a low-rate parallel fourier domain beamforming …...2016/09/19 · to be published in the...
TRANSCRIPT
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
Copyright notice:
c© 2016 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any
current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new
collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other
works.
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: [email protected]
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.
REFERENCES
[1] M. Tanter and M. Fink, “Ultrafast imaging in biomedical ultrasound,”IEEE Trans. Ultrason., Ferroelectr., Freq. Control, vol. 61, no. 1, pp.102–119, Jan. 2014.
[2] D. G. Manolakis, V. K. Ingle, and S. M. Kogon, Statistical and AdaptiveSignal Processing: Spectral Estimation, Signal Modeling, Adaptive
Filtering, and Array Processing, ser. Artech House Signal ProcessingLibrary. Artech House, Inc., 2005.
[3] M. F. Schiffner and G. Schmitz, “Plane wave pulse-echo ultrasounddiffraction tomography with a fixed linear transducer array,” in Acoust.
Imaging, ser. Acoust. Imaging, A. Nowicki, J. Litniewski, and T. Ku-jawska, Eds. Springer Netherlands, 2012, vol. 31, pp. 19–30.
[4] ——, “Pulse-echo ultrasound imaging combining compressed sensingand the fast multipole method,” in 2014 IEEE Int. Ultrasonics Symp.
(IUS), Chicago, IL, Sep. 2014, pp. 2205–2208.[5] ——, “Fast pulse-echo ultrasound imaging employing compressive
sensing,” in 2011 IEEE Int. Ultrasonics Symp. (IUS), Orlando, FL, Oct.2011, pp. 688–691.
[6] K. Gedalyahu, R. Tur, and Y. C. Eldar, “Multichannel sampling of pulsestreams at the rate of innovation,” IEEE Trans. Signal Process., vol. 59,no. 4, pp. 1491–1504, Apr. 2011.
[7] M. Mishali, Y. C. Eldar, O. Dounaevsky, and E. Shoshan, “Xampling:Analog to digital at sub-Nyquist rates,” IET Circuits, Devices Syst.,vol. 5, no. 1, pp. 8–20, Jan. 2011.
[8] A. Burshtein, M. Birk, T. Chernyakova, A. Eilam, A. Kempinski, andY. C. Eldar, “Sub-Nyquist sampling and Fourier domain beamformingin volumetric ultrasound imaging,” IEEE Trans. Ultrason., Ferroelectr.,Freq. Control, vol. 63, no. 5, pp. 703–716, May 2016.
[9] T. Blu, P.-L. Dragotti, M. Vetterli, P. Marziliano, and L. Coulot, “Sparsesampling of signal innovations,” IEEE Signal Process. Mag., vol. 25,no. 2, pp. 31–40, Mar. 2008.
[10] M. F. Schiffner and G. Schmitz, “Compensating the combined effects ofabsorption and dispersion in plane wave pulse-echo ultrasound imagingusing sparse recovery,” in 2013 IEEE Int. Ultrasonics Symp. (IUS),Prague, Jul. 2013, pp. 573–576.