A Theoretical Framework for Performance Characterization of Elastography: The Strain Filter
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164 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 44, no. 1, january 1997
A Theoretical Framework for Performance
Characterization of Elastography:
The Strain Filter
Tomy Varghese, Member, IEEE, and Jonathan Ophir
Abstract—This paper presents a theoretical framework
for performance characterization in strain estimation, which
includes the effect of signal decorrelation, quantization er-
rors due to the finite temporal sampling rate, and electronic
noise. An upper bound on the performance of the strain
estimator in elastography is obtained from a strain filter
constructed using these limits. The strain filter is a term
used to describe the nonlinear filtering process in the strain
domain (due to the ultrasound system and signal process-
ing parameters) that allows the elastographic depiction of
a limited range of strains from the compressed tissue. The
strain filter predicts the elastogram quality by specifying
the elastographic signal-to-noise ratio (SN Re ), sensitivity,
and the strain dynamic range at a given resolution. The
dynamic range is limited by decorrelation errors for large
tissue strain values, and electronic noise for low strain val-
ues. Tradeoffs between different techniques used to enhance Fig. 1. The block diagram of the strain filter, indicating the filtering
elastogram image quality may also be analyzed using the of the tissue strains by the strain filter to predict the dynamic range
strain filter. and respective SN Re at a given resolution in the elastogram. The
tissue strain measured using the ultrasound system is obtained by a
quasi-static tissue compression restricted by the mechanical bound-
ary conditions. The contributions of the signal processing and ultra-
I. Introduction
sound system parameters are indicated as inputs into the strain filter.
Improvements in estimator performance due to other techniques are
ltrasonic techniques for measuring the elasticity
U of compliant tissue generally rely on the estimation of
the strain [1]–[5]. Elastography, a technique of estimating
explained by the enhancement in the correlation coefficient (reduced
signal decorrelation).
the axial strain using differential displacements of the tis-
sue elements due to tissue compression, was proposed by range of elastic moduli. A distribution of strains in the
Ophir et al. [1]. medium is caused by the quasi-static compression under
This paper introduces the strain filter concept for the certain mechanical boundary conditions. The input to the
performance characterization of the strain estimator in strain filter is the actual tissue strain characterized by
elastography. The behavior of the upper bound of the an infinite dynamic range and SN Re , fine (denoted by
SN Re as a function of axial tissue strain forms a band- where epsilon is a very small number) resolution and
pass filter in the strain domain. The strain domain refers sensitivity. However, the interaction of the actual tissue
to the entire range of strains present in the compressed tis- strain with the ultrasound system and signal processing
sue. The strain filter consists of a graphical and analytical parameters corrupts the elastogram obtained, in the sense
representation of the allowable range of strain values and that it now has a finite dynamic range, SN Re , sensitivity,
their resulting SN Re for a given elastographic resolution. and resolution. The strain filter predicts the elastographic
The width of the strain filter specifies the dynamic range, image quality (SN Re , sensitivity, and dynamic range at a
and its height the respective SN Re value of the estimated given resolution) in terms of the signal processing and sys-
strain. tem parameters used to obtain the elastogram. The strain
A block diagram elucidating the strain filter concept filter therefore provides a quantitative assessment of the
is presented in Fig. 1. Elastography uses small mechani- elastogram quality, in terms of the four parameters de-
cal compressions on soft biological tissue that have a wide scribed above.
In addition, the strain filter makes it possible to select
Manuscript received November 27, 1995; accepted July 15, 1996.
This work was supported in part by NIH grants R01-CA38515, R01- the appropriate ultrasound system and signal processing
CA60520, and P01-CA64579, and by a grant from Diasonics Corp., parameters to obtain the best possible elastogram under
Santa Clara, CA. given tissue conditions. In subsequent sections of this pa-
T. Varghese and J. Ophir are with the Ultrasonics Laboratory, De-
partment of Radiology, University of Texas Medical School, Houston, per we discuss the effects of the various ultrasound sys-
TX 77030 (e-mail: tomyv@msrad3.med.uth.tmc.edu). tem parameters such as pulse center frequency and band-
0885–3010/97$05.00 c 1997 IEEEvarghese and ophir: a theoretical famework for performance characterization of elastography 165 width, and signal processing parameters (window size and the CRLB at poor SNRs (
166 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 44, no. 1, january 1997
In this paper, the strain variance is computed from the the total tissue strain (st ) and the lower bound on the
TDE variance using the expression derived by Walker and strain estimation standard deviation (σZZLB ) are substi-
Trahey [17], as long as the ZZLB coincides with the CRLB. tuted in (3):
The decay in the value of the correlation coefficient with st
tissue compression has been modeled using an analytic ex- SN ReU B = . (4)
σZZLB
pression derived by Meunier and Bertrand in [19] to esti- Incorporating the modified ZZLB expression for the
mate decorrelation effects due to rotation, translation, and TDE variance (see Appendix B) into σZZLB using (2),
biaxial deformation of the elastic tissue elements. The the- we obtain:
oretical development of the strain filter is presented in the
(sT )2
next section.
∆t , BT SN RC < γ 0
T6Threshold
2 γ < BT SN RC < δ 0
0
σZZLB ≥
2 2σBB
T ∆t , δ 0 < BT SN RC < µ0 (5)
III. Development of the Strain Filter
0 0
T hreshold µ < BT SN R < η
2σCRLB
2
C
0
Axial strain s is the displacement gradient, which may T ∆t , η < BT SN RC
be estimated from two adjacent time delay estimates sepa-
where η 0 , µ0 , δ 0 , and γ 0 are the modified thresholds (B-
rated by a time interval ∆t [1], assuming a constant speed
2) scaled by the factor T 2∆t . (5) shows the three distinct
of sound in the tissue, viz: 2
operating regions for σZZLB , depending on the value of
τ2 − τ1 BT SN RC . A distinct threshold region is observed between
s= , (1)
∆t the CRLB and the Barankin bound; however, the variance
increases exponentially in this threshold region (see Ap-
where τ1 is the time delay estimate at time t, and τ2 is the pendix B). Accurate estimation of the strain is possible
time delay estimate at time t + ∆t for a data window with only within the CRLB.
duration T . The minimum variance of the time delay estimator is
Since the strain estimate is obtained from a linear com- given by the CRLB [13]–[17]. The CRLB for time delay
bination of two random variables (time delay estimates estimation has been adapted for partially correlated sig-
separated by ∆t), the variance of the strain estimator de- nals by Walker and Trahey in [17], and is given by:
pends on the variance of the time delay estimator. Assum-
ing stationarity, the variance of the strain estimate (σS2 ) is 3 1 1
2
σCRLB = 2∼ (1 + ) −1
2
expressed in terms of the variance of the time delay esti- 2π T (B 3 + 12Bfo2 ) ρ2 SN R2
mates (σt2 ) in [20], and is given by: (6)
where fo is the center frequency, B is the bandwidth, ρ
2στ2
σS2 ≥ . (2) is the correlation coefficient, and the SNR term represents
T ∆t only the contribution due to electronic noise (SN RS ). This
(2) illustrates that for a given window size and overlap, closed-form expression obtained for signals with a rectan-
strain variance is reduced when the variance in the time gular spectrum assumes that the variance of the time delay
estimate is bounded by the CRLB. The Barankin bound
delay estimate is minimized. The resolution in the elas- 2
togram is reduced with an increase in T . Large overlap- exceeds the CRLB by a factor of 12 fB0 [13],[14], and is
ping windows also generate correlated errors which bias given by:
the strain estimate. In addition, an optimal window size 2
exists, where the strain estimation variance is minimum, 2 fo 2
σBB = 12 σCRLB (7)
with an increase in the variance observed as the window B
size is increased or decreased [6, pp. 122, 21]. Therefore The lower bound on the variance of the strain estimate
the strain estimation variance cannot simply be reduced 2
(σZZLB ) is obtained by substituting the bound on the vari-
in the limit as T → ∞, since στ2 also increases. ance of the TDE obtained using (6) and (7) into (5).
The variation of SN ReU B (4) with tissue strain is de-
A. The Strain Filter in Elastography fined as the strain filter. Three distinct regions constitute
the strain filter, which depends on the appropriate lower
A measure of elastographic image quality was described 2
bound that contributes to σZZLB (5).
[6],[7] in terms of the mean to standard deviation ratio
(SN Re ) of the elastogram: B. Effect of Decorrelation Due to Strain
µs on the Correlation Coefficient
SN Re = , (3)
σs
Decorrelation errors increase with tissue strain, causing
where µs and σs are, respectively, the mean and standard a decay in the values of the correlation coefficient. The cor-
deviation of the strain estimates in a region of uniform relation coefficient with motion compensation due to ax-
elasticity. The upper bound of the SN Re is obtained when ial deformation of elastic tissue for a 2-D Gaussian modelvarghese and ophir: a theoretical famework for performance characterization of elastography 167
has been derived by Meunier and Bertrand in [19], and is
given by:
√ 2 (α−1)2
2 αβ −1 f
ρ= p e 2 ( σu ) α2 +1 (8)
2(α2 + 1)(β 2 + 1)
where f is the spatial frequency in cycles/mm (f = 2fco
where c is the speed of sound in tissue = 1.54 mm/s), and
σf is the standard deviation of the Gaussian envelope in
2σ 1
cycles/mm (σu = cf , and σf = 2πσ t
, where σt is the spa-
tial standard deviation and σf is the standard deviation of
the Gaussian envelope in the frequency domain), α repre-
sents the axial compression where α = 1 − s, and s is the
tissue strain. The corresponding lateral expansion is de-
noted by β (with the incompressibility constraint αβ = 1).
The value of the correlation coefficient is substituted in the
expression for the CRLB (6) to model decorrelation effects.
Expressions for the CRLB [13]–[17] have been derived
for flat bandlimited signal and noise spectra. However, the
Fig. 2. The correlation coefficient and the standard deviation of the
correlation coefficient in [19] is derived for a Gaussian- strain estimate for increasing strain values. The standard deviation
shaped spectrum rather than a rectangular spectrum. A of the strain estimate has been normalized by the standard deviation
reasonable approximation was obtained by Céspedes et al. value at 8% strain of 0.0058 µs to enable plotting both the standard
[11] using a rectangular spectrum centered at the Gaus- deviation of the strain estimate and the correlation coefficient on the
sian center frequency with the same mean square ampli- same graph.
tude value as the Gaussian spectrum. The equivalent noise
spectral bandwidth [22, pp. 141] is defined by: with the Barankin bound, the performance of the strain
R∞ filter drops sharply, with a further drop in performance
P (f ) df √
B= 0 = 2πσ f (9) observed as the variance coincides with the constant vari-
P (f ) |max ance level. Note that the strain filter obtained using the
where B is the bandwidth of a rectangular spectrum with ZZLB has a smaller dynamic range than the strain filter
the same total power and peak amplitude as the Gaussian obtained under the optimistic assumption that the strain
pulse spectrum P (f ). estimation variance is always bounded by the CRLB.
Using the following typical signal parameters, T = 1 The range of strains that can be reliably estimated us-
mm (1.3 µs), fo = 5 MHz, B = 3 MHz (60% bandwidth), ing the elastogram determines the dynamic range of the
SN RS = 100 (40 dB), and interval between strain esti- strain filter. The dynamic range of the strain estimator in
mates ∆z = 0.5 mm (∆t = 0.66 µs). The lower bound decibels is defined as follows:
on the standard deviation of the strain estimator (normal-
smax
ized by its value at 8% strain) is plotted along with the DR = 20 log , (10)
smin
corresponding correlation coefficient in Fig. 2, for increas-
ing strain values. Note from Fig. 2 that the lower bound where smax is the maximum strain and smin is the mini-
on the standard deviation (σZZLB ) increases dramatically mum strain at a specified SN Re level in the strain filter.
for strain values >0.5%. This has been previously noted The quantity smin also defines the sensitivity of the strain
by O’Donnell et al. [5] and Céspedes [6, pp. 115]. They filter. A 1% tissue strain corresponds to a decibel value of
show that the standard deviation of the strain for a 1-D −40 dB. The dynamic range estimated for a −3 dB cutoff
simulation increases dramatically due to decorrelation as- level of the strain filter (SN Re level of 10) predicts a 30 dB
sociated with large strain values (>3%) for a fixed corre- dynamic range (observed from Fig. 3) for the strain filter
lation integration time. For the same reason, past work in obtained using the ZZLB bound, compared to the 40 dB
elastography has used small (≤ 2%) applied strains [1],[2]. dynamic range predicted using the CRLB. However, the
Decorrelation errors increase with tissue strain, reduc- dynamic range obtained experimentally for a single com-
ing the value of the correlation coefficient, and causing pression agrees closely with the dynamic range predicted
2
σZZLB to move from the CRLB to the Barankin bound using the ZZLB.
or the constant variance level as shown in (5). The three The slope of the strain filter for low strain values is de-
distinct regions in (5) are observed in plots of the strain termined primarily by electronic noise contributions, with
filter as shown by the curves in Fig. 3. Figure 3 shows decorrelation noise contributing to the progressive flatten-
the strain filter obtained for a 3 MHz rectangular band- ing of the curve. The plateau in the strain filter is caused
width using the ZZLB (CRLB for strains ≤10%, Barankin by increasing decorrelation errors; however, the variance
bound for strains >10% and 30%). As the lower bound coincides this region. At high strain values, the precipitous drop in168 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 44, no. 1, january 1997
Fig. 3. The strain filter illustrating the distinct regions of strain es- Fig. 4. A group of three strain filters, showing the distinct regions of
timation obtained using (5), along with the strain filter obtained strain estimation. These filters correspond to bandwidths of 2.5 MHz,
under the optimistic assumption that the strain estimation variance 3 MHz, and 3.5 MHz, respectively, with a 5 MHz center frequency
is always bounded by the CRLB adapted for partially correlated and T = 1.3 µs. Note the improvement in SN Re and dynamic range
signals. Note that the dynamic range of elastography may be deter- obtained with an increase in the signal bandwidth.
mined from the width of the strain filter, and the heights give the
respective SN Re at every strain level.
resolution levels. In general, as resolution increases, the
maximum value of the SN Re and the dynamic range of
performance is caused by decorrelation noise, which deter- the strain filter decrease. The dynamic range, sensitivity,
mines the slope of the graph in this region. Improvements and SN Re , along with the resolution, provide a complete
in the dynamic range can be obtained either by reduc- characterization of the noise properties in the performance
ing electronic noise errors for low values of the strain, or of the strain estimator in elastography.
reducing the rate of decorrelation for large strain values. Simulation experiments in the next section are used to
Decorrelation errors can be reduced using a combination confirm the validity of the theoretical strain filter model
of small compressions that allow successful step-wise tem- developed in this section.
poral stretching. For example, temporal stretching of the
post-compression signal [6]–[8], MA [5],[9], and [10], and
SMA [9],[10] improves the dynamic range and SN Re of IV. Simulation
strain estimation without sacrificing resolution.
The shape of the strain filter also depends on the vari- A simulation experiment to quantify the performance of
ation of T , f0 , and B terms in the denominator of (6). For the strain estimator for a 1-D tissue mechanical model is
example, the performance of the strain filter for different presented in this section. The 1-D model used in the simu-
values of the bandwidth is shown in Fig. 4. With an in- lation accounts only for the decorrelation of the echo-signal
crease in the system bandwidth, there is an improvement due to the axial component of the strain. The theoretical
in the maximum SN Re value, sensitivity, and dynamic model can be adapted for the 1-D case by setting β = 1
range of the strain filter. In other words, signal decor- in (8). In the simulation of the ultrasound system, a 1-D
relation reduces with an increase in the bandwidth. The sampled array is used to insonify a 2-D point scatterer
strain filters obtained at different center frequencies are medium.
presented in Fig. 5. Note the improvement in the sensitiv-
ity, dynamic range, and SN Re obtained with increasing A. Method
pulse center frequency. The maximum attainable SN Re
is approximately proportional to the square of the center The pre- and post-compression echo-signals are gener-
frequency. ated using a transducer with a center frequency of 5 MHz
The resolution of elastography depends on T and ∆t, and −3 dB band-width of 3 MHz (pulse standard devi-
and is limited only by the correlation length of the ultra- ation of 0.27 µs). The A-scan was sampled at 50 MHz.
sound system. The resolution in the elastogram improves Cross-correlation analysis was performed using a 1 mm
with a decrease in the value of T or an increase in sys- (1.3 µs) overlapping window with 0.5 mm (0.66 µs) over-
tem bandwidth. All the performance plots of the strain lap between consecutive windows.
filter are plotted for a constant value of the resolution. A The transducer is modeled as a 1-D sampled aperture
family of performance curves can be obtained at different composed of point subtransducer elements equally spacedvarghese and ophir: a theoretical famework for performance characterization of elastography 169
Fig. 5. A group of strain filters for center frequencies of 3.5 MHz, 5 Fig. 6. Plot of the theoretical strain filter superimposed on the strain
MHz, and 7.5 MHz, respectively, with a 60% bandwidth and T = filters obtained using a 1-D simulation before ( o o o ) and after
1.3 µs. Note the improvement in SN Re and dynamic range obtained ( x x x ) temporal stretching (motion compensation).
with an increase in the pulse center frequency. The maximum attain-
able SN Re is approximately proportional to the square of the center
frequency. as a percentage; SN Re is plotted, along the y-axis. The
theoretical curve of the strain filter for the 1-D case is also
by λ/2. Each subtransducer element is modeled as a point shown in the figure.
source or receiver with a two-way Gaussian transfer func- Observe from Fig. 6 that both the strain filter curves ob-
tion. The scattering medium is modeled as a 2-D array of tained from the simulation are completely bounded by the
point scatterers. The scatterer density in the media was theoretical strain filter curve. The theoretical strain filter
set to 48 scatterers/pulse width. The elastic target is as- curve forms the upper bound, which determines the attain-
sumed to have a Poisson’s ratio ≈ 0. This implies that able experimental or simulation performance. For strain
lateral and elevational decorrelation effects due to scat- values >2%, decorrelation errors cause the drop in the
terer motion are ignored. The applied stress is assumed value of the estimated strain. The CRLB bound on the
to propagate uniformly so the localized stress is constant variance is no longer applicable for strain values >2%, as
throughout the medium. The displacement of each scat- shown by the simulation experiment, since signal decorre-
terer is a function of the applied strain, and is modeled by lation introduced due to tissue compression dramatically
considering an equivalent 1-D spring system described in increases the variance in the strain estimate.
[6]. The spring constant is a function of the Young’s mod- Inaccurate estimation of the time-delay (increased vari-
ulus of the tissue. The applied strain is the same as the ance) is primarily due to the detection of false peaks and
tissue strain for a uniformly elastic homogenous medium. jitter [17]. The simulation results presented in this section
The pre-compression A-line is obtained from the ran- are obtained using the basic normalized correlation coeffi-
domly distributed scatterers. Each scatterer location is cient function. Non-linear processing to remove phase am-
then changed depending on the compressive force and a biguities (contributing to the reduction of false peaks [17])
post-compression A-line generated. The stretched post- was not performed. Sophisticated algorithms that incorpo-
compression A-line is generated by applying a linear rate additional processing to remove false peaks can there-
stretch factor on the post-compression signal. The process fore significantly improve the performance of the strain
is repeated for 28 different lateral locations in the sim- estimator.
ulated phantom to obtain independent A-line pairs (lat-
eral step > beamwidth of the transducer). The strain val-
ues are computed from the individual A-line pairs. Time- V. Conclusion
delay estimation is performed using the normalized cross-
correlation function, with the strains computed using (1). This paper presents a theoretical framework for quanti-
tative assessment of the quality of elastograms. This frame-
B. Results work is described as a strain filter that is typically a band-
pass filter in the strain domain. This filter allows only a
Fig. 6 shows the mean SN Re value and its standard restricted range of strain values to be included in the elas-
deviation (error bars), derived from 28 independent sim- togram. The deviation of the strain filter from an ideal
ulations, before and after temporal stretching. The x-axis all-pass characteristic in the strain domain is due to the
represents the total applied compressive strain expressed ultrasound system parameters, the finite value of the sono-170 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 44, no. 1, january 1997
graphic SNR, and the effects of signal decorrelation. Signal of the correlation coefficient (used in this paper) and a de-
decorrelation determines the largest value of strain that is rating factor. The derating factor accounts for the decay in
accurately estimated, while SN RS determines the smallest the correlation coefficient with increased window lengths.
measurable strain value. The dynamic range of the system The strain filter described in this paper is therefore more
is thus limited on the low end by electronic noise effects, optimistic in its prediction of the strain estimator perfor-
and on the high end by signal decorrelation, resulting in a mance. In addition, the strain filters obtained at varying
bandpass filter in the strain domain. depths in tissue also would be different. The 2-D and non-
The range of strains over which the CRLB specifies stationary properties of the strain filter will be discussed in
the variance of the time delay defines the optimum per- a later publication. It has also been demonstrated that un-
formance range for the strain estimator. The formulation der certain conditions, i.e. optimal multicompression with
of the CRLB by Walker and Trahey does not address temporal stretching [23], the bandpass characteristic of the
the implications of decorrelation causing the deviation of strain filter changes into a more desirable, high-emphasis
the time delay variance from the CRLB for large tissue characteristic.
strains. Application of the ρ − SN Rρ relationship devel-
oped by Friemel [18] and Céspedes et al. [12] clearly de-
marcates the regions over which the CRLB, the Barankin Acknowledgments
bound, and the constant variance level are applicable. Sig-
nal decorrelation may be reduced using a combination of The authors would like to thank Dr. Ignacio Céspedes
small compressions that allow successful temporal stretch- for the many discussions, and Dr. Michel Bertrand and Dr.
ing, (e.g., temporal stretching described by Céspedes and S. Kaiser Alam for their helpful comments.
Ophir [6]–[8], and SMA described by Huang et al. [9],[10]).
Temporal stretching is currently used to correct for decor-
relation only along the axial direction. Lateral and ele- Appendix A
vational decorrelation errors are more difficult to correct,
since the scatterers move out of the beam for large strains. A. Combining SN RS and SN Rρ to obtain SN RC
Multicompression averaging in conjunction with temporal
stretching of the strain estimates obtained from several The expression for SN RC which is used to compute
small compressions, increases both the SN Re and the dy- the post integration SNR and the thresholds in the ZZLB
namic range of strain estimation without sacrificing resolu- is presented here. The correlation coefficient is converted
tion. The improvements in the elastogram obtained using into an SNR measure [7],[18], that is given by:
these techniques may be quantitatively predicted using the
theoretical strain filter model developed in this paper. ρ
SN Rρ = . (A.1)
The strain filter concept developed in this paper pro- 1−ρ
vides a graphical framework for characterizing elastogra- The variation of SN Rρ with a linear variation in ρ is
phy. The strain filter quantified the elastogram image qual- illustrated in Fig. 7. Note from (A-1) that for ρ = 1,
ity (−3 dB dynamic range 30 dB using the ZZLB and SN Rρ = ∞, dropping exponentially to an SN Rρ value of
maximum elastogram SN Re ≈ 23 dB for a single compres- 22.46 dB for ρ = 0.93. The composite value of the SN R,
sion), for the signal parameters specified in Section III. In combining the contributions of SN RS and SN Rρ is given
a similar manner, different elastographic signal processing by [13],[14]:
techniques can now be compared and their performance
quantified using the dynamic range, sensitivity, resolution, SN RS SN Rρ
and maximum SN Re obtained from the strain filter. The SN RC = . (A.2)
1 + SN RS + SN Rρ
shape of the strain filter defines the quality of the elas-
togram; a narrow, low filter will cause elastographic arti- This expression for SN RC incorporates both the electronic
facts such as image noise and low dynamic range to plague noise level and the decrease in SN R caused by increase in
the elastogram. Thus, the design of the optimal strain fil- signal decorrelation with strain. From (A.2) we observe
ter for a given situation is of utmost importance to the that SN RC will always be bounded by the smallest value
production of quality elastograms. of either SN RS or SN Rρ .
The signal decorrelation parameter and shape of the
strain filter vary with changes in the cross-correlation win-
dow length (resolution) used to obtain the time-delay. A Appendix B
family of strain filter curves can be obtained at different
resolution levels. The effect of the window length param- B. Lower Bound on the Variance of
eter on signal decorrelation has not been analyzed in this the Time Delay Estimator
paper. The dependence of the correlation coefficient on the
finite window length used in the cross-correlation analysis The modified ZZLB derived by Weinstein and Weiss
is discussed in [21]. The expression for the effective correla- [13],[14] gives the tightest lower bound on the TDE vari-
tion coefficient is presented as a product of the peak value ance. We present here the results for bandpass signals withvarghese and ophir: a theoretical famework for performance characterization of elastography 171
characterized by the constant variance level of (T )2 /12,
which corresponds to the variance of a random variable
uniformly distributed between −sT /2 < τ < sT /2. In the
threshold region, the TDE variance increases exponentially
with the post integration SNR.
The thresholds given in (B.2) are computed for the fol-
lowing typical signal parameters: T = 1 mm (1.3 µs), fo
= 5 MHz, B = 3 MHz (60% bandwidth), and SN RS =
40 dB. Assuming SN Rρ 40 dB, SN RC ≈ 40 dB. The
numerical values of the thresholds η, µ, δ, and γ and in
(B-2) are 13.52, 0.78, 0.59, and 0.46 (22.6, −1.1, −2.3, and
−3.4 when expressed in dB), respectively, at a post inte-
gration SNR value of 43.86 dB. The reader is referred to
the papers by Weinstein and Weiss [13],[14] for a detailed
description and derivation of the ZZLB and the thresholds
for narrow and wide-band systems.
Fig. 7. Plot of the non-linear variation of SN Rρ with the correlation
coefficient. References
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2 2 2 reduction in elastograms,” Abstract, AIUM, 1995.
6 f0 −1 B [10] T. Varghese, J. Ophir, and E. I. Céspedes, “Noise reduction in
η= 2 ϕ elastography using temporal stretching with multicompression
π B 24f02
2 averaging,” Ultrasound Med. Biol., vol. 22, no. 8 pp. 1043–1052,
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π B estimation of subsample time delays of digitized echo signals,”
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[12] E. I. Céspedes, J. Ophir, and S. K. Alam, “The combined ef-
γ ≈ 0.46 , (B.2) fect of signal decorrelation and random noise on the variance
of time delay estimation,” IEEE Trans. Ultrason., Ferroelect.,
where fo is the center frequency, ϕ−1 (y) ispthe in- Freq. Contr., vol. 44, no. 1, pp. 220–225, 1997.
R∞
verse of ϕ(y) = √12π y e−µ /2 dµ, and (ζ/2)ϕ( ζ/2) =
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[16] C. H. Knapp and G. C. Carter, “The Generalized correlation Jonathan Ophir received his B.S.E.E.
method for estimation of time delay,” IEEE Trans. Acoust., (summa cum laude), M.S.E.E., and Doctor
Speech, Sig. Proc., vol. ASSP-24, pp. 320–327, 1976. of Engineering degrees from the University of
[17] F. W. Walker and E. G. Trahey, “A fundamental limit on de- Kansas, Lawrence Kansas in 1971, 1973, and
lay estimation using partially correlated speckle signals,” IEEE 1977, respectively. His Doctoral dissertation
Trans. Ultrason., Ferroelect., Freq. Contr., vol. 42, pp. 301–308, describes an early implementation of digital
1995. scan conversion techniques for diagnostic ul-
[18] B. H. Friemel, “Real-time ultrasonic two-dimensional vector ve- trasound. During 1976–1977 he worked as a
locity estimation utilizing speckle-tracking algorithms: Imple- project engineer for Philips Ultrasound Inc.,
mentation and limitations,” Ph.D Dissertation, Duke Univer- where he developed a commercial version of
sity, 1994. their first Digital Scan Converter. He then
[19] J. Meunier and M. Bertrand, “Ultrasonic texture motion analy- spent three years as an Assistant Professor of
sis: theory and simulation,” IEEE Trans. Ultrason., Ferroelect., Radiology at the University of Kansas Medical School in Kansas City,
Freq. Contr., vol. 14, pp. 293–300, 1995. where he was involved in the development of prototype sonographic
[20] E. I. Céspedes, M. F. Insana, and J. Ophir, “Theoretical bounds contrast agents, ultrasound phantoms, and instrumentation for ul-
on strain estimation in Elastography,” IEEE Trans. Ultrason., trasonic tissue characterization. From 1980 until the present time,
Ferroelect., Freq. Contr., vol. 42, no. 5, pp. 969–972, 1995. he has been with the University of Texas Medical School at Hous-
[21] T. Varghese, M. Bilgen, J. Ophir, and M. F. Insana, “Multireso- ton, where he is currently Professor of Radiology. He is also Adjunct
lution strain filter in elastography,” Submitted to IEEE Trans. Professor of Electrical Engineering at the University of Houston. His
Ultrason., Ferroelect., Freq. Contr., 1996. current field of interest is elastography, the imaging of the elastic
[22] J. S. Bendat and A. C. Piersol, Random Data: Analysis and properties of soft tissues. Dr. Ophir has contributed more than 85
Measurement, 2nd ed., New York: John Wiley, 1986. papers to peer-reviewed scientific journals, and holds 15 US and for-
[23] T. Varghese and J. Ophir, “Performance optimization in elas- eign patents. He is a Fellow of the American Institute of Ultrasound
tography: Multicompression with temporal stretching,” (in in Medicine, past Chairman of their standards committee, and past
press), Ultrason. Imag., 1996. member of the board of governors. He is a member of the Editorial
Board of Ultrasound in Medicine and Biology, and is Associate Edi-
tor of Ultrasonic Imaging. In 1992 he received the Terrance Matzuk
Award from the American Institute of Ultrasound in Medicine, and
in 1995 he was recognized as Inventor of the Year by the mayor of
Tomy Varghese (S’92–M’95) received the the city of Houston, Texas. In 1995 he also received an honorary
B.E. degree in instrumentation technology commission as an Admiral in the Texas Navy by Governor George
from the University of Mysore, India, in 1988, W. Bush.
and the M.S. and Ph.D. in electrical engineer-
ing from the University of Kentucky, Lexing-
ton, KY in 1992 and 1995, respectively. From
1988 to 1990 he was employed as an Engineer
in Wipro Information Technology Ltd., India.
He is currently a post-doctoral research asso-
ciate at the Ultrasonics Laboratory, Depart-
ment of Radiology, The University of Texas
Medical School, Houston. His current research
interests include detection and estimation theory, statistical pattern
recognition, tissue characterization using ultrasound, and signal and
image processing applications in medical imaging. Dr. Varghese is a
member of the IEEE and Eta Kappa Nu.You can also read
