# 3D Reconstruction of a Single Bubble in Transparent Media Using Three Orthographic Digital Images
## 1 Introduction  
Studying the hydrodynamics of a single bubble growth and detachment in motionless fluids always attracts researchers [1; 2]. The literature of a single bubble generated in Newtonian fluids has been studied for decades and this well-known classic work was published, [3]. Despite the importance of this problem, many applications are dealing with bubbles in non-Newtonian fluids, and bubbles in such fluids have not been thoroughly studied. Recently, some studies focused on the dependence of the single bubble shape on their dynamics such as velocity discontinuity, as well as on the contact angle, viscosity, and surface tension [4; 5]. To this end, the 3D bubble shape reconstruction is vital for the further analysis of hydrodynamic interactions among a single bubble and surrounding liquid. There exists a significant body of scientific research discussing the calculation of bubble size and its formation. Early researchers used the balance formula of buoyancy and surface tension to calculate the volume of a single bubble [6]. Then, the Rayleigh-Plesset equation [7; 8] and Young-Laplace equation [9; 10] were utilized to predict the formation of bubbles. Many numerical simulation techniques, including the boundary integral method [11], the finite element analysis method [12] have also been applied to forecasting bubble formation. In general, the results of theoretical calculations and numerical simulations need to be verified by accurate experimental data. Therefore, a method which could precisely extract 3D bubble formation from orthographic images can be employed to validate the rationality of various theoretical calculation methods and numerical simulation techniques. With the progressive development of technology, advanced measuring equipment has been employed to study the bubble visualization. Although the distribution of bubbles can be obtained by destructive measurements such as probes [13], invaded probes interfere in the motion of bubbly flows to some extent. Therefore, noninvasive measurement techniques such as electrical tomography [14], interference technology [15], laser tomography [16], and optical imaging [17] have widely been utilized since they have no effect on fluid structure. Among these techniques, the optical imaging technology can completely and conveniently capture the transient changes of a single bubble with high speed cameras and thus has gradually become the common prevailing method for reconstructing the 3D shape of a single bubble in a sparse bubbly flow. [PERSON] et al. [18] used the equivalent diameter as the characteristic length of a single bubble, supposing that the bubble resembled a sphere. They found that as the ratio of the maximum diameter to the minimum diameter of the bubble in the projected image rises, the measurement accuracy of the equivalent diameter method decreases dramatically. [PERSON] et al. [19] presents a hybrid method that uses one camera to reconstruct 3D bubble shape, assuming that bubbles are truncated spheres. In fact, the wall effects and force effects lead to bubble deformation as it develops or moves, so these methods show large errors in reconstructing the seriously deformed bubble. [PERSON] et al. [20] calculated the 3D shape of a single bubble by assuming that the bubble is analogous to an ellipsoid. They employed two orthogonal cameras to capture the images of a single bubble rising in still water and utilized image processing to extract the projected contours of the bubble from the front and side images. Ten two-dimensional parameters were extracted from the two fitted ellipses of the projected contours, and nine three-dimensional parameters of the ellipse model, through which a bubble was finally reconstructed, were then calculated. [PERSON] et al. [21] utilized a camera for front view and another camera for side view. Both of the perpendicular images of cameras are analyzed by image processing, to get the bubble contours at x-z plane and y-z plane.