
Figure BDA0003711927450000047
Figure BDA0003711927450000048
wherein, x = cos (θ),
Figure BDA0003711927450000049
is an m-th order n-th order associated Legendre function, P n (x) Is an n-th Legendre polynomial;
step (3.2): based on the radius of the reconstruction point calculated in the step (2.3)
Figure BDA00037119274500000410
And (3.1) calculating the spherical function of the reconstructed point obtained in the step
Figure BDA00037119274500000411
Solving the spherical harmonic coefficient a by adopting a spherical harmonic coefficient calculation formula nm Specifically, it is represented as:
Figure BDA0003711927450000051
further, the step (4) comprises:
step (4.1): application ofGeneralized Fourier series on spherical surface, and calculating the surface function with expansion series n
Figure BDA0003711927450000052
Figure BDA0003711927450000053
Wherein,
Figure BDA0003711927450000054
is a curved surface function for describing the surface topography of the particles; a is a nm The spherical harmonic coefficient when the expansion series is n;
Figure BDA0003711927450000055
the spherical function is a spherical function when the expansion series is n;
step (4.2): using curved surface functions
Figure BDA0003711927450000056
Calculating three-dimensional reconstruction coordinates of the aggregate;
step (4.3): and generating parameters required by visualization according to the structure of the wrl format.
Further, in the step (4.2), based on the surface curved function
Figure BDA0003711927450000057
Calculating the three-dimensional reconstruction coordinates (x ', y ', z ') of the aggregate by adopting a second coordinate conversion formula, wherein the second coordinate conversion formula specifically comprises the following steps:
Figure BDA0003711927450000058
Figure BDA0003711927450000059
Figure BDA00037119274500000510
wherein x ', y', z 'constitute three-dimensional reconstruction coordinates (x', y ', z') of the aggregate.
Compared with the prior art, the invention has the following advantages and beneficial effects:
compared with the prior art that the three-dimensional scanning data are directly calculated or the sphere harmonic reconstruction based on the voxel data is carried out by applying the CT technology, the aggregate three-dimensional morphology sphere harmonic reconstruction method based on the three-dimensional point cloud data provided by the invention adopts the sphere harmonic reconstruction based on the point cloud data, firstly, the data acquisition is simpler, and secondly, the low-memory-capacity compression ratio is realized by converting the storage mode of the three-dimensional point cloud data into the storage mode of the sphere harmonic coefficient, so that the calculation of the morphology parameters is simplified while a large amount of memory space is saved.
Drawings
FIG. 1 is a flow chart of steps of the method for reconstructing the spherical resonance of the three-dimensional shape of the aggregate based on three-dimensional point cloud data.
FIG. 2 is a diagram illustrating the initial condition and the cutoff condition for extracting the three-dimensional coordinate array in the format wrl of the present invention.
FIG. 3 is a schematic diagram of a Cartesian coordinate system converted to a spherical coordinate system by a vector method according to the present invention.
FIG. 4 (a) is a schematic structural diagram of the three-dimensional scanner of the present invention before acquiring the aggregate for spherical resonance reconstruction;
fig. 4 (b) is a schematic structural diagram of the three-dimensional scanner of the present invention after acquiring the aggregate and reconstructing the spherical resonance.
FIG. 5 is a schematic diagram of the memory occupancy of the aggregate before and after spherical harmonic reconstruction in the invention.
Detailed Description
In the description of the present disclosure, it should be noted that the terms "center", "upper", "lower", "left", "right", "vertical", "horizontal", "inner", "outer", and the like indicate orientations or positional relationships based on the orientations or positional relationships shown in the drawings, and are only for convenience of describing and simplifying the present disclosure, but do not indicate or imply that the referred device or element must have a specific orientation, be constructed and operated in a specific orientation, and thus, should not be construed as limiting the present disclosure.
Furthermore, the terms "first," "second," and "third" are used for descriptive purposes only and are not to be construed as indicating or implying relative importance. Also, the use of the terms "a," "an," or "the" and similar referents do not denote a limitation of quantity, but rather denote the presence of at least one. The word "comprising" or "comprises", and the like, means that the element or item appearing before the word, includes the element or item listed after the word and its equivalent, but does not exclude other elements or items. The terms "connected" or "coupled" and the like are not restricted to physical or mechanical connections, but may include electrical connections, whether direct or indirect.
In the description of the present disclosure, it is to be noted that the terms "mounted," "connected," and "connected" are to be construed broadly unless otherwise explicitly stated or limited. For example, the connection can be fixed, detachable or integrated; can be mechanically or electrically connected; they may be connected directly or indirectly through intervening media, or they may be interconnected between two elements. The specific meaning of the above terms in the present disclosure can be understood in specific instances by those of ordinary skill in the art. In addition, technical features involved in different embodiments of the present disclosure described below may be combined with each other as long as they do not conflict with each other.
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
Examples
As shown in fig. 1, the present embodiment provides a method for reconstructing a spherical harmonic of an aggregate three-dimensional topography based on three-dimensional point cloud data, the method includes the steps of:
step (1): extracting three-dimensional point cloud coordinates of aggregate from three-dimensional data generated by three-dimensional scanning, sfM and other methods;
in this embodiment, the step (1) specifically includes:
step (1.1): converting the format of the three-dimensional data into a preset format;
step (1.2): judging the position of the three-dimensional coordinate array of the stored aggregate according to the structure of the preset format;
step (1.3): and extracting the three-dimensional point cloud coordinates of the aggregate according to the initial condition and the end condition of the extraction of the preset three-dimensional coordinate array.
In practical application, three-dimensional data obtained by three-dimensional scanning is first converted into wrl format by using format conversion software, such as Meshlab and the like. And judging the position of the three-dimensional coordinate array of the stored aggregate according to the wrl format structure, and further extracting the three-dimensional point cloud coordinate of the aggregate by using the preset initial condition and the preset end condition of the three-dimensional coordinate array extraction.
Step (2): surface parameterization, namely converting the aggregate three-dimensional point cloud data into a spherical coordinate system from a Cartesian coordinate system, and taking a polar angle theta and an azimuth angle according to a Gaussian orthogonal point
Figure BDA0003711927450000071
The regular increment is obtained based on an angle approximation algorithm to obtain a radius value of a reconstruction point
Figure BDA0003711927450000081
In this embodiment, the step (2) specifically includes:
step (2.1): moving the origin of coordinates to the inside of the particles, selecting a unit vector in the positive direction of the z axis as an initial vector for polar angle calculation, selecting a unit vector in the positive direction of the x axis as an initial vector for azimuth angle calculation, and calculating polar angles, azimuth angles and radius values of corresponding point cloud coordinate points according to point cloud coordinates;
step (2.2): setting the polar angle theta to be in the value range of [0, pi]Azimuth angle
Figure BDA0003711927450000082
Has a value range of [0,2 pi](ii) a Combined gauss-legendreCalculating a product formula, namely calculating Legendre polynomial zero points by using a Newton iterative algorithm, taking the zero points of Legendre polynomials as Gaussian orthogonal points, and calculating the weight of the Gaussian orthogonal points by combining the Legendre polynomials; dividing the polar angle and the azimuth angle into g parts with unequal intervals according to the solved Gaussian orthogonal point; obtaining reconstructed selection g 2 Polar angle θ (i) and azimuth angle of point
Figure BDA0003711927450000083
Figure BDA0003711927450000084
nP n (x)＝(2n-1)xP n-1 (x)-(n-1)P n-2 (x) (formula 2)
(1-x 2 )P n '(x)＝nP n-1 (x)-nxP n (x) (formula 3)
Figure BDA0003711927450000085
x i ＝x k+1 ，(lim(x k+1 -x k ) = 0) (formula 5)
Figure BDA0003711927450000086
Figure BDA0003711927450000087
Figure BDA0003711927450000091
Wherein P is n (x) Is Legendre polynomial of degree n, P' n (x) Is the reciprocal of a Legendre polynomial of degree n, x k For the value of x, corresponding to the k-1 iteration in Newton iterations i Zero point of Legendre polynomial, w i Is highOrthogonal point x i Polar angle θ (i) and azimuth
Figure BDA0003711927450000092
Represents a correspondence x i ,x j A polar angle and azimuth angle value, where i, j e (1,2,3,4.... G); formula 1 is the series and differential expression of the Legendre polynomial, formula 2 and formula 3 are the recurrence formulas of the Legendre polynomial and its derivatives, formula 4 and formula 5 are the zero points of the Legendre polynomial calculated by using Newton's iterative algorithm, formula 6 is the Gaussian Legendre orthogonal point weight algorithm, and formula 7 and formula 8 are the dividing methods of polar angle and azimuth angle.
Step (2.3): using an angle approximation algorithm to reconstruct the polar angle and the azimuth angle obtained by the calculation in the step (2.1) and the polar angle theta (i) and the azimuth angle of the point selected in the step (2.2)
Figure BDA0003711927450000093
Respectively making difference, taking the radius of the minimum point of the sum of the absolute values of the difference values as the radius of the reconstruction point, traversing all the reconstruction points to obtain the radius values of all the reconstruction selection points describing the surface appearance of the aggregate particles
Figure BDA0003711927450000094
And (3): calculating a spherical harmonic coefficient based on a Legendre polynomial and a recursive algorithm;
in this embodiment, the step (3) specifically includes:
step (3.1): setting reconstruction series n, and calculating the spherical function of the reconstruction point in the step (2.2) by using the legendre function and the legendre polynomial
Figure BDA0003711927450000095
Figure BDA0003711927450000096
Figure BDA0003711927450000097
Where x represents the cosine of the polar angle theta reconstructing the selected point, i.e. x = cos (theta),
Figure BDA0003711927450000098
is an m-order n-order connected Legendre function m ∈ (-n, -n +1,... 0.... N-1,n), P n (x) Is an n-th order legendre polynomial.
Step (3.2): based on the radius of the reconstruction point calculated in the step (2.3)
Figure BDA0003711927450000099
And (3.1) calculating the spherical function of the reconstructed point
Figure BDA0003711927450000101
Solving the spherical harmonic coefficient a by adopting a spherical harmonic coefficient calculation formula nm ；
Figure BDA0003711927450000102
Wherein the upper formula is spherical harmonic coefficient a nm The formula is calculated by the calculation of the formula,
Figure BDA0003711927450000103
can be directly selected when calculating the value of
Figure BDA0003711927450000104
Intermediate polar angle θ (i) and azimuth angle
Figure BDA0003711927450000105
The weight w (i) w (j) at the gaussian orthogonal point.
And (4): the spherical harmonic coefficient a obtained in the step (3) is applied nm Function of and sphere
Figure BDA0003711927450000106
Fourier series summation is carried out to obtain a curved surface function representing the shape of the aggregate
Figure BDA0003711927450000107
Using curved surface functions
Figure BDA0003711927450000108
And visualizing the spherical harmonic reconstruction result.
In this embodiment, the step (4) specifically includes:
step (4.1): the generalized Fourier series on the spherical surface is used to solve the curved surface function with the expansion series of n
Figure BDA0003711927450000109
Figure BDA00037119274500001010
Wherein
Figure BDA00037119274500001011
Is a curved surface function for describing the surface topography of the particles; a is a nm The spherical harmonic coefficient when the expansion series is n;
Figure BDA00037119274500001012
is a spherical function with the expansion order n.
Step (4.2): using curved surface functions
Figure BDA00037119274500001013
Calculating three-dimensional reconstruction coordinates of the aggregate;
step (4.3): and generating parameters required by visualization according to the structure of the wrl format.
The invention is further described below with reference to the accompanying drawings:
referring to fig. 1, a method for reconstructing a spherical harmonic of an aggregate three-dimensional morphology based on three-dimensional point cloud data includes the steps:
(1) Executing preprocessing work:
(1) converting the stl format data obtained by three-dimensional scanning into wrl format by MeshLab software;
(2) judging the position of the aggregate three-dimensional coordinate array according to the structure in the format of wrl; as shown in fig. 2, the basic structure is wrl format, and the data included in the point array is the three-dimensional coordinates of the aggregate.
(3) Traversing each line of the wrl file, and giving the initial condition of three-dimensional coordinate array extraction: the last element of the segmentation string is equal to point, and the next line has a "[" character; the end conditions are as follows: and traversing from the beginning of extracting the three-dimensional coordinates to the next time, and finishing the extraction of the three-dimensional coordinate array when the character of ']' is encountered for the first time.
(2) Spherical harmonic reconstruction:
(1) as shown in fig. 3, the origin of coordinates is moved to the inside of the particle, the unit vector in the positive direction of the z-axis is selected as the initial vector for calculating the polar angle θ, and the unit vector in the positive direction of the x-axis is the azimuth angle
Figure BDA0003711927450000111
Calculating the polar angle theta' and azimuth angle of the corresponding point cloud coordinate according to the point cloud coordinate
Figure BDA0003711927450000112
And a radius value r ', wherein a polar angle theta' and an azimuth angle of a corresponding point cloud coordinate are calculated according to the point cloud coordinate
Figure BDA0003711927450000113
Specifically, a first coordinate transformation formula is adopted for the radius value r', and the first coordinate transformation formula is expressed as:
Figure BDA0003711927450000114
Figure BDA0003711927450000115
Figure BDA0003711927450000116
wherein a is a unit vector pointing to the positive direction of the x axis by taking the origin of coordinates as a starting point, c is a unit vector pointing to the positive direction of the z axis by taking the origin of coordinates as a starting point, d is a vector from the origin of coordinates O to an aggregate surface point M, b is a projection of the vector d on an x0y plane, and x, y and z are coordinates of the aggregate point cloud in a Cartesian coordinate system.
(2) Setting the polar angle theta to be in the value range of [0, pi]Azimuth angle
Figure BDA0003711927450000117
Has a value range of [0,2 pi]Calculating the zero point of Legendre polynomial by combining a Gaussian-Legendre product formula and applying a Newton iterative algorithm, taking the zero point of Legendre polynomial as a Gaussian orthogonal point, and calculating the weight of the Gaussian orthogonal point by combining the Legendre polynomial; dividing the polar angle and the azimuth angle into g (g) with unequal intervals according to the solved Gaussian orthogonal point>=2 n) parts; obtaining a polar angle theta (i) and an azimuth angle selected by reconstruction
Figure BDA0003711927450000121
i,j∈(1,2,3,4......g)；
(3) Calculating a point cloud coordinate to obtain a polar angle theta' and an azimuth angle by using an angle approximation algorithm
Figure BDA0003711927450000122
Polar angle theta and azimuth angle with reconstructed selected points
Figure BDA0003711927450000123
Respectively making difference, taking the absolute value of the difference value and the radius of the minimum point as the radius of the reconstruction point
Figure BDA0003711927450000124
Traversing all reconstruction points to obtain the radius of all reconstruction points
Figure BDA0003711927450000125
Figure BDA0003711927450000126
Figure BDA0003711927450000127
Wherein theta' (l) is a polar angle calculated by the coordinates of the aggregate point cloud,
Figure BDA0003711927450000128
calculated azimuth (L epsilon (1,L), L is the total number of point clouds obtained by scanning), theta 'for the aggregate point cloud coordinates' min The polar angle of the point with the minimum sum of absolute values of the differences between the polar angle and the azimuth angle of the reconstruction point in all the point clouds,
Figure BDA0003711927450000129
the azimuth angle of the point with the minimum sum of the absolute values of the differences between the polar angle and the azimuth angle of the reconstructed point in all the point clouds is obtained.
(4) Setting the reconstruction number of stages to n (n)>= 10), calculate the spherical function of the reconstruction point using equation 9 to equation 10
Figure BDA00037119274500001210
(5) Solving spherical harmonic coefficient a by using formula 11 nm 。
(3) And (3) post-treatment work:
(1) by using equation 12, a curved function with an expansion order of n is obtained
Figure BDA00037119274500001211
(2) Based on surface function
Figure BDA00037119274500001212
Calculating the three-dimensional reconstruction coordinates (x ', y ', z ') of the aggregate by adopting a second coordinate conversion formula, wherein the second coordinate conversion formula specifically comprises the following steps:
Figure BDA00037119274500001213
Figure BDA00037119274500001214
Figure BDA00037119274500001215
wherein x ', y', z 'constitute the three-dimensional reconstruction coordinates (x', y ', z') of the aggregate.
(3) With reference to fig. 2, parameters required for visualization are generated according to the wrl format structure, and the calculated connection mode between the three-dimensional coordinates and the points is output to the wrl text.
As shown in fig. 4 (a) and 4 (b), the aggregate and spherical harmonic reconstruction visualization results obtained by the three-dimensional scanner are shown. As shown in fig. 5, in the embodiment, the storage mode of the three-dimensional point cloud data is converted into the storage mode of the spherical harmonic coefficient, the file compression rate is less than 1%, the storage space is greatly saved, and the surface function is reconstructed to obtain the morphology parameters more conveniently;
the above embodiments are preferred embodiments of the present invention, but the present invention is not limited to the above embodiments, and any other changes, modifications, substitutions, combinations, and simplifications which do not depart from the spirit and principle of the present invention should be construed as equivalents thereof, and all such changes, modifications, substitutions, combinations, and simplifications are intended to be included in the scope of the present invention.

Claims (9)

1. An aggregate three-dimensional shape spherical resonance reconstruction method based on three-dimensional point cloud data is characterized by comprising the following steps:
step (1): extracting three-dimensional point cloud coordinates of aggregate from the three-dimensional data;
step (2): surface parameterization, namely converting the aggregate three-dimensional point cloud data into a spherical coordinate system from a Cartesian coordinate system, and taking a polar angle theta and an azimuth angle according to a Gaussian orthogonal point
Figure FDA0003711927440000011
Based on angle approximation algorithm to obtain radius value of reconstruction point
Figure FDA0003711927440000012
And (3): method for solving spherical harmonic coefficient a based on Legendre polynomial combined with recursive algorithm nm ；
And (4): using the spherical harmonic coefficient a nm Function of and sphere
Figure FDA0003711927440000013
Fourier series summation is carried out to obtain a curved surface function representing the shape of the aggregate
Figure FDA0003711927440000014
Using curved surface functions
Figure FDA0003711927440000015
And visualizing the spherical harmonic reconstruction result.
2. The method for reconstructing the spherical harmonic of the three-dimensional morphology of the aggregate based on the three-dimensional point cloud data according to claim 1, wherein the step (1) specifically comprises the following steps:
step (1.1): converting the format of the three-dimensional data into a preset format;
step (1.2): judging the position of a three-dimensional coordinate array of the stored aggregate according to a structure in a preset format;
step (1.3): presetting initial conditions and finishing conditions for extracting the three-dimensional coordinate array, and extracting three-dimensional point cloud coordinates of the aggregate.
3. The method for reconstructing the spherical harmonic of the three-dimensional morphology of the aggregate based on the three-dimensional point cloud data as claimed in claim 2, wherein in the step (1.1), the preset format is wrl format in Meshlab.
4. The method for reconstructing the spherical resonance of the three-dimensional morphology of the aggregate based on the three-dimensional point cloud data as claimed in claim 2, wherein the initial conditions for extracting the three-dimensional coordinate array are as follows: the last element of the segmentation character string is equal to point, and the next line has an extraction starting character;
the three-dimensional coordinate array extraction end condition is as follows: and traversing from the beginning of extracting the three-dimensional coordinates to the bottom, and finishing the extraction of the three-dimensional coordinate array when the character is extracted and ended in the first time.
5. The method for reconstructing the spherical resonance of the three-dimensional morphology of the aggregate based on the three-dimensional point cloud data according to claim 1, wherein the step (2) specifically comprises the following steps:
step (2.1): moving the origin of coordinates to the inside of the particles, selecting a unit vector in the positive direction of the z axis as an initial vector for polar angle calculation, selecting a unit vector in the positive direction of the x axis as an initial vector for azimuth angle calculation, and calculating polar angles, azimuth angles and radius values of corresponding point cloud coordinate points according to point cloud coordinates;
step (2.2): setting the polar angle theta to be in the value range of [0, pi]Azimuth angle
Figure FDA0003711927440000021
Has a value range of [0,2 pi](ii) a Calculating Legendre polynomial zero points by combining a Gaussian-Legendre product formula and applying a Newton iterative algorithm, taking the zero points of Legendre polynomials as Gaussian orthogonal points, and calculating the weight of the Gaussian orthogonal points by combining Legendre polynomials; dividing the polar angle and the azimuth angle into g parts with unequal intervals according to the solved Gaussian orthogonal point; obtaining reconstructed selection g 2 Polar angle θ (i) and azimuth angle of point
Figure FDA0003711927440000022
Step (2.3): applying an angle approximation algorithm to reconstruct the polar angle and the azimuth angle calculated in the step (2.1) and the polar angle theta (i) and the azimuth angle of the point selected by reconstruction
Figure FDA0003711927440000023
Respectively making difference, taking the radius of the minimum point of the sum of the absolute values of the difference values as the radius of the reconstruction point, traversing all the reconstruction points to obtain the radius values of all the reconstruction selection points describing the surface appearance of the aggregate particles
Figure FDA0003711927440000024
6. The method for reconstructing the spherical harmonic of the three-dimensional morphology of the aggregate based on the three-dimensional point cloud data as claimed in claim 5, wherein the polar angle θ' and the azimuth angle of the corresponding point cloud coordinate are calculated according to the point cloud coordinate
Figure FDA0003711927440000025
And the radius value r' specifically adopts a first coordinate conversion formula, wherein the first coordinate conversion formula is expressed as follows:
Figure FDA0003711927440000026
Figure FDA0003711927440000027
Figure FDA0003711927440000028
wherein a is a unit vector pointing to the positive direction of the x axis by taking the origin of coordinates as a starting point, c is a unit vector pointing to the positive direction of the z axis by taking the origin of coordinates as a starting point, d is a vector from the origin of coordinates to an aggregate surface point M, b is a projection of the vector d on an x0y plane, and x, y and z are coordinates of the aggregate point cloud in a Cartesian coordinate system.
7. The method for reconstructing the spherical resonance of the three-dimensional morphology of the aggregate based on the three-dimensional point cloud data according to claim 1, wherein the step (3) specifically comprises the following steps:
step (3.1): setting reconstruction series n, calculating reconstruction point-sphere function by using legendre function and legendre polynomial
Figure FDA0003711927440000031
Reconstructing a point-sphere function
Figure FDA0003711927440000032
The method specifically comprises the following steps:
Figure FDA0003711927440000033
Figure FDA0003711927440000034
Figure FDA0003711927440000035
wherein x = cos (θ),
Figure FDA0003711927440000036
is an m-th order n-th order associated Legendre function, P n (x) Is an n-th Legendre polynomial;
step (3.2): radius based on calculated reconstruction points
Figure FDA0003711927440000037
And (3.1) calculating the spherical function of the reconstructed point
Figure FDA0003711927440000038
Solving the spherical harmonic coefficient a nm Specifically, it is represented as:
Figure FDA0003711927440000039
8. the method for reconstructing the spherical harmonic of the three-dimensional morphology of the aggregate based on the three-dimensional point cloud data as claimed in claim 1, wherein the step (4) specifically comprises:
step (4.1): the generalized Fourier series on the spherical surface is used to solve the curved surface function with the expansion series of n
Figure FDA00037119274400000310
Figure FDA00037119274400000311
Wherein,
Figure FDA00037119274400000312
is a curved surface function for describing the surface topography of the particles; a is nm The spherical harmonic coefficient when the expansion series is n;
Figure FDA00037119274400000313
the spherical function is a spherical function when the expansion series is n;
step (4.2): using curved surface functions
Figure FDA00037119274400000314
Calculating three-dimensional reconstruction coordinates of the aggregate;
step (4.3): and generating parameters required by visualization according to the structure of the wrl format.
9. The method for reconstructing the spherical harmonic of the three-dimensional morphology of the aggregate based on the three-dimensional point cloud data as claimed in claim 8, wherein in the step (4.2), the method is based on a surface function
Figure FDA0003711927440000041
Polar angle theta and azimuth angle
Figure FDA0003711927440000042
Calculating the three-dimensional reconstruction coordinates (x ', y ', z ') of the aggregate by adopting a second coordinate conversion formula, wherein the second coordinate conversion formula specifically comprises the following steps:
Figure FDA0003711927440000043
Figure FDA0003711927440000044
Figure FDA0003711927440000045
wherein x ', y', z 'constitute the three-dimensional reconstruction coordinates (x', y ', z') of the aggregate.
CN202210722175.8A 2022-06-24 2022-06-24 Aggregate three-dimensional morphology spherical harmonic reconstruction method based on three-dimensional point cloud data Pending CN115222914A (en)

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Publication number Priority date Publication date Assignee Title
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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN116385642A (en) * 2023-03-31 2023-07-04 浙江大学 Three-dimensional scalar information compression reconstruction method based on spherical Shearlet
CN116385642B (en) * 2023-03-31 2023-09-12 浙江大学 Three-dimensional scalar information compression reconstruction method based on spherical Shearlet

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