J˝rgen Arendt Jensen, Member, IEEE, and Peter...

ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 45, no. 3, may 1998 837

A New Method for Estimationof Velocity Vectors

Jørgen Arendt Jensen, Member, IEEE, and Peter Munk

Abstract—The paper describes a new method1 for de-termining the velocity vector of a remotely sensed objectusing either sound or electromagnetic radiation. The move-ment of the object is determined from a field with spatialoscillations in both the axial direction of the transducerand in one or two directions transverse to the axial direc-tion. By using a number of pulse emissions, the inter-pulsemovement can be estimated and the velocity found fromthe estimated movement and the time between pulses. Themethod is based on the principle of using transverse spa-tial modulation for making the received signal influencedby transverse motion. Such a transverse modulation can begenerated by using apodization on individual transducerarray elements together with a special focusing scheme. Amethod for making such a field is presented along with asuitable two-dimensional velocity estimator. An implemen-tation usable in medical ultrasound is described, and simu-lated results are presented. Simulation results for a flow of1 m/s in a tube rotated in the image plane at specific angles(0, 15, 35, 55, 75, and 90 degrees) are made and character-ized by the estimated mean value, estimated angle, and thestandard deviation in the lateral and longitudinal direction.The average performance of the estimates for all angles is:mean velocity 0.99 m/s, longitudinal S.D. 0.015 m/s, andlateral S.D. 0.196 m/s. For flow parallel to the transducerthe results are: mean velocity 0.95 m/s, angle 0.1�, longi-tudinal S.D. 0.020 m/s, and lateral S.D. 0.172 m/s.

I. Introduction

It is a common problem to measure the velocity of amoving object, when the object is observed with a prob-

ing field. Equipment of this kind is used in medical ultra-sound to measure the velocity of blood flow noninvasively.Here an ultrasound beam is emitted by a transducer. Theultrasound field then interacts with the blood and givesrise to a scattered field, which is received by the trans-ducer. Emitting a second pulse then yields a received signalthat is displaced in time compared to the first signal, dueto the movement of the blood between pulse emissions.Repeating this a number of times yields signals suitablefor velocity estimation. Sampling the received response ata fixed time after each pulse emission, corresponding to afixed depth in tissue, results in an audio signal, with anaudio frequency proportional to both the blood velocity

Manuscript received October 6, 1997; accepted January 7, 1998.This work has been supported by B-K Medical A/S, Gentofte, Den-mark and by grant EF 632 from the Academy of Technical Sciencesof Denmark.

The authors are with the Department of Information Technology,Technical University of Denmark, DK-2800 Lyngby, Denmark (e-mail: [email protected]).

1A patent is pending on the method described in this paper.

along the ultrasound beam [1] and the emitted puls’ RFfrequency. This was used by Baker [2] to devise a systemfor displaying the velocity distribution at the depth of sam-pling. A Fourier transform of the sampled signal will yieldthe distribution as frequency is proportional to velocity.This technique also can be used to display velocity imagesby estimating the mean velocity at a particular depth, asdescribed by Kasai et al. [3]. Here the ultrasound beam isemitted a number of times in one distinct direction, andthe velocities along that direction are found by samplingthe received signal for a number of depths and then es-timate the mean velocities for the corresponding depths.The beam direction is then changed. The measurementprocedure is repeated, and the velocities are found alongthe other directions. An image of velocity is then made,and continuously updated over time. The velocity can befound through an autocorrelation approach, as describedby Kasai et al. [3] and Namekawa et al. [4]. Another tech-nique is to use cross-correlation as described by Dotti etal. [5] and Bonnefous et al. [6]. A general description of ul-trasound velocity estimation systems can be found in [1].

Radar systems also use the pulse-echo principle for esti-mating velocity of a moving object. A series of radar pulsesis emitted, and the received signals are recorded. The sig-nals from a specific distance are compared, and the veloc-ity is calculated from the movement of the object betweenpulses, the speed of light, and the time between pulse emis-sions. This is, for example, used for finding the velocity ofairplanes, missiles, or ships, described by Skolnik [7].

The pulse movement principle has also been employedin sonar for finding the velocity of different objects. This isdone by the same methods as mentioned above for medicalultrasound scanners with appropriate adaptations.

A major problem with these velocity estimation tech-niques is that only the velocity component in the beamdirection, i.e., toward or away from the transducer, canbe found. Velocity components perpendicular to the beampropagation direction cannot be measured. A number ofapproaches have sought to remedy this in diagnostic med-ical ultrasound. Two consecutive ultrasound images aremeasured in the speckle tracking approach as described byTrahey et al. [8]. The movement of a regional pattern fromone image to the next is found through two-dimensionalcross-correlation, and the velocity vector for the region isdetermined from the displacement of the region and thetime between the images. The technique needs two images,which makes data acquisition slow, and precludes the useof averaging. The image acquisition also makes this tech-nique difficult to use for full three-dimensional velocity es-

0885–3010/98$10.00 c© 1998 IEEE

838 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 45, no. 3, may 1998

timation. The two-dimensional correlation necessitates ahigh number of calculations, and erroneous velocities canevolve due to false maxima in the correlation function [9].

Another approach is to use two transducers or aper-tures emitting two beams crossing each other in the regionof interest, whereby the velocity can be found in two in-dependent directions [10]. The velocity vector can then befound through a triangulation scheme. The variance, andhence the accuracy, of the transverse component of thevelocity is affected by the angle between the two beams.The small angle between beams at large depths in tissueresults in a high measurement variance, i.e., a low preci-sion. The use of two transducers or a single large arrayalso makes probing between the ribs of a person difficultand can result in loss of contact for one of the transducers.

Other techniques like beam-to-beam correlation [11]and changes in signal bandwidth [12] have been suggested.None of these techniques have, however, attained a perfor-mance that warrants commercial implementation.

The paper proceeds along the following lines: Section IIintroduces a new method for vector velocity estimationbased on the observations from the introduction. Themethod uses a special probing field, and how to producesuch a field is detailed in Section III. A suitable estima-tion algorithm for finding the vector velocity is describedin Section IV, and a simulation of the performance is givenin Section V.

II. A New Method for Estimation of

Vector Velocity

The current systems estimate the velocity from the cor-relation between consecutively received signals. This ispossible because the received signals have an oscillatorynature, which makes it possible to perform the phase-shiftestimation of the autocorrelation approach or the time-shift estimation of the cross-correlation approach. Thephase shift estimation is particularly efficient because itonly needs one complex set of samples for each pulse-echoline to find the velocity magnitude and sign. Very few cal-culations (seven per sample) are needed for the estimationof the velocity [1]. The measurement of this set of samplescan be done directly on the radio frequency data by justtaking a sample at the depth of interest for the in-phasesignal and a second sample 1/(4f0) seconds later for thequadrature signal [1], [13]. Here f0 is the center frequencyof the emitted pulse. Velocity estimation is, thus, possiblefrom the in-phase and quadrature samples, when they em-anate from a signal with oscillations in the direction of thevelocity component.

These two observations make it, in principle, possible todevise a measurement set-up for finding the velocity vec-tor. A probing field is generated with oscillations in eachdirection for which the velocity component is of interest.Measurements of the in-phase and quadrature signals arethen made for each pulse emission. These measurementsare then used by an estimator to yield the velocity com-ponents in the preferred directions, thereby obtaining the

velocity vector [14].An illustration of the signal generation in a traditional

velocity estimation system is shown in Fig. 1. An 8 cy-cle, f0 = 3 MHz pulse is used for emission, and Gaussianapodization of the transducer and delay focusing are usedduring transmission and reception for generating the field.A snapshot of the received voltage at a fixed time is shownin the upper left graph. The graph shows the voltage re-ceived from the transducer, when a single scatterer is atthe position indicated in the graph at the time of the snap-shot. Dark areas correspond to negative voltages and lightareas to positive voltages. Gray corresponds to zero. Thus,it is a depiction of the voltage that will result from sam-pling at the time of the snapshot. A moving scatterer willtraverse the field and give rise to a sampled signal thatis a function of the spatial field, the sampling time, andthe scatterer’s velocity. The dark lines indicate the voltagewhen traversing the field in the axial or the lateral direc-tion. A purely axial velocity gives rise to a received signalthat oscillates. If the time between pulses is Tprf and thescatterer’s axial velocity is vz, then the movement betweenpulses is Tprfvz. At the center position this will result ina sampled signal as shown in Fig. 2. Here the frequency ofthe received signal is:

fz =vzλ/2

=2vzcf0 (1)

where λ = c/f0 is the wavelength of the emitted pulse,and c is the speed of sound. Note that it is a pulse-echomeasurement, hence λ/2 is used in (1). The frequency is:

fz =|~v| cos θλ/2

=2|~v| cos θ

cf0 (2)

if the velocity is at an angle θ with respect to the di-rection of the propagating ultrasound field. This also willintroduce a modulation of the received signal from the lat-eral response of the field. For a purely transverse motion(θ = π), a signal equal to the lateral response shown inthe top right graph of Fig. 1 will be received, and no ve-locity can be detected because the frequency spectrum ofthe sampled signal is centered around zero frequency.

In order to make the sampled signal influenced by trans-verse motion, a lateral modulation of the field must be in-troduced. This is shown in Fig. 3 for an ideal field with an8-cycle axial oscillation and a 4-cycle lateral modulation.The received voltage at a fixed time as a function of spatialposition is shown along with the axial and lateral responseas a function of different scatterer positions. The sampledresponse for a purely lateral movement is also shown ingraph in Fig. 4(d). It can be seen how the signal is af-fected by the lateral movement, and it can, thus, be usedfor velocity estimation. The frequency for this signal is:

fx =vxdx

(3)

where dx is the periodic distance of the lateral modulationand vx is the lateral velocity. The received sampled sig-nal is now affected by both axial and lateral movements

jensen and munk: estimation of velocity vectors 839

Lateral position [mm]

Axi

al p

ositi

on [m

m]

Conventional spatial response

−4 −2 0 2 4

0

0.5

1

1.5

2

2.5−4 −2 0 2 4


Vol

tage

Lateral responses

0

0.5

1

1.5

2

2.5

Axial responses

Axi

al p

ositi

on [m

m]

Voltage

Fig. 1. Ideal received voltage from a single scatterer as a function of spatial position for a traditional ultrasound system.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10−3

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time [s]

Vol

tage

[v]

Fig. 2. Received voltage from a single scatterer traversing the fieldin the axial direction. The velocity vz is 0.25 m/s, Tprf is 100 µs,and the transducer center frequency f0 is 3 MHz.

as shown in Fig. 4 for different velocity directions. Thisdemonstrates how the lateral modulation of the field re-sults in a modulation of the received sampled signal, whichthen contains information about both axial and lateral ve-locity.

As mentioned in the introduction, the in-phase andquadrature signals must be measured to find the sign ofthe velocity. Four measurements then are needed in thenew approach for finding the direction and magnitude ofthe velocity vector. Thus, it is necessary to employ a spa-tial quadrature sampling in which the two measurementsgive the in-phase and quadrature samples for the lateraldirection. This can be done by having two probing fields inwhich the lateral modulation of the quadrature field is 90◦

phase-shifted compared to the in-phase field’s lateral mod-ulation. This is generated by having one emit beamformerand two receive beamformers working in parallel. Samplesare acquired from the two received signals at time 2d0/cand 2d0/c + 1/(4f0) to obtain the four measurements forboth time and spatial quadrature sampling. These sam-ples are then fed into the estimator to yield the velocitycomponents.

Another processing option is to make a compensationfor the axial velocity before determining the lateral veloc-ity. The received lateral signals are affected by the move-ment in the axial direction of the beam. The velocity inthe direction of the beam, therefore, is determined first,and the effect of the axial movement is compensated for inthe received lateral signal, so that only transverse move-ment gives rise to a variation in the compensated signal.Standard techniques then can be used for finding the ve-locity or the velocity distribution, as used in a conventional



Axi

al p

ositi

on [m

m]

Modulated spatial response

−4 −2 0 2 4

0

0.5

1

1.5

2

2.5−4 −2 0 2 4


Vol

tage

Lateral responses

0

0.5

1

1.5

2

2.5

Axi

al p

ositi

on [m

m]

Voltage

Axial responses

Fig. 3. Received voltage from a single scatterer as a function of spatial position from a field that is modulated in the lateral direction.

system for finding the axial velocity. Such an estimator isdescribed in Section IV.

The lateral modulation can be generated in a multitudeof ways. It can be done through the emitted field or fromprocessing the received signals in the receive beamformer.It also can be made by a combination of transmit andreceive beamforming. The most flexibility is afforded bymainly making the lateral modulation in the receive pro-cessing, because it then can be dynamically adapted to along range of depths. The reception processing also makesit possible to employ the method in passive sonar, wherethe sound is received directly from the observed object.

The generation of the lateral modulation can be ob-tained both from apodization of the transducer array ele-ments and from the phasing of the beam. An example ofhow such fields can be constructed is given in Section III.

A number of measurements must be obtained for esti-mating the velocity. For two-dimensional velocity estima-tion, the samples for the axial velocity are, in the imple-mentation described in this paper, obtained using the tra-ditional method by acquiring the in-phase and quadraturesamples. The transverse velocity component is obtainedfrom measurements using two laterally oscillating fieldsthat are 90 degrees phase-shifted relative to each other inthe lateral direction, thereby obtaining a spatial quadra-ture sampling. The use of quadrature beams enables the

measurement of the magnitude as well as the sign of thetransverse velocity component.

A possible set-up for two-dimensional velocity estima-tion is presented in Fig. 5. An emit beamformer generatesa field suitable for both axial and lateral velocity estima-tion. Three beamformers are then used for generating thereceived signals for estimating the axial and lateral veloc-ities. The first beamformer (c) performs conventional fo-cused beamforming for estimating the axial velocity usingphase shift estimation. The two other beamformers (a) and(b) focus their beams to generate the in-phase and quadra-ture signals for the lateral velocity estimation. One set ofmeasurements results from these two beams for each pulseemission. The samples from the axial and lateral beamsare fed into the estimator to yield the velocity vector.

III. Field Generation

The transverse spatial oscillation of the field can be gen-erated in a multitude of ways. Different strategies can beapplied, and the choice mainly is determined by the num-ber of oscillations needed in the lateral direction. The lat-eral modulation of the field can be generated by a singlearray transducer using special beamforming during trans-mit and/or receive. According to linear system theory, the


Fig. 4. Received sampled signal for different axial and lateral ve-locities: (a) ~v = (v, 0), (b) ~v = (cos 30◦, sin 30◦)v, (c) ~v =(cos 45◦, sin 45◦)v, (d) ~v = (0, v). The velocity magnitude v is0.25 m/s, fprf is 10 kHz, the transducer center frequency is 3 MHz,and dx is 2 mm.

Fig. 5. Block diagram of the main components of the system.

transmit and receive beamforming can be interchanged togenerate the resulting field with similar lateral modulationat a certain depth.

The lateral modulation can be generated throughapodization or through steering parts of the beams, sothat they interact and generate the lateral modulation,or it can be a combination of the two. Apodization func-tions control the vibration amplitude of the different arrayelements. Many apodization schemes will lead to a trans-versely oscillating field, the use of functions with two sep-arate peaks across the aperture being a typical example.The beam steering can be done either as plane-wave inter-action or as other forms of focusing at or near the depthfor measuring the velocity vector.

A starting point for developing the lateral modulationis to use the theory for continuous wave (CW) fields. Here

the radial pressure field variation in the far field regionis given by the Fourier transform of the emitting aperturefunction [15]. Thus, it is possible to give a direct predictionof possible aperture functions.

In designing the measurement situation, it must benoted that the combined field characteristics depend onboth the transmitted field and the processing of the re-ceived signal. For the far field approximation, where thereis no radial phase variation due to diffraction, the pulse-echo radiation pattern can be found by multiplying thetransmit pattern with the receive sensitivity pattern.

A. Transmit Apodization Function

Traditional velocity estimation systems uses focusing ofthe emitted field, and this pulsed field will change shapeas a function of depth. This change is not taken into con-sideration when the velocity component in the radial di-rection is calculated. To have a spatially oscillating field,which only depends on the applied receive beamforming,nondiffracting beams must be generated in transmit. Thisensures that the modulation pattern is independent of thedepth, and that the velocity components being measuredcan become spatially uncorrelated.

Nondiffracting beams, thus, must be used in order togenerate a laterally modulated field with a constant spa-tial frequency measured at a straight line across the beam,orthogonal to the direction of propagation. If diffractingbeams are used, a constant spatial frequency can be mea-sured along spherical lines of different radii dependent ofthe depth, as changes in the phase-fronts of the pulse thenare avoided during propagation. Such phase changes willaffect the estimator as will curved phase fronts because thelateral frequency is changed through the field for a curvedphase front.

Nondiffracting beams exist only in theory for infinitelylarge apertures as can be seen from a solution to the lin-ear wave equation [16]. In practice the beams of interestare only diffraction-free for a limited range because of thefinite aperture used [17]. It has been shown [18], that thefields generated by axicon transducers are similar to thenondiffracting beams proposed by Durnin [16]. The axiconfocusing, originally presented by McLeod [19], uses a lin-early increasing time delay function from the edge to thecenter of the aperture. From a focusing point of view, theaxicon focusing creates phase coherence along a focal line,whereas quadratic focusing has phase coherence at a pointin space. Axicon focus also is referred to as conical focus.

In the replica pulse method [20], the diffraction effectsfor a plane piston are explained intuitively as the result ofthe edge wave. The edge wave emanates from the abruptchange in surface velocity at the aperture edge and canbe avoided by apodizing the aperture. As stated before,the CW pressure far field can be calculated as the spatialFourier transform of the aperture’s velocity function. It isknown from signal analysis that the Fourier transform ofthe Gaussian function is a Gaussian function. This intu-itively indicates that the field from a Gaussian apodized


aperture most likely does not change shape in the nearfield. When the far field is reached, it still maintains theGaussian shape with decreasing amplitude and increasingmain lobe width as the depth increases.

The amplitude of the CW radially-symmetric pressurefield for a Gaussian-apodized piston is [21]:

p(ρ, z) = Aσ(0)σ(z)

exp(− ρ2

σ2(z)

)(4)

where

ρ = (x2 + y2)1/2 (5)

and

σ(z) =λ

πσ(0)

√z2 +

π2

λ2 σ4(0). (6)

The distance from the center axis is ρ, A is a constantof proportionality, λ is the wavelength in the surround-ing medium, and σ(0) is the half-width of the amplitudeprofile at the surface of the transducer. The z-axis is inthe propagation direction, and x, y are transverse to thebeam axis. The beam half-width is defined as the radialdistance ρe from the center of the disc at which the ampli-tude drops to 1

e of its maximum. The field properties arecontrolled by σ(z); and (6) shows that the maximum valuefor p(ρ, z) exists for z = 0 and that it decays slowly with-out any oscillations, thus preserving a Gaussian shape atall depths. On the basis of these observations the transmitapodization function is chosen to be Gaussian.

B. Receive Apodization Function

The receive beamforming must be designed to createthe lateral spatial modulation. Determining the receiveapodization function is somewhat more complicated, andsome direct relation between the transducer characteris-tics and the field pattern is needed. The far-field CW ra-diation can be calculated by a Fourier transform of thetransducer’s front-face velocity distribution [15]; this canbe employed in a derivation of the needed apodization pat-tern. The derivation is based on one-dimensional observa-tions. A comparison to a full three-dimensional pulse-echosimulation with the derived field is presented later.

The relation between the front face velocity distributionr(ξ) at the lateral aperture position ξ and the far-fieldradiated pressure R(x) at the lateral field position x is [15]:

R(x) = k1

∞∫−∞

r(ξ) exp(−j 2π

λzxξ

)dξ (7)

when using a far-field para-axial approximation. Here λis the wavelength, z is axial distance to the field point,and k1 is a constant of proportionality that is neglectedduring the following derivation. The integral correspondsto a Fourier transform with a scaling of x by 1/(λz).

The lateral modulation must contain a number of os-cillations and should be bounded to have a finite probingfield. The lateral modulation can thus be described by alateral oscillation multiplied by a rectangular windows as:

R(x) = rect(L) cos(2πfxx) (8)

where x is lateral distance, fx is spatial frequency in thelateral direction, and rect(L) denotes a rectangular func-tion of width L centered around x = 0. The distance be-tween the lateral peaks is dx = 1/fx, and the width of thefield is L. The desired pattern thus consists of two termsthat should be generated by the transducer through thefront face apodization function.

Let the velocity distribution of the aperture r(ξ) be aconvolution of two functions r1(ξ) and r2(ξ):

r(ξ) = r1(ξ) ∗ r2(ξ) (9)

then the radiation pattern becomes:

R(x) = F {r(ξ)} = F {r1(ξ) ∗ r2(ξ)} = R1(x)R2(x)(10)

when spatial Fourier transformation is denoted F , and ∗denotes convolution. The Fourier transform of the rectan-gular function of width L is:

r1(ξ) = F {rect(L)} =L

zλ

sin(πξ Lzλ

)πξ Lzλ

. (11)

A rectangular field, thus, can be obtained with an aperturefunction of a sinc. The width of the field is determined bythe spacing of the zero crossings in the sinc function. Thefirst zero crossing ξ0 is at:

ξ0 =zλ

L. (12)

A wide rectangle thus gives a zero close to the peak atξ = 0 in the sinc function.

The Fourier transform of the cosine term is:

r2(ξ) =1

2zλ

[δ

(ξ

zλ+ fx

)+ δ

(ξ

zλ− fx

)](13)

and combining this with (11) gives:

r(ξ) = r1(ξ) ∗ r2(ξ) =

L

2zλ

sin(π(ξzλ + fx

)L)

π(ξzλ + fx

)L

+sin(π(ξzλ − fx

)L)

π(ξzλ − fx

)L

.(14)

Thus, to generate a lateral oscillation, the aperture musthave a velocity distribution consisting of two sinc func-tions. This aperture function will give a field symmetricacross the acoustic axis of the aperture. The spatial fre-quency fx of the lateral modulation is determined by thedistance between the peaks of the two sinc functions and is:

fx =ξtzλ

=1dx

(15)


Array transducer

x

z

x

z

Array transducer

Fig. 6. Alignment of the pulses through delay focusing.

where ξt is the positive position of the maximum of thesinc function.

The requirement for the velocity estimation is two simi-lar fields phase shifted 90 degrees in relation to each otherin the lateral direction. During the receive process, it ispossible with an array transducer and a number of beam-formers to generate a number of beams in parallel. Bothfields, therefore, can be generated in parallel through asimple phasing and apodization before final summation.The two fields can be generated by shifting them by aphase angle θ and −θ, which tilts the beams by ±θ fromthe z-axis [22]. Let a linear phase shift exp(jkθξ) be addedto an aperture apodization r(ξ) by:

rtilt(ξ) = r(ξ) exp(jkθξ). (16)

Here θ is the phase angle and k = 2π/λ is the wave num-ber.

C. Calculation of Delay Factors

The derivation made above was based on CW theory.Thus, a wave emanating from any point on the aperturecan interfere with a wave from any other point on theaperture at any given position in space. The pulse lengthis in theory infinitely long. In the pulsed mode the pulselength is finite; therefore, the wave fronts must be alignedto interfere at the point of interest. This is illustrated inFig. 6. For weak focusing this also has the consequencethat the relation made in the far field can be found at thefocus plane [22].

Conical focusing is used to direct the energy toward themeasuring point by aligning the phase fronts for the shortpulse used. The two probing beams needed for calculatingthe lateral velocity component govern the choice of theangles θ, α1 and α2 defined in Fig. 7. The cone anglesα1 and α2 are the angles between the wave fronts andthe transducer surface. The position of the center of theoscillating field is controlled by θ.

With reference to Fig. 7, the array element delays arecalculated assuming plane wavefronts. The variable OF

α2

θγ1

α1

γ2

OF

EL

D

D

1

2

Z

X

B

C

FP

ZF

Array transducer

Array elements

Phase front

for wave Phase front

for wave

Fig. 7. Definition of geometry for finite plane wave focusing.

indicates the center position of the sinc-function with ref-erence to the aperture center, ZF is the depth of interest,EL is the distance from the center of the element to thecenter of the aperture, and FP indicates the offset of thefocal point from the transducer center axis.

The following trigonometric relations are used:

sinα1 =D1

EL(17)

cosα1 =B

OF(18)

cos θ =ZF

C(19)

sin θ =FP

C(20)

tan θ =FP

ZF(21)

γ1 =π

2− α1 − θ (22)

cos γ1 =B

C(23)

cos(π

2− α

)= sinα (24)

sin(a+ b) = sin a cos b+ cos a sin b. (25)

Using (18), (20), and (22) in (23) gives:

cos(π

2− α1 − θ

)=OF

FPcosα1 sin θ (26)

and then applying (24) and (25) yields:

cosα1 sin θ(OF

FP− 1)

= sinα1 cos θ. (27)

Now α1 can be found using (21)

α1 = arctan(FP

ZF

(OF

FP− 1))

. (28)


The delay in time for the in-phase lateral receive beam-forming can be found as

D1 =1c·EL · sin

[arctan

(1ZF

(OF − |FP |))]

(29)

where c is the speed of sound. The delays D2 for thequadrature lateral receive beamforming are found for anegative value of FP. Thus:

D2 =1c·EL · sin

[arctan

(1ZF

(OF + |FP |))]

.(30)

The focus point FP for beam 1 and 2 must be chosen,so that the radiation patterns are symmetrically placedaround the axis. Using the value:

FP = ±dx8

(31)

gives beams that are shifted one quarter of the lateral pe-riode relative to each other, which in turn makes themsuitable for the in-phase and quadrature measurements inthe lateral direction.

D. Calculation of the Lateral Modulation Frequency

The value to be used for FP can be found from (15) and(31). Because the method used for finding (15) is based onFraunhofer far-field assumption, two other methods of cal-culating the lateral wave length using geometric approxi-mations are examined. This is done in order to ensure aproper validation of the lateral wavelength.

First the wavelength is calculated from two plane wavesintersecting each other at an angle of 2α with α = α1 = α2.The second approach uses the intersection of two sphericalwaves. The actual lateral frequency will be in the range ofthese two results, since both calculations are approxima-tions to the correct three-dimensional pressure field.

The plane wave interference is shown in Fig. 8. Theplane waves are represented by:

p1(t, x, z) = P1 exp(−jωt) exp (jk (xnx + znz))(32)

and

p2(t, x, z) = P2 exp(−jωt) exp (jk (znz − xnx))(33)

where P1 and P2 are the pressure amplitudes of the waves,and ω is its angular frequency. Here nx and nz are theprojections of the propagation direction to the respectiveaxis, i.e.:

nx = sinαnz = cosα.

2αα λ

α

d /2x

λ/(2 Cos α)

z

xα α

Array transducer

Fig. 8. Intersection of two finite plane waves.

The resulting pressure pt is, assuming P0 = P1 = P2,

pt(t, x, z) = p1(t, x, z) + p2(t, x, z)= P0 exp (−jωt) [exp (jk (xnx + znz)

+ exp (jk (znz − xnx)))]= 2P0 exp (−jωt) exp (jkznz) cos (kxnx) .

(34)

This is a traveling wave in the z direction with an ampli-tude modulation in the x direction. It must be noted thatthe resulting wave travels with a phase velocity of:

cz,phase =c

nz. (35)

From the close-up in Fig. 8, the distance between lateralpeaks dx can be found from:

tanα =λ

4 cosαdx4

(36)

yielding

dx =λ

sinα. (37)

The longitudinal wave length is:

λz =λ

cosα. (38)


r

r

r1

2

ϕ

P

OFOF

Array transducer

Array elements

Fig. 9. Geometry for interaction of spherical waves.

Using (28) for FP = 0 then:

dx =λ

sin[arctan

(OFZF

)] . (39)

Typically ZF > 2 · OF which results in an error of lessthan 10% using:

sin[arctan

(OF

ZF

)]≈ OF

ZF(40)

which gives a much simpler expression for calculating dx:

dx = λZF

OF. (41)

This result corresponds to that in (15).For the derivation using spherical waves, two simple ra-

diators are placed at the centers of the sinc-functions. Thegeometry is depicted in Fig. 9, and the gray dots indicatethe origin for the spherical waves. The pressures of thesetwo spherical waves can be written as:

p1 (r1) = A(r1) exp (−j (ωt− kr1))p2 (r2) = A(r2) exp (−j (ωt− kr2)) .

(42)

The relation between r, r1, and r2 is:

r1 (r) = r2 +OF 2 − 2rOF cosϕ

r2 (r) = r2 +OF 2 + 2rOF cosϕ.(43)

The total pressure at a given location r is:

ptot(r) = A(r1 (r)) exp (−j (ωt− kr1 (r)))+A(r2 (r)) exp (−j (ωt− kr2 (r))) . (44)

In order to simplify the problem, it is assumed that r �2OF , thus:

r1 ≈ r −OF cosϕr2 ≈ r +OF cosϕ

(45)

and with A (r) ≈ A (r1) ≈ A (r2) the total pressure createdby the two interacting spherical waves becomes:

ptot(r) = A(r) exp(−jωt) ·[exp (jk (r −OF cosϕ)) + exp (jk (r +OF cosϕ))]= A (r) exp (−jk (ct− r)) ·

[exp (−jkOF cosϕ) + exp (jkOF cosϕ)]= 2A (r) exp (−jk (ct− r)) cos (kOF cosϕ) .

(46)

The spatial frequency for the radial field is not constant.The width ∆ϕ of the first lobe for the direction ϕ is definedby the angles ϕ01 and ϕ02 of the zeros for cos(kOF cosϕ) =0 given by:

kOF cosϕ01 =π

2kOF cosϕ02 = −π

2. (47)

The width is:

∆ϕ = |ϕ01 − ϕ02| . (48)

The radial wavelength λr is governed by:

λr/2 = r ·∆ϕ. (49)

When kOF > 2OF � λ the angles will be very close toπ/2, and this small difference is denoted δ. Hereby:

ϕ01 =π

2− δ

ϕ02 =π

2+ δ

|ϕ01 − ϕ02| = 2δ

and

cos(π

2− δ)

= sin δ

and

sin δ ≈ δ for |δ| � 1

then

∆ϕ =π

kOF=

λ

2OF(50)

and the radial wavelength λr is given at the depth of in-terest by:

λr = ZF · λ

OF. (51)

This result corresponds to that in (41) and (51), show-ing that the three methods for the stated conditions giveapproximately the same results. The lateral wavelengthis thus determined by the depth, the distance between thesinc peaks, and the fundamental RF frequency of the emit-ted pulse.


E. Field Simulation Example

Using a nonfocused Gaussian apodization for the trans-mit beam and two sinc functions for apodization alongwith axicon plane wave focusing for receive beamforming,results in a field with a sensitivity that oscillates spatiallyin the transverse direction. The receive beamforming, thatgenerates the lateral modulation by apodization and planewave focusing can be controlled dynamically by moving thecenter and the scaling of the sinc function to control thelateral wavelength and the number of oscillations. The ad-vantage of this approach, instead of transmit focusing andapodization, is the ability to obtain a controlled spatiallyoscillating field for all depths.

The basic principle derived from the above was exam-ined for a pulsed system using the Field simulation pro-gram [23]. The form of the transmit field basically shouldbe independent of the depth because a traveling Gaussianplane wave is nondiffracting [21]. A linear array transducerwith 64 elements with a width of 0.41 mm, height 5.0 mm,and a pitch2 of 0.51 mm was used. An 8 cycle, 3 MHz pulsewas used in the simulation.

The emitted pressure fields for nonfocused and focusedGaussian apodized transducers are shown in Figs. 10and 11. The pulsed field is calculated for specific depthsto illustrate the evolution of the pulse shape as it travels.

To create a similar illustration of the receive sensitivityin Fig. 12, the setup for the receive beamforming is usedin a transmit simulation. Hereby the construction of thelateral oscillation can be illustrated, although this is onlycreated during receive beamforming. The apodization anddelay values used are shown in Fig. 13.

The number of zeros from the top of the sinc-functionto the edge is 5, and the offset (OF) of the sinc top fromthe center of the tranducer is 18 times the pitch. The in-terference of the two waves is clearly seen in Fig. 12.

The resulting pulse-echo sensitivity fields for a depth of70 mm are shown in Fig. 14 for the in-phase and quadra-ture channel, respectively. The displacement for the pres-sure fields of one-fourth of the lateral wavelength can beobserved.

For the configuration chosen, dx is:

dx = λZF

OF=

15403 · 106

7018 · 0.51

= 3.9 · 10−3 [m] .

The distance found by inspection of the actual generatedacoustical field was 4 mm (Fig. 15), which is in very goodagreement with the predicted value.

The sinc is calculated to have a zero at a fictional ele-ment adjacent to the edge element in order to reduce edgewaves to a minimum. The number of zeros from the top ofthe sinc to the edge of the aperture determines the band-width of the transversal oscillation. Truncating the sinccorresponds to multiplication with a rectangular window.The effect on the field is a convolution of the ideal fieldwith the Fourier transform of the window.

2The pitch is the distance from the center of one element to thecenter of the adjacent element.

Fig. 10. Emitted pressure field for unfocused Gaussian beam as afunction of depth.

To display the lateral wavelength, the envelope of theaxial direction is calculated for an emitted pressure field.The envelope is calculated using the Hilbert transform ofthe time signal. The lateral modulation at one point intime in the middle of the pulse field are shown in Fig. 16.The result is calculated for different numbers of zeros(number of zeros = 3, 5, 7) in the sinc apodization func-tion. This demonstrates the effect of the width of the sincfunction on the lateral field oscillation. A narrower mainlobe (more zeros) in the sinc function results in more os-cillations with a higher amplitude in the lateral field. Thisshows that a lateral modulation suitable for transverse ve-locity estimation can be generated, which was the centralgoal here. A broad, nondiffracting emit field was used, andthe lateral modulation was solely created through receivefocusing and apodization. This yields a fairly broad lateralfield, but this should not be seen as a central limiting fac-tor for the approach. An emit focusing can be introducedto narrow the probing field and make dx smaller, and thisis the topic of further research. More information aboutthe field generation can be found in [14].


Fig. 11. Emitted pressure field for focused Gaussian beam as a func-tion of depth.

IV. Estimation Algorithm

The estimator must determine both the axial and lateralvelocity. A compensation for the axial velocity must bedone before determining the lateral velocity. Therefore, theaxial velocity is determined first.

The signal from the axial beamformer is passed on tothe axial velocity processor, which samples the signal atthe time t = 2d/c, where d is the depth in tissue and c is thespeed of sound. A second quadrature sample is acquired attime t = 2d/c+ 1/(4f0), where f0 is the center frequencyof the emitted pulse. One set of samples is taken for eachpulsed field received, and the samples for line number i aredenoted x(i) and y(i). The first signal has the number i =0. The axial velocity is found by using the equation [1], [3]:

vz = − c

4πTprff0×

arctan

(∑Nc−2i=0 y(i+ 1)x(i)− x(i+ 1)y(i)∑Nc−2i=0 x(i+ 1)x(i) + y(i+ 1)y(i)

)(52)

where Tprf is the time between pulse emissions from thearray, and Nc is the number of pulse-echo lines in the samedirection used in the estimator.

The axial velocity is used for selecting the samples fromthe left and right signals from the two other beamformers.

Fig. 12. Emitted pressure field for two interacting finite plane wavesused for creating a transverse modulation.

10 20 30 40 50 60−0.5

0

0.5

1

Element number

Nor

mal

ized

apo

diza

tion center of aperture

10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

Element number

Nor

mal

ized

tim

e de

lay center of aperture

Fig. 13. Time delay and amplitude apodization values used on theindividual transducer elements for creating the lateral modulationduring receive beamforming.



Axi

al p

osi

tion [m

m]

Fig. 14. Received voltage for a single scatterer as a function of itsposition at a depth of 70 mm. The in-phase voltage is shown on thetop and the quadrature voltage is shown on the bottom.

Fig. 15. Voltage response as a function of lateral position for thepulsed field used in the example.

The samples taken from the left signals, denoted gl(t), aregiven by:

xl(i) = gl

(2dc− 2vzTprf

ci

)(53)

so as to compensate for the influence from the axial move-ment of the blood. Correspondingly, samples taken fromthe right signals, denoted gr(t), is given by:

yr(i) = gr

(2dc− 2vzTprf

ci

). (54)

These samples enter the estimator given by:

vx = − c

2πTprffx×

arctan

(∑Nc−2i=0 yr(i+ 1)xl(i)− xl(i+ 1)yr(i)∑Nc−2i=0 xl(i+ 1)xl(i) + yr(i+ 1)yr(i)

)(55)

Distance from center line [mm]

Five zeros

Three zeros

One zero

No

rma

lize

d e

nve

lop

e d

ete

cte

d la

tera

l se

nsi

tivity

Fig. 16. Lateral response with oscillations as a function of zeros inthe sinc apodization function.

where fx is the frequency of the laterally oscillating trans-ducer field at the particular depth. Then vx is the trans-verse velocity.

V. Performance

The functionality of the method is examined for two-dimensional velocity vector measurement and is docu-mented by simulations. The simulation is performed usingthe impulse response method developed by Tupholme [24]and by Stepanishen [25] in the implementation developedby Jensen and Svendsen [23]. The high accuracy of thisapproach, when compared to measurements, is describedin Jensen [26]. The paper showed that the simulated pres-sure values were within 1% of the measured ultrasoundfields. The simulation approach is applicable for pulsedfields and is used for three-dimensional modeling of theresponse from a collection of scatterers.

The simulated situation is shown in Fig. 17. A vesselof 10 mm diameter is placed 70 mm from the center ofthe transducer array, i.e., on the axis of the transducer.The vessel contains plug flow (all blood scatterers havethe same velocity), and the 15,000 scatterers in the ves-sel have a Gaussian scattering amplitude distribution withzero mean value and unit variance. This ensures fully de-veloped speckle in the response from the blood model. Thesimulation is done for constant velocity of 1 m/s and avarying angle (θ) for the flow vector. The angles used are0, 15, 35, 55, 75, and 90 degrees. The estimator uses 50RF lines for finding an estimate and the experiment hasbeen repeated 20 times for each angle. The 64-element ar-ray transducer specified in Section III-D was used duringthe simulation.

Fig. 5 shows the implementation of the method appliedhere for the measurement of blood velocity in two dimen-


x

Array transducer

z

θ

vx

vz

v

lateral velocity

axial velocity

Fig. 17. Definition of axial and lateral velocity for the computer ex-periment.

sions. It consists of a generator or pulser, 1; an emit beamformer, 2; a linear array ultrasound emitting transducer, 3;a linear array ultrasound receiving transducer, 5; three re-ceive beam formers, 6a, 6b, and 6c working in parallel andreceiving signals from the receiving transducer. The beam-formed signals are fed into the estimator, 7; that calculatesthe velocity vector. The pulser, 1, generates a pulsed volt-age signal with a number of sinusoidal oscillations at a fre-quency of f0 in each pulse, that is used in the emit beamformer, 2. The emit beam former, 2, is capable of indi-vidually attenuating and delaying the signals to each ofthe elements of the transducer array, 3. In this simulationno delay is introduced during emission, and in Fig. 18 theattenuation values are shown as a function of element num-ber in the transducer. This is done to emit a broad fieldthat can be used for forming all three beams in parallel. Alinear array transducer is used for both emitting and re-ceiving the pulsed ultrasound field. The emitted field fromthe transducer is scattered by the blood in the blood ves-sel, and part of the scattered field is received by the lineararray transducer. The signals from the individual elementsare passed on to two of the receive beam formers, i.e., 6aand 6b in Fig. 5. The signals from the elements are indi-vidually scaled in amplitude and individually delayed andare finally summed to yield a single output signal fromeach receive beamformer. The first receive beamformer,6a in Fig. 5, generates the left signal and the second re-ceive beamformer, 6b in Fig. 5, generates the right signal.Fig. 13 shows both the delay values and the correspondingamplitude scaling factors for both receive beamformers.

Apodiz

ation v

alu

e

Fig. 18. Amplitude scaling factors or equivalently, apodization, usedfor the emit beam former.

The third receive beam former, 6c in Fig. 5, generates thesignal for estimating the axial component.

The result of the simulation is shown in Fig. 19. Thetrue velocity vectors are indicated by the small arrows.The gray ellipses for each velocity vector estimate indi-cate the standard deviations for both the axial estimationand the lateral estimation, respectively. The lateral stan-dard deviation is the semi-major axis, and the axial stan-dard deviation is the semi-minor axis. The mean values ofthe estimates are illustrated by the circles at the centersof the ellipses. The average performance for all angles is:mean velocity 0.99 m/s, longitudinal S.D. of the estimates0.015 m/s and lateral S.D. of the estimates 0.196 m/s. Forflow parallel to the transducer the results are: mean veloc-ity 0.95 m/s, angle 0.1◦, longitudinal S.D. 0.020 m/s, andlateral S.D. 0.172 m/s.

The velocity estimation has only been done at a fixeddistance from the transducer in the simulation. Due tothe use of a nonfocused field, it is possible to dynamicallychange the focusing of the three receive beam formers togenerate the spatially oscillating field at other depths forthe same emitted field, as was demonstrated in Section III.

The example described here only estimated the velocityin a plane, but the method can be changed to give the fullthree-dimensional velocity vector. A two-dimensional ma-trix transducer must then be used as described by Smithet al. [27]. The same emission field can be used because itis unfocused. An extra set of receive beam formers mustthen be employed to make the velocity estimation in they-direction perpendicular to both the z- and x-directions.

VI. Summary

A new method for velocity vector estimation has beenpresented. It uses a single array transducer for measuringthe axial and transverse velocity, which makes it conve-


Fig. 19. Resulting standard deviations for the estimated mean veloci-ties. The true velocity vectors are indicated by the small arrows. Thegray ellipses for each velocity vector estimate indicate the standarddeviations for both the axial estimation and the lateral estimation,respectively. The lateral standard deviation is the semi-major axis,and the axial standard deviation is the semi-minor axis. The meanvalues of the estimates are illustrated by the circles at the centers ofthe ellipses.

nient to use a small aperture window. The axial and lat-eral velocities are estimated, and the sign of the lateralvelocity is determined by using two receive beamformers.

The method uses a number of consecutive pulse emis-sions and compares the received signals to minimize theeffects from media dependent distortions like attenuationand refraction. The method thus only uses the differencefrom pulse to pulse to determine the velocity.

Only two measurements are taken from each beam fordetermining the lateral velocity, so that a very modestamount of calculations must be performed to estimate thetransverse velocity. Employing a standard autocorrelationapproach makes the method robust in terms of noise inthe measurement process, since this estimator is unbiasedfor white noise.

The method can be expanded to find the full velocityvector by using a third beam former, then all the com-ponents of the three-dimensional velocity vector can beestimated.

Acknowledgments

The authors would like to thank Søren Kragh Jespersen,Paul F. Stetson, Jens E. Wilhjelm, and Ole Trier Ander-sen, all at the Department of Information Technology, whostruggled through the many versions of this paper, founderrors, and came up with numerous improvements.

References

[1] J. A. Jensen, Estimation of Blood Velocities Using Ultrasound:A Signal Processing Approach. New York: Cambridge Univ.Press, 1996.

[2] D. W. Baker, “Pulsed ultrasonic Doppler blood-flow sensing,”IEEE Trans. Sonics Ultrason., vol. SU-17, pp. 170–185, 1970.

[3] C. Kasai, K. Namekawa, A. Koyano, and R. Omoto, “Real-timetwo-dimensional blood flow imaging using an autocorrelationtechnique,” IEEE Trans. Sonics Ultrason., vol. 32, pp. 458–463,1985.

[4] K. Namekawa, C. Kasai, M. Tsukamoto, and A. Koyano, “Re-altime bloodflow imaging system utilizing autocorrelation tech-niques,” in Ultrasound ’82, R. A Lerski and P. Morley, Eds.,New York: Pergamon Press, 1982, pp. 203–208.

[5] D. Dotti, E. Gatti, V. Svelto, A. Ugge, and P. Vidali, “Blood flowmeasurements by ultrasound correlation techniques,” EnergiaNuclear, vol. 23, pp. 571–575, 1976.

[6] O. Bonnefous, P. Pesque, and X. Bernard, “A new velocity es-timator for color flow mapping,” Proc. IEEE Ultrason. Symp.,1986, pp. 855–860.

[7] M. I. Skolnik, Introduction to Radar Systems. New York:McGraw-Hill, 1980.

[8] G. E. Trahey, J. W. Allison, and O. T. von Ramm, “Angleindependent ultrasonic detection of blood flow,” IEEE Trans.Biomed. Eng., vol. BME-34, pp. 965–967, 1987.

[9] J. A. Jensen, “Artifacts in velocity estimation using ultrasoundand cross-correlation,” Med. Biol. Eng. Comp., vol. 32/4, Suppl.,pp. s165–s170, 1994a.

[10] M. D. Fox, “Multiple crossed-beam ultrasound Doppler ve-locimetry,” IEEE Trans. Sonics Ultrason., vol. SU-25, pp. 281–286, 1978.

[11] H. F. Routh, T. L. Pusateri, and D. D. Waters, “Preliminarystudies into high velocity transverse blood flow measurement,”Proc. IEEE Ultrason. Symp., 1990, pp. 1523–1526.

[12] V. L. Newhouse, D. Censor, T. Vontz, J. A. Cisneros, and B. B.Goldberg, “Ultrasound Doppler probing of flows transverse withrespect to beam axis,” IEEE Trans. Biomed. Eng., vol. BME-34,pp. 779–788, 1987.

[13] P. A. Magnin, “Doppler effect: History and theory,” Hewlett-Packard J., vol. 37, pp. 26–31, 1986.

[14] P. Munk, “Estimation of the 2-D flow vector in ultrasonic imag-ing: a new approach,” M.S. thesis, Dept. Information Technol-ogy, Technical University of Denmark, Lyngby, Denmark, 1996.

[15] J. W. Goodman, Introduction to Fourier Optics, 2nd ed. NewYork: McGraw Hill, 1996.

[16] J. Durnin, “Exact solutions for non-diffracting beams. I: thescalar theory,” J. Opt. Soc. Amer., vol. 4, pp. 651–654, 1987.

[17] J. Durnin and J. J. J. Miceli, “Diffraction-free beams,” Amer.Phys. Soc. J., vol. 58, pp. 1499–1501, 1987.

[18] J. A. Campbell, “Generation of a nondiffracting beam with fre-quency independent beamwidth,” J. Acoust. Soc. Amer., vol.88, pp. 2467–2477, 1990.

[19] J. H. McLeod, “The axicon: A new type of optical element,” J.Opt. Soc. Amer., vol. 44, pp. 592–597, 1954.

[20] A. Freedman, “Acoustic field of a pulsed circular piston,” J.Sound Vibration, vol. 170, pp. 495–519, 1994.

[21] D. K. Hsu, F. J. Margetan, M. D. Hasselbusch, S. J. Wormley,H. S. Hughes, and D. O. Thompson, “Technique for nonuniformpoling of piezoelectric element and fabrication of gaussian trans-ducers,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol.37, pp. 404–410, 1990.

[22] B. D. Steinberg, Principles of Aperture and Array System De-sign. New York: Wiley, 1976.

[23] J. A. Jensen and N. B. Svendsen, “Calculation of pressure fieldsfrom arbitrarily shaped, apodized, and excited ultrasound trans-ducers,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol.39, pp. 262–267, 1992.

[24] G. E. Tupholme, “Generation of acoustic pulses by baffled planepistons,” Mathematika, vol. 16, pp. 209–224, 1969.

[25] P. R. Stepanishen, “Transient radiation from pistons in an infinteplanar baffle,” J. Acoust. Soc. Amer., vol. 49, pp. 1629–1638,1971.

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[27] S. W. Smith, H. G. Pavy, and O. T. von Ramm, “High-speed ul-trasound volumetric imaging system—Part I: Transducer designand beam steering,” IEEE Trans. Ultrason., Ferroelect., Freq.Contr., vol. 38, pp. 100–108, 1991.

Jørgen Arendt Jensen (M’93) received his M.S. degree in electri-cal engineering in 1985 and the Ph.D. degree in 1989, both from theTechnical University of Denmark. He received the Dr.Techn. fromthe university in 1996.

Dr. Jensen has published a number of papers on signal processingand medical ultrasound and the book Estimation of Blood VelocitiesUsing Ultrasound, Cambridge University Press in 1996. He has beena visiting scientist at Duke University, Stanford University, and theUniversity of Illinois at Urbana-Champaign. He is currently full pro-fessor of Biomedical Signal Processing at the Technical University ofDenmark at the Department of Information Technology.

Peter Munk was born in Grenaa, Denmark, 1963. He received theBachelor of Engineering degree (electrical) in 1987 and his M.Sc.degree in 1996 from the Technical University of Denmark (DTU).He currently is a Ph.D. candidate at the Department of InformationTechnology, DTU.

His research interests are in acoustics and signal processing(space-time and time-frequency) for application within ultrasoundimaging and flow measurements.

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