012801-1 CHINESE OPTICS LETTERS / Vol. 9, No. 1 / January 10, 2011
Application of phase retrieval algorithm in reflective
tomography laser radar imaging
Xiaofeng Jin (
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, Jianfeng Sun (
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), Yi Yan (
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), Yu Zhou (
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Key Laboratory of Space Laser Communication and Testing Technology, Shanghai Institute of Optics and Fine Mechanics,
Chinese Academy of Sciences, Shanghai 201800, China
∗
Corresponding author: jxf2008@siom.ac.cn
Received June 9, 2010; accepted August 11, 2010; posted online January 1, 2011
We apply ph ase retrieval method to align projection data for tomographic reconstruction in reflective
tomography laser radar imaging. In our experiment, the target is placed on a spin table with an unknown,
but fixed, axis. The oscillatory motion of the target in the incident direction of the laser pulse is added at
each view to simulate the real satellites random motion. The experimental simulation results demonstrate
the effectiveness of this method to improve image reconstruction quality. Future research also includes
the development of p rojection registration based on phase retrieval for targets with more complicated
structure.
OCIS codes: 280.3640, 100.3020, 100.5070.
doi: 10.3788/COL201109.012801.
Reflective tomography is one of the most effective high-
resolution imaging methods in la ser radar imaging sys-
tems. The range-resolved laser reflective tomography
imaging laser radar was first introduced by Parker
et al.
[1−4]
Several years later , Matson et al. began ex-
ploring the technique of using the HI-CLASS coherent
laser radar system to obtain reflective images by carrying
out a heterodyne system analysis, deriving and validat-
ing imaging signal-to-noise ratio (SNR) expressions, and
so on
[5−11]
.
The range to the target in incoherent direct detection of
range-resolved reflective tomography cannot be measured
with sufficient accuracy to align the intensity projections
to an appropriate center of rotation. This would bring
serious artifacts in the final image s reconstructed. How-
ever, displacement of the individual projection in the in-
cident direction of laser pulse only produces linear phase
errors in their Fourier transforms. In theory, the Fourier
modulus of each projection is unaffected by the misalign-
ments in the incident direction. We can infer the Fourier
modulus of the target from the misaligned projections
by using the Fourier slice theorem for tomographic re-
construction. The phase retrieval algo rithm developed
by Fienup
[12,13]
can be used for the object image r e cov-
ery. Ford et al. first introduced this iterative technique
to automatically align simulated projection data for re-
flective tomography
[10]
. However, in their presentation,
the projections were required to satisfy the assumptions
associated with transmission tomography by computer
simulation. In other words, they simulately absorbed,
rather than reflected, the signal. Afterwards, Matson
et al. mentioned the phase retrieval algorithm
[11]
; how-
ever, the imaging results of the phase retrieval algo rithm
in reflective tomography has not bee n reported. In this
letter, we present the first image r e c onstruction results of
a target by using phase retrieval a lgorithm in the range -
resolved reflective tomography techniques.
A brief review of mathematical foundations, in-
cluding filtered back-projection and Radon-Fourier
transform
[1−3]
, is first provided for reflective tomograpic
reconstruction. Let f (x, y) denote the image to be re-
constructed, and L
r,φ
denote the s olid line r = xcos φ +
ysin φ (Fig. 1(a))
p(r, φ) =
Z
L
r,φ
f(x, y)ds, (1)
where s represents the solid line along L
r,φ
, p(r, φ) is the
projection of the target f (x, y ) at angle φ, and the vari-
able r denotes the spatial variable along the integration
path in the φ direction.
Using back-projection, the reconstructed image g(x, y)
is given by
g(x, y) =
m
X
i=1
F
−1
1
n
ecF
1
[p(r, φ)]
o
∆φ, (2)
where φ
i
is the angle of the ith projection, ∆φ is the sam-
pling angular separation, m is the total number of pro-
jections, F
1
and F
−1
1
denote the one-dimensional (1D)
Fourier transform and the inverse Fourier transform op-
erators; r represents space variable; ec is the filter func-
tion, which is the product of a window function and the
magnitude of the spatia l frequency
[1,2,5,6,16,17]
.
The Radon-Fourier transform method is bas e d on
Fourier s lice theorem, which states that the 1D Fourier
transform of a projection is a slice through the two-
dimensional (2D) Fourier transform of the target. In
mathematics, the Fourier slice theorem can b e expressed
as
Fig. 1. Diagram of reflective tomography.
1671-7694/2011/012801(4)
c
2011 Chinese O ptics Letters