Can someone summarize the materials and methods section for me please?

Application of complex discrete wavelet transfor

m

in classification of Doppler signals using

complex-valued artificial neural network

Murat Ceylan a, Rahime Ceylan a, Yüksel Özbay a,*, Sadik Kara b

a Selcuk University, Department of Electrical & Electronics Engineering,

Engineering and Architecture Faculty, 42075 Konya, Turkey

b Fatih University, Biomedical Engineering Institue, Department of

Electrical & Electronics Engineering, 34500 Istanbul, Turkey

Received 9 July 2007; received in revised form 14 April 2008; accepted 24 May 2008

Artificial Intelligence in Medicine (2008) 44, 65—7

6

http://www.intl.elsevierhealth.com/journals/aiim

KEYWORDS

Complex wavelet

transform;

Complex-valued

artificial neural

networks;

Atherosclerosis;

Carotid artery;

Doppler signals

Summary

Objective: In biomedical signal classification, due to the huge amount of data, to

compress the biomedical waveform data is vital. This paper presents two different

structures formed using feature extraction algorithms to decrease size of feature set

in training and test data.

Materials and methods: The proposed structures, named as wavelet transform-com-

plex-valued artificial neural network (WT-CVANN) and complex wavelet transform-

complex-valued artificial neural network (CWT-CVANN), use real and complex discrete

wavelet transform for feature extraction. The aim of using wavelet transform is to

compress data and to reduce training time of network without decreasing accuracy

rate. In this study, thepresented structureswereapplied to theproblemofclassificatio

n

in carotid arterial Doppler ultrasound signals. Carotid arterial Doppler ultrasound

signalswereacquired fromleft carotidarteries of38patients and40healthyvolunteers.

The patient group included 22males and 16 femaleswith an established diagnosis of the

early phase of atherosclerosis through coronary or aortofemoropopliteal (lower extre-

mity) angiographies (mean age, 59 years; range, 48—72 years). Healthy volunteerswere

young non-smokers who seem to not bear any risk of atherosclerosis, including 28males

and 12 females (mean age, 23 years; range, 19—27 years).

Results and conclusion: Sensitivity, specificity and average detection rate were

calculated for comparison, after training and test phases of all structures finished.

These parameters have demonstrated that training times of CVANN and real-valued

artificial neural network (RVANN) were reduced using feature extraction algorithms

without decreasing accuracy rate in accordance to our aim.

# 2008 Elsevier B.V. All rights reserved.

* Corresponding author. Tel.: +90 332 223 20 48; fax: +90 332 241 06 35.

E-mail addresses: yozbay@selcuk.edu.tr, yuksel.ozbay@gmail.com (Y. Özbay).

0933-3657/$ — see front matter # 2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.artmed.2008.05.003

mailto:yozbay@selcuk.edu.tr

mailto:yuksel.ozbay@gmail.com

http://dx.doi.org/10.1016/j.artmed.2008.05.00

3

1. Introduction

There are no reliable blood tests for diagnosis of

atherosclerosis. When the symptoms developed,

angiography is taken into account as the gold stan-

dard to detect and quantify the stenosis. Since

angiography is invasive, noninvasive ultrasonic Dop-

pler sonography is mostly endorsed. Recent

advances in the Doppler imaging technique made

it possible to appraise the temporal and spatial flow

characteristics in the different portions of the arter-

ial system, such as aorta, carotid and peripheral

arteries [1—6].

Furthermore, recent advances in the field of

artificial neural networks (ANNs) have made them

attractive for analyzing signals. The application of

ANNs has opened a new area for solving problems

not reasonable by other signal processing techni-

ques [7,8]. Applications of ANNs in the medical field

include photoelectric plethysmography pulse wave-

form analysis [9], diagnosis of myocardial infarction

[10], electrocardiogram analysis [11] and differan-

tation of assorted pathological data [12]. However,

to date, neural network analysis of Doppler signals is

a relatively new approach [13—18]. It is expected

that complex-valued artificial neural networks

(CVANN) whose parameters (weights, threshold

values, inputs and outputs) are all complex num-

bers, will have applications in fields dealing with

complex numbers such as telecommunications [19],

speech recognition, signal and image processing [20]

with the Fourier transformation. When using the

existing method for real numbers, we must apply

the method individually to their real and imaginary

parts. On the other hand, CVANN allow us to directly

process data.

In this paper, we propose two novel cascade

structures, called wavelet transform-complex-

valued artificial neural networks (WT-CVANN)

and complex wavelet transform-complex-valued

artificial neural networks (CWT-CVANN). Real

and complex wavelet transforms are used to

reduce the number of input samples in training

and test data. The basic idea in using wavelet

transform is to eliminate unnecessary features by

compressing Doppler signals. So, we propose using

complex-valued neural network for more efficient

classification of Doppler signals. In the implemen-

ted architectures, CVANN is integrated with fea-

ture extraction algorithms. These architectures

are composed of two subnetworks. The first sub-

network, which includes real and complex wavelet

transforms, is responsible for the compression of

signal. The second subnetwork performs the clas-

sification task using the compressed data. The WT-

CVANN and CWT-CVANN presented in this study

were trained and tested with Doppler signals

obtained from healthy and unhealthy subjects.

WT-CVANN and CWT-CVANN both achieved a cor-

rect classification rate of 100% in classification of

Doppler signals. Moreover, training time of CVAN

N

and processing complexity were reduced consid-

erably.

2. Material and methods

Doppler signals used in this study were acquired by

Toshiba PowerVision 6000 Doppler Ultrasound Unit in

the Radiology Department of Erciyes University Hos-

pital [2]. Before the data was recorded, a color and

pulsed Doppler ultrasound examination of the left

carotid artery was performed in order to exclude the

presence of a hemodynamically significant stenosis.

A linear ultrasound probe of 10 MHz was used to

transmit pulsed ultrasound signals to the proximal

left carotid artery.

2.1. Spectral analysis of carotid arterial

Doppler signals

Diagnostic significance of spectral analysis of Dop-

pler signals in arterial investigation is evolvement of

quantitative parameters of Doppler flow signals

based on spectral analysis, which can be used for

diagnostic intentions in arterial obstructive disease.

Doppler shift frequency, which is directly propor-

tional to the blood flow speed, is subjected to

spectral analysis [21].

In this study, acquired Doppler data was divided

with 50% overlap and windowed with a Hamming

window in frames of 256 data points as used in [6].

Afterwards, as seen in Fig. 1, power spectral density

of every window was calculated using Welch’s

method. Therefore, number of samples for every

subject was reduced to 129.

2.1.1. Welch method–—averaging modified

periodogram for spectral analysis

In the Welch method, L data sections of lengthM are

overlapped and the periodograms are computed

from the L windowed data sections. Also, the per-

iodograms are normalized by the factor U to com-

pensate for the loss of signal energy owing to the

windowing procedure. In fact U equates to 1=k

1=

2

2 ,

where k2 is the factor on the biasing effect of data

windows as necessary to compensate for this reduc-

tion in signal energy [22]. Thus,

U ¼

1

M

X

M�1

n¼

0

w2ðnÞ (1)

66 M. Ceylan et al.

The Welch power density spectral estimate,

PWE( f), is therefore

PWEð fÞ ¼

1

L

X

L�1

j¼0

P jð fÞ (2)

The expected value of the Welch estimate is

E½PWEð fÞ� ¼

1

L

X

L�1

j¼0

E½P jð fÞ� ¼ E½P jð fÞ� (3)

that is the same as the expected value of the

modified periodogram [22].

2.2. Real discrete wavelet transform

(DWT)

In its most common form, the DWTemploys a dyadic

grid (integer power of two scaling in a and b) and

orthonormal wavelet basis functions and exhibits

zero redundancy. Actually, the transform integral

remains continuous for the DWT but is determined

only on a dicretized grid of a scales and b locations

[23]. In practice, the input signal is treated as an

initial wavelet approximation to the underlying

continuous signal from which, using resolution algo-

rithm, the wavelet transform and inverse transform

can be computed discretely, quickly and without loss

of signal information. A natural way to sample the

parameters a and b is to use a logarithmic discretiza-

tion of a scale and link this, in turn, to the size of the

steps taken between b locations. To link b to a, we

move in discrete steps to each location b, which are

proportional to the a scale. This kind of discretization

of the wavelet has the form [23]:

cm;nðtÞ ¼

1

ffiffiffiffiffiffi

am0

p c

t� nb0a

m

0

am0

� �

(4)

where the integers m and n control the wavelet

dilation and translation, respectively; a0 is a speci-

fied fixed dilation step parameter set at a valued

greater than 1, and b0 is the location parameter

which must be greater than zero. A common chooses

for discrete wavelet parameters a0 and b0 are 2 and

1, respectively. This power-of-two logarithmic scal-

ing of both the dilation and translation steps is

known as the dyadic grid arrangement. The dyadic

grid is perhaps the simplest and most efficient dis-

cretization for practical purposes and lends itself to

the construction of an orthonormal wavelet basis.

Substituting a0 = 2 and b0 = 1 into Eq. (4) we see that

the dyadic grid wavelet can bewritten compactly, as

[23]:

cm;nðtÞ ¼ 2�m=2

cð2�mt� nÞ (5)

Real discrete wavelet transform is formed a

filter bank included low-pass and high-pass filters

(see Fig. 2). In Fig. 2, D1 and A1 are outputs of the

first high-pass filter and low-pass filter. In this

paper, the real discrete wavelet coefficients of

Doppler signals were computed using the MATLA

B

software package. Among the various wavelet

bases, the Haar wavelet is the shortest and sim-

plest basis and it provides satisfactory localization

of signal characteristics in time domain; hence it

is ideal for short time signals analysis. Therefore,

the Daubechies-2 wavelet that is the generalized

Haar wavelet was chosen as the mother wavelet in

this study [23].

2.3. Complex discrete wavelet transform

(CWT)

Wavelet techniques are successfully applied to var-

ious problems in signal processing. Data compres-

sion [24], classification [25,26] and denoising [27]

are only some examples. It is perceived that the

wavelet transform is an important tool for analysis

and processing of signals. In spite of its efficient

computational algorithm, the wavelet transform

suffers from three main disadvantages.

Classification of Doppler signals using complex-valued artificial neural network 6

7

Figure 1 The PSDs of healthy and unhealthy with ather-

osclerosis subjects.

Figure 2 The filter bank for discrete wavelet transform.

2.3.1. Limitations of wavelet transform

Although the standard DWT is a powerful tool, it has

three major disadvantages that undermine its appli-

cation for certain signal processing tasks [28,29].

2.3.1.1. Shift sensitivity. A transform is shift sen-

sitive, if the shifting in time, for input signal causes

an unpredictable change in transform coefficients.

It has been observed that the Standard DWT is

seriously disadvantaged by the shift sensitivity that

arises from down samplers in the DWT implementa-

tion [28,30]. Shift sensitivity is an undesirable prop-

erty because it implies that DWT coefficients fail to

distinguish between input signal shifts.

2.3.1.2. Poor directionality. An m-dimensional

transform (m > 1) suffers poor directionality when

the transform coefficients reveal only a few feature

orientations in the spatial domain. Wavelet trans-

form has been poor directional selectivity for diag-

onal features. Because the wavelet filters are

separable and real.

2.3.1.3. Absence of phase information. For a com-

plex-valued signal or vector, its phase can be com-

puted by its real and imaginary projections. Phase

information is valuable in many signal processing

applications [31] such as in image compression and

power measurement [32,33].

Most DWT implementations use separable filter-

ing with real coefficient filters associated with real

wavelets resulting in real-valued approximations

and details. Such DWT implementations cannot pro-

vide the local phase information. All natural signals

are basically real-valued, hence to avoid the local

phase information, complex-valued filtering is

required [34,35].

Recent research in the development of CWTs can

be broadly classified in two groups; redundant CWTs

(RCWT) and non-redundant CWTs (NRCWT). Stan-

dard DWT decimates and gives N samples in trans-

form domain for the same N samples of a given

signal. While the redundant transform gives M sam-

ples in transform domain for N samples of given

input signal (where M > N) and hence it is expensive

by the factor M/N. The NRCWT follows the design

aim to approach towards N samples in transform

domain for a given N input samples [28,29].

The RCWT include two almost similar CWTs. They

are denoted as dual-tree DWT (DT-DWT)-based CWT

(see Fig. 3) with two almost similar versions namely

Kingsbury’s and Selesnick’s [36]. In this paper, we

used Kingsbury’s CWT [28,36] for feature extraction

of Doppler signals.

2.4. Complex-valued artificial neural

network (CVANN)

Recently, there has been an increased interest in

applications of the CVANN to process complex sig-

nals [37—39]. In this study, a complex back-propa-

gation (CBP) algorithm has been used for pattern

68 M. Ceylan et al.

Figure 3 Dual-tree complex discrete wavelet transform.

recognition. We will first give the theory of the CB

P

algorithm as applied to a multilayer CVANN. Fig.

4

shows a model neuron used in the CBP algorithm.

The input signals, weights, thresholds, and out-

put signals are all complex numbers. The activity Yn

of neuron n is defined as

Yn ¼

X

m

WnmXm þ Vn (6)

where Wnm is the complex-valued (CV) weight con-

necting neuron n and m, Xm is the CV input signal

from neuron m, and Vn is the CV threshold value of

neuron n. To obtain the CVoutput signal, the activity

value Yn is converted into its real and imaginary

parts as follows:

Yn ¼ x þ iy ¼ z (7)

where i denotes

ffiffiffiffiffiffiffi

�1

p

. Although various output func-

tions of each neuron can be considered, the output

function used in this study is defined by the following

equation:

fCðzÞ ¼ fRðxÞ þ i fRðyÞ (8)

where fR(u) is called the activation function of neural

network. One of the difficulties encountered in

applying the CBP algorithm to the complex domain

involves the appropriate choice of activation func-

tion. For a practical implementation of the complex

multilayer perceptron, it is necessary that the acti-

vation function be bounded. Several researchers

developed a set of properties that a complex activa-

tion function must satisfy in order to be useful in a

multilayer perceptron trained with the back-propa-

gation algorithm [40]. Complex activation function

that used in this study is a superposition of real and

imaginary logarithmic sigmoids, as shown by

fRðuÞ ¼

1

1þ expð�uRÞ

þ j

1

1þ expð�uIÞ

(9)

Summary of CBP algorithm:

(1) Initialisation

Set all the weights and thresholds to small

complex random values.

(2) Presentation of input and desired (target) out-

puts

Present the input vector X(1), X(2), . . ., X(N)

and corresponding desired (target) response

T(1), T(2), . . ., T(N), one pair at a time, where

N is the total number of training patterns.

(3) Calculation of actual outputs

To obtain the complex-valued output signal,

the activity value Yn is converted into its real

and imaginary parts as Eq. (7).

(4) Calculation of the stopping criteria with respect

to Eq. (10) [38].

If this condition is satisfied, algorithm is

stopped and weights and biases are frozen:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

X

p

X

N

n¼1

jT ð pÞ

n � O

ð pÞ

n j2

v

u

u

t ¼ 10�1 (10)

where T

ð pÞ

n and O

ð pÞ

n are complex numbers and

denote the desired and output value, respec-

tively. The actual output value of the neuron n

for the pattern p, i.e. the left side of (Eq. (11))

denotes the error between the desired output

pattern and the actual output pattern. N

denotes the number of neurons in the output

layer.

(5) Adaptation of weights and thresholds

We will use Wml for the weight between the

input neuron l and the hidden neuronm, Vnm for

the weight between the hidden neuron m and

the output neuron n, um for the threshold of the

hidden neuronm, and gn for the threshold of the

output neuron n. Let Il, Hm, On denote the out-

put values of the input neuron l, the hidden

neuron m, and the output neuron n, respec-

tively. Let also Um and Sn denote the internal

potentials of the hidden neuron m and the

output neuron n, respectively. Um, Sn, Hm,

and On can be defined, respectively, as

Um ¼

P

lWmlIl þ um, Sn ¼

P

mVnmHm þ gn, Hm

= fc(Um), and On = fc(Sn). Let d

n = Tn � On

denote the error between the actual pattern

On and the target pattern Tn of output neuron n.

We will define the square error for the pattern p

as E p ¼ ð1=2Þ

PN

n¼1 jTn � Onj2, where N is the

number of output neurons.

We can show that the weights and the thresholds

should be modified according to the following equa-

tions [38]:

DVnm ¼ �e

@E p

@Re½Vnm�

� ie

@E p

@Im½Vnm�

(11)

Dgn ¼ �e

@E p

@Re½gn�

� ie

@E p

@Im½gn�

(12)

DWml ¼ �e

@E p

@Re½Wml�

� ie

@E p

@Im½Wml�

(13)

Classification of Doppler signals using complex-valued artificial neural network 69

Figure 4 A model neuron used in the complex-BP algo-

rithm.

Dum ¼ �e

@E p

@Re½um�

� ie

@E p

@Im½um�

(14)

Eqs. (11)—(14) can be expressed as

DVnm ¼ HmDgn (15)

Dgn ¼ e

Re½dn�ð1� Re½On�ÞRe½On�

þi Im½dn�ð1� Im½On�ÞIm½On�

� �

(16)

DWml ¼ Il Dum (17)

Dum ¼ e

ð1� Re½Hm�ÞRe½Hm�

�

X

n

Re½dn�ð1� Re½On�Þ

Re½On�Re½Vnm�

þIm½dn�ð1� Im½On�Þ

Im½On�Im½Vnm�

0

B

B

@

1

C

C

A

2

6

6

6

6

4

3

7

7

7

7

5

� ie

ð1� Im½Hm�ÞIm½Hm�

�

X

n

Re½dn�ð1� Re½On�Þ

Re½On�Im½Vnm�

�Im½dn�ð1� Im½On�Þ

Im½On�Re½Vnm�

0

B

B

@

1

C

C

A

2

6

6

6

6

4

3

7

7

7

7

5

(18)

where z̄ denotes the complex conjugate of a com-

plex number z.

3. The results of numerical

experiments

A significant number of data used for applications

are naturally available in representations that are

difficult to learn. Transforming the data into a

more appropriate representation can facilitate

the learning process. For instance, using a smaller

number of parameters, which are often called

features, to represent the signal under study is

particularly important for recognition and diag-

nostic purposes. Given any set of features for data

representation, it is therefore important to esti-

mate the difficulty of learning the underlying

concepts using that training data. The learning

system should then seek to transform the repre-

sentations into a space that is easier for learning

purposes [41]. The studies in the literature

show that the degree of difficulty in training a

neural network is inherent in the given set of

training examples. By developing a technique

for measuring this learning difficulty, they devise

a feature construction methodology that trans-

forms the training data and attempts to improve

both the classification accuracy and computa-

tional times of artificial neural network (ANN)

algorithms. The fundamental notion is to organize

data by intelligent preprocessing, so that learning

is facilitated [41,42]. For this purpose, in this

paper, a classification and feature extraction-

based approach is adopted for classifying Doppler

signals.

3.1. The proposed structures and

training/test data

Training and test data set used in this study are

carotid arterial Doppler ultrasound signals acquired

from left carotid arteries of 38 patients and 40

healthy volunteers. The subjects had no clinical

and echocardiographic evidence of valvular disease

or heart failure. The patient group included 22

males and 16 females with an established diagnosis

of the early phase of atherosclerosis through cor-

onary or aortofemoropopliteal (lower extremity)

angiographies (mean age, 59 years; range, 48—72

years). Healthy volunteers were young non-smokers

who seem to not bear any risk of atherosclerosis,

including 28 males and 12 females (mean age, 23

years; range, 19—27 years). The hardware used in

recording of Doppler signals and recording system’s

features were given in studies of Kara and Latifoğlu

[5] and Ceylan et al. [6] as detailed.

Doppler signals were recorded in 2 s at 44,100 Hz,

there were 88,200 samples in one segment of train-

ing/test data. Samples (88,200) are too many for

taking a better performance of classification.

Accordingly, firstly, power spectral densities (PSD)

of Doppler signals were calculated using Welch

method, therefore 88,200 samples in one segment

of training and test data were reduced to 129 using

Hamming window with 256 data points. In this study,

data set was separated into two subsets. Each set

includes 20 healthy subjects and 19 unhealthy sub-

jects. First subset was used for training and remain-

ing second subset was used for testing. Then, the

same procedure was performed changing used sub-

sets. So, twofold cross-validation was done for

obtaining a better network generalization.

Obtained training and test errors were averaged.

In this study, three structures were formed using

two different feature extraction methods, real dis-

crete wavelet transform and complex discrete

wavelet transform. These structures were WT-

CVANN, WT-RVANN and CWT-CVANN (Fig. 5). Here,

classification tasks were performed by complex-

valued artificial neural network and real-valued

artificial neural network for comparison.

In the first structure, WT-CVANN, feature vectors

of training and test patterns whose lengths are 129

samples, were calculated using real discrete wavelet

transform. Three different feature vectors with dif-

ferent lengths (66 samples, 34 samples and 18 sam-

ples) were formed to present as inputs to CVANN. FFT

values of obtained feature vectors were calculated

for arising of real and imaginary components. Finally,

the new training sets included FFT results (66

samples � 39 subjects, 34 samples � 39 subjects

and 18 samples � 39 subjects) were classified using

70 M. Ceylan et al.

the CVANN. The networks trained by these training

setswerenamedasWT-CVANN1,WT-CVANN2andWT-

CVANN3, respectively. The complex-valued back pro-

pagation algorithm was used for training of the net-

works. In training phase, the weights and biases of

CVANN was initialised with small random complex

numbers. An error goal (stopping criteria threshold of

10�1) was specified (see Eq. (11)). The training ofWT-

CVANN was stopped when the error goal was

achieved. After that, the performance of WT-CVANN

was tested by presenting test subjects. The optimum

numbers of hidden nodes were determined as 12 via

experimentation for all networks with the highest

classification accuracy of 99%. Learning rate was

chosen as 0.7 for WT-CVANN1 and WT-CVANN2 in

training via experimentation, it was chosen as 0.9

forWT-CVANN3. The optimumnetwork was chosen as

WT-CVANN2, so the optimum number of input sam-

ples obtained with real discrete wavelet transform

was found as 34 (Fig. 6).

In the second structure, WT-RVANN, feature vec-

tors of training and test patterns whose lengths are

129 samples, were calculated using real discrete

wavelet transform. Then, FFT of feature vectors

were calculated. The new training sets included

FFT results (66 samples � 39 subjects, 34

samples � 39 subjects and 18 samples � 39 sub-

jects) used by real discrete wavelet transform were

classified using the RVANN by accepting real and

imaginary components as different two inputs

(Fig. 7a). On the other hand, complex-valued neural

networks allow us to directly process data (Fig. 7b).

The networks trained by these training sets were

named as WT-RVANN1, WT-RVANN2 and WT-RVANN3,

respectively. The real-valued back propagation

algorithm was used for training of the networks.

In training phase, the weights and biases of RVANN

was initialised with small random real numbers. An

error goal was specified as 10�1. The training of WT-

RVANN was stopped when the error goal was

achieved. After that, the performance of WT-RVANN

was tested by presenting test subjects. The opti-

mum numbers of hidden nodes were determined as 6

for WT-RVANN1 and WT-RVANN2, while it was deter-

mined as 4 for WT-RVANN3 with the highest classi-

fication accuracy of 99% via experimentation.

Learning rates were chosen as 0.9, 3.0 and 5.0 for

WT-RVANN1, WT-RVANN2 and WT-RVANN-3 in train-

Classification of Doppler signals using complex-valued artificial neural network 71

Figure 5 The block representation of (a) WT-CVANN, (b) WT-RVANN and (c) CWT-CVANN.

ing via experimentation, respectively. The optimum

network was chosen as WT-RVANN2, so the optimum

number of input samples obtained with real discrete

wavelet transform was found as 34 (Fig. 6).

In the third structure, CWT-CVANN, feature vec-

tors of training and test patterns whose lengths are

128 samples (first 128 samples of 129 samples), were

calculated using complex discrete wavelet trans-

form. In forming of feature vectors, dual-tree com-

plexdiscretewavelet transform (CWT) [36]wasused.

The sourcecodesofdual-treecomplexwavelet trans-

form is taken from web. Three different feature

vectors with three different lengths (32 samples,

16 samples and 8 samples) were formed to present

as inputs toCVANN. Finally, theobtainednewtraining

sets formed by CWT (32 samples � 39 subjects, 16

samples � 39 subjects and 8 samples � 39 subjects)

were classified using the CVANN. These networks

trained by obtained new training sets were named

as CWT-CVANN1, CWT-CVANN2 and CWT-CVANN3,

respectively. Training and testing processes were

performed as like first structure. The optimum num-

bers of hidden nodes were determined as 4, 30 and 6

via experimentation for CWT-CVANN1, CWT-CVANN2

andCWT-CVANN3with thehighest classificationaccu-

racy of 99%, respectively. Learning rateswere chosen

as 4, 3 and 2 for CWT-CVANN1, CWT-CVANN2 and WT-

CVANN3, respectively. The optimum network was

chosen as CWT-CVANN3, so the optimum number of

input samples obtained with real discrete wavelet

transform was found as 8 (Fig. 6).

3.2. Test results

After the training phase, all of the networks were

tested with the remaining patterns by using twofold

cross-validation. As noted, the trained network with

optimum parameters was used in the test to achieve

best results. The test results for all of networks are

shown at Table 1. It was shown that the best test

results forWT-CVANNs andWT-RVANNswere obtained

by 34 input samples. Although WT-RVANN2 achieved

to less test error than WT-CVANN2, the both of two

networks obtained 100% sensitivity, specificity and

average detection rate. However, the optimum

number of input sample was found as ‘8’ by the

CWT-CVANN structure in this study. So, the optimum

CWT-CVANN structure was expressed as CWT-

CVANN3. In this case, considering test results, the

CWT-CVANN structure produced more good results

than WT-CVANN. But, if the number of iteration was

considered, the best results were obtained by WT-

CVANN2 and WT-RVANN2 structures.

In this study, as seen in Table 1, RVANN and CVANN

was trained and tested to classify Doppler signals.

According to obtained results, the training time and

test error of RVANN were higher than those of

CVANN, while the number of hidden nodes in RVANN

was less than that of CVANN. Training and test errors

given in Table 1 were calculated according to pub-

lished paper of Özbay et al. [43].

72 M. Ceylan et al.

Figure 7 Presenting of complex-valued inputs to RVANN

and CVANN. (a) For RVANN and (b) for CVANN.

Figure 6 Optimum number of real and complex discrete

wavelet coefficients for WT-CVANN, WT-RVANN and CWT-

CVANN architectures.

For evaluation of network’s performance, sensi-

tivity, specificity and average detection rate for

optimum networks and RVANN/CVANN were deter-

mined. As seen in Table 2, all of networks in this

study were achieved 100% sensitivity, specificity and

average detection rate.

Although numerical errors were obtained

(Table 1), proposed methods were classified all

subjects, successfully, as seen in Table 2, because

numerical errors only indicate convergence of

actual outputs to target outputs. These error values

were mentioned about distance of targets and

actual outputs but classification success was eval-

uated using proposed algorithms in Section 3.3.

3.3. Calculation of training and test errors

Method we used to calculate the numbers of correct

and incorrect classified complex-valued data and

real-valued data are given below in detail. Further-

more, the performances of the ANN algorithms were

calculated using measurements of sensitivity, spe-

cificity and average detection rate [45].

3.3.1. Calculation of number of correct and

incorrect classified complex-valued data

We developed an algorithm to evaluate the classi-

fication results of WT-CVANN outputs for training

and test data in complex plane. Desired values are

coded ‘‘i’’ and ‘‘1 + i’’ for healthy and unhealthy

data, respectively. The number of correct classified

data in WT-CVANN was calculated according to the

following algorithm [44,45]:

In this algorithm, if output of the node is

0 � output of WT-CVANN � 0.5 then this output is

Classification of Doppler signals using complex-valued artificial neural network 73

Table 1 Training and test results for all structures

Method Optimum

architecture

Optimum

learning rate

Averaged

iteration

numbers

Training time

(averaged

second)

Training error

(% averaged)

Test error

(% averaged)

RVANN 258:10:2 0.1 120 13.65 0.25 1.11

CVANN 129:80:1 1.0 4.5 8.51 0.02 0.04

WT-CVANN1 66:12:1 0.7 16.5 1.04 0.09 0.48

WT-CVANN2 34:12:1 0.7 22 1.45 0.1 0.36

WT-CVANN3 18:12:1 0.9 7 1.11 0.15 0.47

WT-RVANN1 132:6:2 0.9 36 1.56 0.25 0.73

WT-RVANN2 68:6:2 3.0 6.5 1.07 0.06 0.06

WT-RVANN3 36:4:2 5.0 10 1.21 0.22 0.59

CWT-CVANN1 32:4:1 4.0 15.5 1.642 0.23 1.285

CWT-CVANN2 16:30:1 3.0 13.5 1.888 0.243 0.816

CWT-CVANN3 8:6:1 2.0 71 7.351 0.256 0.257

Table 2 The comparative representations of test results to belong to the optimum structures and RVANN/CVANN

Measurement of

classifier performance

RVANN CVANN WT-CVANN2 WT-RVANN2 CWT-CVANN3

Sensitivity, % (SEN) 100 100 100 100 100

Specificity, % (SPE) 100 100 100 100 100

Average detection

rate, % (ADR)

100 100 100 100 100

determined as ‘‘0’’; if output of the node is

0.5 � output of WT-CVANN � 1 then this output is

determined as ‘‘1’’ for real and imaginary parts of

WT-CVANN outputs. The graphical representation of

classification regions can be seen in Fig. 8.

3.3.2. Calculation of number of correct and

incorrect classified real-valued data

We used an algorithm to evaluate the classification

results of WT-RVANN outputs for training and test

data in real plane. Desired values are coded ‘‘0’’ and

‘‘1’’ for healthy and unhealthy data, respectively.

The number of correct classified data in WT-RVANN

was calculated according to the following algorithm

[45]:

4. Conclusions and discussion

ANN is a practicle and valuable tool in the medical

field area for the development of decision support

systems. The actual implementation of ANN analysis

of Doppler signals involves several stages of varying

complexity. Acquisition of data during a routine

Doppler ultrasound examination by means of

tape-recording or employing directly the digital,

takes rather short time and does not excessively

prolong duration the examination. A substantial

amount of training data whichmust be preprocessed

off-line into a suitable format for the presentation

to ANN are required. Following the existing trans-

form methods (Fast Fourier transform, complex

wavelet transform, etc.) used for real numbers,

the conventional classification method must be

applied to the new outcoming complex numbers’

real and imaginary parts separately. However,

CVANNs allow us automatically the advantage of

capturing good rotational behaviour of complex

numbers.

In this paper, the CWT-CVANN andWT-CVANN have

been developed and presented to classify Doppler

signals. In these systems, real/complex discrete

wavelet transform for feature extraction were used

to make an existing CVANN system more effective. A

comparative assessment of the performance of

RVANN, CVANN, WT-CVANN, WT-RVANN and CWT-

CVANN show that more reliable results are obtained

with the WT-CVANN for classification of Doppler

signals. CVANNs are still able to generalize with

good accuracy. However, they take longer time to

train. The aim in developing WT-CVANN and CWT-

CVANN was to achieve better results with relatively

few signal features. All of the structures succeeded

to classify Doppler signals with 100% sensitivity,

specificity and accurarcy rate.

In this study, complex discrete wavelet transform

was used firstly with CVANN in application of bio-

medical signal classification and 100% correct clas-

sification rate and 99% accuracy rate were achieved.

We hope that the performance of proposed net-

works will be better, if the number of healthy and

unhealthy subjects used in training and test data are

increased. In future studies, complex-valued wave-

let neural network [45] can be used in spite of

CVANN for classification of the feature vectors

formed with real and complex discrete wavelet

transforms.

In this study, the results show that a new expert

system developed for the interpretation of the car-

otid artery Doppler signals using presented struc-

tures. Proposed new structures have advantages

over conventional methods such as fast diagnosis,

operating convenience and cost effectiveness. This

74 M. Ceylan et al.

Figure 8 The graphical representation of classification

regions for complex-valued data.

system has better clinical application over others,

especially for earlier survey of the population.

For the future studies, other classification meth-

ods (support vector machine, combined NN,

genetic-trained ANN, etc.) can be used to classify

Doppler signals, and obtained results can be com-

pared with the proposed method in this study.

Acknowledgment

This work is supported by the Coordinatorship of

Selcuk University’s Scientific Research Projects.

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76 M. Ceylan et al.

- Application of complex discrete wavelet transform in classification of Doppler signals using �complex-valued artificial neural network

Introduction

Material and methods

Spectral analysis of carotid arterial Doppler signals

Welch method-averaging modified periodogram for spectral analysis

Real discrete wavelet transform (DWT)

Complex discrete wavelet transform (CWT)

Limitations of wavelet transform

Shift sensitivity

Poor directionality

Absence of phase information

Complex-valued artificial neural network (CVANN)

The results of numerical experiments

The proposed structures and training/test data

Test results

Calculation of training and test errors

Calculation of number of correct and incorrect classified complex-valued data

Calculation of number of correct and incorrect classified real-valued data

Conclusions and discussion

Acknowledgment

References