## 4 Discussion  
The three-dimensional position of intracellular organelles in living cells contains crucial information to understand biological system, such as the interaction between the vesicles and cytoskeletal network [25; 26]. Since the depth of living cell is larger than the axial region that  
Figure 6: **(A)** Quantum-dot-labeled vesicle on each focal plane at three different positions of the objective lens. ROI size of each image is 17\(\times\) 17 pixels. The image sets enclosed within the orange boxes were analyzed to evaluate axial position of the vesicle. Here, \(h_{1}\) and \(h_{2}\) represent the positions of the two focal planes according to the axial position of the objective lens: \(P_{0}\)\(P_{1}\), or \(P_{2}\); **(B)** Concatenated axial positions of the vesicle over time. Only the \(I_{det}\) range, which had a detection error smaller than 2 nm, was chosen to evaluate the axial position of the vesicle; **(C)** Vesicle trajectory reconstructed in three dimensions.  
conventional biplane microscopy can usually cover, extension of trackable range with fine detection resolution is required, in the case where users adopt biplane imaging method for the simplicity of the system. In the present work, we extended the depth of trackable range of bifocal microscopy by combining the ratiometric method of dual-focus optics and objective lens oscillation. For appropriate application of the suggested method, the aspects that users should consider are the speed of the piezo actuator on which the objective lens is mounted, which determines the imaging frequency, and the speed of the target particle. Vesicles are mostly transported by cytoskeletons; the speeds of an endocytic vesicle on microtubules by dynein and on actin filaments by myosin VI in vivo are 1 \(\upmu\)m/s and less than 60 nm/s, respectively [27; 28; 29; 30; 31; 32]; Therefore, an imaging frequency of approximately 10 Hz or higher can prevent users from misinterpreting or missing the target vesicle in the cell.  
One of the major benefits of the extended dual-focus imaging method is that the advantages of robustness in detection performance of ratiometric method is preserved. As the conventional dual-focus optics features simplicity in calculation with no computation load unlike those encountered in parametric methods, the proposed method also exploits this simplicity, by directly implementing the process of conventional dual-focus optics scheme.  
Additionally, extending the axial range by oscillating the objective lens does not cause additional intensity loss, which is inevitable in many multiplane imaging systems, because of splitting optical paths and introducing relay lenses [33]. Since the intensity issue is critical for vesicle tracking tasks in the field of live cell imaging, in which the intensities of the biomarkers attached to the target particle are not always captured with sufficient photons [34; 35], conserving the quantity of photons in each image plane makes the suggested method promising for various live cell imaging conditions. Therefore, with the proposed method, it is expected that small vesicles which are not detectable in other multiplane imaging systems can be captured and tracked. Hence, a large dataset of vesicle trajectories in a single cell could be constructed, for further analysis using, for example, a machine learning approach [36].  
Furthermore, the expandability can be considered to be another merit of the proposed method. Although we presented an example of a vesicle that traveled along an axial range of approximately from 0 \(\upmu\)m to 2 \(\upmu\)m in axial range as an example case, it is possible to expand the trackable range, simply by adding more steps in the objective lens oscillation. Since the desired detection accuracy range on the calibration curve can be determined based on the suggested concept of conversion resolution, users can easily customize the extended dual-focus imaging method according to their required depth and desirable localization accuracy.