research article face recognition using mlp and rbf neural...

Research ArticleFace Recognition Using MLP and RBF NeuralNetwork with Gabor and Discrete Wavelet TransformCharacterization A Comparative Study

Fatma Zohra Chelali and Amar Djeradi

Speech Communication and Signal Processing Laboratory Faculty of Electronics and Computer Science University of Science andTechnology Houari Boumedienne (USTHB) PO Box 32 16111 Algiers Algeria

Correspondence should be addressed to Fatma Zohra Chelali chelali zohrayahoofr

Received 13 September 2014 Revised 1 December 2014 Accepted 16 December 2014

Academic Editor Marco Perez-Cisneros

Copyright copy 2015 F Z Chelali and A Djeradi This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Face recognition has received a great attention froma lot of researchers in computer vision pattern recognition andhumanmachinecomputer interfaces in recent years Designing a face recognition system is a complex task due to the wide variety of illuminationpose and facial expression A lot of approaches have been developed to find the optimal space in which face feature descriptors arewell distinguished and separated Face representation usingGabor features and discrete wavelet has attracted considerable attentionin computer vision and image processing We describe in this paper a face recognition system using artificial neural networks likemultilayer perceptron (MLP) and radial basis function (RBF) where Gabor and discrete wavelet based feature extraction methodsare proposed for the extraction of features from facial images using two facial databases the ORL and computer vision Goodrecognition rate was obtained using Gabor and DWT parameterization with MLP classifier applied for computer vision dataset

1 Introduction

Face recognition has received a great attention from a lot ofscientists in computer vision pattern recognition andhumanmachine computer interfaces in recent years

Face is the most common biometric identifier used byhumans this domain is motivated by the increased interestin the commercial applications of automatic face recognition(AFR) as well as by the emergence of real-time processorsAutomatic face recognition (FR) is also one of themost visibleand challenging research topics in computer vision machinelearning and biometrics [1]

Face recognition has become important because of thepotential value for applications and its theoretical challengesToday face recognition technology is being used to combatpassport fraud support law enforcement identify missingchildren and identify fraud There are so many challengesin face recognition some of them are (1) illumination vari-ances (2) occlusions and (3) different expressions and poses[2]

There are many approaches to evaluate face images Themethods used in face recognition can be generally classi-fied into image feature based and geometry feature basedmethods In feature geometry based approach recognitionis based on the relationship between human facial featuressuch as eye(s) mouth nose and face boundary and subspaceanalysis approach attempts to capture and define the face asa whole The subspace method is the most famous techniquefor face recognition In this method the face is treated as two-dimensional pattern of intensity variation [3]

Although facial images have a high dimensionality theyusually lie on a lower dimensional subspace or submanifoldTherefore subspace learning and manifold learning methodshave been dominantly and successfully used in appearancebased FR The classical Eigenface and Fisherface algorithmsconsider only the global scatter of training samples andthey fail to reveal the essential data structures nonlinearlyembedded in high-dimensional space [1]

Face representation using Gabor features has attractedconsiderable attention in computer vision image processing

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 523603 16 pageshttpdxdoiorg1011552015523603

2 Mathematical Problems in Engineering

base

ApplyDWT

Construct feature DWT vectors for

each class

MLPRBFclassifier

DWT-NNRR ()

Gabor-NNRR ()

classifierMLPRBF

Define the parametersof Gabor filters

Convolution of Gwith image I(x y)

in gray levelfiltered images

Originalimage

Filteredimage

Facial image data

G 120590x 120590y f and 120579

Figure 1 An overview of face recognition system

pattern recognition and so on The principal motivation touse Gabor filters is biological relevance that the receptive fieldprofiles of neurons in the primary visual cortex of mammalsare oriented and have characteristic spatial frequencies

Gabor filters can exploit salient visual properties suchas spatial localization orientation selectivity and spatialfrequency characteristics [4]

The Gabor filter was first introduced by David Gabor in1946 and was later shown as models of simple cell receptivefields Since the Gabor features are extracted in local regionsthey are less sensitive to variations of illumination expres-sion and pose than the holistic features such as Eigenface andRandomface [1]

Face recognition usingGabor filterswas firstly introducedby Yang and Zhang and soon proved to be a very effectivemeans in human facial features extraction [1]

We are interested in our work to develop a recognitionsystem using Gabor filters and discrete wavelet transform asfeature extraction Amultilayer perceptron (MLP) and radialbasis function (RBF) neural network classifier are used toclassify an individual from two face databases the ORL andcomputer vision databases Figure 1 describes our system

We investigate then the best way to characterize the facialimages efficiency is evaluated by computing recognition rate(RR) for each parameterization and classifier

The paper is presented as follows Section 2 presents theface characterization using Gabor filters and discrete wavelettransform DWT Section 3 presents simulation results andconclusion is drawn in Section 5

2 Feature Face Extraction andFace Parameterization

The most important step in face recognition system is thefeature extraction It permits us to reduce dimensionalityand characterize the face image using a vector descriptorwhich minimizes intraperson dissimilarities and maximizesthe extraperson difference This task is difficult because ofvariations of pose age expression and illumination and soforth

In our work we use Gabor filters and discrete wavelettransform to describe our face and to build a face recognitionsystem using the multilayer perceptron (MLP) and radialbasis function (RBF) neural network classifier Comparisonis done for each classifier and facial database to decidethe optimal characterization We describe first the Gaborfunction and discrete wavelet transforms to characterize ourfacial images

21 Gabor Filter Representation

211 Gabor Wavelet A Mathematical Overview Gabor filterdefined by Dennis Gabor is widely used in image analysispattern recognition and so forth Gabor filters present twointeresting properties frequency localization and selectivityin orientation

A lot of studies demonstrate that representation byGabor wavelet for image analysis is robust to variation inillumination and facial expressions

Gabor filter in 2D was introduced in biometric researchbyDaugman for iris recognition Lades et al usedGabor filterfor face recognition using ldquoDynamic Link Architecturerdquo

The Gabor wavelet which captures the properties oforientation selectivity spatial localization and optimallylocalized in the space and frequency domains has beenextensively and successfully used in face recognition [5]Daugman pioneered the using of the 2D Gabor waveletrepresentation in computer vision in 1980s [6 7]

Gabor wavelets (filters) characteristics for frequency andorientation representations are quite similar to those ofhuman visual system These have been found appropriatefor texture representation and discrimination This Gaborwavelet based extraction of features directly from the gray-level images is successful and has widely been applied totexture segmentation and fingerprint recognition The com-monly used Gabor filters in face recognition area [3 4] aredefined as follows [6]

Ψ119906V (119885) =

1003817100381710038171003817119896119906V1003817100381710038171003817

2

1205902exp(

minus1003817100381710038171003817119896119906V

1003817100381710038171003817

2

1198852

2 lowast 1205902)[119890119894119896119906V119885 minus 119890

minus1205902

2

]

(1)119885(119909 119910) is the point of coordinates (119909 119910) in image space

119906 and V define the orientation Gabor frequency and 120590 is thestandard deviation of the Gaussian

120583 and V define the orientation and the scale of the Gaborfilters and 119896

120583V is defined as the following form

119896120583V = 119896V119890

119895120593120583 (2)

Mathematical Problems in Engineering 3

x

y

Ψ120583

Figure 2 Gabor filter [2]

119896V = 119896max119891V and 120593

119906= 1205871205838 119896max is the maximum

frequency and 119891 is the spacing factor between kernels in thefrequency domain Usually 120590 = 2120587 119896max = 1205872 and 119891 = radic2

The Gabor wavelet representation of a face image isobtained by doing a convolution between the image and afamily of Gabor filters as described by (3)The convolution ofimage 119868(119911) and a Gabor filterΨ

120583V(119911) can be defined as follows[6]

119865120583V (119911) = 119868 (119911) lowast Ψ

120583V (119911) (3)

where 119911 = (119909 119910) lowast denotes the convolution operatorand 119865

120583V(119911) is the Gabor filter response of the image withorientation119906 and scale V [6] Figure 2 illustrates this principle

The Gabor kernels in (1) are all self-similar since theycan be generated from the same filter the mother wavelet byscaling and rotating via the wave vector 119896

120583VEach kernel is a product of a Gaussian envelope and

a complex plane wave These representation results displayscale locality and orientation properties corresponding tothose displayed by the Gabor wavelets [6]

Each kernel is a product of a Gaussian envelope anda complex plane wave and can be separated into real andimaginary parts Hence a band of Gabor filters is generatedby a set of various scales and rotationsAdetail ofGabor filterscalculations or design can be described later

Gaborwavelet of an image 119868 is then a convolution productof 119868 with a set of Gabor filters with different frequency andorientations

The convolution of the image 119868 with Gabor Ψ119906V(119911) is

defined by

119866119906V (119885) = 119868 (119885) lowast Ψ

119906V (119911) (4)

Figure 3 presents an example of facial representation inGabor wavelet using 40 Gabor filters

212 Design of Gabor Filter Gabor filterworks as a band-passfilter for the local spatial frequency distribution achievingan optimal resolution in both spatial and frequency domainsThe 2DGabor filterΨ

119891120579(119909 119910) can be represented as a complex

sinusoidal signal modulated by a Gaussian kernel function asfollows [4 8]

Ψ119891120579(119909 119910) = exp[minus1

21199092

120579119899

1205902119909

+1199102

120579119899

1205902119910

] exp (2120587119891119909120579119899) (5)

where

[119909120579119899

119910120579119899

] = [sin 120579119899

cos 120579119899

minus cos 120579119899

sin 120579119899

][119909

119910] (6)

120590119909 120590119910are the standard deviations of the Gaussian enve-

lope along the119909- and119910-dimensions119891 is the central frequencyof the sinusoidal plane wave and 120579

119899the orientation The

rotation of the 119909-119910 plane by an angle 120579119899will result in a Gabor

filter at the orientation 120579119899 The angle 120579

119899is defined by

120579119899=120587

119901(119899 minus 1) (7)

119899 = 1 2 119901 and 119901 isin 119873 where 119901 denotes the numberof orientations

Design ofGabor filters is accomplished by tuning the filterwith a specific band of spatial frequency and orientation byappropriately selecting the filter parameters the spread of thefilters 120590

119909 120590119910 radial frequency 119891 and the orientation of the

filter 120579119899[4]

The important issue in the design of Gabor filters for facerecognition is the choice of filter parameters

The Gabor representation of a face image is computed byconvolving the face image with the Gabor filters Let 119891(119909 119910)be the intensity at the coordinate (119909 119910) in a grayscale faceimage its convolution with a Gabor filterΨ

119891120579(119909 119910) is defined

as [4]

119892119891120579(119909 119910) = 119891 (119909 119910) lowast Ψ

119891120579(119909 119910) (8)

where lowast denotes the convolution operator The responseto each Gabor kernel filter representation is a complexfunction with a real part Re119892

119891120579(119909 119910) and an imaginary

part Im119892119891120579(119909 119910) The magnitude response 119892

119891120579(119909 119910) is

expressed as

10038171003817100381710038171003817119892119891120579(119909 119910)

10038171003817100381710038171003817= radicRe2 119892

119891120579(119909 119910) + Im2 119892

119891120579(119909 119910) (9)

We use the magnitude response 119892119891120579(119909 119910) to represent

our facial imagesThe Gabor filter is basically a Gaussian modulated by a

complex sinusoid described by (5) and (6)All filters could be produced by rotation and dilatation of

a mother wavelet A majority of face recognition approachesbased on Gabor filters used only the magnitude of theresponse or a fusion of the magnitude and the real partbecause of the variation of the phase with local patterns andit can be considered irrelevant

The response of the Gabor filter depends on the followingparameters

(a) The orientation 120579 it describes the orientation of thewavelet and characterizes the angles of contour andline images


(a) (b)

Figure 3 Example of facial representation in Gabor wavelet of face (a) amplitude response (b) phase response with 40 Gabor filters (5 scalesand 8 orientations)

(b) The frequency 119891 it specifies the wavelengths of thefunction wavelets with large wavelength are sensitiveto progressive illumination changes in the imagewhereas low wavelengths are sensitive to contours

(c) The standard deviation 120590 this parameter defines theradius of the Gaussian The size of the Gaussiandetermines the number of pixels of image taken intoaccount for the convolution operator

The set of Gabor filters is characterized by a certainnumber of resolutions orientations and frequencies knownas ldquocharacteristicsrdquo

Figure 4 shows the response of the Gabor filter whenvarying the total parameters After a lot of experiments it hasbeen proved that the following parameters 120590

119909= 2 120590

119910= 4

120579 = 1205873 119891 = 16 are relevant for face parameterization andbuild our face recognition system that will be described infuture sections

Gabor filters are used in contour detection this parametercan be used to differentiate an individual from anotherThey represent also the characteristic points in the humanface They are widely used in texture characterization and incontour detection

A simple Gabor filter can be expressed as the product ofa Gaussian and a complex exponential The resultant filteredimage IG can be illustrated for the choice orientation andstandard deviations by Figure 5

We use only the magnitude information without thephase the module of the filtered image shows local informa-tion for the details of the image (frequency decomposition)and for the orientation (directional information) Magnitudeanalysis permits us to detect local structures of interest Thefiltered image presents the characteristic features of a personlike lip eyes and so forth

Face recognition usingGabor filterswas firstly introducedby Lades et al [9] and soon proved to be a very effectivemeans in human facial features extraction Wang et al [10]

proposed a face recognition algorithm combined vectorfeatures consisting of the magnitude of Gabor PCA andclassification SVM [6]

The Gabor wavelet is a continuous filter making the filterprocess computationally demanding Awavelet is a waveformof limited duration that has an average value of zero Thelimited duration of the wavelet enables preservation of spatialinformation in the signal [11]

The following paragraph presents an overview of discretewavelet analysis followed by the application for face parame-terization

22 Wavelet Analysis The discrete wavelet transform isanalogous to the Fourier transform with the exception thatthe DWT uses scaled and shifted versions of a wavelet Itdecomposes a signal into a sumof shifted and scaledwaveletsThe DWT kernels are very similar to Gabor kernels andexhibit properties of horizontal vertical and diagonal direc-tionality Also the DWT possesses the additional advantagesof sparse representation and nonredundant computationwhich make it advantageous to employ the DWT over theGabor transform for facial feature extraction [11] DWThas been successfully used for denoising compression andfeature detection applications in image processing to providefaster performance times and achieve results comparable tothose employing Gabor wavelets [11]

As described in [11] we present a brief review of continu-ous and discrete wavelet relevant to our analysis 2D discretewavelet decomposition is then described and applied for facerecognition

The continuous wavelet transform (CWT) between asignal 119904(119905) and a waveletΨ(119905) is mathematically defined as [11]

119862 (119886 119887) =1

radic119886int 119904 (119905) Ψ(

119905 minus 119887

119886)119889119905 (10)

where 119886 is the scale 119905 is the time and 119887 is the shift


0

5

10

0

5

0

05

1

minus05

(a)

0

5

10

0

5

0

05

1

minus05

(b)

02

46

0

5

10

0

05

1

minus05

(c)

02

46

0

5

10

0

05

1

minus05

(d)

Figure 4 The response of the filter using the parameters (a) 120590119909= 2 120590

119910= 4 120579 = 1205873 119891 = 16 (b) 120590

119909= 2 120590

119910= 4 120579 = 21205873 119891 = 16 (c)

120590119909= 4 120590

119910= 2 120579 = 1205873 119891 = 16 and (d) 120590

119909= 4 120590

119910= 2 120579 = 21205873 119891 = 16

TheDWT is obtained by restricting the scale 119886 to powersof 2 and the position 119887 to integer multiples of the scales andis given by

119888119895119896

= 119862(1

2119895119896

2119895) = 2

1198952

int

+infin

minusinfin

119909 (119905) Ψ (2119895

119905 minus 119896) 119889119905 (11)

where 119895 and 119896 are integers and Ψ119895119896

are orthogonal babywavelets defined as

Ψ119895119896

= 21198952

Ψ(2119895

119905 minus 119896) (12)

Substituting (12) into (11) yields

119888119895119896

= int

+infin

minusinfin

119909 (119905) Ψ119895119896119889119905 (13)

The restricted choice of scale and position results in a sub-sample of coefficients Baby wavelets Ψ

119895119896have an associated

baby scaling function given by

120601119895119896(119905) = 2

1198952

120601 (2119895

119905 minus 119896) (14)

The scaling function or dilation equation can be expressed interms of low-pass filter coefficients ℎ

0(119899) as

120601 (119905) = sum

119899

ℎ0(119899)radic2120601 (2119905 minus 119899) (15)

In addition the wavelet function itself can be expressed interms of high-pass filter coefficients ℎ

1(119899) as

Ψ (119905) = sum

119899

ℎ1(119899)radic2120601 (2119905 minus 119899) (16)

Also a signal 119909(119905) can be represented by the scaling andwavelet functions

119909 (119905) = sum

119896

cA1(119896) 120601119895minus1119896

(119905) + sum

119896

cD1(119896) Ψ119895minus1119896

(119905) (17)

Here cA1(119896) are the approximation coefficients at level 1 and

cD1(119896) are the detail coefficients at level 1 Equation (17) rep-

resents the analysis of a signal that can be repeated numeroustimes to decompose the signal into lower subspaces as shownin Figure 6 [11]


1

05

0

minus050 20 40 60

Response of the filter G

Filtered image

1

05

0

minus055

4

3

2

1 02

46

8104

Figure 5 The response of the filter using the following parameters 120590119909= 2 120590

119910= 4 120579 = 1205873 119891 = 16

The detail and approximation coefficients are found usingthe filter coefficients ℎ

0(119899) and ℎ

1(119899)

cA1(119896) = sum

119899

cA0(119899) ℎ0(119899 minus 2119896) (18)

Equation (18) shows that the level 1 approximation coeffi-cients cA

1can be found by convolving cA

0with the low-pass

filter ℎ0(119899) and downsampling by 2 (indicated by darr2) which

simply means disregarding every second output coefficientSimilarly the level 1 detailed coefficient cD

1 can be found

by convolving cD0with the high-pass filter ℎ

1(119899) and down-

sampling by 2 Therefore the DWT can be performed usingthe filter coefficients ℎ

0(119899) and ℎ

1(119899) The process is applied

repeatedly to produce higher levels of approximation anddetail coefficients as shown in Figure 6 Each time the filteroutputs are downsampled the number of samples is halvedThe original signal 119909(119905) is assumed to start in the subspace1198810 Therefore the level 0 approximation coefficients are the

discrete values of the signal 119909(119899) [11]

2

2

2

2

x(n)

h0(n)h0(n)

h1(n)

h1(n)

cA1

cA2

cD1

cD2

Figure 6 DWT using filter banks

23 Discrete Wavelet Transform (DWT) Applied for FaceRecognition Discrete wavelet transform (DWT) is a suitabletool for extracting image features because it allows theanalysis of images on various levels of resolution Typicallylow-pass and high-pass filters are used for decomposing theoriginal image


2

2L

H L

H

LH

2 LL2 LH

2 HL

2 HH

Rows Columns

N2

ap hd

vd dd

M2NlowastMNlowastM2

N2lowastM2

Figure 7 First level of decomposition of a facial image

The low-pass filter results in an approximation image andthe high-pass filter generates a detail image The approx-imation image can be further split into a deeper level ofapproximation and detail according to different applications[12]

Suppose that the size of an input image is119873 times119872 At thefirst filtering in the horizontal direction of downsampling thesize of images will be reduced to 119873 times (1198722) After furtherfiltering and downsampling in the vertical direction foursubimages are obtained each being of size (1198732) times (1198722)The outputs of these filters are given by [12]

119886119895+1

[119901] =

119899=+infin

sum

119899=minusinfin

119897 [119899 minus 2119901] 119886119895[119899]

119889119895+1

[119901] =

119899=+infin

sum

119899=minusinfin

ℎ [119899 minus 2119901] 119886119895[119899]

(19)

where 119897[119899] and ℎ[119899] are coefficients of low-pass and high-passfilters respectively

Accordingly we can obtain four images denoted asLL HL LH and HH The LL image is generated by twocontinuous low-pass filters HL is filtered by a high-pass filterfirst and a low-pass filter later LH is created using a low-passfilter followed by a high-pass filter HH is generated by twosuccessive high-pass filters [12]

Figure 7 presents the first level of decomposition appliedfor a facial image of dimension119873 lowast119872 pixels where

LL describes the approximation (ap in Figure 7)LH and HL describe the horizontal and verticaldetailsHH describes the diagonal detail

After first level wavelet decomposition the output imagesbecome input images of second level decomposition

Every subimage can be decomposed further into smallerimages by repeating the above procedure The main featureof DWT is the multiscale representation of a functionBy using the wavelets a given image can be analyzed atvarious levels of resolution Since the LL part contains mostimportant information and discards the effect of noises andirrelevant parts we extract features from the LL part of thefirst level decomposition The reasons are that the LL partkeeps the necessary information and the dimensionality of

Figure 8 1st level decomposition

the image is reduced sufficiently for computation at the nextstage [12]

One major advantage afforded by wavelets is the abilityto perform local analysis that is to analyze a localized areaof a larger signal In wavelet analysis we often speak aboutapproximations and detailsThe approximations are the high-scale low-frequency components of the signalThe details arethe low-scale high-frequency components

A lot of tests have been carried we adopt the Haar db2db4 db8 and bior22 wavelet with first level decompositionThe DWT feature vector is assigned for each individualand for the whole database containing facial images Thechoice of wavelet function is very important task in featuresextraction The effectiveness of recognition system dependson this selection

Figure 8 shows the first level decomposition using Haarwavelet a facial image size from computer vision databaseis reduced to (100 lowast 90) with 1st level decomposition Thedimension becomes (50 lowast 45) with 2nd level of decomposi-tion

The graph in Figure 9 describes the distribution of theresultant signal or images (approximation and detail) with


2

15

1

05

0

minus05400

300200

1000 0

100200

300

Distribution of LL LH HL and LL

Figure 9 Distribution of 1st level with Haar DWT

first level decomposition using Haar wavelet It is clearthat a major energy of the approximation information isconcentrated in the region with red colour whereas thehorizontal and vertical details appear with a little energy inblue colour

From our experiments and after a lot of tests only firstlevel decomposition is sufficient to describe correctly ourface image and to build a face recognition system with goodrecognition rate

The DWT is able to extract the horizontal and verticaldetails of a facial image However unlike the Gabor waveletswhich can be oriented to any angle the 2DDWT is restrictedin terms of orientation to horizontal diagonal and direc-tionality (0∘ 45∘ and 90∘) Since features of the face in a2D grayscale image are a union or combination of infinitelyoriented lines the ability to filter at several orientations isnecessary to achieve acceptable face recognition performance[11]

The DWT analysis applied for face recognition has a lotof advantages compared to Gabor analysis the first benefityields with dimensionality reduction which permits us lesscomputational complexity whereas the second advantageconcerns the image decomposition in frequency domain intosubbands at different scales

In this paper we examine the effects of different featureextraction methods like DWT and Gabor on the face recog-nition system and compare with other approaches like DCTfeature extraction and PCA or LDA spatial feature reductionrealized and discussed in a previous work [13]

3 Simulation Results

31 FaceDatabases Our experimentswere performedon twoface databases computer vision and ORL database

First we used the computer vision dataset it containsfrontal images of 395 individuals and each person has 20frontal images This dataset contains images of people ofvarious racial origins mainly of first year undergraduatestudents so the majority of individuals are between 18 and20 years old but some older individuals are also present

Figure 10 Examples from computer vision database [14]

Figure 11 Some examples of acquisition with different face orienta-tion and with different illuminations from ORL database [15]

Images for one person differ from each other in lighting andfacial expressionThe images are 256-colour level with size of200 lowast 180 Some individuals are wearing glasses and beardsThe total number of images is 7900 Samples of the databaseare shown in Figure 10

Wavelet has got a great attention for face recognition thewavelet transform has emerged as a cutting edge technologyThe discrete wavelet transform (DWT) is a linear invertibleand fast orthogonal operation Wavelets are a good math-ematical tool for hierarchically decomposing occupationsWithout degrading the quality of the image to an unaccept-able level image compression minimizes the size in bytes of agraphics file The reduction in file size permits more imagesto be stored in a certain amount of disk or memory space [3]

We used also ORL database for analysing the perfor-mance of our classifier this database contains a set of faceimages taken at the Olivetti Research Laboratory (ORL) inCambridge University UK We use 200 images of 20 indi-viduals For some subjects the images were taken at differenttimes which contain quite a high degree of variability in light-ing facial expression (openclosed eyes smilingnonsmilingetc) pose (upright frontal position etc) and facial details(glassesno glasses) All images are 8-bit grayscale of size 112times92 pixels Samples of theORLdatabase are shown in Figure 11

32 Face Characterization Our face recognition system istested first on the computer vision database We use 200images of 10 subjects In the following experiments tenimages are randomly selected as the training samples andanother ten images as test images Therefore a total of 100images (10 for each individual) are used for training andanother 100 for testing and there are no overlaps between thetraining and testing sets

For ORL database we select randomly 100 samples (5 foreach individual) for training The remaining 100 samples areused as the test set


In our face recognition experiments we extract featuresof facial images from different wavelet transforms (HaarDaubechies and Biorthogonal) by decomposing face imagein LL subbands at first level The wavelets used are Haar db2db4 db8 and bior22 In addition we apply Gabor waveletin order to characterize our facial images Feature vectorsextracted from each method are fed into a MLP and RBFneural network classifier to perform efficiency of our facerecognition

Before applying our face recognition system based onMLP and RBF neural network we have applied the correla-tion criterion between training and test features first for HaarDWT and Gabor features for primary analysis

Figure 12 shows variation of correlation for some individ-uals with the 10 test images from the computer vision datasetWe noticed that we have a good separation (correlationcoefficient is greater than a threshold) for the first the secondand the fifth individual and bad separation for the third andthe sixth individual A threshold of 08 was chosen for thecoefficient of correlation to decide the correct classificationThe discrimination is clear for some individuals whereasresults obtained were not very satisfactory a recognition rateof 725 was obtained

We have computed correlation between training and testfeature vectors on the computer vision dataset using tenclasses and features from Gabor and Haar DWT Figure 13shows recognition rate obtained for each class a globalrecognition rate of 72 to 73 was obtained We noticed badclassification for the third the sixth and the eight classes

In addition the investigation of number of trainingimages when varied from 5 to 10 for computer vision datasetpermits us to perform better recognition rate for 10 facialimages in training and testing phase Figure 14 shows thecorrect classification for each class a global recognition rateof 72 was obtained with 10 facial images whereas only 67was obtained with 5 training images for each class

However the correct classification rate depends on thenumber of variables in each class which will improve thecorrect classification rate for a large number of trainingimages for each individual

We conclude from this analysis that the choice of 10images for training and test is more efficient to characterizethe total database and to perform an appreciable correctclassification only by applying the correlation criterion

We investigate now the use of a neural network clas-sifier to build a face recognition system based on discretewavelet and Gabor characterizationThe following paragraphdescribes a brief theory on the classifiers used in ourwork anddiscusses the simulation results obtained

33 Face Recognition System Using Artificial Neural NetworkNeural networks are feedforward and use the backpropaga-tion algorithm We imply feedforward networks and back-propagation algorithm (plus full connectivity) While inputsare fed to the ANN forwardly the ldquobackrdquo in backpropagationalgorithm refers to the direction to which the error istransmitted [16]

Face recognition is achieved by employing a multilayerperceptron with backpropagation algorithm and radial basis

function classifier For the MLP classifier the first layerreceives the Gabor features or the DWT coefficients with firstlevel decomposition The number of nodes in this layer isequal to the dimension of the feature vector incorporating theDWT or Gabor features The number of the output nodes isequal to the number of the registered individuals Finally thenumber of the hidden nodes is chosen by the user

The features test matrix is defined with variables calledtarget the target matrix has the same dimension as thetraining matrix The network is trained to output a 1 in thecorrect position of the output vector and to fill the rest of theoutput vector with 0rsquos

The MLP NN trained present fast convergence and thetraining process terminated within 4000 to 6000 epochsdepending on the facial database and the size of featuredescriptors with the summed squared error (SSE) reachingthe prespecified goal (10-3)

We used log-sigmoid and tang-sigmoid functions as atransfer function at all neurons (in hidden layer and outputlayer) log-sigmoid function is ideal for face recognitionsystem using Gabor filters and wavelet decomposition

In order to show the importance of processing elementswe trained our MLP classifier with variable hidden unit from10 to 100 For a small number of neurons (10 to 20) in thehidden layer we observed large mean squared error (MSE)so low accuracy The MLP generalizes poorly After sim60 to70 neurons MSE came back to the levels of a system withonly 10 neurons in the hidden layerWhen there are toomanyneurons poor performance is a direct effect of overfittingThe system overfits the training data and does not performwell on novel patterns

In addition we have used an RBF classifier to decidethe correct classification An RBF neural network shown inFigure 15 can be considered a mapping 119877119903 rarr 119877119904

Let 119875 isin 119877119903 be the input vector and let 119862

119894isin 119877119903 (1 le 119894 le 119906)

be the prototype of the input vectors [16] The output of eachRBF unit is as follows

119877119894(119875) = 119877

119894(1003817100381710038171003817119875 minus 119862119894

1003817100381710038171003817) 119894 = 1 119906 (20)

where sdot indicates the Euclidean norm on the input spaceUsually the Gaussian function (Figure 15) is preferred amongall possible radial basis functions due to the fact that it isfactorizable

119877119894(119875) = exp[minus

1003817100381710038171003817119875 minus 1198621198941003817100381710038171003817

2

1205902

119894

] (21)

where 120590119894is the width of the 119894th RBF unitThe 119895th output 119910

119895(119875)

of a neural network is

119910119894(119875) =

119906

sum

119894=1

119877119894(119875) lowast 119908 (119895 119894) (22)

where 1198770 = 1 119908(119895 119894) is the weight or strength of the 119894threceptive field to the 119895th output [16]

According to [16] (21) and (22) show that the out-puts of an RBF neural classifier are characterized by alinear discriminant function They generate linear decision


0 100 200 300 400 500 600 700 800 900 10000

01

02

03

04

05

06

07

08

09

1

Training images

Reco

gniti

on ra

teRR for 5th class

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 1st class

minus02

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 2nd class

minus02

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 3rd class

minus020 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 6th class

minus02

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 4th class

minus02

Figure 12 Variation of correlation for some individuals with the 10 test images from the computer vision dataset for Haar representation


0

20

40

60

80

100

120

Haar-corr-cv datatsetGabor-corr-cv datatset

Clas

s1

Clas

s2

Clas

s3

Clas

s4

Clas

s5

Clas

s6

Clas

s7

Clas

s8

Clas

s9

Clas

s10

Figure 13 Recognition rate obtained for Gabor and Haar usingcorrelation analysis applied for computer vision dataset

0

20

40

60

80

100

120

5 training images10 training images

RR (

)

Clas

s1

Clas

s2

Clas

s3

Clas

s4

Clas

s5

Clas

s6

Clas

s7

Clas

s8

Clas

s9

Clas

s10

Figure 14 Recognition rate obtained for Gabor using correlationanalysis applied for computer vision dataset when varying thenumber of training images from 5 to 10

boundaries (hyperplanes) in the output space Consequentlythe performance of an RBF neural classifier strongly dependson the separability of classes in the 119906-dimensional spacegenerated by the nonlinear transformation carried out by the119906 RBF units [16]

According to Coverrsquos theorem on the separability ofpatterns where in a complex pattern classification problemcast in a high-dimensional space nonlinearly is more likelyto be linearly separable than in a low-dimensional space thenumber of Gaussian nodes 119906 gt= 119903 where 119903 is the dimensionof input space On the other hand the increase of Gaussian

12

1

08

06

04

02

120590i

minus4 minus3 minus2 minus1 1 2 3 4x

f1(x)

0 minus Ci

Figure 15 Radial basis function

units may result in poor generalization because of overfittingespecially in the case of small training sets [16] It is importantto analyze the training patterns for the appropriate choice ofRBF hidden nodes

Geometrically the key idea of an RBF neural network isto partition the input space into a number of subspaces whichare in the form of hyperspheres Accordingly clusteringalgorithms (119896-means clustering fuzzy-means clustering andhierarchical clustering) which are widely used in RBF neuralnetworks are a logical approach to solve the problems [16]

However it should be noted that these clusteringapproaches are inherently unsupervised learning algorithmsas no category information about patterns is used Whileconsidering the category information of training patterns itshould be emphasized that the class memberships not onlydepended on the distance of patterns but also depended onthe Gaussian widths [16]

To perform the efficiency of our proposedmethod appliedfor ORL and computer vision CV database we comparedrecognition rate (RR) obtained for each database and for eachparameterization Section 34will describe the face identifica-tion experiments using theMLP classifierwhereas Section 35presents the simulation results for the RBF classifier

34 MLP Neural Network Classifier for Face RecognitionOnce feature extraction is studied the novel vectors arefed into a MLP and RBF classifier to decide the correctclassification rate

After Gabor filtering we construct our training matrixconsisting of the module of filtered image The characteristicfeatures vectors are fed into a MLP neural classifier withits architecture We use only the magnitude information forGabor characterization

Using Haar discrete wavelet and the ORL database forexample the images of size 112 lowast 92 pixels we retain theapproximation vector of size (56 lowast 46) the resultant vectordescriptor using the reshape function by Matlab permitsus to have a novel vector of size 2576 of all the imagesin ORL database The approximation vector of db2 db4db8 and bior22 is 2679 2891 3339 and 2784 respectivelyTable 1 summarizes the total vector size of feature vectors


Table 1 Size of feature vector with DWT approach

Method (wavelet) Haar db2 db4 db8 bior22Size of feature vector (ORL 112 lowast 92) 2576 2679 2891 3339 2784Size of feature vector (CV 200 lowast 180) 9000 9090 9579 10379 9384

obtained with each wavelet and for the two databases usedin our work

After calculating the DWT and Gabor transform thesefeature vectors are calculated for the training set and thenused to train the neural network this architecture is calledDWT-MLP (DWT-RBF) and Gabor-MLP (Gabor-RBF)

The performance of Gabor-MLP or DWT-MLP approachis evaluated by using the recognition rate (RR) standarddefined by

119877 (119894) =119873119894

119873 (23)

where119873 is the total number of test sample tests and119873119894is the

number of test images recognized correctlyThe choice of feature extraction is a very important step

and then the application of a classifier such as MLP or RBFpermits us to obtain good results for a development of anindividual identification process

Various transfer functions were tested for training thenetwork and averageminimumMSEon training ismeasuredlog-sigmoid function is the most suitable transfer functionThe MLP neural network is trained using learning rulesnamely conjugate gradient (CG) TRAINSCG is a networktraining function that updates weight and bias values accord-ing to the scaled conjugate gradient method Finally networkis tested on training and testing dataset

341 Searching for the Optimal NN Topology

Number of Hidden Neurons In our experiments we demon-strate how we decided on the optimal number of neurons forthe hidden layer

We attempted to find the optimal NN topology orarchitecture by training and testing MLPs whose numberof neurons in hidden layer varies from 10 to 100 we keepthe number of maximum iterations constant (from 3000 to6000) and the MSE training as presented in Table 2 We usedsigmoid functions as transfer functions

For a small number of neurons (10 to 20) in the hiddenlayer we observed large MSE low accuracy and generallylarge partialityTheMLPs were not flexible enough to capturethe underlying process and due to large bias they generalizepoorly We faced similar generalization performance evenwhen we used too many neurons in the hidden layer Aftersim90 neurons MSE came back to the levels of a system withonly 10 neurons in the hidden layer

By adding more and more units in the hidden layer thetraining error can be made as small as desired but generallyeach additional unit will produce less and less benefits Withtoo many neurons poor performance is a direct effect ofoverfittingThe system overfits the training data and does notperform well on novel patterns

Table 2 The optimal neural network topology

Parameter Value Fixedvariable

Dataset Feature vectors (DWT orGabor) Fixed

DWT coefficients Haar db2 db4 bior22Gabor Variable

Training algorithm Gradient descent Fixed

Transfer function Both layers uselog-sigmoid Fixed

Number of neurons inhidden layer 10 20 30 40 90 100 Variable

Minimum trainingMSE 0001 Fixed

Maximum epochs 3000 to 6000 Variable

0

20

40

60

80

100

120

0 50 100 150

RR

Number of neurons in hidden layerGabor-MLP cv dataset ()Gabor-MLP ORL dataset ()

Figure 16 Recognition rate with Gabor-MLP for computer visionand ORL database

Table 2 summarizes the optimal neural network topologyor architecture found in our experiments to achieve bestrecognition rate

Figure 16 shows recognition efficiency obtained for bothcomputer vision and ORL database with Gabor waveletcharacterization and MLP neural network classifier Goodrecognition rate about 96 to 97was obtained for a numberof neurons superior to 60 for the two databases

Figure 17 shows the variation of recognition rate whenvarying the number of neurons in hidden layer with db2 db4db8 and bior22 DWT with MLP classifier


0

20

40

60

80

100

120

0 20 40 60 80 100 120

RR-db2RR-db4RR-Haar

bior22db8

Number of neurons

Figure 17 Recognition rate with DWT-MLP for computer visiondatabase when varying number of neurons in hidden layer

0

10

20

30

40

50

60

70

80

90

0 20 40 60 80 100 120

Haardb2db4

db8bior22

RR (

)

Figure 18 Recognition rate with DWT-MLP for ORL databasewhen varying number of neurons in hidden layer

The best recognition rate about 96was obtained for db8and bior22 wavelet We conclude for this preliminary resultthat db8 andBiorthogonal 22wavelet performbetter than theHaar wavelet good parametrization can be achieved with thetwo discrete wavelets only with 20 to 30 neurons in hiddenlayer

In addition results obtained for computer vision datasetare superior to those obtained for ORL dataset where thebest recognition rate obtained is about 81 Figure 18 showsefficiency of our recognition system for the different discretewavelet obtained with ORL dataset studied in our work

We also noticed that for 20 to 40 neurons in hidden layerwe have no convergence with the MLP algorithm for ORLdataset this can be explained by the fact that the number ofneurons is not enough to train theMLP classifier In addition

0

1

2

3

4

5

6

7

0 20 40 60 80 100 120

Haardb2db4

db8bior22

Number of neurons

t(m

)

Figure 19 Recognition time obtained for ORL database usingvariable DWT with MLP classifier

facial images in ORL dataset present a lot of variabilities inorientation and in facial expressions

All these parameters complicate the task of recogni-tion and classification whereas the advantage of DWTparametrization is essentialy the reduction of feature extrac-tion and time computation compared to Gabor waveletThe classification time required for achieving recognitionface system is presented in Figure 19 in order to providespeed comparisons of the different discrete wavelet transformcharacterization Simulation results are computed on a PCintel core i3 with a 12 GHz processor

It is clear that time recognition is faster for the differentDWT transform when a number of neurons are superiorthan 80 we noticed from Figure 19 that corresponds to afast convergence of the MLP classifier We conclude therelationship between the optimal convergence of the neuralnetwork classifier and time required for this task

35 RBF Neural Network Classifier for Face RecognitionSimulations were carried with MATLAB 76 and using somepredefined functions to train our RBF NN classifier For RBFneural network simulation a lot of tests have been carriedFirst we create an RBF function classifier using newrbefunction to design a radial basis network with zero error onthe design vectorsThis function creates a two-layer networkthe first layer has radbas neurons and calculates its weightedinput with dist and its net input with netprod The secondlayer has purlin neurons and calculates its weighted inputwith dotprod and its net inputs with netsum Both layers havebiases

Newrbe sets the first layer weights to 119883rsquo where 119883 isour training vector descriptors (Gabor filtered images anddiscrete wavelet decomposition) and the first layer biases areall set to 08326spread resulting in radial basis functionsthat cross 05 at weighted input of +minus spread Spread is thesmoother function approximation (neural network toolbox)


0

20

40

60

80

100

0 50 100 150 200 250

CV-RR-Haar RBF

Figure 20 Recognition rate with Haar-RBF for computer visiondatabase

0

10

20

30

40

50

60

70

80

90

0 2000 4000 6000 8000

ORL-RR-Haar RBFSpread

Figure 21 Recognition rate with Haar-RBF for ORL database

For our simulation we calculate recognition rate whenvarying the spread parameter from 10 to 250 for computervision database where variation is from 200 to 5600 for ORLdatabase We note that spread parameter depends on thequality of analyzed image and the database Figures 20 and21 show the variation of recognition rate when varying thespread parameter

It is clear that recognition rate is about 95 for computervision database obtained with a spread parameter equal to 90whereas it is about 83 obtained for spread equal to 5200

We conclude the following

(1) The use of wavelets is a good means to characterizethe data with high speed compared to Gabor filters

(2) The Gabor wavelet can extract the object features indifferent orientations and different scales on the con-trary the two-dimensional wavelets subband coeffi-cients extract information in only two directions

(3) The MLP neural network is a good classifier for faceidentification using the conjugate gradient

Table 3 Recognition rate for proposed method

RR ()classifier

Wavelet Gabor waveletbior22 MLP Haar-RBF MLP RBF

CVdatabase 96 95 95 94

ORLdatabase 81 83 83 85

Table 3 summarizes efficiency of the characterized methodand classifier used in our work Recognition performance ishighly dependent on the facial dataset

4 Discussion

Simulation results showed the efficiency of the proposedmethod about 94 to 96was obtained with Gabor filters andMLPRBF classifier for computer vision database whereasabout 85 was obtained for ORL dataset

We conclude that the best results are obtained for com-puter vision database in additionMLP canperformwell withGabor parameterization

We also conclude thatMLP classifier performs better thanRBF where 94 of correct classification was obtained forGabor parameterization whereas for bior22 and db8 waveletdecomposition it can achieve 96 with the two-classifierneural network

We can explain these results that computer visiondatabase was obtained in controlled environment with a littlevariation in illumination and pose whereas ORL databasepresents a lot of variation in pose illumination and facialexpression All these parameters decrease the efficiency of ourmethod

The results obtained show the importance of designingneural network for face recognition In addition featureextractions like Gabor filters and wavelet decompositionperformwell for the total databaseWe conclude an improve-ment in face recognition when using the Gabor parameteri-zation for ORL dataset but time required for classification islong compared with discrete wavelet This can be explainedby the following the Gabor wavelet can extract the objectfeatures in different orientations and different scales whereasthe two-dimensional wavelets subband analysis extractsinformation in only two directions Both Gabor filters andwavelet decomposition characterize the facial images wherethe best recognition rate was obtained for Biorthogonal 22wavelet and MLP classifier

41 Comparisons with Other Approaches Many face recogni-tion approaches have been performed on the computer visiondatabase In order to compare the recognition performancewe choose some recent approaches tested under similarconditions for comparison Approaches are evaluated onrecognition rate

The performance of the Gabor-MLP and DWT-RBFalgorithm in comparison with some existing algorithms forface recognition is listed in Table 4 It is evident from the table


Table 4 Comparison between DCT-MLP algorithm and otherstatistic methods (computer vision database)

Method PCA LDA DCT +MLP Gabor-MLP DWT-MLP

Recognitionaccuracy(CV)

60[13] 90 [13] 96 [13] 95 96

that our proposed method achieves high recognition rate aswell as high training and recognition speed

Table 4 presents results in terms of correct classificationrate obtained in our previous work [13] and compared withour approach for computer vision dataset

It is clear from Table 4 that our proposed approachachieves high recognition rate It can be seen that the DWT-MLP algorithm exhibits better performance than other linearclassifiers like PCA and LDA

We conclude that the MLP neural network with only onehidden layer performs well compared to the RBF classifier forthe two databases We also find the methods extracting thefeature DCT are good for the recognition system whereasthe time required is long (about 5mn for 100 training images)compared to DWT extraction and classification

A lot of studies have investigated the use of discretewavelet and other statistical reduction methods like PCAfor ORL computer vision and FERET database Resultsobtained are contradictory Zhang et al present in their work[17] a face recognition system by combining two recentlyproposed neural network models namely Gabor waveletnetwork (GWN) and kernel associativememory (KAM) intoa unified structure called Gabor wavelet associative memory(GWAM) using three popular face databases that is FERETdatabase Olivetti-Oracle Research Lab (ORL) database andAR face database A recognition rate about 98was obtainedwith ORL database and 96 for AR dataset [17]

AlEnzi et al apply different levels of discrete wavelettransform (DWT) in different levels and the two-dimensionalprincipal component (2DPCA) to compute the face recogni-tion accuracy processing it through the ORL image database[18] From their experiments and combining 2-level DWLtechnique with 2DPCA method the highest recognitionaccuracy (945) was obtained with a time rate of 428 sec

Authors in [19] propose a face recognition scheme thatcombined wavelet transform and 2-DPCA applied for ORLdataset They compared the recognition performances ofvarious wavelets at 8 levels 1 to 8 Accuracy with only 2-DPCA is 85 whereas the average accuracy of 2-DPCA withwavelet in subbands 2 and 3 is 945

Jain and Bhati [20] develop a face recognition systemusing neural network with PCA to reduce the dimensionalityin wavelet domain For face detection Haar wavelet is used toform the coefficient matrix and PCA is used for extractingfeatures These features were used to train the artificialneural networks classifier applied for ORL dataset The bestrecognition rate obtained was about 8111

From our overall analysis and comparison we agree withJain and Bhati where a similar correct classification rate about

81 to 83 was obtained for ORL dataset The facial imagesvaried in facial expression and illumination PCA reductioncan be applied in futurework to investigate the effect of spatialreduction

5 Conclusion

In this paperGabor anddiscretewavelet based feature extrac-tionmethods are proposed for the extraction of features fromfacial images Face recognition experiments were carriedout by using MLP and RBF classifier We have used twofacial databases the ORL and computer vision We haveimplemented and studied two neural networks architecturestrained by different characteristic feature vectors like discretewavelet transform (Haar db2 db4 db8 and bior22) andGabor wavelet

Simulation results showed the efficiency of the proposedmethod about 94 to 96was obtained with Gabor filters andMLP classifier for computer vision database whereas about83 to 85 was obtained with discrete wavelet decompositionusing the approximation feature descriptor for ORL databaseResults obtained depend first on the quality of stored imageson the classifier and also on the characterization method

The presented MLP model for face recognition improvesthe face recognition performance by reducing the input fea-tures and performs well compared to the RBF classifier Sim-ulation results on computer vision face database show thatthe proposed method achieves high training and recognitionspeed as well as high recognition rate about 96 Anotherclassifier like SVM can be applied for comparison with ourmethod and other parameterizations like contourlets or otherdiscrete wavelets could be investigated to perform betterrecognition rate In future work we plan to apply curveletsanalysis which is better at handling curve discontinuities

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M Yang and L Zhang ldquoGabor feature based sparse represen-tation for face recognition with gabor occlusion dictionaryrdquo inProceedings of the 11th European Conference on Computer Vision(ECCV rsquo10) pp 448ndash461 Springer Crete Greece 2010

[2] S Kalaimagal and M Edwin Jaya Singh ldquoFace recognitionusing gabor volume based local binary patternrdquo InternationalJournal of Advanced Research in Computer Science and SoftwareEngineering vol 2 no 6 2012

[3] A Venus M Alfiras and F Alsaqre ldquoFace recognition algo-rithm using two dimensional principal component analysisbased on discrete wavelet transformrdquo in Digital InformationProcessing and Communications V Snasel J Platos and E El-Qawasmeh Eds vol 188 of Communications in Computer andInformation Science pp 426ndash438 Springer Berlin Germany2011

[4] A Bhuiyan and C H Liu ldquoOn face recognition using gaborfiltersrdquo International Journal of Computer Information Systemsand Control Engineering vol 1 no 4 pp 837ndash842 2007


[5] J Wang and J Cheng ldquoFace recognition based on fusion ofGabor and 2DPCA featuresrdquo in Proceedings of the 18th Interna-tional Symposium on Intelligent Signal Processing andCommuni-cation Systems pp 1ndash4 IEEE Chengdu China December 2010

[6] F Bellakhdhar K Loukil and M Abid ldquoFace recognitionapproach using Gabor Wavelets PCA and SVMrdquo InternationalJournal of Computer Science Issues vol 10 no 2 p 201 2013

[7] L Zhao C Yang F Pan and J Wang ldquoFace recognition basedon Gabor with 2DPCA and PCArdquo in Proceedings of the 24thChinese Control and Decision Conference (CCDC rsquo12) pp 2632ndash2635 May 2012

[8] T Andrysiak and M Choras ldquoImage retrieval based onhierarchical Gabor filtersrdquo International Journal of AppliedMathematics and Computer Science vol 15 no 4 pp 471ndash4802005

[9] M Lades J C Vorbrueggen J Buhmann et al ldquoDistortioninvariant object recognition in the dynamic link architecturerdquoIEEE Transactions on Computers vol 42 no 3 pp 300ndash3111993

[10] X-M Wang C Huang G-Y Ni and J-G Liu ldquoFace recogni-tion based on face Gabor image and SVMrdquo in Proceedings of the2nd International Congress on Image and Signal Processing (CISPrsquo09) pp 1ndash4 IEEE Tianjin China October 2009

[11] M Meade S C Sivakumar and W J Phillips ldquoComparativeperformance of principal component analysis Gabor waveletsand discrete wavelet transforms for face recognitionrdquoCanadianJournal of Electrical and Computer Engineering vol 30 no 2pp 93ndash102 2005

[12] F Y Shih C-F Chuang and P S P Wang ldquoPerformance com-parisons of facial expression recognition in JAFFE databaserdquoInternational Journal of Pattern Recognition and Artificial Intel-ligence vol 22 no 3 pp 445ndash459 2008

[13] F Z Chelali and A Djeradi ldquoFace recognition system usingdiscrete cosine transform combined with MLP and RBF neuralnetworksrdquo International Journal of Mobile Computing andMultimedia Communications vol 4 no 4 pp 1ndash35 2012

[14] ldquoComputer Vision Science Research Projects data setrdquo httpcmpfelkcvutczsimspacelibfaces

[15] ORL face databases httpwwwface-recorgdatabases[16] M J Er W Chen and S Wu ldquoHigh-speed face recognition

based on discrete cosine transform and RBF neural networksrdquoIEEE Transactions on Neural Networks vol 16 no 3 pp 679ndash691 2005

[17] H Zhang B Zhang W Huang and Q Tian ldquoGabor waveletassociative memory for face recognitionrdquo IEEE Transactions onNeural Networks vol 16 no 1 pp 275ndash278 2005

[18] V AlEnzi M Alfiras and F Alsaqre ldquoFace recognition algo-rithm using two dimensional principal component analysisbased on discrete wavelet transformrdquo in Digital InformationProcessing and Communications V Snasel J Platos and E El-Qawasmeh Eds vol 188 of Communications in Computer andInformation Science pp 426ndash438 Springer Berlin Germany2011

[19] K S Kinage and S G Bhirud ldquoFace recognition based on two-dimensional PCA on wavelet subbandrdquo International Journal ofRecent Trends in Engineering vol 2 no 2 pp 51ndash54 2009

[20] S Jain and D Bhati ldquoFace recognition using ANN with reducefeature by PCArdquo International Journal of Scientific Engineeringand Technology vol 2 no 2 pp 595ndash599 2013

Submit your manuscripts athttpwwwhindawicom

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


base

ApplyDWT

Construct feature DWT vectors for

each class

MLPRBFclassifier

DWT-NNRR ()

Gabor-NNRR ()

classifierMLPRBF

Define the parametersof Gabor filters

Convolution of Gwith image I(x y)

in gray levelfiltered images

Originalimage

Filteredimage

Facial image data

G 120590x 120590y f and 120579

Figure 1 An overview of face recognition system

pattern recognition and so on The principal motivation touse Gabor filters is biological relevance that the receptive fieldprofiles of neurons in the primary visual cortex of mammalsare oriented and have characteristic spatial frequencies

Gabor filters can exploit salient visual properties suchas spatial localization orientation selectivity and spatialfrequency characteristics [4]

The Gabor filter was first introduced by David Gabor in1946 and was later shown as models of simple cell receptivefields Since the Gabor features are extracted in local regionsthey are less sensitive to variations of illumination expres-sion and pose than the holistic features such as Eigenface andRandomface [1]

Face recognition usingGabor filterswas firstly introducedby Yang and Zhang and soon proved to be a very effectivemeans in human facial features extraction [1]

We are interested in our work to develop a recognitionsystem using Gabor filters and discrete wavelet transform asfeature extraction Amultilayer perceptron (MLP) and radialbasis function (RBF) neural network classifier are used toclassify an individual from two face databases the ORL andcomputer vision databases Figure 1 describes our system

We investigate then the best way to characterize the facialimages efficiency is evaluated by computing recognition rate(RR) for each parameterization and classifier

The paper is presented as follows Section 2 presents theface characterization using Gabor filters and discrete wavelettransform DWT Section 3 presents simulation results andconclusion is drawn in Section 5

2 Feature Face Extraction andFace Parameterization

The most important step in face recognition system is thefeature extraction It permits us to reduce dimensionalityand characterize the face image using a vector descriptorwhich minimizes intraperson dissimilarities and maximizesthe extraperson difference This task is difficult because ofvariations of pose age expression and illumination and soforth

In our work we use Gabor filters and discrete wavelettransform to describe our face and to build a face recognitionsystem using the multilayer perceptron (MLP) and radialbasis function (RBF) neural network classifier Comparisonis done for each classifier and facial database to decidethe optimal characterization We describe first the Gaborfunction and discrete wavelet transforms to characterize ourfacial images

21 Gabor Filter Representation

211 Gabor Wavelet A Mathematical Overview Gabor filterdefined by Dennis Gabor is widely used in image analysispattern recognition and so forth Gabor filters present twointeresting properties frequency localization and selectivityin orientation

A lot of studies demonstrate that representation byGabor wavelet for image analysis is robust to variation inillumination and facial expressions

Gabor filter in 2D was introduced in biometric researchbyDaugman for iris recognition Lades et al usedGabor filterfor face recognition using ldquoDynamic Link Architecturerdquo

The Gabor wavelet which captures the properties oforientation selectivity spatial localization and optimallylocalized in the space and frequency domains has beenextensively and successfully used in face recognition [5]Daugman pioneered the using of the 2D Gabor waveletrepresentation in computer vision in 1980s [6 7]

Gabor wavelets (filters) characteristics for frequency andorientation representations are quite similar to those ofhuman visual system These have been found appropriatefor texture representation and discrimination This Gaborwavelet based extraction of features directly from the gray-level images is successful and has widely been applied totexture segmentation and fingerprint recognition The com-monly used Gabor filters in face recognition area [3 4] aredefined as follows [6]

Ψ119906V (119885) =

1003817100381710038171003817119896119906V1003817100381710038171003817

2

1205902exp(

minus1003817100381710038171003817119896119906V

1003817100381710038171003817

2

1198852

2 lowast 1205902)[119890119894119896119906V119885 minus 119890

minus1205902

2

]

(1)119885(119909 119910) is the point of coordinates (119909 119910) in image space

119906 and V define the orientation Gabor frequency and 120590 is thestandard deviation of the Gaussian

120583 and V define the orientation and the scale of the Gaborfilters and 119896

120583V is defined as the following form

119896120583V = 119896V119890

119895120593120583 (2)


x

y

Ψ120583


119896V = 119896max119891V and 120593





119865120583V (119911) = 119868 (119911) lowast Ψ

120583V (119911) (3)









defined by

119866119906V (119885) = 119868 (119885) lowast Ψ

119906V (119911) (4)





Ψ119891120579(119909 119910) = exp[minus1

21199092

120579119899

1205902119909

+1199102

120579119899

1205902119910

] exp (2120587119891119909120579119899) (5)

where

[119909120579119899

119910120579119899

] = [sin 120579119899

cos 120579119899

minus cos 120579119899

sin 120579119899

][119909

119910] (6)






119899is defined by

120579119899=120587

119901(119899 minus 1) (7)




filter 120579119899[4]



119891120579(119909 119910) is defined

as [4]

119892119891120579(119909 119910) = 119891 (119909 119910) lowast Ψ

119891120579(119909 119910) (8)


119891120579(119909 119910) and an imaginary


119891120579(119909 119910) is

expressed as

10038171003817100381710038171003817119892119891120579(119909 119910)

10038171003817100381710038171003817= radicRe2 119892

119891120579(119909 119910) + Im2 119892

119891120579(119909 119910) (9)








(a) (b)






119909= 2 120590

119910= 4












119862 (119886 119887) =1

radic119886int 119904 (119905) Ψ(

119905 minus 119887

119886)119889119905 (10)



0

5

10

0

5

0

05

1

minus05

(a)

0

5

10

0

5

0

05

1

minus05

(b)

02

46

0

5

10

0

05

1

minus05

(c)

02

46

0

5

10

0

05

1

minus05

(d)


119910= 4 120579 = 1205873 119891 = 16 (b) 120590

119909= 2 120590

119910= 4 120579 = 21205873 119891 = 16 (c)

120590119909= 4 120590

119910= 2 120579 = 1205873 119891 = 16 and (d) 120590

119909= 4 120590

119910= 2 120579 = 21205873 119891 = 16


119888119895119896

= 119862(1

2119895119896

2119895) = 2

1198952

int

+infin

minusinfin

119909 (119905) Ψ (2119895

119905 minus 119896) 119889119905 (11)



Ψ119895119896

= 21198952

Ψ(2119895

119905 minus 119896) (12)


119888119895119896

= int

+infin

minusinfin

119909 (119905) Ψ119895119896119889119905 (13)




120601119895119896(119905) = 2

1198952

120601 (2119895

119905 minus 119896) (14)


0(119899) as

120601 (119905) = sum

119899

ℎ0(119899)radic2120601 (2119905 minus 119899) (15)


1(119899) as

Ψ (119905) = sum

119899

ℎ1(119899)radic2120601 (2119905 minus 119899) (16)


119909 (119905) = sum

119896

cA1(119896) 120601119895minus1119896

(119905) + sum

119896

cD1(119896) Ψ119895minus1119896

(119905) (17)





1

05

0

minus050 20 40 60


Filtered image

1

05

0

minus055

4

3

2

1 02

46

8104


119910= 4 120579 = 1205873 119891 = 16


0(119899) and ℎ

1(119899)

cA1(119896) = sum

119899

cA0(119899) ℎ0(119899 minus 2119896) (18)



0with the low-pass



1 can be found


1(119899) and down-


0(119899) and ℎ




2

2

2

2

x(n)

h0(n)h0(n)

h1(n)

h1(n)

cA1

cA2

cD1

cD2




2

2L

H L

H

LH

2 LL2 LH

2 HL

2 HH

Rows Columns

N2

ap hd

vd dd

M2NlowastMNlowastM2

N2lowastM2




119886119895+1

[119901] =

119899=+infin

sum

119899=minusinfin

119897 [119899 minus 2119901] 119886119895[119899]

119889119895+1

[119901] =

119899=+infin

sum

119899=minusinfin

ℎ [119899 minus 2119901] 119886119895[119899]

(19)














2

15

1

05

0

minus05400

300200

1000 0

100200

300




































119894isin 119877119903 (1 le 119894 le 119906)


119877119894(119875) = 119877

119894(1003817100381710038171003817119875 minus 119862119894

1003817100381710038171003817) 119894 = 1 119906 (20)


119877119894(119875) = exp[minus

1003817100381710038171003817119875 minus 1198621198941003817100381710038171003817

2

1205902

119894

] (21)


119895(119875)


119910119894(119875) =

119906

sum

119894=1

119877119894(119875) lowast 119908 (119895 119894) (22)




0 100 200 300 400 500 600 700 800 900 10000

01

02

03

04

05

06

07

08

09

1

Training images

Reco

gniti

on ra

teRR for 5th class

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 1st class

minus02

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 2nd class

minus02

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 3rd class

minus020 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 6th class

minus02

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 4th class

minus02



0

20

40

60

80

100

120


Clas

s1

Clas

s2

Clas

s3

Clas

s4

Clas

s5

Clas

s6

Clas

s7

Clas

s8

Clas

s9

Clas

s10


0

20

40

60

80

100

120


RR (

)

Clas

s1

Clas

s2

Clas

s3

Clas

s4

Clas

s5

Clas

s6

Clas

s7

Clas

s8

Clas

s9

Clas

s10




12

1

08

06

04

02

120590i


f1(x)

0 minus Ci















119877 (119894) =119873119894

119873 (23)



















0

20

40

60

80

100

120

0 50 100 150

RR







0

20

40

60

80

100

120

0 20 40 60 80 100 120

RR-db2RR-db4RR-Haar

bior22db8

Number of neurons


0

10

20

30

40

50

60

70

80

90

0 20 40 60 80 100 120

Haardb2db4

db8bior22

RR (

)





0

1

2

3

4

5

6

7

0 20 40 60 80 100 120

Haardb2db4

db8bior22

Number of neurons

t(m

)








0

20

40

60

80

100

0 50 100 150 200 250

CV-RR-Haar RBF


0

10

20

30

40

50

60

70

80

90

0 2000 4000 6000 8000










RR ()classifier





4 Discussion












60[13] 90 [13] 96 [13] 95 96











5 Conclusion






References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










x

y

Ψ120583


119896V = 119896max119891V and 120593





119865120583V (119911) = 119868 (119911) lowast Ψ

120583V (119911) (3)









defined by

119866119906V (119885) = 119868 (119885) lowast Ψ

119906V (119911) (4)





Ψ119891120579(119909 119910) = exp[minus1

21199092

120579119899

1205902119909

+1199102

120579119899

1205902119910

] exp (2120587119891119909120579119899) (5)

where

[119909120579119899

119910120579119899

] = [sin 120579119899

cos 120579119899

minus cos 120579119899

sin 120579119899

][119909

119910] (6)






119899is defined by

120579119899=120587

119901(119899 minus 1) (7)




filter 120579119899[4]



119891120579(119909 119910) is defined

as [4]

119892119891120579(119909 119910) = 119891 (119909 119910) lowast Ψ

119891120579(119909 119910) (8)


119891120579(119909 119910) and an imaginary


119891120579(119909 119910) is

expressed as

10038171003817100381710038171003817119892119891120579(119909 119910)

10038171003817100381710038171003817= radicRe2 119892

119891120579(119909 119910) + Im2 119892

119891120579(119909 119910) (9)








(a) (b)






119909= 2 120590

119910= 4












119862 (119886 119887) =1

radic119886int 119904 (119905) Ψ(

119905 minus 119887

119886)119889119905 (10)



0

5

10

0

5

0

05

1

minus05

(a)

0

5

10

0

5

0

05

1

minus05

(b)

02

46

0

5

10

0

05

1

minus05

(c)

02

46

0

5

10

0

05

1

minus05

(d)


119910= 4 120579 = 1205873 119891 = 16 (b) 120590

119909= 2 120590

119910= 4 120579 = 21205873 119891 = 16 (c)

120590119909= 4 120590

119910= 2 120579 = 1205873 119891 = 16 and (d) 120590

119909= 4 120590

119910= 2 120579 = 21205873 119891 = 16


119888119895119896

= 119862(1

2119895119896

2119895) = 2

1198952

int

+infin

minusinfin

119909 (119905) Ψ (2119895

119905 minus 119896) 119889119905 (11)



Ψ119895119896

= 21198952

Ψ(2119895

119905 minus 119896) (12)


119888119895119896

= int

+infin

minusinfin

119909 (119905) Ψ119895119896119889119905 (13)




120601119895119896(119905) = 2

1198952

120601 (2119895

119905 minus 119896) (14)


0(119899) as

120601 (119905) = sum

119899

ℎ0(119899)radic2120601 (2119905 minus 119899) (15)


1(119899) as

Ψ (119905) = sum

119899

ℎ1(119899)radic2120601 (2119905 minus 119899) (16)


119909 (119905) = sum

119896

cA1(119896) 120601119895minus1119896

(119905) + sum

119896

cD1(119896) Ψ119895minus1119896

(119905) (17)





1

05

0

minus050 20 40 60


Filtered image

1

05

0

minus055

4

3

2

1 02

46

8104


119910= 4 120579 = 1205873 119891 = 16


0(119899) and ℎ

1(119899)

cA1(119896) = sum

119899

cA0(119899) ℎ0(119899 minus 2119896) (18)



0with the low-pass



1 can be found


1(119899) and down-


0(119899) and ℎ




2

2

2

2

x(n)

h0(n)h0(n)

h1(n)

h1(n)

cA1

cA2

cD1

cD2




2

2L

H L

H

LH

2 LL2 LH

2 HL

2 HH

Rows Columns

N2

ap hd

vd dd

M2NlowastMNlowastM2

N2lowastM2




119886119895+1

[119901] =

119899=+infin

sum

119899=minusinfin

119897 [119899 minus 2119901] 119886119895[119899]

119889119895+1

[119901] =

119899=+infin

sum

119899=minusinfin

ℎ [119899 minus 2119901] 119886119895[119899]

(19)














2

15

1

05

0

minus05400

300200

1000 0

100200

300




































119894isin 119877119903 (1 le 119894 le 119906)


119877119894(119875) = 119877

119894(1003817100381710038171003817119875 minus 119862119894

1003817100381710038171003817) 119894 = 1 119906 (20)


119877119894(119875) = exp[minus

1003817100381710038171003817119875 minus 1198621198941003817100381710038171003817

2

1205902

119894

] (21)


119895(119875)


119910119894(119875) =

119906

sum

119894=1

119877119894(119875) lowast 119908 (119895 119894) (22)




0 100 200 300 400 500 600 700 800 900 10000

01

02

03

04

05

06

07

08

09

1

Training images

Reco

gniti

on ra

teRR for 5th class

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 1st class

minus02

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 2nd class

minus02

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 3rd class

minus020 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 6th class

minus02

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 4th class

minus02



0

20

40

60

80

100

120


Clas

s1

Clas

s2

Clas

s3

Clas

s4

Clas

s5

Clas

s6

Clas

s7

Clas

s8

Clas

s9

Clas

s10


0

20

40

60

80

100

120


RR (

)

Clas

s1

Clas

s2

Clas

s3

Clas

s4

Clas

s5

Clas

s6

Clas

s7

Clas

s8

Clas

s9

Clas

s10




12

1

08

06

04

02

120590i


f1(x)

0 minus Ci















119877 (119894) =119873119894

119873 (23)



















0

20

40

60

80

100

120

0 50 100 150

RR







0

20

40

60

80

100

120

0 20 40 60 80 100 120

RR-db2RR-db4RR-Haar

bior22db8

Number of neurons


0

10

20

30

40

50

60

70

80

90

0 20 40 60 80 100 120

Haardb2db4

db8bior22

RR (

)





0

1

2

3

4

5

6

7

0 20 40 60 80 100 120

Haardb2db4

db8bior22

Number of neurons

t(m

)








0

20

40

60

80

100

0 50 100 150 200 250

CV-RR-Haar RBF


0

10

20

30

40

50

60

70

80

90

0 2000 4000 6000 8000










RR ()classifier





4 Discussion












60[13] 90 [13] 96 [13] 95 96











5 Conclusion






References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)






119909= 2 120590

119910= 4












119862 (119886 119887) =1

radic119886int 119904 (119905) Ψ(

119905 minus 119887

119886)119889119905 (10)



0

5

10

0

5

0

05

1

minus05

(a)

0

5

10

0

5

0

05

1

minus05

(b)

02

46

0

5

10

0

05

1

minus05

(c)

02

46

0

5

10

0

05

1

minus05

(d)


119910= 4 120579 = 1205873 119891 = 16 (b) 120590

119909= 2 120590

119910= 4 120579 = 21205873 119891 = 16 (c)

120590119909= 4 120590

119910= 2 120579 = 1205873 119891 = 16 and (d) 120590

119909= 4 120590

119910= 2 120579 = 21205873 119891 = 16


119888119895119896

= 119862(1

2119895119896

2119895) = 2

1198952

int

+infin

minusinfin

119909 (119905) Ψ (2119895

119905 minus 119896) 119889119905 (11)



Ψ119895119896

= 21198952

Ψ(2119895

119905 minus 119896) (12)


119888119895119896

= int

+infin

minusinfin

119909 (119905) Ψ119895119896119889119905 (13)




120601119895119896(119905) = 2

1198952

120601 (2119895

119905 minus 119896) (14)


0(119899) as

120601 (119905) = sum

119899

ℎ0(119899)radic2120601 (2119905 minus 119899) (15)


1(119899) as

Ψ (119905) = sum

119899

ℎ1(119899)radic2120601 (2119905 minus 119899) (16)


119909 (119905) = sum

119896

cA1(119896) 120601119895minus1119896

(119905) + sum

119896

cD1(119896) Ψ119895minus1119896

(119905) (17)





1

05

0

minus050 20 40 60


Filtered image

1

05

0

minus055

4

3

2

1 02

46

8104


119910= 4 120579 = 1205873 119891 = 16


0(119899) and ℎ

1(119899)

cA1(119896) = sum

119899

cA0(119899) ℎ0(119899 minus 2119896) (18)



0with the low-pass



1 can be found


1(119899) and down-


0(119899) and ℎ




2

2

2

2

x(n)

h0(n)h0(n)

h1(n)

h1(n)

cA1

cA2

cD1

cD2




2

2L

H L

H

LH

2 LL2 LH

2 HL

2 HH

Rows Columns

N2

ap hd

vd dd

M2NlowastMNlowastM2

N2lowastM2




119886119895+1

[119901] =

119899=+infin

sum

119899=minusinfin

119897 [119899 minus 2119901] 119886119895[119899]

119889119895+1

[119901] =

119899=+infin

sum

119899=minusinfin

ℎ [119899 minus 2119901] 119886119895[119899]

(19)














2

15

1

05

0

minus05400

300200

1000 0

100200

300




































119894isin 119877119903 (1 le 119894 le 119906)


119877119894(119875) = 119877

119894(1003817100381710038171003817119875 minus 119862119894

1003817100381710038171003817) 119894 = 1 119906 (20)


119877119894(119875) = exp[minus

1003817100381710038171003817119875 minus 1198621198941003817100381710038171003817

2

1205902

119894

] (21)


119895(119875)


119910119894(119875) =

119906

sum

119894=1

119877119894(119875) lowast 119908 (119895 119894) (22)




0 100 200 300 400 500 600 700 800 900 10000

01

02

03

04

05

06

07

08

09

1

Training images

Reco

gniti

on ra

teRR for 5th class

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 1st class

minus02

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 2nd class

minus02

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 3rd class

minus020 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 6th class

minus02

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 4th class

minus02



0

20

40

60

80

100

120


Clas

s1

Clas

s2

Clas

s3

Clas

s4

Clas

s5

Clas

s6

Clas

s7

Clas

s8

Clas

s9

Clas

s10


0

20

40

60

80

100

120


RR (

)

Clas

s1

Clas

s2

Clas

s3

Clas

s4

Clas

s5

Clas

s6

Clas

s7

Clas

s8

Clas

s9

Clas

s10




12

1

08

06

04

02

120590i


f1(x)

0 minus Ci















119877 (119894) =119873119894

119873 (23)



















0

20

40

60

80

100

120

0 50 100 150

RR







0

20

40

60

80

100

120

0 20 40 60 80 100 120

RR-db2RR-db4RR-Haar

bior22db8

Number of neurons


0

10

20

30

40

50

60

70

80

90

0 20 40 60 80 100 120

Haardb2db4

db8bior22

RR (

)





0

1

2

3

4

5

6

7

0 20 40 60 80 100 120

Haardb2db4

db8bior22

Number of neurons

t(m

)








0

20

40

60

80

100

0 50 100 150 200 250

CV-RR-Haar RBF


0

10

20

30

40

50

60

70

80

90

0 2000 4000 6000 8000










RR ()classifier





4 Discussion












60[13] 90 [13] 96 [13] 95 96











5 Conclusion






References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

5

10

0

5

0

05

1

minus05

(a)

0

5

10

0

5

0

05

1

minus05

(b)

02

46

0

5

10

0

05

1

minus05

(c)

02

46

0

5

10

0

05

1

minus05

(d)


119910= 4 120579 = 1205873 119891 = 16 (b) 120590

119909= 2 120590

119910= 4 120579 = 21205873 119891 = 16 (c)

120590119909= 4 120590

119910= 2 120579 = 1205873 119891 = 16 and (d) 120590

119909= 4 120590

119910= 2 120579 = 21205873 119891 = 16


119888119895119896

= 119862(1

2119895119896

2119895) = 2

1198952

int

+infin

minusinfin

119909 (119905) Ψ (2119895

119905 minus 119896) 119889119905 (11)



Ψ119895119896

= 21198952

Ψ(2119895

119905 minus 119896) (12)


119888119895119896

= int

+infin

minusinfin

119909 (119905) Ψ119895119896119889119905 (13)




120601119895119896(119905) = 2

1198952

120601 (2119895

119905 minus 119896) (14)


0(119899) as

120601 (119905) = sum

119899

ℎ0(119899)radic2120601 (2119905 minus 119899) (15)


1(119899) as

Ψ (119905) = sum

119899

ℎ1(119899)radic2120601 (2119905 minus 119899) (16)


119909 (119905) = sum

119896

cA1(119896) 120601119895minus1119896

(119905) + sum

119896

cD1(119896) Ψ119895minus1119896

(119905) (17)





1

05

0

minus050 20 40 60


Filtered image

1

05

0

minus055

4

3

2

1 02

46

8104


119910= 4 120579 = 1205873 119891 = 16


0(119899) and ℎ

1(119899)

cA1(119896) = sum

119899

cA0(119899) ℎ0(119899 minus 2119896) (18)



0with the low-pass



1 can be found


1(119899) and down-


0(119899) and ℎ




2

2

2

2

x(n)

h0(n)h0(n)

h1(n)

h1(n)

cA1

cA2

cD1

cD2




2

2L

H L

H

LH

2 LL2 LH

2 HL

2 HH

Rows Columns

N2

ap hd

vd dd

M2NlowastMNlowastM2

N2lowastM2




119886119895+1

[119901] =

119899=+infin

sum

119899=minusinfin

119897 [119899 minus 2119901] 119886119895[119899]

119889119895+1

[119901] =

119899=+infin

sum

119899=minusinfin

ℎ [119899 minus 2119901] 119886119895[119899]

(19)














2

15

1

05

0

minus05400

300200

1000 0

100200

300




































119894isin 119877119903 (1 le 119894 le 119906)


119877119894(119875) = 119877

119894(1003817100381710038171003817119875 minus 119862119894

1003817100381710038171003817) 119894 = 1 119906 (20)


119877119894(119875) = exp[minus

1003817100381710038171003817119875 minus 1198621198941003817100381710038171003817

2

1205902

119894

] (21)


119895(119875)


119910119894(119875) =

119906

sum

119894=1

119877119894(119875) lowast 119908 (119895 119894) (22)




0 100 200 300 400 500 600 700 800 900 10000

01

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on ra

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minus02

0 100 200 300 400 500 600 700 800 900 1000

0

02

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1

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gniti

on ra

te

RR for 2nd class

minus02

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

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Reco

gniti

on ra

te

RR for 3rd class

minus020 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 6th class

minus02

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 4th class

minus02



0

20

40

60

80

100

120


Clas

s1

Clas

s2

Clas

s3

Clas

s4

Clas

s5

Clas

s6

Clas

s7

Clas

s8

Clas

s9

Clas

s10


0

20

40

60

80

100

120


RR (

)

Clas

s1

Clas

s2

Clas

s3

Clas

s4

Clas

s5

Clas

s6

Clas

s7

Clas

s8

Clas

s9

Clas

s10




12

1

08

06

04

02

120590i


f1(x)

0 minus Ci















119877 (119894) =119873119894

119873 (23)



















0

20

40

60

80

100

120

0 50 100 150

RR







0

20

40

60

80

100

120

0 20 40 60 80 100 120

RR-db2RR-db4RR-Haar

bior22db8

Number of neurons


0

10

20

30

40

50

60

70

80

90

0 20 40 60 80 100 120

Haardb2db4

db8bior22

RR (

)





0

1

2

3

4

5

6

7

0 20 40 60 80 100 120

Haardb2db4

db8bior22

Number of neurons

t(m

)








0

20

40

60

80

100

0 50 100 150 200 250

CV-RR-Haar RBF


0

10

20

30

40

50

60

70

80

90

0 2000 4000 6000 8000










RR ()classifier





4 Discussion












60[13] 90 [13] 96 [13] 95 96











5 Conclusion






References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

05

0

minus050 20 40 60


Filtered image

1

05

0

minus055

4

3

2

1 02

46

8104


119910= 4 120579 = 1205873 119891 = 16


0(119899) and ℎ

1(119899)

cA1(119896) = sum

119899

cA0(119899) ℎ0(119899 minus 2119896) (18)



0with the low-pass



1 can be found


1(119899) and down-


0(119899) and ℎ




2

2

2

2

x(n)

h0(n)h0(n)

h1(n)

h1(n)

cA1

cA2

cD1

cD2




2

2L

H L

H

LH

2 LL2 LH

2 HL

2 HH

Rows Columns

N2

ap hd

vd dd

M2NlowastMNlowastM2

N2lowastM2




119886119895+1

[119901] =

119899=+infin

sum

119899=minusinfin

119897 [119899 minus 2119901] 119886119895[119899]

119889119895+1

[119901] =

119899=+infin

sum

119899=minusinfin

ℎ [119899 minus 2119901] 119886119895[119899]

(19)














2

15

1

05

0

minus05400

300200

1000 0

100200

300




































119894isin 119877119903 (1 le 119894 le 119906)


119877119894(119875) = 119877

119894(1003817100381710038171003817119875 minus 119862119894

1003817100381710038171003817) 119894 = 1 119906 (20)


119877119894(119875) = exp[minus

1003817100381710038171003817119875 minus 1198621198941003817100381710038171003817

2

1205902

119894

] (21)


119895(119875)


119910119894(119875) =

119906

sum

119894=1

119877119894(119875) lowast 119908 (119895 119894) (22)




0 100 200 300 400 500 600 700 800 900 10000

01

02

03

04

05

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1

Training images

Reco

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teRR for 5th class

0 100 200 300 400 500 600 700 800 900 1000

0

02

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08

1

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Training images

Reco

gniti

on ra

te

RR for 1st class

minus02

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 2nd class

minus02

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

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Reco

gniti

on ra

te

RR for 3rd class

minus020 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 6th class

minus02

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 4th class

minus02



0

20

40

60

80

100

120


Clas

s1

Clas

s2

Clas

s3

Clas

s4

Clas

s5

Clas

s6

Clas

s7

Clas

s8

Clas

s9

Clas

s10


0

20

40

60

80

100

120


RR (

)

Clas

s1

Clas

s2

Clas

s3

Clas

s4

Clas

s5

Clas

s6

Clas

s7

Clas

s8

Clas

s9

Clas

s10




12

1

08

06

04

02

120590i


f1(x)

0 minus Ci















119877 (119894) =119873119894

119873 (23)



















0

20

40

60

80

100

120

0 50 100 150

RR







0

20

40

60

80

100

120

0 20 40 60 80 100 120

RR-db2RR-db4RR-Haar

bior22db8

Number of neurons


0

10

20

30

40

50

60

70

80

90

0 20 40 60 80 100 120

Haardb2db4

db8bior22

RR (

)





0

1

2

3

4

5

6

7

0 20 40 60 80 100 120

Haardb2db4

db8bior22

Number of neurons

t(m

)








0

20

40

60

80

100

0 50 100 150 200 250

CV-RR-Haar RBF


0

10

20

30

40

50

60

70

80

90

0 2000 4000 6000 8000










RR ()classifier





4 Discussion












60[13] 90 [13] 96 [13] 95 96











5 Conclusion






References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2

2L

H L

H

LH

2 LL2 LH

2 HL

2 HH

Rows Columns

N2

ap hd

vd dd

M2NlowastMNlowastM2

N2lowastM2




119886119895+1

[119901] =

119899=+infin

sum

119899=minusinfin

119897 [119899 minus 2119901] 119886119895[119899]

119889119895+1

[119901] =

119899=+infin

sum

119899=minusinfin

ℎ [119899 minus 2119901] 119886119895[119899]

(19)














2

15

1

05

0

minus05400

300200

1000 0

100200

300




































119894isin 119877119903 (1 le 119894 le 119906)


119877119894(119875) = 119877

119894(1003817100381710038171003817119875 minus 119862119894

1003817100381710038171003817) 119894 = 1 119906 (20)


119877119894(119875) = exp[minus

1003817100381710038171003817119875 minus 1198621198941003817100381710038171003817

2

1205902

119894

] (21)


119895(119875)


119910119894(119875) =

119906

sum

119894=1

119877119894(119875) lowast 119908 (119895 119894) (22)




0 100 200 300 400 500 600 700 800 900 10000

01

02

03

04

05

06

07

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1

Training images

Reco

gniti

on ra

teRR for 5th class

0 100 200 300 400 500 600 700 800 900 1000

0

02

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06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 1st class

minus02

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 2nd class

minus02

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 3rd class

minus020 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 6th class

minus02

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 4th class

minus02



0

20

40

60

80

100

120


Clas

s1

Clas

s2

Clas

s3

Clas

s4

Clas

s5

Clas

s6

Clas

s7

Clas

s8

Clas

s9

Clas

s10


0

20

40

60

80

100

120


RR (

)

Clas

s1

Clas

s2

Clas

s3

Clas

s4

Clas

s5

Clas

s6

Clas

s7

Clas

s8

Clas

s9

Clas

s10




12

1

08

06

04

02

120590i


f1(x)

0 minus Ci















119877 (119894) =119873119894

119873 (23)



















0

20

40

60

80

100

120

0 50 100 150

RR







0

20

40

60

80

100

120

0 20 40 60 80 100 120

RR-db2RR-db4RR-Haar

bior22db8

Number of neurons


0

10

20

30

40

50

60

70

80

90

0 20 40 60 80 100 120

Haardb2db4

db8bior22

RR (

)





0

1

2

3

4

5

6

7

0 20 40 60 80 100 120

Haardb2db4

db8bior22

Number of neurons

t(m

)








0

20

40

60

80

100

0 50 100 150 200 250

CV-RR-Haar RBF


0

10

20

30

40

50

60

70

80

90

0 2000 4000 6000 8000










RR ()classifier





4 Discussion












60[13] 90 [13] 96 [13] 95 96











5 Conclusion






References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2

15

1

05

0

minus05400

300200

1000 0

100200

300




































119894isin 119877119903 (1 le 119894 le 119906)


119877119894(119875) = 119877

119894(1003817100381710038171003817119875 minus 119862119894

1003817100381710038171003817) 119894 = 1 119906 (20)


119877119894(119875) = exp[minus

1003817100381710038171003817119875 minus 1198621198941003817100381710038171003817

2

1205902

119894

] (21)


119895(119875)


119910119894(119875) =

119906

sum

119894=1

119877119894(119875) lowast 119908 (119895 119894) (22)




0 100 200 300 400 500 600 700 800 900 10000

01

02

03

04

05

06

07

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1

Training images

Reco

gniti

on ra

teRR for 5th class

0 100 200 300 400 500 600 700 800 900 1000

0

02

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08

1

12

Training images

Reco

gniti

on ra

te

RR for 1st class

minus02

0 100 200 300 400 500 600 700 800 900 1000

0

02

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08

1

12

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Reco

gniti

on ra

te

RR for 2nd class

minus02

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

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Reco

gniti

on ra

te

RR for 3rd class

minus020 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 6th class

minus02

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 4th class

minus02



0

20

40

60

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100

120


Clas

s1

Clas

s2

Clas

s3

Clas

s4

Clas

s5

Clas

s6

Clas

s7

Clas

s8

Clas

s9

Clas

s10


0

20

40

60

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RR (

)

Clas

s1

Clas

s2

Clas

s3

Clas

s4

Clas

s5

Clas

s6

Clas

s7

Clas

s8

Clas

s9

Clas

s10




12

1

08

06

04

02

120590i


f1(x)

0 minus Ci















119877 (119894) =119873119894

119873 (23)



















0

20

40

60

80

100

120

0 50 100 150

RR







0

20

40

60

80

100

120

0 20 40 60 80 100 120

RR-db2RR-db4RR-Haar

bior22db8

Number of neurons


0

10

20

30

40

50

60

70

80

90

0 20 40 60 80 100 120

Haardb2db4

db8bior22

RR (

)





0

1

2

3

4

5

6

7

0 20 40 60 80 100 120

Haardb2db4

db8bior22

Number of neurons

t(m

)








0

20

40

60

80

100

0 50 100 150 200 250

CV-RR-Haar RBF


0

10

20

30

40

50

60

70

80

90

0 2000 4000 6000 8000










RR ()classifier





4 Discussion












60[13] 90 [13] 96 [13] 95 96











5 Conclusion






References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























119894isin 119877119903 (1 le 119894 le 119906)


119877119894(119875) = 119877

119894(1003817100381710038171003817119875 minus 119862119894

1003817100381710038171003817) 119894 = 1 119906 (20)


119877119894(119875) = exp[minus

1003817100381710038171003817119875 minus 1198621198941003817100381710038171003817

2

1205902

119894

] (21)


119895(119875)


119910119894(119875) =

119906

sum

119894=1

119877119894(119875) lowast 119908 (119895 119894) (22)




0 100 200 300 400 500 600 700 800 900 10000

01

02

03

04

05

06

07

08

09

1

Training images

Reco

gniti

on ra

teRR for 5th class

0 100 200 300 400 500 600 700 800 900 1000

0

02

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06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 1st class

minus02

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 2nd class

minus02

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 3rd class

minus020 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 6th class

minus02

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 4th class

minus02



0

20

40

60

80

100

120


Clas

s1

Clas

s2

Clas

s3

Clas

s4

Clas

s5

Clas

s6

Clas

s7

Clas

s8

Clas

s9

Clas

s10


0

20

40

60

80

100

120


RR (

)

Clas

s1

Clas

s2

Clas

s3

Clas

s4

Clas

s5

Clas

s6

Clas

s7

Clas

s8

Clas

s9

Clas

s10




12

1

08

06

04

02

120590i


f1(x)

0 minus Ci















119877 (119894) =119873119894

119873 (23)



















0

20

40

60

80

100

120

0 50 100 150

RR







0

20

40

60

80

100

120

0 20 40 60 80 100 120

RR-db2RR-db4RR-Haar

bior22db8

Number of neurons


0

10

20

30

40

50

60

70

80

90

0 20 40 60 80 100 120

Haardb2db4

db8bior22

RR (

)





0

1

2

3

4

5

6

7

0 20 40 60 80 100 120

Haardb2db4

db8bior22

Number of neurons

t(m

)








0

20

40

60

80

100

0 50 100 150 200 250

CV-RR-Haar RBF


0

10

20

30

40

50

60

70

80

90

0 2000 4000 6000 8000










RR ()classifier





4 Discussion












60[13] 90 [13] 96 [13] 95 96











5 Conclusion






References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 100 200 300 400 500 600 700 800 900 10000

01

02

03

04

05

06

07

08

09

1

Training images

Reco

gniti

on ra

teRR for 5th class

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 1st class

minus02

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 2nd class

minus02

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 3rd class

minus020 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 6th class

minus02

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

1

12

Training images

Reco

gniti

on ra

te

RR for 4th class

minus02



0

20

40

60

80

100

120


Clas

s1

Clas

s2

Clas

s3

Clas

s4

Clas

s5

Clas

s6

Clas

s7

Clas

s8

Clas

s9

Clas

s10


0

20

40

60

80

100

120


RR (

)

Clas

s1

Clas

s2

Clas

s3

Clas

s4

Clas

s5

Clas

s6

Clas

s7

Clas

s8

Clas

s9

Clas

s10




12

1

08

06

04

02

120590i


f1(x)

0 minus Ci















119877 (119894) =119873119894

119873 (23)



















0

20

40

60

80

100

120

0 50 100 150

RR







0

20

40

60

80

100

120

0 20 40 60 80 100 120

RR-db2RR-db4RR-Haar

bior22db8

Number of neurons


0

10

20

30

40

50

60

70

80

90

0 20 40 60 80 100 120

Haardb2db4

db8bior22

RR (

)





0

1

2

3

4

5

6

7

0 20 40 60 80 100 120

Haardb2db4

db8bior22

Number of neurons

t(m

)








0

20

40

60

80

100

0 50 100 150 200 250

CV-RR-Haar RBF


0

10

20

30

40

50

60

70

80

90

0 2000 4000 6000 8000










RR ()classifier





4 Discussion












60[13] 90 [13] 96 [13] 95 96











5 Conclusion






References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

20

40

60

80

100

120


Clas

s1

Clas

s2

Clas

s3

Clas

s4

Clas

s5

Clas

s6

Clas

s7

Clas

s8

Clas

s9

Clas

s10


0

20

40

60

80

100

120


RR (

)

Clas

s1

Clas

s2

Clas

s3

Clas

s4

Clas

s5

Clas

s6

Clas

s7

Clas

s8

Clas

s9

Clas

s10




12

1

08

06

04

02

120590i


f1(x)

0 minus Ci















119877 (119894) =119873119894

119873 (23)



















0

20

40

60

80

100

120

0 50 100 150

RR







0

20

40

60

80

100

120

0 20 40 60 80 100 120

RR-db2RR-db4RR-Haar

bior22db8

Number of neurons


0

10

20

30

40

50

60

70

80

90

0 20 40 60 80 100 120

Haardb2db4

db8bior22

RR (

)





0

1

2

3

4

5

6

7

0 20 40 60 80 100 120

Haardb2db4

db8bior22

Number of neurons

t(m

)








0

20

40

60

80

100

0 50 100 150 200 250

CV-RR-Haar RBF


0

10

20

30

40

50

60

70

80

90

0 2000 4000 6000 8000










RR ()classifier





4 Discussion












60[13] 90 [13] 96 [13] 95 96











5 Conclusion






References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119877 (119894) =119873119894

119873 (23)



















0

20

40

60

80

100

120

0 50 100 150

RR







0

20

40

60

80

100

120

0 20 40 60 80 100 120

RR-db2RR-db4RR-Haar

bior22db8

Number of neurons


0

10

20

30

40

50

60

70

80

90

0 20 40 60 80 100 120

Haardb2db4

db8bior22

RR (

)





0

1

2

3

4

5

6

7

0 20 40 60 80 100 120

Haardb2db4

db8bior22

Number of neurons

t(m

)








0

20

40

60

80

100

0 50 100 150 200 250

CV-RR-Haar RBF


0

10

20

30

40

50

60

70

80

90

0 2000 4000 6000 8000










RR ()classifier





4 Discussion












60[13] 90 [13] 96 [13] 95 96











5 Conclusion






References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

20

40

60

80

100

120

0 20 40 60 80 100 120

RR-db2RR-db4RR-Haar

bior22db8

Number of neurons


0

10

20

30

40

50

60

70

80

90

0 20 40 60 80 100 120

Haardb2db4

db8bior22

RR (

)





0

1

2

3

4

5

6

7

0 20 40 60 80 100 120

Haardb2db4

db8bior22

Number of neurons

t(m

)








0

20

40

60

80

100

0 50 100 150 200 250

CV-RR-Haar RBF


0

10

20

30

40

50

60

70

80

90

0 2000 4000 6000 8000










RR ()classifier





4 Discussion












60[13] 90 [13] 96 [13] 95 96











5 Conclusion






References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

20

40

60

80

100

0 50 100 150 200 250

CV-RR-Haar RBF


0

10

20

30

40

50

60

70

80

90

0 2000 4000 6000 8000










RR ()classifier





4 Discussion












60[13] 90 [13] 96 [13] 95 96











5 Conclusion






References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













60[13] 90 [13] 96 [13] 95 96











5 Conclusion






References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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