accurate 3d imaging using uwb radar
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Accurate UWB Radar 3-D Imaging Algorithm for Complex Boundarywithout Wavefront Connections
Journal: Transactions on Geoscience and Remote Sensing
Manuscript ID: TGRS-2008-00376
Manuscript Type: Regular paper
Date Submitted by theAuthor:
25-Jun-2008
Complete List of Authors: KIDERA, Shouhei; Kyoto University, Graduate School of InformaticsSakamoto, Takuya; Kyoto University, Graduate School of InformaticsSATO, Toru; Kyoto University, Graduate School of Informatics
Keywords:Radar imaging, Radar resolution, Radar signal processing,Ultrawideband radar, Inverse problems, Imaging
Transactions on Geoscience and Remote Sensing
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IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. XX, NO. Y, MONTH 2008 100
Accurate UWB Radar 3-D Imaging Algorithm for
Complex Boundary without Wavefront ConnectionsShouhei Kidera, Member, IEEE, Takuya Sakamoto, Member, IEEE, and Toru Sato, Member, IEEE,
Abstract—Ultra-wide band (UWB) pulse radars have immea-surable potential for a high-range resolution imaging in the nearfield and can be used for non-contact measurement of precisiondevices with specular surface or identifying and locating thehuman body in security systems. In our previous work, wedeveloped a stable and high-speed 3-dimensional (3-D) imagingalgorithm, Envelope, which is based on the principle that atarget boundary can be expressed as inner or outer envelopesof spheres, which are determined using antenna location andobserved ranges. Although Envelope produces a high-resolutionimage for a simple shape target that may include edges, it requiresan exact connection for observed ranges to maintain the imaging
quality. For complex shapes or multiple targets, this connectionbecomes a difficult task because each antenna receives multipleechoes from many scattering points on the target surface. Thispaper proposes a novel imaging algorithm without wavefrontconnection to accomplish high-quality and flexible 3-D imagingfor various target shapes. The algorithm uses a fuzzy estimationfor the direction of arrival (DOA) using signal amplitudes andrealizes direct mapping from observed ranges to target points.Several comparative studies of conventional algorithms clarifythat our proposed method accomplishes accurate and reliable3-D imaging even for complex or multiple boundaries.
Index Terms—UWB pulse radars, accurate and stable 3-Dimaging, complex boundary, multiple targets, DOA estimation,wavefront connection
I. INTRODUCTION
UWB pulse radars have great potential for use in super-
resolution imaging, which is required in near field sens-
ing applications such as target identification and self location
by robots or automobiles. They can be applied to surveillance
or security systems for intruder detection or aged care, where
an optical camera has the serious problem of intruding on
privacy in bathrooms or living rooms. They are also suit-
able for non-contact measurement of reflector antennas or
aircraft bodies that have high-precision and specular surfaces.
Although various kinds of radar algorithms based on data
synthesis have been proposed, such as synthetic aperture radar
(SAR) [1], time reversal [2] and other optimization algorithms[3]–[5], they all require intensive computation, and are hardly
applicable to the above applications. Contrarily, the high-
speed 3-D imaging algorithm SEABED achieves direct and
non-parametric imaging based on reversible transforms BST
(Boundary Scattering Transform) and IBST (Inverse BST)
between the time delay and target boundary [6]–[9]. However,
imaging using SEABED becomes unstable for noisy data be-
cause the range derivative in BST can enhance the fluctuation
The authors are with the Dept. of Communications and Computer Engineer-ing, Graduate School of Informatics, Kyoto University, Kyoto, Japan. E-mail:[email protected]
of small range errors. To produce a more stable image, we
have already proposed a real-time 3-D imaging algorithm
named Envelope [10], [11]. This method uses an envelope
of spheres, which are determined with antenna locations and
observed ranges, to create a stable image without requiring
derivative operations. It has been confirmed that this method
robustly reconstructs a high-resolution 3-D image for objects
of simple shape, including those with edges when combined
with the range compensation method termed SOC (Spectrum
Offset Correction) [11]. However, the image obtained with
Envelope becomes unstable for complex boundaries becauseit requires an appropriate range connection. For a complex
surface, this connection is often difficult because each antenna
observes multiple echoes, and there are too many candidates
for the connections. A connection algorithm using a Kalman
filter has been developed to track exact ranges in cluttered
situations [12]. Furthermore, Hantscher et al. have developed
an iterative wavefront extraction method for multiple targets,
which recursively subtracts scattered waveforms to resolve
multiple echoes [13]. However, once the range connections
fail, there is non-negligible inaccuracy in the images obtained
by these conventional algorithms. A global optimization al-
gorithm based on waveform matching has been developed
[14], yet it still requires a long calculation time. In any event,all conventional algorithms specific to either SEABED or
Envelope have a substantial problem in that they are extremely
sensitive to inappropriate connections of wavefronts.
This paper proposes a novel algorithm based on direct group
mapping from observed ranges to target boundary points with-
out having wavefront connection. This algorithm involves a
fuzzy estimation for the direction of arrival (DOA) using signal
amplitudes, which eliminates the range connecting procedure.
The idea is based on a simple principle yet it remarkably
enhances stability and accuracy even in complex boundary
extraction. First, the algorithm for a 2-D model is presented for
simplicity, and it is then extended for a 3-D model. This paper
also presents comparative studies using several conventionalalgorithms, such as SAR and Fourier transform. The numerical
simulations indicate that our proposed method has a significant
advantage in accurate and stable imaging even for complex
shape or multiple targets.
II. 2-D PROBLEM
A. System Model
The upper diagram in Fig. 1 shows the system model. It
assumes that the target has an arbitrary shape with a clear
boundary, and that the propagation speed of the radio wave
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Omni-directional
antenna
0
1
2
3
-1 0 1 2
0
1
3
-1 0 1 2
( x,z)
Target boundary
Observing Imaging
r-space
Quasi wavefront
d-space
X
ε1
Z 3 Z 2 Z 1
Z 4 ε 0
Omni-directional
antenna
Z 3 Z 2 Z 1
Z 4
Z 3
Z 2
Z 1
Z 4
X
x
z
Z
Fig. 1. Relationship between target boundary in r-space (upper) and quasiwavefront in d-space (lower).
is a known constant. An omni-directional antenna is scanned
along the x axis. We use a mono-cycle pulse as the transmitting
current. R-space is defined as the real space in which the target
and antenna are located, and is expressed by the parameters
(x, z). The parameters are normalized by λ, which is the
central wavelength of the pulse. We assume z > 0 for
simplicity. s(X, Z ) is defined as the received electric field
at the antenna location (x, z) = (X, 0). s(X, Z ) is defined as
the output of the Wiener filter with the transmitted waveform,
where Z = ct/(2λ) is expressed by the time t and the speed
of the radio wave c. We connect the significant peaks of
s(X, Z ) as Z for each X and refer to this curve (X, Z ) as
quasi wavefront. D-space is defined as the space expressed by
(X, Z ). Fig. 1 shows the relationship between (x, z) in r-space
and (X, Z ) in d-space. The transform from d-space to r-space
corresponds to the imaging, dealt with in this paper.
B. Conventional Algorithms
1) SAR technique for near field imaging: The SAR tech-
nique is the most useful in radar imagery and is based on
signal synthesis [1]. In the near field case, the distribution
image I (x, z) obtained using SAR is expressed as
I (x, z) =
∞
−∞
s
X,
(x − X )2 + z2
dX. (1)
The target boundary can be extracted from its focused image
I (x, z). An example of applying SAR is presented here. The
target boundary is assumed to be sinusoidal curve and is
expressed as z = 2.0 − 0.2cos(2πx). Fig. 2 shows the
output of the Wiener filter and the extracted quasi wavefront
(X, Z ), where each signal is received at 101 locations for
−2.5 ≤ X ≤ 2.5. Fig. 3 shows the estimated image with
SAR, and the target boundary is highlighted. Although this
-0.6-0.4-0.200.20.40.6
X
Z ’
-2 -1 0 1 2
1
2
3s ( X,Z’)Quasi wavefront
Fig. 2. Output of the Wiener filter from the sinusoidal target boundary.
-0.6-0.4-0.200.20.40.6
x
z
-2 -1 0 1 2
1
2
3
True
I ( x, z)
Fig. 3. Estimated image I (x, z) with SAR.
method accomplishes a stable imaging even for a complex
target boundary, the spatial resolution is insufficient to extract
a clear boundary. Furthermore, this method requires search-
ing operations for the entire assumed region (x, z) and the
calculation time is more than 60 sec using a 2.8 GHz Xeon
processor. It is not applicable to the assumed applications, in
terms of resolution and calculation time.2) SEABED: We have already proposed a non-parametric
imaging algorithm, known as SEABED, that drastically short-
ens the calculation time for target boundary extraction [7],
[8]. It uses the reversible transform BST (Boundary Scattering
Transform) between the point (x, z) in r-space and the point
(X, Z ) in d-space. IBST (Inverse BST) is expressed as
x = X − Z∂Z/∂X
z = Z
1 − (∂Z/∂X )2
. (2)
This transform provides a strict solution for the assumed
inverse problem. The observed range and its derivative are
directly transformed to the target boundary by IBST. We con-
firm that this method achieves high-speed and non-parametricimaging of a simple target. However, for a complex one,
the image produced by SEABED is quite unstable, as shown
in Fig. 4, where the same data as in Fig. 2 is used. This
is because IBST uses the derivative of the quasi wavefront.
The small range fluctuations due to multiple interferences are
enhanced by the range derivative ∂Z/∂X , even if the condition
|∂Z/∂X | ≤ 1 is used. IBST also requires an appropriate
wavefront connection to calculate an accurate derivative. For
the complicated quasi wavefront shown in Fig. 2, this opera-
tion is extremely difficult. To overcome this difficulty, several
methods for finding an appropriate wavefront connection have
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1
2
3
-2 -1 0 1 2
z
x
True
Estimated
Fig. 4. Estimated image with SEABED.
X
( x,z)
z
x
Targetboundary
( x,z)
Z
Z
xp( X i)
Z i
X X’ X i
Fig. 5. Relationship between target boundary and envelopes of circles in2-D problem.
been developed, such as adaptive tracking with a Kalman
filter [12], iterative wavefront subtraction for multiple target
recognition [13], and using a global optimization algorithm
based on waveform matching [14]. However, they all have a
substantial problem in that if the wavefront connection fails,
there is serious image distortion because an incorrect range
connection causes large derivative errors.3) Envelope: To suppress the instability due to fluctuation
in the range derivative, a stable and rapid imaging algorithm
Envelope has been developed [10]. This algorithm uses the
principle that the target boundary is expressed as the envelope
of the circles, with a center point (X, 0), and radius Z . Fig. 5
illustrates the relationship between the target boundary and the
envelopes of circles. Fig. 5 shows that the envelopes of the cir-
cles should circumscribe or inscribe to the target boundary, ac-
cording to the sign of ∂x/∂X = 1−Z∂ 2Z/∂X 2−(∂Z/∂X )2,
which denotes the target curvature. This method approximates
a target region (x, z) for each (X, Z ) as
maxνX(Xi−X)<0 xp(X i) ≤ x ≤ minνX(Xi−X)>0 xp(X i)z =
Z 2 − (x − X )2
, (3)
where ν X = sgn(∂x/∂X ), and X i is a searching variable.
xp(X i) is defined as the intersection point between the circles,
determined with (X, Z ) and (X i, Z i), as shown in Fig. 5.
Eq. (3) enables group mapping from the points (X, Z ) to the
points (x, z) without derivatives. Thus, the instability caused
by range fluctuation is suppressed.
It is confirmed that this method achieves high-speed and
stable imaging for a simple target boundary. However, as
shown in Fig. 6, the image produced by Envelope is unstable
1
2
3
-2 -1 0 1 2
z
x
True
Estimated
Fig. 6. Estimated image with Envelope.
1
2
3
-2 -1 0 1 2
z
x
True
Estimated
Fig. 7. Estimated image with Fourier transform and IBST.
for the complex one. This is because Envelope requires appro-
priately connected quasi wavefronts to obtain a stable image.
Complex targets, in general, have many scattering points on
their surfaces, and each antenna observes many ranges. The
connection procedure is a complicated problem because each
point of (X, Z ) has multiple connecting candidates around
itself. If the connection fails, the image estimated by Envelope
has non-negligible errors because it uses an incorrect envelope
of circles with mistakingly determined ν X .
4) IBST with Fourier Transform: An imaging algorithm
using Fourier transform and IBST realizes stable imaging
because it does not require wavefront connections. The range
inclination ∂Z/∂X can be calculated with the received 2-D
data s(X, Z ). In this method, ∂Z/∂X for each point (X, Z )is approximated as
∂Z/∂X −k̂X/k̂Z , (4)
where k̂X and k̂Z are calculated as
(k̂X , k̂Z) = arg maxkX ,kZ
|S (kX , kZ)(X,Z)|. (5)
Here, S (kX , kZ)(X,Z) is the 2-D Fourier transform of thespatially filtered signal around (X, Z ) given as
S (kX , kZ)(X,Z) =
∞
−∞
∞
−∞
Γ(X , Z ; X, Z )s(X , Z )
· e− j(kXX+kZZ
)dX dZ , (6)
where the gate function Γ(X , Z ; X, Z ) is defined as
Γ(X , Z ; X, Z ) = exp
−
(X − X )2 + (Z − Z )2
2σ2F
. (7)
The derivative for each quasi wavefront can be stably calcu-
lated using the pulse waveform and amplitude. Fig. 7 shows
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X X i
Z Z i
Target
( x,z)
X k
Z k
x
θ(q , q
i
)
θopt (q)
Z j
X j
Fig. 8. Relationship between the target boundary and the convergence orbitof the intersection points.
the image estimated by combining the 2-D Fourier transform
and IBST for the same data in Fig. 2. σF = 0.2λ is set.
The figure shows that the obtained image has many inaccurate
points and the correct target boundary is hardly reconstructed.
This is because the accuracy for ∂Z/∂X is affected by
waveform deformations caused by interferences from multiple
scatterers. Moreover, the method requires 240 sec for imaging,which is not suitable for the real-time application.
C. Proposed Algorithm
1) Principle of the Proposed Algorithm: This section de-
scribes a proposed imaging algorithm to resolve the problems
of the conventional algorithms. The proposed algorithm is
based on the simple principle that a target boundary point
should exist on a circle, with a center at (X, 0) and radius
of Z . Thus, each point (x, z) can be calculated using the
corresponding angle of arrival. For the stable calculation of
the angle, the membership function is defined as,
f (θ, q,qi) = exp
−
{θ − θ(q, qi)}2
2σ2θ
, (8)
where q = (X, Z ), qi = (X i, Z i) and θ (q, qi) denotes
the angle from the x axis to the intersection point of the
circles, with parameters (X, Z ) and (X i, Z i). Fig. 8 shows
the relationship between the intersection points of the circles
and the angle of arrival. This method uses another principle
that the intersection point converges to the true target point
(x, z), when (X i, Z i) moves to (X, Z ) along an exact quasi
wavefront. For the exactly connected quasi wavefront, which
satisfies a continuous and single-valued function of X , the
following proposition holds.
Proposition 1: If |X − X i| ≤ |X − X j | and (X − X i)(X −X j) ≥ 0 are satisfied, then,
|x − xp(X i)| ≤ |x − xp(X j)|, (9)
where x = X − Z∂Z/∂X .Proposition 1 is proved in the Appendix. This proposition
states that as X i moves to X , the distance between x of the
target point and xp(X i) decreases. θ (q,qi) then converges
to the true angle of arrival. According to these conditions, the
evaluation value F (θ;q) for the angle estimation is introduced
SEABED Envelope Proposed
Derivative Operation Required Not Required Not Required
Wavefront Connection Required Required Not Required
TABLE IREQUIRED PROCEDURES IN EACH METHOD.
as,
F (θ; q) =
N qi=0
s(X i, Z i)f (θ, q,qi) e−
(X−Xi)2
2σ2X
, (10)
where the constants σθ and σX are empirically determined,
and N q is the number of the quasi wavefront. The weight
function exp
−(X − X i)2/2σ2X
in Eq. (10) yields a con-
vergence effect of intersection points to the angle estimation.
The optimum angle of arrival for each q is calculated as,
θopt(q) = arg maxθ
F (θ; q). (11)
The target boundary (x, z) for each quasi wavefront (X, Z ) isexpressed as x = X + Z cos θopt(q) and z = Z sin θopt(q).
This method realizes direct mapping from the points of quasi
wavefront to the points of target boundary without wavefront
connections, or derivative operations. Thus, the instability
occurring in the conventional algorithms can be substantially
avoided with this method.
2) Procedure of the Proposed Method:
Step 1). Extract quasi wavefront q, that satisfies
∂s(X, Z )/∂Z = 0,
s(X, Z ) ≥ α maxZ
s(X, Z ). (12)
The parameter α is determined empirically.
Step 2). Calculate F (θ;q) in Eq. (10) and obtain θopt(q)in Eq. (11), where η(q) is set as
η(q) = F (θopt(q);q). (13)
Step 3). Calculate the point on the target boundary (x, z)as
x = X + Z cos θopt(q)z = Z sin θopt(q)
. (14)
Step 4). Carry out step 2) to 3) for all points on the quasi
wavefront, and obtain each target point.
Step 5). Remove the target points that satisfy,
η(q) ≤ β maxi
η(qi). (15)
β is empirically determined.
Step 5) suppresses a false image caused by random noises.
This method is based on a simple procedure that avoids the
difficulty of connecting the quasi wavefront. Table. I shows the
required procedures for each method. The processes “Deriva-
tive Operation” and “Wavefront Connection” yield instabilities
and difficulties for imaging in the case of complex boundaries,
as described in Sec. II-B. The proposed method does not
require these procedures, and instability can be substantially
resolved.
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1
2
3
-2 -1 0 1 2
z
x
True
Estimated
Fig. 9. Estimated image with the proposed method.
0
50
100
150
N u m b e r o f e s t i m a t e d p o i n t s
ε
Envelope
SEABED
Proposed
10-1
100
10-3
10-2
Fourier + IBST
Fig. 10. Error distribution for each method at the complex target.
2
2.5
-1 -0.5 0 0.5 1
Z
X
EstimatedTrue
Fig. 11. Enlarged view for the quasi wavefront in Fig. 2.
D. Performance Evaluation of the Numerical Simulation
1) Complex Boundary: Fig. 9 shows the image estimated
by the proposed method for the same data as those in Fig. 2.
σX = 1.0λ, σθ = π/50 α = 0.2 and β = 0.3 are set. This
result verifies that the proposed method significantly enhances
the stability of the estimated image, even for a complex target
boundary. Fig. 10 shows the distribution of the error , which
is defined as,
= minx x − xi
e2. (16)
Here, x and xie express the location of the true target point
and that of the estimated point, respectively. The figure reveals
that the number of estimated points with ≥ 2.0 × 10−1λis significantly less than the numbers for other algorithms.
Also, we introduce as the mean value of for all estimated
points. It shows = 0.845λ for SEABED, = 0.215λ for
Envelope, = 0.129λ for Fourier+IBST and = 0.075λ for
the proposed method. This result quantitatively shows that the
proposed method has a significant advantage for the accurate
imaging.
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100 120 140 160 180
N o r m a l i z e d e v a l u a t i o n v a l u e
θ [degree]
True angle
F (θ; q)
f (θ, q , q i)
Fig. 12. Evaluation example for f (θ, q,qi) and F (θ;q) at q = (0.0, 2.05).
-1
-0.5
0
0.5
1
X
-2 -1 0 1 20
1
2
3
4Quasi wavefront
Z ’
s ( X,Z’)
Fig. 13. Output of the Wiener filter and the extracted quasi wavefront forS/N =20 dB.
Here, we discuss the effectiveness of the proposed method.
Fig. 11 is an enlarged illustration of the quasi wavefront in
Fig. 2. Fig. 12 shows each value of f (θ, q, qi) and F (θ; q)for the case of q = (0.0, 2.05). We recognize a maximum for
θ 115◦ around the true angle, in spite of the local inclination
of the quasi wavefront ∂Z/∂X being roughly estimated as 0,which corresponds to θ = 90◦. This is because each angle of
arrival can be calculated from the global distribution of the
quasi wavefront, and this avoids inaccuracy in the angle of
arrival due to the derivative or miss-connection of the quasi
wavefront. It is shown that the calculation time of the proposed
method is within 0.2 sec for processing on a Xeon 2.8 GHz
computer, and this time is suitable for real-time operation. The
high-speed imaging is possible because the method uses only
the ranges and amplitudes, and this decreases the calculation
cost for the boundary extraction when compared with SAR
and other data synthesis algorithms.
Next, the application example of a noisy situation is pre-
sented. Fig. 13 shows the output of the Wiener filter and theextracted quasi wavefront for S/N = 20 dB. Here, S/N is
defined as the ratio of peak instantaneous signal power to the
averaged noise power after applying the matched filter. As
shown in Fig. 13, there are many incorrect range points due
to the random noise. Fig. 14 shows the target image obtained
with the proposed method, and a stable image is produced
even for the noisy environment. The accuracy for each method
is quantitatively compared for various noisy cases using the
evaluation value µ, which is defined as the root mean squares
of . Fig. 15 shows the relationship between S/N and µ for
each method. The figure verifies that the proposed method
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1
2
3
-2 -1 0 1 2
z
x
True
Estimated
Fig. 14. Estimated image with the proposed method for S/N = 20 dB.
0.1
1
10 20 30 40 50 60 70
µ
S/N [dB]
SEABED
Fourier+ IBST
Proposed
Envelope
Fig. 15. µ for each S/N at the complex target boundary.
Quasi wavefront
-1
-0.5
0
0.5
1
X
Z ’
-2 -1 0 1 21
2
3
4 s ( X,Z’)
Fig. 16. Output of the Wiener filter and the extracted quasi wavefront forthe small circle and concave boundary.
provides an accurate and stable image in noisy situations, and
µ holds within 0.15λ for S/N ≥ 20 dB.
2) Multiple Boundaries: We show that the proposed algo-
rithm achieves stable imaging, where multiple boundaries with
large and small curvatures are intermingled. Fig. 16 shows
the output of the Wiener filter in the case of the concave
boundary and small circle. Fig. 17 shows the produced image
by the proposed method. The figure confirms that the imageexpresses both target boundaries and this method is applicable
to general multiple target boundaries. This is because the
evaluation value in Eq. (10) approximately reflects the physical
scattering model. The left and right diagrams in Fig. 18 show
the relationship between received power and the density of
intersection points for a small circle. As shown in Fig. 18,
the power of each received signal becomes small because
the target has large curvature; however, the intersection points
converges to the small region, and this increases the evaluation
value F (θ; q) in Eq. (11). Fig. 19 shows the same relationship
as in Fig. 18, for a concave boundary. As shown in this
1
2
3
4
-2 -1 0 1 2
z
x
TrueEstimated
Concave boundary
Small circle
Fig. 17. Estimated image with the proposed method for the small circle andconcave boundary.
x
z ( Z )
x ( X )
4
0
1
2
3
-3 -2 -1 0 1 2 3
Small power
Quasi wavefront
Target boundary
300
1
2
3
4
-3 -2 -1 0 1 2 3
Dense
Intersection point
z
Fig. 18. Relationship between receiving power (left) and density of intersection points (left) for small circle.
0
1
2
3
4
-3 -2 -1 0 1 2 3
Target Boundary
Quasi Wavefront
Large power
Quasi wavefront
Target boundary
x
z ( Z )
x ( X )0
0
z
0
1
2
3
4
-3 -2 -1 0 1 2 3
Intersection pointIntersection point
Sparce
Fig. 19. Same relationship in Fig. 18 for concave boundary.
figure, the density of the intersection points decreases when
the curvature radius of the target boundary is close to that of
circle because the intersection points are less converged on
the circle for q. However, the power of each received signal
becomes relatively large because the scattering paths converge
to each antenna location. This example verifies the relevance
of the evaluation value in Eq. (11) in the reconstruction of
various target boundaries.
III. 3-D PROBLEM
A. System Model
Fig. 20 shows the system model for a 3-D problem. The
target model, antenna, and transmitted signal are the same of
those assumed in the 2-D problem. The antenna is scanned
along the plane, z = 0. We assume a linear polarization in the
direction of the x-axis. R-space is expressed by the parameter
(x, y, z). We assume z > 0 for simplicity. s(X, Y, Z ) is
defined as the received electric field at the antenna location
(x, y, z) = (X,Y, 0). s(X, Y, Z ) is defined as the output of
the Wiener filter with the transmitted waveform. We connect
the significant peaks of s(X, Y, Z ) as Z for each X and Y ,
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Target boundary
ε0
z
x
( X,Y,0)
( x,y,z)
Z
0
y
Omni-directionalantenna
Fig. 20. System model in 3-D problem.
Y
Z ’
-2 -1 0 1 20
1
2
3
-1
-0.5
0
0.5
1
X
Z ’
-2 -1 0 1 20
1
2
3Quasi wavefront Quasi wavefront
s ( X,Y,Z’)
Fig. 21. Output of the Wiener filter and extracted quasi wavefronts at X = 0(left) and Y = 0.6 (right).
and call this surface (X, Y, Z ) a quasi wavefront. D-space is
defined as the space expressed by (X, Y, Z ).
B. Conventional Algorithms
1) SEABED: SEABED algorithm for 3-D problems has
been developed. It achieves real-time and nonparametric 3-D
imaging with IBST [8]. The IBST from the quasi wavefront
(X, Y, Z ) to the target boundary (x, y, z) is formulated as
x = X − Z∂Z/∂X y = Y − Z∂Z/∂Y
z = Z
1 − (∂Z/∂X )2 − (∂Z/∂Y )2
. (17)
This transform give us a direct solution for the clear boundary
extraction. An application example of SEABED is presented
as follows. We assume a complex target boundary with si-
nusoidal surfaces as shown in Fig. 20. The quasi wavefront
is extracted from the received data calculated by the FDTD
(Finite Difference Time Domain) method. The left and right
hand sides of Fig. 21 show the output of the Wiener filter
and the extracted quasi wavefront (X, Y, Z ) at X = 0 and
Y = 0.6, respectively. We take the received signal at 51locations for −2.5 ≤ x, y ≤ 2.5. Fig. 22 is the 3-D view
of the extracted quasi wavefront. Fig. 23 shows the estimated
3-D image and its cross-section at x = 0 and y = 0.6,
respectively, using SEABED. The figure shows that there is
a non-negligible fluctuation of the estimated points because
the image estimated by SEABED strongly depends on the
accuracy of the derivative of Z , which is quite sensitive to
small range errors due to interference.
2) Envelope: Envelope method for 3-D problems has been
proposed to realize stable and high-speed 3-D imaging with an
envelope of spheres [11]. Similar to 2-D problem, This method
Z
-2-1
01
2
-2-1
012
0.5
1
1.5
2
2.5
X
Y
Quasi wavefront
Fig. 22. Extracted quasi wavefront for the 3-D complex target.
-2-1
0 1-2
-10
12
1
z
x y
2
True
-2 -1 0 1 2
1
2
z
-2 -1 0 1 2
1
2
z
2
Estimated
y
x=0
y=0.6
x
Fig. 23. Estimated image with SEABED for the 3-D complex target.
calculates the target boundary (x, y, z) for each (X, Y, Z ) as
maxνX(Xi−X)<0
x3dp (X i) ≤ x ≤ min
νX(Xi−X)>0x3dp (X i)
maxνY (Y i−Y )<0
y3dp (Y i) ≤ y ≤ minνY (Y i−Y )>0
y3dp (Y i)
z = Z 2 − (x − X )2 − (y − Y )2
, (18)
where X i and Y i are searching variables and ν Y =sgn(∂y/∂Y ). y3dp (Y i) is defined as the intersection point
between the projected circles of two spheres determined by
(X, Y, Z ) and (X, Y i, Z i) on the plane x = X . x3dp (X i) is
defined similarly on the plane y = Y . Eq. (18) determines an
arbitrary target boundary without derivative operations, which
can suppress the instability caused by small range errors.
Fig. 24 shows the image obtained with Envelope from the same
views as those in Fig. 23. It is confirmed that the fluctuation of
the estimated points is suppressed without the use of the range
derivative. However, there are significant image distortions
because the miss-connection of the quasi wavefront produces
an incorrect envelope of spheres, which results in the large
errors for the estimated regions. This determination requires
a correct wavefront connection for the quasi wavefront, and
it often becomes more difficult than that in a 2-D problem
because each point must be correctly connected along both
the x and y axes.
C. Proposed Imaging Algorithm
To resolve the previous problems, we extend the proposed
method to 3-D modeling. In this model, the orbit of intersec-
tion points between two spheres for (X, Y, Z ) and (X i, Y i, Z i)becomes a circle. The projected curve of this circle on z = 0
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T ru e E st im at ed
-2-1
0 1
2
-2-1
01
2
1
2
z
x y
1
2
-2 -1 0 1 2
1
2
y
z
1
2
-2 -1 0 1 2
1
2
x
z
2
Estimated
y
x=0
y=0.6
x
Fig. 24. Estimated image with Envelope for the 3-D complex target.
x
y
( X,Y ,0)
Z
( X i ,Y i,0)
Z i
L i
z
( x,y)
d ( x,y, q ,qi )3d 3d
Fig. 25. Intersection line Li of two spheres on z = 0 plane.
becomes a straight line. We define this line as Li. Fig. 25
shows the intersection line Li of two spheres on the z = 0plane. Here, each angle of arrival corresponds to the location
for (x, y) for the assumption z ≥ 0. This method determines
the target location (x, y) to simplify the calculation for 3-D
boundary extraction. The membership function for (x, y) is
defined as
f
x,y,q3d, q3di
= exp
−
d
x,y,q3d,q3di2
2σ2d
, (19)
where q3d = (X, Y, Z ), q
3di = (X i, Y i, Z i), and
d
x,y,q3d, q3di
denotes the minimum distance between the
projected line Li and (x, y). We use the extended principle
that if (X i, Y i, Z i) moves to (X, Y, Z ) along an exact quasi
wavefront, (x, y) converges to that of the true target point. For
stable locating of (x, y), the evaluation value F 3d
x, y; q3d
is introduced as,
F 3d
x, y; q3d
=N qi=0
s(X i, Y i, Z i)f
x,y,q3d,q3di
e−
D(q3d,q3di )2
2σ2D
, (20)
where Dq3d, q3di
=
(X − X 2i ) + (Y − Y 2i ), σd and
σD are empirically determined. The x and y coordinates of
the target boundary for each quasi wavefront q3d are then
calculated asx(q3d), y(q3d)
= arg max
x,yF 3d
x, y; q3d
. (21)
2
22
T ru e E st ima ted
-2-1
0 1
2
-2-1
01
2
1
2
z
x y
-2 -1 0 1 20
1
2
y
z
1
2
-2 -1 0 1 2
1
2
x
z
1
2
2
Estimated
y
x=0
y=0.6
x
Fig. 26. Estimated image with the proposed method for the 3-D complextarget.
0
250
500
750
1000
N u m b e r o f e s t i m a t e d p o i n t s
ε
10-1
100
10-2
Envelope
SEABED
Proposed
Fig. 27. Error distribution for each method at the 3-D complex target.
The z coordinate of each target point is given by z(q3d) = Z 2 − {x(q3d) − X }
2− {y(q3d) − Y }
2. The method elim-
inates the connecting procedures of the quasi wavefront, which
can avoid instability due to the failure of range connections.
Thus, it achieves a direct mapping from the all points of the
quasi wavefront to the points of the target boundary withoutgrouping.
D. Performance Evaluation in Numerical Simulation
This section presents an application example of a 3-D
problem for the proposed method. Fig. 26 illustrates the image
estimated by the proposed method. σd = 0.1λ and σD = 0.5λare set. The method remarkably enhances the stability and
accuracy for 3-D complex target imaging. This is because
it does not require a wavefront connection, and eliminates
instability due to an inappropriate range connection. Further-
more, the proposed method makes use of the distribution of
the quasi wavefront along not only the x- and y- axes but alldirections to obtain an accurate target point. Fig. 27 shows the
distribution of , defined in Eq. (16), for the images estimated
using each method. This figure quantitatively shows that the
proposed method increases the number of the estimated points
with ≤ 1.0 × 10−1λ. The ratio of the estimated points to
the total number with ≤ 2.0 × 10−1λ is around 96.8%,
which is a significant improvement when compared with that
of SEABED (58.8%) or Envelope (73.1%). Also, the proposed
method obtains = 0.091λ, which is superior to those in
SEABED ( = 0.215λ) and Envelope ( = 0.151λ). The
calculation time for this method is around 50 sec for a Xeon
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1
1.2
1.4
1.6
1.8
2
x
y
-2 -1 0 1 2
-2
-1
0
1
2
z
Fig. 28. True target contour image.
z
1
1.2
1.4
1.6
1.8
2
x
y
-2 -1 0 1 2
-2
-1
0
1
2
-2 -1 0 1 2
1
2
x
y
z z x=0
True
Estimated
Fig. 29. Estimated contour image with SEABED after smoothing (left) andits cross-section view at x = 0 (right).
2.8 GHz processor, because it requires a 2-D search for the
assumed region for (x, y) for each quasi wavefront.
Next, the smoothing examples for the obtained images
are presented to clearly show that our algorithm offers a
higher-quality 3-D image compared with the images for the
conventional algorithms. Here, we apply the simple smooth-
ing algorithm by combining an extended median filter and
the Gaussian function. First, we select the estimated points
(xi, yi, zi), which are included in the preselected region for
(x, y), and zi is updated with other points included in this
region as,
zupi =
zmed,
zi ≥ (1 − ξ)minj zj + ξ maxj zj ,zi ≤ (1 − ξ)maxj zj + ξ minj zj
zi, (otherwise)
,
(22)
where zmed is the median value for z among the selected target
points. ξ is empirically determined. Second, the z coordinate
of the selected region (x, y) is smoothed with the Gaussian
function as,
z(x, y) =
i zupi exp
− (x−xi)
2+(y−yi)2
2σ2I
i exp
− (x−xi)2+(y−yi)2
2σ2I
. (23)
The left and right hand sides of Fig. 29 shows the estimatedcontour image and its cross-section along x = 0 using
SEABED, respectively. Fig. 30 shows the image estimated
with Envelope from the same view as that in Fig. 29.
σI = 0.1λ and ξ = 0.2 are set. These figures show that
the smoothed images with both methods hardly reconstruct
a correct target boundary, and the characteristic of target
boundary is lost by the smoothing of inaccurate points. Con-
trarily, Fig. 31 shows the image estimated by the proposed
method from the same view as that in Figs. 29 and 30. The
figure confirms that the image smoothing is effective for the
proposed method, and the target boundary can be accurately
1
1.2
1.4
1.6
1.8
2
x
y
-2 -1 0 1 2
-2
-1
0
1
2
-2 -1 0 1 2
1
2
x
y
z z x=0
True
Estimated
Fig. 30. Smoothed image with Envelope from the same view in Fig. 29.
1
1.2
1.4
1.6
1.8
2
x
y
-2 -1 0 1 2
-2
-1
0
1
2
-2 -1 0 1 2
1
2
x
y
z z
x=0
True
Estimated
Fig. 31. Smoothed image with the proposed method from the same view inFig. 29.
reconstructed even for complex targets. This result shows that
there is a remarkable advantage in using the proposed method
in complicated surface imaging. Also, it shows = 0.081λfor SEABED, = 0.097λ for Envelope and = 0.060λ for
the proposed method, and this evaluation quantitatively proves
the effectiveness of our proposed method.
IV. CONCLUSION
We proposed a novel imaging algorithm without wavefront
connections for complex shape targets. First, we discussed the
characteristic of the images estimated by the conventional al-gorithms as SAR, SEABED, Envelope and IBST with Fourier
transform. Next, we presented a stable and high-speed imaging
algorithm with a fuzzy estimation of the angle of arrival, which
does not require wavefront connection. It was verified that
the proposed method remarkably enhances the stability and
accuracy of the estimated image even for a complex target
boundary. It was also shown that the calculation time for
the proposed method 2-D model is within 0.2 sec, which is
appropriate for real-time operation.
We also extended the proposed algorithm to 3-D model-
ing, and made statistical calculation for the x and y target
coordinates. It was confirmed that this method accomplishes
stable and accurate imaging even for 3-D complex targets.The calculation time for this method is around 50 sec, and
it is important in our future work to enhance the speed of
the imaging. For simple target boundaries such as trape-
zoidal or spherical targets, there are advantages in using the
conventional algorithm Envelope in terms of real-time and
super-resolution imaging. It is promising to select or combine
appropriate algorithms for the assumed application.
ACKNOWLEDGMENT
This work is supported in part by the Grant-in-Aid for
Scientific Research (A) (Grant No. 17206044) and the Grant-
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in-Aid for JSPS Fellows (Grant No. 19-497).
APPENDIX
PROOF OF PROPOSITION 1
Here, we utilize the following proposition, which has been
proved in [10].
Proposition 2: If ∂x/∂X > 0 holds for all (X, Z ), each
target boundary (x, z) satisfies,
(x − X )2 + z2 ≥ Z 2, (24)
where an equal sign holds at only one point of (X, Z ).
Here, the target point is defined as (x(X i), z(X i)), which sat-
isfies x(X i) = X i − Z iZ iXi, and z(X i) = Z i
1 − (Z iXi
)2.
Substituting (x(X i), z(X i)) to Eq. (24) gives
Z 2i + (X i − X )2 − Z 2 − 2Z iZ iXi(X i − X ) ≥ 0, (25)
Contrarily, the derivative of xp(X i) for X i is expressed as,
∂xp(X i)
∂X i=
Z 2i + (X i − X )2 − Z 2 − 2Z iZ iXi(X i − X )
2(X − X i)2.
(26)From Eq. (25),
∂xp(X i)
∂X i≥ 0, (27)
holds. Also, if ∂x/∂X > 0, then Eq. (3) gives
(X − X i)(x − xp(X i)) ≥ 0. (28)
Thus,
xp(X j) ≤ xp(X i) ≤ x, (X j ≤ X i ≤ X )x ≤ xp(X i) ≤ xp(X j), (X ≤ X i ≤ X j)
, (29)
is proved. For ∂x/∂X < 0, the following relationship also
holds with the similar approach,
xp(X j) ≤ xp(X i) ≤ x, (X ≤ X i ≤ X j)x ≤ xp(X i) ≤ xp(X j), (X j ≤ X i ≤ X )
. (30)
Eqs. (29) and (30) correspond to the Proposition 1.
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PLACEPHOTOHERE
Shouhei Kidera received B.E. degree from KyotoUniversity in 2003 and M.E. and Ph.D. degrees fromGraduate School of Informatics, Kyoto Universityin 2005 and 2007, respectively. He is currently a re-search fellow of the Japan Society for the Promotionof Science (JSPS) at Department of Communica-tions and Computer Engineering, Graduate School of Informatics, Kyoto University. His current researchinterest is in UWB radar signal processing. He is amember of the Institute of Electronics, Information,and Communication Engineers of Japan (IEICE) and
the Institute of Electrical Engineering of Japan (IEEJ).
PLACEPHOTOHERE
Takuya Sakamoto was born in Nara, Japan in 1977.
Dr. Sakamoto received his B.E. degree from KyotoUniversity in 2000, and his M.I. and Ph.D. degreesfrom Graduate School of Informatics, Kyoto Univer-sity in 2002 and 2005, respectively. He is an assistantprofessor in the Department of Communicationsand Computer Engineering, Graduate School of In-formatics, Kyoto University. His current researchinterest is in signal processing for UWB pulse radars.He is a member of the Institute of Electronics,Information, and Communication Engineers of Japan
(IEICE), and the Institute of Electrical Engineering of Japan (IEEJ).
PLACEPHOTOHERE
Toru Sato received his B.E., M.E., and Ph.D. de-grees in electrical engineering from Kyoto Univer-
sity, Kyoto, Japan in 1976, 1978, and 1982, respec-tively. He has been with Kyoto University since 1983and is currently a Professor in the Department of Communications and Computer Engineering, Grad-uate School of Informatics. His major research inter-ests have been system design and signal processingaspects of atmospheric radars, radar remote sensingof the atmosphere, observations of precipitation us-ing radar and satellite signals, radar observation of
space debris, and signal processing for subsurface radar signals. Dr. Satowas awarded Tanakadate Prize in 1986. He is a fellow of the Institute of Electronics, Information, and Communication Engineers of Japan, the Societyof Geomagnetism and Earth, Planetary and Space Sciences, the Japan Societyfor Aeronautical and Space Sciences, the Institute of Electrical and ElectronicsEngineers, and the American Meteorological Society.
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