## 1 Introduction  
Viewing an airborne scene through a wavy water-air interface using a submerged camera creates a virtual periscope, which is of great significance in both the military application field and marine biology research [1; 2; 3; 4]. Unlike other underwater imaging systems, the difficulties in such imaging scenarios mainly come from the water-air interface (WAI). Water surface fluctuations are complex and random movements, which will lead to irregular geometric distortion and motion blur in the image, causing the distortion of airborne scenes. Removing such distortions from an instantaneous distorted image is challenging since the shape of the interface is not known a priori and must be estimated simultaneously with the real scene image. In previous works, recovering images distorted by a wavy water surface mainly relied on high-resolution video streams [5; 6; 7; 8; 9; 10; 11; 12]. These methods always require high complexity and large data sets, so they are difficult to apply to real-time observation scenarios. However, in application scenarios such as path planning and obstacle avoidance for underwater vehicles, aerial target detection, recognition and tracking, etc., recovery methods for instant images are particularly important. Previous studies show that the distorted scene image can be effectively recovered by reconstructing the shape of the water-air interface [13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25]. [PERSON] et al. [15; 16] emphasized first the estimation of the shape of the distorted water surface and then reconstructing the scenes. They proposed a method to estimate the water surface by analyzing the sky brightness; in their image system, an upward-looking submerged camera was used to capture the panoramic above-surface scene. The incident light from above will gradually disappear as the normal surface deviates from the viewing line of sight. Assuming that it was completely dark underwater and the sky was uniform, the brightness of the sky thus determined the surface radial slope, and they used a harmonic wave model to estimate the water surface and inverse ray tracing to reconstruct a distortion-free image. [PERSON] et al. [17] studied the recovery method of water surface distortion, and then designed an experimental setup. They first utilized multiple illumination sources (red to illuminate the water surface and blue for the underwater object) to capture an image of the object and a glitter pattern of the water surface simultaneously. Then, the slope of the water surface was estimated using the glitter pattern [18]. Finally, the geometrical distortion effects of the underwater object image were eliminated by multiple accumulations of short-exposure images [19; 20; 21; 22]. [PERSON] et al. [23] added an additional imaging sensor to measure the wavy water surface in real time. It works as an adaptive optics system but utilizing the sun as a guide star. The wavefront sensor consists of a pinhole array imaging the sun onto a diffuser plane directly behind it and a camera capturing the distribution of images of the sun on the diffuser plane. These pinholes have an extremely narrow field of view, so the water surface on their own line of sight is small enough to be assumed isoplanatic. Then, sampled normal vectors of the water surface were deduced from the position of each sun image through pinholes and used to estimate the water surface. [PERSON] et al. [24] considered the problem of multi-view stereo through a dynamic refractive interface. They used multiple cameras along a wide baseline to observe a scene under uncorrelated distortions and recover sparse point clouds. [PERSON] et al. [25] developed a method for recovering the single instantaneous images of underwater objects distorted by surface waves. They first determined an instantaneous shape of the wavy sea surface using the characteristics of the sun glints, and then corrected an image distorted by the wavy surface through reverse projection.