## Abstract

We propose a method to obtain a computer-generated hologram that renders reflectance distributions of individual mesh surfaces of three-dimensional objects. Unlike previous methods which find phase distribution inside each mesh, the proposed method performs convolution of angular spectrum of the mesh to obtain desired reflectance distribution. Manipulation in the angular spectrum domain enables its application to fully-analytic mesh based computer generated hologram, removing the necessity for resampling of the spatial frequency grid. It is also computationally inexpensive as the convolution can be performed efficiently using Fourier transform. In this paper, we present principle, error analysis, simulation, and experimental verification results of the proposed method.

© 2016 Optical Society of America

## 1. Introduction

Computer generated hologram (CGH) is a technique to synthesize a complex optical field that corresponds to a target three-dimensional (3D) object by numerical calculation. Depending on the primitives constituting the 3D object, the CGH techniques can be classified into point light source based methods [1,2], light ray based methods [3–5], and triangular mesh based methods [6–14]. Among these, the mesh-based method is efficient in generating CGH for large objects and compatible with current computer graphics as most computer graphics use polygon based representation. The mesh-based CGH techniques are again divided into two approaches according to the way the angular spectrum is treated. In one approach, the angular spectrum of each triangular mesh is first obtained in uniform local spatial frequency grid by taking discrete Fourier transform in the local plane containing the mesh, and the corresponding angular spectrum in the global coordinates is obtained by re-sampling the angular spectrum in the local coordinates [11–14]. This re-sampling of the angular spectrum may cause some artifacts in the reconstructed image. In another method called fully-analytic mesh based CGH, the angular spectrum in the global coordinates is obtained directly from the analytic formula of a reference triangle, removing the necessities of re-sampling [6–10].

In order to represent the 3D object surface realistically, it is important to render the angular reflectance distribution of individual mesh correctly. The reflectance distribution defines the ambient, diffusive, and specular reflection property of the mesh according to illumination on the mesh and the observing direction. As the CGH reconstructs 3D images not at a fixed viewing direction but within a viewing angle, it is necessary to encode the angular reflectance distribution to the single CGH.

Several techniques have been proposed to realize the reflectance distribution in the mesh based CGH. In one method, the surface phase function corresponding to the desired angular reflectance distribution of each mesh is obtained so that the reflectance distribution fits the spectral shape. The finite-difference time-domain method or microfacet modeling method have also been proposed to find the surface phase function [11–14]. However, these methods obtain the surface phase function of each mesh in discrete data in the local plane, and thus they cannot be applied to the fully-analytic method. In the fully-analytic method, a diffusiveness control has been proposed by dividing each mesh into smaller ones and assigning random phase constants to them [6]. The general reflectance distribution realization, however, has not been proposed in the fully-analytic method.

In this paper, we propose a method to realize the general reflectance distribution of each mesh in the fully-analytic mesh based CGH. We first explain our initial idea that realizes the general reflectance distribution by accumulating the angular spectrums of different carrier waves, instead of finding the surface phase functions of individual mesh. Then we propose a convolution-based method that reduces the computation time of the initial accumulation based method with negligible error. The proposed convolution based method does not find the surface phase function of each mesh, but taking a convolution of the angular spectrum of the mesh having uniform surface phase with a kernel that represents the desired reflectance distribution. The convolution is performed in the global coordinates and thus it does not require any resampling of the angular spectrum. The proposed method is computationally not heavy as the convolution can be performed effectively using Fourier transform. The convolution of the angular spectrum in the global coordinates brings little error in the reconstruction but it is hardly visible.

In the following sections, we explain the principle of the proposed method and present the error analysis. Finally we verify the feasibility of the proposed method by numerical and optical reconstructions.

## 2. Principle

#### 2.1 Review of fully-analytic mesh based CGH for single carrier wave

In the fully-analytic mesh based CGH [6–10], the angular spectrum in the global hologram plane *G*(*f _{x}*,

*f*) is related to the angular spectrum in the local mesh plane

_{y}*G*(

_{l}*f*,

_{xl}*f*) and the analytic formula of a reference mesh’s angular spectrum

_{yl}*G*(

_{r}*f*,

_{xr}*f*) by

_{yr}**f**= [

_{x,y}*f*]

_{x}f_{y}^{T},

**f**= [

_{xl,yl}*f*]

_{xl}f_{yl}^{T},

**f**= [

_{xl,yl,zl}*f*]

_{xl}f_{yl}f_{zl}^{T}=

**Rf**,

_{x,y,z}*f*= {(1/

_{z}*λ*)

^{2}-

*f*

_{x}^{2}-

*f*

_{y}^{2}}

^{0.5}, and

*f*= {(1/

_{zl}*λ*)

^{2}-

*f*

_{xl}^{2}-

*f*

_{yl}^{2}}

^{0.5}which are spatial frequencies in the global and local planes.

**R**and

**c**are 3 × 3 rotation matrix and 3 × 1 shift vector which satisfy

**r**=

_{xl,yl,zl}**Rr**+

_{x,y,z}**c**where

**r**and

_{x,y,z}**r**are 3 × 1 position vectors in the global and local coordinates.

_{xl,yl,zl}**A**and

**b**are 2 × 2 affine matrix and 2 × 1 shift vector between the triangle in local plane and the reference triangle. The shifted local spatial frequency

**f′**in Eq. (2) is given by

_{xl,yl}**u**and

_{xl}**u**are unit vectors of

_{yl}*x*and

_{l}*y*axis represented in the global coordinates and

_{l}**u**= [sin

_{c}*θ*sin

_{x}*θ*(1- sin

_{y}^{2}

*θ*– sin

_{x}^{2}

*θ*)

_{y}^{0.5}]

^{T}is the unit vector of the carrier wave with arbitrary angles

*θ*and

_{x}*θ*as shown in Fig. 1.

_{y}From Eqs. (1)-(3), one can find the relationship between the angular spectrum of an arbitrary triangular mesh in the global coordinate and that of the reference triangle for a specific carrier wave. Hereafter, let us denote *G*(**f _{x,y}**) and

*G*(

_{l}**f**) in Eqs. (1) and (2) as

_{xl,yl}*G*(

**f**;

_{x,y}*θ*) and

_{x},θ_{y}*G*(

_{l}**f**;

_{x,y}*θ*), respectively, in order to emphasis the carrier wave considered in the calculation.

_{x},θ_{y}#### 2.2 Initial idea of angular spectrum accumulation for desired reflectance distribution

In the fully-analytic mesh based method, the reference triangular mesh is assumed to have a uniform phase inside the mesh to enable analytic representation of the angular spectrum. As the complex field inside the reference triangle is uniform, each mesh whose angular spectrum is given by Eqs. (1)-(3) acts like a transparent window which is spatially deformed from the reference triangle and illuminated by the carrier wave. Therefore the diffraction angle of each mesh is very narrow around the carrier wave direction **u _{c}** in the reconstruction.

Using the very narrow diffraction angle of each triangle, in this paper we propose the realization of the reflectance distribution by accumulating the angular spectrums of different carrier waves. The reflectance distribution of the mesh is given by angular distribution of the diffraction intensity. Therefore the desired reflectance distribution can be achieved by

*I*(

*θ*) and

_{x},θ_{y}*ϕ*(

*θ*) are the diffraction intensity and the phase bias for

_{x},θ_{y}**u**direction, respectively and

_{c}*H*(

**f**) is the angular spectrum of the mesh with the desired reflectance distribution. Figure 2 illustrates the idea of Eq. (4). If the reflectance distribution of a mesh is given as shown in Fig. 2(a), the calculation of the corresponding angular spectrum can be achieved by accumulating the angular spectrums for individual carrier waves

_{x,y}**u**with proper weights {

_{c}*I*(

*θ*)}

_{x},θ_{y}^{0.5}as shown in Fig. 2(b). Note that the realization of the reflectance distribution using angular spectrum accumulation has not been proposed in previous literatures to the authors’ best knowledge. Because numerous individual angular spectrums should be calculated separately for just a single mesh, however, this initial idea is not computationally efficient. In the following section, we present an approximated but efficient modification of this initial idea.

#### 2.3 Main proposal of convolution based method for desired reflectance distribution

In order to reduce the computation time, we propose a convolution based implementation of our initial idea presented in previous section. From Eqs. (1) and (2), the angular spectrum of a triangle in global coordinate for a specific carrier wave can be written by

**b**between the given triangle and the reference triangle in the local plane can be made to be a zero vector by setting the local coordinate properly so that one of the vertices of the given triangle coincides with a vertex of the reference triangle in the local plane. For instance, if (0,0) is one vertex of the reference triangle, then by setting

**c**= -

**Rr**where

_{1}**r**is one of the global position vector of a vertex of the given triangle,

_{1}**b**can be made to be the zero vector.

With **b** = **0**, Eq. (5) can be written by

*B*(

**f**;

_{x,y}*θ*,

_{x}*θ*) which is the only term dependent on the carrier wave direction is given by

_{y}**f**= [

_{xl,yl,zl}*f*]

_{xl}f_{yl}f_{zl}^{T}=

**Rf**is considered and R

_{x,y,z}_{mn}is the corresponding element of the rotation matrix

**R**. The rotation matrix

**R**defines the rotation transformation between the global coordinate and the local coordinate and it can be easily shown that

**R**is given by

**R**= [

**u**]

_{xl}^{T}u_{yl}^{T}u_{zl}^{T}^{T}where

**u**,

_{xl}**u**, and

_{yl}**u**are unit vectors of

_{zl}*x*,

_{l}*y*and

_{l}*z*axis represented in the global coordinate. Therefore Eq. (7) reduces to

_{l}In the proposed method, we approximate *B*(**f _{x,y}**;

*θ*,

_{x}*θ*) to

_{y}*B*(

**f**;0,0) for a normal carrier wave, i.e.

_{x,y}*θ*= 0,

_{x}*θ*= 0, by [(1/λ)sin

_{y}*θ*(1/λ)sin

_{x}*θ*]

_{y}^{T}in global coordinate. Therefore, once we calculate

*B*(

**f**;0,0), then $\tilde{B}({f}_{x,y};{\theta}_{x},{\theta}_{y})$ for arbitrary carrier wave direction can be obtained simply by shifting it by corresponding amount without additional calculation. Even though $\tilde{B}({f}_{x,y};{\theta}_{x},{\theta}_{y})$in Eq. (9) is not the same as

_{x,y}*B*(

**f**;

_{x,y}*θ*,

_{x}*θ*) in Eq. (8), the error is negligible as will explained in the next section.

_{y}Using $\tilde{B}({f}_{x,y};{\theta}_{x},{\theta}_{y})$ instead of *B*(**f _{x,y}**;

*θ*,

_{x}*θ*) in Eq. (6), the angular spectrum with desired reflectance distribution given by Eq. (4) can be written as

_{y}**f**and

_{x,y}*D*(

**f**) is the convolution kernel containing reflectance distribution information which is given by

_{x,y}In summary, the proposed convolution based method is as followings. For each triangular mesh, the term in the first parenthesis of the last line of Eq. (10) and *B*(**f _{x,y}**;0,0) are calculated in the uniform global spatial frequency grid and stored separately. Then

*B*(

**f**;0,0) is convoluted with

_{x,y}*D*(

**f**). Finally, the result of the convolution is multiplied with the term in the first parenthesis of Eq. (10) to yield the angular spectrum

_{x,y}*H*(

**f**) which contains the desired reflectance distribution.

_{x,y}Note that the convolution is performed in the uniform global spatial frequency grid and hence the convolution based method does not require any re-sampling or interpolation of the angular spectrum. Also note that the convolution can be performed effectively by taking Fourier transforms of *B*(**f _{x,y}**;0,0) and

*D*(

**f**). Therefore the proposed convolution based method does not increase the computational load significantly. Finally, the convolution kernel

_{x,y}*D*(

**f**) can represent any reflectance model, which makes the proposed method versatile. Figure 3 shows one example of the

_{x,y}*B*(

**f**;0,0),

_{x,y}*D*(

**f**), and

_{x,y}*H*(

**f**) when Phong reflectance model is used.

_{x,y}#### 2.4 Error analysis of the convolution based method

The proposed convolution based method uses the $\tilde{B}({f}_{x,y};{\theta}_{x},{\theta}_{y})$ of Eq. (9) which is an approximation of *B*(**f _{x,y}**;

*θ*,

_{x}*θ*) of Eq. (8). In order to see their difference, we calculate

_{y}**f**, and ${\tilde{f}}_{xy;{\theta}_{x}{\theta}_{y}}$ in the argument of

_{xy;θxθy}*G*in Eqs. (8) and (9) for the same global uniform and rectilinear spatial frequency grid (

_{r}*f*,

_{x}*f*) at a few different orientations of the triangular mesh, i.e. [

_{y}**u**]

_{xl}^{T}u_{yl}^{T}^{T}. The carrier wave direction is set to be

*θ*= 1.524° and

_{x}*θ*= −1.524°. In left parts of Figs. 4(a)-4(f), [

_{y}**u**]

_{xl}^{T}u_{yl}^{T}^{T}

**f**and [

_{xy;θxθy}**u**]

_{xl}^{T}u_{yl}^{T}^{T${\tilde{f}}_{xy;{\theta}_{x}{\theta}_{y}}$}are represented as yellow and red grids, respectively, on a

*G*(

_{r}**A**

^{-T}

**f**) background. In the right parts of Figs. 4(a)-4(f), the difference $\Delta f=\Vert {\left[\begin{array}{cc}{u}_{xl}^{T}& {u}_{yl}^{T}\end{array}\right]}^{T}{f}_{xy;{\theta}_{x}{\theta}_{y}}-{\left[\begin{array}{cc}{u}_{xl}^{T}& {u}_{yl}^{T}\end{array}\right]}^{T}{\tilde{f}}_{xy;{\theta}_{x}{\theta}_{y}}\Vert $is shown. In the left parts of Figs. 4(a)-4(f), it can be seen that the red and yellow grids overlap mostly even though the mismatch increases as the inclination of the triangular mesh surface increases. The maximum difference Δ

_{xr,yr}*f*found at the extreme case where the triangular mesh plane is orthogonal to the hologram plane as shown in Fig. 4(f) is around sin

^{−1}λΔ

*f*= 0.15° which is not significant.

More importantly, as shown in the left parts of Figs. 4(a)-4(f), the mismatch happens where *G _{r}* has low value so that the mismatch has little impact on the result. Note that most power of

*G*, i.e. the angular spectrum of a reference triangle having uniform amplitude and phase is concentrated around the origin with its maximum at

_{r}*G*([0 0]

_{r}^{T}). From Eqs. (8) and (9), it can be seen that the difference between

**f**, and ${\tilde{f}}_{xy;{\theta}_{x}{\theta}_{y}}$ lies in their

_{xy;θxθy}*z*components. By applying binomial expansion to the square root terms, the difference in the

*z*components of

**f**, and ${\tilde{f}}_{xy;{\theta}_{x}{\theta}_{y}}$ can be written as [sin

_{xy;θxθy}*θ*(

_{x}*f*-sin

_{x}*θ*/λ) + sin

_{x}*θ*(

_{y}*f*-sin

_{y}*θ*/λ)]. Therefore the difference in the

_{y}*z*components is zero at (

*f*,

_{x}*f*) = (sin

_{y}*θ*/λ, sin

_{x}*θ*/λ) or ${f}_{xy;{\theta}_{x}{\theta}_{y}}={\tilde{f}}_{xy;{\theta}_{x}{\theta}_{y}}={\left[\begin{array}{ccc}0& 0& 0\end{array}\right]}^{T}$where the

_{y}*G*has its maximum value. As (

_{r}*f*,

_{x}*f*) deviates from (sin

_{y}*θ*/λ, sin

_{x}*θ*/λ), the difference increases but the value of

_{y}*G*also decreases, lowering the impact of the mismatch. Therefore, the approximation used in the proposed convolution based method can be justified.

_{r}## 3. Simulation

Before the experiment, we confirm the principle of the convolution based method by a simulation. A hologram is generated using the convolution based method with 3000 pixel × 3000 pixel resolution, 1 um pixel pitch and 532 nm wavelength. As shown in Fig. 5, a 3D duck object having 0.75 mm lateral size and 5.8 mm axial depth is used in the hologram generation. In the reconstruction, the hologram is numerically propagated to an observation plane at 20 mm distance from the reconstructed object. A virtual circular lens of 3 mm diameter is placed in the observation plane and the intensity pattern in the output plane which is apart from the virtual lens by 24 mm is calculated from the propagated optical field in the observation plane. The focal length of the virtual circular lens is adjusted to form the image of the reconstructed object in the output plane. The lateral position of the virtual circular lens in the observation plane is shifted by various amounts to see the change of the imaged object in the output plane with different observation directions.

Figure 6(a) is the reconstructed image when uniform phase is assigned to all meshes of the duck object. The reconstructed object is seen clearly but only within narrow observation angle as expected. Figure 6(b) shows the reconstructed image when the hologram is generated with diffuse component of the Phong reflection model by the convolution based method. The valid observation angle where the reconstructed image is seen is much enlarged in comparison with the Fig. 6(a) without notable artifacts. Note that the speckle patterns seen in Fig. 6(b) is natural consequence of the diffuse reflection not the artifact of the convolution based method. Figure 6(c) shows the results when the hologram is generated with ambient, diffuse, and specular components of the Phong reflection model by the convolution based method. In addition to the enlargement of the valid observation angle, the change of the specular reflection is clearly seen according to the observation direction, confirming the principle of the proposed method.

## 4. Verification by experiment

In the first experiment, we confirm that the proposed convolution based method can render the desired reflectance distribution. We also compare the convolution based method with the accumulation based method of section 2.2 in order to confirm that the convolution based method has negligible error as analyzed in section 2.4 while it requires much less computation time.

Two holograms of a cube object with a given reflectance distribution are generated by the accumulation based method and convolution based method, respectively. The reconstructed images of each hologram are observed from different directions. The generated hologram has 1080 pixel × 1080 pixel resolution and is loaded to a SLM having 8 um pixel pitch. Due to limited space-bandwidth product of the SLM used in the experiment, however, the viewing angle is narrow and thus it is not easy to see change of the intensity of each triangular mesh of the reconstructed cube according to the observing direction within the viewing angle.

In order to demonstrate observing direction dependency of each mesh of the reconstructed cube more clearly, it is assumed that two groups of the meshes of the cube object have strong and sharp specular reflection to different directions without ambient or diffuse reflections. For this, two carrier waves are assigned to two groups of the meshes of the cube object in the accumulation based method. In the case of the convolution based method, as shown in Fig. 7(a), a centered delta function is used as a convolution kernel *D*(**f _{x,y}**) for one group of the meshes and a shifted delta function is used as the convolution kernel

*D*(

**f**) for other mesh group. The angular separation between two specular reflections is 0.5716° in the experimental setup. In the hologram generation, the size of the cube object is set to be 7.6mm and the generated hologram is multiplied with a slanted plane wave before it is loaded to the SLM in order to separate the DC component in the reconstruction.

_{x,y}Figure 7(b) shows the generated angular spectrums and experimental results. The reconstructed cube object is captured by a camera at 1 m distance from the reconstructed image with slightly different capturing directions. As shown in Fig. 7(b), two groups of the meshes are observed only at the corresponding directions as expected. The observed image in the middle of two observing directions shows two groups of the meshes together as their diffraction angles overlap. Even though the used reflectance model, i.e. two distinct delta convolution kernels, not realistic one, the experimental result clearly shows that both of accumulation and convolution based methods can generate desired reflectance distribution. It is also observed from Fig. 7(b) that the difference in the reconstructed images of the accumulation based method and the convolution based method is not noticeable visibly. Together with the fact that the whole cube image seen in the middle of two directions shows no discrepancy between two groups of the meshes, we can confirm that the artifact caused by the approximation used in the convolution based method is negligible as analyzed in section 2.4.

In order to verify that the convolution based method is faster than the accumulation based method, we compare the computation time of two methods. In the comparison, holograms are generated for two objects, i.e. a single triangular mesh and a cube object of 24 meshes with 1 × 1, 15 × 15 and 31 × 31 carrier waves, respectively. As shown in Table 1, in the single carrier wave case having very narrow viewing angle, the calculation time of the convolution based method is longer than that of the accumulation based method because of the convolution process. As the number of the carrier wave increases, however, the computation time of the accumulation based method increases proportionally while that of the convolution based method is not affected, maintaining fast computation time. Note that in order to represent arbitrary reflectance distribution, the equivalent number of the carrier waves should be large to angularly sample the viewing angle range sufficiently. Therefore it is confirmed that the convolution based method is more computationally efficient than the accumulation based method as expected.

In the second experiment, we generate the holograms of a duck object using the convolution based method with different reflectance models and observe the reconstructed images from the same direction. The size of the duck object is 7.8mm and the reconstructed image was captured at 20 cm distance by a camera. As in the first experiment, the hologram generated by the proposed convolution based method is multiplied with a slanted plane wave to separate the DC term before it is loaded to the SLM.

Figure 8 shows angular spectrum, numerical reconstruction, and the optical reconstruction results when different reflectance models are used. In Fig. 8(a), a hologram *G*(*f _{x}*,

*f*;

_{y}*θ*= 0,

_{x}*θ*= 0) with a single normal carrier wave is used, reconstructing non-diffusive image. The angular spectrum of non-diffusive case is limited around the carrier wave direction, and thus the edge area having high spatial frequency is relatively emphasized in the optical reconstruction. On the contrary, in Fig. 8(b), hologram for the duck object with highly diffusive surface is generated by the convolution based method and reconstructed. The convolution kernel

_{y}*D*(

**f**) for all meshes object in this case is uniform amplitude and random phase function. The amplitude of the convolution kernel was given by inner product of the mesh normal and the illumination direction to represent the diffusive component in the Phong reflectance model. From Fig. 8(b), it can be observed that the diffusive surface with proper shading is optically reconstructed successfully. Finally, Fig. 8(c) shows the result when ambient and specular reflection components of the Phong model are also considered in the convolution kernel

_{x,y}*D*(

**f**) of the proposed method. Compared with the Fig. 8(b), bright highlight on the body of the reconstructed duck image is observed in Fig. 8(c), confirming that the desired reflectance model is rendered as expected.

_{x,y}## 5. Conclusion

In this paper, we propose a method to incorporate the desired reflectance distribution of 3D objects into the fully-analytic mesh based CGH. The proposed method convolves the angular spectrum of a triangular mesh having uniform amplitude and phase with a kernel having the desired reflectance distribution in the global spatial frequency grid. As the convolution is performed not in the local spatial frequency grid but in the global spatial frequency grid, the proposed method does not require additional re-sampling or interpolation, preserving the virtue of the fully-analytic mesh based method. The convolution operation can be performed effectively by using Fourier transform, making the proposed method be computationally efficient. Finally, any reflectance model can be represented using the convolution kernel, making the proposed method versatile. The approximation used in the proposed method is negligible as the error contribution caused by the approximation is small. The principle and the feasibility of the proposed method is verified by optical experiments showing angular intensity distribution of reconstructed meshes corresponding to the reflectance models and appearance of the reconstructed image with diffusive and specular components of Phong reflection model.

## Funding

The Cross-Ministry Giga KOREA Project, Ministry of Science, ICT and Future Planning (MSIP), Korea [GK16D0100, Development of Telecommunications Terminal with Digital Holographic Table-top Display]; ITRC support program, MSIP [IITP-2015-R0992-15-1008]; Basic Science Research Program, National Research Foundation of Korea (NRF) [2013-061913].

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