Digital Holography & 3D Imaging 2022, Cambridge (UK), Tu1A.4 (2022.08.02)

Rendering of Transparent Objects in Large-Scale
Polygon-Based Computer Holography
Hirohito Nishi* and Kyoji Matsushima
Department of Electrical and Electronic Engineering, Kansai University, 3-3-35 Yamate-cho, Suita, Osaka 564-8680, Japan*Corresponding author e-mail: h_nishi@kansai-u.ac.jp

Abstract: A novel method for rendering transparent objects in polygon-based computer holography.
The method uses rotational transform to simulate field propagation inside objects. A full-parallax
large-scale CGH is demonstrated to show validity of the method. © 2022 The Authors
1. Introduce
Full-parallax computer-generated holograms (FP-CGH) can reconstruct 3D object without any conflicts of depthsensation, such as vergence-accommodation conflict. For the last decade, the polygon-based method [1] has
successfully created large-scale FP-CGHs, composed of several tens billion or sometimes a hundred billion pixels[2]. In this technique, 3D objects formed with polygonal surfaces are reconstructed in very impressive deep 3Dscenes. Not only diffusive surfaces but also specular surfaces can be rendered with smooth shading [3]. We also
proposed a technique to render transparent objects [4], but the technique has two major constraints. First, the object
must not have self-occlusion, i.e., concave. Second, to keep the computation time practical, the back face of theobject must be a single planar polygon parallel to the hologram plane, as shown in Fig. 1 (a). We propose a new
method to remove the latter restriction in this paper.
2. Rendering Transparent Object in Polygon-Based Method
2.1. Principle
As shown in Fig. 1 (b), the background field is refracted at the boundary of the back face polygon, and thenpropagates inside the object. The field reaching the front face polygon is refracted again at the boundary. Thisrefraction effect can be simulated by changing the wavelength used in the numerical propagation. However, forexample, the field through polygon P0 is affected by all fields through the back face polygons. This means that thetotal number of numerical calculations needed for the whole field is given by the product of the number of the front
and back face polygons. This is very time-consuming when the object is composed of many polygons.
In the new method, we divide the object at a plane, called the object plane, where the cross section takes themaximum, as shown in (c). All incident background fields are once propagated to the object plane and superposed.
The field at P0 is obtained from the intermediate field in this case. Thus, the total number of propagations is givenby the addition of the number of the back and front face polygons, and much less than that in the case of (b).
2.2. Calculation of Wavefield Transmitted at Surface Boundary
The polygon-based method uses three coordinates systems: global coordinates ( xˆ, yˆ , zˆ) , parallel and tilted localcoordinates ( xˆn , yˆn , zˆn ) and ( xn , yn , zn ) of polygon Pn, where the local coordinates share the origin. The polygon,i.e., the boundary is placed in the ( xn , yn ,0) plane of the tilted local coordinates. Figure 2 shows the procedure forcalculating the refracted light through Pn. 0 and 1 are wavelengths outside and inside the object, respectively.
The incident wavefield of Pn is calculated from the background field fn ( xˆn , yˆn ; 0 ) as follows:
Front face
polygons
Back face
polygonBack face
polygons
Background
field

Front face
polygons

Back face
polygons

Front face
polygons

P0

P0ẑ
λ0

λ1

λ0

λ1ẑ
λ0

λ1

Objectplane(a)(b)(c)
Fig. 1. The principle of simulation of field propagation inside a transparent object. (a) The limited object shape in the previous work [4].
Calculation without (b) and with (c) the intermediate object plane.

Digital Holography & 3D Imaging 2022, Cambridge (UK), Tu1A.4 (2022.08.02)

f n ( xn , yn ; 0 ) = R  f n ( xˆn , yˆ n ; 0 ) ,(1)where R{}
 represents the rotational transform of a wavefield [5], as shown in Fig. 2 (b). The wavefield in thepolygon face is masked as in (c) in order to remove the field outside the polygon Pn as follows:fn( xn , yn ; 0 ) = fn ( xn , yn ; 0 ) An ( xn , yn ) ,(2)where An ( xn , yn ) is a binary mask function that takes 0 and 1 outside and inside Pn, respectively. When the field
passes the boundary, the wavelength of the wave field switches from 0 to 1 . The wavefield is again rotated toobtain the field in the plane parallel to the hologram, as shown in (d).

f n ( xˆn , yˆ n ; 1 ) = R −1 f n ( xn , yn ; 1 ) ,(3)where R −1{}
 is inverse rotational transform. Finally, all fields are superposed in the object plane.
yˆ nynxˆnBackgroundfield
λ0

λ1Pnf n ( xˆn , yˆ n ; 0 )

yˆ n

yn

λ1
λ0 xn
f n ( xn , yn ; 0 )

λ1
λ0 x

n

λ0

λ1xˆn

f n ( xn , yn ; 1 )f n ( xˆn , yˆ n ; 1 )

(a)
(b)
(c)
(d)
Fig. 2. Procedure for calculating the wavefield passing through polygon Pn.

Superposed total field fobj ( xˆ, yˆ ; zˆobj , 1 ) is again masked by the cross section at the object plane.
 ( xˆ, yˆ; zˆobj , 1 ) = f obj ( xˆ, yˆ; zˆobj , 1 ) Aobj ( xˆ, yˆ ) ,fobj(4)where zˆobj is the position of the object plane and Aobj ( xˆ, yˆ ) is a mask function of the cross section, called the silhouette ( xˆ, yˆ ; zˆobj , 1 ) and usingmask. Each field passing the front face polygons can be obtained by propagating the field fobj
almost the same procedure as that shown in Fig. 2. Here, note that the field propagating outside the object must be
calculated separately.
3. Optical reconstruction
We created a large-scale FP-CGH composed of 4.2 G pixels, using the proposed method. Table 1 and Figure 3 show
the parameters and the optical reconstruction. The object model is a sphere with a diameter of 16 mm.
Table 1. Parameters of the created CGH.
Number of pixels (Nx  Ny)
65,536  65,536
Pixel pitches (Px  Py)
1 m  1 m
Number of polygons
2400
Refraction index of object
1.5
Reconstruction wavelength632.8 nm4. Conclusion

Fig. 3. Optical reconstruction of the created CGH.

We proposed an improved technique to render transparent objects without the restriction of the back-side shape. Wecalculated and created a FP-CGH. The optical reconstruction confirms correctness of the proposed method.
5. References[1] K. Matsushima, “Computer-generated holograms for three-dimensional surface objects with shade and texture,” Appl. Opt. 44, 4607-4614(2005).
[2] K. Matsushima, Introduction to Computer Holography, Sect. 1.3 (Springer, 2020).
[3] H. Nishi, and K. Matsushima, “Rendering of specular curved objects in polygon-based computer holography,” Appl. Opt. 56, F37-F44 (2017).
[4] H. Nishi, and K. Matsushima, “Rendering of transparent objects in polygon-based computer holography,” Information Photonics 2017, IP21PM-1-8 (2017).
[5] K. Matsushima, “Formulation of the rotational transformation of wave fields and their application to digital holography,” Appl. Opt. 47,
D110-D116(2008).