A Low-Rate Parallel Fourier Domain Beamforming Method forUltrafast Pulse-Echo Imaging

Martin F. Schiffner and Georg Schmitz

Chair of Medical Engineering, Ruhr-Universitat Bochum, D-44801 Bochum, Germany

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To be published in the proceedings of the IEEE International Ultrasonics Symposium (IUS), Tours, France, 2016.

A Low-Rate Parallel Fourier Domain Beamforming

Method for Ultrafast Pulse-Echo Imaging

Martin F. Schiffner and Georg Schmitz

Medical Engineering, Ruhr-Universitat Bochum, D-44801 Bochum, Germany, Email: martin.schiffner@rub.de

Abstract—We present a Fourier domain beamforming methodfor ultrafast pulse-echo imaging that significantly reduces thenumber of acquired samples. In its basic form, our proposedmethod combines the low-rate in-phase and quadrature samplingof the received analog radio frequency (RF) signals and theelectronic receive focusing in the temporal Fourier domain.Using both synthetic and in vivo measurement data, the basicform reduced the numbers of acquired samples by 69.9%

and 74.97%, respectively, compared to the prevalent fusionof the conventional high-rate sampling of the received analogRF signals and the electronic receive focusing in the timedomain. Moreover, the basic form reduced imaging artifacts. Inits advanced form, our proposed method additionally imposesa finite rate of innovation model on the analytical focused RFsignal. The sparse recovery of the associated model parameters byℓq-minimization, q ∈ [0; 1], enabled reductions in the numbers ofacquired samples by 74.39% and 78.79%, respectively, whilethe main image features were preserved.

I. INTRODUCTION

In pulse-echo ultrasound imaging (UI), the attribute ultrafast

refers to image acquisition over a large field of view (FOV) at

rates in the kilohertz range [1]. Ultrafast UI is enabled by the

combination of incident waves that transfer acoustic energy

into relatively large regions of the FOV, e.g. diverging waves

or steered plane waves (PWs), and the simultaneous analog-to-

digital conversion of the radio frequency (RF) signals received

by hundreds of transducer elements. Conventional beam-

forming methods for ultrafast UI typically follow Shannon’s

sampling theorem and implement the delay-and-sum protocol.

The precise realization of the calculated time delays requires

interpolation and high temporal sampling rates exceeding twice

the upper frequency bound. Hence, large numbers of samples

have to be acquired, transferred, and processed per unit time.

The received RF signals, however, exhibit a bandpass

character [2, p. 44] and may be adequately represented by

only a few coefficients in the temporal Fourier domain. This

compressed representation has been exploited by advanced

methods for image recovery in ultrafast UI, e.g. the fil-

tered backpropagation [3] or inverse scattering [4], [5]. The

Fourier coefficients can be efficiently acquired using two

approaches. First, the Fourier coefficients can be calculated

from the complex-valued envelopes provided by the in-phase

and quadrature (IQ) sampling of the analog RF signals at a low

rate corresponding to the signals’ effective bandwidth. Second,

the Fourier coefficients can be measured from the analog RF

The first author gratefully acknowledges the partial financial support of thiscontribution by the Chair of Medical Engineering, Ruhr-Universitat Bochum,D-44801 Bochum, Germany.

signals using a combination of the multichannel sampling

scheme [6] and special hardware [7]. Both approaches sig-

nificantly reduce the numbers of samples to be acquired and

transferred per unit time.

Recently, the concept of compressed beamforming [8] was

incorporated into conventional UI based on the emission of

focused sound beams. While maintaining adequate image qual-

ity, this concept enables an additional reduction in the numbers

of acquired samples by imposing a finite rate of innovation

(FRI) model [9] on the focused signal to be recovered. This

model exhibits only a few parameters per unit time that can

be estimated from only a few Fourier coefficients. However,

compressed beamforming has not been applied to ultrafast UI.

We present a low-rate parallel Fourier domain beamforming

method for ultrafast pulse-echo UI that significantly reduces

the number of acquired samples, while the image quality is

maintained or even improved. The proposed method can be

applied to both two- and three-dimensional UI and can be

adapted to various types of incident waves. In this contribution,

we exclusively consider steered PWs. In Section II, we present

the theoretical derivations underlying the proposed method.

In Section III, we briefly describe the implementation of the

proposed method on a graphics processing unit (GPU). The

method is validated experimentally in Section IV and the

results are summarized in Section V.

II. PROPOSED BEAMFORMING METHOD

A. Scan Configuration and Data Acquisition Protocol

We consider a pulse-echo scan configuration including a

linear transducer array consisting of Nel ∈ N elements with

the center coordinates

M ={

rc,l ∈ R2 :rc,l =

(

l − 2−1(Nel − 1))

δs,xex,

l ∈ [Nel − 1]0}

,(1)

where δs,x ∈ R+ denotes the element pitch and [Nel − 1]0 ={0, 1, . . . , Nel − 1}.

We sequentially emit Nin ∈ N ultrasonic waves. For each

sequential wave emission n ∈ [Nin − 1]0, the RF signals

received by the individual transducer elements are denoted

by x(n)l : T

(n)obs ⊂ R

+0 7→ R for all l ∈ [Nel − 1]0. The

common observation time intervals are given by the bounded

sets T(n)obs = [t

(n)lb ; t

(n)ub ), where t

(n)lb , t

(n)ub ∈ R

+0 are the

lower and upper bounds, respectively, and each wave emission

begins at t = 0. The conventional sampling at the rate

fs ∈ R+ yields the number of real-valued temporal samples

N(n)t = ⌊t

(n)ub fs⌋ − ⌊t

(n)lb fs⌋+ 1 for each RF signal x

(n)l .

B. Model for the Observed Radio Frequency Signals

The observed RF signals x(n)l exhibit a bandpass character

and may be written as [2, pp. 44,45]

x(n)l (t) = A

(n)l (t) cos

[

ωct+ ϕ(n)l (t)

]

= Re{

x(n)l (t)ejωct

}

= Re{

x(n)l (t)

}

,(2)

where A(n)l : T

(n)obs 7→ R are the amplitudes, ωc = 2πfc ∈ R

+

is the angular center frequency, ϕ(n)l : T

(n)obs 7→ R are the

phases, x(n)l (t) = A

(n)l (t)ejϕ

(n)l

(t) denote the complex-valued

envelopes, and x(n)l are the associated analytical RF signals.

The IQ sampling of the complex-valued envelopes x(n)l at the

rate 0 < fs,IQ ≪ fs yields the numbers of complex-valued

temporal samples N(n)t,IQ = ⌊t

(n)ub fs,IQ⌋ − ⌊t

(n)lb fs,IQ⌋+ 1.

Owing to the bounded observation time intervals T(n)obs of

the RF signals (2), their complex-valued envelopes x(n)l may

be represented by the Fourier series with respect to the time

interval Tx ⊇ T(n)obs as

x(n)l (t) =

∞∑

ν=−∞

c(n)l [ν]ejωνt (3a)

with the angular temporal frequencies ων = |Tx|−12πν and

the coefficients

c(n)l [ν] =

1

|Tx|

∫

T(n)obs

x(n)l (t)e−jων tdt. (3b)

The exact analysis interval Tx will be specified in Subsection

II-C2. Since the complex-valued envelopes x(n)l are effectively

bandlimited in the interval of temporal frequencies Bx =[−2−1B; 2−1B] ⊂ R, B ∈ R+, the Fourier series (3) may be

truncated using the set of indices associated with the relevant

angular temporal frequencies

VLP ={

ν ∈ Z : |ν| ≤⌊

2−1B |Tx|⌋

}

. (4)

The total number of relevant frequencies amounts to Nω,LP =|VLP| = 2⌊2−1B |Tx|⌋+1 ≈ |Bx||Tx| and is approximated by

the effective time-bandwidth product.

C. Electronic Receive Focusing in the Fourier Domain

Image recovery in ultrafast UI is based on the electronic

receive focusing on a regular lattice

L ={

ri ∈ R2 : ri = r0 + ixδxex + izδzez,

ix ∈ [Nx − 1]0 , iz ∈ [Nz − 1]0 , i = ixNz + iz}

,(5)

where r0 ∈ R × R+, Nx, Nz ∈ N, and δx, δz ∈ R+,

discretizing the desired FOV, and is accomplished in two steps.

Given both a focal point ri ∈ L and the center coordinates

of the receiving elements (1), first, the times-of-flight t(n)tof,l(ri)

have to be computed, and second, the focused signal has to

be synthesized given the complex-valued envelopes (3).

1) Computation of the Times-of-Flight: Let c ∈ R+ denote

the average small-signal sound speed. For a steered PW with

the direction of propagation e(n)ϑ = [e

(n)ϑ,x, e

(n)ϑ,z]

T ∈ S1, the

times-of-flight are calculated as

t(n)tof,l(ri) =

e(n)ϑ ·

(

ri − r(n)ref

)

+ ‖rc,l − ri‖2c

, (6)

where r(n)ref = [r

(n)ref,x, 0]

T ∈ R2 denotes the reference position

with the component r(n)ref,x = rc,0,x for e

(n)ϑ,x ≥ 0 and r

(n)ref,x =

rc,Nel−1,x for e(n)ϑ,x ≥ 0. A similar expression for the times-of-

flight can be obtained for diverging waves.

2) Synthesis of the Analytical Focused Signal: Given the

times-of-flight (6), the analytical RF signal originating from

the focal position of interest ri is given by the coherent sum

y(ri, t) =

Nin−1∑

n=0

Nel−1∑

l=0

a(n)l (ri)x

(n)l

[

t+ t(n)tof,l(ri)

]

, (7)

where the coefficients a(n)l : L 7→ R are receive apodization

weights to be specified depending on the application. A typical

application is the maintenance of a constant F -number. With

the minimum and maximum times-of-flight

t(n)tof,min = min

l,i

{

t(n)tof,l(ri)

}

and t(n)tof,max = max

l,i

{

t(n)tof,l(ri)

}

,

respectively, the focused signal (7) occupies the time interval

Ty =[

minn

{

t(n)lb − t

(n)tof,max

}

; maxn

{

t(n)ub − t

(n)tof,min

}

)

. (8)

The Fourier series representing (7) with respect to the time

interval (8) is given by

y(ri, t) =

∞∑

ν=−∞

c[ν]ejων t (9a)

with the angular temporal frequencies ων = |Ty |−12πν and

the coefficients

c[ν] =

Nin−1∑

n=0

Nel−1∑

l=0

a(n)l (ri)c

(n)l [ν − νc]e

jωνt(n)tof,l

(ri), (9b)

where c(n)l are the Fourier coefficients (3b), if Tx = Ty and

ωc = |Ty |−12πνc. As before, the Fourier series (9) may be

truncated using the set of indices associated with the relevant

frequencies (4).

D. Finite Rate of Innovation Model for the Focused Signal

Reducing the total number of relevant frequencies in (4)

below the effective time-bandwidth product generally results

in undesired artifacts in the truncated series (9). The usage of

Nω,rnd < Nω,LP ≈ |Bx||Tx| randomly distributed frequencies

with the indices

VLP,rnd ={

νm = Πm (VLP) ,m ∈ [Nω,rnd − 1]0}

, (10)

where Πm (VLP) denotes the element of index m in a random

permutation of the set VLP, results in noise-like incoherent

aliasing that can be removed by a model-based nonlinear

recovery procedure. The Fourier coefficients with the indices

(10) may be directly measured from the analog RF signals

using special hardware [6], [7].

The analytical focused RF signal (7) is modeled as the finite

stream of pulses

y(ri, t) =

M−1∑

µ=0

bµ(ri)h(t− µTs), (11)

where bµ ∈ C denotes unknown coefficients, h : R 7→ C is the

known analytical pulse shape, and Ts ∈ R+ is a time interval.

With the model (11), the recovery of the analytical focused

RF signal (7) reduces to the recovery of the M coefficients

bµ. Equating the Fourier coefficients of (11) with respect to

the interval (8) with (9b) yields the Nω,rnd ×M linear system

d[ν]

M−1∑

µ=0

bµ(ri)e−jωνµTs = c[ν] (12)

for all ν ∈ VLP,rnd. Assuming a sparse coefficient vector,

the underdetermined system is solved using a denoising ℓq-

minimization approach, q ∈ [0; 1].

III. PARALLEL IMPLEMENTATION

The synthesis equation for the analytical focused RF signal

(9) lends itself for parallel processing. In this contribution,

we consider the parallel processing of blocks of lattice points.

The synthesis was implemented in C using CUDA (NVIDIA

Corp., Santa Clara, CA, USA). All computations were per-

formed on a Tesla K40c GPU with 32bit single precision.

The Fourier coefficients (3b) and d in (12) were estimated

using zero-padded discrete Fourier transforms (DFTs), which

were realized in parallel using the CUFFT library. The num-

ber of points for the complex-valued envelopes x(n)l was

NDFT,x = |Ty|fs,IQ ∈ N. The interval (8) had to be quantized

accordingly. The pulse shape h in (11) was provided at the rate

fs. The number of points was NDFT,h

= NDFT,xfs/fs,IQ ∈ N.

We set Ts = fs−1 and M = N

DFT,h. The Fourier coefficients

(3b) were stored in the shared memory for each lattice block.

IV. EXPERIMENTAL VALIDATION

We performed two experiments using a linear transducer

array (number of elements: Nel = 128, element pitch: δs,x =304.8 µm). Steered plane waves were emitted sequentially with

Ndir ∈ N directions e(n)ϑ = [cos(ϑn), sin(ϑn)]

T, ϑn = 2−1π+(n − 2−1(Ndir − 1))δϑ for n ∈ [Ndir − 1]0, where δϑ ∈ R+

denotes the angular spacing. The parameters in (5) were Nx =Nz = 512, δx = δz = 4−1δs,x, and r0 = [−255.5δx, 0.5δz]

T.

In each experiment, we compared four methods for image

recovery. The first method combined the conventional sam-

pling of the RF signals at the rate fs with the electronic receive

focusing in the time domain. We did not use interpolation. The

second method combined the IQ sampling at the rate fs,IQ with

the proposed method enabling low-rate image recovery using

the set of indices (4). The third method was identical to the

second method but used the set of random indices (10). The

fourth method was identical to the third method but recovered

the coefficients in (11) by approximately solving (12) using

the orthogonal matching pursuit.

A. In Silico Experiment

In the first experiment, the RF data was synthesized nu-

merically using an enhanced version of the forward model

presented in [10]. The simulated object A consisted of 21point-like scatterers embedded in a homogeneous fluid (speed

of sound: c = 1500m s−1). Its structure mimicked a wire

phantom. We used Ndir = 11 directions with an angular spac-

ing of 2.5◦, i.e. δϑ = 180−12.5π. To investigate the robustness

of our approach against measurement noise, Gaussian white

noise was added. The signal-to-noise ratio was 20 dB. The

lower and upper bounds on the common observation time

were t(n)lb = 0 and t

(n)ub = 82.35 µs for all sequential wave

emissions. These bounds enabled ultrafast acquisition rates.

The conventional sampling of the RF signals at the rate of

fs = 20MHz resulted in N(n)t = 1648 real-valued temporal

samples per signal. The interval Ty for the Fourier series

(9) was Ty = [−67.65 µs; 82.28 µs). The center frequency

was fc = 4MHz and the effective bandwidth at −60dBwas B = 2.8MHz. The IQ sampling of the complex-valued

envelopes at the rate fs,IQ = 3MHz resulted in N(n)t,IQ = 248

complex-valued temporal samples per signal, corresponding to

a reduction of 69.9%. With NDFT,x = 450, the total number

of relevant frequencies in (4) was Nω,LP = 421. We selected

Nω,rnd = 211 uniformly distributed frequencies for each wave

emission, corresponding to a reduction of 74.39%.

The images obtained from object A are shown in Fig. 1. The

conventional sampling in combination with time domain re-

ceive focusing (a) caused artifacts due to the limited precision

of the time delays. The IQ sampling in combination with the

proposed algorithm (b) demonstrated improved image quality.

The usage of random frequencies (c) violated the sampling

theorem and introduced noise-like incoherent aliasing. The

proposed FRI model (11) effectively reduced this aliasing in

(d). The image quality of (d) is comparable to (b).

B. In Vivo Experiment

In the second experiment, we acquired in vivo measurement

data using the linear transducer array L14-5/38 connected to a

SonixTouch Research system (Analogic Corp., Sonix Design

Center, Richmond, BC, Canada) equipped with the SonixDAQ

parallel channel data acquisition device. The objects to be

imaged were a common carotid artery and the adjacent jugular

vein (speed of sound: c = 1540m s−1) of a healthy male

proband in the transverse view. We used Ndir = 41 directions

with an angular spacing of approximately 1.163◦, i.e. δϑ =180−11.163π. The lower and upper bounds on the common

observation time were t(n)lb = 0 and t

(n)ub = 78.48 µs for all

sequential wave emissions. The conventional sampling of the

RF signals at the rate of fs = 40MHz resulted in N(n)t = 3140

real-valued temporal samples per signal. The interval Ty

for the Fourier series (9) was Ty = [−69.10 µs; 78.35 µs).The center frequency was fc ≈ 4.25MHz and the effective

Lateral position x (mm)Lateral position x (mm)

Axia

lposi

tionz

(mm

)A

xia

lposi

tionz

(mm

)

dB-15-15

-15-15

-10-10

-10-10

-5-5

-5-5

00

00

55

55

1010

1010

1515

151500

00

55

55

1010

1010

1515

1515

2020

2020

2525

2525

3030

3030

3535

3535

0

-10

-20

-30

-40

-50

-60

(a) (b)

(c) (d)

Fig. 1. Compound images recovered from object A using time domain receive

focusing with N(n)t = 1648 (a), the proposed algorithm with Nω,LP =

421 (b), the proposed algorithm with Nω,rnd = 211 (c), and the proposedalgorithm in combination with the FRI model (11) with Nω,rnd = 211 (d).The F -number was F = 0. The colormap is in decibel (dB).

bandwidth at −60 dB was B ≈ 4.5MHz. The IQ sampling

of the complex-valued envelopes at the rate fs,IQ = 5MHz

resulted in N(n)t,IQ = 393 complex-valued temporal samples

per signal, corresponding to a reduction of 74.97%. With

NDFT,x = 738, the total number of relevant frequencies in

(4) was Nω,LP = 665. We selected Nω,rnd = 333 uniformly

distributed frequencies for each wave emission, corresponding

to a rate reduction of 78.79%.

The images obtained from object B are shown in Fig. 2.

Each inset image magnifies the region indicated by the white

square. The conventional sampling in combination with time

domain receive focusing (a) caused artifacts due to the limited

precision of the time delays. In comparison to Fig. 1 (a), these

artifacts are less pronounced due to the higher sampling rate fs.

The IQ sampling in combination with the proposed algorithm

(b) demonstrated improved image quality (cf. inset image).

The usage of random frequencies (c) violated the sampling

theorem and introduced noise-like incoherent aliasing. The

proposed FRI model (11) effectively reduced this aliasing,

while the main features are clearly preserved (cf. (b) and (d)).

V. CONCLUSION

We introduced a low-rate parallel Fourier domain beam-

forming method for ultrafast pulse-echo UI. Using both syn-

thetic and in vivo measurement data, the proposed combination

of IQ sampling and electronic receive focusing in the Fourier

domain reduced the numbers of acquired samples by 69.9%and 74.97%, respectively. In comparison to the combination of

conventional sampling and time domain receive focusing, the

proposed combination precisely accounts for the time delays

and reduces artifacts. We achieved additional reductions in the

numbers of acquired samples by imposing an FRI model on the

Lateral position x (mm)Lateral position x (mm)

Axia

lposi

tionz

(mm

)A

xia

lposi

tionz

(mm

)

dB

1 mm1 mm

1 mm1 mm

-15-15

-15-15

-10-10

-10-10

-5-5

-5-5

00

00

55

55

1010

1010

1515

151500

00

55

55

1010

1010

1515

1515

2020

2020

2525

2525

3030

3030

3535

3535

0

-10

-20

-30

-40

-50

-60

(a) (b)

(c) (d)

Fig. 2. Compound images recovered from object B using time domain receive

focusing with N(n)t = 3140 (a), the proposed algorithm with Nω,LP =

665 (b), the proposed algorithm with Nω,rnd = 333 (c), and the proposedalgorithm in combination with the FRI model (11) with Nω,rnd = 333 (d).The F -number was F = 1.6. The colormap is in decibel (dB).

focused signal. Using random Fourier coefficients, which can

be obtained by special hardware [6], [7], reductions of 74.39%and 78.79% were achieved. More complex FRI models and

alternative recovery methods potentially improve the results.

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