OSA Topical Meeting on Digital Holography and Three-Dimensional Imaging 2007, Vancouver, DTuC5 (2007.6).

Acquisition of information-rich images usingsynthetic-aperture digital holography
Tatsuya Nakatsuji

Kyoji Matsushima

Department of Electrical Engineering and Computer Science, Kansai University, Yamate-cho 3-3-35, Suita, Osaka 564-8680, Japan.
nakatsuji@laser.ee.kansai-u.ac.jp

Abstract: A synthetic-aperture technique is applied to the acquisition of information-rich images
in phase-shifting digital holography. Since the images have a large viewing zone, diverse
reconstructions can be obtained in various manners.
©2007 Optical Society of America
OCIS codes: (090.1760) Computer holography; (100.6890) Three-dimensional image processing

1. Introduction
Rich images of objects, accompanied with phase information, can be captured by digital holography, especiallyusing the phase-shifting [1] technique. Such information-rich images will find many applications in the future.
However, generally, there are severe limitations on the size and position of objects in current digital holography due
to the low resolution of current image sensors. For example, if the dimension of an object is several centimeters, itmust be placed several meters from the image sensor. Since object size w is larger, the distance from image sensor dmust be longer (Fig. 1). Although almost all information on the object light can be recorded as a complex-valued
image, almost all recorded light comes from the same direction. In this situation, utilizing the rich information
contained in the image is difficult. Therefore, increasing the angle of visual field θ is a significant problem in digitalholography for display purposes. To increase angle θ, distance d must be reduced, which directly leads to increasing
the angle of viewing zone Ω. As a result, reduction of distance d increases information contained in the captured
images, such as object shapes observed from various angles.
Constraints on size w and distance d are relaxed in the lensless Fourier setup [2] because spatial frequency on thesensor in the setup is low compared with the Fresnel scheme. Therefore, we adopt a lensless-Fourier phase-shifting
setup. We also theoretically analyzed the maximum spatial frequency of the interference fringes on the sensor andrepresent the maximum frequency as functions of w and distance d. As a result, the area the object can occupy isdefined, i.e., if an object is placed within the area, numerical reconstruction can be obtained without aliasing error.
Therefore, we refer to this as the “alias-free area”. Information-rich images with largest angle θ are recorded using
the alias-free area.
A lensless-Fourier setup causes another big advantage. In this setup, maximum spatial frequency on the sensor
does not depend on sensor size D. Enlarging sensor size also directly leads to increasing angle Ω. Therefore, weintroduced a synthetic aperture technique [3] into a lensless-Fourier phase-shifting setup to equivalently yield large
sensor size.
In this report, we present the details of adopted techniques introduced to acquire images including richthree-dimensional information with a wide viewing zone and visual field. We also demonstrate numericalreconstructions of an object with a smooth change of visual points to prove extension of the viewing zone.
Furthermore we demonstrate numerical reconstruction in the tilted plane by rotational transformation [4, 5].
Image sensor
Laser (λ=532 nm)

MirrorImage sensor x

ObjectΩw

Beam splitter

θD

ObjectSpatial filterdMirror
Fig. 1

Definition of visual field and viewing zonezone.

Piezo phase shifterFig. 2 Experimental setup

OSA Topical Meeting on Digital Holography and Three-Dimensional Imaging 2007, Vancouver, DTuC5 (2007.6).

2. Experimental setup for synthetic-aperture digital holography
A setup for synthetic-aperture digital holography is shown in Fig. 2. The output beam of a single-mode DPSS laseris split into two arms, and a Piezo phase shifter is inserted in the reference arm. The number of pixels of the imagesensor (Toshiba Terry Corp. CSB4000CL-10A) is 2000×2000 [pixel] and its sampling pitch is 6.0×6.0 [μm]. Thesensor is put on a linear stage to move it in the x-axis direction. Fringe patterns are captured three or four times withdifferent reference phases for each sensor position. A complex-valued image is composed of these captured fringe
patterns based on the principle of phase-shifting digital holography. The sensor is moved within a distance smaller
than sensor size to generate overlap between adjacent images. The exact relative position between adjacent images isprovided using a correlation function between them.
3. Alias-free area and recording big objects
In the lensless-Fourier setup, the reference light is aspherical wave emitted from a point source of light.
Assume that the point source and an object point areplaced at distance d from the image sensor and the point
source is placed on the optical axis while the object pointis placed at distance w from the optical axis. Themaximum spatial frequency of the interference fringe on
the image sensor is given asw/ 2,(1)
f =2
λ d + ( w / 4) 2
where λ is the wavelength. When the sensor pitch is δ, thismaximum frequency must satisfy Nyquist's theorem 2 f ≤ δ −1 . Whenthe position of all object points fulfills this relation, a captured image
has an alias-free property. This alias-free area forms a pyramid in theobject space, and its apex is adjacent to the image sensor, as shown inFig. 2. Any object placed inside the alias-free area is reconstructedwithout aliasing errors.
The numerical reconstruction of a comparatively big object, a modelof a penguin 3.0×4.5 [cm], is shown in Fig. 4. The object is placed 60
cm from the image sensor. Fig. 4 is simply the Fourier transform of thecomplex image calculated from four fringe patterns captured at thesame sensor position (synthetic-aperture technique is not used in thiscase).

Alias-free area

Image sensor
d1
d3
d2

Objects
Fig. 3 Alias-free area in object space

4.5[cm]3[cm]Fig. 4 Numerical reconstruction of bigobject

4. Numerical reconstruction from different angles
The light emitted from objects, as arranged and shown on the left side of Fig. 5, is captured as a merged complex
image g(x, y), composed of five images using the synthetic-aperture technique. In a lensless-Fourier setup,distribution of complex amplitude f (x, y) of the object light is obtained by inverse Fourier transform of g(x, y), asshown in Fig. 5. f (x, y) is given on a plane at distance dR from the sensor, where dR is the distance between thereference point source and the image sensor. We numerically propagate f (x, y) forward for distance d≈dR andcalculate complex amplitude distribution fd(x, y) near theInverse Fourier transform
sensor to change the point of view.
g(x, y)f (x, y)6mm
On distribution fd(x, y), a rectangle pupil whose size andx
position is given by (px, py) and (xe, 0), respectively, isBackward
defined as shown in Fig. 6. f′d(x, y) is a new distribution
propagation f′ｄ (x, y)made by clipping fd(x, y) inside and padding zero outside thepupil. Finally, we again propagate f′d(x, y) backward nearfｄ (x, y)
the object and calculate its amplitude distribution.
30 mm
Forwardpropagation
Examples of the numerical reconstruction of a merged
Objects
Sensorcomplex image g(x, y) recorded at distance dR = 20 [cm] aredR
shown in Fig. 7. The number of pixels of g(x, y) isFig. 5 Procedure for numerical reconstruction
7344×2011, and its dimension is 44.1 × 12.1 mm.
from different points of view
Numerical reconstruction from the left and right points ofOSA Topical Meeting on Digital Holography and Three-Dimensional Imaging 2007, Vancouver, DTuC5 (2007.6).

view is shown in (a) and (b), respectively. The pip of "6" on a die is hiding behind another die in (a), while it clearly
appears in (b).
yxe

5. Numerical reconstruction on a tilted plane
As shown in Fig. 8, planar objects A and B are arranged on the sameplane slanted approximately 70[º] to the sensor. The merged complex
image is composed of 5 individual images captured in dR = 19.5 [cm].
We numerically propagated the merged image to an intermediatedposition between A and B, as shown in Fig. 9(a), and then calculated
the numerical reconstruction on the tilted plane by performing
rotational transformation at 69[º], as shown in Fig. 9(b).

f′ｄ (x, y)
pyx
px
Fig. 6 Definition of pupil

6. Conclusion
In this study, the expansion of a visual field was made byutilizing the alias-free area in lensless-Fourierphase-shifting digital holography. Furthermore, theviewing zone is successfully expanded by using thesynthetic-aperture technique. As a result, information-rich
images were acquired that allowed numericalreconstructions to be obtained from arbitrary points ofview and on an arbitrary tilted plane.

(a)

(b)

7. References[1] I.Yamaguchi and T. Zhang, “Phase-shifting digital holography,”Opt. Lett. 22, 1268-1270 (1997).
Fig. 7 Numerical reconstructions from different[2] C.Wagner, S. Seebacher, W. Osten, and W. Juptner, “Digital
angles: (px, py) = (1.7, 3.5) [mm],recording and numerical reconstruction of lensless Fourier holograms in(a) xe = −9.83 [mm], (b) xe = 9.83 [mm]optical metrology,” Appl. Opt. 38, 4812-4820 (1999).
[3] R. Binet, J. Colineau, and J. Lehureau,“Short-range synthetic aperture imaging at 633 nm
Aby digital holography,” Appl. Opt. 41, 4489-4496(2002).
B[4] K. Matsushima, H. Schimmel, and F. Wyrowski,“Fast calculation method for optical diffraction ontilted planes by use of the angular spectrum of plane
Awaves,” J. Opt. Soc. Am. A20, 1755-1762 (2003).
θ≈70°[5] K. Matsushima, “Rotational transformation forsensor
reconstruction of digital holography and creation ofCGH,” to be presented at the conference on Digital
Fig. 8 Planar objects on tilted plane
Holography and Three-Dimensional Imaging.

(a)

B(b)
B

θ=69°A

sensorFig. 9

Numerical reconstruction of planar objects on tilted plane (b) and its
procedure (a)