Geometric Shape Modelling and Volume Estimation of Dry Bulk Cargo Piles using a Single Image

TL;DR

The paper addresses the challenge of estimating stockpile volumes from a single optical image by introducing a fixed-height cone-based model tied to the material angle of repose, $\tan(\theta_c)=H/b_0$, enabling closed-form volume formulas without 3D reconstruction. It provides a practical workflow and analytical expressions for volume when either $H$ or $\theta_c$ is known, along with uncertainty propagation for footprint measurements. A demonstration on a SkySat image of silica sand piles achieves approximately 95% accuracy in total storage volume, validating the approach for remote-sensing data. The method offers a fast, direct alternative to stereo or LiDAR-based techniques and lays the groundwork for integration with machine learning to handle irregular or reclaimed piles in real-world scenarios.

Abstract

Volume estimation of onshore cargo piles is of economic importance for shipping and mining companies as well as public authorities for real-time planning of logistics, business intelligence, transport services by land or sea and governmental oversight. In remote sensing literature, the volume of pile is estimated by relying on the illumination property of object to construct the geometric shape from a single image, alternatively, stereographic imaging for construction of a digital elevation model from pairs of images. In a fresh perspective, we propose a novel approach for estimating volume from a single optical image in this work where we use the material property, which relates the base dimensions of the pile to its height through the critical angle of repose. In materials literature, often this is well-studied for fixed base and their \textit{in situ} volume estimation for different materials. In this work, however, we mathematically model the geometric shape of the pile through a fixed height model. This is appropriate because the unloading crane arm that forms the pile can rise only up to a certain height and generally moved in the horizontal plane during unloading of the material. After mathematically modelling the geometric shape of regular piles for fixed heights under rectilinear motion of unloader, we provide closed form formula to estimate their volume. Apart from laying the mathematical foundations, we also test it on real optical remote sensing data of an open bulk cargo storage facility for silica sand and present the results. We obtain an accuracy of $95\%$ in estimating the total bulk storage volume of the storage facility. This is a first demonstration study and will be integrated with applied machine learning approaches or current state-of-art approaches in the future for more complex scenarios for estimating dry bulk cargo pile volume.

Figures (6)

• Figure 1: The right circular cone formed by a granular material when released from a single point illustrating the angle of repose denoted as $\theta_c$.
• Figure 2: Cone cross sections for different cases: (clockwise from top to bottom) fixed $\theta_c$, base and height respectively.
• Figure 3: The piles visualized as identical overlapping right-circular cones of same height whose peaks lay in the plane of motion forming a grid pattern as above. In limit, the grid separation distance $\rightarrow 0$.
• Figure 4: (Left to Right) Final form of the fixed height free base models when the granular material is released using a funnel-shaped unloader under static 0D (point discharge), linear (along length dimension only) 1D and rectilinear 2D (along length and breadth dimensions) motion.
• Figure 5: Footprint of fixed height free base models when the granular material is released using a funnel-shaped unloader under (\ref{['fig:fixed_0D']}) (point discharge), (\ref{['fig:fixed_1D']}) (along length dimension only) 1D and (\ref{['fig:fixed_2D']}) 2D (along length and breadth dimensions) motion.