Download 3D locations of the object directly from in

3D locations of the object directly from in-line holograms using the
Gabor transform
Yan Zhang1, De-Xiang Zheng, Jing-Ling Shen, Cun-Lin Zhang
Department of Physics, Capital Normal University
Xisanhuan Beilu 105, Beijing 100037 China
ABSTRACT
In-line holography is a simple way for small object imaging. The Gabor transform is used for object characterization
directly from in-line holograms. Three-dimensional locations of an object can be determined from the Gabor transform
spectrum of its in-line hologram. The relationship between the Gabor angle and the location of the object is established,
Computer simulations and experimental result have been presented to demonstrate the validity of this method for object
charactering. The promising application of this method is particle field analysis.
Keywords: Digital holography and the Gabor transform
1. INTRODUCTION
Digital holography was proposed by Goodman et al. in 1960s1. From then on, this technology was constantly developed,
allowing acquisition of three-dimensional information about an object by using a CCD camera and a computer. In–line
digital holography 2 uses the same beam as the reference wave and object illumination simultaneously, its advantages lie
in the simple experimental set-up, higher signal-to-noise rate (SNR), and providing a simple way for short wavelength
imaging which is an ideal approach for high-resolution imaging, especially for some wavelengths for which there is no
suitable lens or beam splitter. The most promising application of the in-line holography is particle field analysis, such as
the detection and measurement of fog droplets, study of smoke, and research of other kinds of dynamic airborne
aerosols 3. Besides these, its applications also involve deformation analysis 4, object contouring 5, microscopy 6, and
particle measurement 7.
With the improvement of the spatial resolution of CCD cameras and the increasing of computational
performance of the personal computer, digital reconstruction of holograms has been introduced into practice, and many
reconstruction methods have been explored for extracting information of small particles from the in-line hologram. The
Wigner distribution function 8-10 is a time-frequency operator that used for the analysis of non-stationary signals, and
yielding the time variation of the frequency components of a signal; it has been used to extract the information of the
particle directly from the in-line holograms 3. Recently, the wavelet transform 11, 12 and the fractional Fourier transform13
are also used for analyzing the small particles and holograms reconstruction. The Gerchberg-Saxton (GS) 14, 15 and
Yang-Gu (YG) 16 phase-retrieval algorithms are also employed as approaches for digital reconstruction, however, the
prior information about the object should be known; furthermore, more processing time is cost due to the iterative
property of the algorithms. The Gabor transform, which is also proposed by Gabor, is a time-frequency tool for signal
processing, and can present both the frequency and time information of a signal simultaneously, but it has not been
employed to extract information of the object from in-line holograms so far. In this work, the Gabor transform will be
employed to characterize the diffraction pattern of the directly form in-line holograms. The relationship between the
Gabor angle and the longitude distance of the object from the CCD is established. Computer simulations and
experimental result are also presented. It is demonstrated that the Gabor transform is an effective mathematical tool
for extracting information from in-line holograms.
The presentation is organized as follows: in Section 2, we present the definition of the Gabor transform; in
Section 3, we describe some computer simulations to show how to use the Gabor transform to extract information of the
object from diffraction patterns; in Section 4, we present the experimental result abut 3D locations of the object by using
the Gabor transform; and at last, we offer some concluding remarks in Section 5.
1
Further author information: (Send correspondence to Dr. Yan Zhang)
Yan Zhang: Email: [email protected], Fax and Tel: 0086-10-68902178
116
Holography, Diffractive Optics, and Applications II, edited by Yunlong Sheng,
Dahsiung Hsu, Chongxiu Yu, Byoungho Lee, Proceedings of SPIE Vol. 5636
(SPIE, Bellingham, WA, 2005) · 0277-786X/05/$15 · doi: 10.1117/12.570465
2. IN-LINE HOLOGRAPHY AND THE GABOR TRANSFORM
In-line holograms record the far–field diffraction patterns of an object. The typical setup for in-line holograms recording
is schematically shown in Fig. 1, the object with complex amplitude transmittance a(x, y) is coherently illuminated by a
plane wave, and the light propagates along the z direction, a CCD located at distance d from the object records the
diffraction pattern.
Figure 1 Experimental setup for in-line hologram recording.
According to the Fresnel approximation of the free space propagation, the complex field
be given by:
A( x, y ) =
A(x, y)
on the CCD plane can
1
i 2π d
iπ
exp(
) a ( x ', y ') × exp{ [( x − x ') 2 + ( y − y ') 2 ]}dx ' dy ',
iλ d
λ ∫∫
λd
(1)
where λ is the wavelength of the illuminating light, and d is the distance between the object and CCD camera. For
the sake of the brevity, only one-dimensional (1D) case is discussed here, the two-dimensional (2D) case can be
obtained directly. Thus Eq. (1) can be simplified as follows:
A( x) =
1
iλ d
exp(
i 2π d
λ
) ∫ a( x) × exp[
iπ
( x − x ' ) 2 ]dx′ .
λd
(2)
The Gabor transform of a function A(x) is defined as 17:
G ( x2 , v) = ∫ A( x1 )g ( x1 , x2 ) exp(−i 2π x1v )dx1
where
v is the spatial frequency, and
(3)
g ( x1 , x 2 ) is a window function, usually, this function can be selected as a
Gaussian function exp[−( x1 − x2 ) / b] , and b is Gaussian width, which play an important role in the description of
the Gabor transform. The value of b should be selected to be suitable according to the size of object, and we choose
b = 9 × 2000 µ m in this paper.
2
3. COMPUTER SIMULATIONS
A one-dimensional particle is considered as the original object in following computer simulations and its size is
9.0 × 20 µm . The parameters of the system are selected as follows: The wavelength of the illuminating light is
λ = 0.532 µm , the width of the CCD window is 9.0 × 1024 µm , and the number of the pixels is 1024. The distance
between the particle and the CCD is selected as d = 140mm ,. The particle is located in the middle of the input plane. In
calculations, the new hologram A( x) − mean[ A( x )] is used to replace the original hologram A( x ) , for the aim of
eliminating the unimportant direct current, where mean[ A( x)] expresses the mean value of the A( x ) .
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Figure 2(a) represents the hologram of the particle, which is recorded by the CCD camera, and Fig. 2(b)
describes the corresponding Gabor transform spectrum G ( x2 , v) of the new hologram. The direct current (the bright
line which parallels to the abscissa) is quite weak due to the pretreatment of the hologram. As shown in Fig. 2(b), the
angle between the two bright lines is defined as the Gabor angle, and denoted by θ . It can be found that tan(θ / 2) has
a good linear relationship with distance d , as shown by the discrete points in Fig. 3. The linear fitted curve has also
been plotted in the same figure, the relationship between the distance d and tan(θ / 2) can be expressed as:
(4)
tan(θ / 2) = 2.80 × 10 −6 d + 2.28 × 10 −3.
1.1
0.05
0.04
0.03
0.02
-1
Frequency (µm )
Intensity (Arb.Unit)
1.05
1
0.95
0.01
0
-0.01
-0.02
-0.03
0.9
-0.04
-0.05
0.85
-5000
0
-4000 -3000 -2000 -1000
5000
0
1000 2000 3000 4000
X2 (µm)
X1 (µm)
Figure 2 (a) Hologram of a particle. (b) The Gabor transform spectrum of (a).
1.8
data
linear
1.6
1.4
tan(θ/2)
1.2
1
0.8
0.6
0.4
0.2
0
0
1
2
3
d (µm)
4
5
6
5
x 10
Figure 3 Relationship between d and θ .
Thus the distance d can be determined by measuring the Gabor angle in the Gabor transform spectrum of the
hologram. The hologram of three different particles located at different place along the x coordinate is represented in
Fig. 4(a), the sizes of particles are 9.0 × 8µm , 9.0 × 14µm , 9.0 × 20 µm from left to right, respectively. The distance
between the particles and the CCD plane is d = 200mm . The corresponding Gabor transform spectrum is shown in
Fig. 4(b), which can be easily found that the Gabor angle in this figure is larger than that in the Fig. 2(b).
It can also be found that the positions of the particles correspond to the X coordinates of the corresponding
cross points in the Gabor spectrum of its hologram. The intensity of the Gabor spectrum is different due to the different
size, different position, and different absorption of the particles. Furthermore, the size of the particles can be determined
from the Gabor spectrum of the hologram.
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Proc. of SPIE Vol. 5636
4. EXPERIMENTAL RESULT
The experiment has also been carried out to demonstrate the application of the Gabor transform. The setup is the same
as it shown in Fig.1, the wavelength of the illuminating light is λ = 0.532 µ m , the CCD used in the experiment has
1018× 992 pixels with pixel size of 9.0 µm × 9.0µm , only a part of record hologram is used for processing. The
recording distance between the object and the CCD is about 140 mm , and the digital hologram is transmitted to a
computer via a frame grabber that performs 10 bits digitization. A hologram of a fiber recorded by the CCD is shown in
Fig 5(a), however, there are many interference annulus caused by the tiny dust on the surface of the collimating lens, the
illuminating light is not the ideal plane wave. As shown by the black line in Fig. 5(a), only 1D signal along the x
direction is extracted as the location information of the fiber. The corresponding Gabor transform spectrum is shown in
Fig. 5(b). The location of the fiber in the x direction can be directly extracted from the cross of the highlights. In order
to determine the location of the fiber along z direction, the Gabor angle is measured, and the value tan(θ / 2) = 0.40 is
calculated, according to the Eq. (4), the distance d can be obtained as d = 143µ m , which is in good agreement with
experimental value d = 140 µm .
0.05
0.04
0.02
-1
Frequency (µm )
0.03
0.01
0
-0.01
-0.02
-0.03
-0.04
-0.05
-4000 -3000 -2000 -1000
0
1000 2000 3000 4000
X2 (µm)
Figure 4 (a) 1D hologram of three particles. From left to right, the sizes of the particles are 9.0 × 8µm , 9.0 × 14µm , and
9.0 × 20 µm , respectively, and (b) the Gabor transform spectrum of (a).
Figure 5 (a) The experiment obtained hologram of a fiber and (b) the Gabor transform spectrum of the cross line in (a).
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This result demonstrates that this method can effectively extract the location information of the object from the
in-line hologram. However, the SNR is not as optimal as expected. This is due to that the input wave is not a standard
plane wave which influences the SNR. Carefully collimating the input wave, clearing the optical elements, cooling the
CCD, and summing up several frames of holograms can effectively reduce the noise. Generally speaking, it can be
concluded that the location of the object in the z direction can be determined by calculating the Gabor angle well.
5. CONCLUSION
The Gabor transform has been adapted for charactering small particles directly from its in-line hologram. For a given
Gabor angle measured from the Gabor transform spectrum of the diffraction patterns recorded by a CCD camera, the
three-dimensional locations of the object can be determined, i.e. the position of the particle in the x and y directions
and the distance between the object and the CCD plane. The signal to noise rate can be improved by using CCD
cameras with smaller pixel size and higher dynamical rang. Experimental result has also been presented to shown the
validity of this method. Further researches are expected to extend the Gabor transform processing to the case of particle
field analysis.
ACKNOWLEDGES
The work was supported by the Capital Normal University Research Foundation, Beijing science nova program, and the
Natural Science Foundation of Beijing, China (6032006).
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