Information Photonics 2017 (IP'17), Yokohama (Kanagawa), IP-21PM-1-8 (2017.3.21).

Rendering of Transparent Objects in 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, Japannishi@laser.ee.kansai-u.ac.jp.

Abstract: A novel technique is proposed for rendering transparent object in the polygon-based
method. Refraction of light is simulated by the field propagation in the wavelength inside the object.
Rotational transform of wave fields plays an important role to obtain the wave field at the boundary
surface of the object. Simulated reconstruction of a computer-generated hologram calculated by thepropose technique is demonstrated for verifying the validity.
1. Introduction
Computer holography is a 3D imaging technique that does not cause any vergence-accommodation conflict, unlike
conventional 3D displays. We have proposed the polygon-based method and created large-scale computer-generated
holograms (CGH) using the technique, whose number of pixels is over 4 billion [1-3]. In this method, the object isconsidered to be composed of many surface light sources, unlike conventional point-based techniques. This methodfeatures short computation time without any hardware assist. Recently, we proposed several rendering techniques of thepolygon-based CGH, such as rendering of flat and smooth specular surfaces [2,3]. In this paper, we propose a noveltechnique to render transparent objects.
yˆ nTilted local
coordinates

yn

ŷ

xn
zn

Parallel local
coordinates

P0xˆnzˆ  zˆobjzˆnpolygon( xˆ , yˆ ,0)P1

Hologramx̂
Hologramẑ coordinates

Fig. 1. Coordinate systems used in the polygonbased method.

Wavelength
λ0

P3

Wavelength
λ1ẑ

P4
P2

Wavelength
λ0

Fig. 2. Restrictions of the object shape in theproposed technique.

2. Rendering of refraction light in Polygon-Based CGH
Figure 1 shows the three types of the coordinates systems used in the polygon-based method; those are hologram
coordinates ( xˆ, yˆ , zˆ) , parallel local coordinates ( xˆn , yˆn , zˆn ) and tilted local coordinates ( xn , yn , zn ) of polygon n. The
hologram is in the ( xˆ, yˆ ,0) plane of the hologram coordinates. Two local coordinates are defined for each polygon andshare the origin, but these are tilted so that the polygon face is in the ( xn , yn ,0) plane.
The 3D object rendered by the proposed technique has several restrictions, as shown in Fig.2. The back face of theobject must be a single planar polygon placed at zˆ  zˆobj , which is parallel to the hologram. The object must not haveconcave. The cross-section must take the maximum at back face P0, and furthermore the closer section to the hologrammust have smaller cross-section in zˆ  zˆobj .
The wave field f ( xˆ, yˆ ; zˆobj ) entering into the back face P0 is divided into inside and outside wave fields as follows.
finside ( xˆ, yˆ ; zˆobj )  f ( xˆ, yˆ ; zˆobj )  A0 ( xˆ, yˆ ),f outside ( xˆ, yˆ ; zˆobj )  f ( xˆ, yˆ ; zˆobj )  1  A0 ( xˆ, yˆ ) ,(1)where A0 ( xˆ, yˆ ) is a binary mask. The value is 1 inside P0 or 0 outside P0. The outside wave field is propagated to thehologram plane in wavelength 0 of the outside media, which is commonly the same as the vacuum wavelength. The
inside wave field is propagated to the position of each polygon in wavelength 1  0 / nobj , where nobj is the refractive
index inside the object.
Figure 3 shows the procedure for calculation of the wave field propagating through the inside media onto polygon
Pn. The wave field fn ( xˆn , yˆn ; 1 ) in the ( xˆn , yˆn ,0) plane of Pn, as in Fig.3(a), is converted into the field in the polygon
face by using rotational transform [4].

Information Photonics 2017 (IP'17), Yokohama (Kanagawa), IP-21PM-1-8 (2017.3.21).

λ1yˆ nλ0

f n ( xn , yn ; 1 )  R  f n ( xˆn , yˆ n ; 1 ) ,ynxˆnwhere R{}
 stands for the rotational transform and fn ( xn , yn ; 1 ) is thewave field in the ( xn , yn ,0) plane of tilted local coordinates, as in (b). Here, atechnique called spectrum remapping [1-3] is used to reduce computationtime. The wave field in the polygon face is masked as in (c) in order torestrict the field inside the polygon as follows,xnPnPn

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

(a)

yn

An ( xn , yn )xn
Pn
f n ( xn , yn ; 0 )

(c)

(2)

yˆ nxˆnPn
f n ( xˆn , yˆ n ; 0 )(d)Fig. 3. Procedure for calculating thewave field passing through polygon
P n.

(3)
fn( xn , yn ; 1 )  fn ( xn , yn ; 1 )  An ( xn , yn ) ,where An ( xn , yn ) is the binary mask of Pn. When the field crosses theboundary after masking, the wavelength of the wave field changes from 1to 0 . Then, the wave field is again transformed rotationally to obtain thefield in parallel local coordinates as in (d), i.e.,

f n ( xˆn , yˆn ; 0 )  R 1 f n ( xn , yn ; 0 ) ,(4)where R 1{}
 stands for inverse rotational transform. Here not that theoutside wavelength is used in this transform. This field is propagated to thehologram plane. Finally, all fields are superposed in the hologram plane asfollows,f holo ( xˆ, yˆ , 0)   f n ( xˆ, yˆ , 0) ,(5)nwhere fn ( xˆ, yˆ ,0) is the wave field in the hologram plane, which passes the polygon Pn.
3.

Simulation

The CGH of the 3D scene shown in Fig.4 was calculated and reconstructed by simulation using virtual optics [5]. The
parameters used are summarized in Table 1. The Teapot model is cut in half and arranged in contact with a checkeredwall. The simulated reconstruction of this CGH is shown Fig. 5.
25
Table 1. Parameters used for simulation.
Top viewŷ Wall
Number of pixels (Nx × Ny) 32,768 × 32,768
32
Pixel pitches1m × 1mx̂Refraction index of object
1.33
Reconstruction wavelength632.8 nmẑ75
4. Conclusion
We proposed a new technique to render transparent objects in thepolygon-based method. We calculated a CGH by using thetechnique. Simulated reconstruction of the CGH confirms thevalidity of the proposed technique.
This work was supported by JSPS KAKENHI (15K00512)
and MEXT Strategic Research Foundation at PrivateUniversities (2013-2017).

Units: mm

Hologram

Front viewFig. 4. The 3D scene of the CGH.

References[1] K. Matsushima, and S. Nakahara, “Extremely high-definition full-parallax
computer-generated hologram created by the polygon-based method,” Appl.Opt.
48, H54-H63 (2009).
[2] H. Nishi, K. Matsushima, and S. Nakahara, “Rendering of specular surfaces
in polygon-based computer-generated holograms,” Appl. Opt. 50, H245-H252(2011).
[3] H. Nishi, and K. Matsushima, “Rendering of specular curved objects inpolygon-based computer holography,” Appl.Opt., submitted.

Fig. 5. Simulated reconstruction of the CGH
rendered by the proposed technique.

[4] K. Matsushima, “Formulation of the rotational transformation of wave fields and their application to digital holography,” Appl.Opt. 47, D110D116 (2008).
[5] Y. Tsuchiyama, and K. Matsushima, “Full-color large-scaled computer-generated holograms using RGB color filters,” Opt. Express, to bepublished.