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Fourier Domain Beamforming:The Path to Compressed Ultrasound Imaging

Tanya Chernyakova,Student Member, IEEE, Yonina C. Eldar,Fellow, IEEE

Abstract—Sonography techniques use multiple transducer el-ements for tissue visualization. Signals detected at each elementare sampled prior to digital beamforming. The sampling ratesrequired to perform high resolution digital beamforming ar esignificantly higher than the Nyquist rate of the signal and resultin considerable amount of data, that needs to be stored andprocessed. A recently developed technique, compressed beam-forming, based on the finite rate of innovation model, compressedsensing (CS) and Xampling ideas, allows to reduce the numberof samples needed to reconstruct an image comprised of strongreflectors. A drawback of this method is its inability to treatspeckle, which is of significant importance in medical imaging.Here we build on previous work and extend it to a general conceptof beamforming in frequency. This allows to exploit the lowbandwidth of the ultrasound signal and bypass the oversamplingdictated by digital implementation of beamforming in time. Usingbeamforming in frequency, the same image quality is obtainedfrom far fewer samples. We next present a CS-technique thatallows for further rate reduction, using only a portion of th ebeamformed signal’s bandwidth. We demonstrate our methodson in vivo cardiac data and show that reductions up to 1/28 overstandard beamforming rates are possible. Finally, we presentan implementation on an ultrasound machine using sub-Nyquistsampling and processing. Our results prove that the conceptofsub-Nyquist processing is feasible for medical ultrasound, leadingto the potential of considerable reduction in future ultrasoundmachines size, power consumption and cost.

Index Terms—Array processing, beamforming, compressedsensing (CS), ultrasound, sub-Nyquist sampling.

I. I NTRODUCTION

Diagnostic ultrasound has been used for decades to visu-alize body structures. Imaging is performed by transmittinga pulse along a narrow beam from an array of transducerelements. During its propagation echoes are scattered byacoustic impedance perturbations in the tissue, and detectedby the array elements. The data, collected by the transducers,is sampled and digitally integrated in a way referred to asbeamforming, which results in signal-to-noise ratio (SNR)enhancement and improvement of angular localization. Sucha beamformed signal, referred to as beam, forms a line in theimage.

According to the classic Shannon-Nyquist theorem [1], theminimal sampling rate at each transducer element should beat least twice the bandwidth of the detected signal in orderto avoid aliasing. In practice, rates up to4-10 times thecentral frequency of the transmitted pulse are required inorder to eliminate artifacts caused by digital implementation ofbeamforming in time [2]. Taking into account the number oftransducer elements and the number of lines in an image, theamount of sampled data that needs to be digitally processed

is enormous, motivating methods to reduce sampling andprocessing rates.

A possible approach to sampling rate reduction is introducedin [3]. Tur et al. consider the ultrasound signal detected byeach receiver within the framework of finite rate of innovation(FRI) [4]. The detected signal is modeled asL replicas of aknown pulse-shape, caused by scattering of the transmittedpulse from various reflectors, located along the transmittedbeam. Such an FRI signal is fully described by2L parameters,corresponding to the replica’s unknown delays and amplitudes.Based on [4], the relationship between the signal’s Fourierseries coefficients and the unknown parameters is formulatedin the form of a spectral analysis problem. The latter may besolved using array processing methods or compressed sensing(CS) techniques, given a subset of at least2L Fourier seriescoefficients [5], [6]. The required Fourier coefficients canbecomputed from appropriate low-rate samples of the signalfollowing ideas of [3], [7]–[10]. Recent work has developedahardware prototype implementing the suggested sub-Nyquistsystem [11].

The above framework allows to sample the detected signalsat a low-rate, assuming sufficiently high SNR. However, thefinal goal in low-rate ultrasound imaging is to recover a two-dimensional image, obtained by integrating the noisy datasampled at multiple transducer elements. In standard imagingthe integration is achieved by the process of beamforming,which is performed digitally and, theoretically, requireshighsampling rates. Hence, in order to benefit from the ratereduction achieved in [3], one needs to be able to incorporatebeamforming into the low-rate sampling process.

A. Related Work: Compressed Beamforming

A solution to low-rate beamforming is proposed in [5],where Wagner et al. introduce the concept of compressedbeamforming. They show that their approach, applied to anarray of transducer elements, allows to reconstruct a two-dimensional ultrasound image depicting macroscopic pertur-bations in the tissue. To develop their method, the authorsfirst prove that the beam obeys an FRI model, implying thatit can be reconstructed from a small subset of its Fouriercoefficients. However, this required subset cannot be obtainedby the schemes proposed in [3] and [10], since the beamdoes not exist in the analog domain. It is constructed digitallyafter sampling the detected signals. This fundamental obstacleis resolved by transforming the beamforming operator intothe compressed domain. Specifically, Wagner et al. show thatthe Fourier coefficients of the beam can be approximated by

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a linear combination of Fourier coefficients of the detectedsignals. The latter are obtained from the low-rate samples ofthe detected signals, using the Xampling method, proposed in[3], [10] and [11].

Another innovation of [5] regards the approach to beamreconstruction from a subset of its frequency samples. Ratherthan use standard spectral analysis techniques, Wagner et al.view the reconstruction as a CS problem. They demonstratethat CS methodology is comparable to spectral analysis meth-ods and even outperforms the latter when the noise is large.Combining compressed beamforming with CS techniques forsignal recovery, they reconstruct two-dimensional ultrasoundimages, comprised of strong reflectors in the tissue. Significantrate reduction is achieved, while assuming that the number ofreplicas in the FRI model of the beam is small. Such an as-sumption is justified by the fact that only strong perturbationsin the tissue are taken into account. Therefore, the proposedframework allows for robust detection of strong reflectors,butis unable to treat speckle, weak scattered echoes originatingfrom microscopic perturbations in the tissue, which are ofsignificant importance in medical imaging.

B. Contributions

In this paper we build on the results in [5] and showthat compressed beamforming can be extended to a muchmore general concept of beamforming in frequency. Thisapproach to beamforming is applicable to any signal, withoutthe need to assume a structured model. When structure exists,beamforming in frequency may be combined with CS to yieldfurther rate reduction.

The core of compressed beamforming is the relationshipbetween the beam and the detected signals in the frequencydomain, while the notion of “compressed” stems from the factthat the Fourier coefficients of the detected signals can beobtained from their low-rate samples. Here we show that thisfrequency domain relationship is general and holds irrespectiveof the FRI model. This leads to an approach of beamformingin frequency which is completely equivalent to beamformingin time. Beamforming in frequency is equivalent to a weightedaveraging of the Fourier coefficients of the detected signals andcan be performed efficiently by exploiting two facts. First,the frequency domain beamforming operator is defined bythe geometry of the transducer array and does not dependon the detected signals. Hence, the required weights can becomputed off-line and used as a look-up-table during theimaging cycle. In addition, we show numerically, that theseweights are characterized by a rapid decay, implying that theFourier coefficients of the beam can be computed using a smallnumber of Fourier coefficients of the detected signals.

Next, we show that beamforming in frequency domainallows to bypass the oversampling dictated by digital imple-mentation of beamforming in time. Since the beam is obtaineddirectly in frequency, we need to compute its Fourier coeffi-cients only within its effective bandwidth. We demonstratethatthis can be achieved using generalized samples of the detectedsignals, obtained at their Nyquist rate. To avoid confusion,by Nyquist rate we mean the signals effective bandpass

bandwidth, which is typically much lower than its highestfrequency since the detected pulse is normally modulated ontoa carrier and only occupies a portion of the entire bandwidth.Using in vivo cardiac data, we illustrate that beamformingin frequency allows to preserve image integrity with4-10 foldreduction in the number of samples used for its reconstruction.

Further reduction in sampling rate is obtained, similarlyto [5], when only a portion of the beam’s bandwidth isused. In this case beamforming in frequency is equivalentto compressed beamforming. Detected signals are sampledat sub-Nyquist rates, leading to up to28 fold reduction insampling rate. Our contribution in this scenario regards thereconstruction method used to recover the beam from itspartial frequency data. To recover the unknown parameters,corresponding to the FRI model of the beam, Wagner et al.assume that the parameter vector is sparse. The parametersare then obtained as a solution of anl0 optimization problem.Sparsity holds when only strong reflectors are taken intoaccount, while the speckle is treated as noise. To capture thespeckle, we assume that the parameter vector is compressibleand recast the recovery as anl1 optimization problem. Weshow that these small changes in the model and the CSreconstruction technique allow to capture and recover thespeckle, leading to significant improvement in image quality.

Finally, we introduce an implementation of beamformingin frequency and sub-Nyquist processing on a stand aloneultrasound machine and show that our proposed processing isfeasible in practice using real hardware. Low-rate processingis performed on the data obtained in real-time by scanninga heart with a64-element probe. Our approach allows forsignificant rate reduction with respect to the lowest processingrates that are achievable today, which can potentially impactsystem size, power consumption and cost.

The rest of the paper is organized as follows: in SectionII, we review beamforming in time and discuss the samplingrates required for its digital implementation. Following thesteps in [5], we describe the principles of frequency domainbeamforming in Section III, and show that it is equivalentto standard time domain processing. In Section IV we showthat beamforming in frequency allows for rate reduction evenwithout exploiting the FRI model and can be performed at theNyquist rate of the signal. CS recovery from partial frequencydata, implying sampling and processing at sub-Nyquist rates, isdiscussed in Section V. Comparison between the performanceof the proposed method with the results obtained in [5]together with an implementation of beamforming in frequencyand sub-Nyquist processing on a stand alone ultrasound ma-chine are presented in Section VI.

II. CONVENTIONAL PROCESSING INULTRASOUND

IMAGING

Most modern imaging systems use multiple transducerelements to transmit and receive acoustic pulses. This allowsto perform beamforming during both transmission and recep-tion. Beamforming is a common signal-processing technique[12] that enables spatial selectivity of signal transmission or

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reception and is applied in various fields, including wirelesscommunication, speech processing, radar and sonar. In ultra-sound imaging beamforming is used for steering the beam ina desired direction and focusing it in the region of interestinorder to detect tissue structures.

During transmission beamforming is achieved by delayingthe transmission time of each transducer element, whichallows to transmit energy along a narrow beam. Beamformingupon reception is much more challenging. Here dynamicallychanging delays are applied on the signals detected at each oneof the transducer elements prior to averaging. Time-varyingdelays allow dynamic shift of the reception beam’s focalpoint, optimizing angular resolution. Averaging of the delayedsignals in turn enhances the SNR of the resulting beamformedsignal, which is used to form a line in an image. From here on,the term beamforming will refer to beamforming on reception,which is the focus of this work.

A. Beamforming in Time

We begin with a detailed description of the beamformingprocess which takes place in a typical B-mode imaging cycle.Our presentation is based mainly on [13] and [5]. We will thenshow, in Section III, how the same process can be performedin frequency, paving the way to substantial rate reduction.

Fig. 1. M receivers aligned along thex axis. An acoustic pulse is transmittedin a directionθ. The echoes scattered from perturbation in the radiated tissueare received by the array elements.

In the transmit path, a pulse is generated and transmittedby the array of transducer elements. The pulse transmitted byeach element is timed and scaled, so that the superposition ofall transmitted pulses creates a directional beam propagatingat a certain angle. By subsequently transmitting at differentangles, a whole sector is radiated. The real time computationalcomplexity in the transmit path is negligible since transmitparameters per angle are calculated off-line and saved in tables.

Consider an array comprised ofM transceiver elementsaligned along thex axis, as illustrated in Fig. 1. The referenceelementm0 is set at the origin and the distance to themthelement is denoted byδm. The image cycle begins att = 0,when the array transmits an energy pulse in the directionθ.The pulse propagates trough the tissue at speedc, and attime t ≥ 0 its coordinates are(x, z) = (ct sin θ, ct cos θ).A potential point reflector located at this position scatters theenergy, such that the echo is detected by all array elements ata time depending on their locations. Denote byϕm(t; θ) the

signal detected by themth element and byτm(t; θ) the timeof detection. It is readily seen that:

τm(t; θ) = t+dm(t; θ)

c, (1)

where dm(t; θ) =

√

(ct cos θ)2 + (δm − ct sin θ)2 is the

distance traveled by the reflection. Beamforming involvesaveraging the signals detected by multiple receivers whilecompensating the differences in detection time. In that wayweobtain a signal containing the intensity of the energy reflectedfrom each point along the central transmission axisθ.

Using (1), the detection time atm0 is τm0(t; θ) = 2t

since δm0= 0. Applying an appropriate delay toϕm(t; θ),

such that the resulting signalϕm(t; θ) satisfiesϕm(2t; θ) =ϕm(τm(t; θ)), we can align the reflection detected by them-th receiver with the one detected atm0. Denotingτm(t; θ) =τm(t/2; θ) and using (1), the following aligned signal isobtained:

ϕm(t; θ) = ϕm(τm(t; θ); θ), (2)

τm(t; θ) =1

2

(

t+√

t2 − 4(δm/c)t sin θ + 4(δm/c)2)

.

The beamformed signal may now be derived by averaging thealigned signals:

Φ(t; θ) =1

M

M∑

m=1

ϕm(t; θ). (3)

Such a beam is optimally focused at each depth and henceexhibits improved angular localization and enhanced SNR.

Although defined over continuous time, ultrasound imagingsystems perform the beamforming process in (2)-(3) in thedigital domain: analog signalsϕm(t; θ) are amplified and sam-pled by an Analog to Digital Converter (ADC), preceded by ananti-aliasing filter. We next discuss sampling and processingrates required to perform (3).

B. Rate Requirements

Digital implementation of beamforming requires samplingthe signals detected at the transducer elements and transmittingthe samples to the processing unit. The Nyquist rate, requiredto avoid aliasing, is insufficient for digital implementation ofbeamforming due to the high delay resolution needed. Indeed,in order to apply the delay defined in (2) digitally, detectedsignals need to be sampled on a sufficiently dense grid.Typically, the sampling interval is on the order of nanoseconds.Therefore, required sampling rates are significantly higher thanthe Nyquist rate of the signal and can be as high as hundredsof MHz [14].

Due to the impracticality of this requirement, ultrasounddata is sampled at lower rates, typically, on the order of tens ofMHz. Fine delay resolution is obtained by subsequent digitalinterpolation. Interpolation beamforming allows to reduce thesampling rate at the cost of additional computational load re-quired to implement the digital interpolation which effectivelyincreases the rate in the digital domain. The processing, ormore precisely, beamforming rate, remains unchanged as it isperformed at the high digital rate.

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Another common way to improve delay accuracy whilereducing both sampling and beamforming rate is phase-rotation-based beamforming (PRBF) [2]. In this approachcoarse delays, defined by the sampling rate, are followed by avernier control, implemented by a digital phase shift, adjustedfor the central frequency. The phase shifter approximationtoa time delay is exact only at the central frequency, leading toloss in array gain and rise in the sidelobe level. The analysisin [2] shows that the degradation of beam quality can beavoided, provided that the sampling rate is4-10 times thetransducer central frequency. This rule of thumb stems fromthe assumption that typically the transducer central frequencyis approximately twice the radio frequency (RF) bandwidth.RF bandwidth is defined as the distance from the central to thehighest frequency and, hence, is half the bandpass bandwidth.This leads to the conclusion that the sampling rate should beabout 4-10 times the bandpass bandwidth, since, accordingto the analysis in [2], loss in array gain and rise in thesidelobe level are dictated by the ratio between the bandwidthof the signal to the sampling rate. In the sequel, following[2], we denote the rate required to avoid artifacts in digitalimplementation of beamforming, as the beamforming ratefs.

As imaging systems evolve, the amount of elements partic-ipating in the imaging cycle continues to grow significantly.Consequently large amounts of data need to be transmittedfrom the system front-end and digitally processed in real time.Increasing transmission and processing pose an engineeringchallenge on digital signal processing (DSP) hardware andmotivate reducing the amounts of data as close as possible tothe system front-end.

To conclude this section we evaluate the sampling rates andthe number of samples needed to be taken at each transducerelement according to each one of the methods, describedabove. Our evaluation is based on the imaging setup typicallyused in cardiac imaging. We assume a breadboard ultrasonicscanner of 64 acquisition channels. The radiated depthr = 16cm and speed of soundc = 1540 m/sec yield a signal ofduration T = 2r/c ≃ 210 µsec. The acquired signal ischaracterized by a narrow bandpass bandwidth of2 MHz,centered at the carrier frequencyf0 ≈ 3.4 MHz. In orderto perform plain delay-and-sum beamforming with5 nsecdelay resolution, detected signals should be sampled at therateof 200 MHz. Implementation of interpolation beamforming,used in many imaging systems, allows to reduce the samplingrate to 50 MHz, while the required beamforming rate isobtained through interpolation in the digital domain. Hence,each channel yields42000 real valued samples, participatingin beamforming. Rates required by PRBF in this setup, varyfrom 8 to 20 MHz, leading to1680-4200 real valued samples,obtained at each transducer element.

Evidently, processing in the time domain imposes high sam-pling rate and considerable burden on the beamforming block.We next show that the number of samples can be reducedsignificantly by exploiting ideas of sub-Nyquist sampling,beamforming in frequency and CS-based signal reconstruction.

III. B EAMFORMING IN FREQUENCY

We now show that beamforming can be performed equiva-lently in the frequency domain, paving the way to substantialreduction in the number of samples needed to obtain thesame image quality. We extend the notion of compressedbeamforming, introduced in [5], to beamforming in frequencyand show that a linear combination of the discrete Fouriertransform (DFT) coefficients of the individual signals, sampledat the beamforming ratefs, yields the DFT coefficients of thebeamformed signal, sampled at the same rate. This relationshipis true irrespective of the signal structure.

A. Implementation and Properties

We follow the steps in [5] and start from the computation ofthe Fourier series coefficients of the beamΦ(t; θ). As shownin [5], the support of the beamΦ(t; θ) is limited to [0, TB(θ)),whereTB(θ) < T andT is defined by the transmitted pulsepenetration depth. The value ofTB(θ) is given by [5]

TB(θ) = min1≤m≤M

τ−1m (T ; θ), (4)

where τm(t; θ) is defined in (2). Denote the Fourier seriescoefficients ofΦ(t; θ) with respect to the interval[0, T ) by

csk =1

T

∫ T

0

I[0,TB(θ))(t)Φ(t; θ)e−i 2π

Tktdt, (5)

whereI[a,b) is the indicator function equal to1 whena ≤ t < band0 otherwise. Plugging (3) into (5), and after some algebraicmanipulation, it is shown in [5] that

csk =1

M

M∑

m=1

csk,m, (6)

wherecsk,m are defined as follows:

csk,m =1

T

∫ T

0

gk,m(t; θ)ϕm(t; θ)dt, (7)

with

gk,m(t; θ) =qk,m(t; θ)e−i 2π

Tkt,

qk,m(t; θ) =I[|γm|,τm(T ;θ))(t)

(

1 +γ2m cos θ2

(t− γm sin θ)2

)

× (8)

exp

{

i2π

Tkγm − t sin θ

t− γm sin θγm

}

,

andγm = δm/c.The next step is to replaceϕm(t) by its Fourier series

coefficients. Denoting thenth Fourier coefficient byϕsm[n]

and using (8) we can rewrite (7) as

csk,m =∑

n

ϕsm[k − n]Qk,m;θ[n], (9)

whereQk,m;θ[n] are the Fourier coefficients of the distortionfunctionqk,m(t; θ) with respect to[0, T ). According to Propo-sition 1 in [5], csk,m can be approximated sufficiently wellwhen we replace the infinite summation in (9) by a finite sum:

csk,m ≃∑

n∈ν(k)

ϕsm[k − n]Qk,m;θ[n]. (10)

5

The setν(k) depends on the decay properties of{Qk,m;θ[n]}.We now take a closer look at the properties of the Fourier

coefficients ofqk,m(t; θ), defined in (8). Numerical studiesshow that most of the energy of the set{Qk,m;θ[n]} isconcentrated around the direct current (DC) component. Thisbehavior is typical to any choice ofk, m or θ. An examplefor k = 100, m = 14 andθ = 0.421 [rad] is shown in Fig. 2.This allows us to rewrite (10) as

−500 0 500−100

−80

−60

−40

−20

Fourier Series Coefficient Index, n

Mag

nitu

de [d

B]

Fig. 2. Fourier coefficients{Qk,m;θ[n]} of qk,m(t; θ) are characterizedby a rapid decay, where most of the energy is concentrated around the DCcomponent. Herek = 100, m = 14 andθ = 0.421 [rad].

csk,m ≃

N2∑

n=−N1

ϕsm[k − n]Qk,m;θ[n]. (11)

The choice ofN1 andN2 controls the approximation quality.Numerical studies show that20 most significant elements of{Qk,m;θ[n]} contain, on average, more than95% of the entireenergy irrespective of the choice ofk, m or θ. Beamformingin frequency therefore is performed using20 most significantelements in{Qk,m;θ[n]} throughout our work.

Denote byβ, |β| = B, the set of Fourier coefficients of thedetected signal that correspond to its bandwidth, namely, thevalues ofk for whichϕs

m[k] is nonzero (or larger than a thresh-old). Note that (11) implies, that the bandwidth of the beam,βBF , will contain at most(B +N1 +N2) nonzero frequencycomponents. To compute the elements inβBF all we need isthe setβ for each one of the detected signals. In a typicalimaging setupB is of order of hundreds of coefficients, whileN1 andN2, defined by the decaying properties of{Qk,m;θ[n]},are no larger than10. This implies thatB ≫ Ni, i ∈ {1, 2},so B +N1 +N2 ≈ B. Hence, the bandwidth of the beam isapproximately equal to the bandwidth of the detected signals.In addition, it follows from (11) that in order to calculate anarbitrary subsetµ ⊂ βBF of sizeM of Fourier coefficients ofthe beam, we need to know at most(M +N1 +N2) Fouriercoefficients of each one of the detected signalsϕm(t). Theseproperties of frequency domain beamforming will be used inorder to reduce sampling rates.

Equations (6) and (11) provide a relationship between theFourier series coefficients of the beam and the individualsignals. We next derive a corresponding relationship betweenthe DFT coefficients of the above signals, sampled at thebeamforming ratefs. Denote byN = ⌊T · fs⌋ the resultingnumber of samples. Sincefs is higher than the Nyquist rate of

the detected signals, the relation between the DFT of lengthNand the Fourier series coefficients ofϕm(t) is given by [15]:

ϕsm[n] =

1

N

ϕm[n], 0 ≤ n ≤ Pϕm[N + n], −P ≤ n < 00, otherwise,

(12)

whereϕm[n] denote the DFT coefficients andP denotes theindex of the Fourier transform coefficient, corresponding tothe highest frequency component.

We can use (12) to substitute Fourier series coefficientsϕsm[n] of ϕm(t) in (11) by DFT coefficientsϕm[n] of its

sampled version. Plugging the result into (6), we obtain arelationship between Fourier series coefficients of the beamand DFT coefficients of the sampled detected signals:

csk ≃1

MN

M∑

m=1

k−n∑

n=−N1

ϕm[k − n]Qk,m;θ[n] (13)

+

N2∑

n=k−n+1

ϕm[k − n+N ]Qk,m;θ[n]

for an appropriate choice ofn. Sincefs is higher than theNyquist rate of the beam as well, the DFT coefficientsck ofits sampled version are given by an equation similar to (12):

ck = N

csk, 0 ≤ k ≤ Pcsk−N , N − P ≤ k < N0, otherwise.

(14)

Equations (13) and (14) provide the desired relationshipbetween the DFT coefficients of the beam and the DFTcoefficients of the detected signals. Note that this relationship,obtained by a periodic shift and scaling of (11), retains theimportant properties of the latter.

Applying an IDFT on{ck}N−1k=0 results in the beamformed

signal in time. We can now proceed to standard image gener-ation steps which include log-compression and interpolation.

B. Simulations and Validation

To demonstrate the equivalence of beamforming in time andfrequency, we applied both methods on in vivo cardiac datayielding the images shown in Fig. 3. The imaging setup isthat described in Section II-B withfs = 16 MHz. As can bereadily seen, the images look identical.

Quantitative validation of the proposed method was per-formed with respect to both one-dimensional beamformed sig-nals and the resulting two-dimensional image. To compare theone-dimensional signals, we calculated the normalized root-mean-square error (NRMSE) between the signals obtained bybeamforming in frequency and those obtained by standardbeamforming in time. Both class of signals were comparedafter envelope detection, performed by a Hilbert transformin order to remove the carrier. Denote byΦ[n; θj ] the signalobtained by standard beamforming in directionθj , j = 1, ..., J ,and letΦ[n; θj] denote the signal obtained by beamforming infrequency. The Hilbert transform is denoted byH(·). For the

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set ofJ = 120 image lines, we define NRMSE as:

NRMSE =1

J

√

1N

∑Nn=1

(

H(Φ[n; θj ])−H(Φ[n; θj]))2

H (Φ[n; θj ])max−H (Φ[n; θj ])min

,

(15)whereH (Φ[n; θj ])max

andH (Φ[n; θj ])mindenote the max-

imal and minimal values of the envelope of the beamformedsignal in time.

Comparison of the resulting images was performed bycalculating the structural similarity (SSIM) index [16], com-monly used for measuring similarity between two images. Thefirst line of Table I summarizes the resulting values. Thesevalues verify that both 1D signals and the resulting image areextremely similar.

(a) (b)

Fig. 3. Cardiac images constructed with different beamforming techniques.(a) Time domain beamforming. (b) Frequency domain beamforming.

IV. RATE REDUCTION BY BEAMFORMING IN FREQUENCY

In the previous section we showed the equivalence ofbeamforming in time and frequency. We next demonstratethat beamforming in frequency allows to reduce the requirednumber of samples of the individual signals. Reduction canbe achieved in two different ways. First, we exploit the loweffective bandwidth of ultrasound signals and bypass over-sampling, dictated by digital implementation of beamformingin time. This allows to perform processing at the Nyquistrate, defined with respect to the effective bandwidth of thesignal, which is impossible when beamforming is performedin time. As a second step, we show that further rate reductionis possible, when we take into account the FRI structure ofthe beamformed signal and use CS techniques for recovery.

In this section we address rate reduction, obtained bytranslation of the beamforming operator into the frequencydomain. At this stage the structure of the beamformed signalis not taken into account.

A. Exploiting Frequency Domain Relationship

To reduce the rate, we exploit the relationship between thebeam and the detected signals in the frequency domain givenby (11). In Section III-A, we showed that the bandwidth ofthe beam,βBM , contains approximatelyB nonzero frequencycomponents, whereB is the effective bandwidth of the de-tected signals. In order to computeβBM we need a setβ ofnonzero frequency components of each one of the detected

signals. This allows to exploit the low effective bandwidthof the detected signals and calculate only their nonzero DFTcoefficients. The ratio between the cardinality of the setβand the overall number of samplesN , required by standardbeamforming ratefs, is dictated by the oversampling factor.As mentioned in Section II-B, we definefs as 4-10 timesthe bandpass bandwidth of the detected signal, leading toB/N = 1/4-1/10. Assume that it is possible to obtain therequired setβ usingB low-rate samples of the detected signal.In this case the ratio betweenN andB implies potential4-10fold reduction in the required sampling rate.

B. Reduced Rate Sampling

We now address the following question: how do we obtainthe required setβ, corresponding to the effective bandpassbandwidth, usingB low-rate samples of each one of thedetected signals?

Note that sampling is performed in time, while our goal isto extractB DFT coefficients. To this end, similarly to [5], wecan use the Xampling mechanism proposed in [3]. A hardwareXampling prototype implemented by Baransky et al. in [11]is seen in Fig. 4.

Fig. 4. A Xampling-based hardware prototype for sub-Nyquist sampling.The prototype computes low-rate samples of the input from which the setβof DFT coefficients can be computed on the outputs.

The Xampling scheme allows to obtainB coefficients fromB point-wise samples of the detected signal filtered with anappropriate kernels∗(−t), designed according to the transmit-ted pulse-shape and the setβ. The required DFT coefficientsare equal to the DFT of the outputs, therefore, the numberof samples taken at each individual element is equal to thenumber of DFT coefficients that we want to compute. Hence,when we compute all nonzero DFT coefficients of the detectedsignal,4-10 fold reduction in sampling rate is achieved withoutcompromising image quality.

Having obtained the setβ of each one of the detected sig-nals, we calculate the elements ofβBF by low-rate frequencydomain beamforming. Finally, we reconstruct the beamformedsignal in time by performing an IDFT. Note that it is possibleto pad the elements ofβBF with an appropriate number ofzeros to improve time resolution. In our experiments, in orderto compare the proposed method with standard processing, wepaddedβBF with N −B zeros, leading to the same samplinggrid, used for high-rate beamforming in time.

Images obtained by the proposed method, using416 real-valued samples per image line to perform beamforming in fre-quency, and by standard beamforming, using3360 real-valuedsamples to perform beamforming in time, are shown in Fig. 5.Corresponding values of NRMSE and SSIM are reported in the

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second line of Table I. These values validate close similaritybetween the two methods. However, in this case NRMSEis slightly higher, while SSIM is lower, compared to thevalues obtained in Section III-B. Note that these values depictsimilarity between the signals. Differences can thereforebeexplained by the following practical aspect. When we obtainthe set of all nonzero DFT coefficients of the beamformedsignal,βBF , all the signal energy is captured in the frequencydomain. However, the signal obtained by beamforming in time,contains noise, which occupies the entire spectrum. Whenonly the DFT coefficients within the bandwidth are computedin the frequency domain, the noise outside the bandwidth iseffectively filtered out. In the signal obtained by standardbeamforming, the noise is retained, reducing the similaritybetween the two signals.

(a) (b)

Fig. 5. Cardiac images constructed with different beamforming techniques.(a) Time domain beamforming. (b) Frequency domain beamforming, obtainedwith 8 fold reduction in sampling rate.

The entire scheme, performing low-rate sampling and fre-quency domain beamforming, is depicted in Fig. 6. Signals{ϕm(t)}

Mm=1, detected at each transducer element, are filtered

with an appropriate analog kernels∗(−t) and sampled at alow-rate, defined by the effective bandwidth of the transmittedpulse. Such a rate corresponds to the Nyquist rate of thebaseband transmitted pulse. DFT coefficients of the detectedsignals are computed and beamforming is performed directlyin frequency at a low-rate. This framework allows to bypassoversampling dictated by digital implementation of beamform-ing in time and to significantly reduce (up to10-fold) theresulting sampling rate.

ϕ1(t)

ϕM (t)ϕM [n]

ϕ1[n]

s∗(−t)

s∗(−t)

FFT

FFT

Q1

QM

∑1

Mck

Fig. 6. Fourier domain beamforming scheme. The blockQi represents aver-aging the DFT coefficients of the detected signals with weights {Qk,i;θ[n]}according to (13) and (14).

V. FURTHER REDUCTION VIA COMPRESSEDSENSING

We have shown that it is possible to reconstruct a beam-formed signal perfectly from a setβBF of its nonzero DFT

TABLE IQUANTITATIVE VALIDATION OF BEAMFORMING IN FREQUENCY WITH

RESPECT TO BEAMFORMING IN TIME

Method NRMSE SSIMBeamforming in frequency 0.0349 0.9684Beamforming in frequency,reduced rate sampling 0.0368 0.9603

coefficients. In this section we consider further reductioninsampling rate by taking only a subsetµ ⊂ βBF , |µ| = M <BBF = |βBF |, of nonzero DFT coefficients. In this case, asshown in Section III-A,(M+N1+N2) frequency componentsof each one of the detected signals are required, leading toonly (M +N1 +N2) samples in each channel. The challengenow is to reconstruct a beamformed signal from such partialfrequency data.

To this end we aim to use CS techniques, while exploitingthe FRI structure of the beamformed signal. To formulatethe recovery as a CS problem, we begin with a parametricrepresentation of the beam.

A. Parametric representation

According to [5], a beamformed signal obeys an FRI model,namely, it can be modeled as a sum of replicas of the knowntransmitted pulse,h(t), with unknown amplitudes and delays:

Φ(t; θ) ≃

L∑

l=1

blh(t− tl), (16)

whereL is the number of scattering elements in directionθ,{bl}

Ll=1 are the unknown amplitudes of the reflections and

{tl}Ll=1 denote the times at which the reflection from the

lth element arrived at the reference receiverm0. Since thetransmitted pulse is known, such a signal is completely definedby 2L unknown parameters, the amplitudes and the delays.

We can rewrite this model accordingly by sampling bothsides of (16) at the beamforming ratefs and quantizing theunknown delays{tl}Ll=1 with quantization step1/fs, such thattl = ql/fs, ql ∈ Z andN = ⌊T · fs⌋:

Φ[n; θ] ≃

L∑

l=1

blh[n− ql] =

N−1∑

l=0

blh[n− l], (17)

where

bl =

{

bl, if l = ql0, otherwise.

(18)

Calculating the DFT of both sides of (17) leads to thefollowing expression for the DFT coefficientsck:

ck =

N−1∑

n=0

Φ[n; θ]e−i 2π

Nkn = hk

N−1∑

l=0

ble−i 2π

Nkl, (19)

wherehk is the DFT coefficient ofh[n], the transmitted pulsesampled at ratefs. We conclude that recoveringΦ[n; θ] isequivalent to determiningbl, 0 ≤ l ≤ N − 1 in (19).

8

We now recast the problem in vector-matrix notation.Defining anM -length measurement vectorc with kth entryck, k ∈ µ, we can rewrite (19) as follows:

c = HDb = Ab, (20)

whereH is an M × M diagonal matrix withhk as itskthentry,D is anM ×N matrix formed by taking the setµ ofrows from anN ×N DFT matrix, andb is a length-N vectorwith lth entrybl.

Our goal is to determineb from c. We next discuss andcompare possible recovery approaches.

B. Prior Work

As mentioned in Section V-A, the signal of interest iscompletely defined byL the unknown delays and amplitudes.Hence, a possible approach is to extract those values fromthe available setµ of DFT coefficients. To this end, we canview (20) as a complex sinusoid problem. ForM ≥ 2L it canbe solved using standard spectral analysis methods such asmatrix pencil [17] or annihilating filter [18]. Rate reduction isachieved when2L << N , whereN is the number of samplesdictated by the standard beamforming rate.

In the presence of moderate to high noise levels, theunknown parameters can be extracted more efficiently usinga CS approach, as was shown in [5]. Note that (20) isan underdetermined system of linear equations which hasinfinitely many solutions, sinceA is anM ×N matrix withM ≪ N . The solution set can be narrowed down to a singlevalue by exploiting the structure of the unknown vectorb. Inthe CS framework it is assumed that the vector of interest isreasonably sparse, whether in the standard coordinate basis orwith respect to some other basis.

The regularization introduced in [5], relies on the assump-tion that the coefficient vectorb is L-sparse. The formulationin (20) then has a form of a classic CS problem, wherethe goal is to reconstruct anN -dimensionalL-sparse vectorb from its projection ontoK orthogonal rows captured bythe measurement matrixA. This problem can be solvedusing numerous CS techniques, whenA satisfies well-knownproperties such as restricted isometry (RIP) or coherence [6].

In our case,A, defined in (20), is formed by takingKscaled rows from anN × N DFT matrix. It can be shownthat by choosingK ≥ CL(logN)4 rows uniformly at randomfor some positive constantC, the measurement matrixAobeys the RIP with high probability [19]. In order for thisapproach to be beneficial it is important to assume thatL << N . Since random frequency sampling is not practicalfrom a hardware prospective, it is possible instead to sample anumber of frequency bands, distributed randomly throughoutthe spectrum [11]. This approach is implemented in the boardof Fig. 4.

A typical beamformed ultrasound signal is comprised of arelatively small number of strong reflections, correspondingto strong perturbations in the tissue, and many weaker scat-tered echoes, originated from microscopic changes in acousticimpedance of the tissue. The framework proposed in [5] aimsto recover only strong reflectors in the tissue and treat weak

echoes as noise. Hence, the vector of interestb is indeedL-sparse withL << N . To recoverb, Wagner et al. considerthe following optimization problem:

minb

‖b‖0 subject to ‖Ab− c‖2 ≤ ε, (21)

where ε is an appropriate noise level, and approximate itssolution using orthogonal matching pursuit (OMP) [20].

A significant drawback of this method is its inability torestore weak reflectors. In the context of this approach theyare treated as noise and are disregarded by the signal model.As a result, the speckle - granular pattern that can be seenin Fig. 3 - is lost. This severely degrades the value of theresulting images since information carried by speckle is ofmajor importance in many medical imaging modalities. Forexample, in cardiac imaging, speckle tracking tools allow toanalyze the motion of heart tissues and to track effectivelymyocardial deformations [21], [22].

C. Alternative Approach

Fortunately, with a small conceptual change of model, wecan restore the entire signal, namely, recover both strongreflectors and weak scattered echoes.

As mentioned above, a beamformed ultrasound signal iscomprised of a relatively small number of strong reflectionsand many scattered echoes, that are on average two ordersof magnitude weaker. It is, therefore, natural to assume thatthe coefficient vectorb, defined in (20), is compressible orapproximately sparse, but not exactly sparse. This propertyof b can be captured by using thel1 norm, leading to theoptimization problem:

minb

‖b‖1 subject to ‖Ab− c‖2 ≤ ε. (22)

Problem (22) can be solved using second-order methods suchas interior point methods [23], [24] or first-order methods,based on iterative shrinkage ideas [25], [26].

We emphasize that although it is common to view (22) asa convex relaxation of (21), in our case such a substitution iscrucial. It allows to capture the structure of the signal andtoboost the performance of sub-Nyquist processing, as will beshown next, through several examples.

VI. SIMULATIONS AND RESULTS

In this section we examine the performance of low-ratefrequency-domain beamforming usingl1 optimization andcompare it to the previously proposedl0 optimization basedmethod. This is done by applying both methods to stored RFdata, acquired from a healthy volunteer. We then integrate ourmethod into a stand alone ultrasound machine and show thatsuch processing is feasible in practice using real hardware.

A. Simulations on In Vivo Cardiac Data

To demonstrate low-rate beamforming in frequency andevaluate the impact of rate reduction on image quality, we ap-plied our method on in vivo cardiac data. The data acquisitionsetup is described in Section II-B withfs = 16 MHz, leadingto 3360 real valued samples. To perform beamforming in

9

(a) (b) (c)

(d) (e) (f)

Fig. 7. Simulation results. The first row, (a)-(c), corresponds to frame 1, the second row, (d)-(f), corresponds to frame2. (a), (d) Time domain beamforming.(b), (e) Frequency domain beamforming,l1 optimization solution. (c),(f) Frequency domain beamforming, l0 optimization solution..

frequency we used a subsetµ of 100 DFT coefficients, whichcan be obtained from120 real-valued samples by the proposedXampling scheme. This implies28 fold reduction in samplingand14 fold reduction in processing rate compared to standardbeamforming, which requires3360 real-valued samples forthis particular imaging setup. The difference between thesampling and processing rates stems from the complex natureof DFT coefficients. Having computed the DFT coefficients ofthe beamformed signal, we obtain its parametric representationby solving (22). To this end we used the NESTA algorithm[27]. This fast and accurate first-order method, based on thework of Nesterov [28], is shown to be highly suitable forsolving (22), when the signal of interest is compressible withhigh dynamic range, which is particulary true for ultrasoundimaging. An additional advantage of NESTA is that it doesnot depend on fine tuning of numerous controlling parameters.A single smoothing parameter,µ, needs to be selected basedon a trade-off between the accuracy of the algorithm and itsspeed of convergence. This parameter was chosen empiricallyto yield optimal performance with respect to image quality.

The resulting images, corresponding to two different frames,are shown in Figs. 7 (b) and (e). Although the images are notidentical to those obtained by standard beamforming (Figs.7(a) and (d)), it can be easily seen thatl1 optimization, based onthe assumption that the signal of interest is compressible,al-lows to reconstruct both strong reflectors and speckle. Table IIreports corresponding values of NRMSE and SSIM. Althoughthe quantitative values are reduced compared to those obtainedin Sec. IV-B, important information, e.g. the thickness of the

heart wall and the valves, as well as the speckle pattern,essential for tracking tools, are preserved.

We would like to emphasize, that the values of NRMSE andSSIM are provided in order to give a sense of performanceof the proposed method. In practice, unfortunately, thereare no established quantitative measures for the quality ofultrasound images. Validation is typically performed visuallyby sonographers, radiologists and physicians. Furthermore, ourapproach inherently reduces noise so that high similarity withbeamforming in time may not necessarily be advantageous.

TABLE IIQUANTITATIVE VALIDATION OF BEAMFORMING IN FREQUENCY AT

SUB-NYQUIST RATE

Method NRMSE SSIMFrame 1 0.0682 0.7017Frame 2 0.0587 0.6812

To compare the proposed method with the previously de-velopedl0 optimization based approach, we solved (21) withOMP, while assumingL = 25 strong reflectors in eachdirection θ. Resulting images, shown in Figs. 7(c) and (f),depict the strong reflectors, observed in Fig. 7(a) and (b), whilethe speckle is completely lost, degrading the overall image.

B. Implementation on Stand Alone Imaging System

As a next step we implemented low-rate frequency domainbeamforming on an ultrasound imaging system [29]. The lab

10

setup used for implementation and testing is shown in Fig.8 and includes a state of the art GE ultrasound machine,a phantom and an ultrasound scanning probe. In our studywe used a breadboard ultrasonic scanner with 64 acquisitionchannels. The radiated depthr = 15.7 cm and speed of soundc = 1540 m/sec yield a signal of durationT = 2r/c ≃ 204µsec. The acquired signal is characterized by a narrow band-pass bandwidth of1.77 MHz, centered at a carrier frequencyf0 ≈ 3.4 MHz. The signals are sampled at the rate of50MHz and then are digitally demodulated and down-sampled tothe demodulated processing rate offp ≈ 2.94 MHz, resultingin 1224 real-valued samples per transducer element. Linearinterpolation is then applied in order to improve beamformingresolution, leading to2448 real valued samples. Fig. 9 presentsa schematic block diagram of the transmit and receive front-end of the medical ultrasound system being used.

Fig. 8. Lab setup: Ultrasound system, probe and cardiac phantom.

Fig. 9. Transmit and receive front-end of a medical ultrasound system.

At this point of our work, as illustrated in Fig. 10, in-phaseand quadrature components of the detected signals were usedto obtain the desired set of their DFT coefficients. Using thisset, beamforming in frequency was performed according to(13) and (14), yielding the DFT coefficients of the beamformedsignal. In this setup the sampling rate remained unchanged,butfrequency domain beamforming was performed at a low rate.In our experiments we computed100 DFT coefficients of thebeamformed signal, using120 DFT coefficients of each oneof the detected signals. This corresponds to240 real-valuedsamples used for beamforming in frequency. The number ofsamples required by demodulated processing rate is2448.Hence, beamforming in frequency is performed at a rate corre-sponding to240/2448 ≈ 1/10 of the demodulated processing

Fig. 10. Transmit and receive paths of a medical ultrasound system withbeamforming in the frequency domain.

rate. Images obtained by low-rate beamforming in frequencyand standard time-domain beamforming are presented in Fig.11. As can be readily seen, we are able to retain sufficientimage quality despite the significant reduction in processingrate.

(a) (b)

Fig. 11. Cardiac images obtained by demo system. (a) Time domainbeamforming. (b) Frequency domain beamforming, obtained with 10 foldreduction in processing rate.

Our implementation was done on a state-of-the-art system,sampling each channel at a high rate. Data and processing ratereduction took place following DFT, in the frequency domain.However, by implementing the Xampling scheme described inSection IV-B, the set of120 DFT coefficients of the detectedsignals, required for frequency domain beamforming, can beobtained directly from only120 real-valued low rate samples.

VII. C ONCLUSION

In this work we extended the compressed beamformingframework, proposed in [5], to a general concept of beamform-ing in frequency, dual to standard time domain beamforming.We have shown that when performed directly in frequency,beamforming does not require oversampling, essential for itsdigital implementation in time. Hence,4-10 fold reduction insampling rate is achieved by the translation of beamforminginto the frequency domain, without compromising image qual-ity and without involving any additional assumptions on thesignal.

Further reduction in sampling rate is obtained, when onlya portion of the beam’s bandwidth is used. In this case thedetected signals are sampled at a sub-Nyquist rate, leadingto up to 28 fold reduction in sampling rate. In order to

11

reconstruct the beamformed signal from such partial fre-quency data, we rely on the fact that the beamformed signalobeys an FRI model and use CS techniques. To improvethe performance of sub-Nyquist processing and avoid theloss of speckle information, we assumed that the coefficientvector is compressible. This assumption allows to capture bothstrong reflections, corresponding to large perturbations in thetissue, and much weaker scattered echoes, originating frommicroscopic changes in acoustic impedance of the tissue.

Finally, we implemented our frequency domain beamform-ing on a stand alone ultrasound machine. Low-rate processingis performed on the data obtained in real-time by scanning aheart with a64 element probe. The proposed approach allowsfor 10 fold rate reduction with respect to the lowest processingrates that are achievable today.

Our results prove that the concept of sub-Nyquist processingis feasible for medical ultrasound, leading to the potentialof considerable reduction in future ultrasound machines size,power consumption and cost.

ACKNOWLEDGMENT

The authors would like to thank GE Healthcare Haifa andin particular Dr. Arcady Kempinski for providing the imagingsystem and for many helpful discussions. They are grateful toAlon Eilam for his assistance with the implementation of theproposed method on an ultrasound machine.

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