## 1 Introduction  
As a new kind of autonomous underwater vehicle (AUV) combined with propulsion mechanism of fish and robotics, the bionic robot-fish has been widely applied in water quality monitoring, scientific underwater exploration, oceanic supervision and fishery conservation [1; 2; 3; 4], because of several advantages, compared with traditional AUV based on screw propeller, such as low energy consumption, low noise, high propulsion efficiency and high mobility [5]. Nevertheless, the bionic robot-fish still has a large gap in speed, propulsive efficiency and maneuverability in comparison with biological prototype. Therefore, in order to improve the swimming performance of robotic fish, exploring the locomotion patterns of fish through movement observation are the essential issues. Among the available techniques used in observation of fish swimming [6; 7; 8; 9], the vision-based method is considered to be a simple, low cost and easily available one for high precision [10], and has attracted extensive attention from researchers. [PERSON] used a high-speed camera (500 fps) to photograph the swimming and turning behavior of juvenile zebra fish to study the effects of nervous system on fish swimming [11]. [PERSON] used the Kodak high-speed camera (500-1000 fps) to shoot the movement of squid in the shallow water and obtained its swimming strategy at different speed [12]. For obtainingthe actual fish locomotion data, [PERSON] measured the fish swimming modes by camera set up in the upper area of the water tank and five colored markers attached to the fish body from tail to head, and the motion data will be derived from the five markers data [13]. In this system, the marker should contrast sharply with the fish in gray scale. Consequently, [PERSON] proposed a marker less observation method to measure the fish locomotion [14]. However, in the above studies, the information of only one or two planes are concerned, some kinematics data are ignored and accurate three-dimensional information cannot be obtained. To get the three-dimensional (3D) coordinates of fish in the tank, [PERSON] proposed a method that combines a single camera with a waterproof mirror, in which a waterproof mirror is placed at the angle of 45\({}^{\circ}\) on the side of tank for achieving the equivalent of another camera shooting horizontally [15], but because the image captured by the virtual camera is the inverted image, the reconstruction error is large. [PERSON] recorded the fish swimming information in the XY and XZ coordinate surface by two cameras set up, respectively, above vertically and in front of the tank [16], while it still has error when the fish swimming near the front wall of tank. Besides, [PERSON] obtained the 3D information by using a camera and two plane mirrors according to the plane mirror imaging principle [17], which is similar to the [15]. [PERSON] reconstructed the 3D information from 2D images captured by two cameras based on the principle of triangulation [18]. These studies show that vision-based approaches can be easy to design and quick to implement for reconstructing the 3D motion data of fish. However, due to the light refraction that occurs at water-air interface, the reconstruction accuracy is greatly reduced and there is no investigation has been conducted that considers the light refraction for capturing the fish motion. The distortion in 3D reconstruction caused by the refraction should be solved in order to perform more accurate 3D reconstruction. A recent literature review show that the researchers along with the hardware supports allows developing algorithms to do high processing calculation for 3D reconstruction task in many fields [19]. A refractive estimator derived by [PERSON] et al. for sparse two-view geometry show that the proposed refractive method outperforms the perspective method in a controlled lab environment for deep sea applications [20].