Estimating Acoustic Attenuation from Reflected Ultrasound Signals ...

IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-32, NO. 1, FEBRUARY 1984

Estimating Acoustic Attenuation from Reflected Ultrasound Signals: Comparison of SpectralShift and Spectral-Difference Approaches

Abstract-The acoustic attenuation coefficient of soft biological tissue has been observed to have an increasing linear-with-frequency attenuation characteristic with a slope, denoted byp, that varies with the disease condition of the liver. Hence, it would be diagnostically useful to estimate the value of p from reflected ultrasound signals. Two approachesforestimating p areexamined:thespectral-shiftapproach, which estimates p from the downward shift experienced by the propagating pulse spectrum with penetration into the liver, and the spectraldifference approach, which estimatesp from the slopeof the log spectral differences. While the spectral-shift approach requires the propagating pulse to have a Gaussian-shaped spectrum,thespectral-difference methoddoesnotrequirea specificspectral form. A mathematical model is developed to simulate the random ultrasound signals reflected from the liver. The bias and variance properties of the p estimators are determined by using the simulated signals and compared as a function of the data window size. The results indicate that, while the accuracy of both approaches is equivalent forlarge data windows, the frequencyshift approach is more accurate than the spectraldifference approach for most practical cases.

from reflected ultrasound signals will be described, including a new correlation detection method which provides an estimator variance that is smaller thanthat of previouslyproposed methods.

11. ACOUSTICATTENUATIONO F LIVER The acoustic attenuation coefficient of liver tissue has been observed to increase as an approximately linear function of frequency [4]. For a section of liver, having athickness D, the power transfer function, denoted by IH( can be expressed as

f)12,

IH(f)I2 = e-4nPfo,

f>O

(1)

where f is the frequency in Hz and 0is the slope of the acoustic attenuation coefficient. Of diagnosticimportance,the value of 0 has been observed to correlate with the condition of liver tissue; normal livers have been observed to produce values around 0.5 dB/(cm . MHz), whileadvancedcirrhotics may produce values as high as 0.9 [ 5 ].

I. INTRODUCTION OR over the past 20 years, diagnostic ultrasound has provided useful images of the softtissues within the body in a 111. MEASURING FROM DETERMINISTICSIGNALS noninvasive and nontraumatic manner. A piezoelectric transTwo approaches for measuring the 0 value from ultrasound ducer transmits an acoustic pulse into the body which is con- pulses have beenproposed.Thefirst involves exploitinga tained in a narrow beam. Echos from tissue interfaces are defortuitous effect the transfer function of (1) has on a pulse tected by the same transducer and used to generate an image having a Gaussian-shapedenvelope. The second employs log of the structures within the body in a radar-type fashion [ 11. spectraldifferences, in amanner similar to standardsystem However, the appearance of an image produced by a liver havidentification techniques [6]. ing adiffuse disease is not unique to the particular disease To illustrate the first approach, consider an input pulse enpresent: massively fatty-infiltrated livers andthosewithadtering a section of liver of thickness D and an output pulse vancedcirrhosis,conditions that requiredifferenttherapies, emerging from the farend.Lettheinputpulse, p i ( t ) , be a typically produce images that are almost indistinguishable [ 2 ] . sinusoid modulated witha Gaussian envelopehaving a time Moreover, operator-adjustable controls can alter the appearance constant (standard deviation) T: of the image, potentially masking the true diagnostic features, which tendto be subtle for diffuse liver diseases. Forthe p . ( t ) = e-[(t/T)21/2 sin 2nht (2) above reasons, determining accurate quantitative measures of the condition of diffuse liver disease is highly desirable and an where f i will be called the center frequency. The pulse with active research area [3]. This paper will describe one acoustic T = ps and f i = 2.0 MHz, shownin Fig. l(a), is typicalof those produced bydiagnostictransducers [7]. Forsuitably parameter of the liver, the slope of the acoustic attenuation coefficient with frequency, denoted by p, which shows promise large fi, the pulse power spectrum,Pi(f ) , can be expressed as for diagnostic utility. Several methods to estimate the 0value p i ( f,-[2nT(f-fd12 )= ci , f> 0 (3)

F

Manuscript receivedApril 1,1983; revised August 27,1983.This work was supported by the National Science Foundation under Grant ECS-7919601 and an equipment grant from the Picker Corporation. Theauthor is withDepartment of ElectricalEngineering, Becton Center, Yale University, New Haven, CT 06520.

where ci is a constant. The normalized spectra are shown in Fig. 1 as solid curves. After the pulse propagates through the liver, we observethe output pulse p o ( t ) having aspectrum, Pdf), equal to

0096-3518/84/0200-0001$01.000 1984 IEEE

IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-32, NO. 1 , FEBRUARY 1984

2

Frequency (MHz)

Frequency (MHz)

Frequency (MHz)

(dl (e) (f) Fig. 1. Ideal time and spectral waveforms. Amplitude normhized time waveforms: (a) input pulse, (b) unit sampleresponseofminimumphase attenuation fiiter, having linear-with-frequency loss with slopeequal to 5 dB/MHz, and (c) output pulse, convolution of (a) and (b). (d) Normalized Gaussian shaped spectra (solid: input pulse; dotted: output pulse). (e) Normalized log power spectra (solid: input pulse; dotted: output pulse). (f) The loss is the log spectral difference divided by the propagation path D. The slope, p, is the parameter which varies with liver pathology.

proaches to estimate the value of from reflected signals. The first, denoted the spectral-shift approach [7], estimates the P value from the center frequency shift in reflected spectral estimates, corresponding to (7). The second, denoted the spectraldifference approach '[8], estimatesthefromthe slopein the differencebetween log spectralestimates,corresponding to (9). In thispaper, we comparetheperformanceofthe spectral-shift and spectral-difference approaches by analyzing simulated signals. A mathematical model will be developed to incorporate the transformations caused by the liver to produce reflected ultrasound signals. The behavior of the estimation procedures will be illustrated by considering the simplecase of signals reflected from near and far regions of a liver section 5 cm thick. This is analogous to the deterministic case consideredabove,and the 5 cm dimension is a typical dimension observed in the clinical application of these techniques [5].

Iv. PULSESHAPEUSEDINSIMULATIONS To allow for both spectral-shiftandspectral-differenceapproaches to be applied to the same data,theacoustic pulse used in the simulations will be a time-sampled version (1 0 MHz rate) of the Gaussian-shaped pulse given in (2): p . ( k ) = e - [ ( k - 1 1 ) / K ) 2 sin 2nFik, = 0,

(10)

k = 1, 2,. . . , 21

otherwise.

1 N O l 2 Pi(f).

(4) where, for the 10 MHz sampling rate, K = 3.33 and Fi = 0.2. Inmodelingreflected signals, it is helpful to considerthis After subytituting (1) and (3), we find pulse to be the reflection from a normally incident plane in-[2nT(f-fo)I2 terface. Because the propagating pulse is only slowly changed Po(f)=co e , f>O (5) bytheattenuation,its shape is approximatelyconstantfor where eo is a constant and small increments in range (0.5 cm). Such a small range increthemedium willbe denoted as f o = f i - PD/2nT2. (6) ment at the nearsurfaceof the near region, with the corresponding pulse sequence deIn words, the output spectrum maintains the same Gaussian noted by p,(k). A similarregion at the far surface (the far form as that of the input, but has been shifted to a lower cen- region) will have a different propagating pulse shape, p f ( k ) . ter frequency fo, as shown in Fig. 1. The value of can then The signals reflected fromthefar region musttravelback be determined from the downard shift in the center frequency: through the near region and then to the transducer to be detected. Hence, the far region reflection contains all the transP = 2nT2(f i - fo)/D. (7) formations that affected the near region reflection, plus the additional transformation produced by the tissue attenuation Thesecondapproach for measuringemployslogspectral in traversing the roundtrip distance between the near region differences.Afterexpressing (4) in logarithmic form and inand the far region. cluding (l), we have PoU) =

log P i ( f ) - log po(f) = 4nP.P,

f

> 0.

(8)

v. MATHEMATICALMODEL O F THE LIVER

Themathematicalmodelofthe liver employstwo digital That is, the log spectral difference is a linear function of frefilters: an attenuation filter to model its linear-with-frequency quency having a slope proportional, to 0, as shown in Fig. 1. attenuation coefficient, with slope 0, and a random-reflector The P value can then be determined from the slope of the log filter to model the interfaces within the liver encountered by spectral difference: the pulse.

A. Attenuation Filter Since onlythe powertransferfunctionof liver tissue has When analyzing ultrasound signals reflected from within the been measured, given in (l), an assumption must be made rebody, the observed signals are not deterministic, but become garding the phase properties.Recently,it wasargued that a random processesbecause the interfaces within the liver are medium having a linear-with-frequency attenuation coefficient canbe described by aminimum-phasefilter [9]. It was obrandomlyshapedand can bedescribed only inprobabilistic terms. The twodeterministicmethods above suggest ap- served experimentallythatsuchafilteraccuratelypredicted

3

KUC: ESTIMATING ACOUSTIC ATTENUATION

the pulse shape after it propagated through Plexiglas, which 1 has a linear-with-frequency attenuation. For a minimum-phase U U filter, it is well known that the log-magnitude and phase func.e 0 tions are Hilbert a transformpair. To applydiscreteFourier :E 0 E" transforms, thefunction given in (1) was used to generatealini ear log-magnitude function, equal to - 471@)1 FfI, for i = - 127 -1 -1 to 128, where FlZ8= 0.5, corresponding to 5 MHz. A discrete 0 3.2 6.4 0 3.2 6.4 Hilbert transform [ 101 was performed numerically to find the Time ( p s ) Time ( p s ) phase spectrum.The unit sampleresponse was then determined by taking the inverse discrete Fourier transform of the real and imaginary parts derived from the resulting magnitude and phase functions. The /3 value for ourmodel was set equalto 0.5 dB/(cm .MHz), the value observedfornormal livers [ 1I ] . Then, for 5 cm thick tissue, the roundtrip distance (D= 10 cm) traveled by reflected signals will result in an attenuation function having a Frequency (MHz) Frequency (MHz) Frequency (MHz) - 5 dB/MHz slope. The unit sample response corresponding to (4 (dl (e) this loss was calculated using the discrete Hilbert Transform, Fig. 2. Typical simulated data. Amplitude normalized time waveforms and is shownin Fig. l(b). The pulse reflected fromthefar were multiplied with 64-point Hamming window (sampled at 10 MHz region, p f ( k ) , shown in Fig. l(c), is then equal to the convorate); (a) near region, (b) far region, located 5 cm farther in range lutionof p,(k) and thisunitsampleresponse.Thespectra than near region. (c) Normalized power spectral estimates (solid: near region; dotted: far region) demonstrate distortion produced by shown in Fig. 1 were calculated from the pulse waveforms. 0

B. Random-Reflector Filter The reflectors located within the liver are composed of elements of the vascular system, whch are randomly shaped, oriented and distributed in range. For a sonic velocity equal to 0.156 cm/ps (average velocity observed for liver [4]), the signal reflected from a 0.5 cm thick region of liver has a duration equal to 6.4 ps, or 64 samples at the 10 MHz sampling rate. The effect of these reflectorson thepropagating acoustic pulse to produce the reflected signal will be modeled with a randomreflector filter. If r(k) is the signal reflected from a particular region, then

Q

random-reflector filter. (d) Normalized log spectral estimates (solid: vear region; dotted: far region). (e) Loss is estimated from the log power spectral difference divided by roundtrip distance between regions (10 cm). The upward trend of the loss with frequency is evident in the frequencyrange from approximately l to 3 MHz.

different /3 estimationproceduresfor M = 64. We will then consider the effect of varying the value of M . Typical simulated near region and far region windowed data segments are shown in Fig. 2(a) and 2(b). The spectra of the propagating pulses have been estimated by using the modified periodogram,thesquaredmagnitudeof the discreteFourier transform of the windowed data segment [ 131, using a 256point FFT algorithm.Thedistortedspectra and logspectral difference are typical of those observed when analyzing actual reflected data [ 1I] . When thereflected signal is modeledwiththerandomreflector filter, the periodogram can be shown to have approximately uncorrelated values at frequencies separated by twice the reciprocal of the reflected signal duration (the factor of 2 accountsfortheHammingwindow) [ l o ] . Further, ignoring the effects of windowing, the convolutional operation of the random-reflector filter allows one to view the periodogram as amultiplicativelydistorted version of the propagatingpulse spectrum [8] . This random multiplicative,distortioncanbe converted to an additive distortion by expressing the periodogram in logarithmic units. We will take advantage of this resulting signal-in-additive-noise model in our estimation procedures below.

where p ( k ) is the propagating pulse shape in the particular region (i.e., p n in the near region and p f in the far region) and h(k), for k = 1 to 64, is a white Gaussian sequence. The simulatedreflected signal isanonwhite Gaussian process, whose appearance is similar to observed signals. A similar model has beenproposed for modeling the reflectorsequenceobserved in geophysical prospecting [ 121 . Tosimulate signals reflected from sectionsof liver larger than 0.5 cm,thecontinualchangeinthepropagating pulse producedbythedistributedattenuationeffect of tissue is approximated by a piece-wise constantcharacteristic over a 0.5 cm range interval.Foreachincrement of 0.5 cm,the attenuation propagatingpulse is recursivelymodifiedbythe filter, having a slope equal to 0.25 dB/MHz. This attenuated VI. ESTIMATINGp FROM REFLECTEDSIGNALS pulse is then passed throughthe random-reflectorfilter to produce the signal reflected from that range interval. This was To estimate the value of p from reflected signals, we apply implemented by employing the FFT algorithm and an overlap- theapproachesdescribedabove in the deterministic pulse and-add technique [ IO] . analysis to the random reflectedsignals. To isolate an M-point sequence from a longer time series for short-time spectral analysis, we employ an M-point Hamming ' A. Spectral-Shift Estimators data window.Initially, we will observe the behaviorof the For the spectral-shift approach, the p estimate, denoted by


4

-

pSs, is determinedfromthedownwardshiftexperiencedin the far region spectral estimate relative region:

2rT2(Pn- P f )

to that from the near

in a given interval of time [ 171 . If N , is the number of zero crossings observed in M samples, then the zero-crossing estimator of the center frequency, denoted by Pz, is equal to

Pz = NZ/2M.

(1 4)

Fss =

Actingdirectly on the reflected data,thistechnique does not requireaspectraltransformation and,hence, is not afwhere P, is an estimate of the center frequency determined fected by the shape of the applied data window. Itcanbe from the near region data segment, pf is from the far region seen from (14) that the resolution for the zero-crossing techsegment, and D is theroundtripseparationbetweenthe renique is inversely related to the record length. gions. We will considerthreeestimators of thecenter frequency: 1) a correlation technique, that employs theGaussianB. Spectral-Difference Estimator form information, 2) the mean of the spectral estimate, and 3) Forthe spectral-differenceapproach, the 0 value is calcua zero-crossing count analysis. I ) Correlation Method: The situation here is similar to the lated from the slope of difference between the nearregion and radar problem: given a known waveform having an unknown far region log spectra. It can be shown that, while the additive locationinthepresence of independent additivenoise, we distortion to the log spectrum has a distribution that is skewed to negativevalues, the additive distortion to the log spectral would like to estimate its location. The maximum likelihood (ML) estimate of the location is obtained by correlation de- difference has zero mean and is approximately Gaussian 181. tection [ 141 . The detection procedure involves correlating the Then, the ML estimator of the slope of the spectral difference is that produced by the least-square-error solution. Hence, the observed noisy signal with a set of templates, each template estimatorofproducedbythespectraldifferencemethod, being the known signal waveform at a different location. The template that produces the largest correlation value provides denoted by Fsd, is equal to the ML estimate of the signal location. Here, the known wave[ P n ( F k ) - pf ( F k ) l [Fk - Fm 1 form is the parabolic shape of the propagating pulse log specFmin Fk Fmax Fsd = trum, which is corrupted by the additive noise introduced by 477D (Fi - Fm12 therandom-reflector filter.ThiscorrelationprocedureproFmin G Fk Q F m a duces the center-frequency estimate denoted by PC. If PC is (15) calculatzd from the near region (far region) data, then it becomes the valueof P, ($f)in (12). The set of templates is where P, is the near region periodogram, Pf is that from the generatedbycalculatingthe log spectrumofthe pulsesefar region, Fm is the mean frequency of the bandwidth [Fmi,, quence (lo), and translating the spectrum toa different center- F,,,] over which the slope is calculated, and D is twice the frequency location for each template. The resolution to whch thickness.The size of the usefulbandwidth will beseen to PC can be determined is equal to the frequency spacing in the be a function of data windowsize. set of templates, Here the resolution was set by a 256-point FFT. Each template was normalized by setting the maximum C. Initial Comparison to 40 dB and clamping all negative values to zero, producing a To compare the performance of the four p estimation methtemplatedurationapproximatelyequal to 3 MHz. The log ods, a set of 10 000 different reflected signals from the near spectral estimates were also normalized to 40 dB and clamped region and far region were generated, multiplied by a 64-point to zero before being applied to the correlator. Hammingwindow,andanalyzed.Histograms for eachsetof 2) Mean-Frequency Method: Since the center frequency of the estimates were calculated to indicate their respective disa Gaussian-shaped spectrum is also its mean frequency, several tributions, with the results shown in Fig. 3. researchers have proposed the mean of the spectral estimate It can be noted that all the distributions are approximately as the estimate of the center frequency[ 151 , [ 161 . The mean- Gaussian and centered about the true value of p, 0.50 dB/(cm. frequency estimator, denoted by F m , is given by MHz). Thestandarddeviation (SD) is shownforeach histogram. The minimum SD valuewas produced by the corre128 lationmethod, because itincorporatedtheprior knowledge Fkp(Fk) of the Gaussian pulse shape and satisfied the signal-in-additivek=O 128 F m = -~ noisemodel. Themean-frequencymethodproduced an SD p(Fk) value greater than twice that for the correlation method, and k=O the crude zero-crossing method was only slightly less accurate. where P(Fk) is the kth element of the 256-point periodogram The finiteresolutionpropertyofthecorrelationandzerofrom the near or far region. Since the spectral location is cal- crossing methods is evident in the discretenatureoftheir estimator was observed culated,there is no resolutionproblem similar tothat en- histograms.Thespectral-difference to be unbiased for bandwidths G2.0 MHz, forthe64-point countered with the correlation method. data window. (A p estimator will be called unbiasedif its 3) Zero-Crossing Method: One method to estimate the tenexpected value is within 2 percent of the true value.) For the ter frequency with a minimum of hardware is to count the number of zero crossings that the reflected signal experiences 2.0 MHz bandwidth,theaccuracyofthespectral-difference

c

c

-

-

~

KUC: ESTIMATING ACOUSTIC ATTENUATION

5

SD =0.28

Correlation

M

1

8

I

,

U

.

i

i

l

I

i

5

Frequency (MHz)

Beta estimate (dB/cm-MHd

Fig. 3. Histograms of four p estimators normalized to maximum count value. Each histogram represents the results of 10 000 individual 0 estimates. The mean values are all approximately 0.5 dB/cm . MHz. The standard deviation (SD) is given for each histogram.

estimator lies between the correlationandmean-frequency methods. We shall now examine the performance ofthe fourestimators as a functionof data window size. VII.EFFECT

OF

DATA WINDOW SIZE

Sincemostof theestimatorsoperateonthespectra, it would be instructive to examine the effectof the data window size on the spectral shapes. A set of 1000 near region reflections were multipliedby an M-pointHammingwindow, for M equal to 8, 16, 32, 64, 128, and 256. The average of the log spectralestimates was calculatedforeachwindowsize, with results shown in Fig. 4. As expected, the bias error decreases with increasing M. But, forM > 64, the averaged spectra begin to flatten out in the frequencyregions where the signal power is approximately 50 dB below the maximum. This flattening, due the spectral sidelobes in the Hamming window [ 101 , limits the maximumuseful bandwidth here to 3 MHz. To observe the effect of window size on the p estimators, a set of 10 000 near region and far region signal pairs were generated, multiplied by M-point windows, and analyzed.

A. Spectral-Shift Estimators Even though the spectral estimates were significantly biased for M < 64, the correlation and zero-crossing estimators were observed to be unbiasedforall values of M. Since the data window bias is deterministic, this information can be included in the shape of the templates in the correlation method. The template set for a particular window duration was formed by multiplying the pulse sequence by the window and calculating thespectrumofthe weighted pulse sequence. Inthis way,

Fig. 4. Average log spectral estimates as a function of data window size M. Each solid curverepresents average of 1,000 log power spectral estimates of the type shown in Fig. 2(d) as a solid curve. Unbiased power spectrum isshown in dotted curves for comparison.

the template shapes were matched to the expected shape of the periodogram, thus resulting in more accurate and unbiased center-frequency estimates. For the zero-crossing method, the expectedvalueofthecenter-frequencydifference was independent of segment size. Because of the severe spectral distortion for M = 8, the mean-frequency estimates of thecenter frequency were biased toward the mean of the frequency range used in the calculation,producing an expected 0value = 0.42.

B. Spectral-DifferenceApproach Forthismethod,the bias errorinthespectralestimator caused by small data windows will also effect the spectral difference. For M < 32, the difference between biased log spectraproducedflatterlosscurves, biasing the resulting p estiM = 64, an unbiased p estimate was mates toward zero. For obtained for bandwidths 128, this bandwidthincreased to 3 MHz, themaximum usable bandwidth determined above. C. Comparison as a Function o f Window Size The SD values of the unbiased 0 estimators are compared as a function of window size M in Fig. 5. It is noted that the SD values of all the p estimators decrease with increasingM . This decrease occurs because the number of independent points in the spectral estimate increases with the window size, resulting in areduced 0 estimatorvariance.Thelargerwindow sizes provided additional zero-crossing counts from which to estimate thecenter-frequencylocation,and also improved the resolution. For all values of M , the correlation method provides the most' accurate p estimatesbecauseitincorporates the Gaussian shape information. It is interesting to note that, for its simplicity in implementation, the zero-crossing method produces anSD value onlyslightlylarger thanthatforthe mean-frequencyestimator.Unbiasedspectral-difference 0 es-

6


- Correlation - Spectral-difference 0

- Mean-frequency - Zero-crossing

, - . 8

16 32

1

64

.

128 256

Window size (samples)

Fig. 5. Values of standard deviation for unbiased tion of data window size.

c) estimators

asa func-

timates wereobserved for M 2 64. The accuracy of the spectral-difference is equal to the correlation method for M 2 128. It appears that, in terms of the estimator accuracy, the knowledge thattheattenuation is linear-with-frequency is equivalent to the knowledge that the spectrum has a Gaussian shape. Also, for the 3 MHz bandwidth,thecorrelationand the spectral-differenceestimators use the same number of independentrandom variables in their respective /3 estimate calculation. VIII. CONCLUSION

A liver tissue model was developed to allow the comparison of fi estimators in a controlled fashion. The results indicate that for short data window sizes, M < 64, the frequency-shift fl estimators produce more accurate estimates than the spectraldifference estimator, which is plagued by bias errors. Of the spectral-shiftestimators, thecorrelationmethod,whichincorporates the Gaussian-shape information, is themostaccurate. The crude zero-crossing method, which is easy to implement, results in an accuracy only slightly worse than the mean-frequency method. For larger data windows, M Z 128, the accuracy of the frequency-shift correlation and spectraldifference methods are practically equivalent. Under this condition,thespectral-difference method has the advantage, since itdoes not require the Gaussian-shapeassumptionand it can be employed to determine nonlinear loss characteristics of tissue. REFERENCES [ 11 G. B. Devey and P. N. T. Wells, “Ultrasound in medical diagnosis,”

Scientific American, vol. 238, pp. 98-112, May 1978. [2] K.J.W. Taylor, F. S. Gorelick, A. T. Rosenfield, and C.A. Riely, “Ultrasonography of alcoholic liver disease: A histological correlation,” Radiology, vol. 141, pp. 157-162, 1981. [3] M. Linzer and S. Norton, “Ultrasonic tissue characterization,” Ann. Rev. Biophys. Bioeng., vol. 11, pp. 303-329, 1982.

[4] S. A. Goss, R. L. Johnston, and F. Dunn, “Comprehensive compilation of empirical ultrasonic properties ofmammalian tissues,” J. Acoust. Soc. Amer., vol. 64, pp. 423-457,1978. [ 51 R. Kuc, “Clinical application of an ultrasound attenuation coefficient estimation technique for liver pathology characterization,” IEEE Trans. Biomed. Eng., vol. BME-27, pp, 312-319, June 1980. [6] M. Schwartz and L. Shaw,SignalProcessing. New York: McGrawHill, 1975. [7] R. Kuc, M. Schwartz, and L. Von Micsky, “Parametric estimation of the acoustic attenuation coefficient slope for soft tissues,” in Proc. IEEE Ultrason. Symp., 1976, pp. 44-47. [8] R. Kuc and M. Schwartz,“Estimating the acoustic attenuation coefficientslope for liver from reflectedultrasound signals,” IEEE Trans. Sonics Ultrason., vol. SU-26,pp.,353-362,Sept. 1979. [9] R. Kuc, “Digital filtermodels for media having linearwith frequency loss characteristics,” J. Acoust. SOC. Amer., vol. 69, pp. 35-40, Jan. 1981. [ 101 A. Oppenheim and R. Schafer, DigitalSignalProcessing. Englewood Cliffs, NJ: Prentice-Hall, 1975. [ l l ] R. Kuc and K. J. W. Taylor,“Variationofacoustic attenuation coefficientslopeestimates for in vivo liver,” Ultrasound Medicine, Biol., vol. 8, no. 4, pp. 403-412, 1982. [12] J. M. Mendel and J. Kormylo, “New fast optimal white-noise estimators for deconvolution,” IEEE Trans. Geosci Electron., V O ~ .GE-15, 1977, pp. 32-41. [13] P.D. Welch, “The use of the FFT for the estimation of power spectra,” IEEE Trans. Audio Electroucoust., vol. AU-15, pp. 70-74, 1967. [14] P. M. Woodward, Probability and Information Theory with A p plications to Radar. Pergamon Press, 1953. [ 151 C. R. Crawford and A.C. Kak, “Multipath artifacts in ultrasonic transmissiontomography,” Furdue Univ., Lafayette, IN, Tech. Rep. TR-EE 81-43, Dec. 1981. [16] M. Fink, F. Hottier,andJ. F. Cardoso,“Ultrasonic signal processing for in vivo attenuation measurement: Short-time Fourier analysis,” UltrasonicImaging, vol. 5, pp. 117-135, Apr. 1983. [ 171 S. Flax, N. Pelc, G. Glover, F.Gutman,and M. McLachlan, “Spectral characterization and attenuation measurements in ultrasound,” Ultrasonic Imaging, vol. 5, pp. 95-116, Apr. 1983.

Roman Kuc (”75) received the B.S.E.E. degree from the Illinois Institute of Technology, Chicago, IL, in 1968 andthe Ph.D. degree in electrical engineering from ColumbiaUniversity, New York, NY, in 1977. From 1968 to 1975 he was a member of the Technical Staff with Bell Telephone Laboratories, Murray Hill, NJ, engaged in the design of audio recording instrumentation and in developing efficientdigitalspeechcoding techniques. From 1977 to 1979 he was a Research Associate in theDepartment ofElectrical Engineering at Columbia University and in the Radiology Department of St. Luke’s Hospital, New York, NY, where he employedstatistical estimation techniques to extract diagnostic information from reflected ultrasound signals. He is currently pursuing problems in ultrasonic tissue ChardCteriZatiOn and speech recognition as an Associate Professor with the Department of Electrical Engineering at Yale University, New Haven, CT.Heis currently Chairman of the Connecticut Chapter of the IEEE Engineering in Medicine and Biology Societyanda Past Chairman of the Instrumentation Section in the New York Academy of Sciences.