Statistical relationship between ultrasound echo envelope and envelope peak

LTRASONIC

IMAGING

10,

265-274

(1988)

STATISTICAL RELATIONSHIP BETWEEN ULTRASOUND ECHO ENVELOPE AND ENVELOPE PEAK Ping

He

Department of Biomedical Engineering Wright State University Dayton, OH 45435

This study shows that the amplitude information of ultrasound echoes is carried mainly by the envelope peaks (EPs). It is first shown that the EPs in an A-line represent the maximum number of independent amplitude data. It is then demonstrated that the entire envelope could be approximately reconstructed from the EPs. Finally, using the echo data from a is found that there are no significant tissue-mimicking phantom, it differences among the attenuation coefficients estimated from the EPs, the original envelope samples, and the reconstructed envelope samples. The the attenuation results of this study indicate that, in the time domain, coefficient can be most efficiently estimated from the envelope peaks. 0 1988 AcademicPress, Inc. Key words: Attenuation estimation; echo envelope; envelope peak; tissue characterization; ultrasound. I.

INTRODUCTION

In the past two decades, many methods have been proposed for estimating acoustic attenuation of soft tissue using pulse echo techniques in either the time domain [l-3] or frequency domain [4-61. Since ultrasound stochastic signal theory backscattering in soft tissue is a random process, attenuation estimate from echo has been used for deriving the optimal The signals as well as for determining the uncertainty of the estimate. analysis of the estimate uncertainty is of practical importance since a noticeable problem in attenuation estimation of soft tissue is that the results often show a large variation even among normal tissues [1,7]. This large variation makes differential diagnosis based on the attenuation measurement either impossible or unreliable. A quantitative analysis of the estimate variance enables us to determine the required sample volume for a desired measurement precision. It also enables us to compare analytically the performances of different estimation methods. In the frequency domain, the attenuation is estimated from the exponential decay of the echo spectra. The variance of the maximum likelihood estimator for the attenuation coefficient (denoted by 8) using the frequency domain methods has been extensively studied by Kuc [5,7] and by several other researchers [6,8]. One advantage of such an analysis in the frequency domain is that the spectral samples obtained by Fourier transforming the digitized echo samples are mutually independent. Since the frequency resolution is inversely proportional to the time duration of the echo segment, the variance of the 6 estimate obtained from one echo sequence (one A-line] can be determined from the total length of the A-line attenuation and the bandwidth of the echo signal [51. In the time domain, has traditionally been estimated from the exponential decay of the echo amplitudes (echo envelope samples]. In this case, however, the echo envelope samples are not mutually independent. As a matter of fact, the total number of the envelope samples in an A-line, or the time resolution,

265

0161-7346/88 $3.00 Copyright 0 I988 by Academic Press, Inc. AN rights of reproduction in any ,form reserved.

PING HE

can be arbitrarily changed by changing the sample rate. Because of the correlation between envelope samples, the quantification of the variance of attenuation estimation in the time domain is less straightforward than in the frequency domain, although the signal processing itself for attenuation estimation in the time domain is much simpler than that in the frequency domain. A new time domain method for attenuation estimation, called the envelope peak (EP) method, was recently proposed [9,10]. In this method, the attenuation coefficient, 6, is estimated from the exponential decay of the echo envelope peaks (EPs) instead of the echo envelope samples. It has been shown that the EPs in an A-line are mutually independent [ill. As a result, the variance of the /3 estimate can be formulated which was found to have the same dependence on the length of the A-line and the bandwidth of the echo signal as that in the frequency domain [91. Besides simplicity in signal processing, an important advantage of this new time domain method is that the variance of the (3 estimate is much smaller than that of the frequency domain methods under the same scanning conditions [91. It can be shown that the principle of the EP method for estimating /3 from the EPs can also be applied for estimating /3 from the envelope samples. Since the EPs are only a subset of the envelope samples, one may wonder whether some useful echo amplitude information is lost by only using the EPs; whether the variance of attenuation estimation can be further decreased by using all the envelope samples; and finally, whether the (3 value estimated from the EPs will be different from that estimated from the envelope samples. To answer these questions, the statistical relationship between the echo envelope and the EP is further studied and the following results are presented in this paper: (11 The expected envelope peak to peak distance the range (axial) resolution of the pulse-echo in an A-line represent the maximum number of that one can obtain from that echo sequence.

is

approximately equal to Therefore, the EPs amplitude data independent

system.

(2) The entire echo envelope can be roughly reconstructed from the detected EPs. This shows that the amplitude information of the echo signal is mainly carried by the EPs. (3) Using a tissue-mimicking phantom, it was found that there was no statistically significant differences among the ,3 values estimated from the EPs, estimated from the entire envelope samples and from the reconstructed envelope samples. the echo envelope is defined as the In the following discussion, the original echo magnitude of the analytic signal whose real part is signal and whose imaginary part is the Hilbert transform of the echo signal [91. The EPs are defined as the local maxima of the envelopes. The sequence (one Aattenuation coefficient, 6, is estimated from one echo line) only. II.

ESTIMATION OF ATTENUATION ENVELOPE SAMPLES

COEFFICIENT

FROM ENVELOPE

Applying Rice's theory of random electrical echo signal backscattered from soft tissue, it envelope follows a Rayleigh distribution: p(R) where

R is the

= (We1

e

noise [121 can be shown

PEAKS to that

AND FROM ultrasound the echo

-R2/2#

echo envelope

value

and # is

266

the mean

power

of

the

echo.

ATTENUATION

Defining

u = ln(R/fl), p(u)

with

the

the

= .(2u

Similarly, value, the

= 0.060

if we define PDF of v is

p(v)

= 0.35

FROM ENVELOPE AND ENVELOPE PEAK

probability

density

function

(PDF) of u is:

- 0.5 e2"1

mean and standard E [ul

with

ESTIMATION

,

deviation

(s.d.)

as:

vu = 0.636

(31

v = ln(Rp/o), [S]:

where

Rp

m + 2.5)~ - 0.75 c (n + 1) .[(n n=O F(0.5n + 1.751 2"

E [VI = 0.383

,

uv = 0.496

is

the

envelope

peak

ezVl

.

141

(51

The two curves for the PDFs of u and v are depicted in figure 1. It can be seen that v has a narrower distribution than u. From Eqs. (31 and (51, it is also evident that the signal-to-noise ratio (SNR) of v is higher than the SNR of u. The derivations of the maximum likelihood estimate of p from the log EP values in an A-line were presented elsewhere [S,lOl. In brief, if the echo signal is sufficiently narrowband, the envelope may be constructed directly from the echo signal and the EPs are then detected from the envelope. After compensating for the diffraction effects, the log values of the EPs are plotted against their ranges, and 6 can be calculated from the slope of the least squares line. Since the EPs in an A-line are mutually independent, the variance of the ,3 estimate can be found from uv in Eq. (51 and the total number of EPs in the A-line [Sl. If the echo signal is not suff iciently narrowband, a bank of Gaussian narrow-bandpass filters are used to split the original echo signal into a number of narrowband daughter signals. The estimation method is then applied to each daughter signal. The final @ estimate is the weighted average of the daughter 6 estimates [lo]. that

From the definitions of u and v, and from Eqs. (31 and (5), the mean values of In(R) and ln(Rp) have similar relationships

Fig.

267

1

one notices with (I.

Probability density functions of ln(R/flI and ln(Rp/Y-$l. R is the echo envelope value, Rp is the envelope peak value and # is the mean power of the echo.

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Therefore, the above estimation procedure can also be applied to the envelope samples as well. The disadvantage of using envelope samples is that as the number of envelope samples is much larger than the number of EPs in an A-line, the computational load is significantly increased. Since the envelope samples are not mutually independent, the increase in the number of data points does not proportionately decrease the variance of the p estimate. This results in computational inefficiency. In the next section, we will find the maximum number of independent amplitude data in an A-line. III.

RANGE HESOLUTION OF THE PULSE-ECHO SYSTEM AND EXPECTED TO PEAK DISTANCE

In B-scan imaging, it is well known that the axial is determined by the time duration of the incident pulse resolution is determined by the ultrasound beam resolutions define the size of the resolution cell measurement. Since the resolution cell represents the unit, the maximum number of independent echo amplitude should be equal to the number of resolution cell along the length of the A-line divided by the range resolution

AZ = (cT)/2

pulse which

bandwidth of the signal, to is frequency. The range resolution

,

the is (7)

where c is the sound velocity. definition of pulse duration

is

For a decaying the time interval

pulse, between

the conventional the two points

Frequency

Time 2

incident function,

= PO e-4a2$(t-to)2cos12nfo(t-t~)),

where (r is a parameter related to the reference time, and f is the carrier determined by the pulse' duration T:

Fig.

PEAK

(range) resolution and the lateral width. These two in a pulse-echo smallest resolvable data in an A-line the A-line. i.e., of the system.

As a representative example, let us assume that the waveform can be described by the following modulated cosine is depicted on the left side of figure 2: P(t)

ENVELOPE

a. The time-domain waveform of the pulse signal in Eq. (6). b. The power density function of the pulse signal in a.

268

ATTENUATION

ESTIMATION

where the amplitude drops can be calculated as:

The range

T = d-iii-5

/(?ru)

resolution

is

AZ = c/(4.95

FROM ENVELOPE AND ENVELOPE PEAK

to 0.2 of the peak value

It

= So e

N = 2.52

u .

then

equal

T

to (9)

the expected

is a Gaussian

function

which

is

(10) number

of EPs per

second

is

[ill: (11)

envelope

AZ* = c/(5.04

(61,

(81

-(f-f012/20?

that

echo

Eq.

(rl .

has been shown

The expected

Using

.

The power spectrum of the pulse in Eq. (6) depicted on the right side of figure 2: S(f)

[131.

peak to peak distance

is therefore

u') .

(121 L

to AZ-. From Eqs. (91 and (121, we can see that AZ is approximately equal Although in this example a particular pulse waveform was used which has a Gaussian power spectrum, and a somewhat arbitrary definition of the pulse duration was adopted, the close relationship between the range resolution and the average peak to peak distance is clearly indicated by Eqs. (91 and (121. The above discussion shows that the number of EPs, which are mutually in an A-line is approximately equal to the maximum number of independent, independent echo amplitude data in that A-line. From the previous discussion (Eqs. (31 and (511, we also know that the SNR of the log envelope peak is higher than that of the envelope. For these reasons, we conclude that if a time domain, amplitude method is to be used, the attenuation can be most efficiently estimated from the envelope peaks. IV.

RECONSTRUCTION OF ENVELOPE FROM ENVELOPE PEAKS

The discussion in the last section indicates that on the average, each resolution cell produces one envelope peak. Consequently, two consecutive peaks, or two consecutive valleys in the echo envelope waveform, conveniently define the axial size of the resolution cell. Since the envelope samples within the range of the resolution cell are highly correlated, it is possible to use the envelope peak value to predict the values of the surrounding envelope samples. Figure 3 demonstrates the reconstruction of the envelope from EPs. Figure 3(a) shows a segment of echo signal (300 samples) obtained in a tissue-mimicking phantom (for graphic purposes, the rectified signal was plotted) and (bl shows the envelope waveform. From this envelope, 32 EPs were detected and plotted in (cl. From each peak, the left and right branches of the envelope waveform were reconstructed based on Eq. (61 after removing the cosine part. In using Eq. (61, Po and to were replaced by the magnitude and position of the peak, respectively, and the parameter Q was calculated from the distances between that peak and its neighboring peaks by using Eq. (121. The final envelope waveform reconstructed from these 32 EPs was plotted in cdl. Comparing the waveforms in (b) and cdl, some discrepancies are observable, especially around the valleys. The main

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C.

200

. . .

160

. .

.*

120

.

.

. .

l

.

l .

.

80

.

Z

-

. . . .

.

40 .

l

* . .

0 0

30

60

90

izo

180

240

300

0

3*

60

90

Sample Number

b.

0

120

180

Sample

Number

120

180

240

300

240

3n,,

d.

30

60

90

1EO

Sample Fig.

3

180

z‘ld

300

0

30

60

90

Sample

Number

Number

a. A segment of rectified echo signal obtained from a tissuemimicking phantom. The segment contains 300 echo samples. b. The envelope of the echo signal in a. c. 32 envelope peaks detected from the envelope in b. d. The reconstructed envelope from the envelope peaks in c.

is the distortion of the incident pulse source for these discrepancies in multiple reflections. waveform due to coherent interference were the main features of the original waveform Nevertheless, reconstructed. In the next section, it will be further demonstrated that as these discrepancies are far as attenuation estimation is concerned, negligible. V.

ATTENUATION ESTIMATION RECONSTRUCTED ENVELOPE

FROM ENVELOPE PEAKS,

ORIGINAL

ENVELOPE,

AND

In this section, we compare the results of attenuation estimation from the EPs, from the original envelope samples and from the reconstructed envelope samples. The detailed experimental setup and calibration procedure

270

ATTENUATION ESTIMATION

FROM ENVELOPE AND ENVELOPE PEAK

were described elsewhere [9,10]. In brief, a total of 20 A-line echo signals were obtained from a tissue-mimicking phantom which had a 8 value of 0.69 (dB/cm MHz) measured by the substitution method. The length of each A-line was 40 mm. The minimum distance between two A-lines was 10 mm. The transducer used had a 13 mm aperture and a 76 mm focal length. The center frequency of the incident pulse was 2.8 MHz and the s.d. was 0.74 MHz. A Sony/TEK 390AD digitizer was used to sample the echo signal at a sample rate of 20 MHz with a IO-bit resolution. The measurements were performed in a 50-gal water tank and the distance between the transducer and the front surface of the phantom was 60 mm. Since the bandwidth of the incident pulse the estimated B value would be was too wide for the narrowband assumption, biased if the estimation was made directly from the original echo signal [91. To improve the accuracy of /3 estimation, five digital Gaussian bandpass filters were used for split-spectrum processing 1101. The center frequencies of these filters were: 1.73, 2.27, 2.80, 3.33, and 3.86 MHz. The s.d. of each filter was 0.16 MHz. For each A-line, the echo signal was first bandpass-filtered to produce five daughter signals. For each daughter signal, the diffraction effects were compensated for by dividing each sample value by the calibration value for that frequency band at that range 1101, and the envelope and EPs were detected. From the EPs, a new envelope was reconstructed using the procedure described in the previous section. The linear regression method was then applied to the three sets of data: the original envelope samples, the detected EPs, and the reconstructed envelope samples. After averaging the 8 values estimated from the five daughter signals, three final 13 estimates were obtained in each A-line: B(EP), P(OEl, and 8(REl representing 8 estimated from the EP, the original envelope and the reconstructed envelope, respectively. The 60 6 estimates obtained from all 20 A-line signals are listed in table I. When a t-test was applied to the paired differences between any two columns of the data in the table, it was found that there was no significant difference (p > 0.2) between any two estimation methods. The sample means of B(EP1, P(OE1, and B(RE) as listed in table I are all very close to the 8 value (0.69 dB/cm MHz) measured using the substitution method. In the table, the sample s.d. of the three 8 estimates are also presented. As a reference, the theoretical s.d. for ,3(EP) was calculated as 0.068 (dB/cm MHz) using the formula in reference [9]. VI.

DISCUSSION AND CONCLUSION

This study shows that as a subset of echo envelope samples, the EPs have an unique property. That is, the EPs from an A-line represent the maximum independent amplitude data that one can obtain in the A-line. Consequently, the envelope peaks carry essentially all the useful amplitude information of the entire set of the envelope samples. This conclusion is supported by the two demonstrations in this study: 11 the entire set of the envelope samples can be approximately reconstructed from the EPs; 21 the B estimates obtained from the EPs, the original envelope samples and the reconstructed envelope samples have approximately the same value. Because of this property of the EPs, the tissue attenuation can be efficiently estimated from the EPs. In the phantom experiment conducted in this study, the ratio between the numbers of EPs and envelope samples is about 1:lO for the original echo signal. After narrow-bandpass filtering, is the ratio increased to 1:50 for the daughter signals. Therefore, by using EPs instead of envelope samples for attenuation estimation, the computational load is reduced significantly. the

As mentioned earlier, s.d. of the 8 estimate

due to the correlation obtained from the

271

between envelope envelope samples

samples, cannot be

PING HE

Table

I.

Attenuation coefficient estimates obtained from the envelope from the original envelope samples - 6(OEl, peaks - B(EP), and from the reconstructed envelope samples - @(BE) A-line number

B (EP)

B (OE) (dB/cm MHz1

,8 (BE)

1 2 3 4 5

0.7610 0.8771 0.6737 0.6665 0.7420

0.7579 0.8195 0.6776 0.6751 0.6740

0.6486 0.7689 0.6689 0.7036 0.7128

6 7 8 9 10

0.6796 0.5827 0.6523 0.7971 0.6369

0.7263 0.5851 0.6842 0.7832 0.5828

0.6874 0.5754 0.6510 0.7663 0.6640

11 12 13 14 15

0.6944 0.6481 0.6467 0.7098 0.6669

0.7122 0.6695 0.6702 0.6237 0.6840

0.6935 0.6417 0.6734 0.6954 0.7258

16 17 18 19 20

0.6460 0.8815 0.7912 0.6088 0.6867

0.6244 0.8323 0.7746 0.6801 0.6610

0.7656 0.8110 0.7496 0.7361 0.6271

x mean ** s.d

0.703

0.695

0.698

0.082

0.070

0.057

* The 6 value measured using ** The s.d. of 6(EP) predicted

formulated maximum smaller from the provide envelope argument.

the substitution by the theory

method is 0.068

is 0.69 dB/cm MHz dB/cm MHz.

directly. However, since the EPs in an A-line represent the EP has a number of independent amplitude data, and since the log the s.d. of the B estimate obtained s.d. than the log envelope, EPs (this s.d. can be predicted theoretically 19,101) should a lower bound for the s.d. of the 8 estimate obtained from the samples. The data presented in table I seem to support this

from the An unexpected result is that the sample s.d. of the B estimate and also EPs is 20 percent larger than its theoretically predicted value, larger than the s.d. of 8 either estimated from the original envelope samples or from the reconstructed envelope samples. This increase in s.d. #en the for ,tl(EP) seemed to be related to the split-spectrum processing. without estimation method was directly applied to the original echo signal bandpass filtering, the s.d of B(EP1 was 0.067 (dB/cm MHz1 and was indeed were not reported smaller than that of 8(OE) and 8(BEl (the full results here because the means of the three 6 estimates were all biased due to the wide bandwidth of the original echo signal [lo]). Two possible reasons for

272

ATTENUATION ESTIMATION

FROM ENVELOPE AND ENVELOPE PEAK

of B(EP) in the experiment may be given. Firstly, this increase in s.d. the power spectra of the bandpass filters are Gaussian functions, which can neither strictly not overlap each other nor completely fill in the area under the power spectrum of the original echo signal. As a result, some is lost useful information in spectral splitting. Secondly, after narrow-bandpass filtering, the number of EPs in each daughter signal it becomes quite small (about 20). For such a small number of random data, is possible to have a large deviation between the measured and expected properties of the sample statistics. To better understand this matter, further study is needed to investigate how the s.d. of the /3(EP) is affected by the number of filters, the bandwidth of the filters and the overlapping between the spectra of the filters.

ACKNOWLEJXMENT This work was supported in part by the Ohio Challenge Grant Program under grant 660764 .

Board

of

Regent

Research

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