Proceedings of ICO Topical Meeting on Optoinfomatics/Information Photonics 2006, pp.99–100,
September 4-7, 2006, St. Petersburg Russia

PERFORMANCE OF THE POLYGON-SOURCE METHOD FOR CREATINGCOMPUTER-GENERATED HOLOGRAMS OF SURFACE OBJECTS
K. Matsushima, Department of Electrical Engineering and Computer Science, Kansai University,
Yamate-cho 3-3-35, Suita, Osaka 564-8680, Japan. E-mail: matsu@kansai-u.ac.jp
Session “In Memoriam of Emmett Leith: Enlightening the World of Holography”, Poster
The polygon-source method was implemented and its performance was measured. The method
was proposed by the author for fast computation of object fields from virtual surface objects
and creation of computer-generated holograms (CGHs).
1. Introduction
A method based on point sources of light has been successfully used for numerical calculation ofobject fields from virtual objects in CGHs for display purposes. However, the point source method is too
slow to create CGHs of surface objects especially for full-parallax holograms, because the surface density of
100 points/mm2 or more is required for forming point sources into a surface. The author has proposed usingsurface sources of light for surface objects instead of point sources1, 2. In the method, an object is composedof small planar surfaces, polygons. A polygon, illuminated by virtual source of light, reflects the light andbehaves itself as a surface source of light, i.e. the polygon source. The light from the polygon source iscalculated by rotational transformation of wave fields using FFT3. The rotational transformation also plays animportant role in a case occlusion occurs4, i.e. in hidden-surfaces removal5.
2. Computational complexity
The rotational transformation is the core of the proposed method and most likely consumes almost allcomputation time. Fig. 1 shows a diagram of computational process of the transformation. The complex
amplitudes s(x, y), given in local coordinates defined for each polygon, is called surface property function,
which keeps property of the polygon such as the shape, texture or diffusiveness. hˆ( xˆ , yˆ ) is also complex
amplitudes given in global coordinates parallel to the hologram. FFT operation (F) must be executed twice
for the rotational transformation of a polygon. Furthermore, it is required for each polygon that coordinatesrotation (R) including interpolation and multiplication by a phase factor P(d) for translational propagation in
Fourier space.

Fourier Coordinates R
space rotation
RealspaceFs ( x, y )Translational
propagationP(d )Complexamplitude-1Fhˆ( xˆ, yˆ )Fig.1 The diagram for calculating the object fieldfrom a polygon source of light
3. Performance
Optical reconstruction of the hologram of a cube is shown in Fig.2 as an example of the proposed
method. The hologram has 3.2 M (≈8192×4096) sampling points. Fig.2 (a)–(b) show amplitude patterns of
the polygons 1 to 3, calculated around the polygons, respectively. The light emitted from three polygons, ofwhich dimensions are 6 mm square, are separately calculated and totaled on the hologram plane.
Measured total computation time was 92 sec in executing by single CPU of Xeon 3.0 GHz. Fig.3
shows the ratios of the CPU time consumed for each numerical process. In the case of the object shown in
Fig. 2, FFT operations are executed 8 times and were supposed to be the most time-consuming process.
However, measured longest numerical process is the coordinate rotation accompanied with interpolation,
which accounts for 44 percent of all.

Proceedings of ICO Topical Meeting on Optoinfomatics/Information Photonics 2006, pp.99–100,
September 4-7, 2006, St. Petersburg Russia

Hologram
8192 x 4096 pixel(2µm x 4µm)(c) Polygon 38192 x 2048 pixel(2µm x 4µm)

31 26mm(b) Polygon 24096 x 4096 pixel(2µm x 4µm)(a) Polygon 14096 x 2048 pixel(2µm x 4µm)

Fig.2 An example of optical reconstruction of a surface object; a cube,and calculated amplitudes of fields (a)–(b), emitted from each polygon.

Total CPU time: 92 sec
Another
24 %Translational
propagation
1%

FFT
31 %

Coordinates rotation(interpolation)
44 %Fig.3 The ratios of CPU time consumed for each numerical process.
4. Conclusion
Performance of algorithm for calculating object fields emitted from polygon sources of light wasmeasured for fast creation of full-parallax computer-generated holograms of surface object. In the algorithm
FFT operations are required twice for calculating the field from a polygon. However, the measured ratio ofCPU time suggests that the bottle neck of the algorithm is not in FFT but in interpolation. Algorithm of FFT
is sufficiently studied and tuned up, while interpolation is not. Therefore, it is every possibility of further
reducing the CPU time.
1.
2.
3.
4.
5.

K. Matsushima: “Computer-Generated Holograms for Three-Dimensional Surface Objects with Shadeand Texture”, Appl. Opt., 44, 4607-4614(2005).
K. Matsushima: “Development of the Surface Diffraction Method for Computer-Generated Display
Holograms”, 7th Intern. Symposium on Display Holography, North Wales, (2006) in press.
K. Matsushima, H. Schimmel, F. Wyrowski: “Fast Calculation Method for Optical Diffraction on Tilted
Planes Using the Angular Spectrum of Plane Waves”, J. Opt. Soc. Am., A20, 1755-1762(2003).
K. Matsushima: “Fast Creation of Digitally Synthetic Holograms of Surface Objects by Use of a WaveOptical Method”, ICO Conf. on Optics & Photonics in Technology Frontier, Tokyo, 467-468(2004).
K. Matsushima: “Exact Hidden-Surface Removal in Digitally Synthetic Full-Parallax Holograms”,
Practical Holography XIX and Holographic Materials XI, SPIE Proc. #5742, San Jose, 25-32(2005).