ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 46, no. 5, september 1999
1057
A Time-Efficient and Accurate Strain
Estimation Concept for Ultrasonic
Elastography Using Iterative
Phase Zero Estimation
Andreas Pesavento, Student Member, IEEE, Christian Perrey, Student Member, IEEE,
Martin Krueger, Member, IEEE, and Helmut Ermert, Senior Member, IEEE
Abstract—In ultrasonic elastography, the exact estimation of temporal displacements between two signals is the
key to estimate strain. An algorithm was previously proposed that estimates these displacements using phase differences of the corresponding base-band signals. A major
advantage of these algorithms compared with correlation
techniques is its computational efficiency. In this paper,
an extension of the algorithm is presented that iteratively
takes into account the time shifts of the signals to overcome
the problems of aliasing and accuracy in the estimation of
the phase shift. Thus, it can be proven that the algorithm is
equivalent to the search of the maximum of the correlation
function. Furthermore, a robust logarithmic compression
is proposed that only compresses the envelope of the signal. This compression does not introduce systematic errors
and significantly reduces decorrelation noise. The resulting
algorithm is a computationally simple and very fast alternative to conventional correlation techniques, and accuracy
of strain images is improved.
ther a parabola [7] or a cosine function [8] has been fitted
to the cross-correlation function to obtain the exact maximum. The accuracy of curvefitting methods is limited by a
bias. Céspedes et al. [9] showed that the cosine fit used in
the so-called Correlation-Interpolation Method (CIM) is
more accurate than parabola fit. In Section II, we present
the theoretical framework of a new method to estimate the
time shift from the phase of the echo signals. Section III
deals with computational issues, which have to be taken
into account when adapting the new method to digital signals. In Section IV, we compare our concept with those of
other groups that also estimate the time shift from signal
phases. The performance of our new algorithm is discussed
in Section V. We compare our algorithm with correlation
techniques and the algorithm presented in [8] and [10] using theoretical analysis and simulations. Among the curvefitting methods, we limit the comparison to the CIM.
I. Introduction
II. Theoretical Framework
ately , the imaging of elastic properties of biological
tissue using ultrasound has become of special interest
because of the large variance of different elastic moduli in
biological tissue and its ability to discriminate pathological tissue. Mainly two concepts exist for the measurement
of elastic tissue properties: sonoelasticity [2] and elastography [3]. In elastography, strain inside the tissue caused
by external pressure is estimated. To improve the SNR of
the strain images in the presence of highly different elastic moduli, the use of multicompression images has been
proposed [4]–[6]. In these systems, a fast algorithm for the
estimation of temporal displacements between two images
is indispensable. Algorithms estimating the time shift from
phases of either the analytic (complex) rf signals or baseband signals are much faster than methods that are seeking the absolute maximum of the cross-correlation function because those methods require multiple integration
and resampling. In the past, however, curve-fitting methods have also been used to estimate subsample time shifts
without a resampling of the signals. In this method, ei-
L
In elastography, the postcompression echo signal is considered to be a compressed and time-shifted version of the
precompression signal:
x2 (t) = x1 (a · t + t0 ).
(1)
For gradient-based strain estimators, the echo signals in a
small temporal region surrounding the temporal position
t = t1 are considered to be only time-shifted, by neglecting
the compression inside the window:
x2 (t) = x1 (t + τ )
(2)
where τ = (a − 1) · t1 + t0 . This depth-dependent time shift
τ has to be calculated. Using crosscorrelation technique,
the time shift between two rf signals, x1 (t) and x2 (t), is
found by searching the maximum of the cross-correlation
function.
In general, the correlation between a(t) and b(t) for
complex or real signals can be calculated from the crosscorrelation function. For purposes of simplicity we denote:
Manuscript received May 28, 1998; accepted January 14, 1999.
The authors are with Department of Electrical Engineering, Ruhr-University Bochum, 44780 Bochum, Germany (e-mail:
Andreas.Pesavento@ruhr-uni-bochum.de).
c 1999 IEEE
0885–3010/99$10.00 ∞
a, b(t) =
−∞
a∗ (t )b(t + t)dt
(3)
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ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 46, no. 5, september 1999
calculation time, an exact interpolation is replaced by a
linear interpolation (see Section III) , the interpolation
can be done more accurately using the base-band signals.
A base-band signal can be calculated from the analytic
signal by:
ab (t) = a+ (t)e−jωm t
Fig. 1. Modified Newton search for the estimation of the time delay.
The root of the phase is iteratively searched. In each step of the iteration, the phase is calculated and approximated by a linear function
with a slope equal to the transducer’s centroid frequency. The intercept of this linear function with the abscissae is the new estimate for
the time delay.
as the cross-correlation function between a(t) and b(t).
In this paper, we denote time shifts with τ and variables of time-dependent function (like correlation functions) with t. Consequently, the cross-correlation function
x1 , x2 (t), is a time-shifted version of the autocorrelation
function x1 , x1 (t):
x1 , x2 (t) = x1 , x1 (t + τ ).
(4)
Because the autocorrelation function has a maximum at
t = 0, the cross-correlation function has a maximum at
t = −τ . Consequently, the conventional cross-correlation
determines this maximum to estimate the time shift. The
phase ϕ(t) of the correlation function of the corresponding
analytic signals x1+ (t) and x2+ (t) has a root at the same
position:
(5)
ϕ(−τ ) = 0 with ϕ(t) = arg x1+ , x2+ (t) .
This is obvious because the autocorrelation function is
positive-real at zero lag. Therefore, to estimate the time
shift, the root of ϕ(t) can be found. In the vicinity of its
root, ϕ(t) is nearly a linear function with the slope of the
transducers nominal centroid frequency ω0 . Taking advantage of this property, a modified Newton iteration can be
used to find the root and thus the time shift within some
iterations:
tn+1
ϕ(tn )
= tn −
ϕ̇(tn )
arg x1+ , x2+ (tn )
ϕ(tn )
= tn −
.
≈ tn −
ω0
ω0
where ωm denotes the modulation frequency. Analytic signals can be calculated from rf data by adding an imaginary
part to the signal. This imaginary part is equal to the signal’s Hilbert transform.
Obviously, the modulation frequency ωm should be chosen close to the centroid frequency of the signals. The
correlation function of the analytic functions x1+ (t) and
x2+ (t) can be evaluated using the base-band signals x1b (t)
and x2b (t):
x1+ , x2+ (t) = ejωm t x1b , x2b (t).
The algorithm is illustrated by an example in Fig. 1. As
we will demonstrate later, the algorithm is able to estimate time shifts with a high subsample accuracy. In Appendix A, we prove the convergence of the proposed algorithm theoretically.
In practice, data sampled at discrete temporal positions
are used. To estimate the correlation function at any temporal position, interpolation is needed. If, for reasons of
(8)
As mentioned previously, in elastography, the time shift is
a depth-dependent and time-dependent function. It has to
be estimated at discrete positions:
t = kTS .
(9)
To obtain a certain resolution, the correlation functions
are estimated using temporal windows of length Tω . The
following algorithm, derived from (6), iteratively estimates
the time shifts from phases of the correlation function. The
discrete temporal position is denoted with k, n denotes the
iteration index, and N denotes the number of iterations.
Note that the estimation of the phase root t = −τ is replaced by the direct estimation of the time shift τ :
τk,0 = τk−1,N
τk,n = τk,n−1

kTs + T2w
1

+
arg e−jωm τk,n−1
ω0

kTs − T2w
x∗1b (t + τk,n−1 /2)
(10)

· x2b (t − τk,n−1 /2)dt .
In practice, we choose:
ωm = ω0 ,
(6)
(7)
(11)
which is expected to be a good guess for the centroid frequency for a wide range.
To estimate the time shifts at a discrete time sample k,
the time shift at the sample k −1 is used as an initial value
of the iteration given by (10), to ensure that the correlation
function is always evaluated in the vicinity of t = −τ . In
elastography, the displacement is a monotonous function
of depth. The choice of the initial value prevents aliasing
in the calculation of the phase. Thus, the algorithm can
estimate arbitrarily large time shifts. The iteration can be
pesavento et al.: strain estimation concept for ultrasonic elastography
stopped after a fixed number of iterations or after a certain
accuracy has been reached.
As mentioned previously, in small regions with a constant elastic modulus, the time shift can be approximated
by a linear function of depth. For gradient-based strain
estimators, the time shift calculated in a window is assumed to refer to the center of the window. However, local
amplitude variations caused by speckle may cause strong
deviations of the estimated time shift from the time shift
in the middle of the window. In the past, this problem was
resolved by adaptive temporal stretching [11] and logarithmic compression of the signals [12]. Although adaptive
temporal stretching is very time-consuming, logarithmic
compression is easily applicable for the base-band signals.
It reduces amplitude variations caused by speckle and variations in backscatter that cause the estimated time shift to
deviate from time shift in the center of the window. Logarithmic compression of rf data, as in [12], also changes
the phase. When using base-band signals, we only compress the envelope and thus the amplitude of the signals,
leaving the phase unaltered:
xb,log (t) = log(1 + c · |xb (t)|) exp j · arg(xb (t))
(12)
where c is a factor that regulates the degree of compression.
1059
Third, we use linear interpolation defined by:
τ
τ τ τ ak+τ /T +
−
ak+τ /T bk
−
T
T
T
T
(17)
where . . . and · · · denote rounding toward −∞ and
+∞, respectively (often called floor and ceil).
B. Implementation of the Algorithm
We propose two versions of the algorithm described in
(10), which are specified as follows.
1. Algorithm 1: To interpolate the base-band signals at
a time t between two sample times kT and (k + 1)T , linear
interpolation is used. This leads to a high accuracy for
base-band signals [13]. However, our proof of convergence
(Appendix A) assumes exact interpolation. Consequently,
the limit of the iteration is no longer the exact time delay.
The number of iterations was set to 6.
2. Algorithm 2: We call this algorithm modified correlation interpolation technique using complex signals (for
reasons explained later). A nearest neighbor interpolation
is used, and only one iteration (N = 1) will be done for
every window.
III. Computational Issues
IV. Comparison with Other Concepts
A. Signal Interpolation
In digital signal processing, the integral in (10) is replaced by a sum over sampled signals. In the integrand
of (10), we need to consider subsample time shifts of the
signals x1b (t) and x2b (t). Therefore, the signals have to be
resampled at different positions. That means if
ak = s(k · T )
(13)
denotes the sampled version of the signal s(t), we need to
calculate a time shifted version
bk = s(k · T + τ ).
(14)
In this paper, three different interpolation methods have
been used to solve this problem. The simplest method we
apply is the nearest neighbor interpolation:
bk = ak+[τ /T ]
(15)
where the brackets [· · · ] denote rounding toward the nearest integer. Note that in this case, the phase-changing term
in (10) has to be changed to ej[τ /T ]ωm T .
The second applied method, exact interpolation, is defined by:
bk = IDFT(DFT(ak )e−jωτ )
(16)
where DFT and IDFT denote discrete Fourier transform
and inverse discrete Fourier transform, respectively. This is
the ideal implementation of a reconstructive interpolation.
As derived, the algorithm searches for phase root and,
thus, for maxima in the autocorrelation function. Consequently, the same results will be obtained with less computational effort. If we proceed no further than one iteration
after estimating the time shift by ϕ(0)/ω0 , regardless of
τ , the algorithm includes a method described in [1]. However, our extension assures that the correlation function is
always evaluated in the vicinity of t = −τ . This is not the
case in [1] where both aliasing and inaccuracies occur. Although special care was taken to reduce this problem, the
algorithm is limited to time shifts of 10 cycles. One cycle
refers to a temporal length of one period of an oscillation
of the transducer centroid frequency (1 cycle = 2π/ω0 ).
The estimation of time shifts by phases of the correlation function has also been implicitly used by de Jong et al.
in the so-called correlation interpolation technique [8], [10].
In this method, the phase is not directly estimated, but,
with the assumption of a constant amplitude of the complex correlation function in the vicinity of t = −τ , it is calculated at three discrete positions (hence, the method implicitly uses nearest neighbor interpolation): t0 = (k −1)T ,
t1 = kT , and t2 = (k + 1)T . The temporal position t1 is
assumed to be the closest available discrete position to the
maximum of the correlation function. The maximum of the
correlation function can then be assumed to be at:
t = t1 +
β
α
(18)
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ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 46, no. 5, september 1999
with
x1 , x2 (t2 ) − x1 , x2 (t0 )
2
+ j sin(α)x1 , x2 (t1 )
x1 , x1 (T )
α = arccos
.
x1 , x1 (0)
β = arg
(19)
With the assumption that the autocorrelation function has
a constant amplitude in the vicinity of t = 0 we can conclude:
(20)
α = arg x1+ , x1+ (T ) = Ω(T ) · T,
which we approximate by ω0 T as an assumption of our algorithm. In the correlation interpolation technique, however, this term is estimated from the autocorrelation function of the rf signals. β gives a directional estimate of the
angle of the complex correlation function at t = t1 . Thus,
the algorithm is very similar to Algorithm 2. However, it is
more reliable to calculate the phase directly from the complex correlation function because the estimations from rf
data may be affected by short window length and noise.
Algorithm 2 is similar to a method described in [14] as
an extension of the algorithm described in [1]. In contrast
to Algorithm 2, the starting value is found by a complex
2-D correlation followed by one iteration, which avoids the
aliasing that occurred in [1].
V. Verification of the Algorithm
A. Simulation of Compressed and
Time Shifted Speckle Phantom Data
echoes. The considered signal-to-noise ratios (total signal
energy divided by the total noise energy) were ∞, 40, 30,
20, and 10 dB. The power spectrum of the noise was assumed to be a rectangular spectrum bandlimited by the
transducer’s 6 dB bandwidth. In practice, this band limitation can be assured in any ultrasonic system by digital
filtering. The two noise signals added to the pre- and postcompression signals were statistically independent of each
other and of the echo signals. Furthermore, for the calculation of time shifts, different window lengths Tw have
been used: 4 cycles (Tw = 8π/ω0 ), 8 cycles, 16 cycles, and
32 cycles.
The time shifts were calculated using six different methods:
Algorithm 1 without logarithmic compression (A1)
Algorithm 1 with logarithmic compression (A1L)
• Algorithm 2 with logarithmic compression (A2L)
• CIM
• Cross-correlation method using exact interpolation
(CCM)
• Cross-correlation method using exact interpolation
with the conventional logarithmic compression proposed in [12]. (CCML)
•
•
For the implementation of the cross-correlation algorithm, exact interpolation was used. The maximum was
searched using the golden search technique [15] in the
vicinity of the expected peak. Note that the exact interpolation (16) does not change the sampling rate, and any
time shift can be evaluated. Hence, our implementation
of the CCM had an accuracy that was only limited by
the working precision of the computations and was taken
as a gold standard for unstretched, noiseless, time-shifted
data. In all calculations, the working precision was 64 bit
(double precision data).
B. Results and Discussion
To demonstrate the usefulness of the algorithm for elastography, we simulated ultrasonic echo data of a homogenous medium under various degrees of compression. We
modeled the echo data of a scattering medium in the
following way. The medium was assumed to be an ensemble of K scatterers k = 1, ..., K possessing the randomly distributed strengths ak and the randomly distributed axial distances to the transducer yk . The echo signal was calculated by superposing K modulated Gaussian
pulses delayed and weighted according to the distances
and strengths of the scatterers. A different degree of compression is considered by increased or decreased distances
of the scatterers. The centroid frequency of the pulse was
7.5 MHz, and fractional bandwidth (6 dB) was 66%. The
sampling frequency was 30 MHz. The simulated a-lines
possessed a length of 1000 samples. The applied strains
were 0.25, 0.5, 1, and 2%, respectively.
To demonstrate the ability of the algorithm to estimate
constant time shifts, in a different measurement, the second a-line was time shifted using exact interpolation (16).
In both simulations, noise was added to the simulated
The results of the simulations are presented in Figs. 2
through 5. In Fig. 2, the standard deviations of the time
delay estimation (TDE) as a function of strain using the
algorithms A1, A1L, A2L, CCM, and CIM are compared
for different window sizes. The SNR of the rf data was
30 dB. Fig. 3 compares the standard deviations of the TDE
for algorithms A1 and A1L with the Cramer-Rao lower
bound (CRLB) of TDE for two unstretched, time-shifted
signals without a strain applied. The CRLB for TDE was
derived by several authors. In [16], different derivations are
reviewed and shown to be identical. In that paper, expressions are given to calculate the CRLB for given signal and
noise spectra. This derivation has been used in this paper.
However, the calculated CRLB are strictly valid for high
SNR and no strain [17] only. Fig. 4 shows the standard
deviation of the TDE for algorithms A1 and A1L as functions of the SNR for different strains and window lengths.
In this figure, the CRLB is also presented (although it may
not be exact because of the applied strain).
Fig. 5 compares the standard deviations of the TDE
pesavento et al.: strain estimation concept for ultrasonic elastography
1061
(a)
(b)
(c)
(d)
Fig. 2. Standard deviation of the TDE as a function of the applied strain for a SNR of 30 dB and a window length of 4 cycles (a), 8 cycles (b),
16 cycles (c), and 32 cycles (d).
obtained with algorithm A1L (with the modified logarithmic amplitude compression), CCML (the conventional logarithmic amplitude compression) and A1 (with no amplitude compression) as a function of the strain for a window
length of 4 and 32 cycles, respectively.
Fig. 6 shows elastograms of a sponge phantom with a
hard inclusion produced by injecting agar-agar. The transducer had a center frequency of 7.2 MHz. The elastograms
were calculated using algorithms A1, A1L, and CCML. A
window length of 3.9 µs was used to estimate time delays.
Window overlap was 91%. The strain was estimated from
time delays by a least square estimator [18] using nine successive time delay samples.
The noise of the TDE is influenced by different noise
sources:
•
Noise caused by a limited SNR of the echo signals. The
lower bound of the standard deviation of this noise
is the CRLB. The standard deviation increases with
decreasing SNR. This noise is called rf noise in this
paper.
• So-called decorrelation noise caused by the axial compression of the signals. This noise is a systematic error
caused by the estimation of a depth-dependent time
shift in a window of a finite length. It is supposed
to be reduced by logarithmic amplitude compression.
We will show that this noise increases with increasing
strain and window length.
• Other systematic errors caused by the nonlinearity introduced by logarithmic compression, inaccurate interpolation, or quantization. These errors are referred
to as systematic errors in this paper.
Because algorithm A1 and A1L reach the CRLB for unstretched data (Fig. 3), the systematic errors are negligible
for these algorithms. Consequently, the linear interpola-
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ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 46, no. 5, september 1999
Fig. 3. Comparison of the standard deviation of the TDE using algorithm A1 and A1L with the CRLB for unstreteched, time-shifted
data (no strain applied).
tion is sufficiently accurate, and the errors introduced by
the logarithmic compression are far less than the rf noise
for a realistic SNR of the echo data (≤ 40 dB). This can
also be seen in Fig. 2, where the results of algorithm A1
and CCM are always very close. CCM is known to reach
the CRLB for sufficiently accurate interpolation [9]. Note
that we used an exact reconstructive interpolation in our
implementation of CCM.
Because strain was applied in the simulation for Fig. 2,
decorrelation noise can occur as well. Decorrelation noise
becomes significant in comparison with the rf noise for increasing strains or increasing window lengths. As logarithmic compression reduces the decorrelation noise, algorithm
A1L has significantly lower standard deviations than CCM
and algorithm A1. The influence of decorrelation noise is
demonstrated in Fig. 4, where the standard deviation of
the TDE is displayed as a function of the SNR. For high
SNR, the standard deviation is nearly constant, because
the TDE noise is caused by the decorrelation noise. Because decorrelation noise increases with increasing strain,
the range in which the standard deviation of the TDE is
constant is larger for higher strain [Fig. 4(c) and (d)] than
for lower strain [Fig. 4(a) and (b)]. Moreover, in this range,
the standard deviation is reduced by the logarithmic compression [Fig. 4(b) and (d)]. For low SNR, however, the
lower bound of the standard deviation is the rf noise. Both
estimators approximately reach the CRLB in this region.
Fig. 2(a) shows that, for algorithm CIM and A2L, systematic errors are significant for small window lengths.
These systematic errors introduced by the inaccurate interpolation of the true maximum are relatively independent of the strain. For algorithm A2L, these systematic
errors are lower than those for CIM.
Fig. 5 shows that adding the conventional logarithmic
compression to algorithm CCM introduces significant systematic errors for small window lengths [Fig. 5(a)] or small
strains. Hence, in the elastogram of the sponge phan-
tom algorithm, CCML [Fig. 6(c)] introduces significant
errors inside the hard inclusion. For large window lengths
[Fig. 5(b)] or large strains, the decorrelation noise masks
these errors. Decorrelation noise is reduced by both kinds
of logarithmic compression. Hence, in Fig. 5(b), the standard deviations of both algorithm A1L and CCML are
lower than those of algorithm A1 and CCM, respectively.
Remember that CCM and A1 always have approximately
the same performance. This is also valid for the soft regions
in the elastogram of the sponge phantom [Fig. 6(c) and
(d)]. However, the best results are always achieved with algorithm A1L, using the modified logarithmic compression.
Furthermore Fig. 5 shows that increasing the window size
does not always decrease the standard deviation. The rf
noise is decreased with smaller window lengths, but decorrelation noise is increased and cannot fully be eliminated
with logarithmic compression. The optimal window length
depends on the SNR and the applied strain.
The accuracy of the iterative algorithm was achieved
within some iterations. Fig. 7 experimentally demonstrates
the convergence properties of the algorithms (note that a
theoretical proof is given in Appendix A). The mean of
the changes of the time shifts ∆n = τn+1 − τn over several time-shift estimations is plotted versus the iterations
index n. ∆n is plotted in decibels relative to ∆0 . For the
simulation, a window length of 4 cycles (i.e., 0.4 mm for a
7.5 MHz transducer) and a constant time shift was applied.
The plot shows that the difference to the limit is reduced
by approximately 20 dB per iteration step. However, this
factor is a function of the SNR, of the strain, and of the
difference between ω0 and the centroid frequency of the
signals in the observation window. Appendix B shows a
comparison of calculation times of the iterative algorithms
and conventional cross-correlation techniques.
VI. Conclusion
An iterative algorithm has been presented that estimates the time shifts from the phase of the correlation
function in ultrasound elastography. It has been proven
analytically that the algorithm converges to the maximum position of the correlation function (Appendix A).
Our iterative algorithm A1 is a faster equivalent to concepts in which the maximum of the correlation function is
searched and reconstructive interpolation is used (CCM).
We showed that the algorithm reaches the CRLB for unstretched, time-shifted signals.
However, in strain estimation, the model of a constant
time shift of the signals is not applicable. This results in
major decorrelation noise of the time-shift estimation. This
error is the main source of noise in elastography. It can
be significantly reduced using a logarithmic compression
of the envelope of the signals (leading to algorithm A1L),
which is possible using complex signals. It has been shown,
that this logarithmic compression does not introduce new
errors, which is the case for the CCML.
Simulations show that the algorithm has an advantage
with respect to accuracy compared with fast, noniterative
pesavento et al.: strain estimation concept for ultrasonic elastography
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(a)
(b)
(c)
(d)
Fig. 4. Comparison of the standard deviations of the TDE using algorithm A1 and A1L with the CRLB for different window lengths of
0.25% strain (a and b) and 1% strain (c and d).
algorithms (e.g., the CIM or the algorithm proposed in
[1]). In these algorithms, the time shift has also been estimated using the phase of the correlation function, but no
iteration has been applied, which introduces systematic errors. Both of our algorithms (A1 and A1L) converge fast.
2-6 iterations lead to a sufficiently high accuracy. Consequently, the proposed algorithm A1L is a computationally
simple and very fast alternative to the cross correlation
method, and accuracy is improved with respect to decorrelation noise.
Appendix A
The convergence of the modified Newton iteration can
be proven. We denote the Fourier transform of x1 (t) with
capital letters:
x1 (t) ⇔ X1 (ω).
(21)
The autocorrelation function using the analytic signals has
the Fourier transform:
|2X1 (ω)|2 for ω > 0,
x1+ , x1+ (t) ⇔
(22)
0
elsewhere.
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ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 46, no. 5, september 1999
(a)
(b)
Fig. 5. Comparison of the standard deviation of the TDE as a function of the applied strain for the modified logarithmic compression, the
conventional logarithmic compression, and no logarithmic compression. The SNR was 30 dB, and a window length of 4 cycles (a) or 32
cycles (b) was used.
(a)
(b)
(c)
(d)
Fig. 6. Comparison of strain images of a sponge phantom with injected agar-agar leading to a hard inclusion (a) obtained with the algorithm
A1 (b), algorithm CCML (c), and algorithm A1L (d). Algorithm CCM yields the same results as algorithm A1. Our new algorithm A1L is
free of the decorrelation noise occurring with A1 and CCM and systematic errors introduced by the logarithmic compression in CCML.
pesavento et al.: strain estimation concept for ultrasonic elastography
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It is obvious that the phase of the correlation function in
(28) is a product of (t + τ ) with an unknown frequency
Ω(t) within the band limits:
ϕ(t) = arg x1+ , x2+ (t) = Ω(t) · (t + τ ).
(30)
We now use (29) for the derivation of an expression that
describes the reduction of the error of the time delay estimation in one of the modified Newton iterations. This
error is εn = (tn+1 + τ ). Without loss of generality, we
assume it to be positive. From (29) we obtain:
Fig. 7. Change of τn in one iteration step as a function of the iteration
index.
ω1 (tn + τ ) ≤ ϕ(tn ) = Ω(tn ) · (tn + τ ) ≤ ω2 (tn + τ ),
(31)
and using (6) and (30) we obtain:
ω1 (tn + τ ) ≤ ω0 · (tn − tn+1 ) ≤ ω2 (tn + τ ).
The cross-correlation function of the analytic signals can
be expressed by an inverse Fourier transform:
1
x1+ , x2+ (t) =
2π
2
=
π
∞
⇒
|X1+ (ω)|2 ejω(t+τ ) dω
(32)
ω1
ω2
− 1 (tn + τ ) ≤ −(tn+1 + τ ) ≤
− 1 (tn + τ )
ω0
ω0
(33)
−∞
∞
|X1 (ω)| e
2 jω(t+τ )
dω.
(23)
0
Equation (23) is now analyzed for band-limited ultrasonic
pulse-echo signals. Those signals are broadband signals.
However, we can always define a lower bandlimit ω1 and
an upper bandlimit ω2 . Furthermore, a transducer has a
nominal centroid frequency ω0 between these limits, and
the fractional bandwidth is much less than 200%:
0 < ω1 < ω0 < ω2 < 2ω0 .
(24)
Using (24), the integral in (23) can be simplified:
2
x1+ , x2+ (t) =
π
ω2
|X1 (ω)|2 ejω(t+τ ) dω.
⇒1−
(25)
ω1
ω2
ω1
(tn+1 + τ )
≥1−
≥
.
ω0
(tn + τ )
ω0
(34)
Equation (34) shows how the error is reduced by one modified Newton iteration. The absolute values of the left-hand
and the right-hand side of (34) both are less than 1. Hence,
the absolute value of the error εn = (tn+1 + τ ) is always
lower than the geometric sequence ε0 q n :
where
εn+1 < ε0 · q n
(35)
ω1 ω2 ,
1
−
q = max 1 −
< 1,
ω0 ω0 (36)
which proves convergence. Note that q < 1 is derived from
(24) and that, in case of a fractional bandwidth of 100%,
q becomes 0.5.
The integral mean value theorem in analysis proves that,
if a continuous function g(x) holds:
g(x) ≥ 0∀x ∈ [a, b],
(26)
a ξ ∈ [a, b] exists such that:
b
b
f (x)g(x)dx = f (ξ)
a
g(x)dx.
(27)
a
Consequently,
2
x1+ , x2+ (t) = ejΩ(t)(t+τ )
π
ω2
|X1 (ω)|2 dω
ω1
(28)
where Ω(t) satisfies
ω1 ≤ Ω(t) ≤ ω2 .
(29)
Appendix B:
Computational Efficiency
A C++ implementation of A1L took approximately 5.5
seconds on a PentiumTM (Intel Corp., American Fork, UT)
200 MHz for the calculation of an elastogram of 300 scans
and 80 samples per scan, using a window length of 64
samples and 75% window overlap.
To compare the computational efficiency of the algorithms, we derived expressions for the approximate number of real-value multiplications for an a-line of NS samples for each algorithm. Table I shows the result for a window overlap of 75%. The derived formulas only include
multiplications needed for interpolation and calculation of
the cross-correlation function. The terms assume the window length in samples n to be small compared with NS .
1066
ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 46, no. 5, september 1999
TABLE I
Comparison of Computational Effort of Several
Algorithms. The Number of Required Multiplications
is Listed.
Method
Multiplications
A1, A1L
CCM-Reconstructive interpolation filtering
CCM-Brute force
A2L
CIM
144
448
3072
16
24
NS
NS
NS
NS
NS
Hence, the derived formulas only depend on NS . Three iterations have been used in A1 and A1L, although two will
be enough in most cases. In [9], a computationally effective
implementation of the CCM using iterative reconstructive
filtering is described. In this method, at least NR = 6 iterations have to be performed to obtain an accuracy of 0.002
cycles. This value is reached in several cases in Fig. 2.
The filter length of the reconstructive interpolation filter was set to 9, which leads to additional errors of the
TDE of approximately 0.0008 cycles in [9]. In addition to
this ideal implementation, the number of multiplications
for the frequently used brute-force search obtaining the
same accuracy is given.
References
[1] M. O’Donnell, A. R. Skovoroda, B. M. Shapo, and S. Y.
Emelianov, “Internal displacement and strain imaging using ultrasonic speckle tracking,” IEEE Trans. Ultrason., Ferroelect.,
Freq. Contr., vol. 41, no. 3, pp. 314–325, May 1994.
[2] R. M. Lerner, S. R. Huang, and K. J. Parker, “Sonoelasticity
images derived from ultrasound signals in mechanically vibrated
tissues,” Ultrason. Med. Biol., vol. 16, pp. 231–239, 1990.
[3] J. Ophir, I. Céspedes, H. Ponnekanti, Y. Yazdi, and X. Li, “Elastography, a quantitative method for imaging the elasticity of biological tissues,” Ultrason. Imaging, vol. 13, no. 2, pp. 111–134,
Apr. 1991.
[4] M. O’Donnell, S. Y. Emelianov, A. R. Skovoroda, M. A. Lubinski, and S. M. Shapo, “Quantitative elasticity imaging,” in Proc.
1993 IEEE Ultrason. Symp., Baltimore, MD, pp. 893–903, 1993.
[5] T. Varghese, and J. Ophir, “Performance optimization in elastography: Multicompression with temporal stretching,” Ultrason. Imaging, vol. 18, no. 3, pp. 193–214, Jul. 1996.
[6] E. E. Konofagou, J. Ophir, F. Kallel, and T. Varghese, “Elastographic dynamic range expansion using variable applied strains,”
Ultrason. Imaging, vol. 19, no. 2, pp. 145–166, Apr. 1997.
[7] S. G. Foster, P. M. Embree, and W. D. O’Brien, Jr., “Flow
velocity profile via time-domain correlation: Error analysis and
computer simulation,” IEEE Trans. Ultrason., Ferroelect., Freq.
Contr., vol. 37, no. 2, pp. 164–175, May 1990.
[8] P. G. M. de Jong, T. Arts, A. P. G. Hoeks, and R. S. Reneman,
“Determination of time motion velocity by correlation interpolation of pulsed ultrasonic echo signals,” Ultrason. Imaging, vol.
12, no. 2, pp. 84–98, Apr. 1990.
[9] I. Céspedes, Y. Huang, J. Ophir, and S. Spratt, “Methods for
estimation of subsample time delays of digitized echo signals,”
Ultrason. Imaging, vol. 17, no. 2, pp. 142–171, Apr. 1995.
[10] P. G. M. de Jong, T. Arts, A. P. G Hoeks, and R. S. Reneman,
“Experimental evaluation of the correlation interpolation technique to measure regional tissue velocity,” Ultrason. Imaging,
vol. 13, no. 2, pp. 145–161, Apr. 1991.
[11] S. K. Alam, J. Ophir, and E. E. Konofagou, “An adaptive strain
estimator for elastography,” IEEE Trans. Ultrason., Ferroelect.,
Freq. Contr., vol. 45, no. 2, pp. 461–472, Apr. 1998.
[12] I. Céspedes and J. Ophir, “Reduction of image noise in elastography,” Ultrason. Imaging, vol. 15, no. 2, pp. 89–102, 1993.
[13] R. W. Schafer and L. R. Rabiner, “A digital signal processing
approach to interpolation,” Proc. IEEE, vol. 61, no. 6, pp. 692–
702, 1973.
[14] N. A. Cohn, S. Y. Emelianov, M. A. Lubinski, and M. O’Donnell,
“An elasticity microscope. Part I: Methods,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 44, no. 6, pp. 1304–1319,
Nov. 1997.
[15] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C. New York, NY: Cambridge Univ.
Press, 1994.
[16] A. H. Quazi, “An overview on the time delay estimate in active and passive systems for target localization,” IEEE Trans.
Acoust., Speech, Signal Processing, vol. 29, no. 3, pp. 527–533,
Jun. 1981.
[17] T. Varghese and J. Ophir, “A theoretical framework for performance characterization of elastography: The strain filter,” IEEE
Trans. Ultrason., Ferroelect., Freq. Contr., vol. 44, no. 1, pp.
164–172, Jan. 1997.
[18] F. Kallel and J. Ophir, “A least-squares strain estimator for
elastography,” Ultrason. Imaging, vol. 19, no. 3, pp 195–208,
Jul. 1997.
Andreas Pesavento (S’98) was born in
Werl, Germany in 1971. He received his M. S.
degree in electrical engineering from Ruhr
University Bochum, Germany in 1996. He is
currently working at the High Frequency Engineering Institute, Ruhr University, Bochum,
Germany. His research interest are medical ultrasonic imaging, especially multidirectional
quantitative imaging and elasticity imaging.
Christian Perrey (S’98) was born in Recklinghausen, Germany in 1970. He received his
M. S. degree in electrical engineering from
Ruhr University, Bochum, Germany in 1998.
working at the High Frequency Engineering
Institute in the field of elasticity imaging. He
is currently working at Siemens Medical Systems, Ultrasound Group. His fields of interest
are electro-acoustics and ultrasonic imaging.
Martin Krueger (S’92–M’97) was born in
Hamburg, Germany in 1967. He received his
M. S. and Ph. D. degrees in electrical engineering from Ruhr University, Bochum, Germany in 1992 and 1997, respectively. He is
currently working at the High Frequency Engineering Institute, Ruhr University, Bochum,
Germany. His research interests are magnetic resonance imaging and medical ultrasonic imaging, especially b-mode imaging simulation, quantitative imaging, reconstructive
imaging, and elasticity imaging.
pesavento et al.: strain estimation concept for ultrasonic elastography
Helmut Ermert (M’79–SM’98) was born in
Hagen, Germany, on April 2, 1941. He received the Dipl.-Ing. degree in electrical engineering and the Dr.-Ing. degree from the
Technical University Aachen, Germany, in
1965 and 1970, respectively. In 1975 he received the Dr.-Ing. habil. degree (Habilitation) from the Engineering Faculty of the University of Erlangen-Nuremberg at Erlangen,
Germany.
From 1966 to 1970 he worked on millimeter wave engineering and microwave ferrites at the Institute of High Frequency Engineering at the Technical University (RWTH) Aachen. From 1970 to 1975 he was engaged in microwave integrated circuits, microwave ferrites, and
microwave measurement techniques at the Institute of High Frequency Engineering of the University of Erlangen-Nuremberg. From
1978 to 1987 he was a professor of electrical engineering in Er-
1067
langen, working on microwave and acoustic imaging. Since 1987 he
has been a professor of electrical engineering and Director of the High
Frequency Engineering Institute at the Ruhr-University in Bochum,
Germany. He is continuing research on measurement techniques, diagnostic imaging, and sensors in the areas of microwaves, optics, and
ultrasound for applications in medicine, nondestructive testing, and
industry. In 1991 he was co-chairman of the 19th International Symposium on Acoustical Imaging in Bochum, Germany.
Dr. Ermert is a member of the VDE/ITG (Germany), the IEEE
(USA), the German Society of Biomedical Engineering (DGBMT),
the German Society of Nondestructive Testing (DGZfP), and the
German Society of Ultrasound in Medicine (DEGUM). From 1989
to 1991 he was an elected member of the Administrative Committee
of the IEEE Ultrasonics, Ferroelectrics, and Frequency Control Society (UFFC, Region 8). From 1995 to 1997 he was president of the
DGBMT.