OSA Topical Meeting on Digital Holography and Three-Dimensional Imaging 2007, Vancouver, DWB4(2007.6).

Rotational transformation for reconstruction of digitalholography and CGH creation
Kyoji Matsushima
Department of Electrical Engineering and Computer Science, Kansai University, Yamate-cho 3-3-35, Suita, Osaka 564-8680, Japan.
matsu@kansai-u.ac.jp

Abstract: Rotational transformation allows the reconstruction of images on an arbitrarily
tilted plane in a digital holography and the creation of arbitrarily tilted polygon light sourcesin CGHs. Examples of transformation applications are demonstrated.
©2007 Optical Society of America
OCIS codes: (090.1760) Computer holography; (260.1960) Diffraction theory

1. Introduction
There are various formulations for spatial propagation or the diffraction of coherent light waves in a free space,including such well-known formulas as Fresnel-Kirchhoff integral, Rayleigh-Sommerfeld integral, and Fresnel andFraunhofer diffraction. However, these share a common restriction: the reference screen in which light fields areobtained as a result of propagation must be parallel to the source plane in which the boundary condition is given.
Recently, we reported another formulation for the spatial propagation of lightyfrom a source plane to a reference plane [1]. The reference plane is allowed to beŷ
arbitrarily tilted to the source plane. We refer to this formula as rotationalx̂transformation, which is a useful technique in every field of optical engineering,
Source plane
especially in digital holography and computer-generated holograms (CGH).
xg ( x, y,0)
A technique of point sources of light is often used to calculate object fields
emitted from virtual objects in CGHs for display purposes. This technique isvery simple but time-consuming because a gigantic number of point sourceszmust be handled to create an object with a certain dimension. Rotationaltransformation provides another technique: polygon sources of light instead ofpoint sources [2]. The transformation also solves the hidden-surface removal
problem of full-parallax CGHs [3]. Furthermore, rotational transformation canf ( xˆ, yˆ ,0)be straightforwardly applied to digital holography. By using this technique, one
can reconstruct images on arbitrarily tilted planes from complex-valued images
Reference planeacquired using digital holography as if the plane is parallel to the image sensor.
Fig. 1. Definition of source and
2. Brief proof that rotational transformation satisfies Helmholtz equationreference planes

Since discussion has been published on rotational transformation based on the physical interpretation of Fourier
frequencies as a wave vector [1], in this report let us describe another proof based on the Helmholtz equation.
Suppose that r = ( x, y, z ) and ν = (u , v, w) are the source coordinates system and Fourier frequencies. Reference
coordinates system rˆ = ( xˆ , yˆ , ẑ ) shares its origin with source one but is tilted, as shown in Fig. 1. In addition, theFourier frequencies in reference coordinates are νˆ = (uˆ , vˆ, wˆ ) . Note that Fourier frequencies are not independent ofeach other, i.e., w = w(u , v) = (λ−2 − u 2 − v 2 )1 / 2 , wˆ = wˆ (uˆ , vˆ) = (λ−2 − uˆ 2 − vˆ 2 )1 / 2 . Coordinates rotation is given asrˆ = Tr and r = T −1rˆ using rotation matrix T. When distribution of the complex amplitude of a light field are
given by g(x, y, 0) in the source plane (x, y, 0), complex amplitude g(x, y, z) in the source coordinates are given by
the theory of angular spectrum of plane waves [4] as follows:g ( x, y, z ) = F −1{G (u, v) exp[i 2πw(u, v) z ]} = ∫∫ G (u, v) exp[i 2π ( xu + yv + zw(u, v))]dudv,(1)G (u, v) = F{g ( x, y,0)} = ∫∫ g ( x, y,0) exp[−i 2π (ux + vy)]dxdy,

where F and F−1 are Fourier and inverse Fourier transformations, respectively. It is proved that this g(x, y, z) is astrict solution of the Helmholtz equation.
When a rotation matrix is provided, the complex amplitude f ( xˆ , yˆ ,0) on reference plane ( xˆ , yˆ ,0) is written as

OSA Topical Meeting on Digital Holography and Three-Dimensional Imaging 2007, Vancouver, DWB4(2007.6).

⎛ a1 a2 a3 ⎞⎟⎜T ≡ ⎜ a 4 a5 a 6 ⎟(2)= ∫∫ G (u , v) exp[i 2π {(a1u + a 4 v + a7 w) xˆ + (a 2u + a5 v + a8 w) yˆ }]dudv,⎜a a a ⎟89⎠
⎝ 7t−1Since T= (Τ ) for any rotation matrix in general, by substituting a1u + a4 v + a7 w = uˆ , a2u + a5 v + a8 w = vˆ , andf ( xˆ , yˆ ,0) = g (a1 xˆ + a 2 yˆ , a 4 xˆ + a5 yˆ , a7 xˆ + a8 yˆ )−1

dudv = J (uˆ , vˆ) duˆdvˆ into Eq. (2), the complex amplitude in the reference plane is rewritten as
F (uˆ , vˆ) = G (a1uˆ + a2vˆ + a3 wˆ (uˆ , vˆ), a4uˆ + a5vˆ + a6 wˆ (uˆ , vˆ)),f ( xˆ , yˆ ,0) = F−1(3){F (uˆ, vˆ) J (uˆ, vˆ) },(4)J (uˆ , vˆ) ≡ (a2 a6 − a3a5 )uˆ / wˆ (uˆ , vˆ) + (a3a4 − a1a6 )vˆ / wˆ (uˆ, vˆ) + (a1a5 − a2 a4 ).

When the parallax condition is fulfilled in the source or reference coordinates, i.e., u , v << w or uˆ , vˆ << wˆ , Eq.
(4) can be reduced to f ( xˆ , yˆ ,0) ∝ F −1 {F (uˆ , vˆ)} . In summary, complex amplitude in the source plane are
Fourier-transformed, and then the coordinates rotation of Eq. (3) is performed in Fourier space. Finally, the rotatedspectrum is inversely Fourier-transformed to complex amplitude in real space.
3. Reconstruction of tilted planar object in digital holography
Light emitted from a planar object slanted at θ is recorded by lensless-Fourier phase-shifting digital holography [5],as shown in Fig. 2. In this case, g(x, y, 0) is the recorded complex amplitude and f ( xˆ , yˆ ,0) is the reconstructed
amplitude image on the tilted plane. Fig. 3 shows the numerical procedure for reconstruction. Spectrum G(u, v) ismultiplied by phase factor p(d ) = exp[i 2πw(u , v)d ] for translational propagation prior to coordinates rotation R ofEq. (3).
Experimental results are shown in Fig. 4. The number of pixels of the image sensor used for recording is
2000×2000, and the sensor pitch is 6.0×6.0 μm. λ = 532 nm. The distance between the object and image sensor isapproximately 19.5 cm. Since lensless-Fourier type digital
Coordinates
holography is used, FFT is necessary for obtaining theFourier TranslationalRpropagationrotation
space
distribution of complex amplitude. The number of sampling-1
and sampling pitches of the reconstructed images areFFP(d)Real2048×2048 and 8.1×8.1 μm, respectively. The planar object
space
Complexis slanted at approximately 70°. Fig. 4(a) is a close-upg ( x, y )f ( xˆ, yˆ )amplitude
photograph of the planar object and (b) is reconstruction
only using backward translational propagation. The whole Fig. 3. Numerical procedure for reconstruction of tilted
object is not emerged due to the depth of focus in bothplanar object(a)(b)ŷx̂θImage sensor

yPlanar object(c)zxFig. 2. Geometry for recording and numericalreconstruction of tilted planar object

Fig. 4. Photograph of object (a), its numerical reconstruction
without (b) and with (c) rotational transformation

OSA Topical Meeting on Digital Holography and Three-Dimensional Imaging 2007, Vancouver, DWB4(2007.6).

Polygon #1

Local
coordinatesyg1(x, y, 0)zx

Fourier Coordinatesrotation
space
Cubicobject^yRealspace13Fg ( x, y )2RTranslational
propagation

FP(d)Complexamplitude-1f ( xˆ, yˆ )Fig. 6. Numerical procedure for polygon source of light^x^zGlobal coordinates

Fig. 5. Example of an object model.

images, while all characters on the object clearly appear in Fig. 4(c) as a result of rotational transformation at θ =71.8°.
4. Creation of polygon light source in CGHs
When hidden surfaces are ignored, a cubic object can be resolved into three polygons, as shown in Fig. 5. In CGHs
created using rotational transformation, each polygon is regarded as a surface source of light. Total object waves arecalculated on the hologram by superimposing object waves emitted from polygons on each other.
Figure 6 shows the procedure for calculating object waves from a polygon. In this case, g(x, y, 0) is given forevery polygon in the local coordinates, which are also defined for each polygon. An example of local coordinates isshown in Fig. 5. g(x, y, 0) is designed so that its amplitude distribution forms a polygon and its phase distribution
plays the role of a diffuser [2]. This g(x, y, 0) is referred to as the property function of the polygon.
The property function of polygon #1 is shown in Fig. 7(a) as an example. The local coordinates of the property
function are rotated to be parallel to the hologram after performing FFT (Fig. 7(c)), and then the complex amplitudeis calculated by inverse FFT after translational propagation in Fourier space.
An example of the optical reconstruction of a CGH created by these procedures is shown in Fig. 8. The numberof pixels is 8192×4096, and pixel pitches are 2×4 μm. λ = 633 nm.
5. Conclusion
Rotational transformation strictly satisfies Helmholtz’s equation and makes it possible to calculate the distributionof complex amplitude on arbitral planes not parallel to the source plane. This technique can be applied to both
digital holography and CGHs.
6. References[1] K. Matsushima, H. Schimmel, and F. Wyrowski, “Fastcalculation method for optical diffraction on tilted planes
by use of the angular spectrum of plane waves,” J. Opt.
Soc. Am. A20, 1755–1762(2003).
[2] K. Matsushima, “Computer-Generated Holograms for
Three-Dimensional Surface Objects with Shade andTexture,” Appl. Optics 44, 4607–4614(2005).
[3] K. Matsushima, “Exact Hidden-Surface Removal in
Digitally Synthetic Full-Parallax Holograms”, in Practical
Holography XIX and Holographic Materials XI, SPIE
Proc. #5742, (SPIE, 2005), pp.25–32.
[4] J. W. Goodman: Introduction to Fourier Optics 2nd ed.,
(McGraw-Hill, 1996) Chap. 3.10.
[5] T. Nakatsuji and K. Matsushima, “Acquisition ofinformation-rich images using synthetic-aperture digitalholography,” to be presented at the conference on Digital
Holography and Three-Dimensional Imaging.

Amplitude Phase(a)g1(x,y,0)(b)
G1(u,v)(c)^ ^F1(u,v)(d)^ ^| f1(x,y,0)|(d = 0)6mm
Fig. 8. Optical reconstruction ofCGH
createdbypolygon light sources

Fig. 7. Process of creating polygon #1