Three Dimensional System and Application 2013 (3DSA2013), Osaka, P4-1(2013.6.28)

Rigorous Light-Shielding for Hidden-Surface
Removal in High-Definition Computer Holography
1

Sachio Masuda, 1Kyoji Matsushima, 2Sumio Nakahara

1

Department of Electrical and Electronic Engineering, 2Department of Mechanical Engineering
Kansai University
Yamate-cho 3-3-35, Suita, Osaka 564-8680, Japan

Abstract—A technique is proposed for applying rigorous lightshielding to high-definition computer holography. This techniquereduces occlusion errors caused in conventional silhouette maskingthat is currently used for hidden-surface removal in high-definition
computer holography.

in the rotational transform. Another one is to optimize thesampling area and interval of the wave-field in Fourier space.
This is also for reducing the number of samplings of wave-fieldin rotational transform.

Keywords-computer holography; computer-generated hologram;
hidden surface removel

II. RIGOROUS LIGHT-SHIELDING
If the field is near paraxial, i.e. if the light propagatesalmost along the optical axis, the silhouette masking is a goodtrade-off between the computational cost and quality of thereconstructed image. However, if viewing zone of thehologram is large enough that the hologram reconstructs theauto-stereoscopic image, the field is no longer paraxial, andtherefore occlusion errors are produced in the reconstructed
image. This occlusion error can be removed by masking thebackground field with polygon itself in the plain in which thepolygon is laid. To realize this processing, the background field,
initially given in the plane parallel to the hologram, must be
rotationally transformed once.

I.
INTRODUCTION
We have presented some high-definition computergenerated holograms (CGH) that are computed by polygonbased method [1, 2] and fabricated by laser lithography.
Optical reconstruction of these high-definition CGHs is being
comparable to that in traditional optical holography. In thishigh-definition computer holography, the technique of thesilhouette masking plays an important role to shield lightbehind the object and prevent the reconstructed object frombeing see-through [3, 4]. This technique provides hidden
surface removal like conventional computer graphics.

A. Rotational transform of wave fields

A drawback of the silhouette method is that the light isshielded by the silhouette-shaped mask in a plane parallel tothe hologram. Therefore, if the CGH has very high physicalresolution and thus large diffraction angles, off-axis lights
cannot be shielded by the silhouette masks and causes
occlusion errors that give unexpected black shadow orsometimes partial phantom images [4, 5]. To remove theocclusion errors in the silhouette masking, we have alreadyproposed more rigorous method for light shielding [6]. In thistechnique, the wave-field is calculated in the same plane as thepolygon by using the numerical technique of rotationaltransform of wave-fields [7, 8] and masked by using thepolygon-shaped masks.

Suppose that the background wave-field hˆ( xˆ, yˆ ) is initiallygiven in a plane ( xˆ, yˆ , zˆ  0) parallel to the hologram. The
coordinates ( xˆ, yˆ , zˆ) is referred to as parallel coordinates in thisreport. The basic rotational transform of the wave-field iswritten as

However, only the principle of this rigorous light-shielding
technique has been verified in the literature. The technique has
never been applied to the actual high-definition CGHs, because
the rotational transform of the wave-field is very timeconsuming in high-definition computer holography in that
number of sampling points reaches to more than billions orsometimes more than ten billions [9, 10].

where the resultant field h(x, y) is given in the plane (x, y, z = 0)of other local coordinates that shares the origin with the parallel
coordinates but the plane (x, y, z = 0) is not parallel to thehologram. This is referred to as the tilted coordinates. Here,F{·} denote Fourier transform as follow:

In this paper, we propose a new technique to apply the
rigorous light-shielding method to high-definition computer
holography. One of the techniques we propose in this paper isto clip the wave-field in order to reduce the sampling number







h( x, y )  Rot hˆ( xˆ, yˆ )

 F 1 R F {hˆ( xˆ , yˆ )}ˆ (uˆ, vˆ)  F{hˆ( xˆ, yˆ )} H







where (uˆ, vˆ) and (u, v) are spatial frequency in the parallel( xˆ, yˆ , zˆ  0) and the tilted plane ( x, y, z  0) , respectively.

Three Dimensional System and Application 2013 (3DSA2013), Osaka, P4-1(2013.6.28)
Suppose that the transform matrix for coordinates rotation isdefined as xˆ  x a1 a 2 a 3  ˆ  T   a a a , yTy,56 4  a zˆ zaa89
 7  

Shieldedbackground field

Inverserotationaltransform

Background field

Rotationaltransform

Shielded fieldMask

Polygonthe coordinates rotation in Fourier space R{·} is given as(a)R{Hˆ  uˆ, vˆ } Hˆ (a1u  a2 v  a3 w(u , v), a4u  a5v  a6 w(u , v ))  H (u, v)

w  u, v    2  u 2  v2 . 

Finally, the wave-field is given in the tilted plane by inverse
Fourier transform as follow:h( x, y)  F 1{H (u, v)}. 

B. Masking fields in the tilted planeFig. 1 shows the procedure for shielding the backgroundfield that should be shielded by a polygon. The mask function
for the polygon must be multiplied by the background fieldgiven in the tilted plane of the polygon. Therefore, as in (a), thebackground field is obtained in the tilted plane from hˆ( xˆ, yˆ ) byrotational transform and then, as in (b), masked as follow:h( x, y)  Rot{hˆ( xˆ, yˆ )} m( x, y) 

where m( x, y) is the binary mask function defined in the tilted
plane, whose shape is the same as the polygon, i.e., the maskfunction is given as follow:1 inside polygonm( x, y )  otherwise
0(c)Figure 1. The procedure for rigorous light-sheilding.

where w(u, v) is spatial frequency in z direction perpendicular
to the tilted plane and given by(b)III. REDUCTION OF COMPUTATIONAL COST
The procedure for the rigorous light-shielding is very time
consuming, because it includes twice rotational transform ofthe background field whose number of samplings is more thanbillions in high-definition CGHs. To reduce the computationalcost, we mainly adopt three techniques described as follow.
A. Clipping the background field
Polygons are commonly much smaller than the area of thebackground field, i.e. only a small part of the background field
is shielded by a polygon. Therefore, the rotational transformshould be performed in a small area instead of the wholebackground field, i.e. the background field should be clipped in
the small area and then the field h( x, y) in the tilted plane
should be calculated by the rotational transform. Here, therequired wave-field for the accurate light-shielding is a portion
which enters into the polygon and is affected by light-shielding.
We determine the required area in the whole background field
by using the maximum diffraction area of the polygon [9], as inFig. 2. Here, the maximum diffraction area is determined byvertex positions of the polygon and the maximum diffraction
angle given by    2x  max  sin 1 

where x and  are the sampling interval of the backgroundfield and the wavelength, respectively.
xPolygon

The masked background field plane is calculated in the parallelby inverse rotational transform as in Fig. 1 (c):hˆ( xˆ, yˆ )  Rot 1{h( x, y)} Rot 1 Rot{hˆ( xˆ, yˆ )}  m( x, y ) .



zClipped area
Background field
Maximum diffraction angle
Figure 2. Clipping area of the background field.

Three Dimensional System and Application 2013 (3DSA2013), Osaka, P4-1(2013.6.28)

Tilted plane

C. Optimizing the sampling window in Fourier space
Figure 5 shows transition of sampling area in Fourier spacefor rotational transform. The sampling area initially forms a
rectangle because of using FFT as in (a). However, as shown in(b), the sampling area of H (u, v) is distorted after applying
coordinates rotation R{} and positioned far from the origin.
Since FFT do not work well in such cases, new shifted
coordinates should be introduced into Fourier space as follow:

Clipping area

Sampling window
Diffractedfield(u, v)  (u  u0 , v  v0 ), 
Background field

Figure 3. The sampling window required for avoiding information loss inthe the tilted plane.

B. Sampling reduction based on Babinet’s principle
As shown in Fig. 3, the sampling window in the tilted planemust include the whole field diffracted by the clipped area ofthe background field. This is necessary for avoiding
information loss of the clipped field. However, the sampling
window must be very large or sometimes infinity whenpolygons have a large tilt angle . To solve this problem, we
use Babinet's principle and partial field propagation technique[11].
According to well-known Babinet’s principle, lightshielding by the mask is equivalent to that by the aperture
given by inverting the mask, as shown in Fig. 4. Here, theaperture function is given in the tilted plane as follow:
A( x, y)  1  m( x, y) where (u0 , v0 )  (a7 / , a8 /  ) . The inverse FFT is performedin the shifted coordinates and the field in the tilted plane isgiven by [7]h( x, y)  F 1{H (u, v)}exp[i 2 (u0 x  v0 y)] After multiplying the mask function we re-defined the masked
field as h( x, y)  F 1{H (u, v)} m( x, y) . As a result, theFourier transform for inverse rotation is written as
F{h( x, y)m( x, y)}  F{h( x, y) exp[i 2 (u0 x  v0 y )]} H (u  u0 , v  v0 ),

(a)

(d)
Shift

Coordinate rotation

(b)

Mask

Partialwave-field(e)InverseCoordinate rotation

ShiftAperture(a)

When we use the aperture function A( x, y) instead of themask function m( x, y) for masking the field, the samplingarea required for light-shielding can be reduced only to thesmall portion of the wave-field, i.e. the area opened in theaperture whose shape is the same as the polygon. By using thistechnique, we can avoid the problem of infinity sampling
window as shown in Fig. 3, because only the sampling windowthat cover the opened area of the aperture is necessary for lightshielding.

Wholewave-field



(b)

Figure 4. Light-shielding by the mask (a) and aperture (b). Thesecan be equivalent to each other by using Babinet’s principle and apartial field propgation [11]

(c)

(f)
Figure 5. Transition of sampling area in Fourier apace.

Three Dimensional System and Application 2013 (3DSA2013), Osaka, P4-1(2013.6.28)
V. CONCLUSION
We proposed a new technique to apply the rigorous lightshielding technique to high-definition computer holography. Itis verified that the leakage light and occlusion error are reducedin the proposed method.

Sampling window

Rotationaltransformation(a)ACKNOWLEDGEMENT
This work was supported by JSPS. KAKENHI (24500133).

Initial sampling area (b)REFERENCES
Figure 6. Optimization of sampling window in Fourier space.

where H (u, v)  F{h( x, y)} . Thus, the sampling area is again
shifted from the origin, as shown in Fig. 5(e). Finally, theinverse coordinates rotation R{} restores the sampling area toa rectangular shape suited for FFT.
Through the entire processes of the rigorous light-shielding,the shape of the sampling area in the Fourier space is strongly
deformed especially in cases of large rotation angle, as shownin Fig. 6. Therefore, it is very important to optimize theposition and size of sampling window. In this study, as in Fig.
6(a), we set the representative points in the initial samplingarea of Fourier space, whose shape and position is given by
FFT, and traced the position of the sampling points throughrotational transform. We determined the optimized sampling
window from the minimum bounding box including the 5050representative points as in (b).
IV.

GENERATION OF HIGH-DEFINITION CGH FOR
COMPARISON WITH SILHOUETTES METHOD

To proof the validity of the proposed method, the field of
the 3D scene shown in Fig. 7 was calculated by using theproposed method. This 3D scene includes a torus object.
Number of the samplings and the sampling interval of thecalculated field are 65,53665,536 and 1 m, respectively. The
torus is composed of 140 polygons. Fig. 8 shows the numericalreconstruction of the field by virtual optics using numericallens. We can verify that the leakage light is reduced in theproposed method.

Polygon model

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[1]yxo40 [mm]Wall paperz200 [mm]Hologram
Figure 7. The 3D scene used for the examination of light-shielding.

(a) Conventional silhouette method(b) This work

Figure 8. Comparison of two light-shielding techniques.