9th IEEE/ACIS International Conference on Computer and Information Science (ICIS2010)
Yamagata, pp.846-851(2010.8, Invited).

Wave-Field Rendering in Computational Holography
The polygon-based method for Full-Parallax High-Definition CGHs

Kyoji Matsushima
Department of Electrical and Electronic Engineering
Kansai University
Suita, Japanmatsu@kansai-u.ac.jp

Wave-field of
polygonal
surface

Spherical wave

Keywords-3D image; holography; rendering; computergenerated hologram; wave-field;

I. I NTRODUCTION
Computational holography, commonly referred to ascomputer-generated holograms (CGH), is the technologythat creates real light from virtual objects. Therefore, 3D
displays based on full-parallax CGHs make it possible
to produce a strong sensation of depth, unlike other 3Ddisplays based on psychophysical techniques such as stereoscopic images [1].
In CGHs, the size and angle of the viewing-zone are themost important parameters for reconstructing high-quality
3D images. Since a large viewing-zone requires high spatial
resolution in CGHs, both lead to an extremely large numberof pixels for CGHs. In addition, the reconstruction of occlusion in 3D scenes is also significant for depth sensation.
However, there is no practical technique to compute theobject light of occluded 3D scenes for CGHs with large
numbers of pixels. As a result, the technology for CGHs
suffered from the fact that true fine 3D images could not be
produced.
For a long time, ray-oriented point-based methods wereused for computing light emitted from virtual objects in
CGHs. Techniques for fast computation of spherical waves
emitted from a point source of light have been undercontinuous development. However, there are two reasonsthat the point-based method is not suitable for the creationof full-parallax high-definition CGHs. Firstly, point-based
methods are commonly very time consuming, because 3D
objects in a large-scale CGH usually have a large surface
area and therefore need an extremely large number of point
sources to form surfaces out of the points, as shown in
Fig. 1 (a). In addition, there is no practical technique forcreating occluded 3D scenes in the point-based method.

SurfacesourcePointsourceAbstract—Techniques of wave-field rendering in computational holography are proposed. These polygon-based techniques are developed for the creation of high-definition CGHs.
The holograms created, reconstruct a true fine 3D image ofoccluded scenes, which creates a strong sensation of depth due
to motion parallax. These computational holograms can beappreciated as works of art.

(a)

(b)

Figure 1. Forming object surfaces by point-based method (a) and polgyonbased method (b).

A polygon-based method has been proposed instead ofpoint-based methods in computational holography [1], [2].
This technique is similar to conventional computer graphics,
because object surfaces are composed of polygons. Eachpolygon is regarded as a surface source of light, as shown in
Fig. 1 (b). Wave-fields emitted from polygons are computedby numerical methods based on wave optics. Although theindividual computation time of the wave-field of a polygonalsurface is longer than that of the spherical wave of a point
source, the total computation time of the polygon-based
method is shorter than that of the point-based methods,
because the number of polygons required to form object
surfaces is much smaller than that of point sources.
Another advantage of the polygon-based method is thesimilarity to computer graphics. Some basic techniques usedin computer graphics, such as shading or texture mapping,can be applied to polygon-based CGHs. Therefore, in thispaper, the computation of wave-fields by the polygon-based
method is referred to as wave-field rendering of virtual
objects.
Here, the terminology wave-field is used as a planar
two-dimensional distribution of the complex-amplitude ofelectric waves of light.

Light source
Spatial distributionof optical intensity
I ( x, y , z )Human eyes

Human eyes

Diffused light
ObjectSurface

Diffused light(b)(a)
Diffuser platexx̂
Human eyeszDiffusedwave field

Incident wave field
Aperture(c)Figure 2. Schematic illustration of the theoretical model of a polygonal surface source of light: (a) Surface object, (b) Spatial distribution of opticalintensity, (c) The theoretical model of a surface source.

II. WAVE - FIELD R ENDERING OF S URFACE O BJECTS
A. Theoretical Model of Surface Sources of Light
We can see real objects illuminated by a light source,
because the object surfaces scatter the light, as shown in Fig.
2 (a). Suppose that an object is composed of many polygonalplanes, i.e. polygons, and each polygon emits a wave-field. A
polygonal surface is regarded as a surface source of light, i.e.
a planar distribution of optical intensity I(x, y, z), as in (b).
This distribution of light is similar to that in an aperture
irradiated by a plane wave, as in (c). The aperture has thesame shape and slant as the polygon. However, a simple
polygonal aperture may not behave as if it is a surface sourceof light, because the aperture size is too large to diffract
incident light, and thus light passing through the aperture is
not diffused.
Figure 2 (c) shows the theoretical model of a surface
source of light for wave-field rendering. Here, the polygonalsurface source is imitated by a virtual (numerical) diffuserthat is mounted in the aperture whose shape and tilt angle
correspond with the polygon.
B. Surface Functions
To compute the wave-field diffracted by the diffuser witha polygonal shape, a surface function is defined for eachindividual polygon in the local coordinate system that is also
specific to the polygon. An example of the surface function
is shown in Fig. 3 (b).
The surface function h(x, y) for a polygon is generally
given in the following form:h(x, y) = a(x, y) exp[iϕ(x, y)],(1)where a(x, y) and ϕ(x, y) are the real-valued amplitude
and phase distribution in the local coordinates. The phasepattern ϕ(x, y) is not visible in principle, because all image
sensors including human retinas can detect only the intensityof light, whereas the amplitude pattern a(x, y) directly
determines the appearance of the polygon. Therefore, while
the properties required for the diffuser should be provided
by the phase pattern, the shape of the polygon is given by
the amplitude pattern.
C. Rendering Methods Similar to Computer Graphics
As for the amplitude pattern, almost all rendering techniques used in computer graphics can be applied to wavefield rendering. For example, the amplitude pattern for flat
shading of polygonal surface sources is simply given as

C u b ic o b je c t

3

G lo b a l
c o o rd in a te s

2

1

y^

L o c a lc o o rd in a te s
y 2

x 2

y 2

z2

A m p litu d e

x^
z^

y^

x 2

(a )

P h a se
^

x

(b )

A m p litu d e(c )Figure 3. An example of wave-field rendering by flat shading: (a) an object model, (b) the surface function of the polygon #2, (c) the amplitude imageof the computed wave field in a plane parallel to the hologram.

Silhouette maskfollows:{a(x, y) =Wave front

A
0

···
···

inside polygon,
outside polygon(2)where A is a constant that determines the brightness of thepolygon. Note that there is a problem peculiar to holograms,
i.e. the observed brightness of the polygon varies dependingon the tilt angle of the polygon in optical reconstruction [2].
This unintended change of brightness can be avoided bygiving a compensated value to the constant A. The surfacefunction shown in Fig. 3 (b) is for flat shading.
When we use other techniques used in computer graphics
such as smooth shading or texture mapping, the amplitude
inside the polygon is not a constant but function of the local
coordinates x and y. The amplitude pattern should be givenby the same techniques as for computer graphics.
The optical reconstruction of a texture-mapped surface
object is shown in Fig. 9 (a) of the following section.

Obstacle

Object plane
(a)

(b)

Figure 4. Schematic illustration of light-shielding by an obstacle (a) anda silhouette mask (b).

Fig. 4 (b). Numerical techniques such as the band-limited
angular spectrum method [6] can be used for the numericalpropagation.

D. A Feel of MaterialIII. WAVE -F IELD R ENDERING OF M UTUALLYO CCLUDED 3D S CENESThe phase pattern ϕ(x, y) in (1) gives a diffusiveness tothe polygon and is not visible in principle. However, thepattern produces a considerable effect on the feel of thematerial in its optical reconstruction. We currently use thediffuser phase pattern proposed when Fourier-type CGHs
were in use [3]. However, we believe that this is not thebest phase pattern, because optically reconstructed polygons
using the phase pattern do not give any feel for the materialof the polygon.
In wave-field rendering, almost all research concerning
expressions on the feel of the material is left for future study.

Occlusion is one of the most important mechanisms in theperception of 3D scenes. To reconstruct occluded 3D scenes,
light behind an obstacle must be shielded by the obstacle,
as shown in Fig. 4 (a). This is one of the most difficult
rendering tasks in computational holography. In the pointbased method, rays from point sources should be traced todetect the ray crossing any obstacles. However, this is a notrealistic method, because an enormous number of rays must
be inspected in high-definition CGHs.

E. Computation of Wave-Fields of Polygons

A. Basic Principle of the Silhouette Method

The wave-field of a polygon is temporally computed ina plane that is parallel to the hologram and placed close
to the polygon in order to reduce computation time [1]. To
compute the wave-field in the plane, surface functions aretransformed by following almost the same procedure as therotational transformation [4], [5]. The wave-field computedin the temporal plane is translationally propagated onto theobject plane that usually slices the object, as shown in

The solution in wave-field rendering is the silhouettemethod [7], [8]. Let us consider a situation in which an
obstacle shields light from behind it, as shown in Fig. 4 (a).
Since the obstacle is also an object, the object plane slices
it and is parallel to the hologram as in (b).
Supposing that a background wave-field emitted fromother objects behind the obstacle is given in the object
plane. In the silhouette method, the background wave-field

(a)

(b)

Figure 5. An example of the silhouette-masked field (a) and the wave-fieldcombined with the object field emitted from the obstacle itself (b).

(a)

ObjectObjectObjectObject(b)
Figure 6. Schematic illustration of algorithm for acceleration of computation of sparse 3D scenes.

is masked with the shape of the silhouette of the obstacle,
as shown in Fig. 5 (a). Then, the wave-field of the obstacle
itself is added into the masked background wave-field in theobject plane, as in (b). This combined wave-field is again
propagated onto the next object plane or the hologram.
The silhouette method is a first approximation of rigorous
light-shielding by the obstacle. Therefore, this does not work
well in self-occluded objects (obstacles). Such an object, atorus for example, may be constructed as a partial phantom
image. Other methods that make it possible to shield light
more exactly have been proposed [8], [9], but these are morecostly in computation time and therefore cannot be applied
to high-definition CGHs.
B. Speed-up of the Silhouette Method in Sparse 3D Scenes
A sparse 3D scene is a scene composed of many small
objects compared with the scene volume. Light-shielding
by the silhouette method is very inefficient in such sparse
3D scenes, because the method requires the same numberof numerical propagations as the number of objects. Since
the propagated wave-fields usually have the same number ofsampling points as the hologram, this computation is verytime consuming in high-definition CGHs. In addition, since
the frame buffers of wave-fields of high-definition CGHs
consume large amounts of memory, the memory must be
segmented. This also leads to an increase in the computationtime.
If the silhouette of an object occupies only a few segmentsin the frame buffer, light-shielding by the silhouette method

Figure 7. Photographs of optical reconstruction of Moai II. The camera
is focused on the near moai (a) and the far moai (b).

6.5

10
7
Units: cm

Figure 8.

13

6.5

The 3D scene of Moai II

can be implemented by off-axis numerical propagation of
the partial field using the shifted-Fresnel [10] or the shifted
angular spectrum methods [11], as shown in Fig. 6. The
silhouette-masked field is obtained using Babinet’s principle,
i.e. the field is computed by subtracting the partial field fromthe full field. Although the mathematical expression of thetechnique to obtain exact light-shielding is very complicatedin scenes that include multiple objects, it has been shownthat the technique can reduce total computation time.
IV. O PTICAL R ECONSTRUCTION OF P OLYGON -BASEDH IGH -D EFINITION CGH S
Optical reconstruction of some CGHs, created by using
wave-field rendering, is shown in this section.
A. Moai II
Figure 7 shows photographs of the optical reconstructionof a CGH, named Moai II. As shown in Fig. 8, the 3D

(a) Left
Figure 11.

(b) Center(c) Right

Photographs of the optical reconstruction of Aqua II. Photographs (a) – (c) are taken from different viewpoints.

Sphere ofpolygonpoint-based methods, because the stars in the 3D scene areexpressed by point sources of light. The number of pixels
is 131, 072 × 65, 536 ≃ 8 G pixels. The pixel pitches are
0.8 µm × 1 µm. Thus, the angle of the horizontal viewingzone is approximately 47◦ .
C. Aqua II

(a)

(b)

Textureimage

Figure 9. Optical reconstruction of The Moon (a) and schematic illustrationof its texture-mapping (b)

6.5

29

Units: cm
10.5
Figure 10.

The 3D scene of Aqua II.

scene of the Moai II includes two 3D objects and a background image. The 3D objects of the Moai II are renderedby using the flat shading and ordinary silhouette method.
The number of pixels and total number of polygons are
65, 536 × 65, 536 ≃ 4 G pixels and 2,440 polygons (frontface only). Since the pixel is a square with the dimensionsof 1 µm × 1 µm and the hologram is reconstructed by red
light, the angle of the viewing zone is approximately 37◦ inboth horizontal and vertical directions.
B. The Moon
The optical reconstruction of a CGH, named The Moon,is shown in Fig. 9 (a). In the CGH, the 3D object is asphere composed of 800 polygons (front face only). An
astrophotograph of the real moon is mapped as the texture
image onto the spherical surface, as shown in (b). Thisis also the example of a hybrid of polygon-based and

An example of a sparse 3D scene is shown in Fig. 10.
The CGH for the scene is named Aqua II, because thescene is modeled after an aquarium. The 3D scene includes
10 objects such as fish or seaweed, whose sizes are small
compared with the scene volume. If we used the ordinary
silhouette method, 10 times the numerical propagations
would be required for light-shielding. In the case where weassume that a PC with 16 processor cores and 96 G Bytes
memory is used for calculations and the CGH has the same
number of pixels as The Moon i.e. approximately 8 G pixels,
the computation time is estimated to be approximately 85
hours.
However, the actual computation time was 35.5 hours, because the new speed-up algorithm given in the section III-B
was used for the calculations. The optical reconstruction isshown in Fig. 11.
V. C ONCLUSIONS
Algorithms for wave-field rendering are proposed forcomputational holography. These polygon-based techniques
are developed to create high-definition CGHs that have
109 ∼ 1010 pixels in resolution and viewing angles over
45 degrees. The resulting holograms reconstruct true fine
3D images of occluded scenes, which create a strong sensation of depth due to motion parallax. As a result, thesecomputational holograms can be appreciated as works ofart.
ACKNOWLEDGMENTS
The author would like to thank Dr. S. Nakahara for his
efforts in fabricating CGHs. The author also appreciates
the mathematical formulation of the speed-up algorithm insparse 3D scenes, which was provided by Mr. M. Nakamura, a graduate student. The mesh data for the moai
object was provided courtesy of Yutaka Ohtake by the
AIM@SHAPE Shape Repository. This work was supportedby the JSPS.KAKENHI (21500114).

R EFERENCES[1] K. Matsushima and S. Nakahara, “Extremely highdefinition full-parallax computer-generated hologram
created by the polygon-based method,” Appl. Opt.,
vol. 48, pp. H54–H63, 2009. [Online]. Available:
http://ao.osa.org/abstract.cfm?URI=ao-48-34-H54[2] K. Matsushima, “Computer-generated holograms for threedimensional surface objects with shade and texture,” Appl.
Opt., vol. 44, pp. 4607–4614, 2005.
[3] R. Bräuer, F. Wyrowski, and O. Bryngdahl, “Diffusers indigital holography,” J. Opt. Soc. Am., vol. A8, pp. 572–578,
1991.
[4] K. Matsushima, “Formulation of the rotational transformationof wave fields and their application to digital holography,”Appl. Opt., vol. 47, pp. D110–D116, 2008.
[5] K. Matsushima, H. Schimmel, and F. Wyrowski, “Fast calculation method for optical diffraction on tilted planes by useof the angular spectrum of plane waves,” J. Opt. Soc. Am.,
vol. A20, pp. 1755–1762, 2003.
[6] K. Matsushima and T. Shimobaba, “Band-limited angular
spectrum method for numerical simulation of free-space
propagation in far and near fields,” Opt. Express,
vol. 17, pp. 19 662–19 673, 2009. [Online]. Available: http://www.opticsexpress.org/abstract.cfm?URI=oe-1722-19662[7] K. Matsushima and A. Kondoh, “A wave optical algorithm
for hidden-surface removal in digitally synthetic full-parallax
holograms for three-dimensional objects,” SPIE Proc. Practical Holography XVIII, vol. #5290, pp. 90–97, 2004.
[8] A. Kondoh and K. Matsushima, “Hidden surface removal in
full-parallax CGHs by silhouette approximation,” Systems andComputers in Japan, vol. 38, pp. 53–61, 2004.
[9] K. Matsushima, “Exact hidden-surface removal in digitally
synthetic full-parallax holograms,” SPIE Proc. Practical
Holography XIX and Holographic Materials XI, vol. #5742,
pp. 25–32, 2005.
[10] R. P. Muffoletto, J. M. Tyler, and J. E. Tohline, “Shifted Fresnel diffraction for computational holography,” Opt. Express,
vol. 15, pp. 5631–5640, 2007.
[11] K. Matsushima, “Shifted angular spectrum method for offaxis numerical propagation,” (to be published).