Mean-scatterer spacing estimates with spectral correlation
TomyVargheseand KevinD. Donohue
Departmentof ElectricalEngineering,Universityof Kentucky,
Lexington,Kentucky40503
(Received31 August1993;revised11 May 1994;accepted
20 July 1994)
An ultrasonicbackscattered
signal from material comprisedof quasiperiodicscatterersexhibit
redundancyover both its phase and magnitudespectra.This paper addressesthe problem of
estimatingmean-scattererspacingfrom the backscatteredultrasoundsignal using spectral
redundancy
characterized
by the spectralautocorrelation
(SAC) function.Mean-scatterer
spacing
estimates
arecompared
for techniques
thatusethecepstmmandthe SAC function.A-scanmodels
consistof a collectionof regularscatterers
with Gammadistributed
spacings
embedded
in diffuse
scatterers
with uniformdistributed
spacings.
The modelaccounts
for attenuation
by convolvingthe
frequencydependentscatteringcenterswith a time-varyingsystemresponse.Simulationresults
indicatethat SAC-basedestimatesconvergemore reliably over smalleramountsof data than
cepstrum-based
estimates.
A major reasonfor the performance
advantageis the use of phase
informationby the SAC function,while the cepstrumusesa phaseless
powerspectraldensitythat
is directlyaffectedby the systemresponse
andthe presence
of diffusescattering
(speckle).
An
exampleof estimatingthe mean-scatterer
spacingin liver tissuealsois presented.
PACS numbers:43.60.Cg, 43.80.Cs
INTRODUCTION
This work
models
soft tissue as an acoustic medium
containing
two typesof scatterers.
The firstcomponent
is the
Ultrasonicbackscattered
signalsfrom biologicaltissue diffusescatterers,which are uniformly distributedthroughcontaininformationregardingresolvablescattererstructures. out the tissueand are of a sufficientnumberto generatean
The information,however,is corruptedby echoesfrom dif-
echosignal
withcircular
Gaussian
statistics.
TM Thesecond
fuse(unresolvable)
scatterers.
• Thispaperexamines
charac- classof scatterershave regularityassociatedwith them and
terizingtissuemicrostructure
variationswith thespectralcorrelationcomponents
of thebackscattered
signal.The spectral
autocorrelation
(SAC) functionprovidesan estimateof the
mean-scatterer
spacingthatusesphaseinformationto reduce
the degradation
causedby the systemeffectsand the presenceof diffusescatterers.
2-7 It hasbeenshownthatincreasing regularityof the scattererspacinggivesrise to welldefinedlocal maximaamongthe correlationcomponents
in
theSACfunction.
4'5Thispaperdemonstrates
superior
performancefor spectralcorrelationmethodsover methodsthat
simplyusethepowerspectral
density(PSD).
The mean-scatterer
spacingparameterhas been pro-
posed
fortissue
characterization.
Fellingham
andSommer
8
usedmean-scatterer
spacingto differentiatebetweennormal
and cirrhoticliver. Their algorithmwas basedon the detectionof spectralpeaksin the PSD. The cepstrum
wasinitially
contributeto the specularechoesin the backscattered
ultrasoundsignal.This papermodelsthe spacings
betweenthese
regularscattererswith a Gamma distribution.The Gamma
distributionprovidesa flexiblemethodfor simulatingparametric regularityin the scattererspacing.Landini and
Verrazini
•3demonstrated
thatbothaverage
interval
spacing
and the varianceof the spacingcouldbe parametricallydescribedusinga Gammadistribution.The meanandvariance
of the scattererspacingswere estimatedfrom the periodicity
andattenuation
of thecepstralpeaks.The aim of thispaperis
to demonstrate
the advantages
of usingspectralredundancy
in estimatingthe mean-scatterer
spacing.This paper comparesthe estimationperformance
enhancement
attainedus-
inga spectral
redundancy
technique
(SAC),relativeto a cep-
stralanalysis
technique
using
thePSD.
m
Abriefdescription
ofthemean-scatterer
spacing
estima-
usedfortissue
characterization
byKuc9 in orderto reduce
tors are discussed in Sec. I. Section II discusses the simula-
the systemeffectsin the PSD. Performance
comparisons
between cepstral methods for detecting the mean scatterer
tion experimentandanalyzesthe relativemeritsof the techniquesusedin determiningthe scattererspacing.Application
spacing
m showed
thattheAR modelperforms
betterthan of these methods to in vivo data from liver tissue is discussed
FI•Tmethods
forsmaller
gatelengths.
Wagner
etal.n'•2pro- in Sec.IV. Finally,Sec.V summarizes
the significance
of the
posedthe use of three scattererfeaturesto classifytissue
architecture:
the mean-scatterer
spacing,theratio of specular
resultsobtainedto tissuecharacterization
usingultrasound.
to diffuse scatterer intensities, and the fractional standard
deviationin the specularscatterers.
Intensity(squaredenvelope) imageswere usedto estimatethe individualcontributionsof diffuseandregularscatteringto the overallintensity.
The scattererspacingwas estimatedfrom the power spectrum of the intensityby a weightedthresholding
procedure.
3504 d. Acoust.Soc. Am. 96 (6), December1994
I. THEORY
The tissuemodelconsistsof a sparsecollectionof randomlydistributed,
weakscattering
particlesthatinteractwith
0001-4966/94/96(6)/3504/12/$6.00
¸ 1994 AcousticalSocieb/ofAmerica 3504
theincidentpulseonlyonce(multiplescattering
is assumed window,andJ represents
the amountof overlapbetweenthe
to be negligible).The tissuescatterers
withinthe ultrasonic uth andu+lth tapereddatasegment
(i.e., (l-l/J)X100 is
beam field can be written as
x(t)= • a•(t-•,,)+ •] v,,(t-O.).
n=l
thepercent
overlap
between
segments
for l<-•-J<oo).
LetN•
(1)
n=l
wheret is a timeaxis(relatedto thedistartce
by thevelocity
of thepulse),No is thetotalnumberof diffusescatterers,
u,
denotesthe reflectivityof the nth diffusescatterer,0,•representsthedelayassociated
with thenth diffJsescattering
center, N s is the total numberof regularscatterers,
a, denotes
the reflectivityof the nth regularscatterer.
and r, represents
the delayassociated
with the nth regularscatteringcenter.
The attenuation
of the propagating
ul'.rasound
pulsedependson the scatteringandabsorption
propertiesof the tissue.The impulseresponse
dueto the systemandpropagation
path(toandfromthescatterer
field),is represented
by a time
varyingsystemresponsefunctionh(t,r). The impulseresponsechangeswith time due to the frequencydependent
attenuationof the pulseas it propagatesthroughthe tissue.
The A-scan corresponding
to the scattere]field can now be
written as
y(t)=•h(t,r)
• a,(r-r,)
(Ns
+•
v.(T- 0•) clr,
(2)
where •, the variableof integration,represents
the system
responseaxis at a given time t.
For nonstationary
signalanalysis,the periodogram
and
the SAC functionsare computedoversmallwindowedsegmentsof the A-scan with a data taperingWelch window,
usingtheWelch-Bartlett
technique.
is Detectability
of the
mean-scatterer
spacingrequiresthe presenceof at leasttwo
regularscatterers
within the window (i.e., T>A, where A is
the mean-regularscattererspacingand T is the window
andN o denotethe numberof regularanddiffusescatterers
that contributeto the A-scan segmentover the intervalT.
TheFouriertransform
of Eq. (3) withrespect
to r, at a given
time t, can be written as
Yr f;t-u
=H(f;t) • A,(f )e
+•
V.(f )e
(4)
n=l
wheref is frequency,H(f ) andY(f ) are the Fouriertrans-
formsof h(r) andy(r), respectively,
andA•(f ) andV•(f )
denotethe frequencydependentscatteringstrengthof scat-
terera,(r) and v•(r), respectively.
A. The spectral autocorrelation function
The phaseand magnitudespectrumof backscattered
energy from regularlyspacedscatterers
containsinformation
relatedto thespacingmeanandvariance.While thePSD can
be usedto characterize
thescatterer
spacing,it reliesonlyon
the magnitudespectrum,which is directly affectedby the
systemresponse
anddiffusescattered
energy.The SAC funi:tion,however,usesinformationin thephasespectrum,
which
resultsin local maximain the bifrequencyplanewhere the
mean contributionsof the diffuse scattererenergyis zero
(thisis not the casefor the PSD). As a result,the SAC function providesa more robustestimatethan the PSD for scatterer spacingin the presenceof diffusescattering.These
points are illustratedin this sectionby definingthe SAC
functionandderivinga relationship
betweenthe regularand
diffuse scatterer distributions and local maxima in the SAC
function.
TheSACfunction
2 isdefined
overa bJfrequency
plane
by
Sr(f•,f2;t) =E[Yr(f• ;t)¾•(f2;t)]
length).Adjacentoverlappingdata segmmtsare averaged
(Welch-Bartlett
technique
•8)to obtainthePSDandSAC
functionestimates
from theA scan.A singlewindowedsegment is denotedby
L+I
u
-L2
Yr fi;t-u
where Yr(') is the Fouriertransformof the windowed
A-scansegment,
andY•-( ß ) is itscomplex
conjugate.
The
= y r-t+uy wr(r),-•'•<r•<:•
-,
0,
(3)
elsewhere
whereYr(') represents
a tapereddatase,;mentcenteredat
t-uTIJ of finite durationT, wt(-) represents
the tapered
3505 J. Acoust.Soc.Am.,Vol.96, No. 6, December1994
approximation
to the expectedvalueis denotedby the average of L +1 local segments.The relationshipbetweenthe
SAC functionandthe scattererconfigurations
is seenby substitutingin Yr(') fromEq. (4), andassuming
theattenuation
of the pulseover the regionof interest,definedby t andL is
negligible
T. Vargheseand K. D. Donohue:
Meanscattererspacingestimates 3505
[ N;
Sr(f•,f2;t)=H(f•;t)H*(f2;t)XE[
• • An(fOA•*(f2)e
-j2•rff•n-f2r=)+
• • An(f•)Vm*(f2)
e-j2•r(lh•n-f2ø
[n=l ra=l
n=l ra=l
ND NS
ND ND
Vn(fl )Vm*
(f 2)e-J2*r(11Onf20m
)
n=l
m=l
n=l
Underthe assumption
thatthe diffusescatterer
sizesand
positions
areuncorrelated
with theregularscatterers,
theexpectedvalueof thesummation
cross-product
termsbetween
thediffuseandregularscatterers
in Eq. (6) reduces
to zero
Sr(fl,f2;t)
-- H(f, ;t)H* (f2 ;t)
X E
The ith scattererpositionis represented
by i equalintervals
fromthebeginning
of thesegment.
Thischangeof variables
givesa goodapproximation
to the actualspacingwhen the
scattererspacingvarianceis small. For the extreme case
wherethescatterer
spacing
is constant,
Ai = A, Eq. (8) canbe
expressedas:
ST(fl,f2;t)
=H(f i ;t)H*(f • ;t)
• A•(fl)A•*(f2)e
-12•(f•-f:•)
n=l
m=l
E
V•(f•)Vm*
(f2)e-j2•r(1,
0n-I20,.)
n=l
(6)
ra=l
Z An(fl)Am*(f2)
e-j2*ra(fln-f2m)
ra=l
(7)
m=l
+ • E[Vn(f,)Vn*(f2)]B(f•-f2)
ß
Sincethe diffusescatterersizesandpositionsareuncorrelated with each other, cross-scattererterms in the second
(10)
n=l
summationterm becomezero {i.e., g[v•(fOVm*(f2)
X e-J2•r(1•ø•-12ø'"))=O,
tn-•n].In addition,
if thediffuse
The regularscatterer
summation
in Eq. (10) indicates
whenf•n-f2rn equalsintegermultiples
of 1/Afor all tn and
interest,
thetermswheref•f2 arealsouncorrelated?n, the individualtermsin the summationare in phaseand
localmaximaoccurat thesepoints.The maximaoccurboth
Thus,Eq. (7) canbe rewrittenas
alongthePSD axis,wheref• =f2 = k/A, andthroughout
the
rest of the SAC function,where f•-f2=k/A
(for k
Sr(f•,f2;t)
=+-1,2,3;...). When the regularscattererspacingsare randomly
(Gamma)distributed,
similarlocal maximaoccur.
=H(f • ;t)H*(f 2;t)
These maxima, however,are less pronouncedfor greater
scatterer
process
is wide-sense
stationary
overthe regionof
variance
in thescatterer
spacings.
6
In the cepstralapproachfor estimatingmean-scatterer
spacing,
thesystemeffectis reduced
fromthescatterer
componentby takingthelogarithm
of Eq. (10) (to change
multiplicative
relationships
to additive),
andthenapplying
high-
E[•=i
• A,(fl)A•*(f2)e
-'2'•(f'•-f2•)
m=l
E[ V,(f,) V•*(f2)16(L - f2)
(8)
passfiltering?Thediffusecomponent,
however,
is not
separated
fromtheregularcomponent
in thisprocess,
since
theyremaintogetherin the argument
of the logarithmterm.
regionof the SAC function
Sincethefrequencycomponents
of the diffusescatterer On the otherhand,the spectral
wheref• v•f2 containsno contribution
from the diffusescatechosignalare uncorrelated,
they contributeonly to the diagonalof theSAC function,whichis thePSD of thediffuse terers.Therefore,when the SAC functionis computedfrom
component.
Thus,theexpected
valueof the diffusescatterer the data, the off-diagonaltermsresultin higherregular-tosignalratiothanthe diagonalterms(PSD
components
do not contribute
to the off-diagonalcompo- diffusescatterer
nents of the SAC function.
components)
aftersufficientaveraging.
A normalization
for the SAC functionprovidesa way of
To createa variablerelatedto the scattererspacing,the
minimizingthe amplitudevariationsdue the systemand
followingsubstitution
is made
propagation
effects.The normalization
is givenby
•'t
A•=
w-.
l
3506 d.Acoust.Soc.Am.,Vol.96, No.6, December1994
(9)
P(f•,f2)=S(L,f2)/x/S(f•
,f•)S(f2,f2),
(11)
T. Varghese
andK. D. Donohue:
Meanscatterer
spacingestimates 3506
whereP(fl ,f2) denotesthe correlationcoefficient.Note the
resultingdiagonalof the normalizedSAC functionis unity.
Thus,all the informationregardingscatten:rspacingis in the
off-diagonalcomponents
of the normalizedSAC function,
and amplitudevariationsin the spectrumdue to systemseffects are eliminated.
Normalization
wasnotusedpreviously
by Varghese
and
tionshipbetweenthe PSD and thescatterers
is obtainedfrom
Eq. (8) by substituting
f• =f2=f, to obtain
Pr(f ) = H(I; t)H* (f;t)
x e
•, A•(f)A•*(f)e
Donohue,
4 wherethesmallest
scatterer
spacing
wasdetermined graphicallyby the spectralcorrelationpeak farthest
away from the PSD diagonal.In the caseof very regular
tissueotherspectralpeaksoccurcloserto the PSD diagonal
that correspond
to largerscattererspacing'a
harmonicallyrelatedto the smallerspacing.Due to the amplitudevariation
in the spectrumfrom the systemresponse,
spectralcorrelationpeakscloserto thePSD diagonalhavelargermagnitudes
as a resultof their corresponding
spectralzomponents
being
closerto the centerof the systembandwidth.Sincethe normalizationusedin Eq. (11) dividesthe spectralcorrelation
peak magnitudeby the squareroot of the productof the
magnitudes
of its corresponding
components
alongthePSD,
the magnitudevariationsfrom the systemare effectivelyremoved,and correlationpeakswhich remain are due to the
coherencein the phaseinformation.
It shouldbe notedthat the magnitude:
variationsin the
PSD diagonaldue to the scatterspacingsare alsoeliminated
by this normalization.
The phaseinformation,however,is
sufficientto generatelocal maximain the bifrequency
plane
for detectingthescatterer
spacing.Thus,normalization
of the
SAC functionallowsfor consistent
detections
of thespectral
correlation
peaklocationirrespective
of thedistancefromthe
PSD.A simplenumericalalgorithmcanbe usedto detectthe
maximumpeakin the off-diagonalcomponents
(i.e., visual
inspection
of the SAC functionis not necessary).
Sincenormalization
is performedin the frequencydomain and the systemis band limited, useful information
aboutthe scatterer
spacingis determined
by thepulsewidth.
To avoidamplifyinglow signal-to-noise
(SNR) spectralregions,normalizationis performedonly in the regionwhere
significantenergyis presentin the PSD (typicallythe 3-6
dB regionof the receivedsignalspectrum).
Spuriouspeaks
will occurif low SNR regionsare includedin the normalizationwindow(i.e.,usuallybelowthe 10-dEbandwidth
of the
received
pulse).Thesepeakswill resultin scatterer
spacings
+
e[v(l )v2(I )] .
.3)
Usingthechangeof variablesin Eq. (9), with a constant
scatterer
spacingAi=A, Eq. (13) reduces
to
Pr(f )=H([;t)H*(f;t)
x e
)e
+
)] .
04)
We canseef[omEq. (14), for theregularscatterers,
that
iff is a multipleof l/A, all thesummation
te•s arein ph•e
andlocalmaximaoccurin the PSD. Theseperiodicpeaksin
thePSD havebeenuseddirectlyfor dete•ining the average
scatterer
spacing
inveryregular
tissue.
8-m'13
C. The cepstrum
Periodicityin the PSD produces
a dominantpeakin the
cepstrum,which is usedto determinethe scattererspacing.
The cepstrumis the mostcommonlyusedmethodto detect
the scattererspacing.The cepslrumis definedas the Fourier
transformof the logarithmof thePSD. The logarithmof Eq.
(14) resultsin
log PT(f ) = 1og[H(f;t)H*(f;t)]
+log E •] •] An(
f )A•*(f)e j2•a/(n-m)
rt=l
m--I
being detectednear or beyondthe resolutionlimit of the
systemwhen no regularscattererspacingis presentin the
+
tissue.
B. The power spectrum
Classicalmethodsof estimatingthepowerspectrum
are
basedon the periodogram.
The PSD is definedby
P•(f )=E[IYr(f;t)l 2]
--L+I •" Yrf;t-u• ,
u=-L
2
(15)
•pstral processing
assumes
slowvariationsfor the systemresponse
[firstlog te• in Eq. (15)] with respeato the
variations
in theregularscatterer
spacing[second
logte• in
Eq. (15)]. As a result,the log of the systemresponse
is reduced(usinga high-pass
filter)andtheinverseFouriertransform is taken.The scattererspacingperiodicitynow m•irests itself as pea• at integermultiplesof •, with the
dominantpeakat 5. The dominantpeakis usedto determine
themean-scatterer
spacing
intissue.
løObse•ealsofromEq.
(12) (15) thatthe PSD resultingfrom the diffusescatterersadds
whereL + 1 is the numberof segments
axeraged.The rela3507 J. Acoust.Soc.Am.,Vol.96, No. 6, December1994
E[V.(f)V2(f)] .
directlyto the regularscatterer
component
withinthe argumentof the logarithm.As a result,simplespectralfiltering
T. Vargheseand K. D. Donohue:Meanscattererspacingestimates 3507
will notdirectlyreducethisdegradation
of the cepstrum
for
distribution
for regularscattering.
Scattererspacings
were
simulatedfrom caseswith a very irregularspacing(large
The mean-scatterer
spacingis computedfrom the loca- standard
deviation),
to cases
withalmostdeterministic
spaction (At) of the dominantpeak in the cepstrumestimated ings (corresponding
to Gammaordersfrom 50-1000, refrom the PSD. For the SAC function, the location of the
spectively).
Thediffusecomponent
wasmodeled
usinga unidominantpeakamongthe off-diagonal
spectralcomponents form distributionfor scatterers(size 10 /zm) with the
at (f•,f2),
correspondsto a frequency difference scattererdensitychosento simulateGaussianstatisticsfor
(Af=f•-f2)
that is usedto computethe meanscatterer thescatterer
number(about15-20 scatterers
perresolution
spacing
cell usinga Poisson
distribution).
The absorption
coefficient
regularscatterers.
V
waschosen
as0.94-dB
cm-] (corresponds
to measured
val-
1
d=2A-•
=5 VAt,
(16)
uesfor livertissue)anda diffuse-differential
scattering
coef-
ficient
of9X10-4 cm-1 sr-1 forbackscattering
atanangle
of
where V denotesthe velocityof propagationof the ultra180ø,measured
at 3 MHz?
soundpulse.
The pulseparameters
are usedto obtainthe initial sysThe finitebandwidth,of the interrogating
pulseimposes temresponse
h(t) whichis modifiedasthepulsepropagates
limitationson the averagescattererspacingwhich can be
throughthe microstructure
due to the frequencydependent
resolvedby theimagingsystem.The mean-scatterer
spacing attenuation.The interrogatingpulse was simulatedwith a
is resolved,if the correlation
lengthof the systemimpulse centerfrequencyof 3.5 MHz and a Gaussianshapedenveresponse
is shorterthanthe spacingbetweenthe individual lope.The propagation
velocityof thepulsethroughbiologiscatterers.For scatterersnot resolvedby the system,local
cal tissuewas set at 1540 m/s.The bandwidthof the pulse
maximacorresponding
to scattererspacingis not observed.
shouldbe at leastI/A, in orderto observespectral
peaksdue
Simulation
results for the case of unresolvable scatterers
haveshownrelationships
betweenthe averagespectralcorrelation(averageof all correlation
termmagnitudes
in off-
diagonal
region)andthe unresolvable
scatterer
density.
22
When no regularityis presentin scatterers,the dominant
peaktendsto occurnearthefirstoff-diagonal
(dueto leakage
from the PSD).
This sectionpresented
differentmethodsusedto detect
the scattererspacing.Cepstraltechniques
requirehigh-pass
filteringto reducethesystemeffectfromthetissuesignature,
whichinvolvessettinga cutofffrequencyfor the filter.The
cutoff frequencycanbe set to passthe minimumdetectable
scatterer
spacing(i.e., on the orderof the pulsewidth) or
higher,if the rangeof scattererspacingof interestis known.
The SAC functiondoesnot requirehigh-passfiltering,however,a frequencywindowmustbe appliedbeforenormalization to ensureinformationis takenonly from the spectral
regionof the imagingsystem.The estimationtechniqueusing the SAC functiondetectsthe locationof the dominant
off-diagonalcomponent,and usesthis informationto computethescatterer
spacing.
Simulation
resultspresented
in the
next sectionshow the performanceimprovementobtained
usingthe SAC functionto detectperiodicitiesin thepresence
of diffuse backscatterand varying regularityin the regular
scatterers.
to the distribution of the scatterers in the microstructure. The
simulatedpulsehad a 3-dB bandwidthof 1.9 MHz, which
limited the minimumdetectablescattererspacingto 0.41
mm.
The regularscatteringcomponentwas simulatedwith a
scatterer
spacingof 0.51 mm.A gatelength(T) of 4 mm or
128 data pointswas used to form the PSD and the SAC
functionestimates.The mean-scatterer
spacingwas estimatedfrom the FFF cepstrum.The cepstrumwas filtered
usinga high-pass
filter with cutofffrequency1.5 MHz to
reducesystemeffects.The scattererspacingwas also estimatedfrom the locationof the largestoff-diagonalpeak of
the SAC function.The simulationwasperformedfor regular
scattererswith different spacing variancesand different
strengthsof the diffusecomponent.
Whenestimatingthe SAC function,with a hoppingwindow, a phasecorrectionfactoris helpfulfor correctingthe
phasedifferencebetweenregularscattererpositionsandthe
centerof the window.For regularlyspacedscatterersthe
relativephasedifferencebetweenthe centerof the window
and the scattererlocationdegradethe coherentsumsin the
off-diagonalregionsof the SAC function.When consecutive
SAC functionsare averaged,the phasedifferencebetween
the centerof the window and the scattererlocationimpedes
the convergenceof the SAC function. This error makes a
significant
differencewhenonlya few regularscatterers
exist
over the data segmentwindow. When the regularscatterers
are moredensewithin the processing
window,the phasedifA-scans with known tissue and signal parametersare
ference
between
a
regular
scatterer
and
the centerof the winsimulatedusingEqs. (2) and (4). The simulatorparameters
dow
is
small,
and
therefore
does
not
significantly
hinderthe
were chosenwithin the rangeof valuesreportedin literature
convergence.
from experimentalresearchwith differenttypesof tissuein
The phasecorrectionfactor for each tapereddata segvitro?-•7 Thesizeof thequasiperiodic
(regular)
scatterers
II. SIMULATION
was constant for all the simulations.
Scatterer sizes influence
the frequencydependence
of the pulsescattering
andattenuationfor the propagatingultrasonicpulse.The scatterersize
for the regularcomponentof the scatteringwas chosenas 80
ttm. The Gamma functionwas usedto describethe scatterer
3508
d. Acoust. Soc. Am., Vol. 96, No. 6, December 1994
ment aligns the regular scattererswith the centerof the window. The properphaseshift is determinedby the distanceof
any regular scattererlocation to the center of the window.
With the phasecorrectionfactorincluded,Eq. (14) can be
written
as
T. Varghese and K. D. Donohue: Mean scattererspacingestimates 3508
8.3
9.6
7
d.5
5.9
7.7
5.9
Fr equemcy
L.
6.4
.5
9.6
8.3
Frequency
(MHz)
(MHz)
0.14
0.15
0.12
0.12
5
0.1
• 0.1
•
• 0.08
0.075
•0.06
'J
cJ
0.05
O. 04
0.025
O. 02
0.5
1.5
2
0.5
Distance
1
1.5
2
Distance
(a)
(b)
FIG.1.Spectral
autocorrelation
function,
and
FFTcepstrum
forscatterers
with
a0.51ram-mean-scatterer
,;pacing
with
tr=3.15%.
(a)Only
regular
scatterers,
(b) witha +3-dB-diffuse
component
relative
to theregular
scatterers.
Sampleplotsof theunnormalized
SACfunction
andthe
cepstrum
arepresented
in Fig. 1 for regularscatterers
with
pr(
f;t_u
•_)
=yT(
f;t_u
•_)eJ2nœ(rt•
' T2),(17)
cr=3.15% (standarddeviationrelativeto the mean-scatterer
Figure1(a)shows
a contour
plotof theSACfuncwhere
r•srepresents
thelocation
of anyregular
scatterer,spacing).
tionandthecepstrum
for regularscatterers
only,whileFig.
l(b) showstheeffectof +3 dB of thediffusecomponent
in
additionto the regularscatterers.
The contourplots are
location
of a regular
scatterer,
which
isused
toobtain
thresholded
to emphasize
thelocalmaxima.Thelightintenarethosethatexceeded
thethreshold
to indicate
Thelinearphase
correction
hctordoesnotaffectthelocation sityregions
of theoff-diagonal
peaks,
it merelyalignsthewindowwith a local maxima.The 0.51-mm-regularscattererspacingis
thescatterer
structure,
andhelpsin thecoherent
phaseaddi- easilyidentified
in thecepstrum
Fig. l(a) and(b).Notethe
tionduringtheaveraging
of consecutive
overlapped
SAC presence
of a periodicity
at 1 mm,whichis harmonically
andPrO) is thephasecorrected
Fouriertransform
of the
windowedA-scansegment(for practicalapplicationthe
maximumvaluewithina datasegment
is assumed
to be the
functions.
Thephase
corrected
SACfunction
using
Eqs.(17)
relatedto the 0.5-mm scattererspacing.The diffusecompo-
and (10) can be written as
nentdirectly
addsto thePSD,aswasdiscussed
in Eq.(15),
Sr(f•,f2;t)-L
+1 •
u=-L
2
andcausesa broadening
of the cepstralpeaks,and an increase
in thenumber
of spurious
peaksasseenin Fig. 1(b).
Prfl;t-u
For theSAC functionthemean-scatterer
spacingis estimated
(18)
3509 d.Acoust.
Soc.Am.,Vol.96,No.6, December
1994
from the locationof the off-diagonal
peakat (7.75, 6.24)
MHz, whichcorresponds
to Af=l.51 MHz. Substitute
this
valueintoEq. (16) to obtaina scatterer
spacing
of 0.51mm.
Thespectral
correlation
peakfor theSACfunction
withthe
T.Varghese
andK.D. Donohue:
Meanscatterer
spacing
estimates3509
50-
0.3
0
Number
Or'Sepnents
5
10
15
20
25
(a)
50-
0.9!
,,qo."i
•0.40.3
5
0
Number
OfSegments
tO
15
20
25
Nmnl•'O•Selmmts
(b)
ø.91
0.6
•0.5
o
0.3
o
5
io
15
20
'e•' ' '
(c)
FIG.2. Mean-scatterer
spacing
andthecoefficient
of variation
forregular
scatterers
withspacing
0.51mm.(a)Standard
deviation
0.016mm(3.15%),
(b)
standarddeviation0.051 ram (9.99%), (c) standarddeviation0.072 mm (14.14%).
diffusecomponent
is at (4.9, 6.4) MHz, whichcorresponds variablesfor the regularscattererspacing.The diffusescattererswere uniformlydistributedthroughoutthe microstructo Af= 1.50 MHz, givinga scatterer
spacingof 0.51 mm.
turewith uniformlydistributedscattererstrengths.
The PSD
Monte Carlo simulationswere usedto comparethe perand
the
SAC
functions
were
computed
using
a
hopping
winformancebetweenthe SAC and FFT cepstrumestimatesof
dow
with
4-mm
length,
and
a
50%
(J--2)
overlap
of
the
the mean-scatterer
spacing.Convergence
plotsfor the scatalongthe A-scan.The spectrum
at
terer spacingestimateused 25 independent
simulated tapereddatasegments
eachwindowlocationwas averagedwith the previousspecA-scans with the same tissueand signal parameters.The
A-scanswere generatedwith gammadistributedrandom tra to obtain an estimate for the PSD and the SAC function.
3510 J. Acoust.Soc.Am.,Vol.96, No. 6, December1994
T. Vargheseand K. D. Donohue:Meanscattererspacingestimates 3510
Boththecomputed
meanscatterer
spacing;
(d) andthecoefficientof variation(CV) of the spacingestimates
are presentedin Figs.2 and3 for increasing
standard
deviations
in
the regularscattererspacingdistribution.
The coefficientof
variationwascomputed
astheratioof the standard
deviation
of the estimatesto the mean estimatecomputedover the
results from the 25 simulated A-scans.
Figure2 presents
themean-scatterer
spacingandthe CV
for onlyquasiperiodic
scatterers
(no diffusescattering),
as a
functionof the numberof A-scansegments
used.The plots
show that the estimatesobtainedusing ':he SAC function
converges
fasterthanthe cepstraltechniqt•e
andhasa lower
CV. In Fig. 2(a) the SAC functionmeart-scatterer
spacing
estimate(with0.=3.15%)converges
within6 averages
(corresponding
to 14.4 mm or nearlyfour full data segments),
while thecepstraltechnique
converges
in about14 averages
with a smallbias(i.e., lessthan10%). Th.: CV alsoreduces
to 0.0% for the SAC functionwithin 6 averages.
A major
sourceof deterioration
in the performance
of the cepstral
technique
with no noise,is dueto the presence
of the harmonicat i mm, whosepeakdominatesfor someA scans.
With an increase
in thestandard
deviationof theregular
scattererspacing,the resultsin Fig. 2(b) and (c) show a
this casereducesbecausethe spuriouspeaksnearthe filter
cutofffrequencyare consistently
detected.This is not an indicationof goodperformance
sincethe estimationis convergingon the incorrectvalue.The diffusecomponent
thus
introducesspuriouspeaksin the cepstrumand reducesthe
contribution
of theregularscatterers
to thetruecepstral
peak.
The cepstraltechnique
in thepresence
of thediffusecomponent,converges
to the truescatterer
spacingonlyfor regular
scatterers with •r=3.15%.
Figure 4 illustratesthe range and effectivenessof the
SAC functionin estimatingthe mean-regularscattererspacing from 0.5 to 1.35mm, with a standarddeviationof 9.99%,
and -3 dB of the diffusecomponent
to regularcomponent
strength.
The SAC functionconverges
to thetruevalueof the
mean-scatterer
spacingsfor all the spacingsin Fig. 4, with
the estimatesconvergingslower as the correlationpeaks
form closerto the PSD. The slowerconvergence
for larger
scattererspacingsresultsfrom greatererrorsin the phase
correctionfactors.Since noiseis includedin thesescans,the
maximumvaluein a givenwindowwill sometimes
be a peak
from the diffusescatterers,
andas mentionedpreviously,this
error more significantlyaffectsthe caseswhen only a few
scatterers
are in theprocessing
window.The CV for the SAC
slower
convergence
to them6an-scatterer
spacing.
In Fig.
function estimates remains within 5%-10%
2(b)theSACfunction
CV iswithin5%after14averages,
for
shown,while the cepstralestimateshavea CV>40%. These
resultsalongwith the resultsin Fig. 3, showthe dependence
of the cepstraltechniqueson the high-passfilter cutoff fre-
d=0.50 mm with 0':=9.99%.In Fig. 2(c) the SAC function
CV is within 10% with d=0.48
mm for cr=14.14%.
In both
for all the cases
Fig.2(b)and(c) theCV for thecepstral
technique
is at about
quency.
40%, with the mean-scatterer
spacingrangingfrom 0.55 to
0.65 mm.Resultsin Fig. 2(b) and(c) indicatetheSAC functionestimate
is moreaccurate
andreliablethanthecepstrum.
Estimationof the scattererspacingin the presenceof
strongdiffusescattering
is presented
in Fig. 3 with the ratio
of the diffuseto regularscatterer
strength:it +3 dB (i.e., +3
dB of thediffusecomponent
is addedto tlheregularcomponentof theA scan).Figure3(a) to (c) showtheeffectof the
changein the microstructure
regularityia the presenceof
diffusescattering.The SAC functionis able to resolvethe
regularscattererspacingevenin the presence
of the diffuse
component
withnosignificant
degradation
in performance
of
This sectionillustratesthe estimationof the regularscattererspacingin the presenceof the diffusecomponent
and
variationin the regularityof the scattererdistribution.The
SAC functionestimates
performbetterthanthe cepstralestimatesin the presence
of the diffusescattering
component.
The SAC functionincludesphasedifferencesbetweenthe
differentspectralcomponents,
which in the presenceof coherentscatterers
produceswell-definedspectralcorrelation
peaks.The spectralcorrelationpeaksare relativelyinsensitive to the diffusecomponents
sincethesescatterers
havea
randomphasedistributionand their contributions
average
out. Cepstraltechniques,on the other hand, do not utilize
phaseinformationand operateon the PSD characterization
of the signal,which is directlyaffectedby the diffusescatteringcomponent.
theSACfunction
[compare
Figs.2(a)and3(a)].In thepresenceof the diffusecomponentthe SAC functionestimate
fluctuates
aboutthe true scattererspacing,,
by 0.03 mm for
0'=9.9%, and by 0.06 mm for 0'=14.14%, with the CV increasing
to 6% and10%,respectively
[coatpare
Fig.2(b)and
(c) to Fig. 3(b) and(c)].
Note the cepstraltechniqueconvergesto a scatterer
spacing
valuecloseto thecutofffrequency
of thehigh-pass
filter for regularscatterers
with 0' between9.9% to 14.14%.
The cepstralestimates
converges
to this regionbecauseof
the contribution
of the spurious
peaksat lowerfrequencies
andthe largerscattererspacingvariance,•vhichreducesthe
cepstralpeakcorresponding
to the regularscatterer
spacing
III. EXPERIMENTAL
RESULTS
In this section data from in vivo scans of liver tissue are
usedto illustratethe mean-scatterer
spacingestimateusing
the SAC functionand cepstralanalysis.For real tissuethe
SAC functionexhibitssignificantspectralcorrelationcomponentsdue to 'theregularityin the tissuestructure.B-scan
imagesof the liver were obtainedusingthe Ultramark9 ultrasoundsystem(ATL, Bothell,WA). Liver scanswereobtainedusinga transducer
with a centerfrequency
f•=3.5
[observe
in Fig.l(b), thepresence
of strong
spurious
peaks MHz, 3-dB bandwidthf3 •=2.0 MHz, anda samplingfrenearvaluesbelow0.51mm].In addition,
thediffusecompo- quencyf• = 12 MHz. The receivedsignalswerelow-passfilnent addsdirectly to the PSD corresponding
to the regular tered (with a cutoff frequencyof 6 MHz) beforebeing
scattererstructures
and causesa broadeningof the cepstral sampledto preventaliasing.
peaksalongwithothereffectsdueto thenonlinear
operation
The basicfunctionalunit of the liver is the liver 1obule,
(logarithm)
on thesumof thetwo components.
The CV in
whichis a cylindricalstructureseveralmm in lengthand
3511 J.Acoust.
Soc.Am.,Vol.96, No.6, Decomber
1994
T.Varghese
andK. D. C,onohue:
Meanscatterer
spacing
estimates3511
5ø:
l
0.91
0.3
0
0
õ
10
Ifi
20
25
(a)
50-
0.9
0.3.... 5 .... l• .... •, .... •5.... z?,"'
(b)
50,
.• 20
..q
-
10
•
0
(c)
FIG. 3. Mean-scatterer
spacing
andthecoefficient
of variation
for regular
scatterers
withspacing
0.51mm,and+3 dB of diffusescattering
present.
(a)
Standard
deviation
0.016mm (3.!5%), (b) standard
deviation
0.051mm (9.99%),(c) standard
deviation
0.072mm (14.14%).
0.8-2
mm in diameter.
betweenadjacentcells, which link to the terminal bile
The human liver contains 50 000-
100 000 individual 1obules. The lobule is constructed around
ducts.2ø
a centralvein, and is composedof hepaticplatesthat radiate
centrifugallyfrom the centralvein-like spokesin a wheel.
The SAC functionandthecepstrumwereobtainedusing
theWelch-Bartletttechniquewith a 50% overlapof the win-
The hepaticplatesare separated
by hepaticsinusoids
which
link the centralvein to the portalvenules.Eachindividual
lobuleis separated
by a septawhichcontainsthe terminal
bile ducts,portalvenules,and the hepaticarterioles.Each
hepaticplateis usuallytwo cellsthick,with bile canaliculi
3512 J. Acoust.Soc.Am.,Vol.96, No. 6, December1994
dowedaxialdatasegments
of duration10.67/as(128 sample
points
oran8-mm-data
segment)?
A setof segments
were
taken from a section of the liver sector scan that corre-
sponded
to a homogeneous
appearance
in the intensityimage.The dimensions
of the sectionwere 16.47 mm in the
T. Vargheseand K. D. Oonohue:Meanscattererspacingestimates 3512
•80
•-
O1 .................
õ
10
15
20
25
•-.-' .•. -' --•. ....
0
$
10
15
20
25
0
5
to
15
20
25
15
20
25
(a)
•80
i0
15
20
25
Number
Of
(b)
o
0
5
10
15
20
•$
Numl• 0• •egments
t0
(c)
Nu•
O(S•ts
F[G. 4. Mean-scatterer
spacingandthecoefficient
of variationfor regularscatterers
with standard
deviationof 9.99%and-3 dB of thediffusecomponent.
{a) Mean-scatterer
spacingof 0.68 rnrn.(b) Mean-scatterer
spacingof 1.02min. (c) Mean-scatterer
spacingof 1.35min.
axial directionand 2.34ø in the lateraldirection(5 adjacent
A-scans).This tissuesectioncorresponds
to an areaof approximately1.64 cm axially and 0.6 cm laterally.The SAC
functionand cepstralestimatesare presen•:ed
in Fig. 5.
The normalizedSAC functionand the cepstrumare obtainedby averaging15 individualSAC functionsandperiodigrams.The largestspectralcorrelationl:,eakat (3.18,2.43)
MHz in Fig. 5(a), corresponds
to a scatterer
spacingof 1.02
min. Two other correlationpeaks at (3.0,2.7) MHz and
(3.56,2.9)MHz correspond
to scatterer
spacings
of 2.56 and
3513
J. Acoust.Sec. Am., Vol. 96, No. 6, December1994
1.16mm,respectively.
The 2.56-mmspacingis harmonically
relatedto the 1.02-mmspacing(i.e., periodicityfrom scatter-
ers clustered2 at a time).Althougha 2.04-ramspacingis
expectedin thiscase,thereis increased
errorassociated
with
thispeakdueto an increased
(doubled)
variancein thespacing and significance
leakagefrom the PSD values.As a result, the peak is not as sharp or as high as the one corre-
sponding to the fundamental scatterer spacing. The
correlationpeakdueto the dominantspacingis about15.6%
strongerthan the peak due to the spacingof 2.56 mm, and
T. Vargheseand K. D. Donehue:Mean scattererspacingestimates 3513
techniqueis comparedto the standardmethodof usingthe
cepstrum
to estimatethescatterer
spacing.
Significant
performanceimprovements
are obtainedusingthe SAC function
whencomparedto the cepstraltechniquein the presenceof
the diffusescatteringcomponent.The cepstraltechniquealso
hasthe disadvantage
of requiringa priori knowledgeof the
cutoff frequencyto reducethe systemeffect from the tissue
signature.The algorithmusing the SAC functiondoes not
requirea high-pass
filter.
The mean-scanerer
spacingsdetectedby the SAC function is limited by the bandwidthof the pulseand the proximity of the spectralcorrelationpeaksto the PSD. The technique usingthe SAC functioncannotdetectspectralpeaks
which are close (larger periodicities)to the main diagonal
(PSD), since the spectralpeak causedby the periodicity
mergeswith the PSD due to the leakagefrom the finite window.This problemcan be mitigatedin the caseof the SAC
function,by usinglargergatelengthsto detectlargerperiodicities.The SAC functionis a robusttechniquethat is relatively insensitiveto presenceof diffuse(unresolvable)
scatterers and the systemeffect. Ccpstral methodsrely on the
assumptionthat the systemeffect is a slowly varyingcomponentand can be reducedby high-passfiltering.However,
for in vivo analysisa total separationof the systemeffect
from the tissue signaturewill not be obtained becauseof
overlyingtissueandtheweightingfrom thescattererposition
within the beamprofile.
'1
.0
2.9
Frequency
3.8
(MHzl
(a)
0.07}'
0.06[
ACKNOWLEDGMENTS
o.o{
0.02[
0'00[
0
This material is basedon work supportedin part by the
1
2
Distance
• ''
National Science Foundation, Grant MIP-8920602, the National Cancer Institute, and National Institutes of 1Iealth,
4
Grant CA52823, and by BRSGS07RR07114-23 awarded by
the Biomedical ResearchSupportGrant Program,Division
(mm)
of Research Resources,National Institute of Health. The authors would also like to thank the Division
of Ultrasound
at
the ThomasJeffersonUniversity Hospitalfor collectingthe
data usedin this paper.
co)
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D. G. Brown,andT J. Hall, "Describing
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The FFT cepstrum
in Fig. 5(b) showsthedominantpeak
dueto a scattererspacingof 1.21 mm, whichis about4.24%
strongerthanthe next peak due to a 1.4-mm-scatterer
spacing. Basedon the convergence
resultsof the previoussection, the SAC function estimateis consideredmore reliable.
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has more spuriouspeaksthat competewith the dominate
peak.
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methodof estimating
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