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Abstract
We measure the image resolution of aerial images formed with an AIRR (aerial imaging by retro-reflection) system. Using three different types of retro-reflectors inside the AIRR system, we measure the aerial image modulation transfer function (MTF) via the slanted knife edge method, showing that the resolution of depends on the angle of incidence on the retro-reflector, the retro-reflector's anisotropy, and the aerial image's floating distance.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Aerial displays [1–3] form floating information screens in the mid-air, allowing for digital signage and touchless aerial interfaces. Since there is no physical contact with the display, an aerial interface has the advantage that it is immune to hygiene issues. There are many different methods to form aerial images, such as employing dihedral corner reflector arrays (DCRA) [4], Fresnel lenses [5], and a wide variety of other devices [6–15]. However, an undesirable pseudo-image is caused on both sides of the aerial image after the double reflection in DCRA and the aerial image in single reflected light. We have proposed using an AIRR (aerial imaging by retro-reflection) system [16,17] to form aerial images, since it solves the problem of the aforementioned pseudo-images and has a wide viewing direction and low cost. Aerial images are used in a number of applications. For example, omni-directional aerial images for observing the behavior of fish [18], and the formation of multiple aerial images from a single light source using a principle of infinity mirror [19]. In addition, aerial displays can be made secure such that they can only be read from the front of the aerial image [20].
When some information is presented on aerial image, the emphasis is placed on the resolution of the aerial image. In AIRR, a retro-reflector is used to focus the light in mid-air, generally with a sheet retro-reflector. However, the critical factors that contribute to the resolution of aerial image using arrayed optical elements are not yet clear. A previous study [21] revealed that the diffraction pattern of retro-reflector influences the sharpness of aerial image, but contrast transfer function (CTF) used to measure the diffraction pattern has a problem that it is influenced by the light source.
The purpose of this study is to reveal the factors that influence the resolution of aerial image. In this study, we focused on incident angle at the retro-reflector, retro-reflector's anisotropy, and floating distance of the aerial image. The ‘'anisotropy'‘ here means the feature that divergence distribution of the retro-reflector changes depending on the rotation. First, the preliminary that the effective reflectance of retro-reflective sheet has angular dependency was shown [22]. A decrease in the amount of light forming the aerial image will cause a decrease in its contrast. Second, in the case of retro-reflective sheet without anisotropy, it is assumed that reflected light diverges due to the multiple reflection and diffraction, which causes a decrease in the resolution of aerial image. In contrast, when the retro-reflective sheet has anisotropy, it is assumed that the influence of multiple reflection and diffraction is small because the retro-reflectable incident angle is narrow on the optical element. Therefore, the resolution of aerial image is expected to increase under anisotropy of retro-reflector. Third, previous study showed that the aerial image of point light source becomes larger as the floating distance of the aerial image increases, i.e., there is a distance dependency in the blur of the aerial image [23]. Therefore, it is expected that the resolution of aerial image decreases as the floating distance of the aerial image increases.
In this study, we quantitatively evaluated the resolution of aerial image by using the modulation transfer function (MTF). The MTF is calculated using the slanted knife edge method [24–26]. Compared with the CTF, this method does not require a test chart for each spatial frequency, and the MTF can be calculated on a single image. A part of this study was presented in JSAP-OSA Joint Symposia 2020 [27] and the 27th International Display Workshops [28]. In this paper, we have conducted additional experiments to evaluate the both effects of incident angle at the retro-reflector without anisotropy and of floating distance on the resolution of aerial images.
2. Principle
2.1 Retro-reflection of retro-reflective sheet
The main feature of the retro-reflective sheet is to retro-reflect the incident light in the direction of the light source. The retro-reflective sheets are widely used in road signs and license plates to improve their visibility to drivers. Figure 1 shows the retro-reflection principle for the two different types of retro-reflective sheet that we consider in this paper: bead-type and prism-type. The bead-type sheet consists of many small spheres. Incident light on the beads is refracted through the surface of the bead, reflected at the mirrored surface at the back of the bead, and refracted again at the bead surface, so that it returns in the direction of the light source. The prism-type sheet consists of many small corner cube arrays. Incident light is sequentially reflected on the three sides of the corner cube array, so that it returns in the direction of the light source. The bead-type retro-reflector accepts incident light from a wide range (about 120°) and retro-reflects it. On the other hand, the prism-type retro-reflector retro-reflects the incident light from the front side of the prism, so the retro-reflectable incident angle is limited.
The principle of aerial image formation with AIRR is shown in Fig. 2. Light rays emitted from a light source are first reflected by a beam splitter, and then impinge on a retro-reflector sheet and return towards the incident direction. The retro-reflected light rays pass through the beam splitter and form the aerial image at the plane-symmetrical position of the light source with respect to the beam splitter. The floating distance is the distance between the beam splitter and the aerial image. Due to the nature of AIRR, as the distance between the light source and the beam splitter increases, the floating distance also increases. Figure 2(b) shows an example of an aerial image that is formed with AIRR.
2.2 Slanted knife edge method
An imaging system’s modulation transfer function (MTF) expresses how well the contrast of the object’s spatial frequencies are reproduced in the spatial frequencies of the image. MTF is used as one of the indices to quantify the sharpness of aerial image. The slanted knife edge technique is a common method for estimating the MTF curve for spatial frequency in one-direction, within a region of interest (ROI) of a recorded edge image. The edge is the image of a slightly tilted knife edge, blocking a uniformly illuminated background.
The procedure for the slanted knife edge method is shown in Fig. 3, demonstrating how the MTF curve is calculated from the initial image. The sampling rate across the edge can be improved by slightly tilting the edge horizontally or vertically with respect to the pixel grid. As shown in Fig. 3(a), the ROI is extracted from the slanted knife edge image and converted to grayscale. The ESF curve is obtained by projecting the slanted edge image ROI onto the projection axis and superimposing it [Fig. 3(b)]. By keeping track of the subpixel-location of the projected pixel centers, the projection ESF data obtains a smaller sampling interval, effectively eliminating the problem of aliasing. Figure 3(c) shows a conceptual diagram of the obtained ESF curve, which contains noise. The obtained ESF curve is denoised using wavelet filtering. Figure 3(d) shows the conceptual diagram of the ESF curve after denoising. Next, the line spread function (LSF) curve is derived by differentiating the denoised ESF curve [Fig. 3(e)]. Finally, the MTF curve [Fig. 3(f)] is derived by performing a Fourier transform on the LSF curve, normalizing it, and taking the absolute value.
Fig. 3. Procedure for the slanted-edge method for calculating the MTF: (a) grayscale image ROI, (b) projection of the slanted edge image, (c) ESF curve before denoising, (d) ESF curve after denoising, (e) LSF curve, and (f) MTF curve.
3. Experiments
3.1 Retro-reflectors used in experiments
Since it is very difficult to create and prepare a retro-reflector by yourself, we prepared three types of commercially available retro-reflectors that are expected to have variations in performance. Figure 4 shows photographs of the three retro-reflectors. The 3M scotchlite 8910 (3M8910) in Fig. 4(a) is a bead-type retro-reflector which is expected to have no anisotropy [29]. The Nikkalite CRG (NCRG) in Fig. 4(b) and RF-Ax in Fig. 4(c) (Nippon Carbide Industries) are prism-type retro-reflectors which are expected to have anisotropy [30]. Micrographs of each retro-reflector are shown in Fig. 5. Both NCRG and RF-Ax are prism type, but the aperture size is largely different. From the viewpoint of geometric optics, the smaller size of the unit element contributes to higher image resolution, which is because the ray shift is small in the situation. But from the viewpoint of wave optics, the larger size of the unit element contributes to higher image resolution, which is because the spread of the wavefront by diffraction is small. By comparing the results of these two prism types retro-reflector, it is also expected that a suggestion can be obtained whether the geometrical or wave optical factor is dominant on the resolution of the aerial image.
Fig. 4. Retro-reflectors used in experiments. (a) 3M scotchlite 8910 (3M8910), (b) Nikkalite CRG (NCRG), and (c) RF-Ax.
3.2 Incident angle to retro-reflector
In this experiment, we investigate how the incident angle on the retro-reflector affects the aerial image resolution. The experimental setup is shown in Fig. 6: light from the light source enters the integrating sphere, and is reflected multiple times throughout the sphere. This generates a uniform intensity distribution, which is then half hidden by a knife edge. The resulting half-moon-shaped aerial image is formed with AIRR. The beam splitter used in this setup is a half mirror (transmittance and reflectance ∼50%). We use a digital camera (Nikon, D5500) to capture a view of the aerial image, and the camera is tilted by 5 degrees with respect to the axis of the knife edge. The recording conditions of the camera were: ISO 400, F-number 4.5, and focal length 35 mm. The floating distance L between the knife edge and the beam splitter is 124 mm.
Fig. 6. Experimental setup to obtain an aerial image of the slanted knife edge method and to investigate how the incident angle at the retro-reflector influences the aerial image resolution. (a) System layout; (b) photograph of the system.
The three 120 mm square retro-reflectors used in this experiment are shown in Fig. 7. As shown in Fig. 6(a), the retro-reflector is rotated by α in order to change the incident angle at the retro-reflector. The initial position of the retro-reflector is set to 0°. Examples of retro-reflector rotation are shown in Fig. 8. The retro-reflector is rotated from 0° to 45° in 15° increments. The MTF is calculated for each of the three types of retro-reflectors at each angle. The exposure time is adjusted so that brightness of the aerial images formed by using the three types of retro-reflectors are the same. As a result, the exposure times for the 3M8910, NCRG, and RF-Ax were 1/25, 1/60 and 1/100 seconds, respectively.
We performed a unit conversion from LSF curve (pixel) to MTF curve (mm) using a test target. In this paper, we use the unit conversion of 0.038 mm/pixel, which is the result of a preliminary experiment using a test target.
3.3 Retro-reflector anisotropy
In this experiment, we investigate the influence of a retro-reflector’s anisotropy on the aerial image resolution. The experimental setup is the same as in Fig. 6 of Sec. 3.2.
Figure 9 shows the three 75 mm radius retro-reflector sheets used in this experiment. Each retro-reflector is rotated in the direction shown in Fig. 6(a). The initial position of the retro-reflector is set to 0°. The retro-reflector is rotated from 0° to 90° in 15° increments in the direction of β shown in Fig. 6(a). Since the incident light is set to enter the same place when the retro-reflector is rotated in the direction of β, the MTF changes depending on the rotation angle if the retro-reflector is anisotropic.
3.4 Floating distance of aerial image
In this experiment, we investigate the influence of the floating distance on the aerial image resolution. The experimental setup is the same as in Sec. 3.2 and Fig. 6.
Figure 10 shows the three 120 mm square retro-reflectors used in this experiment. The distance L between the knife edge and the beam splitter is set to 124 mm, then 224 mm. The MTF is calculated for each of the three types of retro-reflectors for each floating distance of aerial image. We adjusted the exposure time so that brightness of aerial images formed with using the three types of retro-reflectors are the same. As a result, the exposure times for the 3M8910, NCRG, and RF-Ax were 1/25, 1/60 and 1/100 seconds, respectively.
Fig. 10. Retro-reflectors used in the floating distance experiments: (a) 3M8910, (b) NCRG, and (c) RF-Ax.
4. Results
4.1 Incident angle at the retro-reflector
Figure 11 shows views of the slanted knife edge aerial images formed with AIRR, and grayscale images extracted from the aerial images. One can clearly see differences in the aerial image blur for each retro-reflector type. The ESF curves are calculated by projecting the grayscale images of Fig. 11 on the projection axis and denoising the result (Fig. 12). The ESF curve for for the 3M8910 AIRR has a gentle slope (high amount of blur), whereas the ESF curve for the RF-Ax AIRR has a steep slope (low blur). However, the NCRG and RF-Ax have a gentle slope as the incident angle increases. The LSF curves of Fig. 13 are derived by differentiating the denoised ESF curves.
Fig. 12. Calculated denoised ESF curves from the images shown in Fig. 11: (a) 3M8910, (b) NCRG, and (c) RF-Ax.
Fig. 13. Calculated LSF curves from the denoised ESF curves shown in Fig. 12: (a) 3M8910, (b) NCRG, and (c) RF-Ax.
Figure 14 shows the MTF curves of the 3M8910, NCRG, and RF-Ax AIRR systems, respectively. The 3M8910 has the lowest image resolution among the three types of retro-reflectors, as shown in Fig. 14(a). For the 3M8910, there is almost no change in MTF with the incident angle on the retro-reflector. This indicates that when the 3M8910 is used as a retro-reflector, there is no need to consider the placement angle. The NCRG MTF [Fig. 14(b)] decreases significantly when the incident angle on the retro-reflector is 30° or more. This indicates that when the NCRG is used as a retro-reflector, it is necessary to consider the placement angle. Among the three types of retro-reflectors, the MTF of the aerial image formed with RF-Ax as the retro-reflector is the most sensitive to the incident angle on the retro-reflector [Fig. 14(c)]. The aerial image resolution decreases gradually as the incident angle increases up to 30°, and further degrades as it approaches 45°.
The MTF measurement by the slanted knife edge method can measure higher spatial frequency. As a preliminary experiment, the MTF of the camera lens is measured and the results are explained in Appendix A.
4.2 Retro-reflector’s anisotropy
Figure 15 shows views of the slanted knife edge aerial images formed with AIRR in the anisotropy experiment. Figure 16 shows the corresponding denoised ESF and the LSF curves, and Fig. 17 the MTF curves. The 3M8910 MTF [Fig. 17(a)] doesn't depend on the rotation angle of the retro-reflector because it is not changed. The NCRG MTF [Fig. 17(b)], when the spatial frequency is between 0.1 and 0.4 lp/mm, changes depending on the rotation angle of the retro-reflector. For the RF-Ax MTF [Fig. 17(c)], when the spatial frequency is above 0.2 lp/mm, the curve changes slightly depending on the rotation angle of the retro-reflector.
Fig. 16. Calculated (a) denoised ESF curves from the images shown in Fig. 15, and (b) LSF curves from the denoised ESF curves.
4.3 Floating distance of aerial image
Figure 18 shows slanted knife edge aerial images obtained for floating distances of 124 and 224 mm. At this point, it is already clear that the larger floating distance causes an increased blur. Figure 19 shows the denoised ESF curves, and Fig. 20 the corresponding LSFs. The ESF curves for 224 mm distance have a lower grayscale value than the 124 mm distance for all three retro-reflectors compared.
Fig. 18. Aerial images and grayscale images for different retro-reflectors and different floating distances.
Fig. 19. Calculated denoised ESF curves from the grayscale images shown in Fig. 18: (a) 3M8910, (b) NCRG, and (c) RF-Ax.
Fig. 20. Calculated LSF curves from the ESF curves shown in Fig. 19: (a) 3M8910, (b) NCRG, and (c) RF-Ax
Figure 21 shows the MTF curves. The 3M8910 AIRR exhibits almost no change in MTF with floating distance [Fig. 21(a)], indicating that when the 3M8910 is used as a retro-reflector, there is no need to consider the floating distance of the aerial image. For the NCRG [Fig. 21(b)], when the spatial frequency is between 0.1 and 0.4 lp/mm, the MTF decreases as the floating distance of aerial image increases. In particular, there is a large difference in the MTF between 0.2 and 0.4 lp/mm. Above 0.6 lp/mm, there is almost no change in the MTF with floating distance. In the RF-Ax of Fig. 21(c), when the spatial frequency is between 0.2 and 0.7 lp/mm, the MTF decreases as the floating distance of aerial image increases. In particular, the difference in MTF is particularly large at a spatial frequency around 0.4 lp/mm. The amplitude of the MTF curve for approximately 1.8-fold increase in the floating distance is 0.30-fold for NCRG at 0.4 lp/mm and 0.45-fold for RF-Ax at 0.5 lp/mm in spatial frequency. Thus, the 3M8910 forms the most stable aerial image among the three types of the retro-reflectors, regardless of the floating distance of aerial image.
5. Discussion
The 3M8910 bead-type retro-reflector is not influenced by all three factors: incident angle, anisotropy, and floating distance. Thus, there is no anisotropy in the ray shift due to the aperture size and diffracted light spread in the bead-type. Since the aerial image resolution formed with the 3M8910 is not influenced by either geometric- or wave-optical factors, there is no dominant factor. Instead, multiple reflections and scattering have a significant influence on the aerial image resolution. Therefore, the prism-type with anisotropy, which is less influenced by multiple reflections and scattering, can form aerial images with higher aerial image resolution.
The MTFs of the prism-type retro-reflectors, NCRG and RF-Ax, are influenced by all three factors: incident angle, anisotropy, and floating distance. Therefore, there is anisotropy in the ray shift due to the aperture size and the diffracted light spread in the prism-type. The MTFs of RF-Ax for incident angle, NCRG for anisotropy, and RF-Ax for floating distance vary widely. The difference between NCRG and RF-Ax is the size of the optical element, which is larger for RF-Ax. The relationship between each factor and the aerial image resolution is shown in Table 1. The closer the incident angle is to 0°, the aerial image resolution is higher. As the incident angle increases, the amount of ray shift decreases, but the MTF decreases. Therefore, the wave-optical factor is dominant in the incident angle. Anisotropy is a factor in the aerial image resolution of both prism-types, since the MTF changes with the rotation angle. Since the amount of ray shift does not change, the wave-optical factor is the dominant factor of anisotropy. The shorter the floating distance, the aerial image resolution is higher. If the ray shift is dominant factor for the resolution, the dependence of MTF on the floating distance should be low. If diffraction is dominant factor, increasing the floating distance should reduce the MTF. In the result of experiment using prism-type, longer floating distance brought the decrease of MTF, which suggests that the wave-optical factor is dominant in this situation.
Table 1. Relationship between aerial image resolution and incident angle, prism size, and floating distance.
6. Conclusion
We have revealed the factors which influence the resolution of aerial images. In the bead-type, there is no anisotropy in both ray shift due to aperture size and diffracted light spread. Therefore, neither the geometric- nor the wave-optical factors are dominant. The prism-type has anisotropy in both ray shift due to aperture size and diffracted light spread. The experimental results of incident angle, anisotropy, and floating distance show that the wave-optical factor is dominant.
Appendix A
The MTF of the aerial image is corrected by the MTF of the camera lens. Therefore, we measured the MTF of the camera lens as a preliminary experiment.
The experimental setup is shown in Fig. 22. The half-moon shaped light, half hidden by the knife edge, is directly incident on the camera lens. Figure 23 shows the MTF curve of the camera lens. Figure 23(a) shows the MTF of the camera lens in the range of spatial frequency, which is the horizontal axis adopted in this paper, and the MTF at 1.0 lp/mm is about 0.9. This result indicates that the camera lens derived correction is applied when measuring the MTF of aerial image. In Fig. 23(b), the range of the spatial frequency, which is the horizontal axis of Fig. 23(a), is extended to 10.0 lp/mm. This result indicates that the MTF measurement using the slanted knife edge method can be measured at a higher spatial frequency.
Fig. 22. Experimental setup to investigate the MTF of camera lens. (a) System layout; (b) photograph of the system.
Funding
Accelerated Innovation Research Initiative Turning Top Science and Ideas into High-Impact Values (JPMJAC1601); Japan Society for the Promotion of Science (19H04155, 20H05702).
Disclosures
The authors declare no conflicts of interest.
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