Japanese Journal of Applied Physics
Vol. 45, No. 5B, 2006, pp. 4727–4731#2006 The Japan Society of Applied Physics

Accurate Ultrasonic Measurement of Surface Proﬁle Using Phase Shiftof Echo and Inverse Filtering
Chihiro A RIHARA, Hideyuki H ASEGAWA and Hiroshi K ANAI
Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan(Received November 30, 2005; accepted February 28, 2006; published online May 25, 2006)

Atherosclerosis is the main cause of circulatory diseases such as myocardial infarction and cerebral infarction, and it is very
important to diagnose atherosclerosis in its early stage. In the early stage of atherosclerosis, the luminal surface of an arterial
wall becomes rough because of the injury of the endothelium [R. Ross: New Engl. J. Med. 340 (2004) 115]. Conventional
ultrasonic diagnostic equipments cannot detect such roughness on the order of micrometer because of their low resolution of
approximately 0.1 mm. In this study, for the accurate detection of surface roughness, an ultrasonic beam was scanned in the
direction that is parallel to the surface of an object. When there is a gap on the surface, the phase of the echo from the surfacechanges because the distance between the probe and the surface changes during the scanning. Therefore, surface roughness
can be assessed by estimating the phase shift of echoes obtained during the beam scanning. Furthermore, lateral resolution,
which is deteriorated by a ﬁnite diameter of the ultrasound beam, was improved by an inverse ﬁlter. By using the proposed
method, the surface proﬁle of a phantom, which had surface roughness on the micrometer order, was detected, and theestimated surface proﬁles became more precise by applying the inverse ﬁlter. [DOI: 10.1143/JJAP.45.4727]
KEYWORDS: roughness, surface profile, phase shift, inverse filter, point spread function, atherosclerosis

1.

Introduction

Recently, increases in the number of circulatory diseases
such as myocardial infarction and cerebral infarction havebecome a serious problem. Therefore, it is important todiagnose atherosclerosis, which is the main cause of these
diseases, in its early stage. The intima-media thickness isuseful marker for the diagnosis of atherosclerosis.1) In
addition, the evaluation of the elasticity of an arterial wall isalso useful for the diagnosis of early-stage atherosclerosis2)
because there are signiﬁcant diﬀerences between the elastic
moduli of normal arterial walls and those aﬀected byatherosclerosis.3,4)
We developed a method, namely, the phased tracking
method, for measuring the regional strain and viscoelasticity
of an arterial wall.5–15) Although this method is useful for the
diagnosis of atherosclerotic changes of the intima-media
region, the initial step of atherosclerosis development wasreported to be endothelial damage.16) Therefore, for the
diagnosis of early-stage atherosclerosis, it is more eﬀective
to initially evaluate endothelial damage. For the evaluationof endothelial function, several methods have been proposed
for the evaluation of endothelium function including
measurement of changes in diameter17) and intima-media
elasticity15) caused by ﬂow-mediated dilation. In thesemethods, the measured changes in diameter and elasticity
are caused by the response of the media to nitric oxide
generated by the endothelium. Therefore, the region where
the endothelium was injured cannot be identiﬁed.
The luminal surface of a clinically healthy artery iscovered by a layer of endothelial cells and it is smooth. In
the early stage of atherosclerosis, endothelial cells on theluminal surface are damaged,16) edema develops under theendothelium, endothelial cells are separated from the intima,
and then the luminal surface becomes rough. There areseveral studies on the ultrasonic detection of the surface
roughness of objects based on the characteristics of reﬂectedechoes18) or the three-dimensional representation of obE-mail address: kanai@ecei.tohoku.ac.jp

jects.19) Furthermore, there are several studies on thecomputer modeling of an object with a nonplanar but
smooth surface.20,21) However, this computer modeling is not
yet available for an object with a rough surface.22) To detectatherosclerosis in its early stage, the roughness of theluminal surface needs to be detected with an accuracy of
micrometers that cannot be achieved by these methods. In
conventional ultrasonic diagnostic equipments, the lateralinterval of ultrasonic beams is approximately 100 mm, and a
B-mode image is constructed using the amplitude of thereceived echo. Therefore, the spatial resolution of a B-mode
image of an ultrasonic diagnostic equipment is 100 mm in thelateral direction and 150 mm, which corresponds to thewavelength at 10 MHz, in the axial direction of the ultrasonic beam.
In this study, to increase the spatial resolution in themeasurement of surface proﬁle, surface roughness isevaluated using the phase of the echo from the surface ofan object by scanning the ultrasonic beam with a pitch that ismuch smaller than the element pitch of the ultrasonic probe.
When there is a gap on the surface, the phase of the received
echo from the surface is changed by scanning. By estimating
the phase shift of echoes obtained during the scanning,surface roughness is accurately detected in the range of75 mm (180 ) which corresponds to the wavelength at
10 MHz. Furthermore, lateral resolution, which is deteriorated by a ﬁnite beam diameter, is improved by an inverse
ﬁlter which is designed based on a point spread function23) ofthe ultrasonic beam.
2.

Principle

2.1 Fine beam scanning using stage
A conventional linear-type ultrasonic probe of an ultrasonic diagnostic equipment (Aloka SSD-6500) used in thisstudy transmits and receives ultrasonic pulses by scanningwith the ultrasonic beam in the direction that is parallel tothe surface of an object [center frequency f0 ¼ 10 MHz,
scanning pitch L ¼ 0:1 mm, and number of electronicallyscanned beams M ¼ 70 (l1 ; l2 ; . . . ; lm ; . . . ; lM )]. UltrasonicRF echoes are sampled at 40 MHz. B-mode images consist

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Jpn. J. Appl. Phys., Vol. 45, No. 5B (2006)

C. ARIHARA et al.

ultrasonic probexztime (depth)

ultrasonicdiagnostic
equipment

water

stage

ultrasonic probex+∆ x x x–∆ x x x+∆ x
∆x ∆x

x

x–∆ x

objectz(x)∆ z(x–∆ x)
object∆z(x)Fig. 2. Surface proﬁle, zðxÞ, obtained by phase shift of echoes.

controller
Fig. 1. Experimental system for measurement of surface of phantom withultrasound.

of sampled points, which are obtained every 19.25 mm(sound speed c0 ¼ 1; 540 m/s) in the axial direction and
0.1 mm in the lateral direction of the ultrasonic beam. A setof two-dimensional discrete data obtained by one scan iscalled a frame.
Figure 1 shows the schematic diagram of the experimental
system. The depths of the object surface and the focal point
were 12 mm from the surface of the ultrasonic probe.
The object was measured at a frame rate of FR ¼
10 frames/s by moving the probe automatically using a stage
at a constant speed vstg ¼ 50 mm/s for 2 s. The moving
distance for 2 s was 100 mm, which corresponds to theinterval, L, of electronic scanning of the ultrasonic beam, inthe x-direction. Let us deﬁne the location of beam l1 in theﬁrst frame by xð1; 1Þ ¼ 0 mm; the location, xðk; mÞ, of beam
lm at k-th frame is deﬁned as follows:xðk; mÞ ¼ ðm  1Þ  L þ

ðk  1Þ  vstg:
FR

set to be much smaller than 37.5 mm to prevent the aliasingof the estimated phase shift, ðxÞ.
The surface proﬁle, zðxÞ, cannot be obtained when there
are echo-lacking beam positions in the region of interest
because the zðxÞ of each of the two neighboring beams is
integrated with respect to the beam position.
In this experiment, it is diﬃcult for the surface of theultrasonic probe to move in a direction perfectly parallel tothe surface of the object. Therefore, the global angle
between the surface of the ultrasonic probe and the surfaceof the object needs to be compensated. Angle compensatedproﬁle, gðxÞ, was obtained by subtracting the linear line,
which corresponds to the global slope of the surface of theobject relative to that of the probe, from the estimated zðxÞ.
The linear line is obtained using the least-squares method byexperimental data, fðx1 ; z1 Þ; ðx2 ; z2 Þ; . . . ; ðxNdata ; zNdata Þg, as follows [zi ¼ zðxi Þ, i ¼ 1; 2; . . . ; Ndata ]:gðxÞ ¼ zðxÞ  ða^x þ b^Þ;
ð2:4Þ
!
a^
1
¼
^
det
b
0
!
!
!
! 1
Ndata
Ndata
Ndata
Ndata
X
X
X
X
B
1
xi zi 
xi
zi C
B
C
i¼1
i¼1
i¼1
i¼1
B
C
!
!
!
! C;
B
Ndata
Ndata
Ndata
Ndata
B
C
X
X
X
X
@ 
xi
x i zi þ
x2i
zi A

ð2:1Þ

The probe was moved by x ¼ 5 mm during a frameinterval.

i¼1

2.2 Detection of gaps on surface
The quadrature demodulated signal of the received RF
echo was used to measure surface proﬁle with an accuracy of
micrometer-order in the axial direction of the beam. Whenthe phase shift of the received echoes between two
neighboring locations, x and x þ x, is ðxÞ, whichcorresponds to the diﬀerence, zðxÞ, between the distancesfrom the probe to the surface at x and that at x þ x, surface
proﬁle, zðxÞ, is obtained as follows:c0 ðxÞ
zðxÞ ¼
;
ð2:2Þ
4 f
Zx 0
zðxÞ ¼zðx0 Þ dx0 :ð2:3Þ
0Figure 2 shows the method to obtain surface proﬁle, zðxÞ,from the phase diﬀerence of the received echoes between
two neighboring locations. In the experimental system usedin this paper, zðxÞ of less than 37.5 mm, which is a quarterof the wavelength and is the aliasing limit, can be measured.
In this study, the displacement of the probe in the x-axis was

i¼1

i¼1

i¼1

ð2:5Þ
det ¼

Ndata
X
i¼1

!
x2i

Ndata
X
i¼1

!
1 

Ndata
X

!2
xi

;

ð2:6Þ

i¼1

where a^ is the global slope of the linear line, and b^ is thez-intercept. Ndata is the total number of beam positions.
2.3 Improvement of lateral resolution using inverse ﬁlter
Figure 3 shows the block diagram for estimating surface
proﬁle. Beam diameter deteriorates lateral resolution. The
measured surface proﬁle, gðxÞ, of the object is expressed bypoint spread function, hðxÞ, determined by the diameter ofthe ultrasonic beam, actual surface proﬁle, f ðxÞ, noise nðxÞ,and their Fourier spectra, GðuÞ, HðuÞ, FðuÞ, and NðuÞ, asfollows:gðxÞ ¼ hðxÞ  f ðxÞ þ nðxÞ;GðuÞ ¼ HðuÞ  FðuÞ þ NðuÞ;where u is the spatial frequency.

4728

ð2:7Þ
ð2:8Þ

C. ARIHARA et al.

(a)noise surface profile estimatedactualn(x) obtained by surface profile
surface profile
experiment
^f(x)
f(x)
g(x)
m(x)
h(x)
inverse filter
point spreadfunction( 10 -2 )
1.2

N(u)
F(u)

H(u)

G(u)

M(u)

^
F(u)

Fig. 3. Block diagram for estimating surface proﬁle, f^ðxÞ.

0.8
0.6

0.2

1
jHðuÞj2;HðuÞ jHðuÞj2 þ Pn ðuÞ=P f ðuÞ

ð2:10Þ

The estimated surface proﬁle, f^ðxÞ, is obtained by the
inverse Fourier transform of F^ ðuÞ.
3.

1000

1500

2000

distance x [µm]

-20
-40
-60

|H’(u)|2|P(u)|2

-80
-100
0

0.5

1

1.5

2

2.5

3

3.5

4

spatial frequency u [mm-1 ]Experimental Results

3.1 Determination of inverse ﬁlter property
The beam proﬁle was measured by transmitting ultrasound to a stainless wire (diameter: 16 mm) and receiving thereﬂected echo from the wire by moving the x-position of theultrasonic probe. When the diameter of the wire is
suﬃciently small compared with that of the beam, the wireis assumed to be a point scatterer. The solid line in Fig. 4(a)shows the measured beam proﬁle, h0 ðxÞ. The beam width at
half maximum was 500 mm. In eq. (2.8), point spreadfunction, hðxÞ, is assumed to be noise free. Therefore, as
shown by the dashed line in Fig. 4(a), Gaussian function,pðxÞ, which has the minimum root mean squared diﬀerencewith respect to the measured beam proﬁle, h0 ðxÞ, was used as
point spread function, hðxÞ. Figure 4(b) shows the power
spectrum, jH 0 ðuÞj2 , of the measured beam proﬁle, h0 ðxÞ, andthe power spectrum, jPðuÞj2 , of Gaussian function, pðxÞ.
In addition, the signal to noise ratio Pn ðuÞ=P f ðuÞ must be
determined. However, the power spectrum, P f ðuÞ, of theactual surface proﬁle in a body is unknown. Therefore, theaveraged surface proﬁle, gðxÞ, of an object obtained by 10
measurements was used as P f ðuÞ. Then, the surface proﬁle,gn ðxÞ, of a ﬂat object (rubber plate) was measured for 10
times to determine the magnitude of noise. Noise, nðxÞ, wasdeﬁned by the diﬀerence between each gn ðxÞ and theaveraged proﬁle, gðxÞ, as follows:
10
1 X
gn ðxÞ;10 n¼1

ð3:1Þ

nn ðxÞ ¼ gn ðxÞ  gðxÞ:ð3:2Þ

gðxÞ ¼

500

0

ð2:9Þ

where Pn ðuÞ and P f ðuÞ are the averaged power spectra ofnoise and the actual surface proﬁle, respectively. If there isno noise, Pn ðuÞ is 0 and MðuÞ is an idealistic inverse ﬁlter.
The frequency spectrum, F^ ðuÞ, of the estimated surface
proﬁle, f^ðxÞ, is obtained as follows:F^ ðuÞ ¼ MðuÞ  GðuÞ:0(b)
power spectrum [dB]MðuÞ ¼

500 µm0.4

0Inverse ﬁlter, MðuÞ, is obtained by minimizing the meansquared diﬀerence between f ðxÞ shown by the model in
eq. (2.7) and the estimate, f^ðxÞ, as follows:24,25)

h’(x)
p(x)

1.0

normalized space

Frequency Spatial domain
domainJpn. J. Appl. Phys., Vol. 45, No. 5B (2006)

Fig. 4. (a) Measured beam proﬁle, h0 ðxÞ, and Gaussian function, pðxÞ,ﬁtted with measured beam proﬁle, h0 ðxÞ. (b) Spectra of measured beam
proﬁle, H 0 ðuÞ, and Gaussian function, PðuÞ.

The power spectrum, Pn ðuÞ, of noise was deﬁned by themean of 10 power spectra, jNn ðuÞj2 , of nn ðxÞ.
Pn ðuÞ ¼

10
1 X
jNn ðuÞj2 :10 n¼1

ð3:3Þ

3.2 Results of basic experiments
An object made of silicone rubber was placed in a watertank. Figure 5(b) shows the surface proﬁle along a black line
shown in Fig. 5(a). The surface proﬁle shown in Fig. 5(b)was measured using a surface proﬁlometer.
Power spectra, Pn ðuÞ, P f ðuÞ, jHðuÞj2 , and jMðuÞj2 , obtained by the above procedure are shown in Fig. 6. jMðuÞj2
was positive in a spatial frequency range of less than
2.0 mm1 , and negative over 2.0 mm1 . The signal wasincreased in a spatial frequency of less than 2.0 mm1 , andwas decreased over 2.0 mm1 . The inverse ﬁlter, jMðuÞj2 ,had the reciprocal property of the spectrum, HðuÞ, ofthe point spread function in the frequency range wherePn ðuÞ=P f ðuÞ was low.
Figure 7(a) shows a B-mode image obtained by theultrasonic diagnostic equipment. Figure 7(b) shows themeasured surface proﬁle, gðxÞ, of the object in the regionof interest surrounded by the white line in Fig. 7(a). A gap,

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C. ARIHARA et al.

(a)

(a)

distance x [µm]
0

8 cm

2000

4000

6000

distance [mm]

0

8 cm

10

15

(b)

(b)

12

20

10

surface profile [µm]surface profile [µm]

5

8
6
4
2
0
-2
0

500

1000

1500

2000

2500

15
10
5
0

3000

distance x [µm]

-5

0

1000

2000

3000

4000

distance x [µm]Fig. 5. (a) Image of object made of silicone rubber. (b) Surface proﬁle ofobject measured by bump instrument.

Fig. 7. (a) B-mode image. (b) Measured surface proﬁle, gðxÞ.

power spectrum [dB]100by the vertical motion of the probe because of the motion of
the automatic stage.
Figure 8(a) shows surface proﬁles, gðxÞ and f^ðxÞ. The
estimate, f^ðxÞ, of the surface proﬁle was obtained after
applying the inverse ﬁlter to gðxÞ. After inverse ﬁltering, thesurface proﬁle became steeper and the upper base of the
trapezoidal shape became almost ﬂat. Figure 8(b) shows thepower spectra of surface proﬁles, gðxÞ and f^ðxÞ. After inverse
ﬁltering, spatial frequency components less than 2.0 mm1 ,which sharpened the estimated surface proﬁle, f^ðxÞ, wereincreased, and the undesirable components over 2.0 mm1 ,which are caused by movement of the stage, were reduced.
The estimated surface proﬁle, f^ðxÞ, had ripples. Thisphenomenon called the Gibbs phenomenon occurs by
forcing high frequency components to be removed in the
Fourier analysis. Window function is generally used toreduce the ripples by the Gibbs phenomenon. However, in
this study, a rectangular window was used to avoid thedeformation of the surface proﬁle, gðxÞ.

50

0

-50

-100

0

0.5

1

1.5

2

2.5

3

3.5

4

spatial frequency u [mm-1]
Pn(u)
Pf (u)
Pn(u)/Pf (u)|H(u) |2
|M(u)|2
Fig. 6. Inverse ﬁlter property.

4.
which could not be recognized in Fig. 7(a), could be foundeasily in Fig. 7(b). The spatial resolution in Fig. 7(b) wasmuch improved in comparison with that of the conventional
B-mode image. Comparing Fig. 7(b) with Fig. 5(b), thewidth and height of the gap found in gðxÞ were similar to theactual surface proﬁle of the object (width: 2,000 mm, height:
12 mm). Cyclical noise shown in gðxÞ of Fig. 7(b) was caused

Conclusions

In this study, the surface proﬁle of an object made ofsilicone rubber on the micrometer order was measured using
the phase of an RF echo. By applying an inverse ﬁlter thatwas designed on the basis of the point spread function of an
ultrasonic beam to the measured surface proﬁle, theestimated surfaces became more precise than the measured
surface proﬁles without the inverse ﬁlter.

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C. ARIHARA et al.

(a)surface profile [µm]

20
g(x)
^
f(x)

15
10
5
0
-5

0

1000

2000

3000

4000

distance x [µm](b)
power spectrum [dB]

100
50
0

|G(u)|2^ 2|F(u)|

-50

-100

0

0.5

1

1.5

2

2.5

3

3.5

4

-1

spatial frequency u [mm ]Fig. 8. (a) Surface proﬁles, gðxÞ and f^ðxÞ. (b) Power spectra of surface
proﬁles, gðxÞ and f^ðxÞ.

Acknowledgement
The authors would like to thank Drs. Sahashi and Doi formeasurements with a surface proﬁlometer.

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