Summary of a portion of a paper

Application of complex discrete wavelet transform 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—76

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 theproblemofclassification 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: [email protected], [email protected] (Y. Özbay).

0933-3657/$ — see front matter # 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.artmed.2008.05.003

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 CVANN 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

n¼0

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 MATLAB 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 67

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 CBP 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.

References

[1] Evans D. Doppler signal analysis. Ultrasound in Med & Biol 2000;26(suppl. 1):S13—5.

[2] Kara S, Dirgenali F. Detection of atherosclerosis using auto- regressive modelling and principles component analysis to carotid artery Doppler signals. Bio-Medical Materials and Engineering 2006;16/4:269—77.

[3] Polat K, Kara S, Latifoğlu F, Güneş S. A novel approach to resource allocation mechanism in artificial immune recogni- tion system: fuzzy resource allocation mechanism and appli- cation to diagnosis of atherosclerosis disease. Lecture Notes in Computer Science-LNCS September 2006;244—55.

[4] Latifoğlu F, Şahan S, Kara S, Güneş S. Diagnosis of athero- sclerosis from carotid artery Doppler signals as a real-world medical application of artificial immune systems. Expert Systems with Applications November 2007;33/3.

[5] Kara S, Latifoğlu F. A system to diagnose atherosclerosis via wavelet transforms, principal component analysis and arti- ficial neural networks. Expert Systems with Applications March 2007;32(2).

[6] Ceylan M, Ceylan R, Dirgenali F, Kara S, Özbay Y. Classifica- tion of carotid artery Doppler signals in the early phase of atherosclerosis using complex-valued artificial neural net- work. Computers in Biology and Medicine 2007;37(1):28—36.

[7] Miller AS, Blott BH, Hames TK. Review of neural network applications in medical imaging and signal processing. Med Biol Eng Comput 1992;30:449—64.

[8] Baxt WG. Use of an artificial neural network for data analysis in clinical decision making, the diagnosis of acute coronary occlusion. Neural Comput 1990;2:480—9.

[9] Allen J, Murray A. Development of a neural network screen- ing aid for diagnosing lower limb peripheral vascular disease from photoelectric plethysmography pulse waveforms. Phy- siol Meas 1993;14:13—22.

[10] Baxt WG. Application of artificial neural networks to clinical medicine. Lancet 1995;1135—8.

[11] Edenbrandt L, Heden B, Pahlm O. Neural networks for analysis of ECG complexes. J Electrocardiol 1993;26:66—73.

[12] Dybowski R, Gant V. Artificial neural networks in pathology and medical laboratories. Lancet 1995;1203—7.

[13] Baykal N, Reggia JA, Yalabık N, Erkmen A, Beksac MS. Feature discovery and classification of Doppler umbilical artery blood flow velocity waveforms. Computing in Medi- cine & Biology 1996;26:451—62.

[14] Siebler M, Rose G, Sitzer M, Bender M, Steinmetz H. Real- time identification of cerebral microemboli with us feature detection by a neural network. Radiology 1994;3:739—42.

[15] Serhatlioğlu S, Hardalaç F, Guler I. Classification of tran- scranial Doppler signals using artificial neural network. Journal of Medical Systems April 2003;27(2).

[16] Turkoğlu I, Arslan A, Ilkay E. An intelligent system for diagnosis of the heart valve diseases. Expert Systems with Applications 2002;23:229—36.

[17] Turkoğlu I, Arslan A, Ilkay E. An intelligent system for diagnosis of the heart valve diseases with wavelet packet neural networks. Computers in Biology and Medicine 2003; 33:319—31.

[18] Ubeyli ED, Guler I. Neural network analysis of internal carotid arterial Doppler signals: predictions of stenosis and occlusion. Expert Systems with Applications 2003;25:1—13.

[19] Benvenuto N, Marchesi M, Piazza F, Uncini A. A comparison between real and complex-valued neural networks in com- munication applications. In: Kohonen T, Makisara K, Simula O, Kangus J, editors. Proceedings of the international con- ference on neural networks. Espoo, Finland; 1991.

[20] Hirose A, Hiramatsu K. Proposal of complex-valued region- based-coupling segmentation neural networks and the appli- cation to radar imaging systems. In: Proceedings of the IEEE- INNS-ENNS international joint conference on neural net- works, IJCNN 2000, vol. 1; 2000. p. 146—51.

[21] Evans DH, SkidmoreWN. Doppler ultrasound: physics, instru- mentation and clinical applications. England: John Wiley & Sons Ltd.; 1989.

[22] Ifeachor EC, Jervis BW. Digital signal processing–—a practical approach, 2nd ed., England: Prentice Hall; 2002.

[23] Addison PS. Wavelet transforms and the ECG: a review. Physiological Measurement 2005;26:R155—99.

[24] Beylkin G, Coifman R, Rokhlin V. Fast wavelet transforms and numerical algorithms. Communications on Pure and Applied Mathematics 1991;141—83.

[25] Unser M. Texture classification and segmentation using wave- let frames. IEEE Transactions on Image Processing 1995;4: 1549—60.

[26] Neumann J, Steidl G. Dual-tree complex wavelet transform in the frequency domain and an application to signal classi- fication. Technical Report. University of Mannheim; 2003.

[27] Donoho DL. De-noising by soft-thresholding. IEEE Transac- tions on Information Theory 1995;41:613—27.

[28] Shukla PD. Complex wavelet transforms and their applica- tions. PhD Thesis. The University of Strathclyde; 2003.

[29] Fernandes F. Directional, shift-insensitive, complex wavelet transforms with controllable redundancy. PhD Thesis. Rice University; 2002.

[30] Strang G, Hguyen T. Wavelets and filter banks. England: Wellesley-Cambridge Press; 1996.

[31] Oppenheim AV, Lim JS. The importance of phase in signals. Proceedings of IEEE 1981;69(5 (May)):529—41.

[32] Lorenzetto GP, Kovesi P. A phase based image comparison technique. In: Proceedings of the digital image computing: techniques and applications. 1999. pp. 212—7.

[33] Driesen J, Belmasn R. Time-frequency analysis in power measurement using complex wavelets. In: IEEE international symposium on circuits and systems, ISCAS2002. 2002. pp. 681—4.

[34] Lina JM. Image processing with complex daubechies wave- lets. Journal of Math Imaging and Vision 1997;7:211—23.

[35] Bulow T, Sommer G. Hypercomplex signals–—a novel exten- sion of the analytic signal to the multidimensional case. IEEE Transactions on Signal Processing 2001;49:2844—52.

[36] Selesnick IW, Baraniuk RG, Kingsbury NG. The dual-tree complex wavelet transform. IEEE Signal Processing Magazine 2005;22:123—51.

[37] Georgin N, Koutsougeras C. Complex domain backpropaga- tion. IEEE Trans CAS 1992;39:330—4.

Classification of Doppler signals using complex-valued artificial neural network 75

[38] Nitta T. An extension of the back-propagation algorithm to complex numbers. Neural Network 1997;10:1391—415.

[39] Nitta T. An analysis of the fundamental structure of complex- valued neurons. Neural Processing Letters 2000;12:239—46.

[40] Haykin S. Adaptive filter theory. USA: Prentice Hall; 2002. [41] Güler İ, Übeyli ED. ECG beat classifier designed by combined

neural networkmodel. PatternRecognition 2005;38:199—208. [42] Piramathu S, Ragavan H, Shaw MJ. Using feature construc-

tion to improve the performance of neural networks. Man- agement Science 1998;44(3):416—30.

[43] Özbay Y, Ceylan R, Karlik B. A fuzzy clustering neural net- work architecture for classification of ECG arrhytmias. Com- puters in Biology and Medicine 2006;36:376—88.

[44] Özbay Y, Ceylan M. Effects of window types on classification of carotid artery Doppler signals in the early phase of atherosclerosis using complex-valued artificial neural net- work. Computers in Biology and Medicine 2007;37:287—95.

[45] Özbay Y, Kara S, Latifoğlu F, Ceylan R, Ceylan M. Complex- valued wavelet artificial neural network for Doppler signal classifying.Artificial Intelligence inMedicine2007;40:143—56.

76 M. Ceylan et al.