Impact of selective breeding on European aquaculture

IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 47, NO.

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Small Element Array Algorithm for Correcting Phase Aberration Using Near-Field Signal Redundancy-Part I: Principles Yue Li, Member, IEEE Abstract-A near-field, signal-redundancy algorithm for measuring phase-aberration profiles has been proposed previously [1]-[3]. It is designed for arrays with a relatively large element size for which relatively narrow beams are transmitted and received. The algorithm measures the aberration profile by cross-correlating signals collected with the same midpoint position between transmitter and receiver, termed common midpoint signals, after a dynamic near-field delay correction. In this paper, a near-field signalredundancy algorithm for small element arrays is proposed. In this algorithm, subarrays are formed of adjacent groups of elements to narrow the beams used to collect common midpoint signals and steer the beam direction, so that angle-dependent, phase-aberration profiles can be measured. There are several methods that could be used to implement the dynamic near-field delay correction on common midpoint signals collected with subarrays. In this paper, the similarity between common midpoint signals collected with these methods is also analyzed and compared using a so-called corresponding-signal concept. This analysis should be valid for general target distributions in the near field and wide-band signals.

I. INTRODUCTION ultrasound imaging systems, focusing on transmission and reception is performed by assuming that the velocity inside the body is uniform. Unfortunately, the velocity inside the body is not uniform, which may result in increased side lobes and degraded lateral resolution. Many methods have been developed for correcting phase aberrations [1]-[40], such as the nearest-neighbor crosscorrelation algorithm [6], [7] and the translating apertures algorithm [4], [5]. A near-field, signal-redundancy algorithm has been proposed and successfully tested on data collected from tissue-mimicking phantom and from tissue [1]-[3]. This algorithm is based on the signal-redundancy principle: in an active imaging system, common midpoint signals, for which the position midpoints between the transmitter and the receiver are the same, are redundant (identical) for arbitrary target distributions in the far field because the arrival time of an echo from a point target located in the far field is the same for common midpoint signals [l],[as],[45]. However, the arrival time of an echo from a point target located in the near field is different

I

N MEDICAL

Received September 8, 1998; accepted October 4, 1999. The author is with CSIRO Telecommunication and Industrial Physics, West Lindfield, NSW, Australia (e-mail [email protected]).

for common midpoint signals [ 11, [45]. Therefore, common midpoint signals are not redundant for echoes from arbitrary target distributions in the near field, and this effect is termed the near-field effect in this paper. A dynamic nearfield delay correction, which is similar to the shift process in the two-step shift-and-add dynamic focusing process has been proposed to reduce the difference between common midpoint signals for echoes from targets in the near field [l].(Note that the dynamic near-field delay correction is not a dynamic focusing process because “add” is not involved). It has been shown [3] that the dynamic near-field delay correction is necessary for the successful measurement of phase-aberration profiles for a near-field, image formation system. In this algorithm, after common midpoint signals are collected, they are dynamically corrected for the near-field effect and cross-correlated with one another. In a related way, the phase errors are measured from peak positions of these cross-correlation functions. The phase-aberration profile across the array is, then, derived from these measurements. There are some connections between the nearest neighbor cross-correlation; the translating apertures; and the near-field, signal-redundancy algorithms. However, these algorithms are different, and the differences between them have been described [50]. An important issue that influences the performance of the near-field, signal-redundancy algorithm is the degree of validity of the phase screen on the transducer surface model for the effect of aberrators in the medium. For distributed aberrators, a phase-screen model at the transducer surface with the phase-aberration profile depending on directions may be a better approximation than the phase-screen model that has a fixed value for all angles [49]. The large element (2.5 wavelengths) of the array used in [2] transmits and receives signals only in a small angular range; therefore, a single measurement may be enough. For small element arrays, one may have the following problem: each element transmits signals into, and receives echoes from, a wide angular range, and these echoes may experience different phase-aberration values, as shown in Fig. 1; in this case, the peak position of the cross-correlation function between common midpoint signals is not directly related to the phase-aberration value at any particular direction. A narrower beam, which limits the transmitted signals and received echoes to a smaller angular range, may help to make the transmitted and received beams experience approximately a single aberration value and may make it possible to measure a phase-aberration profile. In

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wide beam 0

k

\

targets

1

small element

0

sub-array

c

inhomogeneity

I

n

U

Fig. 1. A small element with a wide beam width cannot be used t o measure angle-dependent phase-aberration profiles.

Fig. 2. Subarrays can be formed to narrow the beam width and steer the beam direction at different angles, which makes it possible to measure angle-dependent phase-aberration profiles.

phase screen

this case, the measured aberration profiles may be different at different beam angles; therefore, several aberration profiles may need to be measured. When using these measured profiles to correct the image, one may use each profile for the correction of the image in the corresponding direction. In this paper, it is proposed that one can group small elements into subarrays to collect common midpoint signals with the required narrow directivity pattern, and the beam angle of subarrays can be steered in different directions to measure the phase-aberration values for each direction, as shown in Fig. 2. The condition is that the phase-aberration values for elements in the same subarray should be considered as identical in all directions. This requirement may limit the maximum size of subarrays and therefore limit the narrowest achievable beamwidth and the maximum spatial frequency of the aberrator that can be successfully measured. These may be better understood from the following discussion. Let us define direct signals in a received signal as signals that propagate along a straight line from a transmission element to a point target and then back to a reception element; indirect signals are any other signals that propagate along a different path from that of direct signals. In a homogeneous medium, the received signal from a point scatterer may be modeled as containing a direct signal only. When distributed aberrators exist, we may be able to model received signals from a point scatterer as the sum of a direct signal and many indirect signals that may be produced at the boundaries between different media. The time period for a direct signal to propagate from a transmitter to a point target and back to a receiver is the actual time it takes to propagate through ~- the different media with different velocities. The energy of a direct signal should ~

0 transmitter

'\, Scattered ray by '\,the phase screen IT1 \,.

IR3

-,

0

Fig. 3. The direct and indirect signals generated by a phase screen located at some distance from the transducer surface.

also be much larger than that of an indirect signal. By assuming that direct signals propagate along straight lines, the refraction effect at boundaries between media with different velocities has been ignored. We assume that in the human body, where the velocity in different soft tissues is generally similar (approximately 1.4 to 1.6 mm.ps-'), refraction effects may be ignored; then, this model should be a good approximation. For example, consider a thin phase screen located at some distance from the transducer surface, as shown in Fig. 3. When a ray passes the phase screen, we assume that most of its energy will propagate continuously in its

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original direction with a time shift introduced by the phase screen; scattered rays, which contain much less energy, are also produced at the phase screen because of phase inhomogeneity. In the received signals, the direct signal is from the ray travelling along straight lines from the transmitter to the point target (DT1) and then to the receiver (DR1). This signal is time shifted twice by the phase screen. The received indirect signals may be generated in three different ways (Fig. 3.): DT2 -+ I T 1 4 I R l ; DT1 4 DR2 -+ IR2; DT2 -+ I T 1 4 IR3 -+ IR4. Because we have assumed that the energy in scattered rays at the phase screen is small, the main indirect signals are IR1 and IR2. With the help of this model, we may be able to explain the advantage of forming subarrays further. A small element transmits and receives signals in a large angular range. It receives not only many direct signals from targets in a large angular range that may experience different phase-aberration values, but also a large number of indirect signals. Both effects decrease the similarity between common midpoint signals. After forming subarrays to narrow the beamwidth, direct signals in the received signal are mainly from point targets in a small angular range, and they are more likely to experience similar phase-aberration values. The narrower beam also causes the received signal to contain fewer indirect signals. Both effects increase the similarity between common midpoint signals. The measured phase-aberration values are those experienced by direct signals from point targets in the beam steering direction. Using these measured aberration profiles in the image formation process to correct for phase aberrations, direct signals are in focus, but indirect signals are not. These indirect signals caused by distributed aberrators may add a noisy background to the image. Note that there is also a background in a well-focused image that is caused by the ‘(tails” of the point-spread function. Indirect signals may increase the relative level of these “tails.” Several methods have been proposed [5],[29]-[36]to correct for the effect of indirect signals for different purposes and under different conditions. In addition to distributed aberrators, multiple scattering is another source of indirect signals that also raises the relative level of the “tails” of the point-spread function. If one replaces the phase screen in Fig. 3 with a scatterer screen, the phase-aberration problem will no longer exist, but the indirect signal problem will remain. Another important issue that influences the performance of the near-field, signal-redundancy algorithm is the degree of similarity between common midpoint signals for arbitrary target distributions. When collecting common midpoint signals with subarrays, there are several methods that may be used to implement the dynamic near-field delay correction, and they result in different degrees of similarity between common midpoint signals. The similarity between common midpoint signals collected with these methods is analyzed in the next section and compared using a so-called “corresponding signal” concept. Based on the result, a phase-aberration correction algorithm for small element arrays is proposed in Section 111.

X

T

Fig. 4. Geometry for theoretical analysis.

11. PRINCIPLES Consider a three-element transducer array as shown in Fig. 4. The three elements are labeled U , c, and d , respectively. For simplicity, a two-dimensional analysis is performed here. The element width and the array pitch are h. It is assumed that the element size in the elevation direction is zero. If there is a point target located at (RT,&) with unity reflectivity coefficient, ignoring the spherical spreading, the obliquity factor, and the attenuation effects, the received signal ycc(t) at element c when element c is also the transmitter is

] (t

-

p

_ _h2

-

RIT

+

R z T ) dAxldAx2,

(1)

CO

h -2

and the signal y u d ( t ) collected at the upper element U when the lower element d is the transmitter is

2

2

_h

_h

where p ( t ) is the reflected pulse; CO is the sound propagation velocity in the medium; R ~ TR, ~ TR, U ~ and , R ~ are T the distances from X I , x2, x u , and xd on the elements to

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Echo from point target

:hvA !

I I

I I

II

I

Echo from point target 11 Signal I

I

I

common-midpoint signal pair A

I

I

I

Signal I1 II

!

II

common-midpoint signal pair B Fig. 5. Two common midpoint signal pairs A and B. Each signal contains two echoes from two point targets. Signals I and I1 are different in both match signal pairs. The question is which pair of common midpoint signals is more suitable for phase-aberration measurements.

the target position, respectively; and A X , =Xu -

h; Axd = xd

+ h; Ax1

= x1;

Ax2 = 2 2 .

(3)

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mon midpoint signal pair is more similar or which method should be used for phase-aberration measurement. Some work can be found in the literature that uses the cross-correlation coefficient as an indicator of the similarity between two signals. The van Cittert-Zernike Theorem has been applied to ultrasound in [8] to analyze the correlation between two matched signals collected in the nearest-neighbor, cross-correlation algorithm. (Matched signals are defined as the two signals that are cross-correlated in an algorithm; they are generally different.) An extended van Cittert-Zernike Theorem developed in [4] included more transmission-reception geometries in the cross-correlation coefficient expression, and the effect of phase aberrations was also included. When identical effective apertures are used to collect matched signals, (27) of [4] gives a unity cross-correlation coefficient in the absence of phase aberrations. This is only true under far-field approximations, which are used to reduce the near-field (1) to the far-field (2) in [4]. Matched signals collected with identical effective apertures are different for general target distributions in the near field [l],[45]. For the four different methods investigated in this paper, the effective apertures used to collect matched signals are all identical. Therefore, the results derived in [4] and [8]cannot be used in this paper. Instead of deriving an analytic expression for the crosscorrelation coefficient for arbitrary target distributions in the near field and wide-band signals and then deriving analytic relationships between the cross-correlation coefficient and the peak position measurement variance as well as bias, which are difficult or impossible, a different approach is taken in this paper to compare the similarity between common midpoint signals Ycc(t) and & ( t ) collected with four different methods. It is based on the so-called corresponding signal concept, which is discussed subsequently.

dxl, dx2, dx,, and dxd have been changed to dAxl, dAxz,

dAx,, and

d A x d , respectively in (1) and (2) using (3). Each element can be a single transducer or a group of smaller transducer elements. Let Ycc(t) and Y u d ( t ) denote y c c ( t ) and y & ( t ) , respectively, for multiple point targets. For phase-aberration profile measurement using the signal-redundancy method, it is ideal if Y c c ( t ) and & ( t ) have identical shape and the peak position of the crosscorrelation function between Y c c ( t ) and Y & ( t ) is located at zero when there is no phase aberration. Unfortunately Ycc(t) and Y u d ( t ) are different for arbitrary target distributions in the near field, and the peak position of the cross-correlation function between them depends on the target distribution, the shape of the pulse p ( t ) , and how the dynamic near-field delay correction is applied to them. To compare the performance of different methods for implementing the dynamic near-field delay correction, the degree of similarity between the two common midpoint signals Y c c ( t ) and Y u d ( t ) needs to be estimated for each method and compared. For example, Fig. 5 shows two common midpoint signal pairs A and B obtained with different methods. Each signal contains two echoes that are generated by two point targets. The question is which com-

A . Corresponding Signals Common midpoint signals are identical for arbitrary target distributions in the far field because for any echo from a point target in one of the two common midpoint signals, there is an echo from the same point target with the same shape and arrival time in the other common midpoint signal [l].We term these two echoes (pulses) corresponding signals. For multiple point targets, there are many corresponding signal pairs between two common midpoint signals. When common midpoint signals are processed with dynamic, near-field delay corrections, the pulse shape of the two corresponding signals may become different because of the pulse-stretching effect [l].To take the pulsestretching effect into account, corresponding signals can be defined as the same part of the echo (instead of the whole echo) from the same point target [l].For a single point target, if the transducer element has a finite size, y c c ( t ) and y U d ( t ) can still be regarded as the sums of many pulses transmitted an,d received at different points on each element, as shown in (1) and ( 2 ) . To take the element size into account, corresponding signals can be defined (in ad-

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dition to the same part of the echo from the same point target) as signals in y c c ( t ) and y u d ( t ) that are common midpoint signals, and the displacement of their transmitter and receiver positions from the center of each element is also identical:

two echoes with one unit magnitude that generates the same f ( r ,RT,6,) distribution. Therefore, f ( r ,RT,6,) should be proportional to the absolute value of the scattering coefficient a ( R T ,&)l of the point target located at ( R T ,&), the transmission-beam pattern bT (&), and the reception-beam pattern bR (0,) of each element (subarray). f ( r ,RT,&) should also relate to the distances between the point target and the transmitter as well as the B. The Arrival Time Difference Distribution receiver. Because the aperture size is much smaller than Between Corresponding Signals F ( r ) the target range, these distances are replaced with RT in (8). The value of m in (8) depends on the type of geometFor point targets in the near field, the arrival times of rical spreading on transmission and reception. For examcorresponding signals are different, and the difference de- ple, for cylindrical spreading on transmission and spheripends on the point target position. An arrival time differ- cal spreading on reception, m = 3/2. The peak value of ence distribution function F ( r ) between corresponding sig- f ( r ,RT,e T ) should also be proportional to the element nals in Ycc(t)and Y u d ( t ) can be derived, where F ( r ) is the (or subarray) width h. The element width h is the same number of corresponding signal pairs (weighted by their in the four different methods discussed in this paper. The strength) with an arrival time difference in a unity time peak value of f ( r ,RT,6,) has been normalized with a real period around r . From (l),the arrival time t12 (Ax1,Azz) scattering coefficient ao, beam patterns at an angle Oi, tarfor the part of the reflected pulse p (SR/co)in y c c ( t ) ,where get distance Ro, and element width ho in (8). If the unit of -AR < 6R < A R and 2AR is the pulse length in the r is microseconds, the unit of f ( r ,RT,e T ) is megahertz. medium (note that 6R defined here is double the SR value f ( r ,RT,OT)in ( 8 ) is for a single point target located at defined in [l]), is ( R T ,&). For multiple point targets, the total arrival time difference distribution F ( r ) is the sum of many triangles (5) with different heights and widths: and, similarly, in tud

Y u d ( t ) , from

(Axd,A&)

(2),

RdT+RUT =

CO

F ( r )=

+ -.6R CO

(6)

The arrival time difference r between corresponding sigrials is t u d ( a x d ,~ X U -) t12 ( ka , X 2 ) , Where the a X S are related bY the conditions of (4).Assuming RT >> h, the leading terms of a Taylor series expansion yield:

Because a x u and h d range from -h/2 to h/2, the arrival time difference distribution for a single point target located at (RT,e T ) , f ( r ,RT,e T ) , has a triangular shape:

where

2h2 rw = 271 = -sin2 et; CORT

(9)

I I denotes the absolute value, tri [(T - T I ) /T,] is the triangle function, its peak is located at 7 1 , its width is r,, and its peak value is 2/r,; the total area under tri [ ( r- T I ) /T,] is unity. The peak value of the triangle f (7,RT,6,) should be proportional to the amplitude of the received echo from the point target located at ( R T , & ) ,because an echo with two units magnitude may be regarded as

JJ f

( 7 ,RT,0,)

RTdRTdOT.

(10)

In a three-dimensional analysis, the element size in the elevation direction y needs to be considered. Two additional equations = ayuand = ayd should be included in (4) to define corresponding signals. The triangular-shaped arrival time difference distribution caused by a single point target becomes a rectangular pyramid.

ayl

ay2

C. F ( r ) for Four Different Methods of Implementing Dynamic Near-Field Delay Corrections 1. Method I: Without Dynamic Near-Field Delay Correction: Using this method, common midpoint signals collected with subarrays are formed by adding all signals collected with appropriate individual elements together without any delay correction or focusing, and, then, they are directly cross-correlated for the phase-aberration profile measurement. They are equivalent to signals collected as if each subarray is a single large element, which are ycc(t) and Y u d ( t ) in (1) and (2) for a single point target. Therefore, F ( t ) can be obtained from (8)-(10) for this method. Some of the triangles described by (8) are shown in Fig. 6. They are plotted as groups according to the range RT of the point target that generates them. The peak value of triangles in each group is plotted as the same; in reality, l a (RT, 6'T) may also be treated as the average scattering coefficient of targets located in a unit area around (RT, OT),and, for simplicity, ~ ( R T , &is) assumed t o be constant within the bandwidth of the pulse.

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0" 180-'

47,

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____, eT

900

R , < R, triangle peak positions for targets at R - R

Fig. 6. The arrival time difference distribution between corresponding signals without dynamic near-field delay correction.

it depends on target strength, beam pattern, etc. The different peak values for different groups are intended to make it easier to distinguish the different groups in the figure. For each group, the triangle at the right-hand side is generated by a point target located at OT = 90" at that range. The delta-shaped pulses located at T = 0 are generated by point targets located in OT = 0" and 180" directions. When OT increases from 0" to go", the position of the triangle peak moves to the right, and its width increases gradually. When 9T increases from 90" to 180", the position of the triangle peak moves to the left, and its width decreases gradually. For targets located at OT = 0" and 1800, the triangle width is zero. Point targets, located at OT = 0" and 180" directions outside the array, are similar to point targets in the far field; the arrival time of corresponding signals is identical in the two common midpoint signals; therefore, f ( T , RT,19,) is a delta-shaped pulse located at T = 0 for these point targets.

2. Method 11: Single Element Position Dynamic NearField Delay Correction: Using this method, common midpoint signals collected with subarrays are also formed by adding all signals collected with appropriate individual elements together without any corrections, as it is done in Method I, which are y c c ( t ) and yud(t) for a single point target, but, a single element position dynamic near-field delay correction is applied to them by assuming that all signals are transmitted and received at the center of each subarray. For a single point target, the corrected signals YLc(t) and Y h d ( t ) of Y c c ( t ) and Y u d ( t ) become [I]

h

h

where t is replaced with 2Ri/co. Rhi and R-hi are the distances from x = h and -h on the elements to the correction point (Ri,&), respectively. Note that y c c ( t )= yLC(t). In yAc(t),for a particular value Ax1 and Ax2, the part of the reflected pulse p (GR/co) is located at 2R:/co, where Ri is determined by

Similarly, in y h d ( t ) ,for a particular value Ax, and Axd, the part of the reflected pulse p(SR/co) is located at 2RY/co,where RY is determined by

By solving RI and RY from (13) and (14),respectively (Appendix A), using conditions of (4), the arrival time difference r for the part of the pulse p (SR/co)is

The distribution of T is also a triangle for a single point target. 6R is assumed to be in the range -AR < 6R < AR, where 2AR is the pulse length. For AR << RT, the influence of SR is small and, therefore, ignored in the following analysis. This effectively ignores the pulse-stretching effect and assumes that the pulse shape of corresponding signals in yLc(t) and y h d ( t ) are identical. The width r, and peak

LI: SMALL ELEMENT ARRAY ALGORITHM

position

71

35

of the triangular distribution, given by ( 1 5 ) ,are

71 =

h2 ET

-(sin2OT - sin2ei) . CO

The arrival time difference T for the part of pulsep ( 6 R / c o ) (I7) in the two corrected signals is

The beam patterns b, (0,) and bR (0,) in this method should be the same as that in Method I, because in both methods the beam pattern of subarrays is the same as a single large element with the size of the subarray. Because the target distribution, element size, normalization parameters, and r, are also identical in both methods, the F ( r ) differences between Methods I and I1 are caused by the different r1 values. F ( T ) for Method I1 can be obtained from ( 1 6 ) , (17), (8), and (10). Some of the triangles described in (15) are shown in Fig. 7 with the correction angle 0%= 90". Compared with Fig. 6, one can see that the single element position dynamic near-field delay correction moves triangles generated by point targets located at the same range by the same amount; however, triangles generated by point targets located at different ranges are shifted by different amounts. The correction also shifts the peak positions of triangles generated by point targets located at the correction angle 90" in all ranges to the T = 0 position, but the width of each triangle is unchanged. 3. Method 111: Full Element Posztzon Dynamrc NearFaeld Delay Correctzon: In this case, it is assumed that signals can be corrected according to their exact transmitter and receiver positions Axl, Ax2, Axd, and Axu. That is assuming that each subarray is formed by infinite number of point-like elements. This is an ideal case that cannot be realized but is analyzed here for comparison. The corrected signals for a single point target are

- RIT

+

CO

R2T)dAz1dAx2

the position of p ( S R / c o ) in y h d ( t ) is at 2Ry/co, where Ry is determined bv

r = - [h2

RT

sin20,

CO

The distribution of r is still a triangle for a single point target. The width r, and peak position r1 of triangles in (22) are T,

2h2

= -Isin' QT - sin20i

RT

CO

1

and 71

=

h2

-(sin2#, - sin2~

i )

CORT

F ( r ) for Method I11 can be obtained from ( 2 3 ) , ( 2 4 ) , (8), and (10). Because, in this case, the transmitted and received beams by each subarray are dynamically focused at the correction direction 6i,bT (0,) and bR (6,) are different from those in Methods I and I1 even when 8i = 90". Because the aperture size of the subarray is much smaller than focal distances, the focusing effect should be small, and, for simplicity, we may assume that, for 8i = go", bT (8,) and bR (0,) in Method I11 are the same as those in Methods I and 11. Therefore, when 0i = go", F ( t ) differences between Methods I11 and I, and I1 are caused by different r1 or r, values. Some of the triangles described in (22) are shown in Fig. 8 with the correction angle Oi = 90". Compared with Fig. 7 , triangle-peak positions are not changed, but triangle widths have been changed. The width of triangles generated by point targets in the 90" direction is reduced to zero, and the width of triangles generated by point targets in the 0" and 180" directions is increased. (18) 4. Method IV: Dynamic Near-Field Delay Correction with Subarrays: Using this method, the common midpoint signals collected with subarrays are formed by correcting signals collected with individual elements according to the center positions of their transmission and reception elements. Assume that hl is the pitch of the array and that there are n elements in each subarray, then

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T R,= R ,

47, NO. 1,JANUARY 2000

8,= 90'

I

0 Fig. 7. The arrival time difference distribution between corresponding signals with single element position dynamic near-field delay correction.

ei= 90"

z

0

Fig. 8. The arrival time difference distribution between corresponding signals with full dynamic near-field delay correction.

The corrected signals for a single point target are

-

+

R I T RZT

)dAx1dAx2

CO

The position of the part of the reflected pulse p(GR/co)

the position of p (GR/co)in ?&(t) k a t 2Ry/co, where R; (27) is determined by

(Rl,i

(n-1)

Ydd

(2)

l , = - L y

R z , ~ Rldi

Zd=-W-$ -9 -

+

RdT

d~x,dLkd

+ RdT) = 6R.

(30)

By solving R: and Ry from ( 2 9 ) and (30), respectively, and using ( 2 6 ) , the arrival time difference T for the part of pulse p (GR/co)is

(28)

CO

=

where Rzli, Rlzi,Rlui,and Rldiare the distances from x1 = llhl, 5 2 = lzhl, xu = l,hl+ nhl, and xd = ldhl - nhl on the elements to the correction point (Ri, &), respectively.

Rldi) - (%T

3 [sin2 RT CO

n

eT -

(1 -

E)

sin2

(axu- h

a] (1 d )

sin2&'. (31)

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When n = 1 and ho = h, (31) reduces to (15), as expected. Eq. (31) contains two terms, each of which is a triangle. The first triangle is similar to that in (15) except that the triangle is sampled (discrete) in this method. The second triangle is similar to that in (7). The arrival time difference distribution for a single point target in (31) is the convolution of the two triangles:

where

D.Parameters of F ( r ) Useful f o r the Comparison F ( T )has been derived for four different methods of implementing the dynamic near-field delay correction. Next, we need to find a method to compare the similarity between common midpoint signals formed by the four different methods using F ( r ) .We can use the weighted average position (mass center) TO of F ( 7 )

to indicate the relative time shift between common midpoint signals and the square of the weighted average distance 0: to TO (variance) of F ( T )

to indicate the degree of similarity between common midpoint signals. When U: = 0.0 and ro = 0.0, common midpoint signals @ denotes convolution, are identical, and there is no relative time shift between them. If a : = 0.0 and TO # 0.0, common midpoint signals n2h2 are still identical, but there is a relative time shift between TI = 1 (sin2 eT - sin2ei) , (34) CORT ' them. When U: is larger in one method than that in another, common midpoint signals in the second method are more similar than those in the first. If 1701is larger in the first method than that in the second, common midpoint and signals in the second method have a smaller relative time shift than those in the first. This analysis does not consider the arrival time difference between noncorresponding signals because it is not directly related to the similarity F ( r ) using this method can be obtained from (32)-(36) between common midpoint signals. To understand this, and (10). For n = 1, hl = h, f ( T , RT,0,) in (32) becomes consider two pairs of identical common midpoint signals the same as that in Method 11, and for n + CO, nhl = h, I and I1 shown in Fig. 10(a and b), respectively. There f (r,RT,0,) in (32) becomes the same as that in Method are two corresponding signal pairs between signals I and 111, as expected. In this method, the transmitted and re- 11: a - a1 and b - b l . All corresponding signals arrive at ceived beams by each subarray are also steered at the di- the same time, but the separations between subsignals are rection of e,, which is the same as that in Method 111. different in Fig. 10(a and b). F ( T ) is a delta-shaped pulse However, the focusing quality is not as good as that in located at r = 0 in both cases, which indicates that signals Method 111. But, again, because of the small aperture size I and I1 are identical. On the other hand, the arrival time compared with focal distances, the difference should be difference distribution among all subsignals is different in small and, for simplicity, we may assume that b~ (0,) and Fig. 10(a and b). Ignoring the arrival time difference bebR (6,) in this method are the same as those in Method 111. tween noncorresponding signals also implies t hat we define For 8, = 90", b~ ( e T ) and b~ (0,) in this method can also the two common' midpoint signal pairs in Fig l l ( a and b) be considered as the same as those in Methods I and 11. as having the same degree of similarity. Some of the distributions described in (32), as a result of We may use the parameters U: and TO in three possithe convolution between two triangles, are shown in Fig. 9 ble applications. One is to find out for what kind of target with 6, = 90", n = 3, and nho = h. Compared with Fig. 7 distributions, common midpoint signals, collected with the and 8, the peak position of f (7,RT,0,) is not changed, same method, are more similar and with smaller relative but the triangle width has been changed. The width of time shift. We may derive 0: and TO values for different the sampled triangle is zero for point targets located at target distributions and find the optimal target distribugo", and it is only sampled once; the convolution results tion for that method. Another possible application of 0: in a single triangle [the second-group triangles in (31)] cen- and TO is to find out what kind of pulse shapes make comtered at zero point. The width of the second-group trian- mon midpoint signals, collected from the same target disgles becomes zero for targets at 0"and 180". Therefore, the tribution with the same method, more similar and with convolution results are sampled triangles [the first-group smaller relative time shift. A third application of a : and triangle in (31)]. T O , which is the application being discussed in this paper,

~

38

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47, NO. 1,JANUARY 2000

0

z

Fig. 9. The arrival time difference distribution between corresponding signals with subarray dynamic near-field delay correction.

is to find out with which method common midpoint signals, collected from the same target distribution with the same pulse shape, are more similar and with smaller relative time shift. The results may be different for different target distributions and pulse shapes.

E. Comparison of 0; and Four Different Methods From

Values for the

70

(lo), (38) can be rewritten as

where

A (RT,&) =

1

f ( 7 ,RT,&) d r ,

and To = 71

(RT,QT).

(46)

Eq. (44) indicates that 0; of F ( T ) is the weighted (with the area under each triangle) average of the variance of each individual triangle 0: ( R T ,0,) plus the variance of triangle-peak positions. From (44), one can also see that there are two possible ways to reduce the value of a : . One is to reduce the width of individual triangles so that 0: (RT,19,) is reduced, and the other is to make triangles “closer” together. Note that (44) is valid for arbitraryshaped f ( 7 ,RT,&). Eq. (46) indicates that the average position TO of F ( 7 ) is the same as the mean of the average positions 71 ( R T 19,) , of the f ( T , RT,&), as expected. From (8), (32), and (41), it can be easily shown that A (RT,&) can be expressed as

(41)

and the over-bar indicates the following average:

-

JJ B (RT,O T ) A ( R T OT) , RT~ JJ A (RT, Q T ) RT~ R ~ TO T

RT~~T

in all four methods. A ( R T , & ) is identical in Methods I and I1 for the same target distribution because bT (0,) and b~ (0,) are identical in hkthods I and 11. A ( R T OT) , Note that B ( R T ,OT) is not the “variance” o f f ( 7 ,RT,0,) because To is not the mean position ‘of f ( 7 , R T , e T ) . can also be considered as identical in Methods I11 and A (&, 6,) is the total area under f ( T , RT,Q T ) . Let I v because bT (0,) and bR (0,) can be approximated as the same in Methods I11 and IV. The beams b~ (0,) and 71 ( R T ,19,) be the mean position of f ( 7 ,RT, &), b~ (e,) in Methods I and I1 are always steered at 90”. The J 7f ( 7 ,RT, OT) d7 beam direction in Methods I11 and IV, however, is steered 71 (RT,QT) = , (43) at the direction e,, which is variable. Therefore, generally, A (RT,e T ) A ( R T , Q Tin ) Methods I and I1 is different from that in +hat (39) becomes Methods I11 and IV. But, when 0, = go”, A ( & , & ) can be considered as identical in all four methods. ( R ~ , - 7,,~2 0: = + (44) . (42)

~

eT)

where 0 :

0:

’

(RT,OT) is the variance of f

,

( R T eT> =

‘T - T1 (RT’e T ) 1 2

(7,RT,e T ) ,

(7’ RT’

A ( R T ,0,)

dT, (45)

1. Comparison of Methods I and 11: Because A ( R T 0,) , and the width of triangles generated by the same point target are identical in these two methods, the first term in (44) is the same in Methods I and 11. From (9) and (17),

LI: SMALL ELEMENT ARRAY ALGORITHM

39

common-midpoint signal pair A

common-midpoint signal pair B

4 4 signal I a % . . .

b

.................... to 4

a1

t

4

bl signal I1 ,. . . . . . . . . . . . . . . . . . . . . . .

to

t to

pairs

pairs Aarrival-time-difference distribution between corresponding signals

A arrival-time-difference distribution between corresponding signals

t-

It 1 0 ........................

t

,

t

1 -

0

2

pairs arrival-time-difference distribution between all sub-signals a-al, b-bl

+

0 l

0

4

2

0

2

Fig. 10. The arrival time difference distribution between all subsignals is not directly related to the similarity between two common midpoint signals. The arrival time difference distribution between all subsignals is different in (a) and (b), even though the two common midpoint signals I and I1 are identical. On the other hand, the arrival time difference distribution between corresponding signals is a delta-shaped pulse.located a t zero point in both (a) and (b), which indicates that signals I and I1 areidentical. When signals I and I1 are different, the arrival time difference distribution between corresponding signals is no longer a delta-shaped pulse.

the peak position T I J I (RT,8,) of the triangle generated by a point target located at ( R T ,8,) in Method I1 is

where TO,I is the average position in Method I. Then, from (44),

where 7 1 , (RT, ~ 8,) is the peak position of the same triangle in Method I, and

where and C T : , ~are ~ the variances of F ( T ) in Methods I and 11, respectively, and [see (sa), next page]. If E is less than zero for a particular target distribution, that is

(49) From (48) and (49), the average position Method I1 is TOJI = TO,I -

TO,J

(g)(2) RT

sin2&

of F ( r ) in

(50)

and 8i

# 0, then

< .:,I.

40

IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL.

common-midpoint signal pair A a

b

a

signal I

to al

t t M

1

signal I1

...........

I

t t

t

b-bl

0

bl

. . . .I . . . . . . .

, . . . . . . . . . . I. .

t

to

pairs arrival-time-difference distribution between corresponding signals 1 -

.

arrival-time-difference distribution between corresponding signals

fl

t

a1

signal I1

1 a-a1 . . . . . . . .... 0

t. t to

bl

pairs

1, JANUARY 2000

b

t

to

NO.

common-midpoint signal pair B

51

signal I

47,

f

'I;

'I; 0 pairs arrival-time-difference distribution between all sub-signals 2 1 a-a1 b-bl 1 a-bl b-a 1

pairs arrival-time-difference distribution between all sub-signals a-a 1 b-b1

0

0

'

z

Fig. 11. Not considering the arrival time differences between noncorresponding signals implies that the degree of similarity between common midpoint signals in pairs A and B is identical.

E

= sin2

oZ

sin2 O, - 2 sin2 OT

As an example, we may evaluate E for the uniform target distribution la ( R T ,&)I = ao. In this case, from (47), RT and OT become separable in A (RT,OT). Therefore, (53) becomes sin2O, < 2sin2 ot.

(54)

To proceed further, assume a rectangular beam pattern: l b (e,)] ~ = I ~ T (90") I and l b (e,) ~ I = l b (90") ~ I for 90" 81 < QT < 90" 81 where 81 is between 0" and go", and ( b (&)I ~ = I ~ R(&)I = 0.0 for other OT values. Then, (54)

+

-

-

becomes sin2 8,

sin 201

< 1 + -.

261

(55)

Therefore, for 01 < go", E is less than zero for 0, # 0, and, as a result, o:,II < For speckle-generating target distributions, a comparison of the ensemble average value of can be performed by taking ensemble average on both sides of (51) for all possible speckle-generating target distributions. The con-

41

LI: SMALL ELEMENT ARRAY ALGORITHM

dition for the ensemble average of us,II to be less than the ensemble average of is that the ensemble average of E is less than zero. The ensemble average of la (RT, 8 ~ ) in ( (47) is a constant a0 for speckle-generating target distributions. Assuming the denominator in (42) is approximately constant, the condition for the ensemble average of E to be less than zero will be the same as that for the uniform target distribution, that is (54). However, the result for speckle-generating target distributions is a statistic result. Eq. (54) does not rule out the possibility that, for some specific speckle-generating target distributions, E may be larger than zero. For uniform and speckle-generating target distributions, from (51) and (52)) the us difference Aug between Methods I and I1 is

the beam pattern. Eq. (53) and (58) are the conditions for U: and 1 ~ 0 1 values to be reduced by the correction. The meaning of these conditions (and conditions derived below) will be discussed in Section II,F, which shows that it is not difficult to satisfy these conditions.

2. Comparison of Methods 111 and IV: A (RT,8 ~ can ) be considered as identical in these two methods. From (24) and (34), it can be easily shown that the average position 71 (RT,0,) of f (7, RT, 8 ~ in) Methods I11 and IV are identical. Therefore, the second term in (44) is identical for Methods I11 and IV, and, from (46),

ITO,I I

=

ITO,IVI

.

(59)

Next, we compare the first term in (44). From (16) and (23), the variance U: (RT, 8,) of f ( T , RT,OT) in Method I11 is 2 h4 2 ur,III(RT, 8 ~ =) -(sin’ 8~ - Sin28i) .

6c;Rg

When (54) is satisfied, Au: is greater than zero. Aug is also proportional to the variance of RT. Assume that the signal range is from RT = RI to RT = R2 (RI < Rz). Then, for a fixed RI value, when R2 increases, the variance of ~ / R Twill increase (for uniform and speckle-generating target distributions), and, as a result, Au: will increase. This is due to the fact that the near-field delay correction is a “dynamic” correction. For a fixed signal window length R2 - RI, when RI increases, the variance of ~ / R Twill decrease, which results in the reduction of Aug. This is because, for a larger RI value, targets are closer to the far field, and the effect of the dynamic near-field delay correction is reduced. To compare the values of 1701in Methods I and 11, from (461, (48) and (491,

(60)

In Method IV, f ( T , RT, 8,) is the sum of many triangles. From (32) and by applying (44) to f(7, R T , ~ T(instead ) of F ( T ) )it , can be shown that

2 sin’ 8i sin2 - sin4 ~i n2

’1

+ (sin’ 8~ - sin’ ~ i ) .

(61)

From (60), (61), and nho = h, it can be shown that, for 2 > u:,III, it needs

(T,,~~

For uniform and speckle-generating target distributions, RT and 8~ are separable in A (RT, 8 ~ ) and, , as a result, (62) becomes (54); therefore, the condition for u~,,,, < us,Iv is the same as that for us,II < us,I.

3. Comparison of Methods 11 and IV: Generally, A (RT,8,) is different in Methods I1 and IV because the

It can be shown that if

beam pattern is different. Without assuming a specific target distribution and beam pattern, it is difficult to perform a comparison of us and TO for these two methods. Even though it is possible to perform a comparison for 1~0,111< ~ T O , J ~For . uniform target distribution, RT and 8~ uniform and speckle-generating target distributions with are separable variables in A (RT, 8 ~ )As . a result, (58) be- an assumed beam pattern, it is relatively complex and will comes (54). Therefore, for uniform and speckle-generating not be included here. Instead, a comparison is performed target distributions, the condition for < I T O, I / is the below for the special case 8i = 90°, where A (RT,8,) can same as the condition for be considered as identical in Methods I1 and IV. From (17) < u:,~. The dynamic near-field delay correction is designed to and (34), it can be easily shown that the average position reduce the arrival-time difference between corresponding 71 (RT,8 ~ o)f f ( 7 ,RT,8,) in Methods I1 and IV are idensignals generated by point targets around the correction tical. Therefore, from (45), angle. Whether or not this correction will reduce the values of U: and 1701depends on the target distribution and 1~0,111= I.ro,~vI. (63)

~TO,I I

42

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The second term in (44) is also identical in Methods I1 and IV. From (8) and (23), the variance o: ( R T 0,) , of f ( 7 ,RT,OT) in Method I1 is

From (61), (64), and nho = h, it can be shown that, for oT,Iv 2 < u : , ~it~ needs ,

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in Fig. 9, it does not reduce the widths of these triangles to zero, as is done in Method 111. However, triangle widths have been reduced compared with those using Methods I and 11. Therefore, the ability of these methods to reduce the arrival time difference between corresponding signals generated by point targets around the correction angle is in the following order (from high to low) Methods 111, IV, 11, and I.

2. Dynamic Near-Field Delay Corrections Increase the Similarity Between Common Midpoint Signals Only af Condition (53) or (62) is Satisfied: The shift of the small arrival time difference region from around 0" and For n.> 1 and 6'i # O", (65) becomes (62). For uniform and 180" to around the correction angle does not guarantee speckle-generating target distributions, it reduces to (54). that common midpoint signals will become more similar. oT,II 2 < o:,~ is only true when the target distribution and the beam pattern satisfy (53). From (53), one can see that 4. A Summary of the Comparison Results: if targets only exist at a single range RT = Ro, (53) is not When (53) is satisfied, o:,lI < o:,I; when (62) is sat- satisfied and a:,II = CT:,~.This is because Method I1 moves isfied, IT:,,,^ < o:,lv; for d i = 90" and when (53) and the triangular-shaped distributions generated by point tar(62) are satisfied, o:,III < gets located at the same range in Method I by the same < 0 9 , ~<~o:,,. ITO,I I = /TO,IV/; /TO,II < /TO,I/ when (58) is satisfied; amount without changing their width (Fig. 6 and 7); therefor di = 90", ~TO,II/ = ITO,I I = ~TO,IVI, and, if (58) is fore, F ( 7 ) has identical shape in Methods I and 11. For triangular-shaped distributions generated by point targets also satisfied, 170,iI> /TO,II/ = ITOJII = 170,rvI. For uniform and speckle-generating target distribu- located at different ranges in Method I, Method I1 moves tions, conditions (53), (58), and (62) reduce to (54). them by different amounts. It shifts the peak positions of triangles generated by .point targets located at the correcF. Discussions tion angle di in all ranges to T = 0 as shown in Fig. 7 where 8i = 90". Therefore, it decreases the distances between The following conclusions can be reached from previous triangles generated by point targets around the correction angle at different ranges and increases the distances beanalyses. tween triangles generated by targets at other directions. If echoes from the region around the correction angle dom1. Dynamac Near-Faeld Delay Correctaons Increase the Samalaraty Between Echoes from Darectaons Around inate in received signals, the second term in (44) will be the Correctaon Angle an Common Madpoant Sagnals: As smaller in Method I1 than that in Method I. (The first term shown in Fig. 6, without any near-field delay correction is the same in both methods.) Therefore, o:,l < o:,~. The (Method I), which is equivalent to correcting at 0" and condition that echoes from the region around the correc180" directions, the arrival time difference between cor- tion angle are dominant in received signals is not a necesresponding signals generated by point targets located in sary condition for a:,Il < o:,l because it can also be true the 0" and 180" directions at any range is zero. For point when echoes from the region around the correction angle targets in the 90" direction, the arrival time difference be- do not dominate in the received signals. The sufficient and is (53). For uniform tween corresponding signals is large. The single element necessary condition for a:,II < position dynamic near-field delay correction (Method 11) and speckle-generating target distributions, (53) becomes moves the direction of small arrival time difference be- (54), and echoes from the region around the 90" direction tween corresponding signals from 0" and 180" directions (the beam direction) are generally dominant in received to the correction direction 8, (8,= 90" in Fig. 6 through signals. Therefore, if 8i = 90", ~ 3 , <~ a~:,I. ' Because the 9), as shown in Fig. 7. The full element position dynamic left-hand side of (54) is independent of the correction an< o:,~ is true for d i = 90", it should also be near-field delay correction (Method 111)not only moves the gle, if o:,l peak positions of triangles generated by point targets lo- true for other 8, values. In conclusion, (53) is generally cated in the correction direction 8, at all ranges to the zero satisfied for speckle-generating target distributions. Note position, it also moves the small triangle width directions that for some target distributions, the similarity between from around 0" and 180" to around the correction angle common midpoint signals may be reduced by the dynamic e,, as shown in Fig. 8. The subarray dynamic near-field near-field delay correction. For example, if there are only delay correction (Method IV) is an approximation of the targets in the 0" and 180" directions, which is not likely, full element position dynamic near-field delay correction then, the two common midpoint signals are redundant in (Method 111). Even though it moves the peak positions of Method I (Fig. 6), and they are no longer redundant after triangles generated by point targets located in the correc- the dynamic near-field delay correction (Fig. 7). tion direction at all ranges to the zero position, as shown As it is in Method 11, Methods I11 and IV also move

43

LI: SMALL ELEMENT ARRAY ALGORITHM

the triangular-shaped distributions generated by point targets located at the same range in Method I by the same amount. However, the width of triangles has also been changed in Methods I11 and IV ( f ( T , RT,6 ~in)Method IV may no longer have a triangular shape), as shown in Fig. 8 and 9. Method I11 moves the small triangle width region from around 0" and 180" directions to around the cor-' rection angle 8,. It reduces the width of triangles generated by point targets in the correction angle to zero and increases the width of triangles generated by point targets around 0" and 180"directions. Method IV does similar things as Method I11 does, but it does not reduce the width of triangles generated by point targets in the correction angle to zero, and it does not increase the width ) by point targets around 0"and of f (7,RT,8 ~ generated 180" directions, which does not have a triangular shape, as much as Method I11 does. Because the second term in (44) is the same in Method I11 and IV, the method that gives a smaller first term gives a smaller o: value. If echoes from the region around the correction angle dominate in the received signals, the method that gives smaller triangle width around the correction angle gives a smaller o: value; therefore, o:,III < Again, the condition that echoes from the region around the correction angle are dominant in received signals is not a necessary condition. The sufficient and necessary condition is (62). For speckle-generating target distributions, echoes from the region around the correction angle (the beam direction) generally dominate in received signals in Methods I11 and IV; therefore, a:,IIl < o:,Iv are generally true for specklegenerating target distributions. Because A (RT,8 ~ is ) generally different in Methods I1 and IV, the comparison of these two methods is performed for 8,= 90" only, where A (RT,&) is identical in < o:,II when Methods I1 and IV. The condition for 2 6, = 90" is the same as the condition for o:,III < o.r,Iv, which is (62). This is expected because the second term in (44) is the same in Method I1 and IV for 6, = 90" and Method I1 does not reduce the width of those triangles generated by point targets around the correction angle. For speckle-generating target distributions, when 8, = go", echoes from the region around the correction angle will dominate in both Method IV and 11; therefore, or,IV 2 < o:,Ir is true in this case. When 6%# 90", echoes from the region around the correction angle are still dominate in Method IV, but they are no longer dominant in < should also be true for Method 11; therefore, 8, # 90". Because it is complex to compare Methods I1 and IV for 6, # 90" analytically, this is not included in this paper.

03,~~.

3. 1r01 is Also Important to the Phase-Aberration Profile Measurement: ( T O (indicates the relative time shift between two common midpoint signals and is related to the 1 bias of phase-aberration measurements. A small ( ~ 0 value is helpful to make a successful phase-aberration measurement. As shown in Fig. 6 and 7, Method I1 moves the peak positions of the triangular-shaped distributions gen-

erated by point targets located at the correction angle in all ranges in Method I to T = 0. Therefore, if echoes from the region around the correction angle dominate in received signals, 1~0,111< ( ~ 0 ~ This 1 . condition is also not a necessary condition. The sufficient and necessary condition is (58). It has been shown that /TO/ is the same in Methods I11 and IV. This is because the average position off ( T , RT,6 ~in) these two methods is identical, as shown in Fig. 8 and 9. For 6i = 90", TO/ in Method I1 is also the same as that in Methods I11 and IV. For speckle-generating target distributions, when 6, # 90", echoes from the region around the correction angle are still dominant in Methods IV and 111, but they are no longer dominant in Method 11; therefore, I T O, I I ( = < < ~ T O Jshould be true in this case.

/TO,IVI

~T~,III

4. To Make a Successful Measurement of AngleDependent Phase-AberrationProfiles, Echoes from the Region Around the Correction Angle Should Dominate in Common Midpoint Signals: To measure angle-dependent phase-aberration profiles, it is required that echoes from the region around the correction angle are dominant in received signals. Otherwise, even if the similarity between common midpoint signals is improved, the measured aberration profile may not correspond to the correction angle, and, therefore, cannot be used correctly. This requirement is not a necessary condition for reducing the values of 1701 and 0.: However, it becomes a necessary condition for a successful measurement of angle-dependent phaseaberration profiles. This condition is satisfied by specklegenerating target distributions. 5. For Large Element Arrays, Method 11 is Preferred: The comparison results of Methods I and I1 are useful for selecting a method for large element arrays. For example, the element size is greater than two wavelengths. In this case, the beam direction cannot be steered. However, a single element position dynamic near-field delay correction can be performed; when (53) is satisfied, it will increase the similarity between common midpoint signals, and, when (58) is satisfied, it will also reduce the absolute value of the relative time shift between them. Because the beam direction remains at the 90" direction, it is only suitable for measuring phase-aberration profiles around the 90" direction. On the other hand, signals collected with large element arrays are only used to form image pixels around the 90" direction. Therefore, Method I1 should be selected for the measurement [11. 6. For Small Element Arrays, Method IV is Preferred: Because Method 111 cannot be realized, a method has to be chosen from Methods I1 and I11 for small element arrays. The element size is about one-half of a wavelength. Method IV is better than Method I1 for at least two reasons. First, the beam-steering angle of subarrays in Method IV is at the correction angle; therefore, it helps to make echoes from the region around the correction direction dominate in common midpoint signals, which makes

44

IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL.

it possible to measure angle-dependent phase-aberration profiles. Second, it has been shown that when Oi = go", 0; and 1701 are also smaller in Method IV if echoes from the region around the correction direction dominate in common midpoint signals, which is true for speckle-generating target distributions.

111. SMALLELEMENT ARRAYALGORITHM

.

The implementation of Method IV includes the following steps. First, a data set y k l ( t ) is collected by transmitting on all elements, one at a time, and receiving at several elements for each transmission, where k is the transmitter element index and 1 is the receiver index. The number of receiving elements for each transmission depends on the number of elements in each subarray. Second, common midpoint signals y& ( s ,O,, t ) and yld ( s ,O,,t ) , where s is the subarray index, are formed with the subarray correction method at correction angle O,, as described in Section 11. Assume that each subarray has n transducer elements, the total number of subarrays is S , and the total number of transducer elements in the,array is N = Sn, as shown in Fig. 12(a). Then, sn

sn

and

.

sntn

sn-n

where (68) is true. The correction angle O i , which is also the beam direction of subarrays, should step through the angular range of the image, so that a set of common midpoint signals is obtained for each angle. The increment of each step may depend on the beam width. The third step is to measure the peak position Ars (&) ( s = 2 , 3 , . . . S - 1) of -cross-correlation functions between common midpoint signals yLc ( s ,Oi,t ) and ydd ( s ,Oi, t ) .Then, the phase-aberration profiles across the array at all directions T~ (&) can be derived using (28) in [l].The derived phase-aberration value for each subarray should be assigned to all elements in the subarray to obtain TZ(Oi) from rs (Oi). The derived profiles at different angles may have different undetermined linear components. For a small aperture array or an image pixel, which is far from the transducer, where the angle of an image pixel to all elements in the array is about the same, this angle-dependent undetermined linear term generally does not influence the focusing. But it may cause image distortion if the linear term is very different for different angles. For a large aperture array or an image pixel, which is near the transducer, however,

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phase-aberration values measured at different angles may be used at the same image pixel, and, therefore, if possible, it is important to estimate the linear terms, when they are very different for different angles. However, it is usually difficult to estimate-these linear terms. Therefore, another method is proposed here to solve this problem. 'It measures a phase-aberration profile for each image line instead of each angle, as shown in Fig. 12(b). In this case, y& ( s ,O i , t ) and yLd ( s ,Oi,t ) are still as expressed in (66) and (67), but tkz( s ,& , t ) in (68) becomes (69). That is, each subarray forms a beam along each image line, and a phase-aberration profile measurement is made from common midpoint signals formed for each image line. In this case, the focus quality of each pixel is not influenced by the undetermined linear phase-aberration profiles. However, the image may still be distorted if the undetermined linear terms are very different at different image line angles. The number of elements in each subarray should be chosen so that the required directivity or beam width, which is determined by the estimated changing rate of the phaseaberration value with angles, is achieved. However, this is also difficult to estimate. In the experimental work described in the second part of the paper [51], a trial-anderror approach has been used by forming common midpoint signals with several subarray sizes. When forming beams using each subarray for the phase-aberration measurement, resolution is not critical. Therefore, one can use apodization to concentrate more signal energy around the correction angle by reducing energy leakage to side lobes. This is helpful to make echoes.from the region around the correction angle be dominant in common midpoint signals. Because the aperture of each subarray is relatively small, the effective aperture concept [41]-[47] may be valid for beam patterns. Therefore, one may also use only part of the available signals (one signal for each midpoint) with appropriate weighting to form a beam that has the desired beam pattern. This can reduce the computation load of this algorithm. But the signal-to-thermal noise ratio will be lower than that when all of the signals are used.

IV. SUMMARY In this paper, a subarray algorithm using near-field signal redundancy has been proposed to measure angledependent phase-aberration profiles on an array that has a pitch and element size about one-half of a wavelength. The main reason for forming subarrays is to obtain a narrower beam by increasing the element size so that the phaseaberration value experienced by the transmitted beam as well as the received echoes can be approximated as a single value at each subarray under more situations. The condition is that the phase-aberration values for elements in the same subarray should be able to be considered as identical in all directions. Four different methods may be used to implement the dynamic near-field delay correction on common midpoint

LI: SMALL ELEMENT ARRAY ALGORITHM

45

peak position

sub-array

0

s= 1

'

3

2

0

0

0

0

0

S

o o ' o o o o

transducer k,l=1 2 3 n=3

AT&)

peak position

sub-arrav . I

4l correction direction

4

0

0

4

0 0 0

0

+ l S=

1

2

3

0

0

0

S

0 0 0

T$

n n n n n n r-n--in onnclonooo ocloO 00. oonooooooo

transducer k,1=123

0

0

0 0 0 0

(N-1)/2 n=3 N is odd

N

@i

&

imageline

Fig. 12. (a) Steering angle-based measurement; (b) image line-based measurement.

+/(;)2+(l-sn+q)

h2

2 +/(;)2+

(1-1--

N-1 2

)

'h2 -

(1 -1-

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IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL.

signals collected with subarrays. The similarity between common midpoint signals collected with these methods has also been analyzed and compared in this paper. The analysis is based on the corresponding signal concept. The weighted average position ro of the arrival time difference distribution between corresponding signals F ( r ) has been used to indicate the relative time shift between two common midpoint signals, and the square of the weighted average distance r? of F ( r ) to ro has been used to indicate the similarity between two common midpoint signals. This analysis is valid for targets in the near field and wide-band signals. The subarray dynamic near-field delay correction method has been shown to be a better choice compared with the single element position dynamic near-field delay correction for forming common midpoint signals collected with subarrays. In Section I1 of this paper [51], this algorithm is tested using data collected from a tissuemimicking phantom with a non-isoplanatic aberrator. APPENDIXA: DERIVING (15) BY SOLVINGRL

AND

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APPENDIXB: ro WHEN A CONSTANT PHASE ERROR OCCURSACROSSTHE ARRAY From (9), (17), ( 2 4 ) , (34), and (46), it can be seen that ro > 0 in Method I, and ro < 0 in Methods 11, 111, and IV. However, the experimental results described in Section I1 of the paper [51] show that the value of ro for Methods I1 and IV is still greater than zero. This is because of the effect of the lens in front of the transducer that is used to focus the beam in the elevation direction. The lens is made of a material with a velocity < 1.54 mm.ps-l. This introduces a constant delay aberration across the array Ar(one-way, Ar > 0). To take this into account in the arrival time difference distributions between corresponding signals, one just needs to replace 6R with 6R 2coAr in (15), ( 2 2 ) , and (31). Note that (7) is not influenced by Ar For Method 11, ignoring pulse-stretching effect, (15) becomes

+

Rr

The variables in (14) can be expressed as Then, the peak positions of triangles becomes

R-h, = d h 2

+ R: + 2hR, cos 0,; 71 =

RdT = d x z

+ R$

-

2XdR.t COS d ~ .(AI)

h2 CORT

h2

-(sin2 @T - sin2e,) + -Ar

sin20,.

R$

(B2)

For 0, = go",

R, = R: makes these variables satisfy (14). From the approximation of

71

= --

h2

CORT

COS' QT

h2 + -A7sin2

R$

Bi;

(B3)

therefore, and similar approximations of R-h,, R U ~and , RdT, (14) becomes

+

R;~ - (RT B ) R; + C = o

(A31

where

and

h2 c=sin2~ i . 2 By choosing the correct answer from the two solutions of (A3) and from RT >> B ,

+

+

2R; = ~ R T 6R - (xu xd) Cos& x; + x2 sin2 - h2 sin2 8, + sin2 8,. RT 2R$ 2RT

+-

It can be shown that ro is still the same in Methods 11,111, and IV when 0, = 90". Therefore, for ro > 0 in Methods 11, 111,and IV, the delay aberration Ar introduced by the lens needs to satisfy (B5).

ACKNOWLEDGMENT (A6)

From (13), (3), (4), and (A6), (15) is obtained. Similar procedures are used to derive ( 2 2 ) and (31).

The author thanks Don Price, Laurie Wilson, David Carpenter, and John Wendoloski for helpful discussions and anonymous reviewers for their constructive comments that have helped to improve the manuscript.

47

LI: SMALL ELEMENT ARRAY ALGORITHM

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[44] R. J . Kozick and S . A. Kassam, “Synthetic aperture pulse-echo imaging with rectangular boundary arrays,” I E E E Trans. Image Processzng, vol. 2, no. 1, pp. 68-79, Jan. 1993. [45] C. R. Cooley and B. S. Robinson, “Synthetic focus imaging using partial datasets,” in 1994 Ultrason. Symp., Cannes, France, pp. 1539-1542. [46] G. R. Lockwood, P. C. Li, M. O’Donnell, and F. S. Foster, “Optimizing the radiation pattern of sparse periodic linear arrays,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 43, no. 1, pp. 7-14, Jan. 1996. [47] G. R. Lockwood and F. S. Foster, “Optimizing the radiation pattern of sparse -periodic two-dimensional arrays,” ” . IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 43, no. 1, pp. 15-19, Jan. 1996. [48] J . W. Goodman, Introduction to Fourier Optics. San Francisco, CA:.McGraw-Hill, Inc., 1968. [49] Y . Li and B. Robinson, “Phase-aberration-correction algorithm for phase-array transducers using near-field signal redundancy,” in 1997 IEEE Int. Ultrason. Symp., Toronto, Ontario, Canada, pp. 1729-1732. [50] Y . Li and R. Gill, “A comparison of matched signals used in three different phase-aberration correction algorithms,” in 1998 IEEE Int. Ultrason. Symp., Sendai, Japan, pp. 1707-1712. [51] Y . Li and B. Robinson, ‘Small-element-array ,algorithm for correcting phase-aberrations using near-field signal redundancyPart 11: Experimental results,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 47, no. 1, pp. 49-57, Jan. 2000.

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Yue Li (M’90) received the B.Sc. degree in physics from Beijing University, China, in 1982; the M.S. degree in physics from Institute of Acoustics, Academia Sinica, Beijing, China, in 1985; and the Ph.D. degree in Ultrasonics from Drexel University, Philadelphia, PA, in 1990. He was a research postdoctoral fellow in Drexel University from 1990 t o 1992. Since 1992, he has been with Division of Telecommunication and Industrial Physics, CSIRO, Australia, where he is currently a principal research scientist. His current research interest includes phase-aberration correction, blood flow measurement, and three-dimensional underwater ultrasound imaging.