A Cuboid Volume Measuring Method Based on a Single RGB Image

Abstract

We should estimate the camera’s pose when utilizing a camera to measure volume. Traditional methods based on monocular camera lack depth, so they can’t estimate a camera’s pose from image to spatial features. And the principle of binocular cameras usually has errors in the centimeter range in volume. Considering these problems, this paper proposes a new method to measure the cuboid volume using a monocular camera, relying on geometric relationships about angles from 2D to 3D. This thesis has two advantages: firstly, the camera pose parameters, i.e. orientation and position, are easily recovered from the correspondence between a corner and a nonsingular point of a view; secondly, experiments using OpenCV present good results, and the real images show the method to be extremely promising. Volume measurement can be achieved with just one image, requiring low computing power of the computing device, easy to implement, at low cost, and with errors that can be down to the millimeter level, improving the measurement accuracy.

Appendix A

As shown in Fig. 3, we take $|m_{0}q_{i}| = |m_{0}q_{j}| = 1; q_{i}q_{i}^{'} \perp \overrightarrow{l_{i}}^{'}, q_{j}q_{j}^{'} \perp \overrightarrow{l_{j}}^{'}$, so

$$\begin{aligned} |q_{i}^{'}q_{j}^{'}|^{2} = |q_{i}q_{j}|^{2} - (|q_{j}q_{j}^{'}| - |q_{i}q_{i}^{'}|)^{2} = 2 - (sin \theta _{j} - sin \theta _{i})^{2} \end{aligned}$$

(12)

In $\triangle q_{i}^{'}m_{0}m_{j}^{'}$

$$\begin{aligned} \begin{aligned} |q_{i}^{'}q_{j}^{'}|^{2}&= |m_{0}q_{i}^{'}|^{2} + |m_{0}q_{j}^{'}|^{2} -2|m_{0}q_{i}^{'}||m_{0}q_{j}^{'}|cos \phi _{ij} \\ {}&= (cos\theta _{i})^{2} + (cos\theta _{j})^{2} - 2cos\theta _{i}cos\theta _{i}cos\phi _{ij} \end{aligned} \end{aligned}$$

(13)

From Eqs. (12) and (13), we can get

$$\begin{aligned} tg\theta _{i}tg\theta _{j} + cos\phi _{ij} = 0,(i \ne j; i,j = 1,2,3) \end{aligned}$$

(14)