Free-viewpoint images captured using phase-shifting
synthetic aperture digital holography
Tatsuya Nakatsuji and Kyoji Matsushima*
Department of Electrical and Electronic Engineering, Kansai University, Yamate-cho 3-3-35, Suita, Osaka 564-8680, Japan*Corresponding author: matsu@kansai-u.ac.jp
Received 15 October 2007; revised 6 February 2008; accepted 6 February 2008;
posted 7 February 2008 (Doc. ID 88556); published 18 April 2008

Free-viewpoint images obtained from phase-shifting synthetic aperture digital holography are given forscenes that include multiple objects and a concave object. The synthetic aperture technique is used toenlarge the effective sensor size and to make it possible to widen the range of changing perspective in the
numerical reconstruction. The lensless Fourier setup and its aliasing-free zone are used to avoid aliasing
errors arising at the sensor edge and to overcome a common problem in digital holography, namely, anarrow field of view. A change of viewpoint is realized by a double numerical propagation and by clipping
the wave field by a given pupil. The computational complexity for calculating an image in the given perspective from the base complex-valued image is estimated at a double fast Fourier transform. The experimental results illustrate the natural change of appearance in cases of both multiple objects and aconcave object. © 2008 Optical Society of America
OCIS codes:
090.1995, 110.6880.

1. Introduction

The explosive evolution of computer technology has
increasingly accelerated digital signal processing(DSP) in the field of optics. A great deal of memory
is required for handling light wave fields correctly.
However, the amount of memory installed in personal computers increases from year to year, with theresult that desktop computers with high-performance CPUs and gigabytes of memory enable us
to handle wave fields sampled over a certain areaof the cross section of field propagation.
This yields the possibility that various effects forwave fields, created by optical components such as
lenses, mirrors, prisms, or specific optical devices,can be replaced by DSP. Digital holography provides
an acquisition system of wave fields emitted fromreal objects and is therefore the most important technique for the DSP of light. For example, the focusing
device in imaging equipment, composed of various
lenses controlled mechanically, can be replaced by
DSP. As a further example, images on planes that0003-6935/08/19D136-08$15.00/0© 2008 Optical Society of America
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are almost perpendicular to the image sensor can
be clearly reconstructed by using a specific computational process of captured wave fields [1].
As the phase distribution of object fields captured
by digital holography is also useful for shape measurement and deformation analysis, much of the literature on digital holography focuses on this field,
especially on holographic microscopy [2]. In thispaper, however, we explore the possibility of digitalholography as an acquisition method for free-viewpoint images of scenes with self- and mutual occlusion. Here, the term occlusion, widely used in thefield of computer graphics and three-dimensional(3D) imaging, describes the situation in which an object further from the viewpoint is hidden behind
other objects closer to the viewpoint. Occlusion is
one of the most important mechanisms in the perception of 3D scenes. Free-viewpoint images of occluded
objects have been reported in some methods and devices based on ray optics, such as synthetic aperture
integral imaging [3,4]. As far as we know, however,the natural reconstruction of 3D scenes withoccluded objects has not yet been reported in digitalholography. In computer-generated holograms fordisplay purposes, dealing with occlusion is a problem

of hidden-surface removal, and methods such as thelayered hologram [5], ray tracing [6], the silhouettemask [7,8], and the tilted mask [9] have been reported. Superimposing occlusion into the wave field
captured by using digital holography has also been
proposed for 3D scene reconstruction [10].
The reason that natural reconstruction of occlusion is difficult in digital holography is that the resolution of image sensors currently available is too low
and the sensor size too small to capture 3D scenes
with occlusion. Low spatial resolution of the image
sensor restricts the geometry of the object space; i.
e., small objects must be placed far from the sensor
to maintain the aliasing-free condition required by
Nyquist’s theorem. As the object distance d increases
with the increasing lateral extent of the object w as
shown in Fig. 1, the angle of the visual field θ is generally small in digital holography. Furthermore, agreater object distance leads to a smaller angle φas illustrated in Fig. 1. This angle, defined by the distance d and the size of the image sensor W, is referred
to as the viewing-zone angle by analogy to a 3D display. In digital holography, typical values for thevisual field and viewing-zone angles are approximately ∼2° and ∼1°, respectively (assuming a sensor
pitch of 10 μm, sensor size of 20 mm, and object distance of 1 m). Therefore, numerical reconstruction ofa different perspective is not easy.
To overcome these problems and create free-viewpoint images in digital holography, we adopt threetechniques: synthetic aperture [11,12], phase shifting [13], and a lensless Fourier setup [14,15]. In thisinvestigation, the synthetic aperture technique isused to extend the effective sensor size. This technique was first proposed to improve the spatial resolution in centimeter band radars. In the same vein,
superresolution has been reported when the synthetic aperture technique is used in digital holography [11,12]. However, our reason for using thetechnique differs from that of superresolution, andthe object size captured is considerably larger thanthat given in previous reports.
The phase-shifting lensless Fourier setup and thealiasing-free zone presented in this paper make itpossible to maximize the visual field and minimize
the object distance. These techniques, which use aspherical reference wave, also play an important rolein avoiding the aliasing error arising at the sensor

edge, because the fringe frequency is almost constantover the sensor surface in a spherical reference wave,
unlike that in a plane wave. As a result, the viewingzone angle can be increased from φ to φ0 by extending
the sensor area, as shown in Fig. 1.
Our final aim is varied digital processing of opticalwave fields. The wave field must be captured in awide cross section of field propagation to accomplish
this. Numerical reconstruction of free-viewpoint
images is an example of the DSP of optical wave
fields.

Fig. 1. Definition of the visual field in capturing wave fields andthe viewing zone in numerical reconstruction.

Fig. 2. (Color online) Definition of the coordinate systems andschematic geometry used in numerical reconstruction.

2.

Reconstruction of Free-Viewpoint Images

The coordinate system defined for the numerical reconstruction of digital holograms is shown in Fig. 2.
The plane ðx; y; 0Þ is referred to as the reconstruction
plane on which a point light source that emits thereference wave is placed and on which wave fields
are obtained. The surface of the image sensor isplaced at a distance dR from the reconstruction
plane.
A.

Numerical Reconstruction in a Lensless Fourier Setup

Suppose that the wave field of the object wave is given by f ðx; y; 0Þ in the reconstruction plane andOðxs ; ys Þ on the surface of the image sensor. Whenthe point source is at the origin, the reference wave
is represented by the normalized parabolic phasefunction as follows:
x2s þ y2sRðxs ; ys Þ ¼ exp ik
;
2dR


ð1Þ

where k ¼ 2π=λ is the wavenumber and λ is the wavelength. The product Oðxs ; ys ÞR ðxs ; ys Þ is captured
by using a phase-shifting technique and is represented by using a Fresnel transform of f ðx; y; 0Þ as
follows:
OR ðxs ; ys Þ ¼kf ðx; y; 0Þ exp iðx2 þ y2 Þ
2dR


2π
ðxxs þ yys Þ dxdy
× exp iλdRZ Z

¼ F ff ðx; y; 0Þϕ ðx; yÞgu¼xs =λdR ;v¼ys =λdR ; ð2Þ

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k
2
2
ϕðx; yÞ ¼ exp iðx þ y Þ ;
2dR

ð3Þ

where F f·g represents the Fourier transform and u
and v are Fourier frequencies with respect to x and y,
respectively. The ϕðx; yÞ represents a phase correction factor. The wave field of the object is obtainedby rearranging Eq. (2) after the Fourier transform:f ðx; y; 0Þ ¼ F fOR ðxs ; ys Þgus ¼x=λdR ;vs ¼y=λdR × ϕðx; yÞ;ð4Þ
where us and vs are Fourier frequencies with respect
to xs and ys , respectively.
In the numerical calculation, the sampling pitchesof the wave field f ðx; y; 0Þ are given by
Δx ¼

λdR
;
N x δxΔy ¼

λdR;
N y δyð5Þwhere δx and δy are the sensor pitches and N x and N y
are the number of pixels in the sensor used for capturing the interference fringe.
B. Numerical Reconstruction of Different Perspectives

The amplitude image jf ðx; y; 0Þj gives a reconstructed
image of the object. However, the viewpoint of the object cannot be changed in the image jf ðx; y; 0Þj. To
give the image the ability to vary the viewpoint,
the wave field f ðx; y; 0Þ should be propagated numerically to a plane around the image sensor. This is theforward propagation shown in Fig. 3.
The angular spectrum of plane waves method [16]is used for this numerical propagation; i.e., the wave
field in the plane at z ¼ dP is given byf ðx; y; dP Þ ¼ P dP ff ðx; y; 0Þg¼ F 1 fF 0 ðu; vÞ exp½i2πðλ2  u2  v2 Þ1=2 dP g;ð6Þ

where P d f·g represents the propagation operator at a
distance d and F 0 ðu; vÞ ¼ F ff ðx; y; 0Þg is the spectrum in the reconstruction plane. Note that the sampling pitches of the wave field are not changed in the

numerical propagation by Eq. (6). Therefore, the propagation distance dP should be selected carefully sothat the extent of the wave field is not greater than
the sampling area at z ¼ dP . If the distance dP is too
great, degradation arises in the final reconstructed
images as a result of the aliasing error in the numerical calculation of Eq. (6).
The wave field f ðx; y; dP Þ is the base complexvalued image for yielding a series of images of theobject from different viewpoints. Changing the viewpoint is done simply by multiplying the base complex
image by a pupil function as follows:^f ðx; y; dP ; xe ; ye Þ ¼ f ðx; y; dP Þpðx; y; xe ; ye Þ:

The definition of the pupil function used in this report is shown in Fig. 4. The pupil is a rectangle pupil,the size and position of which are given by px × py andðxe ; ye Þ, respectively.
The wave field ^f ðx; y; dP ; xe ; ye Þ clipped by the pupil
is propagated backward as follows:^f ðx; y; 0; xe ; ye Þ ¼ P d f^f ðx; y; dP ; xe ; ye Þg;B

3.

Capturing Object Waves

A.

Experimental Setup

The lensless Fourier setup in synthetic aperture digital holography is shown in Fig. 5. The output of thesingle-mode diode-pumped solid-state laser with awavelength of 532 nm is split into two paths. The
piezo phase shifter and the spatial filter are inserted
into the reference path to form a phase controlled
spherical reference wave. Here, dR is the distance between the reference point source and the image sensor, as defined in Fig. 2. The object is also irradiatedby a spherical wave formed by the objective lens. The
object wave and the reference wave are combined in
the cube beam splitter and yield an interference

py

xe

pz

ye
x

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ð8Þ

where dB ð≥ 0Þ is the distance of backward propagation chosen to adjust the focus. The amplitude imagej ^f ðx; y; dP  dB ; xe ; ye Þj is the final image from theviewpoint at ðxe ; ye ; dP Þ.

yFig. 3. (Color online) Principle of numerical reconstruction fromdifferent viewpoints.

ð7Þ

Fig. 4. Definition of a pupil.

fringe on the sensor surface. A CMOS-type image
sensor is used to capture the fringe pattern. The
number of pixels in the image sensor is 2000 ×2000 pixels, and its sampling pitch is 6:0 μm × 6:0 μm.
As the image sensor is moved translationally inthis experiment, instead of rotating the object asin [11], the sensor is installed on a motor-controlled
translational stage, and its position is shifted perpendicular to the optical axis. The resolution and accuracy of the motor-controlled stage are 2 and 15 μm,
respectively. The distribution of the complex amplitude is obtained by a four-step phase shift in thisexperiment. Four fringe patterns are therefore captured for each position of the sensor, and then a complex-valued image is composed of these captured
fringe patterns by using the phase-shifting digitalholography technique [13].
B. Aliasing-Free Zone

To maximize the visual field in the lensless Fourier
setup, we use a technique referred to as the aliasingfree zone. The aliasing-free zone is a safe zone defined in the object space. When an object placed inside the aliasing-free zone is captured by the lensless
Fourier setup, the reconstructed images are not degraded by the aliasing error.
The maximum fringe frequency should be estimated in a lensless Fourier setup to define the aliasing free-zone. Figure 6 shows the theoretical model
used in the estimation. For simplicity, we ignore
the y axis and consider the 2D coordinates ðx; zÞ inthe estimation that follows. Furthermore, it is assumed that the point object P is on the same plane
as the reference point source R and placed at a distance w=2 from the optical axis. When point S is on
the sensor surface and Δθ is the angle between line
segments PS and RS, the wavelength of the fringe
caused by the interference between the point sources
P and R is given as
2π
Λ¼;
jkP  kR j

ð9Þwhere kP and kR are x elements of the wave vectors offields propagating from the point sources P and R to
the point S, respectively.

Fig. 5. (Color online) Experimental setup for capturing object
waves by phase-shifting synthetic aperture digital holography.

Fig. 6. Theoretical model for estimating the maximum spatial
frequency of the interference fringe on the sensor surface.

Obviously, the interference fringe has a maximum
frequency when the angle Δθ is at its maximum.
When the point S is at the position x ¼ w=4, Δθreaches the maximum angle. In this case, sincekR ¼ kP ¼ sinðΔθ=2Þ, the minimum value of thefringe wavelength is given byΛmin ¼λ;
2 sinðΔθ=2Þ

ð10Þ

where sinðΔθ=2Þ is given by the geometry as follows:
w=4
sinðΔθ=2Þ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ :2d þ ðw=4Þ2

ð11Þ

To avoid aliasing errors, the sensor pitch δ must
1
satisfy Nyquist’s condition 2Λ1
min ≤ δ . As a result,the object size w must be limited as follows:
4λdw ≤ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ :16δ2  λ2

ð12Þ

If the size of an object placed at a distance d is lessthan w given by the above relation, aliasing errors do
not arise. Therefore, the zone fulfilling condition (12)is referred to as the aliasing-free zone. This zone isshaped as a triangular pyramid in the object space
as shown in Fig. 7. We can obtain a maximum visual
field by placing an object inside the aliasing-free zone
and minimizing the distance from the image sensor
for the given object.

Fig. 7. (Color online) Aliasing-free zone.
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4. Experimental Results
A. Calibration of the Distance of the Reference Point
SourceA fairly large object was captured by the lensless
Fourier setup without the synthetic aperture technique in order to verify the visual field of the constructed capturing system. An example of theamplitude image jf ðx; y; 0Þj is shown in Fig. 8. The
dimension of the object is 3 cm × 4:5 cm. Since thealiasing-free zone is given by w ≤ 0:089d in the experimental setup, the minimum object distance is
50 cm. Therefore, the object was placed at approximately 60 cm from the sensor to allow for a marginof error.
The distance between the reference point source
and the image sensor dR affects both the sampling
pitch of the wave field in Eq. (5) and the phase correction factor in Eq. (3). Since the precision of numerical propagation is strongly affected by both, theaccuracy of dR is important. If dR is not accurate,
a clear final image cannot be obtained from the proposed method, because numerical propagation isused twice for generating the image from a given
viewpoint. We calibrated dR by exact comparison between the dimension of the actual object and the extent of the object on the reconstructed amplitude
image jf ðx; y; 0Þj. Exact sampling pitches are measured first by the comparison, and then an exact
dR is obtained from Eq. (5). Once the distance dR
is calibrated, it is no problem to replace and shift
the object within a certain range. In the results from
Fig. 8, the calibrated distance dR and sampling pitchof the wave field f ðx; y; 0Þ are 61:7 cm and 26:7 μm,
respectively.
An example of an amplitude image jf ðx; y; dP Þj ofthe propagated wave field is shown in Fig. 9. Here,the calibrated distance and the sampling pitches
are 61:9 cm and 26:8 μm, respectively. Figure 10
shows the geometry of the object. Since the reconstruction plane coincides with the plane of the letter“K,” no numerical propagation is required for reconstruction in Fig. 9(a), whereas the wave field is propagated to the plane at dP ¼ 6:0 cm in Fig. 9(b).

Fig. 8. Numerical reconstruction without numerical propagationand the synthetic aperture technique. The number of pixels is2048 × 2048 pixels.
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Fig. 9. (Color online) Numerical reconstruction without synthetic
aperture technique. The amplitude image of the wave field (a) without and (b) with numerical propagation. The propagation distanceis dP ¼ 6:0 cm. Each image contains 2048 × 2048 pixels.

Clear images are reconstructed by accurate numerical propagation.
B.

Stitching Complex-Valued Images

Multiple complex-valued images captured by using
the phase-shifting technique are stitched up andcombined into a large complex-valued image. Figure 11 is an example of a stitched complex-valued image composed of 5 × 5 segments. The number ofpixels of the individual segments is approximately
1330 × 1330 pixels. This segment size is smaller
than the sensor size, because all segments are captured at intervals less than the sensor size to givean overlap between adjacent segments and enable
them to be stitched exactly. The correlation function
is calculated for each pair of adjacent segments to detect the exact overlap position [11]. The positioning
error of the motor-controlled stage, obtained from thecorrelation function, is approximately 40 μm.
C.

Reconstructed Images from Different Viewpoints

Examples of amplitude images j ^f ðx; y; 0; xe ; ye Þj reconstructed from different viewpoints are shown in
Fig. 12. The object is a nut. The stitched complexvalued image with the number of samplings 7346 ×7334 is embedded in 8192 × 8192 sampling grids toperform the fast Fourier transform. The calibrated
distance dR is 18:6 cm, and thus the pixel pitch of

Fig. 10. Schematic geometry of the object reconstructed in Fig. 9.

Fig. 11. Example of a synthetic complex-valued image stitchedfrom 5 × 5 segments.

the wave field f ðx; y; 0Þ after the fast Fourier transform is 2:01 μm. The wave field is again embedded
in 32; 768 × 32; 768 sampling grids to avoid any aliasing errors in the numerical propagation and thenpropagated forward to a plane at a distance dP ¼
20 cm from the reconstruction plane. The propagatedwave field is masked by a pupil aperture with dimensions 3:2 × 3:2 mm and then propagated backward to
the reconstruction plane. Finally, the region containing the object is clipped from the wave field.
Amplitude images reconstructed from different
perspectives are shown in Fig. 12. Viewpoints of theimage move from left (xe ¼ 16:5 mm) in Fig. 12(a) toright (xe ¼ þ16:5 mm) in Fig. 12(c). The change ofdegree of self-occlusion is observed in the toruslike
shape of the nut. A multimedia file is provided to demonstrate the free-viewpoint reconstruction.
Other examples of the numerical reconstruction ofmultiple objects with mutual occlusion and the geometry of the objects are shown in Figs. 13 and 14,
respectively. Almost all the parameters for capturing
and reconstructing are the same as those in Fig. 12,
but the viewpoint moves vertically in 13(a) and 13(c)as well as horizontally in 13(a) and 13(b). The change
in occlusion is demonstrated more clearly than in
Fig. 12, because mutual occlusion occurs in this example. The pips of the six pattern on a die are allhidden behind another die in 13(a) and partially hidden in 13(c), while all of the pattern is clearly shown

Fig. 13. (Multimedia online; ao.osa.org) Amplitude images
jf ðx; y; 3:0 cm; xe ; ye Þj of mutually occluded objects from different
viewpoints (associated movie file, 1:0 Mbyte).

in 13(b) and 13(d). A multimedia file demonstrating
this change in viewpoint is also provided.
5.

Discussion

Numerical reconstruction of different perspectives isdependent not only on the viewpoint ðxe ; ye Þ, but also
on the pupil size px × py and distance dP . The pupil
size gives the depth of focus of the reconstructed
images. A smaller pupil gives a deeper depth of focus,
but this most likely causes speckle problems. Amplitude images reconstructed for different pupil sizes
are shown in Figs. 16(a)–16(c) below. Speckle noise
increases significantly with decreasing pupil size.
This is one of the well-known properties of image formation from coherent light.
The depth of focus is actually given by angles
Δφx;y ≈ px;y =dP rather than the pupil size itself, as
shown in Fig. 15(a). When a pupil is placed at a distance d0P in the line of sight and the size is given as2
p0x × p0y ¼ ðpx × py Þd02
P =dP ;

ð13Þ

the pupil seems to play the same role as the pupil atthe distance dP . However, this causes the problem ofnarrowing the visual field. Amplitude images reconstructed with pupils at different distances, but in thesame line of sight, are shown in Figs. 16(d)–16(f). The
appearance of the object does not change noticeably,
but the visual field decreases as the pupil distance dP
decreases. Furthermore, in the case of a small dis-Fig. 12. (Multimedia online; ao.osa.org) Amplitude images jf ðx; y; 0; xe ; ye Þj of a self-occluded object from different viewpoints (associatedmovie file, 499 Kbytes).
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Fig. 14. (Color online) Schematic geometry of the mutually occluded objects reconstructed in Fig. 13.

tance as in Fig. 16(f), the image is degraded becauseof the small number of sampling points within
the pupil.
The entire object can be reconstructed if the whole
object field reaches the pupil. This is illustrated inFig. 16(b), where, since the object field does not diffuse over the maximum diffraction angles given asθ0x;y ¼ sin1 λ=ð2Δx;y Þ in the x and y directions, theobject field extends fully in region A and partiallyin region B. As a result, we adopt the criterion thatthe pupil position (i.e., the center of the pupil) should
be at least inside region A. The minimum pupil distance can, therefore, be obtained from the simple geometrical analysis given below:dP ≥2xe;max þ wx;
2 tan θ0x2ye;max þ wy;
2 tan θ0y

ð14ÞFig. 15. (Color online) Visual field in the numerical reconstruction from different viewpoints.

where xe;max , ye;max , and wx;y are the absolute valuesof the maximum pupil position and the lateral extents of the object space, respectively. In Figs. 13
and 14, where the lateral extents of the object space
are wx ¼ 1:3 cm and wy ¼ 0:6 cm, the minimum pupil
distance obtained from Eq. (14) is approximately
17 cm. Thus, dP ¼ 20 cm was adopted for the numerical reconstruction.
6.

Conclusion

Free-viewpoint images for self- and mutually occluded objects were obtained by using phase-shifting
synthetic aperture digital holography. The capture ofobject waves using digital holography commonly suffers from narrowness of both the viewing zone andthe visual field. In this report, the viewing zone isenlarged by extending the effective sensor size, using

Fig. 16. Numerical reconstruction using different pupils: (a)–(c) Only the pupil size varies; (d)–(f) the pupil distance differs, but all pupils
are placed in the same line of sight and have the same angle φx;y as that used in Fig. 13.
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a synthetic aperture technique. A phase-shifting
lensless Fourier setup and aliasing-free zone are also
used to enlarge the visual field and to avoid aliasing
errors caused by extending the sensor size.
Reconstruction of different perspectives is obtained by digital signal processing (DSP) of captured
wave fields, involving a double numerical propagation and clipping the wave field by the pupil placed
at a given viewpoint. In this method, a double fast
Fourier transform is the computational complexity
required for reconstructing an amplitude image fromthe base wave field.
Experimental results show that free-viewpoint
images obtained from the proposed method are capable of reconstructing natural occlusion for both selfand mutually occluded objects. Free-focus images area natural result in digital holography. This reportshows that free-viewpoint images as well as freefocus images are also possible in digital holography.
Various methods of DSP of light are expected in the
future.
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