Publication : SPIE Proc. Practical Holography XVIII and Holographic Materials X, SPIE#5290, San Jose, California, 90–97(2004).

1A Wave Optical Algorithm for Hidden-Surface Removal
in Digitally Synthetic Full-Parallax Holograms
for Three-Dimensional Objects
K. Matsushima and A. Kondoh
Department of Electrical Engineering, Kansai University
Yamate-cho 3-3-35, Suita, Osaka 564-8680, JapanABSTRACT
A novel algorithm is presented for hidden-surface removal in digitally synthetic holograms. The algorithm isable to work with full-parallax holograms and remove obstructed ﬁelds in the object wave emitted from threedimensional (3-D) surface objects. This algorithm is initially discussed as a rigorous procedure to obtain ﬁelds
behind a tilted planar surface by using the rotational transformation of wave ﬁelds, and ﬁnally results in thesilhouette approximation for reduction of computation time. Reconstructions of holograms created by using thealgorithm are demonstrated.
Keywords: computer-generated hologram, digitally synthetic hologram, hidden-surface removal, surface model,
silhouette approximation

1. INTRODUCTION
Algorithms for hidden-surface removal found in literatures on digitally synthetic holograms are either the geometrical method1, 2 or the wave optical method based on layered model.3 The former is suitable for the ray
tracing in the point-source model4, 5 as shown in Fig. 1 (a) but usually much time-consuming algorithm, andtherefore it can handle only horizontal-parallax-only holograms. The latter originated in A. W. Lohmann, of
which scheme is shown in Fig. 1 (b), has a deﬁnite advantage of relatively short computation time owing toutilization of FFT for numerical propagation and is capable of removing ﬁelds behind obstacles in full-parallax
holograms. The major drawback of the conventional wave-optical method is that the objects must be sliced
in some planes parallel to the hologram, and therefore it is diﬃcult to apply standard rendering algorithms
necessary for realistic reconstruction of 3-D objects, such as shading and texture mapping.
This report is intended to propose a novel wave-optical algorithm for hidden surface removal. The algorithm
is based on a technique to calculate complex amplitude upon tilted plane by the rotational transformation of wave
ﬁelds6 and makes it possible to reconstruct a 3-D image with natural overlapping and shading in full-parallax
holograms.

2. THE WAVE FIELD BEHIND A TILTED SURFACE
2.1. Lohmann’s layered hologram
The principle of Lohmann’s layered hologram and the coordinate system used in this work is shown in Fig. 2. In
the coordinates, a hologram is placed in the plane (x̂, ŷ, 0) and ẑ-axis corresponds to the direction of the object
wave. It is assumed that incidnet light on a plane V, which is parallel to the hologram at ẑ = ẑ0 , is given ash(x̂, ŷ). Furthermore, a binary function m̂(x̂, ŷ) shows whether a location (x̂, ŷ) on the plane V is included insidethe object or not, and is deﬁned as a mask function:0(inside object)(1)m̂(x̂, ŷ) =1(otherwise)
Further author information: (Send correspondence to Matsushima)
Matsushima: E-mail: matsu@kansai-u.ac.jp, Telephone/Fax: +81-6-6368-0933

Publication : SPIE Proc. Practical Holography XVIII and Holographic Materials X, SPIE#5290, San Jose, California, 90–97(2004).

2P o in t s o u rc e

O b je c t
O b s ta c le

O b je c t

H o lo g ra m

H o lo g ra m

(a )

(b )

Figure 1. Conventional algorithms for hidden-surface removal in digital holograms; (a) geometrical methods; (b) thelayered method.

yVO b je c tm ( x , y )h ( x , y )H o lo g ra mx

z 0OzFigure 2. The principle of the layered method and the coordinate system used in this work.

This function represents a section of the object. As the incident ﬁelds is masked with the section on the plane
V, the ﬁeld behind V is obtained asĥ (x̂, ŷ) = m̂(x̂, ŷ)ĥ(x̂, ŷ).
(2)Moreover, when it is assumed that the section of object emits the light o(x̂, ŷ), the amount ﬁeld behind the plane
V is given by ĥ (x̂, ŷ) + o(x̂, ŷ). One can calculates incident ﬁelds upon next plane including the next section ofobject by calculating propagation of the total ﬁeld ĥ (x̂, ŷ) + o(x̂, ŷ). Thus, when this procedure is started fromthe farthest section from hologram and iterated plane by palne to the hologram, we can calculate the object
wave without the ﬁeld propagating inside the object or hiding behind obstacles.

2.2. Obstruction of field propagation by tilted planes
It is necessary to handle obstruction of a ﬁeld by a tilted plane for removing hidden surface of surface-modeled
objects composed of small planar surfaces. To consider the procedure for calculating the obstructed ﬁeld, let us
assume that a tilted planar fragment P, which is a part of object surface, hides an incident light ĥ(x̂, ŷ) behind
the P. We can calculate the obstructed light ĥ (x̂, ŷ) by the following procedure, as shown in Fig. 3.

Publication : SPIE Proc. Practical Holography XVIII and Holographic Materials X, SPIE#5290, San Jose, California, 90–97(2004).

h (x , y )h  ( x , y )h ¢( x , y )m ( x , y )3h  ¢ ( x , y )

P
(a )

(b )

(c )

Figure 3. The basic procedure for hidden surface removal by employing a wave-optical method. The incident field isobtained upon the object surface by the coordinate rotation in Fourier domain (a) and masked behind the fragment (b).
Finally, the fields obstructed are calculated on a plane parallel to the hologram (c).

Step 1: Calucation of the incidnet field in a tilted coordinatesh(x, y) as well as ĥ(x̂, ŷ) is an incident ﬁeld but it is given in a tilted coordinates (x, y, z), in which the fragment
P is laid upon the (x, y, 0) plane. A method reported in Ref.6 can be used to obtain the ﬁelds h(x, y) from thesource ﬁelds ĥ(x̂, ŷ) given on the plane parallel to the hologram as summarized bellow.
Spectrum of the ﬁeld in the tilted plane is obtained asH(u, v) = R{Ĥ(û, v̂)},(3)where Ĥ(û, v̂) is Fourier spectrum in the source parallel plane and given asĤ(û, v̂) = F {ĥ(x̂, ŷ)},and R{·} stnads for change of variables for coordinates rotation as follows:⎛⎛ ⎞⎛ ⎞⎞â1 â2 â3uû⎝ v̂ ⎠ = T ⎝ v ⎠ , T = ⎝ â4 â5 â6 ⎠ .
wŵâ7 â8 â9It is noted that the Fourier frequencies are not independent for each other and the relations are given asŵ(û, v̂) =λ−2 − û2 − v̂ 2 ,λ−2 − u2 − v 2 .
w(u, v) =(4)

(5)

(6)
(7)

Complex amplitudes h(x, y) on the tilted plane can be obtained by the following inverse Fourier transformationby using the paraxial approximation.6h(x, y) = F −1 {H(u, v)}.
(8)
As a result, the rotational transformation of the incident ﬁelds is rewritten ash(x, y) = F −1 RF{ĥ(x̂, ŷ)}.

(9)Consequently, twice FFT and once change of variable are necessary for a rotational transformation of the ﬁeld.
It is noted that the change of variable for rotational transformation requires an interpolation of the ﬁeld because
the transformation causes distortion of the equidistant sampling grid.

Publication : SPIE Proc. Practical Holography XVIII and Holographic Materials X, SPIE#5290, San Jose, California, 90–97(2004).

4H o lo g ra mh n ( x , y )S n + 1 ( x , y )h n¢ ( x  , y  )yo n ( x , y )VnS n ( x , y )Vxn + 1z n + 1z nz

Figure 4. Synthesis of the object wave with a wave-optical masking in the silhouette approximation.

Step 2: Masking the incident field on the tilted plane
As shown in Fig. 3(b), the ﬁelds behind the fragment P is given in the tilted plane as follows:h (x, y) = h(x, y)m(x, y),(10)where the mask function m(x, y) is again zero in a location included in the fragment P and unity in otherwise.
Step 3: Inverse rotational transformation of the masked fieldObject waves that propagate almost along ẑ-axis and reach the hologram are signiﬁcant for synthesis of object
wave. Thus, the ﬁeld h (x, y) must be rotated inversely so as to obtain a ﬁeld upon the plane parallel to thehologram. As a result, the ﬁelds obstructed by the fragment P is given byĥ (x̂, ŷ) ==F −1 R−1 F {h (x, y)}F −1 R−1 F F −1 RF{ĥ(x̂, ŷ)}m(x, y) ,(11)where R−1 {·} stands for change of variables for inverse rotation.

2.3. The silhouette approximation
Computation time is too long to apply all steps described above to each fragment of the object surface, because
four times FFT and twice interpolation is required for handling a fragment. It appears as if it is possible tointegrate Step 1 and 3, which is rotation and inverse rotation of a ﬁeld, and reduce computation time. However,
the integration is not easy because of non-linearity of eqs. (6) and (7). Therefore, an approximation should beintroduced to the integration.
Fourier frequency ŵ(û, v̂) is expanded into power series of λ−1 .
ŵ(û, v̂)  λ−1 [1 − (λû)2 /2 − (λv̂)2 /2] + · · ·(12)If the sampling distance of hologram is much lager than the wavelength, i.e., if the spectral bandwidth of ĥ(x̂, ŷ)
is far narrower than λ−1 , the ﬁeld is considered to propagate in almost ẑ direction. Therefore, it can be assumed

5Publication : SPIE Proc. Practical Holography XVIII and Holographic Materials X, SPIE#5290, San Jose, California, 90–97(2004).

Table 1. Parameters used to synthesize object waves and fabricate holograms.

Number of pixels

8192 × 4096

Pixel size

2 µm × 4 µm

Reconstruction wavelength632.8 nmthat û, v̂  λ−1 and eq. (12) approximate to ŵ(û, v̂) ∼ λ−1 . In that cases, the ﬁeld behind the fragment is given
on a plane parallel to the hologram as follows (see Appendix A):ĥ (x̂, ŷ) = m(â1 x̂ + â4 ŷ, â2 x̂ + â5 ŷ)ĥ(x̂, ŷ).

(13)This equation means that the incident ﬁeld is simply masked by m(â1 x̂ + â4 ŷ, â2 x̂ + â5 ŷ), which corresponds
to a silhouette of the fragment given on a plane parallel to the hologram. We refer to this as the silhouetteapproximation.

3. RECURRENCE FORMULA FOR SYSNTHESIZING THE OBJECT WAVEAs shown in Fig. 4, assume that there are planar fragments P1 , · · · , PN in object space where the index is assignedin the order in which the farther fragment has smaller index, and also assume that there are planes V1 , · · · , VN
parallel to the hologram at ẑ1 , · · · , ẑN , which intersect corresponding fragments with same index. When maskfunctions Sn (x̂, ŷ) are deﬁned as silhouettes of fragments given upon Vn , and the subtotal ﬁeld emited fromfragments located further than Vn , i.e. incident ﬁeld to Vn , is given as hn (x̂, ŷ) upon the plane Vn , the ﬁeld
right behind the silhouette mask is obtained byhn (x̂, ŷ) = Sn (x̂, ŷ)hn (x̂, ŷ).

(14)

On the next plane Vn+1 , the ﬁeld on (x̂, ŷ) emitted from the surface fragment Pn itself is superimposed onthe ﬁeld Fdn {hn (x̂, ŷ)}, which is the numerically propagated ﬁeld of hn (x̂, ŷ). Here, Fd {·} denotes a propagation
operator for a distance of d and dn = |ẑn+1 − ẑn |. As a result, the next subtotal ﬁeld on Vn+1 , including all ﬁeld
emitted from fragments located further than zn+1 , is wirtten as
Fdn {hn (x̂, ŷ)} + on (x̂, ŷ)

hn+1 (x̂, ŷ) =(15)Fdn {hn (x̂, ŷ)Sn (x̂, ŷ)} + on (x̂, ŷ).

=The starter value of this recurrence formula, which is the ﬁeld emitted from the furthest fragment, is given as
h2 (x̂, ŷ) = o1 (x̂, ŷ).

(16)

The object wave is calculated by the recurrence formula (15) in sequence from the starter value, and ﬁnal ﬁeld
upon the hologram is given as follows:
hHologram (x̂, ŷ) =F|ẑN | {hN (x̂, ŷ)SN (x̂, ŷ)} + oN (x̂, ŷ),(17)

where N is the number of fragments constructing the object and ẑN +1 = 0.

4. FABRICATION AND OPTICAL RECONSTRUCTION
Some holograms, of which object waves were synthesized by using eq.(15), were fabricated and optically reconstructed. The common parameters used for calculation and fabrication is shown in Table 1.
Fig. 5 (a) shows a reconstruction of the hologram of a cube placed at ẑ = 100[mm]. The width of the object
is 12mm. In synthesis of the object wave, the object ﬁeld on (x̂, ŷ) is calculated by the point source method ina density of 2500 point/cm2 . Fig. 5 (b) is of three-dimensional characters “KU”, of which total width is 12mm

Publication : SPIE Proc. Practical Holography XVIII and Holographic Materials X, SPIE#5290, San Jose, California, 90–97(2004).

= 6

> Figure 5. Optical reconstruction of holograms of 3-D objects, in which hidden-surfaces are removed by the proposed
method; (a) a cube, (b) 3-D characters “KU”.

=  >  = ? ?  @  @  @  @ 

Figure 6. Optical reconstructions of the hologram of a torus.

Publication : SPIE Proc. Practical Holography XVIII and Holographic Materials X, SPIE#5290, San Jose, California, 90–97(2004).

= 7

> Figure 7. Optical reconstructions of a 3-D object constructed of many small planes.

again, but the characters are placed at ẑ = 50[mm] and the density is 10000 point/cm2 . Noted that all of theobjects are shaded by ﬂat shading method.
Fig. 6 (a)–(d) show optical reconstructions of the identical hologram of a torus placed at ẑ = 150[mm], of
which diameter is 10mm. The object is constructed of 256 planar fragments and all fragments are composedof point sources in 10000 point/cm2 . Heights of the viewpoint is increasing in (a) to (d) and the reconstructed
image varies with the heights of the viewpoint, i.e., the reconstructed image has vertical parallax. Fig. 7 (a) and(b) also show reconstructions of a three-dimensional object constructed of many small planar fragments. The
height of the viewpoint in (a) is diﬀerent from that in (b) again.

APPENDIX A. THE SILHOUETTE APPROXIMATION
Spectrum of the masked ﬁeld in tilted coordinates is given by Fourier transform of eq.(10):H  (u, v) ==F {h (x, y)}
R{Ĥ} ∗ M (u, v),(18)where M (u, v) is spectrum of the mask function. As Step 3 of Sec.2.2 shown in Fig. 3(c), complex amplitudes of
the ﬁeld on the plane parallel to the hologram is obtained by rotating the spectrum.
ĥ (x̂, ŷ) ==F −1 R−1 {H  (u, v)}
F −1 R−1 {R{Ĥ} ∗ M (u, v)}.

When the inverse matrix of eq.(5) is deﬁned as⎛a1
T−1 = ⎝ a4
a7a2
a5a8

⎞
a3a6 ⎠ ,a9(19)(20)

and assume that only the lowest order in the expansion (12) is signiﬁcant i.e. higher orders can be ignored, thechange of variables is written as follows:u =v =α(û, v̂) = a1 û + a2 v̂ + a3 /λβ(û, v̂) = a4 û + a5 v̂ + a6 /λ.

(21)Moreover, the inverse functions are given asû= α(u, v)−1 = â1 u + â2 v + â3 /λv̂= β(u, v)−1 = â4 u + â5 v + â6 /λ.

(22)

Publication : SPIE Proc. Practical Holography XVIII and Holographic Materials X, SPIE#5290, San Jose, California, 90–97(2004).

8Eq.(19) is rewritten by use of above functions.
ĥ (x̂, ŷ) =Ĥ(α−1 (u , v  ), β −1 (u , v  ))M (α(û, v̂) − u , β(û, v̂) − v  ) exp[i2π(ûx̂ + v̂ ŷ)] dûdv̂du dv  .

×(23)The variables (û, v̂) of the integration is changed withuv==α(û, v̂) − u ,β(û, v̂) − v  ,(24)and furthermore, (u , v  ) is changed to (û , v̂  ) by eq. (21). As a result, the ﬁeld on the plane parallel to thehologram is given byĥ (x̂, ŷ) ==|a1 a5 − a2 a4 ||â1 â5 − â2 â4 |×

M (u, v) exp[i2π[(â1 x̂ + â4 ŷ)u + (â2 x̂ + â5 ŷ)v]]dudv

×Ĥ(û , v̂  ) exp[i2π(x̂û + ŷv̂  )]dû dv̂ m(â1 x̂ + â4 ŷ, â2 x̂ + â5 ŷ)ĥ(x̂, ŷ).

(25)

ACKNOWLEDGMENTS
Authors acknowledge S. Yamanaka and H. Miyauchi for their assistance in fabricating holograms. This work
was partly supported by Kansai University High-Technology Research Center.

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