Research Article Statistical Fractal Models Based on GND ...downloads.hindawi.com/journals/bmri/2013/656391.pdf · Research Article Statistical Fractal Models Based on GND-PCA and

Hindawi Publishing CorporationBioMed Research InternationalVolume 2013 Article ID 656391 8 pageshttpdxdoiorg1011552013656391

Research ArticleStatistical Fractal Models Based on GND-PCA and ItsApplication on Classification of Liver Diseases

Huiyan Jiang1 Tianjiao Feng1 Di Zhao1 Benqiang Yang2 Libo Zhang2 and Yenwei Chen3

1 Software College Northeastern University Shenyang 110819 China2 Radioactive Branch PLA General Hospital of Shenyang Military Region Shenyang 110016 China3Department of Information Science and Engineering Ritsumeikan University Shiga 5258577 Japan

Correspondence should be addressed to Huiyan Jiang hyjiangmailneueducn

Received 1 July 2013 Accepted 31 August 2013

Academic Editor Tai-hoon Kim

Copyright copy 2013 Huiyan Jiang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A new method is proposed to establish the statistical fractal model for liver diseases classification Firstly the fractal theory is usedto construct the high-order tensor and then Generalized 119873-dimensional Principal Component Analysis (GND-PCA) is used toestablish the statistical fractal model and select the feature from the region of liver at the same time different features have differentweights and finally Support Vector Machine Optimized Ant Colony (ACO-SVM) algorithm is used to establish the classifier forthe recognition of liver disease In order to verify the effectiveness of the proposed method PCA eigenface method and normalSVM method are chosen as the contrast methods The experimental results show that the proposed method can reconstruct livervolume better and improve the classification accuracy of liver diseases

1 Introduction

Liver cancer is a common disease in daily life and is oftendiagnosed when it is advanced and very few liver cancerpatients can be cured So it is necessary for us to diagnose livercancer as early as possible ComputerAidedDiagnosis (CAD)technology is established with the development of computergraphics technology image processing technology and pat-tern recognition technology Since CT images increase theburden of doctors research of this kind of technology isurgently needed In recent years scientists have researchedseveral typical variable models such as Snake Model Thesemethods are more suitable for objects with smooth boundaryand do not use the valuable prior knowledge Active TextureModel (ATM) which is evolved from them can reflect thetexture feature of the object The roughness of surface inmedical images is an important factor to distinguish amonglesions so ATM cannot represent object features well onlywith the texture model of gray feature

In order to solve the problems as above we proposed thestatistical fractal model based on the feature of gray level andfractal dimension The statistical fractal model can be better

used in the analysis of medical images such as diseases recog-nition but the construction of statistical appearance modelis a challenging task when the number of training samples ismuch fewer than the number of dimensions of data

Principal Component Analysis (PCA) method [1] is afamous method used in the subspace recognition and it isone of the classical methods based on statistical feature Butthis method has two problems The first is that the originalspace structure of image is damaged in the vectorizationprocess The second is that it may cause the dimensiondisaster when we transfer the image into a vector So we needmore large space to calculate the covariancematrix of imagesIn order to solve these problems we use the Generalized 119873-dimensional PCA [2] to learn subspace in this paper

Support Vector Machine (SVM) [3] is commonly usedto train a classifier And the factor to affect the classificationperformance is the parameters used in SVM So we use AntColony Optimization (ACO) algorithm to optimize SVMparameters and then we use Directed Acyclic Graph DAG[4] to multiclassify liver diseases

As above for protecting the special space structure infor-mation of liver images and solving the dimension disaster

2 BioMed Research International

Yes

Feature weighed

Calculation of the core tensor

Construction of liver diseasesclassifier

the core Calculation of

No

Recognition of liverdiseases

Feature weighed

Based on ACO-SVM

Calculation of optimal U

k = k + 1

Bi (i =

tensor Bj (j =

U(i)opt = U

(i)

k

Input M images of the training

setof

Segmentation

liver region

Feature selection based on fractal

and gray leveldimension tensor

Initial values

determined by

Maximization of the cost

of the testingset

Segmentation

region

Feature selectionbased on fractaland gray level

dimension tensor

function C998400

Input S imagesof liver

Construction of N-

A(i)test (j =

Update U(i)

kConstruction of N-

the first Jn

k = 0 U(n)0

whose column is |U(i)

k+1minus U

(i)

k| lt 120576

1 middot middot middot S) 1 middot middot middot S)

1 middot middot middotM)

(i = 1 middot middot middot n n = 4)A(i)train (i = 1 middot middot middotM)

Figure 1 The main flow of the proposed method

problem we extracted the gray feature and fractal featureto establish the high-order tensor of liver volume and con-structed the statistical fractalmodel withGND-PCAmethodFor improving recognition accuracy SVMoptimized byACO(ACO-SVM) was used to recognize liver diseases images

This paper is organized as follows Section 2 introducesthe proposed method firstly we will introduce some knowl-edge of PCA and tensor then we will show the constructionof high-order tensor and finally we introduce the method ofGND-PCA for the construction of statistical fractal modeland ACO-SVM [5] for classification In Section 3 we presentthe construction of liver images after GND-PCA and theresults of classification Section 4 concludes the works in thispaper

2 Materials and Methods

In this section we will introduce some background knowl-edge about GND-PCA method firstly We mainly presentthe method of PCA 2D-PCA and ND-PCA and the basicknowledge of tensor And then we will introduce our methodof construction of statistical fractal model The main flow isshown in Figure 1 The process of the proposed method isdescribed as follows

(1) Liver images preprocessing Firstly we segment theCT images of abdomen to gain the liver region of

image and then we calculate the fractal dimension ofliver image

(2) High-order tensor construction At first we collecta group of fractal features and gray level features(pixels) and then we combine them into a newdataset

(3) Statistical fractal model establishment In this paperwe use the method of Generalized 119873-dimensionalPCA (GND-PCA) to establish the statistical fractalmodel for classification

(4) Liver diseases classification based on ACO-SVMAfter we obtain the statistical fractal model by GND-PCAwe treat the core tensors as samples and thenweuse SVM optimized by ACO to classify liver diseases

21 PCA Method and Its Extension PCA is an application of119870-119871 conversion in statistics The purpose of PCA is to lowerthe dimension of data through finding a linear mapping Themapping meets the following conditions

(1) The error of sample reconstruction is minimized(2) The mapping of sample set in low dimension space

has the maximum variance(3) The correlation among samples is erasedTurk and Pentland proposed the famous method named

eigenface to realize PCA Suppose that we have 119870 training

BioMed Research International 3

Ntimes N image

N2 vector

Figure 2 2D matrix to vector

samples 1198681 1198682 119868

119870 Firstly we transfer these samples into

vectors shown in Figure 2 The 119873 times 119873 image is transferredinto a column vector that is to say the training samples1198681 1198682 119868

119870are transferred into 119883

1 1198832 119883

119870 119883119894is a

column vector with the dimension of 119899 (119899 = 1198732) Each 119883

119894

is in the space of 119899 dimension According to the knowledgeof linear algebraic 119883

119894can be expressed by 119899 basis in the

119899 dimension space If we express 1198831 1198832 119883

119870by only

one vector obviously we should use the average value 119898

of 1198831 1198832 119883

119870 We rename 119898 as the zero-dimension

expression of sample datasets Doing as above is useful andeasy but the shortcoming of it is that it can not show thedifference between samples So the second step of PCA is tocentralize the training sample sets 119868

1 1198682 119868

119870 and then we

find the 1-dimension to 119889 dimension expression of the newsample sets

Compared with PCA 2D-PCA uses the 2-dimensionimage matrix directly for feature extraction It can calculatethe covariance matrix accurately with less time Imaginingthat there is an 119899 dimension column vector which is nor-malized we can project any image (119898 times 119899) to it 119884 = 119860119883and get the 119898 dimension image eigenvector The separatingcapacity of 119883 can be measured by the total divergence ofthe projected samples and the total divergence of projectedsamples can be expressed by the trace of the covariance of thereflected eigenvector 119869(119883) = tr(119878

119909) 119878119909is the covariance of

the projected eigenvector of training sample and tr(119878119909) is the

trace of 119878119909The purpose ofmaximization of 119869(119883) is to find the

mapping direction and the final total divergence of mappingsample is the largest

The advantage of 2D-PCA method is that we do notneed to transform images into vectors and we can use theimages themselves directly to deal with data informationand find a group of basis which can express the originalsamples best Moreover the eigenfactor is a matrix not avector which PCAmethod needs It keeps the space structureof the original images And it does not only wipe off thecorrelation between the samples effectively but also wipeoff the correlation between the rows in one sample Butthe method has shortages too and the mapping coefficientmatrix is large and wastes lots of memory space because ofthe ignorance of the difference between the columns in onesample

Alternative 2D-PCA is proposed to overcome theseproblems as above The method can solve the problem ofignorance of the difference between the columns in onesample but also cannot solve the problem of the largecoefficient matrix and the difference between both columnsand rows As a result the G2D-PCAmethod is proposed andthis method considers the correlation between both columnsand rows The mapping function is 119862 = 119885

119879119860119883 and it can be

seen as mapping to the rows first and then to the columns orto the columns first and then to the rows At the same time theiteration ideology is proposed by G2D-PCA to obtain betterresults

ND-PCA is proposed for modeling of high-dimensiondata This method is based on HOSVD At the same time wetreat the data as a high-dimension tensor The method cansolve the problemof high cost effectively but it also has a largecoefficient matrix as 2D-PCA method

22 The Basic Knowledge of Tensor Tensor can be treated asthe expansion of matrix Vector is a first-order tensor andmatrix is a second-order tensor So if we stack up severalmatrixes with the same dimension we obtain the cubic arraynamed third-order tensor The analysis of high-order tensoruses the math operation as follows [6]

Suppose that 119883 is an 119872-order tensor 119883 isin 1198771198731times1198732timessdotsdotsdottimes119873

119872 and 119873

119894is the dimension of tensor 119883 The element of 119883 is

defined as 11988311989911198992119899119872

for 1 le 119899119894le 119873119894 1 le 119894 le 119872 The tensor

product is defined as follows

(119883 otimes 119884)1198991times1198992timessdotsdotsdottimes119899

119872times1198991015840

1times1198991015840

2timessdotsdotsdottimes119899

1015840

119872

= 1198831198991times1198992timessdotsdotsdottimes119899

119872

1198841198991015840

1times1198991015840


1015840

119872

(1)

We can transfer the 119872-order tensor to a matrix byextending the 119873

119889th vector of tensor 119883 and put others after

the 119873119889 The product function of tensor 119883 and matrix 119880 is

shown as follows

(119883times119889119880)1198941times1198942timessdotsdotsdottimes119895times119894

119889minus1timessdotsdotsdottimes119894119872

= sum

119894119889

(1198831198941times1198942timessdotsdotsdottimes119894119889minus1

119880119895times119894119889

) (2)

23 Construction of High-Order Tensor In this paper weconstruct high-order tensors based on fractal theory Firstlywe use the method of box [7] and blanked [8] to calculate4 groups of fractal feature and then we establish high-ordertensors based on the fractal feature and the texture featurepointing to each pixel

231 The Calculation of Fractal Feature We use the methodof blanket and box to calculate the fractal feature of liverimages which are segmented by the doctor The liver imageand its segmentation result are shown in Figure 3

The first fractal feature is obtained by the blanketmethodFirstly we treat the images as a hilly terrain surface whoseheight from the normal ground is proportional to the graylevel of the images Then all points at distance 120576 from thesurface on both sides create a blanket whose thickness is 2120576The estimated surface area is the volume of blanket dividedby 2120576 For different 120576 the blanket area can be iterativelyestimated as follows The covering blanket is defined by its


(a) Original image (b) Segmentation result

Figure 3 Liver segmentation

upper surface 119906120576and the lower surface 119889

120576 and we provide

the gray level function 119892(119894 119895) 1199060(119894 119895) = 119887

0(119894 119895) = 119892(119894 119895) for

120576 = 1 2 3 Blanket surfaces are defined as follows

119906120576(119894 119895) = max119906

120576minus1(119894 119895) + 1

max|(119898119899)minus(119894119895)|le1

119906120576minus1

(119898 119899)

119887120576(119894 119895) = min119887

120576minus1(119894 119895) minus 1

min|(119898119899)minus(119894119895)|le1

119887120576minus1

(119898 119899)

(3)

The volume of the blanket is defined as follows

V120576= sum

119894119895

(119906 (119894 119895) minus 119887120576(119894 119895)) (4)

The surface area can be defined as follows

119860 (120576) =(V120576minus V120576minus1

)

2

119860 (120576) = 1198651205762minus119863

(5)

At last the fractal feature1198631can be described as (6) and

V is the volume of the blanket

1198631= 2 minus log(V120576minusV120576minus1)2119865

120576 (6)

The other fractal features are obtained by the method ofbox It is to treat the gray level image 119865(119877 times 119877) as a box in 3-dimensional fractal curves The image can be separated intoseveral boxes (119878times119878) 120575 = 119878119877 119911 is the gray level of the imagesthe plane surface119883119884 can be separated into several grids Themaximum level and the minimum level of gray level of the

image in the grid (119894 119895) can be treated as the 119896th and the 1stbox 119899

120575(119894 119895) = 119897minus119896+1 and then we calculate the total number

119873120575(119865) = sum119899

120575(119894 119895) and the fractal feature can be defined

as (7) In this paper we obtain 1198632to 1198634by giving different

numbers of the boxes such as 4 8 and 16

119863119861=log119873120575(119865)

log (1120575) (7)

There is a big texture difference in coarse level betweendifferent liver images but the fractal dimension has a smallchange in smooth-faced images and a large change in shaggyimages So the fractal dimension is a useful feature for liverdiseases classification

232 The Construction of High-Order Tensor In this paperwe use 50 groups of liver images of 512 times 512 After we extractfour kinds of fractal features we extract the texture featuresWe use all features we obtained to establish the high-ordertensors

24 The Construction of Statistical Fractal Dimension Basedon GND-PCA We provide a series of zero-mean value 119873-order tensor 119860 isin 119877

1198681times1198682timessdotsdotsdottimes119868

119873 And we need to gain a groupof new119873-order tensor 119861 isin 119877


119873(119869119899lt 119868119899) and 119861 needs

to be closed to the original tensor as much as possible Thenwe define tensor images by the texture and fractal featuresobtained from the segmented liver images We use Tuckermodel [9] to reconstruct 119873-order tensor 119860 by 119880

(119899) 119880(119899) =

119869119899times 119868119899 The reconstruction of three-order tensor is shown in

Figure 4The orthogonal matrix 119880(119899)opt can be obtained by minimiz-

ing the cost function 119862 which is shown as (8)


I1

I2

I3

I1

I2

I3

U(1)

Opt

U(2)

Opt

U(3)

Opt

J1

J2J3

J1

J2

J3

Ai

Bi

times1

times2

times3

Figure 4 Reconstruction of third-order tensor image

In 119860119894isin 1198771198681times1198682timessdotsdotsdottimes119868

119873 119894 = 1 2 119872 119872 is the numberof samples 119860lowast

119894is the reconstructed tensor There are two

methods to minimize the cost function

119862 =

119872

sum

119894=1

1003817100381710038171003817119860 119894 minus 119860lowast

119894

1003817100381710038171003817

2

=

119872

sum

119894=1

10038171003817100381710038171003817119860119894minus 119861119894times1119880(1)times2119880(2)

times sdot sdot sdot times119873119880(119873)10038171003817100381710038171003817

2

(8)

119860lowast

119894= 119861119894times1119880(1)times2119880(2)

times sdot sdot sdot times119873119880(119873)

(9)

The first is to minimize the cost function 119862 directly andwe can calculate the orthogonal matrix by the function 119861

119894=

119860119894times1119880(1)119879

times2119880(2)119879

timessdot sdot sdot times119873119880(119873)119879

But it is difficult to calculatethe functionThe secondmethod is to maximize 1198621015840 shown as(10) and it is easier to calculate In this paper we used thesecond method

1198621015840=

119872

sum

119894=1

1003817100381710038171003817100381710038171003817119860119894times1119880(1)119879

times2119880(2)119879


2

(10)

25 Construction of the Classification of Liver Diseases In thispaper ACO is used to optimize SVM to train a liver diseasesclassifier DAG structure for multiclassification is used todistinguish liver diseases

251 Feature Selection Based on Liver Statistical FractalModelThe samples consist of the core tensor of each tensor Wetransfer the core tensor into a one-dimensional vector usingthe method of nonlinear data dimensionality reduction [10]The training set is 119863 = 119889

1 1198892 119889

119872 (119872 is the total

number of the samples) and the set of features is 119879 =

1199051 1199052 119905

119875 (119875 is the total number of the features)

252 Feature Weighed The number of gray level features weselect is too much and the number of fractal features is fewerthan it So we give a higher weight to the fractal features Aseries of experiments showed that the classification accuracyis much better when the weight of fractal feature is 06 andthe weight of gray level feature is 04

253 Construction of Classifier Based on ACO-SVM Somediseases such as cirrhosis and hepatic cyst are different from

1234

234 123

34 23 12

not 2 not 4

not 4

not 1

not 1

not 3

4 3 2 1

SVM1

SVM2 SVM3

SVM4 SVM5 SVM6

Figure 5 DAG-SVM 119896 = 4

cancer They are usually confused with cancer in CAD SVMis always used in binary classification If we want to classify 4kinds of liver diseases using SVM we should combine severalSVMs In this paper we use the method of directed acyclicgraph (DAG-SVM (119896 = 4)) to realize the multiclassificationof liver diseases and DAG is shown in Figure 5

If we classify 4 kinds of liver diseases we should use 6SVMs119862 is the penalty factor and 120590 is the parameter of kernelfunction In order to optimize these two parameters by ACOalgorithm 119862 and 120590 must be discretized firstly In this paperthe two parameters are discretized according to effective bitswhich are determined by experiences The parameter 119862 and120590 has five effective bits respectively The value of each bit canbe varied from 0 to 9 For119862 its top digit is hundreds place soits value ranges from 0 to 99999 While for 120590 its top digit isones place and thus its value ranges from 0 to 99999

Then heuristic information 120578(119894 119895) is set to 1 Classificationaccuracy is used to evaluate SVMperformance and thereforeΔ120591(119894 119895) = 119876 sdot Acc is used in the global update process Here119876 is pheromone intensity and Acc is maximal classificationaccuracy in each cycle The whole process is executed asfollows

Step 1 Discretizing parameters 119862 and 120590 by the method asabove

Step 2 Initializing pheromone 120591(119894 119895) = 1 and pheromoneincrement Δ120591(119894 119895) = 0

Step 3 Executing search process for the first best path

(1) Laying ants at the origin of coordinates

(2) Putting each ant to next city whose 119909 coordinate isdifferent from the previous visited cities randomly


(3) Modifying pheromone of transfer path for each antaccording to local update rule

(4) Modifying pheromone of the path for the best antaccording to global update rule if all the ants finishvisiting 10 nodes else returning to (2)

Step 4 Laying ants at coordinate origin again

Step 5 Putting each ant to the next city chosen according tostate transition rule

Step 6 Modifying pheromone of transfer path for each antaccording to local update rule If ants finish tour we jump toStep 7 otherwise we return to Step 5

Step 7 Training a SVM classifier with 119862 and 120590 obtained byeach antWe find out the best ant which produced the highestaccuracy and modify pheromone for the best ant accordingto global update rule If the accuracy meets terminationcondition or the times of loop are bigger than the maximumcycle times we jump to Step 8 otherwise we return to Step 4

Step 8 Outputting best 119862 120590 and maximum accuracy

3 Results and Discussion

We select 120 groups of liver images 60 groups are normalliver 20 groups are cirrhosis liver volume 20 groups are can-cer liver volume and 20 groups are hydatoncus liver volumeThere are 50 images in each group The thickness of eachimage is 3mm and the resolution is 512 times 512 In 120 groupsof images we selected a half as training samples the others astesting samples

31 Reconstruction Results after GND-PCA In this paper weuse leave-one-out method to test the generalization ability ofmodels constructedwithout fractal features for liver volumesOne of all images is shown in Figure 6The location of tumoris in the lower left corner of the liver image Firstly one vol-ume is excluded from the training data which is used for theconstruction of the model and then it is reconstructed by thetraining models for checking

The volume is reconstructed from 5 times 5 times 3 to 300 times 300 times30 which is shown in Figure 7 In Figure 7 the first row isreconstruction of slice 3 the second row is reconstructionof slice 13 and the third row is reconstruction of slice 23Column (a) is original liver image column (b) is that thedimension of mode-subspace is 5 times 5 times 3 column (c) is 100 times100 times 10 column (d) is 200 times 200 times 20 column (e) is 300 times

300 times 30 and column (f) is the reconstructed volume usingeigenface by PCA as the contrastive method

Since the dimension of the original volume is 512 times 512 times50 we can calculate the compressing rate for all casesThe compressing rate is 00006 07629 61035 and205994 With the growth of the dimension of mode-subspace reconstruction result is better Because of overfit-ting the method of PCA is worse than GND-PCA

It needs less iteration times using GND-PCA which isshown in Figure 8 and the value of the cost function does not

Figure 6 Liver cancer image

dramatically change after two iterationsTherefore we set theiteration times of GND-PCA as two in our experiment

In Figure 9 it shows the relationship between originalvolume and the reconstructed volume Abscissa a is themode-subspace of 5times 5times 3 b is 100times 100times 10 c is 200times 200times20 d is 300 times 300 times 30 e is 400 times 400 times 40 and f is 512 times

512 times 50The normalized correlation grows with the growth of

mode-subspace size When the mode-subspace size is 512 times

512 times 50 the normalized correlation is 1 It means that wecan reconstruct the original volume without any errors Thenormalized correlation can be defined as (11) 119868(119909 119910 119911) is theoriginal tensor volume and 119868

lowast(119909 119910 119911) is the volume after

reconstruction

119873119862 =

sum119909119910119911

119868 (119909 119910 119911) 119868lowast(119909 119910 119911)

radicsum119909119910119911

1198682 (119909 119910 119911)radicsum119909119910119911

119868lowast2 (119909 119910 119911)

(11)

32 Results of Classification The result of each SVM inACO-SVMmulticlassifier is shown inTable 1 From the table we cansee that the statistical fractal model has better accuracy thanthe statistical texture model without fractal feature

Compared with other classifier ACO-SVM with theweighed fractal feature has better accuracy which is shownin Figure 10 Classifier BPNN is BP neural network and theaccuracy is 6923 Classifier FL is Fisher linear classifierwith 4623 accuracy Classifier KNN is k-Nearest Neighboralgorithm whose accuracy is 4723 Classifier SVM is theconventional SVM with 6268 accuracy Classifier ACO-SVM is the conventional ACO-SVM whose accuracy is8987 Classifier F-ACO-SVM is themethod which is ACO-SVM with fractal features and the accuracy is 9143Classifier WF-ACO-SVM is ACO-SVM with weighed fractalfeature the accuracy is 9306 As Figure 10 shows ACO-SVM does better than others in classification And when weuse weighed fractal feature in our statistical fractal model wecan reach a better accuracy in liver diseases classification


Table 1 Parameters optimization result of SVM for multiclassification using ACO

Classifier Best C Best 120575 NFD Acc FD Acc WFD AccACO-SVM1 62257 13114 9644 9764 9785ACO-SVM2 78396 52349 9802 9852 9864ACO-SVM3 100230 12255 9774 9987 9987ACO-SVM4 14020 02378 9976 9983 9998ACO-SVM5 98469 11424 943 9657 9665ACO-SVM6 87678 10765 9882 9933 9964

(a) (b) (c) (d) (e) (f)

Figure 7 Reconstruction results

1001800

1002000

1002200

1002400

1002600

1002800

1003000

1003200

1 2 3 4 5 6 7

Valu

e of c

ost f

unct

ion

Number of iterations

Figure 8 Convergence of GND-PCA

4 Conclusions

In this paper we have presented the construction of high-order tensors with weighed fractal dimension feature andgray feature And GND-PCA which is a subspace learningmethod has been used to get the core tensor from those high-order tensors and establish the statistical fractal model forthe later classification ACO-SVM has been used to train aliver image classifier As an application for classifying liver

04

05

06

07

08

09

1

Nor

CoR

ate

Mode-subspace basisa b c d e f

Figure 9 Normalized correlation between the original volume andthe reconstructed volumes

diseases the method using statistical fractal models based onGND-PCA and ACO-SVM achieved the better classificationaccuracy because statistical fractal models based on GND-PCA can preserve the information of the original image asmuch as possible and ACO can find the optimal parametersfor SVM In conclusion under the condition of a smallnumber of samples the classifier of this paper can achievethe better recognition accuracy than others such as BPNNthe conventional SVM and the conventional ACO-SVM


6923

4623 4723

6268

8987 9143 9306

0102030405060708090

100

BPN

N FL

KNN

SVM

ACO

-SV

M

F-AC

O-S

VM

WF-

ACO

-SV

M

Accu

racy

()

Classifier

Figure 10 Result of multiclassification using seven classifiers

Therefore the proposed method can improve the classifica-tion accuracy of liver diseases and assist doctors to diagnoseliver diseases

Acknowledgment

The research is supported by the National Natural ScienceFoundation of China (no 61272176 no 60973071)

References

[1] M A Turk and A P Pentland ldquoFace recognition using eigen-facesrdquo in Proceedings of the IEEE Computer Society Conferenceon Computer Vision and Pattern Recognition (CVPR rsquo91) pp586ndash591 June 1991

[2] R Xu and Y W Chen ldquoGeneralized N-dimensional principalcomponent analysis (GND-PCA) and its application on con-struction of statistical appearance models for medical volumeswith fewer samplesrdquo Neurocomputing vol 72 no 10ndash12 pp2276ndash2287 2009

[3] B Liu Z F Hao and X W Yang ldquoNesting support vectormachinte for muti-classification [machinte read machine]rdquo inProceedings of the International Conference onMachine Learningand Cybernetics (ICMLC rsquo05) vol 7 pp 4220ndash4225 August2005

[4] J C Platt N Cristianini and T J Shawe ldquoLarge margin DAGsfor multiclass classificationrdquo Advances in Neural InformationProcessing Systems vol 12 no 3 pp 547ndash553 2000

[5] X Liu H Jiang and F Tang ldquoParameters optimization inSVM based-on ant colony optimization algorithmrdquo AdvancedMaterials Research vol 121-122 pp 470ndash475 2010

[6] D Tao X Li X Wu and S J Maybank ldquoGeneral tensordiscriminant analysis and Gabor features for gait recognitionrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 29 no 10 pp 1700ndash1715 2007

[7] S Peleg J Naor R Hartley and D Avnir ldquoMultiple resolutiontexture analysis and classificationrdquo IEEETransactions onPatternAnalysis andMachine Intelligence vol 6 no 4 pp 518ndash523 1984

[8] N Sarkar and B B Chauduri ldquoAn Efficient differential box-counting approach to compute fractal dimension of imagerdquo

IEEE Transactions on Systems Man and Cybernetics vol 24 no1 pp 115ndash120 1994

[9] L de Lathauwer B de Moor and J Vandewalle ldquoOn the bestrank-1 and rank-(R

1 R2 R

119899) approximation of higher-order

tensorsrdquo SIAM Journal onMatrix Analysis and Applications vol21 no 4 pp 1324ndash1342 2000

[10] H Eghbalnia A Assadi and J Carew ldquoNonlinear methods forclustering and reduction of dimensionalityrdquo in Proceedings ofthe International Joint Conference on Neural Networks (IJCNNrsquo99) vol 2 pp 1004ndash1009 July 1999

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Yes

Feature weighed

Calculation of the core tensor

Construction of liver diseasesclassifier

the core Calculation of

No

Recognition of liverdiseases

Feature weighed

Based on ACO-SVM

Calculation of optimal U

k = k + 1

Bi (i =

tensor Bj (j =

U(i)opt = U

(i)

k

Input M images of the training

setof

Segmentation

liver region

Feature selection based on fractal

and gray leveldimension tensor

Initial values

determined by

Maximization of the cost

of the testingset

Segmentation

region

Feature selectionbased on fractaland gray level

dimension tensor

function C998400

Input S imagesof liver

Construction of N-

A(i)test (j =

Update U(i)

kConstruction of N-

the first Jn

k = 0 U(n)0

whose column is |U(i)

k+1minus U

(i)

k| lt 120576

1 middot middot middot S) 1 middot middot middot S)

1 middot middot middotM)

(i = 1 middot middot middot n n = 4)A(i)train (i = 1 middot middot middotM)

Figure 1 The main flow of the proposed method

problem we extracted the gray feature and fractal featureto establish the high-order tensor of liver volume and con-structed the statistical fractalmodel withGND-PCAmethodFor improving recognition accuracy SVMoptimized byACO(ACO-SVM) was used to recognize liver diseases images

This paper is organized as follows Section 2 introducesthe proposed method firstly we will introduce some knowl-edge of PCA and tensor then we will show the constructionof high-order tensor and finally we introduce the method ofGND-PCA for the construction of statistical fractal modeland ACO-SVM [5] for classification In Section 3 we presentthe construction of liver images after GND-PCA and theresults of classification Section 4 concludes the works in thispaper

2 Materials and Methods

In this section we will introduce some background knowl-edge about GND-PCA method firstly We mainly presentthe method of PCA 2D-PCA and ND-PCA and the basicknowledge of tensor And then we will introduce our methodof construction of statistical fractal model The main flow isshown in Figure 1 The process of the proposed method isdescribed as follows

(1) Liver images preprocessing Firstly we segment theCT images of abdomen to gain the liver region of

image and then we calculate the fractal dimension ofliver image

(2) High-order tensor construction At first we collecta group of fractal features and gray level features(pixels) and then we combine them into a newdataset

(3) Statistical fractal model establishment In this paperwe use the method of Generalized 119873-dimensionalPCA (GND-PCA) to establish the statistical fractalmodel for classification

(4) Liver diseases classification based on ACO-SVMAfter we obtain the statistical fractal model by GND-PCAwe treat the core tensors as samples and thenweuse SVM optimized by ACO to classify liver diseases

21 PCA Method and Its Extension PCA is an application of119870-119871 conversion in statistics The purpose of PCA is to lowerthe dimension of data through finding a linear mapping Themapping meets the following conditions

(1) The error of sample reconstruction is minimized(2) The mapping of sample set in low dimension space

has the maximum variance(3) The correlation among samples is erasedTurk and Pentland proposed the famous method named

eigenface to realize PCA Suppose that we have 119870 training


Ntimes N image

N2 vector


samples 1198681 1198682 119868




1 1198832 119883

119870 119883119894is a


119894




119870by only


of 1198831 1198832 119883



1 1198682 119868

119870 and then we









119879119860119883 and it can be





119872 and 119873


defined as 11988311989911198992119899119872




119872times1198991015840

1times1198991015840


1015840

119872


119872

1198841198991015840

1times1198991015840


1015840

119872

(1)




shown as follows



= sum

119894119889


119880119895times119894119889

) (2)










0(119894 119895) = 119892(119894 119895) for


119906120576(119894 119895) = max119906

120576minus1(119894 119895) + 1

max|(119898119899)minus(119894119895)|le1

119906120576minus1

(119898 119899)

119887120576(119894 119895) = min119887

120576minus1(119894 119895) minus 1

min|(119898119899)minus(119894119895)|le1

119887120576minus1

(119898 119899)

(3)


V120576= sum

119894119895

(119906 (119894 119895) minus 119887120576(119894 119895)) (4)


119860 (120576) =(V120576minus V120576minus1

)

2

119860 (120576) = 1198651205762minus119863

(5)




120576 (6)




119873120575(119865) = sum119899




119863119861=log119873120575(119865)

log (1120575) (7)







119873(119869119899lt 119868119899) and 119861 needs


(119899) 119880(119899) =





I1

I2

I3

I1

I2

I3

U(1)

Opt

U(2)

Opt

U(3)

Opt

J1

J2J3

J1

J2

J3

Ai

Bi

times1

times2

times3






119862 =

119872

sum

119894=1

1003817100381710038171003817119860 119894 minus 119860lowast

119894

1003817100381710038171003817

2

=

119872

sum

119894=1



2

(8)

119860lowast

119894= 119861119894times1119880(1)times2119880(2)


(9)


119894=

119860119894times1119880(1)119879

times2119880(2)119879



1198621015840=

119872

sum

119894=1

1003817100381710038171003817100381710038171003817119860119894times1119880(1)119879

times2119880(2)119879


2

(10)



1 1198892 119889

119872 (119872 is the total


1199051 1199052 119905




1234

234 123

34 23 12

not 2 not 4

not 4

not 1

not 1

not 3

4 3 2 1

SVM1

SVM2 SVM3

SVM4 SVM5 SVM6
































reconstruction

119873119862 =

sum119909119910119911

119868 (119909 119910 119911) 119868lowast(119909 119910 119911)

radicsum119909119910119911

1198682 (119909 119910 119911)radicsum119909119910119911

119868lowast2 (119909 119910 119911)

(11)






(a) (b) (c) (d) (e) (f)


1001800

1002000

1002200

1002400

1002600

1002800

1003000

1003200

1 2 3 4 5 6 7

Valu

e of c

ost f

unct

ion



4 Conclusions


04

05

06

07

08

09

1

Nor

CoR

ate





6923

4623 4723

6268

8987 9143 9306

0102030405060708090

100

BPN

N FL

KNN

SVM

ACO

-SV

M

F-AC

O-S

VM

WF-

ACO

-SV

M

Accu

racy

()

Classifier



Acknowledgment


References











1 R2 R











Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


Ntimes N image

N2 vector


samples 1198681 1198682 119868




1 1198832 119883

119870 119883119894is a


119894




119870by only


of 1198831 1198832 119883



1 1198682 119868

119870 and then we









119879119860119883 and it can be





119872 and 119873


defined as 11988311989911198992119899119872




119872times1198991015840

1times1198991015840


1015840

119872


119872

1198841198991015840

1times1198991015840


1015840

119872

(1)




shown as follows



= sum

119894119889


119880119895times119894119889

) (2)










0(119894 119895) = 119892(119894 119895) for


119906120576(119894 119895) = max119906

120576minus1(119894 119895) + 1

max|(119898119899)minus(119894119895)|le1

119906120576minus1

(119898 119899)

119887120576(119894 119895) = min119887

120576minus1(119894 119895) minus 1

min|(119898119899)minus(119894119895)|le1

119887120576minus1

(119898 119899)

(3)


V120576= sum

119894119895

(119906 (119894 119895) minus 119887120576(119894 119895)) (4)


119860 (120576) =(V120576minus V120576minus1

)

2

119860 (120576) = 1198651205762minus119863

(5)




120576 (6)




119873120575(119865) = sum119899




119863119861=log119873120575(119865)

log (1120575) (7)







119873(119869119899lt 119868119899) and 119861 needs


(119899) 119880(119899) =





I1

I2

I3

I1

I2

I3

U(1)

Opt

U(2)

Opt

U(3)

Opt

J1

J2J3

J1

J2

J3

Ai

Bi

times1

times2

times3






119862 =

119872

sum

119894=1

1003817100381710038171003817119860 119894 minus 119860lowast

119894

1003817100381710038171003817

2

=

119872

sum

119894=1



2

(8)

119860lowast

119894= 119861119894times1119880(1)times2119880(2)


(9)


119894=

119860119894times1119880(1)119879

times2119880(2)119879



1198621015840=

119872

sum

119894=1

1003817100381710038171003817100381710038171003817119860119894times1119880(1)119879

times2119880(2)119879


2

(10)



1 1198892 119889

119872 (119872 is the total


1199051 1199052 119905




1234

234 123

34 23 12

not 2 not 4

not 4

not 1

not 1

not 3

4 3 2 1

SVM1

SVM2 SVM3

SVM4 SVM5 SVM6
































reconstruction

119873119862 =

sum119909119910119911

119868 (119909 119910 119911) 119868lowast(119909 119910 119911)

radicsum119909119910119911

1198682 (119909 119910 119911)radicsum119909119910119911

119868lowast2 (119909 119910 119911)

(11)






(a) (b) (c) (d) (e) (f)


1001800

1002000

1002200

1002400

1002600

1002800

1003000

1003200

1 2 3 4 5 6 7

Valu

e of c

ost f

unct

ion



4 Conclusions


04

05

06

07

08

09

1

Nor

CoR

ate





6923

4623 4723

6268

8987 9143 9306

0102030405060708090

100

BPN

N FL

KNN

SVM

ACO

-SV

M

F-AC

O-S

VM

WF-

ACO

-SV

M

Accu

racy

()

Classifier



Acknowledgment


References











1 R2 R











Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology







0(119894 119895) = 119892(119894 119895) for


119906120576(119894 119895) = max119906

120576minus1(119894 119895) + 1

max|(119898119899)minus(119894119895)|le1

119906120576minus1

(119898 119899)

119887120576(119894 119895) = min119887

120576minus1(119894 119895) minus 1

min|(119898119899)minus(119894119895)|le1

119887120576minus1

(119898 119899)

(3)


V120576= sum

119894119895

(119906 (119894 119895) minus 119887120576(119894 119895)) (4)


119860 (120576) =(V120576minus V120576minus1

)

2

119860 (120576) = 1198651205762minus119863

(5)




120576 (6)




119873120575(119865) = sum119899




119863119861=log119873120575(119865)

log (1120575) (7)







119873(119869119899lt 119868119899) and 119861 needs


(119899) 119880(119899) =





I1

I2

I3

I1

I2

I3

U(1)

Opt

U(2)

Opt

U(3)

Opt

J1

J2J3

J1

J2

J3

Ai

Bi

times1

times2

times3






119862 =

119872

sum

119894=1

1003817100381710038171003817119860 119894 minus 119860lowast

119894

1003817100381710038171003817

2

=

119872

sum

119894=1



2

(8)

119860lowast

119894= 119861119894times1119880(1)times2119880(2)


(9)


119894=

119860119894times1119880(1)119879

times2119880(2)119879



1198621015840=

119872

sum

119894=1

1003817100381710038171003817100381710038171003817119860119894times1119880(1)119879

times2119880(2)119879


2

(10)



1 1198892 119889

119872 (119872 is the total


1199051 1199052 119905




1234

234 123

34 23 12

not 2 not 4

not 4

not 1

not 1

not 3

4 3 2 1

SVM1

SVM2 SVM3

SVM4 SVM5 SVM6
































reconstruction

119873119862 =

sum119909119910119911

119868 (119909 119910 119911) 119868lowast(119909 119910 119911)

radicsum119909119910119911

1198682 (119909 119910 119911)radicsum119909119910119911

119868lowast2 (119909 119910 119911)

(11)






(a) (b) (c) (d) (e) (f)


1001800

1002000

1002200

1002400

1002600

1002800

1003000

1003200

1 2 3 4 5 6 7

Valu

e of c

ost f

unct

ion



4 Conclusions


04

05

06

07

08

09

1

Nor

CoR

ate





6923

4623 4723

6268

8987 9143 9306

0102030405060708090

100

BPN

N FL

KNN

SVM

ACO

-SV

M

F-AC

O-S

VM

WF-

ACO

-SV

M

Accu

racy

()

Classifier



Acknowledgment


References











1 R2 R











Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


I1

I2

I3

I1

I2

I3

U(1)

Opt

U(2)

Opt

U(3)

Opt

J1

J2J3

J1

J2

J3

Ai

Bi

times1

times2

times3






119862 =

119872

sum

119894=1

1003817100381710038171003817119860 119894 minus 119860lowast

119894

1003817100381710038171003817

2

=

119872

sum

119894=1



2

(8)

119860lowast

119894= 119861119894times1119880(1)times2119880(2)


(9)


119894=

119860119894times1119880(1)119879

times2119880(2)119879



1198621015840=

119872

sum

119894=1

1003817100381710038171003817100381710038171003817119860119894times1119880(1)119879

times2119880(2)119879


2

(10)



1 1198892 119889

119872 (119872 is the total


1199051 1199052 119905




1234

234 123

34 23 12

not 2 not 4

not 4

not 1

not 1

not 3

4 3 2 1

SVM1

SVM2 SVM3

SVM4 SVM5 SVM6
































reconstruction

119873119862 =

sum119909119910119911

119868 (119909 119910 119911) 119868lowast(119909 119910 119911)

radicsum119909119910119911

1198682 (119909 119910 119911)radicsum119909119910119911

119868lowast2 (119909 119910 119911)

(11)






(a) (b) (c) (d) (e) (f)


1001800

1002000

1002200

1002400

1002600

1002800

1003000

1003200

1 2 3 4 5 6 7

Valu

e of c

ost f

unct

ion



4 Conclusions


04

05

06

07

08

09

1

Nor

CoR

ate





6923

4623 4723

6268

8987 9143 9306

0102030405060708090

100

BPN

N FL

KNN

SVM

ACO

-SV

M

F-AC

O-S

VM

WF-

ACO

-SV

M

Accu

racy

()

Classifier



Acknowledgment


References











1 R2 R











Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology























reconstruction

119873119862 =

sum119909119910119911

119868 (119909 119910 119911) 119868lowast(119909 119910 119911)

radicsum119909119910119911

1198682 (119909 119910 119911)radicsum119909119910119911

119868lowast2 (119909 119910 119911)

(11)






(a) (b) (c) (d) (e) (f)


1001800

1002000

1002200

1002400

1002600

1002800

1003000

1003200

1 2 3 4 5 6 7

Valu

e of c

ost f

unct

ion



4 Conclusions


04

05

06

07

08

09

1

Nor

CoR

ate





6923

4623 4723

6268

8987 9143 9306

0102030405060708090

100

BPN

N FL

KNN

SVM

ACO

-SV

M

F-AC

O-S

VM

WF-

ACO

-SV

M

Accu

racy

()

Classifier



Acknowledgment


References











1 R2 R











Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology




(a) (b) (c) (d) (e) (f)


1001800

1002000

1002200

1002400

1002600

1002800

1003000

1003200

1 2 3 4 5 6 7

Valu

e of c

ost f

unct

ion



4 Conclusions


04

05

06

07

08

09

1

Nor

CoR

ate





6923

4623 4723

6268

8987 9143 9306

0102030405060708090

100

BPN

N FL

KNN

SVM

ACO

-SV

M

F-AC

O-S

VM

WF-

ACO

-SV

M

Accu

racy

()

Classifier



Acknowledgment


References











1 R2 R











Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


6923

4623 4723

6268

8987 9143 9306

0102030405060708090

100

BPN

N FL

KNN

SVM

ACO

-SV

M

F-AC

O-S

VM

WF-

ACO

-SV

M

Accu

racy

()

Classifier



Acknowledgment


References











1 R2 R











Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology








Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology

Documents

Research Article Statistical Fractal Models Based on GND ...downloads.hindawi.com/journals/bmri/2013/656391.pdf · Research Article Statistical Fractal Models Based on GND-PCA and