Rendering of specular surfaces in polygon-based computer-generated holograms

Hirohito Nishi; Kyoji Matsushima; Sumio Nakahara

doi:10.1364/AO.50.00H245

1. Introduction

Computer-generated holograms (CGHs) reconstruct the light of a three-dimensional (3D) scene, unlike conventional displays, which provide only binocular disparity. CGHs are thus an important candidate for future 3D display technology. However, CGH technology suffers from the twin problems of computational and display limitations. Extremely high display resolution is required to display fine 3D images using CGHs. More than one billion pixels must be displayed with a physical resolution of less than $1 μm$ to create fine 3D images as in classical holography. Electrical devices are unable to meet these requirements at this time, although efforts are continuing toward the realization of true holographic 3D displays. However, brilliant 3D still pictures have recently been realized using CGH technology [1–5]. These still CGHs, fabricated using laser lithography, reconstruct spatial 3D images of occluded 3D scenes in full parallax, which is comparable with the images in classical holography. Because the reconstructed images give almost all of the depth cues, viewers experience a strong sensation of depth that has never been realized using conventional 3D systems.

The other serious problem with CGHs is the long computation time required to compute the full-parallax wave field emitted from occluded 3D scenes. Although CGHs have a long history, few techniques have been proposed for computing or estimation of the light from virtual objects, whose shape and properties are provided by numerical models. The most popular technique at present is the point-based method [6,7]. In this method, an object is assumed to be covered by many point light sources, and spherical waves emitted from these point sources are computed and superposed in the hologram plane. This method is easy to implement but is very time consuming for full-parallax CGHs because point-based methods need gigantic numbers of point sources to create the full 3D surface of the object. Although many techniques for acceleration of this method have been proposed [8–13], it is not easy to use point-based methods to create high-definition full-parallax CGHs of occluded 3D scenes. The other main category used in CGHs is the multiviewpoint projection technique. In this technique, the object fields are yielded by numerous projection images of the object taken from different viewpoints. This technique is applicable to both real scenes [14] and virtual scenes [15–17]. Layered computation models that slice the object may also form a small category among these methods [18,19].

To create extremely high-definition CGHs by laser lithography [1], a polygon-based method is used to compute the object field of the occluded 3D scene [20]. This type of technique also forms a category for computation of CGHs [21–25]. In this method, the object is considered to be composed of small planar surface light sources that have polygonal shapes, and the wave fields emitted by these polygonal sources are calculated and are superposed in the hologram plane. This method speeds up the computation of CGHs considerably because the number of polygons required to form the surface is much smaller than the number of point sources used in the point-based method. The high-definition CGHs created using the polygon-based method can reconstruct fine 3D images and are comparable with classical holograms.

One issue with the polygon-based method as well as with point-based methods is that it cannot reconstruct the large variety of realistic surfaces that can be reconstructed in modern computer graphics (CG). The CGHs created using the method in [20] only reconstruct diffuse surfaces. Thus, the viewers of the hologram do not get any indication of the materials of the surfaces in the optical reconstruction. The theory for realistic rendering of CGHs has been discussed in the literature [26]. This theory is based on a generic model but eventually computes the object fields using a point-based method and does not present any concrete procedure for rendering of realistic surfaces. Two techniques have been proposed for rendering of specular surfaces in the polygon-based method [24,25]. In the former technique, based on a microfacets model, however, it has not been verified whether the specular surfaces are reconstructed in practice. The latter technique uses an atomic force microscope to measure the roughness of real metal surfaces. However, the wave field of the reflected light is computed by using the finite-difference time-domain (FDTD) method, which is an accurate simulation technique for electromagnetic fields, but consumes an enormous amount of computer resources. Therefore, only a small fragment of the possible surfaces can be created by this technique. Computing high-definition CGHs using this method is likely to be impossible. Several studies also propose methods based on the Phong reflection model [27] and the Cook-Torrance reflection model [28]. The former is conceptually similar to the technique proposed in this paper. However, the formulation is incomplete and insufficient to create actual CGHs of 3D objects. The latter method using a special propagation kernel (impulse response) is also too time consuming to create high-definition CGHs because the propagation kernel is dependent on the reflection direction of each individual polygon.

In this paper, we propose a technique for rendering of specular surfaces in the polygon-based method. This technique, based on the Phong reflection model, features sufficiently fast computation to create polygon-based high-definition CGHs. An actual high-definition CGH, fabricated using laser lithography, is demonstrated for verification of the proposed technique.

2. Rendering of Specular Surfaces

In the polygon-based method, all polygons have their own local coordinates [20], as shown in Fig. 1. The local coordinates of the polygon $n$ are given by $(x_{n}, y_{n}, z_{n})$ , where the $z_{n}$ axis is always perpendicular to the polygon surface. Because we concentrate on a given single polygon here, the suffix $n$ is largely omitted in the rest of this section.

Fig. 1. Coordinate system used for formulation.

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A. Surface Function

The theoretical model of the polygonal surface light source is a slanted aperture irradiated by a plane wave behind the aperture, as shown in Fig. 2(a) [20,29]. The aperture has the same shape and slant as the polygon. Also, a diffuser plate is mounted in the aperture because the aperture corresponding to the polygon is commonly too large to diffuse the irradiated light sufficiently. The aperture provided with the diffuser is expressed using a complex function defined in the $(x, y, 0)$ plane of the local coordinates. This complex function is referred to as a surface function. The common form of the surface function is given by

h (x, y) = a (x, y) \exp [i ϕ (x, y)],

where

a (x, y)

is the amplitude distribution that gives the shape, texture, and brightness of the polygon. The phase distribution

ϕ (x, y)

plays the role of the diffuser that spreads the light over the hologram. Thus, the phase distribution is important for rendering of the specular surface because the diffusiveness and direction of the emitted light is determined by

ϕ (x, y)

.

Fig. 2. Schematic illustration of wave fields emitted from the slanted aperture (a) before and (b) after spectral remapping.

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Figure 3(a) shows an example of the surface function for a conventional diffuse surface. Figure 3(b) also shows the Fourier spectrum given by

H (u, v) = F {h (x, y)},

where the symbol

F {\cdot}

stands for a Fourier transform and

u

and

v

are spatial frequencies with respect to

x

and

y

of the local coordinates, respectively. In our polygon-based method, the spectrum is shifted to force the direction of the emitted light into the

Z

axis of the global coordinates and to calculate the wave field in the plane parallel to the hologram [20]. This procedure is called spectral remapping. Although light travels along the global

Z

axis after spectral remapping, as shown in Fig. 2(b), the light is emitted to the normal direction of the aperture before the remapping. The following formulation and discussion are thus given in the local coordinates before the spectral remapping procedure.

Fig. 3. Example of the surface function of (a) a diffuse surface and (b) its Fourier spectrum.

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B. Spectrum of Diffuse and Specular Reflection

Figure 4 schematically illustrates two types of surface reflection: diffuse and specular reflection. The spatial spectra of these two types of reflection are also schematically depicted in Fig. 5. Because the spectrum gives the far-field pattern of the light, diffuse reflection has a broadband spectrum. A quasi-random pattern is thus provided to the phase distribution $ϕ (x, y)$ in rendering of diffuse surfaces so that the spectrum has a broad band as in Fig. 3(b).

Fig. 4. Schematic illustration of (a) diffuse and (b) specular reflection.

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Fig. 5. Schematic illustration of the spatial spectrum of diffuse and specular reflection.

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In specular reflection, the energy of the reflected light is concentrated around the reflection direction, which is not usually coincident with the normal of the polygon. The spectrum is therefore narrow when compared with diffuse reflection and is shifted from the origin, as shown in Fig. 5. When the reflection angle is $θ$ in the $(x, 0, z)$ plane of the local coordinates, the shift magnitude is obtained from the far-field diffraction as follows:

Δ u = \frac{\sin θ}{λ},

where

λ

is the wavelength of the light. The frequency shift is found easily using the far-field consideration, whereas the shape of the spectrum cannot be determined without a reflection model.

C. Spectral Envelope Based on Phong Reflection Model

The fact that specular surfaces have a limited diffusiveness leads to narrowing of the bandwidth of the reflected light. Here the reflected light is represented by the surface function in the polygon-based method. This means that the spectrum of the surface function must be band limited for creation of specular surfaces. We use the Phong model for specular reflection [30] to determine the shape of the spectrum of the phase distribution $ϕ (x, y)$ . The Phong model, illustrated in Fig. 6(a), is the most popular model for rendering of specular surfaces in CG. In the Phong model, the relative brightness of the specular light that a viewer observed in the viewing direction $V$ is given b:

I (V; R) = {(R \cdot V)}^{α},

where

α

is a shininess constant. The vector

R = R_{x} x + R_{y} y + R_{z} z

gives the direction of regular reflection, where

x

,

y

, and

z

are the unit vectors in

x

,

y

, and

z

of the local coordinates. In this model, a larger shininess constant value provides higher specularity, and we note that the brightness is zero if the angle of

R

and

V

is more than

90 °

(

R \cdot V < 0

). The brightness is shown as a function of the viewing angle

φ

in Fig. 6(b).

Fig. 6. Phong model for specular reflection.

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In the far field, the light traveling in the viewing direction can be regarded as a plane wave. We therefore propose that the viewing direction can be interpreted as a unit wave vector as follows:

V = k / k,

where

k

is the wave vector given in the local coordinates and

k = | k | = 2 π / λ

is the wavenumber. Also, the wave vector can be written using the spatial frequencies as follows:

k = 2 π (u x + v y + w z),

where

u

and

v

are again the Fourier frequencies but are limited in

u^{2} + v^{2} \leq 1 / λ^{2}

. The frequency

w

with respect to

z

is not independent of

u

and

v

, and is given by

w = {(λ^{- 2} - u^{2} - v^{2})}^{1 / 2} .

By substituting Eqs. (6) and (7) into Eq. (5), the view vector is rewritten as

V = V (u, v) = λ [u x + v y + {(λ^{- 2} - u^{2} - v^{2})}^{1 / 2} z] .

Thus, the brightness of the specular refection is rewritten as a function of frequency by substituting Eq. (8) into Eq. (4):

I_{1} (u, v; R) = {\begin{array}{lcl} λ^{α} {[R_{x} u + R_{y} v + R_{z} {(λ^{- 2} - u^{2} - v^{2})}^{1 / 2}]}^{α} & \dots & R \cdot V (u, v) \geq 0, \\ 0 & \dots & otherwise. \end{array}

Examples of the shape of

I_{1} (u, 0; R)

are shown in Fig. 7(a). The shape in

θ = 0

is symmetrical with respect to the center frequency, but a little deformation can appear in cases where there is a large refection angle.

Fig. 7. Two types of spectral shapes based on the Phong model for specular reflection.

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To create the specular surfaces using the polygon-based method, the phase factor $\exp [i ϕ (x, y)]$ must be modified so that its spectral envelope is fitted to $I_{1} (u, v; R)^{1 / 2}$ because the brightness of the surface corresponds to the intensity of the reflection field. The modified spectrum of the diffuser is given by

G_{1} (u, v; R) = Φ (u, v) I_{1} {(u, v; R)}^{1 / 2},

Φ (u, v) = F {\exp [i ϕ (x, y)]} .

Figure 8 shows an example of the modified spectrum

G_{1} (u, v; R)

, the spectral envelope

I_{1} (u, v; R)

, and the spectrum of the original diffuser

Φ (u, v)

. Because

| F^{- 1} {G_{1} (u, v; R)} | ≢ 1

, we define the modified phase distribution of the diffuser as follows:

ϕ_{1} (x, y; R) = \arg {F^{- 1} {G_{1} (u, v; R)}},

where

\arg {ξ}

is the argument of

ξ

. The phase factor

\exp [i ϕ_{1} (x, y)]

is not identical to

F^{- 1} {G_{1} (u, v; R)}

, but the spectral envelope is very similar to

I_{1} (u, v; R)

. Thus, the surface function and its spectrum are given by

h_{1} (x, y; R) = a_{s} (x, y) \exp [i ϕ_{1} (x, y; R)],

H_{1} (u, v, R) = F {a_{s} (x, y) \exp [i \arg {F^{- 1} {G_{1} (u, v; R)}}]},

where

a_{s} (x, y)

is the amplitude distribution for specular refection. Here we note that a double Fourier transform is necessary to compute the spectrum for a given specular polygon if the phase

ϕ_{1} (u, v; R)

is used for the diffuser.

Fig. 8. Example of the modified spectrum $G_{1} (u, v; R)$ (a) that is the product of the spectral envelope $I_{1} (u, v; R)^{1 / 2}$ (b) and the spectrum of the original diffuser $Φ (u, v)$ in (c). Here $α = 30$ and $θ = π / 6$ .

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D. Rendering Objects Composed of Specular Surfaces

Each field of the specular polygons is created by modification of the spectral envelope. However, computation using Eq. (14) is time consuming when calculating the whole wave field of objects because each of the polygons comprising the object has its own reflection vector of $R_{n}$ as shown in Fig. 9. The phase distribution must be generated for each polygon if using $ϕ_{1} (u, v; R_{n})$ .

Fig. 9. Reflection in polygon surfaces.

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To reduce the computational effort, the light that is emitted by the specular polygon and that travels in the normal direction should be calculated as a first step. In this case, the surface function based on the Phong model is given by

h (x, y; z) = a_{s} (x, y) \exp [i ϕ_{1} (x, y; z)] .

The surface function emitting the light in the direction of

R

is given by

h_{2} (x, y; R) = h (x, y; z) \exp [i k R \cdot r],

where

r = x x + y y + z z

is a position vector and

\exp [i k R \cdot r] = \exp [i k (R_{x} x + R_{y} y)]

is the plane wave traveling in the

R

direction. In Fourier space, the spectrum of the modified surface function is written as

H_{2} (u, v; R) = F {h_{2} (x, y; R)} = H_{0} (u - \frac{R_{x}}{λ}, v - \frac{R_{y}}{λ}),

H_{0} (u, v) = F {h (x, y; z)} .

In this case, the fast Fourier transform (FFT) is executed only once for each polygon because the phase factor

\exp [i ϕ_{1} (x, y; z)]

is independent of the reflection direction and is thus precomputed. To obtain the spectrum

H_{2} (u, v; R)

, the amplitude distribution

a_{s} (x, y)

multiplied by the precomputed phase as in Eq. (15) is Fourier transformed as in Eq. (18). The direction of the reflected light is given by the shift of Eq. (17). As a result, the computation time is considerably reduced when compared with computing Eq. (13).

By using the same procedure as Eqs. (16) and (17), the spectral envelope of $H_{2} (u, v; R)$ is given by

I_{2} (u, v; R) = F {F^{- 1} {I_{1} (u, v; z)} \exp [i k R \cdot r]} = I_{1} (u - \frac{R_{x}}{λ}, v - \frac{R_{y}}{λ}; z) .

Examples of

I_{2} (u, v; R)

are shown in Fig. 7(b). The spectrum agrees well with

I_{1} (u, v; R)

, especially for the small reflection angle case. Although a little difference is found in the large reflection angle case, we can considerably accelerate the computation of the whole field of specular objects by using the shape

I_{2} (u, v; R)

.

E. Procedure for Rendering Surfaces

The final surface function is basically given by the weighted sum of the surface functions for diffuse reflection and specular reflection, as follows:

h (x, y) = K_{d} h_{d} (x, y) + K_{s} h_{s} (x, y),

where

h_{d} (x, y)

and

h_{s} (x, y)

are the surface functions for diffuse and specular reflection, respectively. The coefficients

K_{d}

and

K_{s}

are the weights of these functions. However, in the proposed method, the specular surface function calculated by Eq. (17) is given in Fourier space instead of real space. Therefore, the weighted sum is carried out in Fourier space as follows:

H (u, v) = K_{d} H_{d} (u, v) + K_{s} H_{2} (u, v; R),

where

H_{d} (u, v)

is the spectrum of the surface function for diffuse reflection. Figure 10 shows an example of generation of this surface function. The specular surface function

h (x, y; z)

with the phase distribution

ϕ_{1} (x, y; R)

of Eq. (15) is Fourier transformed and then shifted as in Fig. 10(a). Here the amplitude distribution for the specular reflection

a_{s} (x, y)

is a binary function and gives only the shape of the polygon because the shading of the object is provided by the diffuse surface. The diffuse surface function is also Fourier transformed and superposed into the spectrum of the specular surface function.

Fig. 10. Example of the procedure for generating a specular surface.

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3. Creation of High-Definition CGH of Specular Object

We created a high-definition CGH, named “The Metal Venus I” after the first polygon-based high-definition CGH, “The Venus” [1], that reconstructs the object with diffusive surfaces. The 3D scene of the CGH, shown in Fig. 11, is almost the same as that of The Venus. The statue of the Venus is $5.7 cm$ in height and is composed of 1396 polygons. The parameters used to compute the CGH are summarized in Table 1. Note that the size of the sampling window for the specular surface function is the same as that for the diffuse surface function, but the sampling interval is smaller than that used for the diffuse function. In the Fourier space, this leads to extension of the sampling window. Because the specular spectrum is shifted prior to the weighted sum, if the sampling windows are not sufficiently large, the spectrum may not overlap with that of the diffuse surface. This is why the sampling interval for the specular surface function is set to a value smaller than that for the diffuse surface.

Fig. 11. 3D scene of The Metal Venus I.

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Table 1. Parameters Used for Creation of The Metal Venus I

View Table

Computation of the CGH was executed using a PC with two Xeon X5680 ( $3.33 GHz$ ) CPUs and $144 Gbytes$ of memory. The total number of CPU cores is 12. The total computation time was approximately $3.6 h$ . The itemized computation times are shown in Fig. 12. The longest part of this time was consumed by computation of the field of the Venus statue. This took 2.6 times longer to compute than the original Venus, which was rendered with only diffusive surfaces. The increased computation time is most likely caused by the separate calculations for specular and diffuse light in the proposed technique.

Fig. 12. Itemized computation time of The Metal Venus I and The Venus. The computation time for The Venus is not the same as that reported in [1] because recent developments in computers and algorithms accelerated the computation.

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The optical reconstruction of The Metal Venus I is shown in Figs. 13(a) and 14. The Venus is shown in Fig. 13(b) for comparison with The Metal Venus I. It is shown that a metallic surface has been reconstructed for The Metal Venus, and the surface brightness clearly varies with respect to the viewpoint.

Fig. 13. Photographs of optical reconstructions of (a) The Metal Venus I (Media1) and (b) The Venus [1] using transmitted illumination of an He–Ne laser. The photographs are taken from different viewpoints.

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Fig. 14. Photographs of optical reconstructions of The Metal Venus I (Media2) using reflected illumination of an ordinary red LED.

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4. Discussion

In Eq. (20), the role of weights $K_{d}$ and $K_{s}$ are different from those in CG. These coefficients represent the brightness ratio in CG, but $H_{s} (x, y)$ and $H_{d} (x, y)$ represent the amplitudes of the electric field of light in this case. The intensity and brightness are proportional to the squares of these functions. Also, the surface brightness in polygon-based CGHs depends on the bandwidth of the phase distribution of the surface function used. For example, if $K_{d} = K_{s}$ , the diffuse surface is reconstructed to be considerably darker than the specular surface. The specular surface emits light in a smaller solid angle than that of the diffuse surface, and thus the brightness of the specular surface is higher than that of the diffuse surface. In this study, these weights are determined by trial and error using a simulated reconstruction technique.

The shininess constant $α$ in the proposed method is inherited from the Phong model. This constant plays a similar role in the Phong model and thus represents the sharpness or directionality of the reflection light. However, the distribution of the reflection light in CGHs is different from that in CG if rendered with the same shininess constant because the spectral shape of $I_{2} (u, v; R)$ is not identical with $I_{1} (u, v; R)$ of the Phong model, especially in large reflection angles, as shown in Fig. 7.

The weight of the ambient light is not explicitly indicated in Eq. (20) because the surface function for ambient light is not used in the polygon-based method. However, the ambient light ratio is involved in the diffuse surface function. In the Metal Venus I, the ambient light has 0.1 times the brightness of the diffuse polygon that is perpendicular to the incident illumination light.

5. Conclusion

A new method for rendering of specular surfaces is proposed for creation of realistic CGHs. The Phong reflection model is used to determine the spectral shape of the reflected light of the specular surfaces. The spectrum of the surface functions used in the polygon-based method is modified so that the envelope fits the spectral shape. In the proposed method, the reflection direction is changed by shifting the spectrum, rather than generating individual surface functions fitted to each polygon, to accelerate the computation. A high-definition CGH named The Metal Venus I was created using the proposed method. Optical reconstruction of The Metal Venus I shows a metallic appearance, and the brightness changes depending on the viewpoint, as intended.

This work was supported by the JSPS.KAKENHI (21500114) and the Kansai University research grants, Grant-in-Aid for Encouragement of Scientists, 2011–2012. The mesh data for the Venus object are provided courtesy of INRIA by the AIM@SHAPE Shape Repository.

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Hologram
Number of pixels ( $N_{x} \times N_{y}$ )	$4.3 \times 1 0^{9}$ ( $65, 536 \times 65, 536$ )
Pixel pitches	$1 μm \times 1 μm$
Dimensions of hologram	$65.5 mm \times 65.5 mm$
Hologram type	Binary amplitude
Reconstruction wavelength	$632.8 nm$
3D Object and Scene
Number of polygons of Venus (front face only)	718
Dimension of Venus ( $W \times H \times D$ )	$26.7 mm \times 57.8 mm \times 21.8 mm$
Dimension of wallpaper ( $W \times H$ )	$65.5 mm \times 65.5 mm$
Distance of Venus from hologram	$150 mm$
Distance of wallpaper from hologram	$300 mm$
Rendering Parameters
Illumination vector in global coordinates $(X, Y, Z)$	$(- 0.75, - 0.25, - 1.0)$
Sampling interval of diffuse surface function	$1 μm \times 1 μm$
Sampling interval of specular surface function	$0.25 μm \times 0.25 μm$
Shininess constant ( $α$ )	10
Weight of diffuse light ( $K_{d}$ )	1.0
Weight of specular light ( $K_{s}$ )	0.8

Rendering of specular surfaces in polygon-based computer-generated holograms

Abstract

1. Introduction

2. Rendering of Specular Surfaces

A. Surface Function

B. Spectrum of Diffuse and Specular Reflection

C. Spectral Envelope Based on Phong Reflection Model

D. Rendering Objects Composed of Specular Surfaces

E. Procedure for Rendering Surfaces

3. Creation of High-Definition CGH of Specular Object

4. Discussion

5. Conclusion

References

Supplementary Material (2)

Cited By

Figures (14)

Tables (1)

Equations (21)

Applied Optics