Reprinted from

REGULAR PAPER

Estimation of the Elevational Distance between Image Planes
by Analysis of Ultrasonic Echoes from Point Scatterers
Atsuhiro Suzuki, Hideyuki Hasegawa, and Hiroshi Kanai

Jpn. J. Appl. Phys. 50 (2011) 07HF09# 2011 The Japan Society of Applied Physics

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REGULAR PAPER

Japanese Journal of Applied Physics 50 (2011) 07HF09
DOI: 10.1143/JJAP.50.07HF09

Estimation of the Elevational Distance between Image Planes
by Analysis of Ultrasonic Echoes from Point Scatterers
Atsuhiro Suzuki1 , Hideyuki Hasegawa1;2 , and Hiroshi Kanai2;1 
1
2Graduate School of Biomedical Engineering, Tohoku University, Sendai 980-8579, Japan
Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan

Received December 13, 2010; accepted February 21, 2011; published online July 20, 2011
There are two approaches to three-dimensional (3D) image reconstruction using a 1D array ultrasonic transducer: mechanical linear scanning andfree-hand scanning. Mechanical scanning employs a motorized mechanism to translate the transducer linearly. However, the large size andweight of the scanning system sometimes make it inconvenient to use. In free-hand scanning, a sensor (e.g., electromagnetic or optical) is
attached to the ultrasonic transducer to measure the position and orientation of the transducer. These techniques are sensitive to the usage
environment. Recently, sensorless free-hand scanning techniques have been developed. Seabra et al. reported sensorless free-hand techniques
for the carotid artery by monitoring the velocity of the ultrasound probe [J. C. R. Seabra, L. M. Pedro, and J. F. Ferandes: IEEE Trans. Biomed.
Eng. 56 (2009) 1442]. This system achieved an accuracy of 2.5 mm [root mean square (RMS) error] of the location. To develop accurate
sensorless measurement, we propose a novel method using the phase shift between ultrasonic RF echoes. In this study, we measured thetransmit-receive directivity of a linear-array transducer using a silicone phantom and estimated the elevational distance between two 2D US
images using the phase shift. An accuracy of 49.9 m in RMS, which is less than that of the previous sensorless free-hand method, could beachieved by the proposed method. # 2011 The Japan Society of Applied Physics

1. Introduction

There are two approaches to three-dimensional (3D) image
reconstruction using a 1D array ultrasonic transducer:
mechanical linear scanning and free-hand scanning.1–4)
Mechanical scanning employs a motorized mechanism totranslate the transducer linearly. During the movement of
the transducer, 2D images are acquired at constant elevational intervals (typically 0.2 mm in 3D carotid ultrasoundimaging).4) 3D imaging by mechanical scanning oﬀers threeadvantages: short imaging time, high-quality 3D images,
and fast reconstruction. However, the large size and weightof the scanning system make it inconvenient to use.5)
To overcome these size and weight issues related tomechanical scanning, free-hand scanning techniques havebeen developed. In free-hand scanning, a position sensor is
attached to the ultrasonic transducer to measure the positionand orientation of the transducer.6) Two types of position
tracking systems are commonly used in 3D free-handscanning, i.e., electromagnetic and optical systems. Electromagnetic position sensors can achieve an accuracy of
0.6 mm [root mean square (RMS) error] in location and0.4 (RMS error) in orientation when the position sensor isused alone (Northern Digital Polaris Vicra). However, underelectromagnetic interference (e.g., highly conductive metals,
ac power cabling and motors), the accuracy of position
tracking deteriorates.7–10) Birkfellner et al. reported that theaccuracy of the electromagnetic system was degraded from
1.8 mm (position sensor alone) to 2:9  1:0 mm in a surgicalenvironment.10) One alternative is to use optical tracking
devices that have better accuracy and precision thanelectromagnetic tracking devices [0.25 mm (RMS error) in
location using the position sensor alone (Northern Digital
Aurora)]. However, the line of sight between the position
sensor and tracking tool should be maintained for successful
optical tracking.11,12)
To overcome the restriction of usage environment andreduce the cost related to free-hand scanning, sensorless
free-hand scanning techniques have been developed.13,14)

E-mail address: kanai@ecei.tohoku.ac.jp

Seabra et al. reported sensorless free-hand techniques for the
carotid artery by monitoring the velocity of an ultrasoundprobe.14) This system achieved an accuracy of 2.5 mm (RMS
error) of location by calibration. To realize an accurate
sensorless method, we propose a novel method using thephase shift between ultrasonic RF echoes. In the present
study, we validated the proposed method by estimating theelevational distance between two 2D ultrasonic (US) imagesof a silicone phantom.
2. Principles
2.1 Estimation of the elevational distance between two
2D US images

In this study, the elevational distance between two 2D US
images was estimated using the phase shift of ultrasonicechoes when a 1D linear array probe was mechanicallyscanned in only the elevational direction. Figure 1(a) showsa typical cross-sectional image of a carotid artery. It ispreferable to avoid the eﬀect of pulsation of the artery in
the estimation of the elevational distance between two 2D
ultrasonic images. Therefore, tissues outside the artery [region
surrounded by the blue line in Fig. 1(a)] are used for theestimation of the elevational distance. In the present study, theelevational distance was estimated from the phase shift ofultrasonic RF echoes measured in the two image planes. In
addition, as shown in Fig. 1(b), we plan to determine therelative position between the two image planes by estimating
the elevational distance using the phase shift at multiple
spatial points and by determining functions representing theimage planes15) using the estimated elevational distances.
Figure 2 shows the deﬁnition of the directions of focus
and an illustration of the acoustic ﬁeld. As shown in Fig. 2, a
10 MHz linear array probe (Aloka UST-5545), whose beamwidth was about 800 m (width at half maximum of acousticpressure proﬁle) in the elevational direction at the focal
point, was used. For simplicity, we assumed that theultrasonic (US) probe was moved in the elevational (y)and axial (z) directions. Figure 3 shows the deﬁnition of the
movement of the US probe. In this study, we assumed thattwo point scatterers were located at position Aðxi ; yi;n ; zi;k Þ [inthe n-th frame (n ¼ 1; 2; . . . ; N) at the i-th lateral beam

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Jpn. J. Appl. Phys. 50 (2011) 07HF09

(a)

al
Axial

E
y

(b)

Lateral
x

on

ati
lev

A. Suzuki et al.

Elevational

5 mmx
Lateralz

Estimated plane^ +c^y = ^ax+bz
Δ y4

z
Axial

14 mm

y

Δz 3
Δ y3
Δ y2

n frame

Δz 2
Δz 1

Δ y1
23 mm

Δz 4

n + 1frame

larger influence of pulsation area
small influence of pulsation area
Fig. 1. (Color online) (a) Typical carotid short-axis B-mode image. (b) Schematic of estimation of movement of two 2D image planes.

(b-1)
x Lateraly Elevational
linear probe
10 MHzz
Axial(a)ultrasonic probey Elevationalx Lateral

electrical focus
5 mm

30 mm
hydrophonewater tank(b-3) xy5Normalized amplitudez
acoustic focus Axial
(b-2)
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0

3 mm

z

automatic
XYZ stage
Normalized
acousticpressure

10

15
Axial
z [mm] 20

0.9
0.7
0.5
0.3

25
30
-1000 -500 0 500 1000
Elevational

500 1000 1500 2000 2500
Time [ns]Δy [ μm]

Fig. 2. (Color online) (a) Deﬁnition of directions of focus. (b) Schematic showing system for measurement of transmitted pulse with a hydrophone,measured transmitted pulse (yi;n ¼ 0 m), and measured acoustic ﬁeld.

(i ¼ 1; 2; . . . ; I) and at the k-th depth point (k ¼ 1; 2; . . . ; K)]
and Bðxi ; yi;n ; zi;ðkþkÞ Þ. When position Oðxi ; yi;n ; 0Þ at thecenter of the array in the n-th frame was moved to position
O0 ðxi ; yi;n þ yi;n ; zi;n Þ in the (n þ 1)-th frame and theelevational displacement yi;n was less than the beam width(0 < yi;n < 400 m), the echo from point Aðxi ; yi;n ; zi;k Þ isshown at position A0 in the ultrasonic image obtained in the(n þ 1)-th frame. In the same way, the echo from positionBðxi ; yi;n ; zi;ðkþkÞ Þ is displayed at position B0 in the (n þ 1)-thframe. In this case, O0 A ¼ O0 A0 and O0 B ¼ O0 B0 . By the
Pythagorean theorem, the relationship among these positions
is expressed as follows:

O0 A2 ¼ y2i;n þ ðOA þ zi;n Þ2 ;ð1Þ

¼ y2i;n þ ðOB þ zi;n Þ2 :

ð2Þ

0

2

OB

By solving these simultaneous equations, the displacementin the elevational direction yi;n and the displacement in theaxial direction zi;n are given as
ðO0 B2  O0 A2 Þ  ðOB2  OA2 Þ;ð3Þ2ðOB  OAÞ
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
y^i;n ¼ O0 A2  ðOA þ ^zi;n Þ2 ;
ð4Þwhere OA and OB correspond to the depths of the k-th and(k þ k)-th depth points zi;k and zi;ðkþkÞ , respectively.

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Jpn. J. Appl. Phys. 50 (2011) 07HF09(b)

LateralElevational yx
Axial

LateralElevational yx
n + 1framen frameultrasonic probe
z
Axial

(a)

A. Suzuki et al.

n + 1frame
ultrasonic probe

n framezO’(x i , y i,n+Δ y i,n, Δz i,n)O’Δ z i,nΔ z i,nO

O(x i , y i,n, 0)

Δ y i,nΔ y i,nA(x i , y i,n, z i,k)main lobe

O’A = O’A’ A
O’B = O’B’A’B

B (x i , y i,n, z i,k + Δk)

B’Fig. 3. (Color online) (a) Deﬁnition of direction of mechanical scanning. (b) Geometrical deﬁnition.

By measuring the phase shifts of ultrasonic echoes, we
can accurately estimate the distances O0 A and O0 B as
O0 A ¼ O0 A0 ¼ zi;k þ uz;A ðiÞ;O0 B ¼ O0 B0 ¼ zi;ðkþkÞ þ uz;B ðiÞ;

ð5Þ
ð6Þ

where uz;A ðiÞ and uz;B ðiÞ are the axial displacements less than
the sampling interval z. By using the sound speed c0 and thesampling frequency fs , the sampling interval z is given asz ¼

c0¼ 12:5 m:
2fs

ð7Þ

In this study, the sound speed was c0 ¼ 1000 m/s in thesilicone phantom and sampling frequency was set at
fs ¼ 40 MHz.
2.2 Estimation of the displacement by the cross spectrum

The displacement of a point scatter between frames (causedby change in the propagation distance) less than thesampling interval z is estimated using the phase shift ofRF signals by the cross spectrum to obtain the change in the
distance O0 A–OA.
The phase shift ð f0 ; xi ; yi;n ; zi;k Þ is calculated from thefrequency spectra Gn ð f0 ; xi ; zi;k Þ and Gnþ1 ð f0 ; xi ; zi;k Þ of the
RF echoes rfn ðxi ; zi;k Þ and rfnþ1 ðxi ; zi;k Þ as16)
exp½ jð f0 ; xi ; yi;n ; zi;k ÞG ð f0 ; xi ; zi;k Þ  Gnþ1 ð f0 ; xi ; zi;k Þ:¼ n
jGn ð f0 ; xi ; zi;k Þ  Gnþ1 ð f0 ; xi ; zi;k Þjð8ÞIn this study, the frequency spectrum Gn ð f0 ; xi ; zi;k Þ is
obtained at the lateral beam position xi and the elevational
position yi;n by extracting an RF signal with a Hanningwindow with a length of (2Wdep þ 1) points in the axial
direction centered at a depth zi;k . The frequency at the
maximum of the power spectrum jGn ð f ; xi ; zi;k Þj2 was usedas the frequency of interest, f0 .

The displacement uz;A ðiÞ is less than the sampling intervalz, therefore, uz;A ðiÞ between rfn ðxi ; zi;k Þ at the elevational
position yi;n and rfnþ1 ðxi ; zi;k Þ at (yi;n þ yi;n ) is given byaccumulating the displacement between two consecutive
frames obtained from the phase shift ð f0 ; xi ; yi;n ; zi;k Þ ofrfnþ1 ðxi ; zi;k Þ from rfn ðxi ; zi;k Þ as18)
udz;A ðiÞ ¼

n1
X
c0
l¼0

4f0

ð f0 ; xi ; yi;l ; zi;k Þ:

ð9ÞSimilarly, the displacement udz;B ðiÞ is given by the phaseshift ð f0 ; xi ; yi;n ; zi;ðkþkÞ Þ of rfnþ1 ðxi ; zi;ðkþkÞ Þ fromrfn ðxi ; zi;ðkþkÞ Þ as
n1
X
c0
ud
ð f0 ; xi ; yi;l ; zi;ðkþkÞ Þ:ð10Þz;B ðiÞ ¼
4f0
l¼0
By substituting eqs. (9) and (10) into eqs. (3) and (4), theelevational displacement y^i;n and the axial displacement^zi;n are expressed as follows:122d½ðzi;k þ k  z þ udz;B ðiÞÞ  ðzi;k þ uz;A ðiÞÞ
2k  z ððzi;k þ k  zÞ2  z2i;k Þ:ð11Þ
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2ð12Þ
zi;n Þ2 :y^i;n ¼ ðzi;k þ udz;A ðiÞÞ  ðzi;k þ ^^zi;n ¼

By averaging y^i;n and ^zi;n for (2Wlat þ 1) beams, themean elevational and axial displacements yi;n and zi;n ofeach region (surrounded by blue lines in Fig. 1) are obtained
as follows:WlatX1yi;n ¼y^ðiþlÞ;n ;ð13Þ
2Wlat þ 1 l¼W

07HF09-3latzi;n ¼

1

Wlat
X

2Wlat þ 1 l¼W^zðiþlÞ;n :

ð14Þ

lat

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Jpn. J. Appl. Phys. 50 (2011) 07HF09(b)x LateralyΔy=0
Elevational

Elevational

z
Axial

y

x

Axial

(a)

A. Suzuki et al.

Lateral

x0

z0 =

x96

x180
water

0 mm

siliconez

siliconexyz θ stage
water tanksilicone sizex × y × z = 50 mm × 60 mm × 15 mm

16 mm
18 mm

10
5
0
-5
0

50 100 150 200 250 300 350 400

Δy i,n [ μm]actual elevational displacement
30

(c)

20
10
0
-10
-20
-30

0

50 100 150 200 250 300 350 400

Δ y i,n [ μm]

(b)

20

u^z,B [ μm]u^ z,A [ μm]

15estimated residual displacement

(a)

20

25

15
10
5
0
-5
0

50 100 150 200 250 300 350 400

Δy i,n [ μm]actual elevational displacement
estimated elevational displacement
Δ ^y i,n [ μm]25estimated axial displacement
Δ ^z i,n [ μm]estimated residual displacement

Fig. 4. (Color online) (a) Schematic of system for basic experiment. (b) B-mode image of the silicone phantom.

700

(d)

600
500
400
300
200
100
0
-100
0

50 100 150 200 250 300 350 400

Δ y i,n [ μm]actual elevational displacement

actual elevational displacement
actual value

estimated value

Fig. 5. (Color online) Estimated displacements of the phantom. (a) Axial displacement u^ z;A . (b) Axial displacement u^ z;B . (c) Axial (vertical) displacements^zi;n . (d) Elevational (horizontal) displacements y^i;n .

3. Experimental Methods

In this study, the elevational distance between two 2D US
images was estimated using the phase shift properties of
echoes from point scatterers by assuming that there are many
point scatterers in the tissue. In the present study, point
scatterers in a target (silicone phantom), which mimics
tissues, were used to determine elevational distance using
the phase shift of received RF echoes.
Figures 4(a) and 4(b) show an experimental system anda 2D US image of the silicone phantom, respectively.
The phantom was made of silicone rubber (sound speed:1000 m/s) containing 5% graphite powder by weight (75{106 m). It was measured at a frame rate of 60 Hz andmoved by an automatic XYZ stage at a constant speed of1200 m/s, resulting in an elevational distance between
images in two consecutive frames of 20 m (typical freehandscanning speed).9) RF echoes were acquired using a 10 MHz
conventional linear-type probe of ultrasonic diagnostic
equipment (Aloka SSD-6500). The sampling frequency of
the RF signal was fs ¼ 40 MHz and the sound speed in waterat 20 degrees was c0 ¼ 1485 m/s (Willard’s experimental
equation).17)

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A. Suzuki et al.

(b)

y

x

Axial

Elevational

Lateral

z

z0 =

x0

x180

x96

estimated elevational displacement
Δ ^y i,n [ μm](1) OA = 4 mm(a)Actual value

700
600
500
400
300
200
100
0
-100

silicone(2) 6 mm(3) 8 mm

3.4 mm

16 mm

4.1 mm0

50 100 150 200 250 300 350 400(2) OA = 6 mm
estimated elevational displacement
Δ ^y i,n [ μm](1) 4 mm

50 100 150 200 250 300 350 400

Δ y i,n [ μm]actual elevational displacement

0 mm

water

0

1200
1000
800
600
400
200
0
-200

Δ y i,n [ μm]actual elevational displacement(3) OA = 8 mm
estimated elevational displacement
Δ ^y i,n [ μm]

18 mm

1600
1400
1200
1000
800
600
400
200
0
-200
0

50 100 150 200 250 300 350 400

Δ y i,n [ μm]actual elevational displacement

Fig. 6. (Color online) Experimental results at diﬀerent distances of OA. (a) Illustration of position of ROI and axial kernel size. (b) Comparison of theaccuracies obtained by diﬀerent depths of OA at a constant axial kernel size (Wdep ¼ 90 points) and distance AB (k ¼ 90 points).

4. Results
4.1 Estimation of elevational distance of the phase shift
between 2D ultrasound images of the silicone
phantom

Figures 5(a)–5(d) show estimation results obtained usingeqs. (9)–(12), respectively. The plots and vertical bars
respectively show the means and standard deviations of
the elevational and axial displacements. The mean andstandard deviation at each position were obtained using
individual elevational and axial displacements along 41
ultrasonic beams. The solid line in Fig. 5(d) shows the actual
elevational displacement which was assigned to the automatic XYZ stage (resolution in the elevational direction:
1 m). In this experiment, the silicone phantom was moved
in only the elevational direction.
Figures 5(a) and 5(b) show the estimated displacements
uz;A and uz;B at positions A and B [in Fig. 3(b)], respectively. In Fig. 5, the axial size of kernels and the distance ofAB were set at 2.2 mm (Wdep ¼ 90 points) and 1.1 mm(k ¼ 90 points), respectively. In Figs. 5(a) and 5(b), theestimated errors increase with the elevational displacement
yi;n .

Figures 5(c) and 5(d) show the estimated elevational displacement y^i;n and the axial displacement ^zi;n , respectively. In the estimation of the elevational distance y^i;n , data
points that give an imaginary number of eq. (12) were notused because scatterers at such points were not considered
to be located at the center of the ultrasonic probe in theelevational direction at the beginning of mechanical scanning.
4.2 Dependence of the accuracy on distance OA atconstant axial kernel size and distance AB

In this section, the dependence of accuracy on the distance
OA was evaluated using a constant axial kernel size and the
distance AB. To evaluate the accuracy of the estimation of
the elevational distance, the RMS error of the estimatedmean elevational displacement yi;n from the actual
elevational displacement was used. The RMS error "meanin the estimated mean elevational displacement yi;n fromthe actual elevational displacement (displacement uy ¼
20 m between consecutive frames) is expressed asvﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
u
N
u1 X
ð15Þ
ðyi;n  uy  nÞ2 ;"mean ¼ t
N n¼1

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(a-2)

estimated elevational displacement
Δ ^y i,n [ μm]ε mean = 49.9

A. Suzuki et al.

μm

(a-3)

600
500
400
300
200
100
0
-100
0

50 100 150 200 250 300 350 400

Δ y i,n [ μm]actual elevational displacement1000

ε mean = 118.2 μ m

estimated elevational displacement
Δ ^y i,n [ μm](a-1)

estimated elevational displacement
Δ ^y i,n [ μm]Jpn. J. Appl. Phys. 50 (2011) 07HF09
ε mean = 247.8 μ m

1200
1000
800
600
400
200
0
-200
0

50 100 150 200 250 300 350 400

Δ y i,n [ μm]actual elevational displacement

(b)

(3)

OA = 4 mm

RMS error
ε mean [μm]

800

240
200
160
120
80
40

600
400
200

(2)

(1)
minimum RMS

0
0.75

-200
0 50 100 150 200 250 300 350 400

AB [mm]Δ y i,n [ μm]actual elevational displacement
Actual value

1.5

2.25 4.5

1.5
3.0
Axial kernel size
(2Wdep +1) Δ z[mm]

Fig. 7. (Color online) Dependences of accuracy on axial kernel size and distance AB at a constant distance of OA (¼ 4 mm). (a) Estimation result at eachaxial kernel size and AB. (a-1) AB (k ¼ 120 points), Wdep ¼ 120 points. (a-2) AB (k ¼ 60 points), Wdep ¼ 60 points. (a-3) AB (k ¼ 30 points),
Wdep ¼ 30 points. (b) RMS error.

where N is the number of frames used for calculation of"mean . N was set at 20 to keep the elevational displacement
less than the elevational beam width.
Figure 6 shows the elevational displacements y^i;n of thephantom at each distance of OA [(1)–(3)]. In Fig. 6, themeans tend to follow the actual proﬁle, and the standard
deviation decreases when the distance OA is smaller.
4.3 Dependence of accuracy on axial kernel size anddistance AB at constant distance OA

In this section, the dependences of the accuracy on the axial
kernel size and the distance AB (¼ k  z) were evaluatedat a constant distance of OA. To evaluate the accuracy of theelevational distance estimation, the RMS error "mean deﬁned
by eq. (15) was used.
Figure 7(a) shows the errors in the elevational displacements fy^i;n g and axial displacements f^zi;n g of the phantom
estimated by the proposed method for each combination of
the distance of AB and the length of the axial kernel size
ð2Wdep þ 1Þ  z. In Fig. 7(a), the means tend to follow theactual proﬁle and the standard deviation is larger when theaxial kernel size and distance AB are larger. Figure 7(b)shows the RMS error "mean at each axial kernel size anddistance AB. As shown in Fig. 7, the RMS error "mean is
reduced by lengthening the size of the axial window and the
distance AB. The least RMS error "mean achieved using anaxial kernel size of 3.0 mm (Wdep ¼ 120 points) and a
distance AB of 1.5 mm (k ¼ 120 points) was 49.9 m.

5. Discussion

In this study, elevational displacements were estimated using
the phase shifts (corresponding to the axial displacement) of
RF signals in two 2D US images. The phase shift due to theelevational displacement was estimated by the cross spectrum technique, and a larger correlation kernel size in theaxial direction was found to achieve better accuracy. Errors
in the estimated elevational displacements were reduced byincreasing the kernel size, and similar errors were obtainedwith kernel sizes greater than or equal to 1.0 mm.
The result that further shortening of the distance OA
resulted in a more accurate RMS error can be describedusing eqs. (11) and (12). For simplicity, eqs. (11) and (12)were rearranged as follows:^zi;n2OAðu^ z;B ðiÞ  u^ z;A ðiÞÞ þ 2ABu^ z;B ðiÞ þ u^ 2z;B ðiÞ þ u^ 2z;A ðiÞ
;
2AB
ð16Þ
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
y^i;n ¼ 2OAðu^ z;B ðiÞ  ^zi;n þ u^ 2z;A ðiÞ þ ^z2i;n Þ:
ð17Þ¼

The estimated displacements u^ z;A ðiÞ and u^ z;B ðiÞ contain noisefrom the experimental system and these terms are multiplied
by a constant distance of OA. Therefore, the shorter the
distance OA is, the less noise aﬀecting estimated values^zi;n and y^i;n is. As shown in Fig. 3, scatterers wereassumed to be located at the center of the i-th beam.

07HF09-6# 2011 The Japan Society of Applied Physics

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Jpn. J. Appl. Phys. 50 (2011) 07HF09

A. Suzuki et al.

Actually, however, point scatterers were located at various
positions relative to the beam position. This discrepancy
between the assumption and the actual situation leads to alarge error and a large standard deviation. In our future
work, we plan to decrease the bias and random errors byselecting valid scatterer positions by referring to the echo
amplitude (the higher the elevational displacement is, thelower the amplitude is).
In this study, the displacement estimates are biased when
the center frequency of the RF echo changed. Such an
apparent change in the center frequency could be caused by
the interference of echoes from scatterers. Therefore, theeﬀect of the center frequency variation on the estimation of
the elevational displacement should be reduced.19)
Finally, in this study, we estimated the elevational andaxial displacements by solving simultaneous equations using
only two data points. Obviously, the accuracy would be
improved using more points. In our future work, we plan todecrease (bias and random) these errors using the leastsquares method and to show the eﬀectiveness of theestimation in the elevational direction.
6. Conclusions

1) S. Ogawa and S. Ueno: Hishinshu Kashika Handobukku (Noninvasive

Visualize Technique Handbook) (NTS, Tokyo, 2007) [in Japanese].
2) I. Akiyama: Acoustic Imaging (Corona, Tokyo, 2010) [in Japanese].
3) G. W. Fischer, I. S. Salgo, and D. H. Adams: J. Cardiothoracic Vasc.

Anesth. 22 (2008) 904.
4) J. S. Suri, C. Kathuria, R. Chang, F. Molinari, and A. Fenster: Advances in

5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)

In this study, we investigated a method of estimating theelevational distance between two 2D ultrasound images. We
achieved an accuracy of the RMS error of 49.9 m under the
optimum conditions, which is less than that of the previous
method. The proposed sensorless free-hand method shows
potential for more accurate estimation of the distance inthe elevational direction using the phase shift properties of
echoes than existing methods.

16)
17)
18)
19)

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Diagnostic and Therapeutic Ultrasound Imaging (Artech House, Norwood,MA, 2008).
A. Fenster, K. Surry, W. Smith, J. Gill, and D. B. Downey: Comput.
Graphics 26 (2002) 557.
R. Weller, T. Pfau, M. Ferrari, R. Griﬃth, T. Bradford, and A. Wilson:
Ultrasound Med. Biol. 33 (2007) 402.
M. Hastenteufel, M. Vetter, H. Meinzer, and I. Wolf: Ultrasound Med.
Biol. 32 (2006) 1359.
D. C. Barratt, A. H. Davies, A. D. Hughes, S. A. Thom, and K. N.
Humphries: Ultrasound Med. Biol. 27 (2001) 957.
D. C. Barratt, A. H. Davies, A. D. Hughes, S. A. Thom, and K. N.
Humphries: Ultrasound Med. Biol. 27 (2001) 1421.
W. Birkfellner, F. Watzinger, F. Wanschitz, R. Ewers, and H. Bergmann:IEEE Trans. Med. Imaging 17 (1998) 737.
T. J. Macgillivray, E. Ross, W. Simpson, and C. A. Greig: Ultrasound Med.
Biol. 35 (2009) 928.
G. Treece, A. Gee, P. Cash, and C. Hamish: Ultrasound Med. Biol. 29(2003) 529.
P. Hassenpﬂug, R. W. Prager, G. M. Treece, and A. H. Gee: UltrasoundMed. Biol. 31 (2005) 1499.
J. C. R. Seabra, L. M. Pedro, J. F. Fernandes, and J. M. Sanches: IEEE
Trans. Biomed. Eng. 56 (2009) 1442.
L. Anton and J. Yamashita: Yasashii Senkei Daisu no Oyo (Easy Linear
Algebra) (Gendai-Sugakusha, Kyoto, 1984) [in Japanese].
H. Kanai: Oto to Shindo no Supekutoru Kaiseki (Spectrum Analysis ofSound and Vibration) (Corona, Tokyo, 2004) [in Japanese].
J. Saneyoshi: Choonpa Binran (Ultrasonic Tehchnical Handbook) (Nikkan
Kogyo Shimbun, Tokyo, 1991) [in Japanese].
H. Kanai, M. Sato, Y. Koiwa, and N. Chubachi: IEEE Trans. Ultrason.
Ferroelectr. Freq. Control 43 (1996) 791.
H. Hasegawa and H. Kanai: IEEE Trans. Ultrason. Ferroelectr. Freq.
Control 55 (2008) 2626.

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