# 3D Reconstruction of a Single Bubble in Transparent Media Using Three Orthographic Digital Images
## 6 Conclusions  
In the present work, we developed a precise method based on orthographic images to reconstruct a 3D bubble shape of a single bubble, which is elliptical in the horizontal direction and has an arbitrary shape in the vertical direction. The proposed reconstruction method demonstrated a surprising degree of accuracy compared with the methods in [PERSON] et al. [21] and [PERSON] et al. [20], and its superiority was notable when the bubble deformation was increased.  
The method in this paper differed from [PERSON] et al. [21] in two points. Firstly, the posture correction reduced the tilt problem caused by bubble oscillation. Second, the projection relationship and the geometric information were used to calculate the major and minor semi-axes of each horizontal ellipse slices. Therefore, instead of using the projection width directly, the method in this paper reduced the negative impact of horizontal deformation and azimuth angle on the reconstruction accuracy.  
According to the above discussion, the following conclusions can be drawn:  
1. The posture correction helps to improve the reconstruction accuracy in this paper, especially for the bubble with a horizontal cross-section remarkably deviating from a circle.
2. The dimensionless parameters of an ellipse in horizontal section (major and minor semi-axes) are independent of the projection width. By using the dimensionless elliptic parameters, the actual major and minor semi-axes of the horizontal elliptical section of each layer can be quickly calculated.
3. The method in [PERSON] et al. [20] is more suitable for reconstructing a bubble similar to an ellipsoid because both of the horizontal and vertical deformations inversely impact its accuracy. The vertical  
Figure 18: The bubble volume and pressure with the time.  
deformation has a little influence on the method in [PERSON] et al. [21], but the horizontal deformation contributes significantly to its accuracy, especially when the ratio of the vertical axis to the horizontal axis is smaller than 0.5 or when the azimuth angle deviates greatly from either 0 or 90\({}^{\circ}\).
4. The method in this paper successfully improves the flaws in methods in [PERSON] et al. [20] and [PERSON] et al. [21]. The method in this paper can better reconstruct the bubble with severe horizontal deformation; in addition, similarly to the method in [PERSON] et al. [21], the vertical deformation has a negligible influence on the reconstruction accuracy of the method in this paper, leading to its higher practicality.
5. The two methods in [PERSON] et al. [20] and [PERSON] et al. [21] are extremely sensitive to the orientation of the bubble since the azimuth angle of the horizontal elliptical slice strongly affects the reconstruction accuracy; however, it does not impact the method in this paper.  
The bubble-shaped samples used in our experiments were considered to be regular bubbles since the bubbles generally representing a larger volume, less distortion, and a regular shape in Newtonian or non-Newtonian fluids with higher viscosity are similar to regular bubbles. Therefore, the reconstruction method developed herein is undoubtedly suitable for such a kind of bubbles. However, the application of the proposed method to reconstructing bubbles with a large horizontal and/or vertical distortion must be further validated. In this way, further exploration needs to be done to study the accuracy of this method when bubbles have both vertical and horizontal deformation. The algorithm should be sufficiently replenished on the problem that cameras are not ideal when perpendicular to each other.  
Investigation, M.H.; methodology, X.Q.; validation, X.Q.; writing--original draft, X.Q.; writing--review and editing, Y.Z. ([PERSON]) and Y.Z. ([PERSON]). All authors have read and agreed to the published version of the manuscript.  
This work was funded by the National Natural Science Foundation of China (Grant No. 51579219).  
The authors declare no conflict of interest.