Deep-learning-enhanced model reconstruction of realistic 3D rock particles by intelligent video tracking of 2D random particle projections

Abstract

Rock assemblies, such as ballasts and cobbles, are typically composed of granular particles of various sizes and shapes that significantly affect their mechanical behavior. Currently, it remains challenging to generate realistic and identical three-dimensional (3D) models for rock particles with a limited number of their individual two-dimensional (2D) projections. This study proposes a novel and effective deep-learning-enhanced approach to achieve the realistic reconstruction of 3D particle models, which includes the following sequential steps: (1) automatically detecting 2D projected images of each 3D particle individually during dynamic scanning of a rock particle assembly; (2) predicting 3D particle size indexes based on 2D images for each particle; (3) generating a corresponding 3D skeleton network from 2D outlines; and (4) reconstructing the particle surface of individual particle by spatial interpolations. Sensitivity analysis and verification are finally performed on both randomized superball particles and real cobbles/ballasts. The results demonstrated that the proposed approach is robust and efficient in reconstructing 3D rock particle models and offers a rapid and low-cost practical solution for convincible numerical studies of granular materials.

Appendices

Appendix 1

To generate the 3D model for a superball particle, the polar representation is used in this study. Given the local spherical coordinate $\left(\theta ,\varphi \right)$ of a point on the surface, the corresponding local Cartesian coordinate $\left(x,y,z\right)$ is expressed as:

$$\left\{ {\begin{array}{*{20}l} {x\left( {\theta ,\varphi } \right) = {\text{Sign}}\left( {{\text{cos}}\left( \theta \right)} \right)a\left| {{\text{cos}}\left( \theta \right)} \right|^{\varepsilon } \left| {{\text{cos}}\left( \varphi \right)} \right|^{\varepsilon } } \hfill \\ {y\left( {\theta ,\varphi } \right) = {\text{Sign}}\left( {{\text{sin}}\left( \theta \right)} \right)b\left| {{\text{sin}}\left( \theta \right)} \right|^{\varepsilon } \left| {{\text{cos}}\left( \varphi \right)} \right|^{\varepsilon } } \hfill \\ {z\left( {\theta ,\varphi } \right) = {\text{Sign}}\left( {{\text{sin}}\left( \varphi \right)} \right){\text{c}}\left| {{\text{sin}}\left( \varphi \right)} \right|^{\varepsilon } } \hfill \\ \end{array} } \right.$$

(45)

where term $\mathrm{Sign}\left(x\right)$ is the signum function defined as:

$$\mathrm{Sign}\left(x\right)=\left\{\begin{array}{l}-1\\ 0 \\ 1\end{array}\right. \begin{array}{l}\quad \mathrm{if} \quad x<0,\\ \quad\mathrm{if}\quad x=0,\\\quad \mathrm{if} \quad x>0.\end{array}$$

(46)

Appendix 2

The $\theta$ and $\varphi$ for a random unit vector in 3D space can be easily generated through the following steps:

• Step 1: Generate a pair of random variables ${\xi }_{1}$ and ${\xi }_{2}$ that are uniformly distributed on [− 1,1].

• Step 2: Calculate ${r}^{2}$ by Eq. (47). If ${r}^{2}<1$, abandon this pair and repeat step 1.

  $${r}^{2}={{\xi }_{1}}^{2}+{{\xi }_{2}}^{2}$$

  (47)

• Step 3: Calculate ${n}_{x},{n}_{y},{n}_{z}$ by the following expression

  $$\left\{ {\begin{array}{*{20}l} {n_{x} = 2\xi _{1} \sqrt[2]{{1 - r^{2} }}} \hfill \\ {n_{y} = 2\xi _{2} \sqrt[2]{{1 - r^{2} }}} \hfill \\ {n_{x} = 1 - r^{2} } \hfill \\ \end{array} } \right.$$

  (48)

Appendix 3

The process to search all the surface points on the 3D particle that corresponds to the projected points in plane M is implemented based on the following steps:

• 1. Randomly select one point ${Q}_{1}$ from plane M such that vector $\mathop \rightharpoonup \limits_{{OQ_{1} }} \left({x}_{q1}, {y}_{q1},{z}_{q1}\right)$ is perpendicular to the projection vector $\mathbf{O}\mathbf{V}$, as shown in Fig. 6b.

• 2. Find the surface point ${P}_{1}$, as shown in Fig. 6c that has a normal vector $\mathop \rightharpoonup \limits_{{N_{1} }}$ parallel to vector $\mathop \rightharpoonup \limits_{{OQ_{1} }}$ as follows: Given a normal vector $\mathop \rightharpoonup \limits_{{N_{1} }}\left({n}_{x},{n}_{y},{n}_{z}\right)$ on the surface, the corresponding local spherical coordinates $\left(\theta ,\varphi \right)$ of ${P}_{1}$ are obtained using the following equations:

  $$\theta =\mathrm{atan}2\left[{\mathrm{Sign}\left({n}_{y}\right)\left|b{n}_{y}\right|}^{\frac{1}{2-{\varepsilon }_{1}}},{\mathrm{Sign}\left({n}_{x}\right)\left|a{n}_{x}\right|}^{\frac{1}{2-{\varepsilon }_{1}}}\right]$$

  (49)
  $$\varphi =\mathrm{atan}2\left[{\mathrm{Sign}\left({n}_{z}\right)\left|c{n}_{z}\right|\mathrm{cos}\theta }^{\left|2-{\varepsilon }_{2}\right|\frac{1}{2-{\varepsilon }_{2}}},{\left|a{n}_{x}\right|}^{\frac{1}{2-{\varepsilon }_{2}}}\right]$$

  (50)

  where the term $\mathrm{atan}2\left(x,y\right)$ is the arctangent function of $\frac{x}{y}$ producing results in the range $(-\uppi ,\uppi ]$. After the local spherical coordinates $\left(\theta ,\varphi \right)$ of ${P}_{1}$ are obtained, the corresponding local Cartesian coordinates $\left(x,y,z\right)$ of ${P}_{1}$ can be computed using Eq. (45).

• 3. Rotate $\mathop \rightharpoonup \limits_{{OQ_{1} }}$ at a discretized angle $\Delta\updelta =\uppi /\mathrm{m}$ around axis $\mathbf{O}\mathbf{V}$ to obtain a new vector $\mathop \rightharpoonup \limits_{{OQ_{i}^{\prime } }}$ and find the corresponding surface point ${P}_{i}$ with $\mathop \rightharpoonup \limits_{{N_{i} }} \left(\updelta \right)$ parallel to $\mathop \rightharpoonup \limits_{{OQ_{i}^{\prime } }}$;

• 4. Repeat (3) until $\updelta =2\uppi$, and connect all acquired points, in turn, to form the full projected outline, as shown in Fig. 6d.