We propose a hologram calculation technique that enables reconstructing a shaded three-dimensional (3D) image. The amplitude distributions of zone plates, which generate the object points that constitute a 3D object, were two-dimensionally modulated. Two-dimensional (2D) amplitude modulation was determined on the basis of the Phong reflection model developed for computer graphics, which considers the specular, diffuse, and ambient reflection light components. The 2D amplitude modulation added variable and constant modulations: the former controlled the specular light component and the latter controlled the diffuse and ambient components. The proposed calculation technique was experimentally verified. The reconstructed image showed specular reflection that varied depending on the viewing position.
© 2012 OSA
Holography  is a three-dimensional (3D) display technique that reconstructs the wavefront of light emitted from an object. Numerous studies have been conducted to develop a 3D holographic display. The 3D holographic display technique calculates a hologram pattern from the 3D shape data of an object and modulates a laser beam using the calculated pattern. Therefore, it can even generate a 3D image of a nonexistent object.
Computer graphics (CG) technology for generating a two-dimensional (2D) image from the 3D shape data of an object has progressed remarkably. The development of shading techniques, which calculate the light reflection on an object surface, has considerably enhanced the realistic appearance of a generated 2D image. Numerous shading techniques have been developed. Among them, both Phong shading  and Blinn shading  are simple and powerful. These techniques calculate light reflection by considering the position of the camera (or the eye), positions of light sources, directions of object surfaces, and the material composition of the objects. The 2D image is calculated from a fixed viewing position. On the other hand, holography provides a 3D image with horizontal and vertical parallaxes, which means that the image changes depending on the viewing position. Therefore, a shading technique appropriate for holography, i.e., one that provides variable reflection depending on the viewing position, must be developed to enhance the realistic appearance of a 3D image generated by an electronic holographic display.
Shading techniques for holography have already been proposed by several authors [4–8]. They used the polygon-based model to represent the shape of a 3D object in order to adapt the shading techniques developed in the CG field. Two methods were proposed: one method controls the phase distributions on polygon planes [4–6] and the other controls the intensity distributions of the Fourier transformed images of polygons [7,8].
Several studies used the integral imaging technique to capture a real object [9–13] because the holographic recording of a real object requires a dark and vibration-isolated environment. A combination of a lens array and a high-resolution image sensor is used to obtain light ray information from a real object. A number of parallax images are generated from the obtained light ray information. Then, they are recorded on a hologram using the holographic stereogram technique . This technique can generate a 3D image that provides variable surface reflection depending on the viewing position. However, 3D images generated by holographic stereograms are more blurred than those generated by holograms.
In the present study, we use the point-based method instead of the polygon-based method to model a 3D object. In the point-based method, a 3D object is represented by an aggregate of object points. A hologram pattern is calculated by superimposing zone plates that generate spherical waves converging to the object points. We propose a shading technique that modulates the 2D amplitude distributions of the zone plates on the basis of the Phong reflection model.
2.1 Zone plate technique
When the point-based method is used to model an object, the zone plate technique [15–17] is usually used for the hologram calculation. As shown in Fig. 1 , an object surface is represented by a number of object points. A zone plate consists of concentric circular fringes that generate a spherical wave that converges to generate a light point, i.e., an object point. Therefore, the superposition of the zone plates provides a hologram distribution that reconstructs a 3D image. The calculation time required for the zone plate technique can be reduced by employing a look-up table of precalculated zone plates .
2.2 Phong reflection model
In the present study, the Phong shading  and zone-plate techniques are combined. Now, we briefly explain the Phong shading model.
The Phong shading model describes the light reflection on an object surface. In Fig. 2 , the object surface is shown and four unit vectors are defined. The normal vector n is normal to the object surface, the light vector l indicates the direction to the light source, and the reflection vector r indicates the direction of the reflected light that obeys the law of reflection. The incidence angle of a ray from the light source onto the surface is denoted by θ. The reflection angle is also given by θ. The view vector v is a unit vector indicating the direction to the camera viewpoint. The angle between the reflection vector r and the view vector v is denoted by α.
The Phong reflection model considers three reflection light components: diffuse, specular, and ambient. These components are discussed below.
The diffuse reflection light component consists of light scattered in all directions, as shown in Fig. 3(a) . The diffuse reflection light intensity is constant in all directions; therefore, it does not depend on the view vector v but depends on the incidence angle θ of the illumination light. When the illumination light intensity is represented by Il, the diffuse reflection light intensity Id is given by
The specular reflection light component consists of light reflected in a range of directions whose center direction is the reflection vector r, as shown in Fig. 3(b). The specular reflection light intensity Is depends on the angle α between the reflection vector r and the view vector v, and it is given by
The ambient reflection light component consists of the sum of light reflections from surrounding objects in the scene. Because the ambient light consists of rays traveling in various directions, as shown in Fig. 3(c), its reflection is independent of direction. If the light intensity of the ambient light is denoted by I0, the intensity of its reflection Ia is given by
Finally, the summation of the three reflection light components gives the total reflection light:
2.3 Zone plate technique with shading
Here, we explain how the Phong shading model is combined with the hologram calculation using the zone plate technique.
As explained previously, a zone plate generates light that converges to an object point, as shown in Fig. 4 . The object point is located on the object surface. We assume that each point on the zone plate emits a ray; we further assume that these rays converge to the object point and then are re-emitted. Therefore, the intensity at the point where a ray is emitted on the zone plate might determine the ray’s intensity. This suggests that the 2D intensity modulation of the zone plate could control the angular intensity distribution of the rays emitted from the object point. Hence, the light reflection on the object surface, which has different intensities for different directions, could be determined. Nonuniform modulation of the zone plate by a local peak distribution might generate specular reflection light, and uniform modulation might generate both diffuse and ambient reflection light.
In the Phong reflection model used for CG, only one view vector v corresponds to each point on an object surface. However, for the hologram calculation, multiple view vectors must correspond to each point. This is because a hologram produces a 3D image that can be viewed from multiple viewing positions. Let us consider a ray that is emitted from a point on the zone plate, passes through the object point, and finally enters the eye of an observer. This ray’s direction corresponds to the view vector to the eye for the Phong reflection model. By considering multiple rays from all points on the zone plate, multiple view vectors can be defined. As shown in Fig. 4, the position on the zone plate from where a ray is emitted is denoted by (x, y) and the coordinate origin is located at the center of the zone plate. The distance between the zone plate and object point is denoted by z. Because the vector from the point on the zone plate to the object point is given by (−x, −y, z), the view vector v is given by
The above view vector is calculated for each point on the zone plate, which is then used in Eq. (4) to obtain the intensity modulation at point (x, y) on the zone plate. In the hologram calculation, the amplitude distributions of the zone plates are superimposed so that the amplitude of the zone plate is multiplied with the square root of the intensity obtained from Eq. (4), which is denoted by m(x, y, z). When the distribution of the zone plate is denoted by g(x, y, z), the distribution g’(x, y, z) of the modulated zone plate is given by the following:
Figure 5 depicts the modulation of the zone plate. The concentric circular fringes show the original zone plate pattern. The diffuse reflection light depends on the light vector l and normal vector n, and it does not depend on the view vector v; therefore, the modulation is uniform throughout the zone plate. The ambient reflection light also does not depend on the view vector v; thus, the modulation is uniform throughout the zone plate. Therefore, both diffuse and ambient reflection light components are generated by a uniformly modulated zone plate. The specular reflection light depends on the view vector v and is maximum in the direction of the reflection vector r so that the center of the modulation peak corresponds to the direction of the reflection vector r. Therefore, the specular reflection light component is generated by a zone plate that is two-dimensionally modulated. The sum of the uniformly modulated zone plate and the two-dimensionally modulated zone plate provides the final zone plate that is used for the hologram calculation with shading.
The effectiveness of the proposed hologram calculation technique for shading holographic 3D images was verified by experiments. The following subsections describe the experimental system and results.
3.1 Experimental system
A 4f optical system with a single-sideband filter, shown in Fig. 6 , was used for the experiments because it can eliminate the conjugate image and zero-order diffraction light that significantly degrade the reconstructed image . This single-sideband technique requires the use of a half zone plate instead of a zone plate for generating an object point [20, 21]. The half zone plate is either the upper half or the lower half of the zone plate. It generates both a spherical wave and a conjugate spherical wave that are spatially separated on the Fourier plane of the 4f optical system so that the single-sideband filter can remove the conjugate component. The zero-order diffraction light, which becomes a sharp peak at the center of the Fourier plane, can also be removed by the filter. In Sec 2.3, we used the entire zone plate for the explanation because it has usually been used to explain the zone plate technique. For the entire zone plate, the center of the concentric circular fringes is located at its center. In contrast, for the half zone plate, it is located at the midpoint of the lower or upper edge of the zone plate.
The focal length of the two Fourier transform lenses constituting the 4f optical system was 150 mm. A reflection-type liquid-crystal spatial light modulator (SLM) was used (LC-R 1080, HoloEye Corp.) The resolution was 1,920 × 1,200 pixels, the pixel pitch was p = 8.1 μm, and the screen size was 0.72 in. For simplicity, a transmission-type SLM is depicted in Fig. 6. Because a reflection-type SLM was used in the experiments, a polarization beam splitter was placed in front of it for illumination with laser light. A He–Ne laser (λ = 632.8 nm) was used as the light source. The magnification of the 4f optical system was unity so that the hologram screen size was 0.72 in. The horizontal and vertical viewing zone angles were 4.5° and 2.2°, respectively. The use of the single-sideband filter halved the vertical viewing zone angle.
3.2 Experimental results
There are several ways to divide an object surface into object points. In the present study, a 3D object was represented by a texture image and a depth image. In the experiments, to clearly show the differences among the light reflection components, only the depth image was used for the hologram calculation. The depth image is shown in Fig. 7 . The resolution was 256 × 192 and the number of gray levels was 256. Therefore, the object points were arranged with a regular interval in the horizontal and vertical directions. The normal vector n at each object point was calculated by computing a vector normal to the plane containing adjacent object points.
First, we verified the generation of the three reflection light components. These three components were generated separately. Figure 8 shows the photographs of the reconstructed images when the lighting parameters were l = (0, 0, 1), Il = 1.0, and Ia = 1.0. Figure 8(a) shows the reconstructed image when only the diffuse reflection light was calculated (ks = 0 and ka = 0). The light intensity changed depending on the inclination of the object surface. Figure 8(b) shows the reconstructed image when only the specular reflection light was calculated (kd = 0, n = 5.0, and ka = 0). Several highlights were observed on the object surface. Figure 8(c) shows the reconstructed image when only the ambient reflection light was calculated (kd = 0 and ks = 0). The reflection light intensity was uniform on the object surface. Second, the three reflection light components were generated simultaneously. The photograph of the reconstructed image with kd = 0.25, ks = 0.6, n = 5.0, and ka = 0.01 is shown in Fig. 9 . Therefore, the three reflection light components used in the Phong reflection model were successfully generated.
Third, the change in the light reflection with the viewing position was verified. Figures 10(a) and (b) show the photographs of the reconstructed images captured from the left and right sides, respectively. The holograms were calculated with material parameters of kd = 0.25, ks = 0.6, n = 10.0, and ka = 0.01 and lighting parameters of l = (0, 0, 1), Il = 1.0, and Ia = 1.0. The value of parameter n was enlarged in order to clarify that the movement of the highlights on the object surface depended on the viewing position.
Next, the position of the light source was changed. Figures 11(a) and (b) show the photographs of the reconstructed images when the holograms were calculated with the light vectors l = (−1, 1, 1) and l = (1, 0, 1), respectively. Both images were captured from the same viewing position. The diffuse and specular reflection light components changed with the light source position. The other lighting parameters were Il = 1.0 and Ia = 1.0. The material parameters were kd = 0.25, ks = 0.6, n = 10.0, and ka = 0.01.
Finally, the results for various material parameters are shown in Figs. 12 –14 . Figure 12 shows the results for different diffuse reflection constants: Fig. 12(a) shows the result when the diffuse reflection was small (kd = 0.1), and Fig. 12(b) shows the result when it was high (kd = 1.0). Figure 13 shows the results when the specular reflection constant was changed. Figure 13(a) shows the result when the specular reflection was small (ks = 0.1), and Fig. 13(b) shows that when it was high (ks = 1.0). Figure 14 shows the results for different values of parameter n that determines the sharpness of the highlights. Figures 14(a), (b), and (c) show the results when n = 1.0, n = 5.0, and n = 10.0, respectively.
We experimentally proved that the proposed hologram calculation technique using the 2D modulation of the zone plate can generate shaded 3D images. However, the feasibility of applying the proposed technique to any reconstructed image needs to be evaluated. The width of the peak modulation used for specular reflection should be larger than the pitch of the concentric fringes of the zone plate. Therefore, it might be difficult to produce an object point that is located far from the screen and emits sharp specular reflection light. In the experiments described in Sec. 3, the distances from the screen to object points were not large because the hologram screen size was small and a large depth was not required for the 3D image.
In the experiments, the object points were arranged at constant intervals in both horizontal and vertical directions. Therefore, the distance between adjacent object points differed depending on the inclination of the object surface, where the object points are located. The distance between object points, i.e., the density of object points, influences the brightness of the object surface. Because the viewing zone angle of the experimental system was small, the 2D arrangement of object points with a constant interval on the hologram screen did not cause unnatural brightness on the object surface. A detailed study should be performed on the method to divide an object surface into object points.
From the experimental results, there are two problems with the reconstructed images: the lack of color and generation of speckles. A color display system could be constructed using the time- or space-multiplexing technique. The former technique generates R, G, and B reconstructed images in a time sequential manner and requires a high frame rate SLM. On the other hand, the latter technique uses three SLMs for the three colors and requires larger system dimensions. The generation of speckles is an intrinsic problem of holography. Speckles are generated in both electronic and optical holography. This is because coherent light is used for recording and reconstructing holograms. Fortunately, numerous studies have been conducted on the elimination of speckles. For instance, our research group proposed a method that uses the time-multiplexing display technique to reduce the spatial interference between object points . This method is also based on the zone plate technique; therefore, it might be combined with the hologram calculation technique proposed in the present study.
The viewing zone angle of the experimental system was small. Therefore, the change in the observed image resulting from the change in the viewing position was small. Numerous techniques for enlarging the viewing zone angle have been proposed. The shading technique proposed in this study should be combined with these techniques.
The 2D modulation of the zone plate based on the Phong shading technique was shown to effectively generate a holographic 3D image with shading. In addition to the Phong shading technique, a large number of shading techniques have been proposed in the CG field, and some of them can also be applied to determine the 2D modulation of zone plates. It is possible to modulate a zone plate with a more complicated 2D distribution than that determined by the Phong reflection shading technique. For example, it is known that using the bidirectional reflectance distribution function (BRDF)  can provide a highly realistic image in the CG field. BRDF is a four-dimensional function that defines how light is reflected on the object surface. It provides a 2D angular intensity distribution of reflection light for the incident light vector l. Therefore, this 2D intensity distribution could be used to modulate the zone plate.
The developed method uses the amplitude modulation of the zone plates. Therefore, an amplitude-modulation SLM should be used to display the hologram distributions obtained by the superposition of the modulated zone plates. Because the amplitude modulation of the zone plates results in a reduction in the light power emitted from the zone plates, the reconstructed image becomes darker. It is known that the use of a phase-modulation SLM to display hologram patterns makes the reconstructed images brighter. Therefore, the proposed method should be modified such that phase-only hologram distributions, which are displayed by phase-modulation SLMs, can be designed.
The modulation of the zone plates usually results in an increase in the spot size of the object points generated by the zone plates, thus decreasing the resolution of the reconstructed images. In the extreme case that the specular reflection is too large and the dynamic range of the SLM is limited, the modulation of the zone plate practically decreases the diameter of the zone plates. This decrease in the zone plate diameter increases the spot size of the object points.
We have proposed a hologram calculation technique that can shade a holographic 3D image. The proposed technique is based on the zone plate technique, and the zone plates were two-dimensionally modulated on the basis of the Phong reflection model. To verify the proposed technique, experiments were conducted. The experimental results showed that the proposed technique effectively shaded reconstructed images.
The present study was supported by a Grant-in-Aid for Challenging Exploratory Research from the Japan Society for the Promotion of Science (JSPS), No. 23656234.
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