一种融合频率扰动的KL展开3D颗粒随机建模方法

doi:10.11996/JG.j.2095-302X.2026040736

图学学报 ›› 2026, Vol. 47 ›› Issue (4): 736-745.DOI: 10.11996/JG.j.2095-302X.2026040736

• 计算机图形学与虚拟现实 • 上一篇下一篇

一种融合频率扰动的KL展开3D颗粒随机建模方法

赵啦啦¹(), 杨亦卓¹, 段晨龙², 郭辰昊¹, 王清龙¹, 王宏都³

¹ 中国矿业大学机电工程学院，江苏徐州 221116
² 中国矿业大学化工学院，江苏徐州 221116
³ 玉溪大红山矿业有限公司，云南玉溪 653405

收稿日期:2025-12-23 接受日期:2026-05-18 出版日期:2026-08-31 发布日期:2026-08-31
通讯作者:赵啦啦，E-mail：lala.zhao@cumt.edu.cn
基金资助:
国家自然科学基金(52075535);国家自然科学基金(52261135540);国家杰出青年科学基金(52125403)

A 3D random particle modeling method integrating KL expansion and frequency perturbation

ZHAO Lala¹(), YANG Yizhuo¹, DUAN Chenlong², GUO Chenhao¹, WANG Qinglong¹, WANG Hongdu³

¹ School of Mechanical and Electrical Engineering, China University of Mining & Technology, Xuzhou Jiangsu 221116, China
² School of Chemical Engineering and Technology, China University of Mining & Technology, Xuzhou Jiangsu 221116, China
³ Yuxi Dahongshan Mining Co., Ltd., Yuxi Yunnan 653405, China

Received:2025-12-23 Accepted:2026-05-18 Published:2026-08-31 Online:2026-08-31
Contact: ZHAO Lala，E-mail：lala.zhao@cumt.edu.cn
Supported by:
National Natural Science Foundation of China(52075535);National Natural Science Foundation of China(52261135540);National Science Fund for Distinguished Young Scholars(52125403)

摘要/Abstract

摘要：

高效建立具有真实颗粒几何与形状特征的随机颗粒模型数据集是开展3D颗粒识别及颗粒仿真等应用的关键问题，针对传统随机建模方法存在的颗粒形态真实性不足以及生成效率较低等问题，提出了一种融合频率扰动的KL (Karhunen-Loeve)展开3D颗粒随机建模方法。首先，对真实矿物颗粒表面进行三维重建与频谱分析，提取颗粒表面频谱特征，构建颗粒表面扰动函数。然后，引入KL展开对离散化的颗粒表面采样数据进行特征分解与相关性分析，通过保留主要特征模态实现主结构形态的随机生成，并将真实颗粒的频率扰动信息融入主结构，生成兼具主结构与局部扰动的3D随机颗粒模型，有效提高颗粒随机建模的真实性与多样性。最后，为验证所提方法的有效性，与2种改进的球谐分析方法进行对比，从几何特征参数(长、宽、高)、形状特征参数(长宽比、球形度、法向扰动角)，以及包含Wasserstein距离和模型生成效率在内的综合指标进行评估。结果表明，所生成颗粒模型的几何及形状特征参数都与真实颗粒最为接近，同时，与真实颗粒间的Wasserstein距离仅为0.397 3，低于基于分形特性的改进球谐分析方法的0.569 4与基于基因变异的改进球谐分析方法的1.066 7，生成相同数量颗粒模型的耗时较低(15.3 s)，在精度与效率之间取得了良好平衡。为3D颗粒识别及仿真等相关应用场景提供了一种新的高效生成具有真实颗粒特征的3D颗粒随机建模方法。

关键词: 3D颗粒模型, KL展开, 频率扰动, 随机建模, 真实颗粒

Abstract:

Efficiently establishing a random particle-model dataset with realistic geometric and shape characteristics is a critical issue for applications such as 3D particle recognition and particle simulation. To address the problems of insufficient morphological realism and low generation efficiency in traditional random modeling methods, a 3D random particle modeling method based on KL (Karhunen-Loeve) expansion integrated with frequency perturbation was proposed. First, three-dimensional reconstruction and spectral analysis were performed on real mineral particle surfaces to extract surface spectral features and construct particle-surface perturbation functions. Then, KL expansion was introduced to perform feature decomposition and correlation analysis on discretized particle-surface sampling data. By retaining the principal feature modes, the random generation of the primary structural morphology was achieved. Furthermore, the frequency perturbation information from real particles was incorporated into the primary structure, generating 3D random particle models with both primary structural characteristics and local perturbation features, thereby effectively improving the realism and diversity of random particle modeling. Finally, to verify the effectiveness of the proposed method, comparisons were conducted with two improved spherical harmonic analysis methods. The evaluation was performed from the perspectives of geometric characteristic parameters (length, width, and height), shape characteristic parameters (aspect ratio, sphericity, and normal perturbation angle), as well as comprehensive metrics including Wasserstein distance and model-generation efficiency. The results demonstrated that the generated particle models exhibited geometric and shape characteristic parameters that were closest to those of real particles. Moreover, the Wasserstein distance between the generated models and real particles was only 0.397 3, which was lower than 0.569 4 for the improved spherical harmonic analysis method based on fractal characteristics and 1.066 7 for the improved spherical harmonic analysis method based on genetic variation. In addition, the proposed method required less time (15.3 s) to generate the same number of particle models, achieving a favorable balance between accuracy and efficiency. The proposed method provided a new and efficient approach for generating 3D random particle models with realistic particle characteristics for applications such as 3D particle recognition and simulation.

Key words: 3D particle models, KL expansion, frequency perturbation, stochastic modeling, real particles

中图分类号:

TP391

赵啦啦, 杨亦卓, 段晨龙, 郭辰昊, 王清龙, 王宏都. 一种融合频率扰动的KL展开3D颗粒随机建模方法[J]. 图学学报, 2026, 47(4): 736-745.

ZHAO Lala, YANG Yizhuo, DUAN Chenlong, GUO Chenhao, WANG Qinglong, WANG Hongdu. A 3D random particle modeling method integrating KL expansion and frequency perturbation[J]. Journal of Graphics, 2026, 47(4): 736-745.

图/表 12

图1 本文方法核心流程

Fig. 1 Proposed method core process

图2 矿物颗粒样本3D扫描实验配置((a) 实验系统组成；(b) 识别误差)

Fig. 2 Mineral particle sample 3d scanning experiment configuration ((a) Experimental system composition; (b) Recognition error)

图3 颗粒3D数据点云对齐过程((a) 过程1；(b) 过程2)

Fig. 3 Particle 3d point cloud alignment process ((a) Process 1; (b) Process 2)

图4 真实颗粒样本模型

Fig. 4 Real particle sample model

图5 三维高斯向量归一化((a) 原始标准三维高斯；(b) 归一化后方向分布)

Fig. 5 3D Gaussian vector normalization ((a) Original standard 3D gaussian; (b) Normalized direction distribution)

图6 异形颗粒模型

Fig. 6 Irregular particle model

图7 3D高斯相关性椭球面(3.5σ等值面)

Fig. 7 3D Gaussian-correlated ellipsoid (3.5σ contour)

图8 3种颗粒模型生成方法对比((a) 基于分形特性的改进球谐分析法；(b) 基于基因变异的改进球谐分析法；(c) 本文方法)

Fig. 8 Comparison of three particle model generation methods ((a) Improved spherical harmonic analysis based on fractal characteristics; (b) Improved spherical harmonic analysis based on genetic variation; (c) Ours)

图9 颗粒特征参数示意图((a) 颗粒模型主惯性轴；(b) 颗粒最小长方体包围盒；(c) 颗粒面法向扰动角)

Fig. 9 Schematic diagram of particle characteristic parameters ((a) Principal axes of the particle model; (b) Minimum cuboid bounding box of the particle; (c) Particle surface normal perturbation angle)

图10 颗粒模型几何特征参数概率密度分布曲线((a) 长度概率密度分布曲线；(b) 宽度概率密度分布曲线；(c) 高度概率密度分布曲线)

Fig. 10 Probability density distribution curve of geometric parameters for a particle model ((a) Length probability density distribution curve; (b) Width probability density distribution curve; (c) Height probability density distribution curve)

图11 颗粒模型形状特征参数概率密度分布曲线((a) 不同颗粒模型长宽比的概率密度分布曲线；(b) 不同颗粒模型球形度的概率密度分布曲线；(c) 不同颗粒模型法向扰动角的概率密度分布曲线)

Fig. 11 Probability density distribution curve of geometric parameters for a particle model ((a) Probability density distribution curves of the aspect ratios of different particle models; (b) Probability density distribution curves of the sphericity of different particle models; (c) Probability density distribution curves of the normal perturbation angles of different particle models)

表1 不同的颗粒随机建模方法综合评价指标对比

Table 1 Comparative analysis of comprehensive evaluation metrics for different particle random modeling methods

方法	Wasserstein距离	颗粒建模用时/s
基于分形特性的改进球谐分析方法	0.569 4	5.79
基于基因变异的改进球谐分析方法	1.066 7	712.56
本文方法	0.397 3	15.30

参考文献 22

[1]	杜欣, 曾亚武, 高睿, 等. 基于CT扫描的不规则外形颗粒三维离散元建模[J]. 上海交通大学学报, 2011, 45(5): 711-715.
	DU X, ZENG Y W, GAO R, et al. 3-D modelling of irregular shape particles for discrete element method based on X-ray tomography[J]. Journal of Shanghai Jiaotong University, 2011, 45(5): 711-715 (in Chinese).
[2]	边学成, 李伟, 李公羽, 等. 基于颗粒真实几何形状的铁路道砟剪切过程三维离散元分析[J]. 工程力学, 2015, 32(5): 64-75, 83.
	BIAN X C, LI W, LI G Y, et al. Three-dimensional discrete element analysis of railway ballast’s shear process based on particles’ real geometry[J]. Engineering Mechanics, 2015, 32(5): 64-75, 83 (in Chinese).
[3]	李典哲, 刘璐, 季顺迎. 基于扩展多面体组合单元的非规则颗粒材料离散元方法[J]. 应用数学和力学, 2023, 44(7): 769-783.
	LI D Z, LIU L, JI S Y. A discrete element method for irregular granular materials based on multi-dilated polyhedron elements[J]. Applied Mathematics and Mechanics, 2023, 44(7): 769-783 (in Chinese).
[4]	罗滔, 李刚, OOI E T, 等. 堆石体宏细观力学特性演化机制的离散元模拟[J]. 武汉大学学报(工学版), 2018, 51(7): 607-612.
	LUO T, LI G, OOI E T, et al. DEM modelling of macro- and meso-mechanisms for rockfill materials[J]. Engineering Journal of Wuhan University, 2018, 51(7): 607-612 (in Chinese).
[5]	李世亮, 方强, 王屹华, 等. 基于动态剪枝的跨域小样本图像生成方法研究[J]. 图学学报, 2026, 47(1): 131-142. DOI
	LI S L, FANG Q, WANG Y H, et al. A dynamic pruning approach for cross-domain few-shot image generation[J]. Journal of Graphics, 2026, 47(1): 131-142 (in Chinese). DOI
[6]	LI F, LIU X Y, YIN Y F, et al. A novel method for particle instance segmentation and size measurement[J]. IEEE Transactions on Instrumentation and Measurement, 2024, 73: 5006817.
[7]	WU X Y, KROELL N, GREIFF K. Deep learning-based instance segmentation on 3D laser triangulation data for inline monitoring of particle size distributions in construction and demolition waste recycling[J]. Resources, Conservation and Recycling, 2024, 205: 107541. DOI URL
[8]	孙禾衣, 李艺潇, 田希, 等. 结合程序内容生成与扩散模型的图像到三维瓷瓶生成技术[J]. 图学学报, 2025, 46(2): 332-344. DOI
	SUN H Y, LI Y X, TIAN X, et al. Image to 3D vase generation technology combining procedural content generation and diffusion models[J]. Journal of Graphics, 2025, 46(2): 332-344 (in Chinese). DOI
[9]	MOLLON G, ZHAO J D. 3D generation of realistic granular samples based on random fields theory and Fourier shape descriptors[J]. Computer Methods in Applied Mechanics and Engineering, 2014, 279: 46-65. DOI URL
[10]	SUN Q, ZHENG J X. Clone granular soils with mixed particle morphological characteristics by integrating spherical harmonics with Gaussian mixture model, expectation-maximization, and Dirichlet process[J]. Acta Geotechnica, 2020, 15(10): 2779-2796. DOI
[11]	SUN Q, ZHENG J X. Realistic soil particle generation based on limited morphological information by probability-based spherical harmonics[J]. Computational Particle Mechanics, 2021, 8(2): 215-235. DOI URL
[12]	刘初升, 赵啦啦, 闫俊霞, 等. 矿物颗粒的随机分形仿真算法[J]. 工程图学学报, 2009, 30(6): 66-70.
	LIU C S, ZHAO L L, YAN J X, et al. Study on random fractal simulation algorithm for mineral particles[J]. Journal of Engineering Graphics, 2009, 30(6): 66-70 (in Chinese).
[13]	陈泽仁, 刘正彬, 关威, 等. 面向播种过程离散元仿真的玉米颗粒建模方法[J]. 农业工程学报, 2024, 40(14): 14-22.
	CHEN Z R, LIU Z B, GUAN W, et al. Maize grain modelling for the DEM simulation of sowing process[J]. Transactions of the Chinese Society of Agricultural Engineering, 2024, 40(14): 14-22 (in Chinese).
[14]	崔泽群, 陈友川, 赵永志, 等. 基于超二次曲面的非球形离散单元模型研究[J]. 计算力学学报, 2013, 30(6): 854-859.
	CUI Z Q, CHEN Y C, ZHAO Y Z, et al. Study of discrete element model for non-sphere particles base on super- quadrics[J]. Chinese Journal of Computational Mechanics, 2013, 30(6): 854-859 (in Chinese).
[15]	王亚楠, 杨波, 赵秀阳. 基于轮廓提取和自适应单元分解的水泥颗粒三维重构[J]. 计算机辅助设计与图形学学报, 2010, 22(1): 152-157.
	WANG Y N, YANG B, ZHAO X Y. Reconstruction of cement particle by contour extraction and adaptive partitioning of unity[J]. Journal of Computer-Aided Design & Computer Graphics, 2010, 22(1): 152-157 (in Chinese).
[16]	WEI D H, WANG J F, NIE J Y, et al. Generation of realistic sand particles with fractal nature using an improved spherical harmonic analysis[J]. Computers and Geotechnics, 2018, 104: 1-12. DOI URL
[17]	WANG X, NIE Z H, GONG J, et al. Random generation of convex aggregates for DEM study of particle shape effect[J]. Construction and Building Materials, 2021, 268: 121468. DOI URL
[18]	ZHOU B, WANG J F, ZHAO B D. Micromorphology characterization and reconstruction of sand particles using micro X-ray tomography and spherical harmonics[J]. Engineering Geology, 2015, 184: 126-137. DOI URL
[19]	ZHOU B, WANG J. Generation of a realistic 3D sand assembly using X-ray micro-computed tomography and spherical harmonic-based principal component analysis[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 2017, 41(1): 93-109. DOI URL
[20]	GIANNIS K, THON C, YANG G Q, et al. Predicting 3D particles shapes based on 2D images by using convolutional neural network[J]. Powder Technology, 2024, 432: 119122. DOI URL
[21]	SHI J J, ZHANG W, WANG W, et al. Randomly generating three-dimensional realistic schistous sand particles using deep learning: variational autoencoder implementation[J]. Engineering Geology, 2021, 291: 106235. DOI URL
[22]	尹晓龙, 沈恩亚, 王建民. 典型三维模型表示方法转换技术综述[J]. 计算机辅助设计与图形学学报, 2025, 37(10): 1666-1680.
	YIN X L, SHEN E Y, WANG J M. A review of conversion techniques for typical 3D model representation methods[J]. Journal of Computer-Aided Design & Computer Graphics, 2025, 37(10): 1666-1680 (in Chinese).

一种融合频率扰动的KL展开3D颗粒随机建模方法

A 3D random particle modeling method integrating KL expansion and frequency perturbation

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