Page 1

Computer-generated holograms for three-dimensional

surface objects with shade and texture

Kyoji Matsushima

Digitally synthetic holograms of surface model objects are investigated for reconstructing three-

dimensional objects with shade and texture. The objects in the proposed techniques are composed of

planar surfaces, and a property function defined for each surface provides shape and texture. The field

emitted from each surface is independently calculated by a method based on rotational transformation

of the property function by use of a fast Fourier transform (FFT) and totaled on the hologram. This

technique has led to a reduction in computational cost: FFT operation is required only once for calculating

a surface. In addition, another technique based on a theoretical model of the brightness of the recon-

structed surfaces enables us to shade the surface of a reconstructed object as designed. Optical recon-

structions of holograms synthesized by the proposed techniques are demonstrated.

Society of America

OCIS codes:

090.1760, 090.2870, 090.1970.

© 2005 Optical

1.

Computer-generated

dimensional (3-D) displays, sometimes called digi-

tally synthetic holograms, are desired media for

creating 3-D autostereoscopic images of virtual ob-

jects. However, the technology suffers from two prob-

lems: the necessity for extremely high spatial

resolution to fabricate or display the holograms, and

long computation times for the creation, especially in

full parallax holograms.

For the past decade, techniques using point sources

of light have been widely used to calculate object

waves.1,2This point source method is simple in prin-

ciple and potentially the most flexible for synthesiz-

ing holograms of 3-D objects. However, because it is

too time consuming to create full parallax holo-

grams,3many methods to reduce the computation

time, including geometric symmetry,4look-up tables,3

difference formulas,5recurrence formulas,6employing

computer-graphics hardware,7and constructing spe-

cial CPUs,8have been attempted.

Point source methods for calculating spherical

Introduction

hologramsforthree-

waves emitted from point sources are commonly ray

oriented. As they trace the ray from a point source to

a sampling point on the hologram, the procedure is

sometimes referred to as ray tracing.2However, there

are also wave-oriented methods to calculate object

fields in which fields emitted from objects defined as

planar segments9,10or 3-D distributions of field

strength11are calculated by methods based on wave

optics. The major advantage of wave-oriented meth-

ods is that they can use a fast Fourier transform

(FFT) for numerical calculations. Therefore the com-

putation time is shorter than for point source meth-

ods, especially in full parallax holograms. However,

the optical reconstruction of accurately rendered 3-D

objects such as a shaded cube, as reported for wave-

oriented methods, was not discussed in the papers

citedabove.Thisissobecauseofalackofwell-defined

procedures to generate object fields for arbitrarily

shaped surfaces that are diffusive and sometimes

have texture. The technique for shading the recon-

structed object according to such design parameters

as the position of the illumination light and the ratio

of the surrounding light is also important in creating

real 3-D images by wave-oriented methods.

In wave-oriented methods, calculating fields are

commonly based on coordinate transformation in

Fourier space.10,11A similar method based on the

Rayleigh–Sommerfeld integral has been reported

within the context of free-space beam propagation.12

Recently, the author reported a more precise formu-

lation and numerical consideration13as an extension

The author (matsu@kansai-u.ac.jp) is with the Department of

Electrical Engineering and Computer Science, Kansai University,

3-3-35 Yamate-cho, Suita, Osaka 564-8680, Japan.

Received 17 June 2004; revised manuscript received 8 March

2005; accepted 21 March 2005.

0003-6935/05/224607-08$15.00/0

© 2005 Optical Society of America

1 August 2005 ? Vol. 44, No. 22 ? APPLIED OPTICS 4607

Page 2

of an angular spectrum of plane waves14in which

remapping the angular spectrum plays an important

role. The remapping also eases the difficulty of cre-

ating object fields in wave-oriented methods.

Inthispapertwotechniquesforsynthesizingobject

fields in surface models are presented for creating

3-D images by use of computer-generated holograms.

The first technique, based on the rotational transfor-

mation of wave fields presented in Ref. 13 and on

remapping of the angular spectrum, provides a

method for synthesis of the object fields. This tech-

nique makes it possible to create diffusive fields of

arbitrarily tilted planar surfaces that have an arbi-

trary shape and texture. Furthermore, another tech-

nique is also presented for avoiding unexpected

changes in brightness of the surfaces of objects. The

technique enables us to render surface objects as the

designers intended.

2.

The coordinate systems and geometry used in this

study are shown in Fig. 1. Objects consist of planar

surfaces that are diffusive and luminous by reflecting

virtual illumination. Each surface has its own two

local coordinates, called tilt and parallel. The tilted

local coordinates defined for the nth planar surface

are denoted rn? ?xn, yn, zn?, defined such that the

planar surface is laid on the ?xn, yn, 0? plane. A com-

plex function hn?xn, yn? is defined on the plane to give

the nth surface such properties as shape, brightness,

diffusiveness, and texture. Thus these complex func-

tions are referred to as the property functions of the

surface.

Parallel local coordinates r ˆn? ?x ˆn, y ˆn, z ˆn? are also

defined for each surface. They share their origin with

the tilted coordinates, but the axes are parallel to

those of the global coordinates. In the global coordi-

nates, denoted r ? ?x ˆ, y ˆ, z ˆ?, the hologram is placed on

the ?x ˆ, y ˆ, 0? plane. All property functions of surfaces

are defined in the following form:

Object Model and the Property Function of Surfaces

hn(xn, yn)?an(xn, yn)?(xn, yn)pn(xn, yn),(1)

where an?xn, yn? is a real function that provides am-

plitudes of the property function to keep the shape

and the texture of the nth surface.

If the property function is defined only as the am-

plitude distribution, the surface yields little diffusive-

ness, as shown in Fig. 2(a). For example, an?xn, yn?

in Fig. 1 is a simple rectangular function; i.e., the

amplitude is constant within the rectangular surface.

This situation is similar to the optical diffraction of a

plane wave by a rectangular aperture; therefore, if

the surface is visible to the naked eye, the light has

not been sufficiently diffracted by the aperture. To

give surfaces large diffusiveness, the amplitude func-

tions must be multiplied by a given diffusive phase:

?(xn, yn)?exp[ik?d(xn, yn)], (2)

where ?d?xn, yn? is a phase that behaves as a numer-

ical diffuser. Random functions are candidates for the

diffusive phase, but full random functions are not

appropriate to the diffusive phase because the ran-

dom phases are discontinuous and have a large Fou-

rier frequency. Thus the random phases cause

speckles in the reconstruction and problems in nu-

merical calculation. In the research reported in this

paper, a digital diffuser proposed for Fourier holo-

grams15is used for phase function ?d?xn, yn?.

If a property function is given by the product of the

amplitude function and the diffusive phase, the car-

rier frequency of the field on the tilted ?xn, yn, 0? plane

is zero. This forces the surface to emit light perpen-

dicularly to the surface, as shown in Fig. 2(b). If the

surface is sufficiently diffusive, a portion of the emit-

ted field may reach the hologram, but high diffusive-

ness results in high computational costs such as the

need for a great number of sampling points. There-

fore the phase of a plane wave propagating perpen-

dicularly to the hologram should be multiplied by the

two factors given above. This plane-wave factor

causes the field to propagate into the hologram, ex-

pressed by

pn(xn, yn)?exp[i(kx, nxn?ky, nyn)],(3)

where kx, nand ky, nare the xnand yncomponents,

respectively, of the wave vector of the plane wave.

The property function given by hn?xn, yn? is trans-

formed into the complex amplitude hˆn?x ˆn, y ˆn? in the

parallel coordinates by the method described in Sec-

Fig. 1.

local coordinates defined for a planar surface.

Geometry and definitions of global coordinates and tilted

Fig. 2.

a diffusive phase, and (c) a diffusive phase multiplied by the phase

of a plane wave propagating to a hologram.

Fields emitted from surfaces with (a) a constant phase, (b)

4608 APPLIED OPTICS ? Vol. 44, No. 22 ? 1 August 2005

Page 3

tion 5 below. When this transformation is written as

hˆn(x ˆn, y ˆn)???x?y?z{hn(xn, yn)},(4)

fields from all surfaces are superimposed upon the

hologram plane as follows:

hˆ(x ˆ, y ˆ)??

n

?dn{hˆn(x ˆn, y ˆn)},(5)

where ?x, ?y, and ?zare rotation angles on each axis

and

?dn? ?

represents translational

through distance dn.

propagation

3.

The details of rotational transformation were already

reported in Ref. 13. However, I summarize it for con-

venience in this section, because rotational transfor-

mation of wave fields is the core of techniques

proposed in this paper. Note that rotation of just a

single planar surface is presented in this section;

therefore the subscript n is omitted in this section.

Rotational Transformation of Property Functions

A.

The Fourier spectrum of a property function is given

in the tilted local coordinates as

Rotation of Coordinates in Fourier Space

H(u, v)??{h(x, y)}

???

??

?

h(x, y)exp[?i2?(ux?vy)]dxdy,

(6)

where u and v denote the Fourier frequencies in the

x and y axes, respectively. The frequency in the z axis

is not independent of u and v and is given by

w?u, v? ? ???2? u2? v2?1?2, where ? is the wave-

length.TherelationamongFourierfrequenciesinthe

parallel local coordinates ?u ˆ, v ˆ, w ˆ ?u ˆ, v ˆ?? is analogous

to this and given by w ˆ ?u ˆ, v ˆ? ? ???2? u ˆ2? v ˆ2?1?2. In

Fourier space, the frequencies ?u, v, w?u, v?? can be

transformedinto

?u ˆ, v ˆ, w ˆ ?u ˆ, v ˆ??

coordinate-transformationproceduresandviceversa.

Suppose that r is transformed into r ˆ by a rotation

matrix T, i.e., r ˆ ? Tr and r ? T?1r ˆ. The frequencies

in the parallel coordinates are given as

by ordinary

u??(u ˆ, v ˆ)?a1u ˆ ?a2v ˆ ?a3w ˆ (u ˆ, v ˆ),

v??(u ˆ, v ˆ)?a4u ˆ ?a5v ˆ?a6w ˆ (u ˆ, v ˆ),(7)

where the rotation matrix is defined as

T?1??

a1 a2 a3

a4 a5 a6

a7 a8 a9?.(8)

Therefore the spectrum in the parallel coordinates is

given by

Hˆ(u ˆ, v ˆ)?H[?(u ˆ, v ˆ), ?(u ˆ, v ˆ)]. (9)

Complex amplitudes of the field are obtained in

parallel coordinates by inverse Fourier transforma-

tion of the spectrum in the paraxial condition13:

hˆ(x ˆ, y ˆ)???1{Hˆ(u ˆ, v ˆ)}

???

??

?

Hˆ(u ˆ, v ˆ)exp[i2?(x ˆu ˆ ? y ˆv ˆ)]du ˆdv ˆ.

(10)

B.

In the actual numerical calculation, to use a FFT one

must sample the spectrum as well as the field at an

equidistant sampling grid within a given sampling

area. The coordinate transformation of Eq. (9), how-

ever, causes distortion and a shift of the sampling

area and points. This distortion and shift are depen-

dent on the way the tilted local coordinates are de-

fined.16

Figure 3(a) shows an example of the definition in

which the parallel coordinates r ˆ are transformed into

the tilted coordinates r by rotation of the coordinates

on the z ˆ axis before the y axis. In this case, rotation

matrix T?1is given by

T?1??

sin ?ycos ?z sin ?ysin ?z

Remapping the Fourier Spectrum

cos ?ycos ?z cos ?ycos ?z ?sin ?y

?sin ?z

cos ?z

0

cos ?y?.

(11)

Figure 3(b) shows the sampling area of Hˆ?u ˆ, v ˆ? for

several rotation angles when h?x, y? is sampled at

intervals of ?x? ?y? 2 ?m in ? ? 633 nm. The

sampling area of H?u, v?, a ?x

distorted and shifted. Therefore, resampling accom-

panied with interpolation is necessary for using an

inverse FFT in calculating the complex amplitudes of

the field. However, simple interpolation is not suffi-

cient because the FFT algorithm does not work effec-

tively for spectra sampled far from zero frequency.

This shift can be interpreted to be a carrier frequency

observed in the parallel local coordinates.

Let us reverse the procedure of rotation of the co-

ordinates and shift the origin of Fourier space ?u, v?

in the tilted coordinates. The origin of Fourier space

?u ˆ, v ˆ? in the parallel coordinates is inversely projected

to the frequencies ?u0, v0? in the tilted coordinates by

matrix (8) as follows:

?1? ?y

?1square, is

u0??(0, 0)?a3/?,

v0??(0, 0)?a6/?.(12)

To ensure that the center of the spectrum in the

parallel coordinates is located at the origin of the

Fourier space after rotation of the coordinates, the

following new shifted Fourier space should be intro-

1 August 2005 ? Vol. 44, No. 22 ? APPLIED OPTICS4609

Page 4

duced in the tilted coordinates:

u??u?u0,

v??v?v0. (13)

The spectrum expressed in shifted Fourier space

?u?, v?? is written as

H?(u?, v?)?H(u??u0, v??v0).(14)

The spectrum in the parallel coordinates is obtained

by remapping spectrum H??u?, v?? onto Fourier space

?u ˆ, v ˆ? as follows:

Hˆ(u ˆ, v ˆ)?H?(u?u0, v?v0)

?H?(?(u ˆ, v ˆ)?u0, ?(u ˆ, v ˆ)?v0),(15)

where the sign for nearly equal means that an inter-

polation is required.

The Fourier spectrum in the shifted Fourier space

is obtained by application of the shift theorem of the

Fourier-transform theory to Eq. (14):

H?(u?, v?)??{h(x, y)exp[?i2?(u0x?v0y)]}. (16)

The exponential factor in brackets in Eq. (16) is at-

tributed to the carrier frequency observed in the par-

allel coordinates, whereas factor p?x, y? of the

property function was introduced to force the emitted

field toward the hologram, canceling the carrier fre-

quency in the parallel coordinates. In fact, the expo-

nential factors of Eq. (16) and p?x, y? cancel each

other out. Equation (16) is rewritten by substitution

of Eq. (3) as follows:

H?(u?, v?)??{a(x, y)?(x, y)exp{i[(kx?2?u0)x

?(ky?2?v0)y]}},(17)

where the subscript n is omitted again. The wave

vector of a plane wave propagating along the z ˆ axis is

expressed by ?0, 0, 2???? in parallel coordinates.

Thus the plane wave in the tilted coordinates is ob-

tained by coordinates rotation by use of matrix (8) as

follows:

kx?2?a3??,

ky?2?a6??. (18)

The spectrum of relation (15) is rewritten by substi-

tution of Eqs. (18) and (12):

H?(u?, v?)??{a(x, y)?(x, y)}.(19)

As a result, the factor p?x, y? is no longer required in

the property function if the spectrum is calculated in

shifted Fourier space ?u?, v??. Therefore let us rede-

fine the property function as

h(x, y)?a(x, y)?(x, y), (20)

and its spectrum as

H(u, v)??{h(x, y)}. (21)

Consequently, the rotational transformation is

summarized as follows: First, one obtains spectrum

H?u, v? of the property function of Eq. (20) by fast

Fourier transformation. The center of the spectrum is

placed at the origin in the Fourier space. Next, the

spectrum in the parallel coordinates is obtained by

remapping spectrum H?u, v?, expressed by substitut-

ing Eq. (12) into relation (15) as follows:

Hˆ(u ˆ, v ˆ)?H(?(u ˆ, v ˆ)??(0, 0), ?(u ˆ, v ˆ)??(0, 0)).

(22)

Finally, the complex amplitudes of the field are ob-

tained in the parallel coordinates by an inverse Fou-

rier transformation of Eq. (10).

4.Holograms of a Single Surface with Texture

A.

The hologram of a single planar surface with texture

was fabricated for verifying the technique described

in Section 5. The planar surface and the hologram are

Single Axis Rotation

Fig. 3.

upon the z ˆ axis before the x axis and (b) resampling areas of the

Fourier spectrum at several rotation angles in the rotation scheme.

Schematic of rotation upon two axes: (a) a plane rotated

4610 APPLIED OPTICS ? Vol. 44, No. 22 ? 1 August 2005

Page 5

sampled at intervals of 2 ?m in the x axis and 4 ?m

in the y axis. The planar surface has sampling points

of 16,384 ? 4096 and a binary texture. Amplitude

distribution a?x, y? is shown in Fig. 4(a). The surface

object was placed at z ˆ ? ?10 cm and rotated only on

theyaxisatanangleof80°,asshowninFig.4(b).The

hologram with 8192 ? 4096 pixels was encoded by a

point-oriented method and fabricated by a special

printer constructed for printing synthetic fringes.17

The hologram is reconstructed by a 633 nm He–Ne

laser, and the reconstructed image is captured by a

digital camera placed at z ˆ ? 10 cm. Figure 5 shows

photographs of a reconstructed image. The camera,

whose focus is fixed, was moved from left to right for

Figs. 5(a)–5(c) and back and forth for Figs. 5(d)–5(f).

The apparent size of the texture pattern changes

when the viewpoint is moved in Figs. 5(a)–5(c), while

the defocused position of the texture pattern changes

when the camera is moved because the focal plane of

the camera moves in Figs. 5(d)–5(f).

B.

Holograms of a single planar surface rotated on two

axes were also fabricated and optically reconstructed.

The method of rotation of two axes is the same as

shown schematically in Fig. 3. The amplitude image

with 8192 ? 4096 sampling points is shown in Fig. 6.

Two-Axis Rotation

The number of sampling points and sampling pitches

of the hologram is the same as in the single-axis

rotation.Opticalreconstructionsofthehologramsare

shown in Fig. 7. Four holograms with different rota-

tion angles were fabricated and reconstructed. The

appearance of texture on the planar surfaces varies

according to the rotation angle.

5.

Its Shading

Three-dimensional objects such as cubes and pyra-

mids can be built from planar surfaces. Thus, one can

synthesize the fields of these objects by superimpos-

ing the fields from the planar surfaces onto the holo-

gram. However, a problem that does not exist when

one is synthesizing a single-surface object is that the

brightness of the reconstructed surface varies, de-

pending on the angle of the surfaces. As a result,

objects are shaded as if an unexpected illumination

were throwing light. In a single-surface object, the

Holograms of a Three-Dimensional Object and

Fig. 4.

plane rotated upon a single axis. (b) Geometry for capturing the

reconstruction. The dimensions of texture of a checker embedded

in the property function are 16.4 mm ? 8.2 mm.

(a) Planar object used for fabricating the hologram of a

Fig. 5.

moving a camera (a)–(c) from left to right and (d)–(f) back and

forth.

Optically reconstructed images of a hologram captured by

Fig. 6.

rotation.

Planar object used for fabricating a hologram in two-axis

Fig. 7.

rotated at several angles.

Optical reconstructions of holograms of planar surfaces

1 August 2005 ? Vol. 44, No. 22 ? APPLIED OPTICS4611

Page 6

change of brightness of a surface is not perceived

because there is nothing to compare with the single

surface in a piece of hologram. This unexpected and

unwanted change of brightness must be resolved if

one is to shade the object as intended.

A.

To compensate for unexpected shading it is necessary

to investigate which parameters govern the bright-

ness of the surface in reconstruction. Figure 8 is a

theoretical model that predicts the brightness of a

surface represented by sampled property function

h?x, y?. Suppose that the amplitude of a property

function of a surface is a constant, i.e., that a?x, y?

? a, and suppose that a2provides optical intensity on

the surface. In such cases, the radiant flux ? of a

small area ?A on the surface is given by

????

?A

Theory of Brightness of Reconstructed Surfaces

|h(x, y)|2dxdy

??A?a2, (23)

where ? is the surface density of the sampling points.

Assuming that the small area emits light within a

diffusion angle in a direction at ?vto the normal

vector, the solid angle corresponding to the diffusion

cone is given as ? ? A?r2, where A ? ??r tan ?d?2is

the section of the diffusion cone at a distance r and ?d

is the diffusion angle of light that depends on diffuser

function ??x, y? of Eq. (1).

According to photometry, brightness of the surface,

observed in a direction at an angle ?v, is given by

L?

d??d?

cos ?v?A. (24)

Assuming that light is diffused almost uniformly, i.e.,

that d??dA ? ??A, the brightness is rewritten by

substitution of d? ? ???A?dA, d? ? dA?r2, and re-

lation (23) into Eq. (24) as follows:

L?

?a2

? tan2?dcos ?v

. (25)

As a result, the brightness of the surface depends

on the surface density of sampling, the diffusiveness

of the diffuser function, and the amplitude of the

surface property function. In addition, the brightness

of the surface is governed by observation angle ?v. In

other words, if several surfaces with the same prop-

erty function are reconstructed from a hologram, the

brightness varies according to the direction of the

normal vector of the surface. This phenomenon

causes unexpected shading.

Inasmuch as only a simple theoretical model has

been discussed so far, relation (25) is only partially

appropriate for measuring the brightness of optically

reconstructed surfaces of real holograms. The bright-

ness given in relation (25) diverges in the limit

?v→ ??2, but an actual hologram cannot produce in-

finite brightness for its reconstructed surface. Thus

relation (25) is not sufficient to compensate for the

brightness. To avoid the divergence of brightness

in relation (25), one should introduce angle factor

?1 ? ????cos ?v ? ?? shown in Fig. 9, instead of

1?cos ?v, a priori. This angle factor is unity in ?v

? 0 and 1 ? 1?? in ?v? ??2. Consequently, the

brightness is given as an expression of

L?

?a2

? tan2?d

(1??)

(cos ?v??),

(26)

where ? is a parameter that plays a role in preventing

the divergence of brightness and in preventing over-

compensation. Because ? is dependent on actual

methods for fabricating holograms, such as encoding

the field or the property of recording materials, it

should be determined experimentally.

B.

The amplitude of a property function that recon-

structs a surface in a given brightness L is obtained

by solution of Eq. (26) for a as follows:

Compensation for Brightness and Shading Objects

a??

L? tan2?d

?

(cos ?v??)

(1??)?

1?2

. (27)

Fig. 8.

property function sampled at an equidistant grid.

Model of brightness of a planar surface expressed by a

Fig. 9. Curves of the angle factor for several values of ?.

4612 APPLIED OPTICS ? Vol. 44, No. 22 ? 1 August 2005

Page 7

However, angle ?vis unknown in synthesizing the

object field, and therefore it seems impossible to com-

pensate for the change of brightness. But holograms

are observed in a direction along the z ˆ axis, i.e., per-

pendicular to the hologram, because the hologram is

usually observed at a distance of more than several

tens of centimeters. Hence it is possible to approxi-

mate ?vby an angle ?nformed between the nth sur-

face and the hologram. Objects are shaded by a

method based on Lambert’s law and the diffused re-

flection model. The brightness of the nth surface, of

which the normal vector forms angle ?^nwith the vec-

tor of illumination, is given by

Ln?L0(cos ?^n?le), (28)

where leis the ratio of the surrounding light to the

illumination and L0is brightness in ?^n? 0 and le

? 0. By substitution of Lnof Eq. (28) into L of Eq. (27)

amplitude anof the nth surface is given as follows:

an?a0?

(cos ?^n?le)(cos ?n??)

1??

?

1?2

, (29)

a0??

L0? tan2?d

? ?

1?2

. (30)

Here, observation angle ?vis replaced by the angle of

the normal vector, ?n.

6.

Objects

First, I fabricated several holograms of the same hex-

agonal prism with which to determine the value of

parameter ?. Figure 10 shows the optical reconstruc-

tion of three holograms. The reconstructed image of

the hologram without compensation for brightness is

shown in Fig. 10(a). The left-hand surface of the hex-

agonal prism, which has the largest angle ?n, is the

brightest of the object surfaces. As shown in Fig.

10(b), the hologram with compensation in ? ? 0 is

contrasted to that in Fig. 10(a). Here, remember that

compensation in ? ? 0 leads to unlimited compensa-

tion. Therefore the surface that forms a large angle

with the hologram is dark as a result of overcompen-

sation. Figure 10(c) is also applicable to a hexagonal

prism whose brightness is compensated for by

? ? 0.5.Differencesofbrightnessdisappearbyproper

compensation for brightness, which dissolves borders

between surfaces.

Figure 11 shows optical reconstruction of 3-D ob-

jects whose brightness is completely compensated for

at ? ? 0.5. In addition, the surfaces are shaded; the

amplitudes of the surfaces are determined by use of

Eq. (29) in given virtual illumination and surround-

ing light. Arrows and numbers in Fig. 11 indicate

Optical Reconstruction of Three-Dimensional

Fig. 10.

? ? 0, 0.5, respectively.

Optical reconstructions of unshaded hexagonal prisms (a) without brightness compensation and (b), (c) with compensation in

Fig. 11.

(a) le? 0 and from the upper left in (b) le? 0.7; a hexagonal prism ?le? 0.5? is shown in (c). Brightnesses of objects are all compensated

for at ? ? 0.5. Arrows and numbers in parentheses define the illumination vector in global coordinates.

Optical reconstructions of 3-D objects shaded with illumination light. Cubes are illuminated from the upper right in

1 August 2005 ? Vol. 44, No. 22 ? APPLIED OPTICS4613

Page 8

illumination vectors in global coordinates. As ex-

pected from the vectors, object surfaces are shaded in

the reconstructions.

7.

The computation time in the proposed techniques is

given by rotational transformation of the surfaces of

the object. According to Ref. 13, the computation time

of rotational transformation is dominated by FFTs,

and FFT operation must be executed twice to rotate a

planar surface. However, most of the inverse FFTs of

Eq. (10) can be omitted from calculating the total

field; just an inverse FFT operation is necessary to

create a hologram because the translational propa-

gation of the field ?d? ? can be carried out in Fourier

space. In the synthesis of holograms described in pre-

vious sections, the method of the angular spectrum of

plane waves14is used for the operation of the propa-

gation. Therefore the total field of Eq. (5) on the ho-

logram is expressed by

Discussion

hˆ(x ˆ, y ˆ)???1??

n

Hˆ

n(u ˆn, v ˆn)exp[i2?w ˆ (u ˆn,v ˆn)dn]?,

(31)

where dnis the distance between the ?x ˆn, y ˆn, 0? plane

of the parallel coordinates and the hologram. Thus

the number of times a FFT is executed is N ? 1, to

calculate the total field of an object composed of N

pieces of planar surface. As a result, one FFT?surface

is approximately estimated as the computational cost

in the proposed techniques.

8.

Full parallax computer-generated holograms of

three-dimensional surface objects were synthesized

by use of a wave-optical method. In this method, an

object is composed of some planar surfaces, and a

complex function defined for each surface retains

such properties as shape, texture, and brightness.

The fields emitted from the tilted surfaces are calcu-

lated by use of the rotational transformation of the

property function and totaled on the hologram.

When surfaces build an object, the change of

brightness that depends on the angle of view causes

unexpected shading of the surface. A theoretical

model with which to predict the brightness of the

reconstructed surface and prevent unexpected shad-

ing has been proposed. This technique allows the

object to be shaded as one intends. Finally, optical

reconstructions of holograms synthesized by use of

the proposed techniques have been demonstrated to

verify the validity of the methods.

Conclusion

This study is partly supported by the Kansai Uni-

versity High Technology Research Center and in

part by Kansai University research grants, includ-

ing a grant-in-aid for encouragement of scientists,

in 2003.

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4614 APPLIED OPTICS ? Vol. 44, No. 22 ? 1 August 2005