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A volumetric display that creates a distortion-free three-dimensional (3D) image in midair is described. The proposed system consists of rotating prism sheets used as an optical scanner and a dihedral corner reflector array (DCRA), which is a distortion-free imaging element. Two prism sheets are arranged in a symmetrical configuration to reduce an unnatural motion parallax caused by optical aberrations. A cross-section of the 3D image is formed by the DCRA in midair and moved by the rotating prism sheets to create a 3D displayable space. A 3D volume image was displayed without image distortion or unnatural motion parallax.
© 2013 Optical Society of America
1. Introduction
A volumetric display system is a type of three-dimensional (3D) display that shows a 3D volume image and satisfies all the criteria of stereoscopic vision, so that it can be categorized as a full parallax multiview 3D display [1,2]. An observer can see the 3D image without special glasses, unlike with conventional binocular displays. The 3D volume image is created by many points of light distributed in a 3D space, so that eye fatigue due to a conflict between vergence and accommodation [3,4] does not occur, in contrast to the case of conventional stereoscopic displays [5,6].
In some volumetric displays, the 3D volume images are created as a stack of two-dimensional (2D) images, which is a method that takes advantage of the persistence of vision [7–13]. These types of volumetric displays, called swept volumetric displays, require a mechanical device that moves the location of the 2D image at a faster rate than the duration of the persistence of vision. Actuality Systems Perspecta [7–9] is one such device that rotates a screen and projects cross-sectional images of a 3D volume image onto the screen. Other volumetric display systems having a rotational mechanism have been proposed, such as using a rotating helical screen [10,11], a rotating light-emitting diode array [12], or a rotating thin photoluminescence sheet [13]. These display systems can create a true 3D image that can be observed from all 360 degrees. However, these displays require complicated systems to project an image onto a rotating screen. In addition, an observer cannot reach their hands into the 3D volume image because there is a rotating screen around the image. Another type of volumetric displays, called static volumetric displays, use a stack of liquid crystal scattering shutters [14], or a fluorescent material to illuminate a spot inside the fluorescent material by using a laser [15–17]. Although they do not require a rotational mechanism such as that used in the swept volumetric display, the number of voxels or the size of an image is restricted by the number of liquid crystal scattering shutters or the size of the fluorescent material, respectively. Again, with this type of display, an observer cannot reach into the 3D volume image.
On the other hand, volumetric displays based on the optical scanning of an image plane [18–21] have been proposed as a floating volumetric display, meaning that a 3D volume image is displayed in midair. This property enables an observer to reach their hands into the 3D volume image. In the proposed display system, a 2D real image is formed in midair by an imaging element and moved by an optical scanner. A floating 3D volume image is created from a stack of the 2D images formed in midair. This system is a type of swept volumetric display, but it does not require a moving 2D display device such as a rotating projection screen, so that the optical system is relatively simple. However, the viewing area and displayable space are restricted by the scanning range of the optical scanner and the effective aperture of the imaging element. In previous studies [18–20], a galvanometer mirror scanner was used as an optical scanner. The scanning range and effective area of the mirror scanner could not be expanded because of the limitations in the mechanical load allowed by its oscillation movement.
In order to widen the displayable space, we proposed to use a rotating prism sheet as the optical scanner for the volumetric display, based on optical scanning of the image plane in our previous studies [22,23]. Since the mechanical load required to rotate a thin prism sheet is lower than the mechanical load required to drive a galvanometer mirror scanner, the scanning range and the effective area could be expanded and the displayable space was widened. However, the position of a 3D volume image changes depending on viewing position, which can be described as an unnatural motion parallax. This problem is caused by the non-linearity of the light-bending angle of a prism sheet and the optical aberration of the concave mirrors used as imaging elements. This non-linearity of a prism sheet is the principal factor in the unnatural motion parallax. Since the bending angle depends on the incident angle, reducing the incident angle dependence of the light bending is required to achieve a natural motion parallax. Suppression of the optical aberration of the imaging element is also an important problem.
In this study, we propose to use a scanning device consisting of an axially symmetric double prism sheet, which reduces the incident angle dependence by cancelling out the ray bending. With this scanning device, an incident ray is scanned without changing its outgoing angle. In addition, we propose the use of a dihedral corner reflector array (DCRA) [24,25], which is a distortion-free imaging element, in place of the concave mirrors. We verified the effectiveness of the proposed optical scanning method by a ray tracing simulation and an experimental display system.
The basis of our volumetric display system is described in Section 2.1. Optical scanning by the rotating prism sheets, which are arranged in a symmetrical configuration, and imaging by the DCRA are explained in Sections 2.2 and 2.3, respectively. Section 3 shows the ray tracing simulation method and the experimental setup of the proposed volumetric display. After the results of the simulation and experimental display are shown, the effectiveness of the proposed system for decreasing the unnatural motion parallax and the image distortion is discussed in Section 4. Finally, Section 5 concludes this paper.
2. Method
2.1. A volumetric display system based on optical scanning of an image plane
Figure 1 shows the principal optical setup of our volumetric display system. The whole display system consists of a 2D display (such as a set comprised of an inclined screen and a projector), an optical scanner, and an imaging element. The 2D display device is set perpendicularly to the x-y plane and is inclined with respect to the x-z plane at an angle θinc in order to have depth information. The angle of the rays from the 2D display device are changed rapidly by the optical scanner. Subsequently, the rays are incident onto the imaging element. As a result, a 2D image displayed on the 2D display device is formed by the imaging element and moved by the optical scanner. Since the formed image contains depth information owing to the inclination of the 2D display device, a stack of the afterimages of the moved 2D image makes a 3D volume image. In order to take advantage of persistence of vision, moving the 2D image faster than the visual persistence time of the human eye, generally about 20 Hz, is required.
Fig. 1 Schematic illustration of the volumetric display system based on optical scanning of an inclined image plane.
In this system, the size of the displayable space is restricted by the scanning range of the optical scanner and the effective size of the 2D display device, the optical scanner, and the imaging element. It is easy to enlarge the size of the 2D display device and the imaging element because they are stably fixed. On the other hand, it is difficult to enlarge the size of the optical scanner because a typical optical scanner that has a wide scanning range requires being physically moved faster than the length of the duration time of the afterimage. For example, a galvanometer mirror scanner has an oscillating movement and a rotating prism sheet, as reported in our previous study [23], has a rotational movement. In these scanning systems, the mechanical load of the rotating prism sheet was less than the load of the galvanometer mirror scanner, so that using the rotating prism sheet in our volumetric display system was effective for enlarging the size of the 3D image. However, it was a problem that the observed motion parallax was unnatural, as the direction of the outgoing ray depended on its incident angle. As such, the unnatural motion parallax can be reduced when the angular dependency of the outgoing ray is removed.
2.2. Optical scanning by rotating symmetrically arranged prism sheets
The proposed optical scanner consists of a double prism sheet with the same deflection angle to reduce the unnatural motion parallax. These prism sheets were arranged at a distance in an axially symmetric configuration as shown in Fig. 2.
Figure 3 shows a diagram of rays passing through the double prism sheet without reflection, rendered by the ray tracing simulation software FRED. Bending of the incident ray transmitted by the first prism sheet is canceled out by the second prism sheet, so that the direction of the incident ray and outgoing ray are equal. In other words, the amount of directional change of the ray is independent of its incident angle. On the other hand, the incident angle dependence of the deflection angle for the rays remains between the first prism sheet and the second prism sheet, which causes image movement and the unnatural motion parallax. The distance an image moves and the amount of unnatural motion parallax increase with increasing distance between the double prism sheets. Therefore, it is required to space the double prism sheet such that the moving distance of the image is large enough to display a volume image, while keeping the observable unnatural motion parallax negligible.
We next describe the unnatural motion parallax in comparison with natural motion parallax. In this case, we only have to discuss rays that enter into the human eye because the visual information perceived by an observer is important. In other words, it is only necessary to analyze rays with a very small spread angle; thus, we use a simplified model of Fig. 3, which uses a principal ray emitted from a point light source. Figure 4 shows examples of ray paths in the simplified model from a point light source L in the cases of no prism sheet, one prism sheet, and a double prism sheet arranged in a symmetrical configuration. We define the amount of motion parallax as a difference of viewing positions at height for two rays with angle and from the point light source L. From Fig. 4 (left), the amount of motion parallax in a natural scene is
Fig. 4 Illustration of principal rays from point light sources and the amount of motion parallax without a prism sheet (left), with one prism sheet (center), and with the proposed setup of a double prism sheet (right).
From Snell’s law, the outgoing angles of the rays refracted by the first prism sheet in Fig. 4 are
where is the refractive index of the prism sheet and is the apex angle of the prism. Therefore, the amounts of motion parallax when the rays pass through the prism sheet(s) are where the thickness of the prism sheet was ignored. To analyze the amounts of motion parallax for ray angle , we assigned values to the other variables in accordance with the experimental setup described in Section 3 and represented the amount of motion parallax as shown graphically in Fig. 5. From this graph, obviously differs from for nearly incident angles, so that the observed motion parallax through the single prism sheet seems to be unnatural. On the other hand, is approximately similar to for , so that the observed motion parallax through the symmetrically arranged double prism sheet seems to be nearly natural in this experimental configuration. However, the difference between and becomes large when the ray angle is below , especially near the numerical aperture of the prism sheet ( for ).Fig. 5 Graph of motion parallax (no prism sheet), (one prism sheet) and (the proposed setup of two prism sheets) versus incident angle . In this case, , [cm], [cm], [cm], [deg], and [deg].
The distance from the light source to its virtual image is
Rotating the double prism sheet in the same direction around an axis perpendicular to their surfaces and at the same speed moves a virtual image of the point light source L in a circular motion of radius l. When the inclined 2D display is put in place of the point light source L, rotating the double prism sheet moves the virtual image of the 2D display in a circular motion without changing its inclination θinc. Figure 6 shows the movement of the 2D image plane using the proposed optical scanning method. The swept volume V created from a stack of the moved virtual image planes iswhere w and h are the width and the height of the screen. Note that there is a blank space in the center of the volume when that is not swept by the moved image plane. On the other hand, the multiply scanned area, i.e. the area swept more than once by the moved image plane, is increased when . To eliminate the blank space and minimize the multiple scanned area, , so thatTherefore, making longer can increase the volume of the 3D displayable space. The floating 3D volume image can be created from the moving image plane by forming a real image using an optical imaging element.Fig. 6 Schematic illustration of image plane movement achieved by rotating the symmetrically arranged prism sheets (a) and the volume of the swept space (b), (c), and (d). Panel (b) is the side view, (c) is the front view, and (d) is the overhead view.
2.3. Imaging by the DCRA
The DCRA [24,25] is a distortion-free imaging element that consists of a number of small square holes whose inner walls are planar mirrors. The principle of imaging by the DCRA is described using a schematic diagram of a 2D array of square holes. Figure 7 shows a schematic diagram and the ray paths for imaging by the DCRA. Since each square hole works as a dihedral corner reflector, some of the incident rays reflected twice at the square mirror hole invert their direction for x-z components. On the other hand, rays pass through the square hole without changing direction for the y component. Therefore, the optical path of the ray is plane-symmetric with respect to the surface of the DCRA; the rays converge at the plane’s symmetric point just by reflection of the planar mirrors. Therefore, a real image is formed without optical aberrations, such as the five Seidel aberrations for imaging with a spherical lens. However, the image is blurred by diffraction produced at the small square hole.
Fig. 7 Imaging by the DCRA. Panel (a) is a schematic diagram, (b) shows the ray paths simulated by FRED, and (c) is an irradiance (power/area) spread function on a plane at the image points.
When the incident rays are reflected once or more than twice, these rays do not converge to form a real image and are observed as stray light, shown in Fig. 7(b). As a result, the stray light causes an undesirable virtual image. Setting the incident angle properly is required because the transmittances of the ray reflected twice and of the others are different from the incident angle. An effective incident angle for double reflection is ± 15 degrees horizontally from the center and from 30 degrees to 50 degrees vertically.
The DCRA can be fabricated by orthogonally combining two strip mirror arrays, in which the mirrors are perpendicular to the device surface, as illustrated in Fig. 8. The DCRA used in this study had a distance between the strip mirrors of 0.5 mm, a thickness of 3 mm, and an effective area of 15 × 15 cm2, so that the number of mirror holes was 90000. When an orthogonal strip mirror DCRA is viewed from a direction perpendicular to the surface of the element, the crossed strip mirror arrays are regarded as a 2D array of square mirror holes. In this study, we used the strip mirror type DCRA because it is easy to make an element that consists of a 2D square hole array, which has a larger area than the conventional DCRA.
Fig. 8 Schematics of the slit-type DCRA used in this study. Panel (a) is an overhead view and (b) is a top view.
3. Optical setup of a ray tracing simulation and an experimental display system
We evaluated the motion parallax of a moving 2D image using an experimental system and another piece of ray tracing software, The Persistence of Vision Raytracer (POV-Ray) [26]. Figure 9 shows the experimental setup of the double prism sheet arranged in a symmetrical configuration. We used acrylic prism sheets, with a refractive index , and an apex angle .. The double prism sheet was supported at intervals of . by four acrylic rods. From Eq. (5), the distance between the light source and its virtual image is for normal incidence . The double prism sheet, arranged in a symmetrical configuration, was attached to a bearing. This was rotated by a servo motor through a timing belt.
Fig. 9 Picture of the double prism sheet arranged in a symmetrical configuration on the bearing with the timing belt. The area of the prism sheet was 10 centimeters square. The thickness of the prism sheet was 2 mm and the prism pitch was 0.3 cm.
POV-Ray was used to simulate the proposed optical scanning system and visualize the motion parallax. An optical setup including a 2D screen and the double prism sheet arranged in a symmetrical configuration was modeled and rendered in POV-Ray as shown in Fig. 10. This 3D space was captured by a virtual camera and the field of view was 46 degrees to mimic the human eye. The light source was seen through (1) the air (no prism sheet), (2) a single prism sheet, and (3) the double prism sheet arranged in a symmetrical configuration, respectively.
Fig. 10 Setup with the symmetrically-arranged prism sheets in POV-Ray. When the screen was seen through the air, the fixed prism sheets were removed. When the screen was seen through one prism sheet, the bottom prism sheet remained in and the upper prism sheet was removed.
We also developed an experimental volumetric display including both the proposed optical scanner and the DCRA as shown in Fig. 11. We used LightCommander (made by LOGIC PD) that can switch a cross-section of a 3D image at a 5000 [Hz] refresh rate, in synchronization with the rotation of the prism sheets. The prism sheets were rotated at a 20 Hz speed, which is the refresh rate of the 3D volume image, so that 250 cross-sectional images were projected onto the 2D screen during one rotation of the optical scanner. Since this display system is sensitive to vibration, the support frames of the rotating prism sheets were fixed firmly on a table. It is not problematic for the observations if the amplitude of vibration is submillimeter. The resolution of the cross-sectional image is 1024 × 768 pixels. Therefore, the displayable space of the 3D volume image has 1024 × 768 × 250 voxels. The real image plane of the 2D screen formed by the DCRA has a size of cm, cm and an inclination angle . From Eq. (7), the 3D displayable volume of the developed volumetric display is approximately 2300 cm3, which is approximately 3.5 times larger than the displayable volume made by a previous volumetric display using a galvanometer mirror scanner [20]. However, the voxel density is lower than the previous volumetric display because the number of voxels is same. A volume image that has a higher voxel density could be displayed by using a projector that has a higher resolution and frame rate.
Fig. 11 Experimental setup of the proposed volumetric display. Panel (a) is a schematic diagram and (b) is a picture taken from the side.
4. Results and discussion
4.1. Motion parallax of the image plane
We observed virtual images of the 2D screen through the double prism sheet at several rotation angles. The observation of the optical scanning is shown in Fig. 12. The image plane of the 2D screen was moved laterally in a circular motion with a radius of 4 cm. Therefore, the validity of the derived equations for optical scanning and assessing the function of a double prism sheet was confirmed by the experimental results. The unnatural motion parallax was not observed. However, stray light is observed as a strip-shaped bluish- white light on the prism sheet, as shown in Fig. 12.
Fig. 12 Movement of the image plane by rotating the symmetrically arranged prism sheets. The blue marker was put on to make rotation of the prism sheets more visible. The strip-shaped bluish-white light is stray light.
Next, the result of the ray tracing simulation performed by POV-Ray is described. Figure 13 shows the rendering result of the ray tracing simulation, the setup of which is illustrated in Fig. 10. We measured the distance of the lateral movement of a light source that arises when changing the viewing position (camera location). From the rendering result, the lateral movement distance was 104 pixels in the case of using one prism sheet and 121 pixels in the case of no prism sheet (the natural motion parallax). Therefore, the observed motion parallax through the single prism sheet was different by approximately 14% from the natural motion parallax. On the other hand, in the case where the symmetrically arranged prism sheets were used the pixel variation was same as the natural motion parallax. Therefore, the proposed optical scanning method is effective in reducing the unnatural motion parallax observed in the volumetric display system using the prism sheet as an optical scanner.
Fig. 13 Results of the POV-Ray simulation. The red points are point light sources. The upper panels were observed from the left and the bottom panels were observed from the right.
4.2. 3D volume display
Figure 14 is the 3D image produced by the volumetric display system shown in Fig. 11. The floating 3D volume image, which was 6 × 6 × 6 cm3 in size and created from 34 cross-sectional images, was displayed without distortion. Natural binocular parallax and motion parallax was also observed. Thus, a volumetric display without distortion or unnatural motion parallax was achieved by use of the proposed optical scanning method.
Fig. 14 3D floating volume image produced by the proposed volumetric display. Panel (a) was captured in a dark room and (b) was captured in a lighted environment (Media 1).
However, an undesirable image was observed at the upper left of the displayed 3D image in Fig. 14. This undesirable image was caused by the stray light described in Section 4.1. To identify where the stray light came from, we again simulated ray paths using FRED. Figure 15 shows the result of a ray tracing simulation including the ray paths in the prism sheet. As shown in Fig. 15, the stray light is due to surface reflection, total internal reflection (TIR), and multiple refraction. In this display system, the rays creating the 3D image and the stray light are emitted from same light source. Therefore, the brightness of the stray light increases with the increasing brightness of the projector, i.e. the contrast of the image is a constant unless the the stray light is reduced. Applying anti-reflective coating to the surface of the prism sheet is a simple solution for suppressing part of stray light. Although they are difficult to make, applying light shields to the vertical planes of the prism sheet can remove the TIR and the multiple refraction. We verified the effectiveness of these stray light suppression methods using FRED, as shown in Fig. 16.
Fig. 15 Diagram of stray light passing through the prism sheet. Red lines show reflected rays, including surface reflection and TIR.
Fig. 16 Diagrams of stray light passing through the prism sheet when stray light suppression methods are applied. (a) Anti-reflective coating. (b) Anti-reflective coating and light shields attached to vertical planes of the prism sheet.
5. Conclusion
We proposed a volumetric display system consisting of a double prism sheet arranged in a symmetric position and a DCRA. The double prism sheet was arranged at a distance in an axially symmetric configuration to reduce the unnatural motion parallax, which is observed in the case of using a single prism sheet. We developed expressions to quantify the unnatural motion parallax using a simplified model. From these expressions, we verified that the unnatural motion parallax is reduced by use of the proposed optical setup. An additional ray tracing simulation from POV-Ray was used to visualize motion parallax. The simulation showed that the motion parallax observed through the double prism sheet was same as the natural motion parallax.
We developed an experimental optical setup for a volumetric display including the double prism sheet. The proposed optical scanning method reduced the unnatural motion parallax to an observationally negligible level. Formation of the moved image plane using the DCRA created a 3D volume image in midair. However, an undesirable image was observed because of stray light occurring within the prism sheet. The effectiveness of stray light suppression achieved by applying anti-reflection coating and light shields was simulated by FRED.
For practical use, it is required to apply a stray light suppression method and to improve performance of the device. It is especially important to improve the refresh rate of the projector to increase the number of cross-sections of the 3D image. Colorization of the 3D image is also required.
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