Hội nghị toàn quốc về Điều khiển và Tự động hoá - VCCA-2011

VCCA-2011

Improvement of Distorted Born Iterative Method for Reconstructing of

Sound Speed

Cải tiến phương pháp lặp Born vi phân áp dụng vào ảnh siêu âm cắt lớp

Tran Duc Tana, Gian Quoc Anh

b

a Vietnam National University/University of Engineering and Technology

b Nam Dinh University of Technology Education

e-Mail: tantd@vnu.edu.vn

Abstract:

Ultrasound imaging is widely used for medical

application. However, conventional ultrasound

imaging uses pulse echo method which can not

resolve structures that are smaller than the wavelength

of the incident wave. Inverse scattering of the

measured signal has a ability to characterize small

structures. The material properties such as sound

contrast are very useful to detect small objects. The

Born Iterative Method (BIM) and Distorted Born

Iterative Method (DBIM) are utilized to build a linear

relationship between the measured data and sound

contrast. DBIM can offer a fast convergence

compared to BIM. However DBIM is easier affected

by noise. This paper proposed a modified DBIM in

order to ensure both fast convergence and noise

reduction. An ensemble of measured ultrasound signal

has been combined to combat with noise. The

simulation result has proved that this method can

enhance the reconstruction’s quality without much

additional complexity.

Tóm tắt Chụp ảnh siêu âm hiện được sử dụng rộng rãi cho các

ứng dụng y tế. Tuy nhiên, phương pháp hiện tại trong

các máy siêu âm là sử dụng các tín hiệu phản hồi có

nhược điểm là khó có thể tái tạo được các cấu trúc có

kích thước nhỏ hơn bước sóng. Kỹ thuật tán xạ ngược

thì lại cho phép thực hiện điều này. Người ta có thể

nhận biết các khối u lạ vì khi tín hiệu siêu âm truyền

qua thì tốc độ truyền sẽ thay đổi. Hai phương pháp lặp

Born (BIM) và lặp Born vi phân (DBIM) được ưa

chuộng bởi cho phép xây dựng mối liên hệ tuyến tính

giữa tín hiệu siêu âm đo được với sự khác biệt tốc độ

siêu âm khi truyền qua khối u. Phương pháp DBIM

cho tốc độ hội tụ nhanh hơn BIM nhưng DBIM lại dễ

chịu ảnh hưởng của nhiễu hơn BIM. Bài báo này vì

thế đề xuất giải pháp cải tiến DBIM để vừa có thể

đảm bảo hội tụ nhanh lại vừa giảm tác động của nhiễu

bằng cách sử dụng tổ hợp tín hiệu siêu âm thu được ở

các đầu đo. Kết quả mô phỏng đã cho thấy chất lượng

khôi phục ảnh đã được cải thiện rõ rệt mà gần như

không ảnh hưởng tới độ phức tạp của thuật toán.

Symbols Symbols Unit Meaning Regularization parameter

Nt Number of transmitter

Nr Number of receiver

h mm Size of pixel

N Number of pixels in one

direction

co( r

) m/s Velocity of the propagating

wave

c1( r

) m/s Velocity of the propagating

wave in the object

r

(rad/m)2 Object function

rpinc Pa Incident field

rp

Pa Total pressure field

rp sc Pa Scatter field

Stopping error

k0 rad/m Wave number

Abbreviation DBIM Distorted Born Iterative Method

BIM Born Iterative Method

RRE Relative Residual Error

SVD Singular Value Decomposition

US Ultrasound

ROI Region of Interest

MSE Mean Square Error

Introduction Conventional medical ultrasound scanners have been

widely used for imaging most of soft tissue in the

human’s body (see Fig. 1). The popular principle of

these kinds of scanners is B-mode imaging which is

based on the reflectivity and scattering amplitude in

the tissues. B-mode scanners can operate in real time

due to their low complexity. On the other hand,

inverse scattering has ability to detect the small size

tissue that B-mode ultrasound can not [4]. Thus, it

will be promising applications in tissue

characterization.

Initial approaches utilized the projection theory that

widely used in X-ray and nuclear tomography [1], [2].

However, these ray-based methods were not suitable

with diffraction properties of ultrasound propagation.

After that, the Born Iterative Method (BIM) based on

first-order Born approximation has been introduced as

one of efficient diffraction tomography approaches

[5]. After that, the same authors has developed

Distorted Born Iterative Method (DBIM) in order to

improve BIM by using Green’s function updated each

798

Hội nghị toàn quốc về Điều khiển và Tự động hoá - VCCA-2011

VCCA-2011

iteration. DBIM has been proved to outperform DIM

by its fast convergence. However, the limitation of

DBIM is that it more sensitive with noise.

H. 1 Illustration of pulse-echo method that using linear

array transducer for obtaining a rectangular cross-

sectional image [3]

In [13], authors performed edge detection during the

iterative process to speed up the convergence and

enhance the reconstruction’s quality. However, its

complexity is high and it is still highly sensitive to

noise.

This paper focus to reduce these disadvantages by

using an ensemble of measured ultrasound signals for

the modified DBIM. At the certain positions of a

transmitter and a receiver, we measured the scattered

pressure in K times instead of once as previous

methods. Thus, the noise can be reduced by averaging

technique while still ensure the fast convergence of

existing DBIM. The paper is organized as following:

Section 2 presents the working principle of DBIM and

modified DBIM when using ensemble measured data.

Consequently, a simulation has been performed in

Section 3 in order to analysis the effective of

proposed scheme. Lastly, Section 4 is remarks and

conclusions.

Material and Method

We set up a measurement configuration of

transmitters and receivers in order to obtain the

scattered data. At an instance, only one transmitter

and one receiver are active to obtain a corresponding

measured data value. This data was processed using

DBIM to reconstruct the sound contrast of scatters. In

this way, we can detect if there is any tissue in this

medium.

2.1 Distorted Born Iterative Method

We assume that there is an infinite space containing

homogeneous medium such as water whose

background wave number is k0. There is also an

object with constant density and a wave number k(r)

put inside this medium (see Fig. 2). Note that k(r)

depends on the space. The wave equation of the

system can be shown as:

rprprp scinc (1)

Where rp

, rp inc , and rp sc are the total

pressure, incident pressure and scattered pressure

fields, respectively.

H. 2 Geometrical and acoustical configuration. The

region of interest (ROI) consists of the object need

to be reconstructed is centered at the origin of a 2D

space and discretized with N×N square pixels of

side h. The number of transmitters and receivers are

Nt and Nr respectively. With Nt transmitter and Nr

receivers, we would have NtNr measurements.

This equation can be expressed in detail:

'''' 0 rdrrGrprrprp inc

(2)

Where G0(.) is the homogenous Green function,

and 20

2krkr

is the object function need to

be reconstructed from scattered data.

One of the effective solutions to solve the equation (1)

by discreting is Method of Moment (MoM). The

pressure in the grid points (see Fig. 2) can be

computed in vector form with size N2×1:

incpDCIp . (3)

And the exterior points give scatter signal:

pDBp sc .. (4)

Where B is the matrix with Green’s coefficient

G0(r,r’) from each pixel to the receiver, C is the

matrix with Green’s coefficient G0(r,r’) among all

pixels, I is identity matrix, and D(.) is an operator

that transform a vector into a diagonal matrix. The

detail of calculation of B and C can be found in [8].

There are two unknown variables p and in

equations (3) and (4). In this case, the first Born

A0

A1

Transmitter

Receiver

799

Hội nghị toàn quốc về Điều khiển và Tự động hoá - VCCA-2011

VCCA-2011

approximation has been applied and the forward

equation (3) and (4) can be rewritten [7]:

.

..

M

pDBp sc

(5)

Where pDBM .. .

For each transmitter and receiver, we will have a

matrix M and a scalar value scp . Realize that

unknown vector has N×N variables which are

equal to the number of pixels in ROI. The object

function can be computed by iterations: )1()1()( nnn OOO (6)

Where )(nO and )1( nO are object functions at

present and previous steps, respectively; O can be

found by solving Tikhonov regularization problem: 2

2

2

2minarg

ssc

Mp (7)

Where sc

p is the (NtNr×1) vector contains the

difference between measured and predicted scattered

ultrasound signals; sM is system matrix (NtNr×N2)

formed by NtNr different matrixes M ; and is the

regularization parameter that needs to be carefully

selected [6].

2.2 Modified Distorted Born Iterative Method

In practice, we can measure the scattered signal

measuresc

p by subtract the total field in case of with

and without object inserted in the medium [9].

However, we can not ignore the noise here which can

be assumed as Gaussian noise. Thus, the exact

formula of vectorsc

p when using NtNr

measurements is:

npppscsc

measure

sc (8)

Where n is the (NtNr×1) noise vector.

From equation (8) we can see that DBIM is more

affected by noise than BIM. By increasing the

numbers of transmitter or receivers, we may only

obtain quick convergent but noise reduction. The time

consuming will even be higher if using large number

of measurements. However, at the certain positions of

a transmitter and a receiver, we can measure the sc

sp in K times. Thus, we will obtain an ensemble

of measured data sc

sp which have KNtNr

components. After that, the measured data of

scattering should be averaged before brought to

modified DBIM:

sci

K

i

averagesc

pK

p 1

1 (9)

This technique is called signal averaring applied in

the time domain in order to increase the signal to

noise ratio (SNR) by the square root of the K. All

three conditions are ensured in this case:

+ Scattered pressure signal and noise are uncorrelated.

+ Signal strength can be controlled at the same value

in the replicate measurements.

+ The measurement noise is random with a mean of

zero and a constant variance.

Consequently, the regularization problem should be:

2

2

2

2

minarg

saveragesc

Mp (10)

After that, the following iteration call DBIM should

be carried out to reconstruct object function :

Algorithm 1. Modified DBIM

1: Choose an initial 0

and inc

pp 0

2: While (n < Nmax) or (RRE < ) do

{

3: Calculate two matrices B and C ; p and scp

correspond to n

using equation (3) and (4).

4: Calculate the vector averagesc

p

5: Calculate RRE correspond to n

using (10)

6: Calculate a new 1

n

by using (5)

7: n=n+1;

}

Where Nmax is the maximum number of iterations,

is stopping error determined by noise floor [11],

and RRE is relative residual error:

measuresc

averagesc

p

p

RRE

(11)

For simulation purposes, measuresc

p can be computed

by applying the ideal object function to (5) and adding

random noise.

Simulations

The modified DBIM reconstruction has been tested by

numerical simulation. The table below summarize the

scenario’s simulation.

Table 1. Simulation parameters

Parameters Values

Frequency of ultrasound signal F=1 Mhz

Number of pixels N×N, N=20

Number of transmitters Nt=40

Number of receivers Nr=20

Noise 5%

800

Hội nghị toàn quốc về Điều khiển và Tự động hoá - VCCA-2011

VCCA-2011

The radius of the cylinder R= 5

Number of ensemble K=5

Speed of sound contrast 2%

We assume that there is a cylindrical tissue in the

center of the medium. Figure 3 shows the ideal object

function that can be expressed as:

Rrif

Rrifcc

r

0

1120

21

2

(12)

Where is radial frequency, 0

0c

k

is the wave

number in reference medium, c1 is the speed of the

sound in the object (cylinder here) and R is radius of

the object.

The incident field for a Bessel beam of zero order in

2-D is given by:

k

inc rrkJrp 00 (13)

Where J0 is the 0th

– order Bessel function and krr

is the distance between the transmitter and the kth

point in the ROI.

05

1015

20

0

10

200

2

4

6

8

x 105

(a)

5 10 15 20

5

10

15

20

(b) H. 3 Ideal object function of simulated scenario (no noise

contamination) in mesh mode (a) and image mode

(b)

In order to prove the high performance of the

proposed scheme, we will compare the

reconstruction’s quality between conventional DBIM

and modified DBIM.

Figures 4 and 5 show the conventional DBIM

reconstruction at the first and fourth iterations,

respectively. After the first BIM iteration, the object

function could not be estimated correctly yet. After

four iterations, this object function has been

reconstructed successfully.

H. 4 Object function at 1st conventional DBIM

reconstruction

H. 5 Object function at 4th conventional DBIM

reconstruction

Figure 6 shows the modified DBIM reconstruction at

the fourth iteration. It is easy to see that the quality of

reconstructed image in this figure is outperformed the

one shown in Fig. 5 because the speckles are great

reduced.

801

Hội nghị toàn quốc về Điều khiển và Tự động hoá - VCCA-2011

VCCA-2011

H. 6 Object function at 4th modified DBIM reconstruction

Figure 7 is the reconstructed comparisons among the

ideal, conventional DBIM (without improvement) and

modified DBIM (with improvement). The curve

reconstructed by modified DBIM can tracked the

ideal with a mean square error (MSE) that is below

10%. Other wise, difference between the ideal and

conventional DBIM is obvious.

H. 7 Comparison between ideal and final reconstructed

object functions (with and without improvement)

Another way to evaluate the performance of the

system with/without improvement, we define the

speed error as:

ideal

tionreconstrucideal

c

cc

Err

(13)

Where ideal

c and tionreconstruc

c are ideal and

reconstructed velocities of ultrasound signal in N2

pixels.

Figure 8 shows error comparison of two schemes with

and without improvement using Monte-Carlo

methodology [12]. It is obvious that by applying the

modified DBIM, the reconstructed quality has been

strongly improved. It also can be seen that after 4th

DBIM iteration, the error performances are saturated

due to noise background. It means that even if we

continue to increase the DBIM iterations, we can not

improve the reconstruct quality anymore.

H. 8 Performance comparison of conventional and

modified DBIM

Conclusions

Sound contrast reconstruction can be applied in

diagnosis to detect cyst, tumor, etc. This paper has

successful by averaging the ensemble of measured

signals and applying to DBIM in order to improve the

reconstructed quality of sound contrast. A Monte-

Carlo simulation of sound contrast reconstruction has

been performed to prove the ability of this method.

The DBIM is chosen here to solve the inverse

scattering problem due to its fast convergent compare

to BIM [10]. Furthermore, its disadvantage which is

noise sensitivity has been solved in this work.

References

[1] J. Greenleaf, S. Johnson, W. Samayoa, and F.

Duck: Algebraic reconstruction of spatial

distributions of acoustic velocities in tissue from

their time-of-flight profiles. Acoustical

Holography, vol. 6, 1975, pp. 71–90.

[2] J. Greenleaf, S. Johnson, S. Lee, G. Herman,

and E. Wood: Algebraic reconstruction of

spatial distributions of acoustic absorption

within tissue from their two-dimensional

acoustic projections. Acoustical Holography,

vol. 5, 1974, pp. 591–603.

[3] Jensen, J.A: Linear description of ultrasound

imaging systems, International Summer School

on. Advanced Ultrasound Imaging, 1999, pp. 1-

74.

[4] W.C. Chew, G.P. Otto, W.H. Weedon, J.H. Lin,

C.C. Lu, Y.M. Wang, and M.Moghaddam:

Nonlinear Diffraction Tomography-The Use of

Inverse Scattering for Imaging. Int. J. Imaging

Sys. Tech., vol. 7, 1996, pp. 16-24.

802

Hội nghị toàn quốc về Điều khiển và Tự động hoá - VCCA-2011

VCCA-2011

[5] A. Devaney: Inversion formula for inverse

scattering within the Born approximation.

Optics Letters, vol. 7, no. 3, March 1982, pp.

111–112.

[6] Gene H. Golub, Per Christian Hansen, and

Dianne P. O'Leary: Tikhonov Regularization

and Total Least Squares. SIAM Journal on

Matrix Analysis and Applications. Vol. 21 Issue

1, Aug. 1999.

[7] R. J. Lavarello and M. L. Oelze: Tomographic

Reconstruction of Three-Dimensional Volumes

Using the Distorted Born Iterative Method.

IEEE Transactions on Medical Imaging, Vol.

28, 2009, pp. 1643-1653.

[8] Lavarello Robert: New Developments on

Quantitative Imaging Using Ultrasonic Waves.

University of Illinois at Urbana-Champaign,

2009.

[9] R. Lavarello and M. Oelze: A study on the

reconstruction of moderate contrast targets using

the distorted born iterative method. IEEE

Ultrasonics Symposium, 2007.

[10] Andrew J. Hesford and Weng C. Chew: Fast

inverse scattering solutions using the distorted

Born iterative method and the multilevel fast

multipole algorithm. Journal of the Acoustical

Society of America, Vol. 128(2), 2010, pp. 679-

690.

[11] Martin, R.: Noise power spectral density

estimation based on optimal smoothing and

minimum statistics. IEEE Transactions on

Speech and Audio Processing, Vol. 9, 2001, pp.

504 – 512.

[12] Kalos, Malvin H.: Whitlock, Paula A. Monte

Carlo Methods. Wiley-VCH. ISBN 978-

3527407606, 2008.

[13] Osama S. Haddadin, Sean D. Lucas, Emad

S.Ebbini, Solution to the inverse scattering

problem using a modified distorted Born

iterative algorithm, Proceedings of IEEE

Ultrasonics Symposium, 1995, pp. 1411 - 1414.

Tran Duc Tan was born in

1980. He received his B.Sc.,

M.Sc., and Ph.D. degrees

respectively in 2002, 2005, and

2010 at the Univer-sity of

Engineering and Technology

(UET), Vietnam National

University Hanoi, Vietnam

(VNUH), where he has been a

lecturer since 2006. He is

author and coauthor of several

papers on MEMS based sensors and their application.

His present research interest is inDSP applications.

Gian Quoc Anh received the

B.S. degree in physics from

VNU-University of Sience in

2003 and M.S. degree in

electronics and

telecommunication

technology from VNU-

University Engineering and

Technology in 2010. He is

currently a lecturer in the

Faculty of Electrical and

Electronics, Nam Dinh

University of Technology Education. His research

interests are applications of digital signal processing

and embedded systems.

803