## 1 Introduction  
Optical see-through head-mounted displays (OST-HMDs) are one of the more promising technologies for achieving augmented reality (AR). Recently, many consumer OST-HMDs, such as HoloLens (Microsoft, Redmond, WA, USA), Magic Leap One (Magic Leap, Plantation, FL, USA), and NrealLight (Nreal, Haidian, Beijing, China), have been released. In the research stage, OST-HMDs have been miniaturized to the scale of an ordinary glasses [1]. However, to the best of our knowledge there has been no similar miniaturization of video see-through HMDs (VST-HMDs). OST-HMDs have a characteristic that allows users to observe the real world directly through a semi-transparent display. Consequently, such devices have the advantage of producing scenes with no time delay aside from the generated virtual objects and no blackout in user vision even when the HMD unexpectedly fails. However, these qualities make it difficult for OST-HMD systems to physically decrease the number of light rays that reach the retina from the real environment and allow a time delay between the image and the real world. Therefore, the realism of virtual objects that require attenuation or shielding from outside light decreases. The aforementioned statement holds for transparent virtual objects (e.g., glass or liquid) that require a significant amount of shielding and attenuation for representation. Representation of transparent objects in computer graphics requires the rendering of optical phenomena such as refraction, reflection, shadow, and caustics [2]. Figure 1 shows the results produced by different displays representing a virtual transparent object in a real environment. The middle image shows the results produced by an ordinary display, whereas the right-hand image shows the results produced by a transmissive display such as an OST-HMD. The transmissive display image shows a virtual transparent object that is not partially optically consistent. However, the shadow and caustics are not accurately represented. To represent the shadow, it is necessary to reduce the luminance of the region in which the light of the environment is occluded by the virtual object. However, OST-HMDs cannot reduce the luminance of the shadow regions and, therefore, cannot accurately represent the light and dark patterns created by shadows and caustics. Another drawback is that the light rays produced by the real environment, which should be occluded by the virtual transparent object, penetrate it directly. Therefore, it is necessary to first erase the area of the real environment occluded by the object in order to accurately represent a virtual transparent object. Since OST-HMDs cannot completely rewrite the real environment, the virtual transparent object must be superimposed over the environment with light rays penetrating the object. Another limitation that we faced when representing virtual transparent objects on stereoscopic displays such as HMDs is that the visual effects caused by eye movements must be reproduced. Real transparent objects must be represented by multiple images--a surface image produced by features such as dirt and a drawn pattern, a refracted image caused by distortion of the background, and a reflected image caused by the reflection of the surrounding environment. For each scenario, the distance between the eyes and the image differs, and the viewer focuses on each image selectively. In this process, the eyes converge (a process hereafter referred to as \"vergence\") and the crystalline lenses adjust (hereafter referred to as \"accommodation\") to enable clear viewing of the target [3]. As a result of the application of these visual functions, the appearance of transparent objects varies depending on the image. For example, while focusing on the surface of a transparent object, disparity and defocus occur in the refracted and reflected images as a result of mismatches between the vergence and accommodation points.