1 Introduction

In aquariums, there are various kinds of tanks used to display sea creatures. Those tanks are mainly made of flat and round glass walls. Although the tanks give a lot of views of things such as fish, plants and rocks, it is known that those views are all seen distorted due to light distortion. Technically, this phenomenon stems from light refraction, or the fact that light rays bend as travelling through the boundaries between different media: water, glass of the tank and air. Light distortion more badly affects the views as you get closer to the tank. You may feel dizziness and faintness at the time. Therefore we previously presented a way of correcting the distortion of such views, or constructing views without distortion [1]. A set of photos are taken at specific positions outside the tank and a distortion-free view is constructed from those photos using the image based rendering technique [2]. In that study we focused on simple rectangular tanks while we are focusing on spherical tanks in this study.

There have been various studies dealing with light distortion. Treibitz et al. study a way of measuring underwater objects with accuracy [3]. Photos taken by an underwater camera are affected by light distortion in the same way as mentioned above. It leads to inaccuracy of measurement on the photos. To deal with this problem, correspondence from each pixel on the photos to a light ray, called ray-map, is created during the calibration process. The map enables accurate measurement of objects on the photos even though the objects are seen distorted. In contrast, Sedlazeck et al. study a way of creating photo-realistic underwater images by modeling light refraction, light scattering and light attenuation [4]. So far, no studies are found attempting to construct observer’s views without distortion. 3D structure estimation of underwater objects studied in [5] could construct those views using the extracted geometric information, but it seems inappropriate to create underwater atmosphere by including drifting dust, ascending air bubbles, tiny creatures and the like. Therefore, in this study, we exploit the image based rendering technique and construct observer’s views without distortion.

The rest of this manuscript is organized as follows. In Sect. 2, a basic idea of correcting the distortion of observer’s views, which look into a spherical tank, is explained. In Sect. 3, it is shown that the distortion is successfully corrected in a real world environment. Finally, we give concluding remarks in Sect. 4.

Fig. 1.
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Our approach to correct the distortion of an observer’s view.

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2 Correction of Distortion

Figure 1 shows the sectional top view of a spherical tank and illustrates how the distortion of an observer’s view is corrected. The observer stands right in front of the straight line Ray1 coming from a fish. Ray2 represents the actual light path. In this situation, the observer does not see the fish in that direction represented by Ray1. Thus, the distortion-free view from the observer needs to be comprised of a set of rays represented by Ray2 rather than Ray1. In order to capture the rays, a camera (hereinafter referred to as a reference camera) is placed at a series of positions and takes photos. Finally, those photos are merged as a single image without distortion.

To be more specific, let \((x_o, \theta _o)\) be the position of the observer and the orientation of Ray1. Let \((x_c, \theta _c)\) be the position of the reference camera and the orientation of Ray2. \(n_1, n_2\) and \(n_3\) are refractive indices of water, acrylic glass and air, respectively. \(R_1\) and \(R_2\) are the inside and outside radius of the spherical tank. Given \((x_o, \theta _o)\), \((x_c, \theta _c)\) is obtained as follows.

$$\begin{aligned} x_c= & {} (x_o+R_2)\frac{n_1}{n_3}\frac{\sin {\theta _o}}{\sin {\theta _c}} - R_2 \end{aligned}$$
(1)
$$\begin{aligned} \theta _c= & {} \theta _o - \arcsin \left( \frac{x_o+R_2}{R_1}\sin {\theta _o}\right) \nonumber \\&\quad +\arcsin \left( \frac{n_1}{n_2}\frac{x_o\!+\!R_2}{R_1}\sin {\theta _o}\right) \nonumber \\&\quad -\arcsin \left( \frac{n_1}{n_2}\frac{x_o\!+\!R_2}{R_2}\sin {\theta _o}\right) \nonumber \\&\quad +\arcsin \left( \frac{n_1}{n_3}\frac{x_o\!+\!R_2}{R_2}\sin {\theta _o}\right) \end{aligned}$$
(2)

Using (1) and (2), each ray represented by \((x_o, \theta _o)\) traces back to another ray represented by \((x_c, \theta _c)\), thus each pixel on the observer’s view can be drawn by picking up the color of the corresponding pixel on the reference camera’s view.

In the next section, we confirm that the observer’s view is drawn without distortion.

Fig. 2.
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A spherical tank made of acrylic glass. Note that no water in the tank to see the right positions of the rocks and plants inside. (Color figure online)

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3 Experiment on Correction of Distortion

Figure 2 shows an experimental setup and Fig. 3 shows the sectional top view. The spherical tank is 300 mm in outside diameter and 3 mm in thickness. To find the effect of light distortion easily, it was decorated with gravel, rocks and artificial plants. The observer stood 50 mm to the surface of the tank and looked in \(15^{\circ }\) off-center. The reference camera was placed at 24 to 35 mm to the tank. It was also angled \(40^{\circ }\) off-center for 24 to 31 mm and \(0^{\circ }\) for 32 to 35 mm. At each position one reference camera’s view, or one reference photo, was taken, therefore a total of 12 reference photos were obtained.

Figure 4(a) shows a view into the empty tank from the observer’s position. Figure 4(b) shows a view at the same position but the tank is filled with water. As expected, the view is distorted compared to (a). Specifically the objects in (b) are skewed towards the right. Figure 4(d) to (f) are three of the 12 reference photos. Note that these photos are all affected by light distortion. Finally as shown in Fig. 4(c), the desired view from the observer was constructed from the reference photos. Enclosed regions in (d), (e) and (f) contributed to the corresponding parts in (c). Thus, (c) is comprised of partial regions of the reference photos.

Finally, it is obviously shown that the distortion found in (b) was successfully corrected in (c), but not totally. Some peripheral part was left gray background. The light gray could be drawn if reference photos at 1 to 23 mm positions were available as well as upward/downward ones. In reality, however, reference photos up to 23 mm positions were unable to be obtained because of no room to place the reference camera closer to the tank. It should also be mentioned that only rightward reference photos were taken and used for simplicity in this experiment. As for the dark gray, none of reference photos was obtained due to total internal reflection. That is, light rays coming from that dark gray region towards the observer cannot penetrate through the acrylic glass, but reflect back inside.

Fig. 3.
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An experimental setup (sectional top view).

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Fig. 4.
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Views from the observer and photos taken by the reference camera in a real world environment. The vertical lines help understand to what extent (b) is distorted, also corrected.

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4 Conclusions

In this study, we discussed a way of correcting the distortion of views that look into spherical tanks. It is widely known that those views are all seen distorted due to light distortion. That is, light rays traveling inside the tank directly towards an observer make a change to their course when passing the boundaries between the different media: water, acrylic glass and air. Therefore, those rays come away from the observer. To capture the rays, a reference camera is placed at specific positions and takes photos. Finally, those photos are merged as the observer’s view without distortion. We conducted an experiment in a real world environment and confirmed that the observer’s view was successfully constructed. However, a peripheral part of the view suffers from total internal reflection, resulting in a blank space.

In future work, we will seek another approach to fill in the blank space by capturing light rays that cannot penetrate through the acrylic glass of the tank due to total internal reflection.