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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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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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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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




F o r P

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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-




F o r P

e e r R

e v i e w


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.


Accurate 3D Imaging Using UWB Radar

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