# A Computer-Aided Diagnosis System for Breast Cancer Using Independent Component Analysis and Fuzzy Classifier

Authors: Ikhlas Abdel-Qader; Fadi Abu-Amara
Journal: Modelling and Simulation in Engineering
 (2008)

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## Abstract

Screening mammograms is a repetitive task that causes fatigue and eye strain since for every thousand cases analyzed by a radiologist, only 3–4 are cancerous and thus an abnormality may be overlooked. Computer-aided detection (CAD) algorithms were developed to assist radiologists in detecting mammographic lesions. In this paper, a computer-aided detection and diagnosis (CADD) system for breast cancer is developed. The framework is based on combining principal component analysis (PCA), independent component analysis (ICA), and a fuzzy classifier to identify and label suspicious regions. This is a novel approach since it uses a fuzzy classifier integrated into the ICA model. Implemented and tested using MIAS database. This algorithm results in the classification of a mammogram as either normal or abnormal. Furthermore, if abnormal, it differentiates it into a benign or a malignant tissue. Results show that this system has 84.03% accuracy in detecting all kinds of abnormalities and 78% diagnosis accuracy.


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## Body

## 1. Introduction

Breast cancer is considered one of
the most common and fatal cancers among women in the USA [1]. According to National
Cancer Institute, 40 480 women died due to this disease and on average every
three minutes one woman is diagnosed with this cancer. Right now there are over
two and a half million women in the US
who have been treated from it [1]. Radiologists
visually examine mammograms to search for signs of abnormal regions. They
usually look for clusters of microcalcifications, architectural distortions, or
masses.Early detection of breast cancer via
mammography improves treatment chances and survival rates [2]. Unfortunately,
mammography is not perfect. False positive (FP) rates are 15–30% due to the
overlap in the appearance of malignant and benign abnormalities while false negative
(FN) rates are 10–30%. A result of FP
is defined to be when a radiologist reports a suspicious change in the breast
but no cancer is found after further examinations. Therefore, it leads to
unnecessary biopsies and anxiety. A result of FN means failure to detect or correctly
characterize breast cancer in a case of which later tests conclude that cancer
is present. Nonetheless, mammography has an overall accuracy rate of 90% [3].CAD algorithms have been developed to
assist radiologists in detecting mammographic lesions. These systems are
regarded as a second reader, and the final decision is left to the radiologist.
CAD algorithms have improved total radiologist accuracy of detection of cancerous
tissues [4]. CADD algorithms are considered as an extremely challenging task
for various reasons. First, the imaging system may have serious imperfections.
Second, the image analysis task is compounded by the large variability in the appearance
of abnormal regions. Finally, abnormal regions are often hidden in dense breast
tissue. The goal of the detection stage is to assist radiologists in locating abnormal
tissues.Many methods have been proposed in
the literature for mammography detection and diagnosis utilizing a wide variety
of algorithms. Chang et al. [5] developed a 3D snake algorithm that finds the
tumor’s contour after reducing the noise levels and followed by an edge enhancement
process. Finally, the tumor’s contour is estimated by using the gradient vector
flow snake. Kobatake et al. [6] proposed the iris filter to detect lesions as
suspicious regions with a low contrast compared to their background. The proposed
filter has the features' extraction ability of malignant tissues. Bocchi et al.
[7] developed an algorithm for microcalcification detection and classification
by which the existing tumors are detected using a region growing method combined
with a neural network-based classifier. Then, microcalcification clusters are
detected and classified by using a second fractal model. Also, Li et al. [8]
developed a method for detecting tumors using a segmentation process, adaptive
thresholding, and modified Markov random fields, followed by a classification
step based on a fuzzy binary decision tree. Bruce and Adhami [9] used the
modulus-maxima technique of discrete wavelet transform as a feature extraction
technique combined with a Euclidean distance classifier. A radial distance
measure of mass boundaries is used to extract multiresolution shape features. Finally,
the leave-one-out and apparent methods are used to test their proposed technique.
Peña-Reyes and Sipper [10] applied a combined fuzzy-genetic approach with new methods
as a computer-aided diagnosis system. Zheng and Chan [11] combined artificial
intelligent methods with the discrete wavelet transform to build an algorithm
for mass detection. Hassanien and Ali [12] proposed an enhanced rough set
technique for feature reduction and classification. Swiniarski and Lim [13] integrated ICA with rough set model for breast-cancer detection. First, features are reduced and extracted using ICA. Then, extracted features are selected using a rough set model. Finally, a rough
set-based method is used for rule-based classifier design.This work is based on integrating PCA, ICA, and fuzzy
classifier to identify and label suspicious regions from digitized mammograms. The
rest of this paper is organized as follows: Section2 presents PCA and ICA algorithms and covers fuzzy logic adaptation as a classifier. The proposed integrated approach is presented
in Section 3. Section 4 presents the experimental results followed by the
conclusions in Section 5.


## 2. Background

### 2.1. PCA

PCA is a decorrelation-based
technique that finds the basis vectors for a subspace in order to select the
most important information. PCA consists of two phases. The first phase findsv uncorrelated and orthogonal vectors;
and the second phase projects the testing data into a subspace spanned by these v vectors [14]. PCA algorithm can be presented as follows:(i)
constructRtrain matrix with dimension N×M, where N
is the total number of training subimages and M is the size of each square subimage;
then, generate its normalized matrix PM×N;(ii)
covariance matrix is constructed using(1)CN×N=PN×MTPM×N;(iii)
letλi and Ei,i=1,2…M, be its eigenvalues and eigenvectors that satisfy the equation CEi=λiEi,
where
λ1≥λ2≥…≥λM≥0; discard of all eigenvalues less than T (a
predetermined threshold) and retain the rest (the principal components) to
produce the reduced matrix RM×vR. T is calculated using (2)T=∑k=1vλk∑q=1Mλq.The given testing dataRtest is projected into
the space spanned by the reduced training matrix RM×vR using (3)Wv×N=(RM×vR)TRtestM×N.


### 2.2. ICA

Higher-order statistics, such as ICA techniques, are used
to compensate for PCA shortcomings. ICA is based on the use of moments and cumulants up to fourth-order to describe any
distribution of a random variable.In general, ICA  is a relatively new technique developed
to find a linear representation of nongaussian data so that the data components
are statistically as independent as possible. ICA has the ability to describe localized
shape variations and it does not require a Gaussian distribution of the data as
in PCA. However, the resulting vectors are not ordered; and, therefore, ICA requires a method for
ordering the resulting vectors.The statistical latent variables
model is used to define ICA.
Assuming that we haven linear mixtures r1…rn of n independent components s1…sn according to (4)r=∑i=1naisiorR=AS.The digital mammographic imageR is considered
as a mixture of linear combination of statistically independent source regions S
where A, the mixing matrix, and its coefficients describe uniquely the mixed
source regions and can be used as extracted features. After estimating the
matrix A and its inverse W (the separating matrix), the independent components
can be estimated using (5)S=WR.


### 2.3. Fuzzy Classifier

Fuzzy logic can be interpreted as the
emulation of human reasoning on computers [15]. Fuzzy rules are more
comprehensible than crisp rules since they can be expressed in terms of linguistic
concepts. The value of the linguistic variable is not a number but a word. For
example, the linguistic variable “size” might have the values “small,” “medium,”
and “large.” Each one of these values is called a fuzzy set when implemented
using fuzzy logic and thus fuzzy sets can be used to model linguistic variables.Fuzzy classifier is ideally suited to
the labeled observed data to provide interpretable solutions. It handles
imprecise data and the resulting fuzzy rules are interpretable, that is, fuzzy
classifier structure can be analyzed through its semantic structure. There are
two different methods for development of fuzzy classifiers; approximate and descriptive fuzzy rule base.Each fuzzy rule is defined using
membership function of fuzzy sets in an approximate fuzzy rule base which is
implemented in this work. Values of the linguistic variable can be described in
terms of numerals using membership functions. The object membership degree to a
fuzzy set defines a membership function. Its domain is the universe of
discourse (all values an object may take) and its range of the interval[0.0,1.0].
A commonly used membership function is the triangular function. Figure 1 shows
a triangular membership function of a fuzzy set “Small.”Figure 1
A triangular membership function of
the fuzzy set “Small.”In Figure1, an object x has a
membership degree of 0.7 to the fuzzy set “Small.” A fuzzy space is defined to
be the set of fuzzy sets that define fuzzy classes for a particular object as
shown in Figure 2.Figure 2
Fuzzy space
of the object of Figure1 that consists of three fuzzy sets: “Small,” “Medium,”
and “Large.”Fuzzy space allows the object to
partially belong to different classes simultaneously. This idea is very useful
in cases where the difference between classes is not well defined. For example,
the objectx has a membership degree of 0.7 to the fuzzy set “Small” and 0.3 to
the fuzzy set “Medium.” Similarly, in mammographic images, the difference
between benign/malignant and normal/abnormal subimages is not well defined. For
example, an abnormal subimage may be classified as benign rather than malignant
which can be described in terms of numerals using membership functions as it
has a membership degree of 0.7 to the fuzzy set “benign” and 0.3 to the fuzzy
set “malignant.” Fuzzy membership functions are easy to implement and their
fuzzy inference engines are fast.In descriptive fuzzy rule base,
linguistic variables are commonly defined by fuzzy if-then rules where labelsAij are used to represent
a discrete set of linguistic fuzzy sets. For example, fuzzy classification
rules that describe each class of subimages may be developed to represent each
class. Fuzzy rules have the form (6)IFantecedentTHENconsequent[weight]. Fuzzy
rules can also be expressed as (7)Ri:Ifx1isAi1and…xtisAitthenY=Classi[weight], where Y represents the decision class
(i.e., normal, abnormal, benign, or malignant) and Aij represents a fuzzy set
for j: 1,…,tth selected feature.


## 2.1. PCA

PCA is a decorrelation-based
technique that finds the basis vectors for a subspace in order to select the
most important information. PCA consists of two phases. The first phase findsv uncorrelated and orthogonal vectors;
and the second phase projects the testing data into a subspace spanned by these v vectors [14]. PCA algorithm can be presented as follows:(i)
constructRtrain matrix with dimension N×M, where N
is the total number of training subimages and M is the size of each square subimage;
then, generate its normalized matrix PM×N;(ii)
covariance matrix is constructed using(1)CN×N=PN×MTPM×N;(iii)
letλi and Ei,i=1,2…M, be its eigenvalues and eigenvectors that satisfy the equation CEi=λiEi,
where
λ1≥λ2≥…≥λM≥0; discard of all eigenvalues less than T (a
predetermined threshold) and retain the rest (the principal components) to
produce the reduced matrix RM×vR. T is calculated using (2)T=∑k=1vλk∑q=1Mλq.The given testing dataRtest is projected into
the space spanned by the reduced training matrix RM×vR using (3)Wv×N=(RM×vR)TRtestM×N.


## 2.2. ICA

Higher-order statistics, such as ICA techniques, are used
to compensate for PCA shortcomings. ICA is based on the use of moments and cumulants up to fourth-order to describe any
distribution of a random variable.In general, ICA  is a relatively new technique developed
to find a linear representation of nongaussian data so that the data components
are statistically as independent as possible. ICA has the ability to describe localized
shape variations and it does not require a Gaussian distribution of the data as
in PCA. However, the resulting vectors are not ordered; and, therefore, ICA requires a method for
ordering the resulting vectors.The statistical latent variables
model is used to define ICA.
Assuming that we haven linear mixtures r1…rn of n independent components s1…sn according to (4)r=∑i=1naisiorR=AS.The digital mammographic imageR is considered
as a mixture of linear combination of statistically independent source regions S
where A, the mixing matrix, and its coefficients describe uniquely the mixed
source regions and can be used as extracted features. After estimating the
matrix A and its inverse W (the separating matrix), the independent components
can be estimated using (5)S=WR.


## 2.3. Fuzzy Classifier

Fuzzy logic can be interpreted as the
emulation of human reasoning on computers [15]. Fuzzy rules are more
comprehensible than crisp rules since they can be expressed in terms of linguistic
concepts. The value of the linguistic variable is not a number but a word. For
example, the linguistic variable “size” might have the values “small,” “medium,”
and “large.” Each one of these values is called a fuzzy set when implemented
using fuzzy logic and thus fuzzy sets can be used to model linguistic variables.Fuzzy classifier is ideally suited to
the labeled observed data to provide interpretable solutions. It handles
imprecise data and the resulting fuzzy rules are interpretable, that is, fuzzy
classifier structure can be analyzed through its semantic structure. There are
two different methods for development of fuzzy classifiers; approximate and descriptive fuzzy rule base.Each fuzzy rule is defined using
membership function of fuzzy sets in an approximate fuzzy rule base which is
implemented in this work. Values of the linguistic variable can be described in
terms of numerals using membership functions. The object membership degree to a
fuzzy set defines a membership function. Its domain is the universe of
discourse (all values an object may take) and its range of the interval[0.0,1.0].
A commonly used membership function is the triangular function. Figure 1 shows
a triangular membership function of a fuzzy set “Small.”Figure 1
A triangular membership function of
the fuzzy set “Small.”In Figure1, an object x has a
membership degree of 0.7 to the fuzzy set “Small.” A fuzzy space is defined to
be the set of fuzzy sets that define fuzzy classes for a particular object as
shown in Figure 2.Figure 2
Fuzzy space
of the object of Figure1 that consists of three fuzzy sets: “Small,” “Medium,”
and “Large.”Fuzzy space allows the object to
partially belong to different classes simultaneously. This idea is very useful
in cases where the difference between classes is not well defined. For example,
the objectx has a membership degree of 0.7 to the fuzzy set “Small” and 0.3 to
the fuzzy set “Medium.” Similarly, in mammographic images, the difference
between benign/malignant and normal/abnormal subimages is not well defined. For
example, an abnormal subimage may be classified as benign rather than malignant
which can be described in terms of numerals using membership functions as it
has a membership degree of 0.7 to the fuzzy set “benign” and 0.3 to the fuzzy
set “malignant.” Fuzzy membership functions are easy to implement and their
fuzzy inference engines are fast.In descriptive fuzzy rule base,
linguistic variables are commonly defined by fuzzy if-then rules where labelsAij are used to represent
a discrete set of linguistic fuzzy sets. For example, fuzzy classification
rules that describe each class of subimages may be developed to represent each
class. Fuzzy rules have the form (6)IFantecedentTHENconsequent[weight]. Fuzzy
rules can also be expressed as (7)Ri:Ifx1isAi1and…xtisAitthenY=Classi[weight], where Y represents the decision class
(i.e., normal, abnormal, benign, or malignant) and Aij represents a fuzzy set
for j: 1,…,tth selected feature.


## 3. Proposed CADD Algorithm

In this section, a computer-aided
detection and diagnosis algorithm of suspicious regions in mammograms is developed.
PCA algorithm is used as a dimensionality reduction module followed by ICA as a feature extraction
module. Finally, a fuzzy classifier is used to classify testing subimages into
normal/abnormal and at a later stage to classify the abnormal subimages into
malignant/benign as a diagnosis system. Figure3 presents the general framework
for this system.Figure 3
Block diagram of the proposed CADD system.


### 3.1. Subimages Generation

MIAS
database has a total of 119 regions of suspicion (ROS) divided into 51 malignant
and 68 benign. Two different sets of abnormal subimages, each set consists of
119 ROS, are cropped and scaled into35×35 and 45×45 pixels based on the center
of each abnormality.Then, five different sets of normal
subimages, each set consists of 119 subimages, are cropped and scaled randomly from
normal MIAS mammograms where two sets of size35×35 and three sets of size 45×45 pixels.Each set of abnormal subimages is
mixed with one set of normal subimages every time and then divided into two
groups; one for training phase and the other group for testing phase as shown in Table1.Table 1
Different sets used to
evaluate the detection algorithm performance.
#Training setTesting setROSNormalTotalROSNormalTotalSize-pixels16059119596011935×3526059119596011935×3536059119596011945×4546059119596011945×4556059119596011945×45Each training set is used to create the
matrixRtrain with dimension N×M where
each row contains a subimage. The training matrix dimensionality is reduced by using
PCA algorithm to generate RR. Then, the covariance matrix is
estimated by using (8)CN×v=RtrainN×MRM×vR.


### 3.2. Unsupervised Learning

Estimation of the separating matrix,W, and the independent source regions, S, is done in an unsupervised manner.
The independent source regions are estimated by using (9), where (RR)T is the transpose of the reduced matrix RR.
The separating matrix, W, is initialized to the identity matrix yielding (9)S=W(RR)T.To reach the maximum statistical independence
ofS, the nonlinear function Φ(S) is used to estimate the marginal probability
density function of S using its central moments and cumulants. Minimum mutual information
algorithm [16] is used to estimate Φ(S) as shown in (10)–(14). Equations (10)
and (11) are used to estimate the ith central moments and cumulants where E is
the expected value and μ is the mean of the current feature r. Equations
(12)–(14) are used to
estimate Φ(S) (∘ indicates the Hadamard product of two matrices) (10)mi=E(r−μ)i,(11)k3=m3,k4=m4−3,(12)Φ(S)=f1(k3,k4)∘S2+f2(k3,k4)∘S3,(13)f1(k3,k4)=0.5k3(4.5k4−1),(14)f2(k3,k4)=1.5(k3)2+16k4(4.5k4−1).Natural gradient descent method [16]
is used to estimate the change of W according to dW/dt=η[I−Φ(S)ST]W, where η(t) is the learning rate and I is the
identity matrix. If dW/dt is not close to zero, W is updated using (15)Wi(t+1)=Wi(t)+dWdt.Finally, selected features resulting from
the training process are estimated using minimum square error method (MSE) [17, 18].(i)
From (8), the training
matrix is reconstructed as (16)Rtrain≈C(RR)T.(ii)
Substitute (9) into (16): (17)Rtrain≈CN×V(RR)T=CN×VAS.(iii)
There,
the reduced dimensionality selected features from the training set are estimated
by(18)Qtrain=CA.Same procedure followed for training
data is used for testing; andRtest
is projected into the
reduced matrix (RR) from the training procedure. The
reduced dimensionality extracted features from the testing procedure are
estimated by using (19)Qtest=RtestRRA.


### 3.3. Fuzzy Classifier Modeling

The matricesQtrain and Qtest contain the reduced dimensionality extracted
features from subimages where each one of size N by v. Each class of
subimages (normal, abnormal, benign, and malignant) is represented by a single
fuzzy rule by aggregating the membership functions of each antecedent fuzzy set
using the information about selected feature values of training subimages.The proposed fuzzy-based
classification algorithm can be summarized as follows.(1)
Four activation functionsμbs,μms,μas,μns, with each one is of size N by 1,
are initialized to 0 where each element of them represents the aggregated
membership functions of the selected feature values for the corresponding
testing subimage. Each one represents the degree of activation of the selected
feature values and so these parameters are defined as
(i)
μbs: represents the degree of activation
for the benign testing subimages,(ii)
μms: represents the degree of activation
for the malignant testing subimages,(iii)
μas: represents the degree of activation for the
abnormal testing subimages, and(iv)
μns: represents the degree of activation for the
normal testing subimages.(2)
Since subimages have different intensities and the goal is to reduce the
variation and the computational complexity, the selected features ofQtrain and Qtest are mapped into a limited range of [r1,r2]
using (20)q(x,y)=r1+(q(x,y)−min(q))(r2−r1)max(q)−min(q).(3)
Using (21), membership functions of fuzzy sets of the testing subimages are
obtained from the product space of the selected features from the training
phase: (21)Aij(xj)=si(xj)s(xj),i=1,…,v;j=1,…,N, where si(xj) represents number of samples of the current feature xj,s(xj) represents the total number of all
samples in the current feature xj, that is, the product space of the
current feature. Also, the subscript (j) is the index for the selected feature
for each training subimage, and (i) is the index for the current
processed sample of the current feature.(4)
The membership functions are normalized by using(22)Aij(xj)=Aij(xj)maxij(Aj(xj)).(5)
The degree of activation of the developed membership functions is computed for
the testing subimages forμas,μns in the detection phase and for μbs,μms in the diagnosis phase by aggregating estimated
membership functions: (23)μi(x)=∑j=1NAij(xj).(6)
There are many methods used in the literature to determine to which class a
subimage belongs (i.e., normal/abnormal or benign/malignant). An efficient one
is the maximum algorithm. It classifies the testing subimage into the class
that has the maximum degree of activation according to (24) where C1 is used as an index of
a testing subimage being identified as normal or abnormal and C2 for being identified
as benign or malignant: (24)C1=max(μas(x),μns(x)),C2=max(μms(x),μbs(x)).


## 3.1. Subimages Generation

MIAS
database has a total of 119 regions of suspicion (ROS) divided into 51 malignant
and 68 benign. Two different sets of abnormal subimages, each set consists of
119 ROS, are cropped and scaled into35×35 and 45×45 pixels based on the center
of each abnormality.Then, five different sets of normal
subimages, each set consists of 119 subimages, are cropped and scaled randomly from
normal MIAS mammograms where two sets of size35×35 and three sets of size 45×45 pixels.Each set of abnormal subimages is
mixed with one set of normal subimages every time and then divided into two
groups; one for training phase and the other group for testing phase as shown in Table1.Table 1
Different sets used to
evaluate the detection algorithm performance.
#Training setTesting setROSNormalTotalROSNormalTotalSize-pixels16059119596011935×3526059119596011935×3536059119596011945×4546059119596011945×4556059119596011945×45Each training set is used to create the
matrixRtrain with dimension N×M where
each row contains a subimage. The training matrix dimensionality is reduced by using
PCA algorithm to generate RR. Then, the covariance matrix is
estimated by using (8)CN×v=RtrainN×MRM×vR.


## 3.2. Unsupervised Learning

Estimation of the separating matrix,W, and the independent source regions, S, is done in an unsupervised manner.
The independent source regions are estimated by using (9), where (RR)T is the transpose of the reduced matrix RR.
The separating matrix, W, is initialized to the identity matrix yielding (9)S=W(RR)T.To reach the maximum statistical independence
ofS, the nonlinear function Φ(S) is used to estimate the marginal probability
density function of S using its central moments and cumulants. Minimum mutual information
algorithm [16] is used to estimate Φ(S) as shown in (10)–(14). Equations (10)
and (11) are used to estimate the ith central moments and cumulants where E is
the expected value and μ is the mean of the current feature r. Equations
(12)–(14) are used to
estimate Φ(S) (∘ indicates the Hadamard product of two matrices) (10)mi=E(r−μ)i,(11)k3=m3,k4=m4−3,(12)Φ(S)=f1(k3,k4)∘S2+f2(k3,k4)∘S3,(13)f1(k3,k4)=0.5k3(4.5k4−1),(14)f2(k3,k4)=1.5(k3)2+16k4(4.5k4−1).Natural gradient descent method [16]
is used to estimate the change of W according to dW/dt=η[I−Φ(S)ST]W, where η(t) is the learning rate and I is the
identity matrix. If dW/dt is not close to zero, W is updated using (15)Wi(t+1)=Wi(t)+dWdt.Finally, selected features resulting from
the training process are estimated using minimum square error method (MSE) [17, 18].(i)
From (8), the training
matrix is reconstructed as (16)Rtrain≈C(RR)T.(ii)
Substitute (9) into (16): (17)Rtrain≈CN×V(RR)T=CN×VAS.(iii)
There,
the reduced dimensionality selected features from the training set are estimated
by(18)Qtrain=CA.Same procedure followed for training
data is used for testing; andRtest
is projected into the
reduced matrix (RR) from the training procedure. The
reduced dimensionality extracted features from the testing procedure are
estimated by using (19)Qtest=RtestRRA.


## 3.3. Fuzzy Classifier Modeling

The matricesQtrain and Qtest contain the reduced dimensionality extracted
features from subimages where each one of size N by v. Each class of
subimages (normal, abnormal, benign, and malignant) is represented by a single
fuzzy rule by aggregating the membership functions of each antecedent fuzzy set
using the information about selected feature values of training subimages.The proposed fuzzy-based
classification algorithm can be summarized as follows.(1)
Four activation functionsμbs,μms,μas,μns, with each one is of size N by 1,
are initialized to 0 where each element of them represents the aggregated
membership functions of the selected feature values for the corresponding
testing subimage. Each one represents the degree of activation of the selected
feature values and so these parameters are defined as
(i)
μbs: represents the degree of activation
for the benign testing subimages,(ii)
μms: represents the degree of activation
for the malignant testing subimages,(iii)
μas: represents the degree of activation for the
abnormal testing subimages, and(iv)
μns: represents the degree of activation for the
normal testing subimages.(2)
Since subimages have different intensities and the goal is to reduce the
variation and the computational complexity, the selected features ofQtrain and Qtest are mapped into a limited range of [r1,r2]
using (20)q(x,y)=r1+(q(x,y)−min(q))(r2−r1)max(q)−min(q).(3)
Using (21), membership functions of fuzzy sets of the testing subimages are
obtained from the product space of the selected features from the training
phase: (21)Aij(xj)=si(xj)s(xj),i=1,…,v;j=1,…,N, where si(xj) represents number of samples of the current feature xj,s(xj) represents the total number of all
samples in the current feature xj, that is, the product space of the
current feature. Also, the subscript (j) is the index for the selected feature
for each training subimage, and (i) is the index for the current
processed sample of the current feature.(4)
The membership functions are normalized by using(22)Aij(xj)=Aij(xj)maxij(Aj(xj)).(5)
The degree of activation of the developed membership functions is computed for
the testing subimages forμas,μns in the detection phase and for μbs,μms in the diagnosis phase by aggregating estimated
membership functions: (23)μi(x)=∑j=1NAij(xj).(6)
There are many methods used in the literature to determine to which class a
subimage belongs (i.e., normal/abnormal or benign/malignant). An efficient one
is the maximum algorithm. It classifies the testing subimage into the class
that has the maximum degree of activation according to (24) where C1 is used as an index of
a testing subimage being identified as normal or abnormal and C2 for being identified
as benign or malignant: (24)C1=max(μas(x),μns(x)),C2=max(μms(x),μbs(x)).


## 4. Experimetal Results

Table2 shows results of the proposed
CADD algorithm against PCA and ICA
algorithms for the same testing data using fuzzy classifier. Algorithm accuracy
is defined as the ratio between number of correctly classified testing subimages
and total number of testing subimages. Results demonstrate that combining ICA and PCA algorithms improves
the total algorithm performance in all testing sets over usage of PCA algorithm
only. PCA algorithm has a best result of 80.67% while 84.03% for the proposed
CADD algorithm as shown in Table 2. The proposed algorithm improved PCA algorithm
accuracy with an average of 8.56% for all tests.Table 2
FP and FN; and total PCA, ICA, and PCA-ICA algorithms accuracy.
SetPCAICACADDPCFPFNAccuracyFPFNAccuracyPCFPFNAccuracy11917.65%25.21%57.14%10.08%40.34%49.58%2515.13%18.48%66.39%22026.05%10.08%63.87%10.08%40.34%49.58%1012.61%18.48%68.91%3510.08%14.29%75.63%10.08%40.34%49.58%59.24%6.73%84.03%4612.61%21%66.39%10.08%40.34%49.58%520.17%10.08%69.75%5511.75%7.58%80.67%10.08%40.34%49.58%67.56%8.41%84.03%Table2 also shows the simulation results
of ICA algorithm versus the proposed CADD algorithm. ICA algorithm has an accuracy of 49.58% in
all testing sets. In contrast, the best result of applying the proposed CADD
algorithm is 84.03%. These results indicate that using PCA algorithm for
dimensionality reduction before ICA algorithm
improves the ICA algorithm accuracy with an average of 50.51%. Results from ICA algorithm show that fuzzy classifier performance
is degraded when no dimensionality reduction module is implemented. A fuzzy
classifier requires features reduction method in order to minimize total number
of membership functions and improves its accuracy. As for ICA algorithm alone, each subimage has larger
number of selected features and therefore fuzzy classifier performance is degraded
in all testing subimages.The experimental results of the
proposed CADD algorithm as a computer-aided diagnosis system are shown in Table3. The best result is 78% where 15 malignant subimages out of 25 are correctly
classified and 31 benign subimages out of 34 are correctly classified.Table 3
Computer-aided diagnosis using CADD algorithm.
SetTraining setTesting setSize-pixelsPCCADD AlgorithmBenignMalignantTotalBenignMalignantTotalFPFNAccuracy134266034255935×35365.1%16.9%78%234266034255945×4568.48%13.55%77.97%This system uses several parameters
that impact the performance and accuracy of results such as the number of selected
principal components, learning rate, and mapping range.


### 4.1. Number of Selected PC

Using PCA algorithm to reduce data
dimensionality as a preprocessing step for ICA algorithm affects the total algorithm
accuracy. In Table4, simulation results on test sets 1–5 (PC indicates
the number of selected principal components) are shown. These results indicate
that selecting less than 11 principal components achieves acceptable results in
all simulations. This means that less than 0.81% of principal components are
selected for subimages of size 35×35 pixels and less than 0.5% of principal
components are selected for subimages of size 45×45 pixels. This is harmony
with all literature that used PCA algorithm for dimensionality reduction.Table 4
Number of selected principal components impact on algorithm
accuracy where learning rate and mapping range of each set are kept fixed.
PCSet no. 1Set no. 2Set no. 3Set no. 4Set no. 5561.35%65.55%84%69.75%81.52%655.46%64.71%79.83%68.07%84.03%762.19%66.39%78.99%68.91%82.35%866.39%66.39%78.99%65.55%78.15%959.66%66.39%70.59%67.23%76.47%1058.82%68.91%80.67%66.39%74.79%1163.03%65.55%69.75%63.87%75.63%1258.82%60.5%72.27%63.03%78.15%1362.19%63.87%70.59%63.87%74.79%1463.87%62.19%73.95%62.19%77.31%1557.98%63.03%69.75%63.03%73.95%1662.19%59.66%68.07%63.03%76.47%1762.19%67.23%72.29%63.87%77.31%1863.03%60.5%71.43%62.19%76.47%1964.71%67.23%72.29%64.71%79.83%2062.19%64.71%79.83%62.19%73.95%2160.5%66.39%74.79%61.35%80.67%2263.03%66.39%78.15%62.19%71.43%2363.03%63.87%80.67%63.87%74.79%2458.82%60.5%73.95%62.19%80.67%2559.66%60.5%68.91%63.03%79.83%


### 4.2. Learning Rate

The learning rate for computing the
change inW for ICA algorithm determines the speed of convergence for dW/dt and it impacts the total algorithm accuracy. Figures
4–8 show learning
rate impact on test sets 1–5. It can be
concluded that choosing a learning rate close to 0.0045 produce acceptable
results for all sets.Figure 4
Learning rate impact on algorithm accuracy
for test set no. 1 where other parameters are kept constant.Figure 5
Learning rate impact on algorithm accuracy
for test set no. 2 where other parameters are kept constant.Figure 6
Learning rate impact on algorithm accuracy
for test set no. 3 where other parameters are kept constant.Figure 7
Learning rate impact on algorithm accuracy
for test set no. 4 where other parameters are kept constant.Figure 8
Learning rate impact on algorithm accuracy
for test set no. 5 where other parameters are kept constant.


### 4.3. Mapping Range

Figures9–13 show the accuracy
of the results versus the mapping range values for all test sets 1–5 and it can be
concluded that choosing a mapping range equal to [0,9] or [0,15] is acceptable
for all testing sets.Figure 9
Mapping range impact on algorithm accuracy for test set no. 1.Figure 10
Mapping range impact on algorithm accuracy for test set no. 2.Figure 11
Mapping range impact on algorithm accuracy for test set no. 3.Figure 12
Mapping range impact on algorithm accuracy for test set no. 4.Figure 13
Mapping range impact on algorithm accuracy for test set no. 5.The proposed system performance is a parameter-dependent
and an investigation of this dependency is outside this presentation but rather
is left for future investigations. Efforts developed earlier such as in [19, 20] can be investigated. Estimating the parameters will continue to be one
of the main disadvantages of algorithms such ICA  where human intervention is needed.In other classification methods such
as in fractal models, [7], a set of 30 mammograms are used that contains single
and clustered microcalcifications. 50 subimages are extracted and divided into
30 subimages for the training phase and 20 subimages for the testing phase. Results
of using two different multilayer subnetworks in neural network-based
classifier indicate that the proposed system has a classification accuracy of
90%. Also, in discrete wavelet transform method [9], a set of 60 mammograms are
used. Masses are segmented manually as a preprocessing step for the
classification system. The proposed system classifies masses into round,
nodular, or stellate. Results indicate a classification accuracy of 83%. In
[13], 330 subimages are cropped and scaled into sizes of 20×20,40×40, and
60×60 pixels form all MIAS mammograms as one subimage from each mammogram.
Results using ICA-Rough indicate a classifications accuracy of 82.22% for subimages
of size 60×60 pixels and for PCA-Rough of 88.57% for subimages of size 40×40 pixels.Furthermore, Table2 shows that each
test set has different algorithm accuracy so cropping size for example has an
impact on the results.


## 4.1. Number of Selected PC

Using PCA algorithm to reduce data
dimensionality as a preprocessing step for ICA algorithm affects the total algorithm
accuracy. In Table4, simulation results on test sets 1–5 (PC indicates
the number of selected principal components) are shown. These results indicate
that selecting less than 11 principal components achieves acceptable results in
all simulations. This means that less than 0.81% of principal components are
selected for subimages of size 35×35 pixels and less than 0.5% of principal
components are selected for subimages of size 45×45 pixels. This is harmony
with all literature that used PCA algorithm for dimensionality reduction.Table 4
Number of selected principal components impact on algorithm
accuracy where learning rate and mapping range of each set are kept fixed.
PCSet no. 1Set no. 2Set no. 3Set no. 4Set no. 5561.35%65.55%84%69.75%81.52%655.46%64.71%79.83%68.07%84.03%762.19%66.39%78.99%68.91%82.35%866.39%66.39%78.99%65.55%78.15%959.66%66.39%70.59%67.23%76.47%1058.82%68.91%80.67%66.39%74.79%1163.03%65.55%69.75%63.87%75.63%1258.82%60.5%72.27%63.03%78.15%1362.19%63.87%70.59%63.87%74.79%1463.87%62.19%73.95%62.19%77.31%1557.98%63.03%69.75%63.03%73.95%1662.19%59.66%68.07%63.03%76.47%1762.19%67.23%72.29%63.87%77.31%1863.03%60.5%71.43%62.19%76.47%1964.71%67.23%72.29%64.71%79.83%2062.19%64.71%79.83%62.19%73.95%2160.5%66.39%74.79%61.35%80.67%2263.03%66.39%78.15%62.19%71.43%2363.03%63.87%80.67%63.87%74.79%2458.82%60.5%73.95%62.19%80.67%2559.66%60.5%68.91%63.03%79.83%


## 4.2. Learning Rate

The learning rate for computing the
change inW for ICA algorithm determines the speed of convergence for dW/dt and it impacts the total algorithm accuracy. Figures
4–8 show learning
rate impact on test sets 1–5. It can be
concluded that choosing a learning rate close to 0.0045 produce acceptable
results for all sets.Figure 4
Learning rate impact on algorithm accuracy
for test set no. 1 where other parameters are kept constant.Figure 5
Learning rate impact on algorithm accuracy
for test set no. 2 where other parameters are kept constant.Figure 6
Learning rate impact on algorithm accuracy
for test set no. 3 where other parameters are kept constant.Figure 7
Learning rate impact on algorithm accuracy
for test set no. 4 where other parameters are kept constant.Figure 8
Learning rate impact on algorithm accuracy
for test set no. 5 where other parameters are kept constant.


## 4.3. Mapping Range

Figures9–13 show the accuracy
of the results versus the mapping range values for all test sets 1–5 and it can be
concluded that choosing a mapping range equal to [0,9] or [0,15] is acceptable
for all testing sets.Figure 9
Mapping range impact on algorithm accuracy for test set no. 1.Figure 10
Mapping range impact on algorithm accuracy for test set no. 2.Figure 11
Mapping range impact on algorithm accuracy for test set no. 3.Figure 12
Mapping range impact on algorithm accuracy for test set no. 4.Figure 13
Mapping range impact on algorithm accuracy for test set no. 5.The proposed system performance is a parameter-dependent
and an investigation of this dependency is outside this presentation but rather
is left for future investigations. Efforts developed earlier such as in [19, 20] can be investigated. Estimating the parameters will continue to be one
of the main disadvantages of algorithms such ICA  where human intervention is needed.In other classification methods such
as in fractal models, [7], a set of 30 mammograms are used that contains single
and clustered microcalcifications. 50 subimages are extracted and divided into
30 subimages for the training phase and 20 subimages for the testing phase. Results
of using two different multilayer subnetworks in neural network-based
classifier indicate that the proposed system has a classification accuracy of
90%. Also, in discrete wavelet transform method [9], a set of 60 mammograms are
used. Masses are segmented manually as a preprocessing step for the
classification system. The proposed system classifies masses into round,
nodular, or stellate. Results indicate a classification accuracy of 83%. In
[13], 330 subimages are cropped and scaled into sizes of 20×20,40×40, and
60×60 pixels form all MIAS mammograms as one subimage from each mammogram.
Results using ICA-Rough indicate a classifications accuracy of 82.22% for subimages
of size 60×60 pixels and for PCA-Rough of 88.57% for subimages of size 40×40 pixels.Furthermore, Table2 shows that each
test set has different algorithm accuracy so cropping size for example has an
impact on the results.


## 5. Concluuding Remarks

A CADD system has been developed and
implemented. Its framework is based on integrating PCA, ICA, and fuzzy logic. The performance of the
proposed CADD is compared against PCA and ICA performance individually. Extensive simulations using 833 subimages are performed.
These results indicate that combining ICA and
PCA algorithms improves PCA algorithm accuracy about 8.56% for all test sets
and ICA algorithm accuracy about 50.51%. The best results are obtained with subimage sizes
of45×45 pixels over the 35×35 size. Using ICA algorithm for feature extraction without
using a preprocessing module of PCA degraded fuzzy classifier performance. ICA takes advantage of the
reduction of dimensionality and noise to produce more accurate and robust
results. Parameter values play a vital role in the system’s performance and
their selection should be investigated to improve system’s robustness. Other membership
functions can be modeled based on mean and standard deviation of selected feature
values.



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*Source: 238305-2008-05-15.xml*