Application of classical and model-based spectral methods to ophthalmic arterial Doppler signals with uveitis disease

Computers in Biology and Medicine 33 (2003) 455– 471www.elsevier.com/locate/compbiomed

Application of classical and model-based spectral methods toophthalmic arterial Doppler signals with uveitis disease

Inan G�uler∗, Elif Derya �UbeyliDepartment of Electronics and Computer Education, Faculty of Technical Education, Gazi University, 06500

Teknikokullar, Ankara, Turkey

Received 28 August 2002; received in revised form 13 December 2002; accepted 9 January 2003

Abstract

In this study, Doppler signals recorded from ophthalmic artery of 75 subjects were processed by PC-computerusing classical and model-based methods. The classical method (fast Fourier transform) and three model-basedmethods (Burg autoregressive, moving average, least-squares modi�ed Yule–Walker autoregressivemoving average methods) were selected for processing ophthalmic arterial Doppler signals with uveitis disease.Doppler power spectra of ophthalmic arterial Doppler signals were obtained by using these spectrum analysistechniques. The variations in the shape of the Doppler spectra as a function of time were presented in theform of sonograms in order to obtain medical information. These Doppler spectra and sonograms were thenused to compare the applied methods in terms of their frequency resolution and the e�ects in determinationof uveitis disease.? 2003 Elsevier Ltd. All rights reserved.

Keywords: Doppler signal; Spectral analysis; Power spectral density; Sonogram; Ophthalmic artery; Uveitis

1. Introduction

Doppler ultrasound has been a widely used technique in clinical applications since the 1980s.Although Doppler ultrasound may be used for the study of various types of motion within thebody, its major use remains the detection and quanti�cation of �ow in arteries and veins. Dopplersystems are based on the principle that ultrasound, emitted by an ultrasonic transducer, is returnedpartially towards the transducer by the moving red blood cells, thereby inducing a shift in frequencyproportional to the emitted frequency and the velocity along the ultrasound beam. Since the scat-terers within the ultrasound beam usually do not move at the same speed, a spectrum of Doppler

∗ Corresponding author. Tel.: +90-312-212-3976; fax: +90-312-212-0059.E-mail address: [email protected] (I. G�uler).

0010-4825/03/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0010-4825(03)00020-9

mailto:[email protected]

456 I . G�uler, E.D. �Ubeyli / Computers in Biology and Medicine 33 (2003) 455– 471

frequencies will be observed. By using spectrum analysis techniques, the variations in the shapeof the Doppler spectra as a function of time are presented in the form of sonograms in order toobtain medical information [1–9]. Doppler ultrasound has proven to be a valuable technique forinvestigation of numerous orbital and ocular conditions. Doppler ultrasonography is a reliable tech-nique, which demonstrates the �ow characteristics and resistance of ophthalmic arteries in variousocular and orbital disorders. The results of the studies in the literature have shown that Dopplerultrasound evaluation can give reliable information on both systolic and diastolic blood velocitiesof the ophthalmic arteries and have supported that Doppler ultrasound is useful in screening certainhemodynamic alterations in ophthalmic arteries [6,10,11].A number of spectral estimation techniques have recently been developed and have been compared

to the more standard fast Fourier transform (FFT) method, or Welch method, for Doppler ultrasonicsignal processing. FFT-based power spectrum estimation methods are known as classical methodsand have been widely studied in the literature [2–9,12–15]. Autoregressive (AR), moving average(MA), autoregressive moving average (ARMA) methods are model-based (parametric) methods andthese model-based methods’ spectra can be computed via di�erent algorithms [2–9,12–15].Uveitis means “in�ammation of the uvea”, or the middle layer of the eye. The uvea consists

of three structures: the iris, the ciliary body, and the choroid. The iris is the colored structuresurrounding the pupil, visible in the front of the eye. The ciliary body is a structure containingmuscle and is located behind the iris which focuses the lens. The choroid is a layer containingblood vessels that line the back of the eye and is located between the inner visually sensitive layer,called the retina, and the outer white eye wall, called the sclera. In�ammation occurring in any ofthese three structures is termed “uveitis”. Pupillary miosis, retinal detachments, retinal vasculitis,optic nerve edema, and keratic precipitates might be observed symptoms during the determinationof uveitis [16,17].In this study, Doppler signals obtained from 75 subjects, 28 of them have su�ered from uveitis,

were examined by taking into consideration of their power spectral density (PSD) estimates andsonograms. Spectral analysis of ophthalmic arterial Doppler signals with uveitis disease was per-formed using the classical and model-based methods. The result of the previous study which appliedFFT, Burg AR, least-squares AR methods to ophthalmic arterial Doppler signals with Behcet diseaseshowed that the least-squares AR method was the most appropriate one [6]. In the present study, FFTand three model-based methods (Burg AR, MA, and least-squares modi�ed Yule–Walker ARMAmethods) were used for spectral analysis of ophthalmic arterial Doppler signals with uveitis disease.Using these spectrum analysis techniques, the time-dependent spectral distributions were visualizedand detailed documentations of the Doppler signals were obtained. These methods were comparedin terms of their frequency resolution and the e�ects in determination of uveitis disease.

2. Materials and method

2.1. Measurements of ophthalmic arterial doppler signals

Doppler ultrasound system is used as a noninvasive method to observe the hemodynamics of theophthalmic artery. Transorbital examinations are performed with a Doppler unit using a 10 MHzultrasonic transducer. The block diagram of the measurement system is shown in Fig. 1. The system

I . G�uler, E.D. �Ubeyli / Computers in Biology and Medicine 33 (2003) 455– 471 457

AnalogDoppler

Unit Recorder

Analog/DigitalInterface

Board

Computer&

Printer

Ultrasonic Transducer

Fig. 1. Block diagram of measurement system.

consists of �ve blocks. These are 10 MHz ultrasonic transducer, analog Doppler unit (DiasonicsSynergy Color Doppler Ultrasonography), recorder, analog/digital interface board (Sound BlasterPro-16 bit), a personal computer with a printer [2–6].In this study, ophthalmic arterial Doppler signals are obtained from 75 subjects. The group is

consisted of 35 females and 40 males with ages ranging from 18 to 60 years and a mean age of35.5 years. A Diasonics Synergy color Doppler ultrasonography is used during examinations andsonograms are taken into consideration. According to examination result, 28 of 75 subjects havesu�ered from uveitis. This group is consisted of 13 females and 15 males with a mean age 37 years(range 23–60).The ultrasonic transducer is applied on a horizontal plane to the closed eyelids using sterile

methylcellulose as a coupling gel. Care is taken not to apply pressure to the eye in order to avoidartifacts. The probe is most often placed at an angle of 60◦ from the midline pointing towardsthe orbital apex. Good and consistent signals can be obtained at 37–42 mm depth. The instrumentdisplays Doppler frequencies as centimeters per second on the y-axis and plotted time in seconds onthe x-axis called as sonograms. In sonograms, time is plotted along the horizontal axis, frequencyalong the vertical axis and the power at a particular frequency and time as the intensity of thecorresponding pixel [10,11]. Ophthalmic arterial signals used in this study belong to healthy subjectsand unhealthy subjects su�ering from uveitis. MATLAB software package was used to form PSDsand sonograms of ophthalmic arterial Doppler signals. FFT, Burg AR, MA, and least-squares modi�edYule–Walker ARMA methods were selected to obtain PSDs and sonograms which represent thechanges in Doppler frequency with respect to time.

2.2. Fast Fourier transform method for spectral analysis

FFT methods such as Welch method are de�ned as classical (nonparametric) methods. Welchspectral estimator is one of the FFT methods and relies on the de�nition of periodogram method.If available information on the signal consists of the samples {x(n)}Nn=1, the periodogram spectralestimator is de�ned as

PPER(f) =1N

∣∣∣∣∣N∑n=1

x(n) exp(−j2�fn)∣∣∣∣∣2

; (1)

where PPER(f) is the estimation of periodogram.In Welch method, signals are divided into overlapping segments, each data segment is windowed,

periodograms are calculated and then average of periodograms is found. {xl(n)}; l = 1; : : : ; S aredata segments and each segment’s length equals to M . In this method, generally overlapping ratio


is taken as 50% (M=2). Welch spectral estimator is de�ned as

Pl(f) =1M1P

∣∣∣∣∣M∑n=1

v(n)xl(n) exp(−j2�fn)∣∣∣∣∣2

and PW(f) =1S

S∑l=1

Pl(f); (2)

where Pl(f) is the periodogram estimate of each signal interval, v(n) is the data window, P is theaverage of v(n) given as P = (1=M)

∑Mn=1 |v(n)|2; PW(f) is the Welch PSD estimate, M is the

length of each signal interval and S is the number of signal intervals.Then, evaluation of PW(f) at the frequency samples basically reduces to the computation of the

following discrete Fourier transform (DFT):

X (k) =N∑n=1

x(n) exp(−j2�N

)nk; k = 0; : : : ; N − 1; (3)

where X (k) is expressed as the discrete Fourier coe�cient, N is the length of available data andx(n) is the input signal on the time domain. The procedure that computes Eq. (3) is called asFFT algorithm. The estimated Welch PSD can be computed by use of the DFT, which in turn ise�ciently computed by the FFT algorithm. Variance of an estimator is one of the measures oftenused to characterize its performance. For 50% overlap and triangular window, variance of the Welchmethod is de�ned as

var(PW(f)) =98Svar(Pl(f)); (4)

where PW(f) is the Welch PSD estimate and Pl(f) is the periodogram estimate of each signalinterval.Since limS→∞var(PW(f)) = 0, Welch method is asymptotically consistent estimator.Welch spectral estimator can be e�ciently computed via FFT and is one of the most frequently

used PSD estimation methods. FFT method is a poor spectral estimator because its variance is highand frequency resolution of FFT method is limited by the available data record duration, independentof the characteristics of the data. In addition, a smeared spectral estimate is a consequence of thewindowing. Owing to these limitations of FFT method, model-based (parametric) spectral estimationmethods, like AR, MA, and ARMA methods, are extremely valuable for data analysis [2–9,12–15].

2.3. AR method for spectral analysis

The model-based methods are based on modeling the data sequence x(n) as the output of a linearsystem characterized by a rational system. In the model-based methods, the spectrum estimationprocedure consists of two steps. Given the data sequence x(n), 06 n6N −1, the parameters of themethod are estimated. Then from these estimates, the PSD estimate is computed. AR method is mostfrequently used model-based method because estimation of AR parameters can be done easily bysolving linear equations. In AR method, data can be modeled as output of a causal, all-pole, discrete�lter whose input is white noise. AR method of order p is expressed as the following equation:

x(n) =−p∑k=1

a(k)x(n− k) + w(n); (5)


where a(k) are AR coe�cients and w(n) is white noise of variance equal to �2. AR(p) model canbe characterized by AR parameters {a[1]; a[2]; : : : ; a[p]; �2}. The PSD is

PAR(f) =�2

|A(f)|2 ; (6)

where A(f) = 1 + a1e−j2�f + · · ·+ ape−j2�fp [2–9,12–15].

2.3.1. The Burg method for AR parameter estimationThe method developed by Burg (1975) for AR parameter estimation that is based on the minimiza-

tion of the forward and backward prediction errors and on estimation of the re�ection coe�cient.The forward and backward prediction errors for a pth-order model are de�ned as

e f ;p(n) = x(n) +p∑i=1

ap; ix(n− i); n= p+ 1; : : : ; N; (7)

eb;p(n) = x(n− p) +p∑i=1

a∗p; ix(n− p+ i); n= p+ 1; : : : ; N: (8)

The AR parameters are related to the re�ection coe�cient kp by

ap; i =

{ap−1; i + kpa∗p−1;p−i; i = 1; : : : ; p− 1;kp; i = p:

(9)

Burg method considers the recursive-in-order estimation of kp given that the AR coe�cients fororder p− 1 have been computed. The re�ection coe�cient estimate is given by

kp =−2 ∑N

n=p+1 e f ;p−1(n)e∗b;p−1(n− 1)∑N

n=p+1 [|e f ;p−1(n)|2 + |eb;p−1(n− 1)|2]: (10)

The prediction errors satisfy the following recursive-in-order expressions:

e f ;p(n) = e f ;p−1(n) + kpeb;p−1(n− 1); (11)

eb;p(n) = eb;p−1(n− 1) + k∗pe f ;p−1(n) (12)

and these expressions are used to develop a recursive-in-order algorithm for estimating the ARcoe�cients. From the estimates of the AR parameters, PSD estimation is formed as

PBURG(f) =ep

|1 +∑pk=1 ap(k)e−j2�fk |2

(13)

where ep = e f ;p + eb;p is the total least-squares error.The Burg method for estimating the parameters of AR method is computationally e�cient and

yields a stable AR method. On the other hand, the Burg method is suboptimal in that it esti-mates the n re�ection coe�cients by decoupling an n-dimensional minimization problem into the none-dimensional minimizations [12–15].


2.4. MA method for spectral analysis

MA method is one of the model-based methods and in this method signal is obtained by �lteringwhite noise with an all-zero �lter. MA method of order q is expressed as the following equation:

x(n) =q∑k=0

b(k)w(n− k); (14)

where b(k) are MA coe�cients and w(n) is white noise of variance equal to �2. MA (q) model canbe characterized by MA parameters {b[1]; b[2]; : : : ; b[p]; �2}. The PSD is

PMA(f) = �2|B(f)|2 (15)

where B(f) = 1 + b1e−j2�f + · · ·+ bqe−j2�fq [12–15].

2.4.1. Estimation of the MA PSDEstimation of the MA spectrum can be done by the reparameterization of the PSD in terms of

the autocorrelation function. For an MA of order q,

r(k) = 0 for |k|¿q: (16)

Owing to this equation, the de�nition of the PSD as a function of {r(k)} turns into a �nite-dimensional spectral model:

PMA(f) =q∑

k=−qr(k)e−j2�fk (17)

Hence estimate of MA PSD is obtained by inserting estimates of autocorrelation functions inEq. (17):

PMA(f) =q∑

k=−qr(k)e−j2�fk : (18)

Spectra with both sharp peaks and deep nulls cannot be modelled by either AR or MA methods. Inthese cases, ARMA model becomes valuable for spectral analysis [12–15].

2.5. ARMA method for spectral analysis

The spectral factorization problem associated with a rational PSD has multiple solutions, withthe stable and minimum phase ARMA model being one of the model-based methods. In ARMAmethod, data can be modeled as output of a causal, pole-zero, discrete �lter whose input is whitenoise. ARMA method of order p; q is expressed as the following equation:

x(n) =−p∑k=1

a(k)x(n− k) +q∑k=0

b(k)w(n− k); (19)

where a(k) are AR coe�cients, b(k) are MA coe�cients and w(n) is white noise of variance equalto �2. ARMA (p; q) model can be characterized by ARMA parameters {a[1]; a[2]; : : : ; a[p]; b[1],


b[2]; : : : ; b[q]; �2}. The PSD is

PARMA(f) = �2∣∣∣∣B(f)A(f)

∣∣∣∣2

; (20)

where A(f) = 1 + a1e−j2�f + · · ·+ ape−j2�fp and B(f) = 1 + b1e−j2�f + · · ·+ bqe−j2�fq [12–15].

2.5.1. Least-squares modi�ed Yule–Walker method for ARMA parameter estimationA reliable method is to construct a set of linear equations and to use the method of least squares

on the set of equations. Suppose that the autocorrelation sequence can be accurately estimated up tolag M , where M ¿p+ q. Then the following set of linear equations can be written:

r(q) r(q− 1) · · · r(q− p+ 1)r(q+ 1) r(q) · · · r(q− p+ 2)

......

r(M − 1) r(M − 2) r(M − p)

a1

a2...

ap

=−

r(q+ 1)

r(q+ 2)

...

r(M)

(21)

or equivalently,

Ra=−r; (22)

Since dimension of R is (M − q)xp and M − q¿p the least squares criterion can be used to solvefor the parameter vector a. The result of this minimization is

a=−(R∗R)−1(R∗r): (23)

Finally, the estimated ARMA power spectrum is [12–15]

PARMA(f) =PMA(f)

|1 +∑pk=1 a(k)e−j2�fk |2

; (24)

where PMA(f) is estimate of the MA PSD and is given in Eq. (18).

2.6. Selection of AR, MA and ARMA model orders

One of the most important aspects of the use in model-based methods is the selection of the modelorder. Much work has been done by various researchers on this problem and many experimentalresults have been given in the literature [12–15]. One of the better known criteria for selecting themodel order have been proposed by Akaike [18], called the Akaike information criterion (AIC), isbased on selecting the order that minimizes Eq. (25) for AR method, Eq. (26) for MA method, andEq. (27) for ARMA method:

AIC(p) = ln �2w + 2p=N (25)

AIC(q) = ln �2w + 2q=N (26)

AIC(p; q) = ln �2w + 2(p+ q)=N (27)

where �2w is the estimated variance of the linear prediction error. Note that the term �2w decreasesand therefore ln �2w also decreases as the orders of AR, MA, and ARMA method are increased.


However, in Eq. (25) 2p=N increases with an increase in p, in Eq. (26) 2q=N increases with anincrease in q, and in Eq. (27) 2(p+ q)=N increases with an increase in p and q. In this situation,a minimum value is obtained for some p in Eq. (25), for some q in Eq. (26), and for some p andq in Eq. (27) [18]. In this study, model orders of AR, MA, and ARMA methods were taken as 10by using Eqs. (25)–(27).

3. Results and discussion

Doppler shift signal can be processed to achieve either a �ow velocity waveform or a Dopplerpower spectrum. Clinically useful information can be extracted from these types of output. A partic-ular physiological or pathological change gives rise to a variation in waveform shape and abnormalDoppler waveforms can be recognized by analyzing waveforms [1–11]. In this study, PSDs ofophthalmic arterial Doppler signals were obtained by using FFT, Burg AR, MA, and least-squaresmodi�ed Yule–Walker ARMA methods. Doppler power spectra describe the distribution of powerwith frequency. Since the velocity components are proportional to the Doppler frequency shifts,the Doppler PSD estimations give shape of the velocity distribution within the ophthalmic artery.PSDs of ophthalmic arterial Doppler signals for healthy subject (subject no: 16), unhealthy subjectsu�ering from uveitis (subject no: 25) are presented in Figs. 2 and 3, respectively.When Doppler PSDs are examined, it is seen that classical method (FFT) has large variance

(Figs. 2(a) and 3(a)). FFT method is based on a �nite record of data, the frequency resolution ofthese methods equal to the spectral width of the window length N , which is approximately 1=N .The principal e�ect of windowing that occurs when processing with FFT is to smear or smooth theestimated spectrum. Owing to smearing FFT method cannot resolve details in the studied spectrumthat are separated by less than 1=N in cycles per sampling interval. For this reason, 1=N is called thespectral resolution limit of the FFT method. Furthermore, this method su�ers from spectral leakagee�ects, due to windowing that are inherent in �nite-length data records. Often, the spectral leakagemasks weak signals that are present in the data. Smearing and spectral leakage are particularly criticalfor spectra with large amplitude ranges, such as peaky spectra. The basic limitation of the FFTmethod is the inherent assumption that the autocorrelation estimate is zero outside the window. Thisassumption severely limits the frequency resolution and the quality of the power spectrum estimatethat is achieved. From another viewpoint, the inherent assumption in the FFT method estimate isthat the data are periodic with period N . Neither one of these assumptions is realistic [1–8,12–15].Model-based methods do not require such assumptions. A model for the signal generation can be

constructed with a number of parameters that can be estimated from the observed data. From themodel and the estimated parameters, PSD can be computed. The modeling approach eliminatesthe need for window functions and the assumption that the autocorrelation sequence is zero outsidethe window. As it is seen from Figs. 2(b) and 3(b), Burg AR PSD estimation method avoid theproblem of leakage and provide better frequency resolution than do the FFT method. AR equationmay model spectra with narrow peaks by placing zeroes of the A-polynomial in Eq. (6) close to theunit circle. This is an important feature since narrowband spectra are quite common in practice. Inaddition, the estimation of parameters in AR signal models is a well-established topic; the estimatesare found by solving linear equations of the system. However, Burg AR method is known to haveseveral disadvantages. First, it exhibits spectral line splitting at high signal-to-noise ratios. Line


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Fig. 2. PSDs of ophthalmic arterial Doppler signals recorded from 30-year-old healthy subject (subject no: 16): (a) FFT,(b) AR, (c) MA, (d) ARMA methods.

splitting means that the spectrum of x(n) may have a single sharp peak, but the Burg methodmay result in two or more closely spaced peaks. For high-order models, the method also introducesspurious peaks. Furthermore, for sinusoidal signals in noise, the Burg method exhibits a sensitivity tothe initial phase of a sinusoid, especially in short data records. This sensitivity is seen as a frequencyshift from the true frequency, resulting in a phase dependent frequency bias [12–15]. MA spectralestimation is valuable when the PSD is characterized by broad peaks and/or sharp nulls [12–15].As it is apparent from Figs. 2(c) and 3(c) smooth ophthalmic arterial Doppler PSDs are obtainedby MA method. Since the MA PSD is based on all-zero model of the data, it is not possibleto use it to estimate PSDs with sharp peaks. Owing to the MA spectral estimator is not a highresolution spectral estimator for processes with narrowband spectral features, MA method has beenfound inappropriate for obtaining ophthalmic arterial Doppler PSDs. Spectra with both sharp peaksand deep nulls cannot be modeled by either AR or MA methods. In these cases, ARMA spectralestimation provides an opportunity to improve on AR and MA spectral estimations. By combiningpoles and zeros, ARMA method provides a more e�cient representation, from the viewpoint of thenumber of model parameters, of the spectrum of a random process [12–15]. When ophthalmic arterial


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Fig. 3. PSDs of ophthalmic arterial Doppler signals recorded from 38-year-old unhealthy subject su�ering from uveitis(subject no: 25): (a) FFT, (b) AR, (c) MA, (d) ARMA.

Doppler PSDs are examined, Burg AR and least squares modi�ed Yule–Walker ARMA methods’(Figs. 2(d) and 3(d)) performance characteristics have been found to be superior to FFT and MAmethods.The selection of the model orders in AR, MA, and ARMA spectral estimators is a critical subject.

Too low-order results in a smoothed estimate, while too large order causes spurious peaks andgeneral statistical instability. In the case of the dimension of autocorrelation matrix is inappropriateand the model orders chosen incorrect, poor spectral estimates are obtained by AR, MA, and ARMAspectral estimators. Heavy biases and/or large variabilities may be exhibited [12–15]. In this study,Akaike Information Criteria [18] was taken as the base for choosing the model order. According toEqs. (25)–(27) model order p was taken as 10 for AR method, model order q was taken as 10 forMA method and model orders p and q were taken as 10 for ARMA method.Peak frequencies and power levels of ophthalmic arterial Doppler signals for healthy subject

(subject no: 16) and unhealthy subject su�ering from uveitis (subject no: 25) are given in Tables1 and 2, respectively. From these tables it is seen that peak frequencies and power levels of


Table 1Peak frequencies and power levels of Doppler signals recorded from healthy subject (subject no: 16)

Method P1=f1 P2=f2

FFT 53.5962/0.0246 28.5536/0.5168AR 66.5010/0 29.1831/0.5400MA Smooth spectrumARMA 65.5811/0 29.1347/0.5890

Table 2Peak frequencies and power levels of Doppler signals recorded from unhealthy subject su�ering from uveitis (subject no:25)

Method P1=f1 P2=f2

FFT Spurious peaksAR 63.8336/0 35.5135/0.2209MA Smooth spectrumARMA 67.7222/0 38.3270/0.1963

Where P1 is the power level of the �rst peak, P2 is the power level of the second peak in dB/Hz and f1 is thefrequency of the �rst peak, f2 is the frequency of the second peak in Hz.

ophthalmic arterial Doppler signals for healthy subject and unhealthy subject su�ering from uveitisare di�erent from each other. According to Table 1 FFT method’s peak frequencies and power levelsare di�erent from AR and ARMA methods’ peak frequencies and power levels. Since Doppler PSDobtained by MA method is smooth, MA method has been found inappropriate for healthy subject’sophthalmic arterial Doppler signals. AR and ARMA methods’ peak frequencies and power levelsare similar for ophthalmic arterial Doppler signals obtained from healthy subject. From Table 2it is seen that FFT method’s spectrum has spurious peaks and does not produce accurate spectralestimates due to limits on resolution. Owing to Doppler PSD obtained by MA method is smooth,MA method has been found unsuitable for unhealthy subject’s (su�ering from uveitis) ophthalmicarterial Doppler signals. AR and ARMA methods’ peak frequencies and power levels of unhealthysubject’s ophthalmic arterial Doppler signals are not so similar to each other as in the case of healthysubject’s ophthalmic arterial Doppler signals. These two methods’ frequencies and power levels areslightly di�erent for unhealthy subject. AR and ARMA methods produce frequency estimates whichare unbiased and nearly attain the Cramer–Rao bound [12–15]. From Tables 1 and 2 it is apparentthat AR and ARMA methods produce the true frequencies as the peaks of the spectral estimatesfor ophthalmic arterial Doppler signals obtained from healthy and unhealthy subjects. In Table 3six di�erent subjects’ peak frequencies and power levels of ophthalmic arterial Doppler signals arepresented. According to this table peak frequencies and power levels of healthy subjects’ ophthalmicarterial Doppler signals are very similar to each other and this similarity is valid for unhealthysubjects’ ophthalmic arterial Doppler signals.FFT, AR, MA, and ARMA methods are compared with the use of statistical tools such as corre-

lation coe�cients (r). The correlation coe�cients between FFT, AR, MA, and ARMA methods are


Table 3Peak frequencies and power levels of Doppler signals recorded from healthy and unhealthy subjects su�ering from uveitis

Subject no. (condition) Method P1=f1 P2=f2

12 FFT 53.5462/0.0238 28.5261/0.5062(healthy) AR 66.5068/0 29.1921/0.5427

MA Smooth spectrumARMA 65.5652/0 29.1395/0.5841

21 FFT Spurious peaks(unhealthy) AR 63.8228/0 35.5059/0.2235


24 FFT 53.5410/0.0221 28.5205/0.5010(healthy) AR 66.5014/0 29.1986/0.5485




35 FFT 53.5396/0.0215 28.5237/0.5065(healthy) AR 66.5009/0 29.1954/0.5496




Where P1 is the power level of the �rst peak, P2 is the power level of the second peak in dB/Hz and f1 is thefrequency of the �rst peak, f2 is the frequency of the second peak in Hz.

calculated with a statistical package (SPSS version 8.0). Ophthalmic arterial Doppler PSD valuesare used for calculation of the correlation coe�cients. The correlation coe�cient is limited with therange [ − 1; 1]. When r = 1 there is a perfect positive linear correlation between the two methods’PSD values, which means that they vary by the same amount. When r = −1 there is a perfectlylinear negative correlation between the two methods’ PSD values, that means they vary in oppositeways (when one of the method’s PSD values increase, the other method’s PSD values decrease bythe same amount). When r = 0 there is no correlation between the two methods’ PSD values (thevalues are called uncorrelated). Intermediate values describe partial correlations. The correlation co-e�cients between FFT, AR, MA, and ARMA methods calculated from ophthalmic arterial DopplerPSD values for healthy subject (subject no: 16) and unhealthy subject su�ering from uveitis (subjectno: 25) are given in Tables 4 and 5, respectively. According to these tables there is a perfect positivelinear correlation between PSD values of the methods. However, from Figs. 2, 3 and Tables 1–3


Table 4The correlation coe�cients between FFT, AR, MA, and ARMA methods for healthy subject (subject no: 16)

Methods FFT AR MA ARMA

FFT 1.000 0.917 0.953 0.926AR 0.917 1.000 0.967 0.997MA 0.953 0.967 1.000 0.972ARMA 0.926 0.997 0.972 1.000

Table 5The correlation coe�cients between FFT, AR, MA, and ARMA methods for unhealthy subject (subject no: 25)

Methods FFT AR MA ARMA

FFT 1.000 0.953 0.953 0.946AR 0.953 1.000 0.997 0.997MA 0.953 0.997 1.000 0.993ARMA 0.946 0.997 0.993 1.000

it is apparent that only AR, ARMA methods’ Doppler PSDs are similar and FFT, MA methods’Doppler PSDs are di�erent from other methods’ PSDs.The variation in the shape of Doppler power spectrum as a function of time can be presented

in the form of a sonogram. In Fig. 4, ophthalmic artery �ow sonograms recorded from 30-year-oldhealthy subject (subject no: 16) are presented. Ophthalmic artery �ow sonograms in Fig. 4(a)–(d) areobtained by using FFT, Burg AR, MA, and least squares modi�ed Yule–Walker ARMA methods,respectively. When these sonograms are inspected, it is seen that there are some di�erences inthe case of �ow spectra. Spectral resolutions are very low in FFT and MA sonograms comparingwith Burg AR and least squares modi�ed Yule–Walker ARMA sonograms. In FFT sonogram somespurious frequencies are seen comparing with Burg AR and least-squares modi�ed Yule–WalkerARMA sonogram. Furthermore, the sonograms of FFT and MA methods are not clear in comparisonwith Burg AR and least-squares modi�ed Yule–Walker ARMA methods’ sonograms. Burg AR andleast-squares modi�ed Yule–Walker ARMA methods o�er good quality sonogram outputs in termsof frequency resolution.In Fig. 5, ophthalmic artery �ow sonograms recorded from 38-year-old unhealthy subject su�ering

from uveitis (subject no: 25) are given. Ophthalmic artery �ow sonogram in (a) is obtained usingFFT method, in (b) using Burg AR method, in (c) using MA method, and in (d) using least-squaresmodi�ed Yule–Walker ARMA method. As it is seen from Fig. 5, in the case of uveitis diseasesystole and diastole are not clear. Some misleading frequencies are seen on the FFT sonograms.The sonograms obtained by using the FFT and MA methods do not have high-frequency resolution.There is a distinct qualitative improvement in the ophthalmic arterial Doppler sonograms of unhealthysubject su�ering from uveitis obtained using Burg AR and least-squares modi�ed Yule–WalkerARMA methods over the FFT and MA methods. In our previous study [6], a qualitative improvementin the appearance of ophthalmic arterial Doppler sonograms of unhealthy subject su�ering fromBehcet disease obtained using the least-squares AR method over the FFT and Burg AR methodswas su�cient for determination of Behcet disease.


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Fig. 4. Ophthalmic arterial Doppler sonograms recorded from 30-year-old healthy subject (subject no: 16): (a) FFT, (b)AR, (c) MA, (d) ARMA.

Feature extraction is a process of pattern recognition and consists of extracting and combiningsalient features of the pattern vector into a feature vector (for example resistivity index or pulsatilityindex). Then decision is given whether such a feature vector is obtained from a normal or abnormalartery. Waveform indices such as resistivity index (RI) and pulsatility index (PI) can be used forevaluation of ophthalmic arterial Doppler waveforms. Both RI and PI are re�ections of the resistanceto �ow, downstream from the point of insonation. Great care must be taken with interpretation ofRI and PI in the clinical setting as they are in�uenced by many factors including proximal stenosis,post stenosis and peripheral resistance. RI and PI are de�ned as

RI = (S − D)=S; (28)

PI = (S − D)=M; (29)

where S is maximum systolic height, D is end diastolic height and M is mean height of thewaveform [1]. Independent-samples t-tests are used to compare RI and PI values of healthysubjects and unhealthy subjects su�ering from uveitis. t-tests are performed with a statistical


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Fig. 5. Ophthalmic arterial Doppler sonograms recorded from 38-year-old unhealthy subject su�ering from uveitis (subjectno: 25): (a) FFT, (b) AR, (c) MA, (d) ARMA.

package (SPSS version 8.0). The level of statistical signi�cance is taken as p=0:05. The results ofindependent-samples t-tests show that there is a statistically signi�cant di�erence between RI andPI values of healthy subjects and unhealthy subjects su�ering from uveitis (p¡ 0:001). FromTable 6 and Figs. 6 and 7 it is seen that there is no overlap in RI and PI values of healthy anduveitis subjects. In this situation, RI and PI values can be used to predict the presence or absenceof uveitis disease. Calculation of RI and PI from ophthalmic arterial Doppler sonograms obtainedby FFT and MA methods is di�cult since systole and diastole are not clear as seen in Figs. 4(a),(c), 5(a), and (c). On the other hand, ophthalmic arterial Doppler sonograms obtained by Burg ARand least-squares modi�ed Yule–Walker ARMA methods are more clear so that calculation of RIand PI will not be di�cult by these two methods.

4. Conclusion

Ophthalmic arterial Doppler signals with uveitis were processed using FFT, Burg AR, MA,and least-squares modi�ed Yule–Walker ARMA methods. Interpretation and performance of these


Table 6RI and PI values of healthy and uveitis subjects

Subject No Condition RI PI

12 Healthy 0.60 1.4016 Healthy 0.61 1.4121 Unhealthy 0.81 1.6024 Healthy 0.59 1.3925 Unhealthy 0.79 1.5930 Unhealthy 0.80 1.6135 Healthy 0.60 1.4142 Unhealthy 0.79 1.6050 Healthy 0.61 1.4156 Unhealthy 0.80 1.62

00.10.20.30.40.50.60.70.80.9

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Fig. 6. Resistivity index values for healthy and uveitis subjects.

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Fig. 7. Pulsatility index values for healthy and uveitis subjects.

methods were compared in terms of their frequency resolution and the e�ects in clinical applications.Since FFT and MA methods have low spectral resolution, these two methods have not been foundappropriate for evaluating ophthalmic arterial Doppler PSDs and sonograms of healthy and unhealthysubjects su�ering from uveitis. Burg AR and least-squares modi�ed Yule–Walker ARMA methods’performance characteristics have been found extremely valuable for analysis of ophthalmic arterial


Doppler signals obtained from healthy and unhealthy subjects su�ering from uveitis, because of theirclear spectra.

References

[1] D.H. Evans, W.N. McDicken, R. Skidmore, J.P. Woodcock, Doppler Ultrasound: Physics, Instrumentation andClinical Applications, Wiley, Chichester, 1989.

[2] N.F. G�uler, M.K. K�ym�k, I. G�uler, Comparison of FFT and AR-based sonogram outputs of 20 MHz pulsed Dopplerdata in real time, Comput. Biol. Med. 25 (1995) 383–391.

[3] N.F. G�uler, M.K. K�ym�k, I. G�uler, Autoregressive-based sonogram output of 20 MHz pulsed Doppler data, Med.Prog. Technol. 21 (1995) 105–110.

[4] I. G�uler, F. Hardala�c, S. M�uld�ur, Determination of aorta failure with the application of FFT, AR and wavelet methodsto Doppler technique, Comput. Biol. Med. 31 (2001) 229–238.

[5] I. G�uler, S. Kara, N.F. G�uler, M.K. K�ym�k, Application of autoregressive and fast Fourier transform spectral analysisto tricuspid and mitral valve stenosis, Comput. Methods Programs Biomed. 49 (1996) 29–36.

[6] I. G�uler, F. Hardala�c, E.D. �Ubeyli, Determination of Behcet disease with the application of FFT and AR methods,Comput. Biol. Med. 32 (2002) 419–434.

[7] P.I.J. Keeton, F.S. Schlindwein, Spectral broadening of clinical Doppler signals using FFT and autoregressivemodelling, Eur. J. Ultrasound 7 (1998) 209–218.

[8] P.I.J. Keeton, F.S. Schlindwein, D.H. Evans, A study of the spectral broadening of simulated Doppler signals usingFFT and AR modelling, Ultrasound Med. Biol. 23 (7) (1997) 1033–1045.

[9] J.Y. David, S.A. Jones, D.P. Giddens, Modern spectral analysis techniques for blood �ow velocity and spectralmeasurements with pulsed Doppler ultrasound, IEEE Trans. Biomed. Eng. 38 (1991) 589–596.

[10] K. Aikimbaev, B. Guvenc, A. Canataroglu, H. Canataroglu, F. Baslamisli, M. Oguz, Value of Duplex and colorDoppler ultrasonography in the evaluation of orbital vascular �ow and resistance in sickle cell disease, Am. J.Hematol. 67 (2001) 163–167.

[11] S. Oruc, E.C. Sener, A comparative study on the e�ects of apraclonidine and timolol on the ophthalmic blood �owvelocity waveforms, Int. Ophthalmol. 23 (1999) 69–73.

[12] J.G. Proakis, D.G. Manolakis, Digital Signal Processing Principles, Algorithms, and Applications, Prentice-Hall,Englewood Cli�s, NJ, 1996.

[13] S.M. Kay, S.L. Marple, Spectrum analysis—a modern perspective, Proc. IEEE 69 (1981) 1380–1419.[14] P. Stoica, R. Moses, Introduction to Spectral Analysis, Prentice-Hall, Englewood Cli�s, NJ, 1997.[15] S.M. Kay, Modern Spectral Estimation: Theory and Application, Prentice-Hall, Englewood Cli�s, NJ, 1988.[16] R.E. Smith, R.A. Nozik, Uveitis A Clinical Approach to Diagnosis and Management, Waverly Press, Inc., Baltimore,

1983.[17] J.P. Dernouchamps, C. Verougstraete, L. Caspers–Velu, M.J. Tassignon, Recent Advances in Uveitis, Kugler

Publications, Amsterdam, 1993.[18] H. Akaike, A new look at the statistical model identi�cation, IEEE Trans. Autom. Control AC-19 (1974) 716–723.

Inan G�uler graduated from Erciyes University in 1981. He took his M.S. degree from Middle East Technical Universityin 1985, and his Ph.D. degree from Istanbul Technical University in 1990, all in Electronic Engineering. He is a professorat Gazi University where he is Head of Department. His interest areas include biomedical systems, biomedical signalprocessing, biomedical instrumentation, and electronic circuit design. He has written more than 90 articles on biomedicalengineering.

Elif Derya �Ubeyli graduated from C� ukurova University in 1996. She took her M.S. degree in 1998, in electronic engi-neering. She is a research assistant at the Department of Electronics and Computer Education at Gazi University. Herinterest area is biomedical signal processing.

Documents

Application of classical and model-based spectral methods to ophthalmic arterial Doppler signals with uveitis disease