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Optik 124 (2013) 4847– 4854

Contents lists available at ScienceDirect

Optik

jou rn al homepage: www.elsev ier .de / i j leo

recursive color image edge detection method using Green’s function approach

ahra Zareizadeh, Reza P.R. Hasanzadeh ∗, Gholamreza Baghersalimiepartment of Electrical Engineering, University of Guilan, Rasht, Iran

r t i c l e i n f o

rticle history:eceived 4 October 2012ccepted 18 February 2013

a b s t r a c t

In this paper, an extended version of image edge detector using Green’s function approach is proposedfor detection of edges in the color vector space field. In the proposed method, the relationship betweenthe Red, Green and Blue components is considered to design a differential operator for detection of edges

eywords:olor imagedge detectionector differential

in color images. By using the proposed operator, partial derivatives of all components of color image cansimultaneously affect on the edge detection process. Therefore the proposed method can preserve thevector nature of color images during the edge processing stages. Also, the proposed method is comparedboth quantitatively and qualitatively with other color edge detectors. Experimental results show that theproposed method can efficiently preserve the edges even when the color images corrupted with differentlevels of noise.

reen’s function.

. Introduction

In computer vision, edge is one of the most important extractedeatures of the image for several applications such as pattern recog-ition, object tracking, and image retrieval [1–6]. Therefore edgeetection operators can play important roles for quality and modal-

ty of the image analysis. Due to the fact that a color image is usuallyrganized based on three components, i.e. Red (R), Green (G) andlue (B), the color image can provide more information than that ofray-level ones. Therefore more detailed edge information is alsoxpected from color edge detection. While in gray-level images

discontinuity in the gray-level function is indicated as an edge,he term “color edge” cannot be clearly defined for color images.his is based on this reality that in a color image, a color vector isssigned to a pixel, while a scalar gray-level is assigned to a pixelf a gray-level image [7].

Techniques used for color image edge detection can be subdi-ided on the basis of their principle procedures into two classes.irst, monochromatic-based techniques which deal with informa-ion from the individual color channel or color vector componenty separating and then combining individual results of each com-onent. Second, vector-valued techniques which treat the color

nformation as color vectors in a vector space represented by aector norm [8].

In monochromatic techniques, several definitions have been

roposed for color edges. A common definition states that an edgexists precisely in the color image if the intensity of image containsn edge [9]. This definition ignores possible discontinuities in the

∗ Corresponding author. Tel.: +98 131 6690270; fax: +98 131 6690271.E-mail address: hasanzadehpak@guilan.ac.ir (R.P.R. Hasanzadeh).

030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved.ttp://dx.doi.org/10.1016/j.ijleo.2013.02.024

© 2013 Elsevier GmbH. All rights reserved.

hue and/or saturation values. A second definition for a color edgestates that an edge exists in the color image if at least one of thecolor components contains an edge [10]. A third monochromatic-based definition for color edges is based on the calculation of thesum of absolute values of the gradients for the three color compo-nents [10]. All aforementioned definitions ignore the relationshipamong vector components.

Since a color image represents a vector-valued function, adiscontinuity of chromatic information can also be defined in avector-valued way. Therefore, techniques that are used for gray-level images should accommodate for color images. In fact, theapplication of gray-level techniques for individual components ofcolor images is often inadequate.

The vector-valued techniques are categorized into multidi-mensional gradient method and vector method [11]. In themultidimensional gradient methods, first, the partial derivative ofthe gradient is computed in the horizontal and vertical directions[12]. Then, a 2 × 2 matrix is formed from the cross product of thegradient vector in each component; the matrices are summed overall channels and the edge magnitude and direction are given by theprinciple eigenvalues and its eigenvectors. Various forms of thisapproach are used in [13–15]. Accordingly, an extended version ofCanny filter by using Jacobin matrix is proposed in [7]. For a typicalcolor image, the vector gradient is less sensitive to noise than thescalar gradient [16].

In the vector-valued methods, the vector nature of color is pre-served throughout the computation. Many vector methods of coloredge detection form their output from some combination of the

vector differences [17,18]. Histograms of vector differences [19],vector projections [20,21] and differential geometry [22,23] aretechniques that are based on vector differences. One of the mostwidely reported classes of edge detectors are based on vector order

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848 Z. Zareizadeh et al. / O

tatistics [24–26] that represent an extension of the morphologi-al edge operators for gray-scale images [27]. These detectors i.e.ector mean, vector median (VM), vector range (VR) and minimumector dispersion (MVD) operate by detecting local minima andaxima in the image function and then combine them linearly in

suitable way in order to produce a response on the edge. Also, anpproach for color vector ordering by principle component analy-is is discussed in [28] that can be used for color edge detection.ollowing such approach, a new sub-pixel color edge detectionlgorithm based on the moment preserving principle has been pro-osed in [29]. This method is fast but it does not detect many truedges in image. Also color edge detectors based on vector ordertatistics are not able to estimate edge direction.

Other vector methods are also presented. In [30,31] edges arextracted by detecting the change of entropy in a window regionf the image. Also, edge points can be defined by the location ofensity minima in multidimensional feature space [32]. A compassdge detector is approached in [11] that uses a circular window. In

proposed method in [33], smoothness of each pixel in color images firstly calculated by means of similarity relation matrix, and thenixels with lower similarity are assigned as an edge. In some ofolor edge detectors fuzzy membership values are used [34–36].

A differential operator based on Green’s function of a match-ng equation is introduced for edge detection of gray-level images37,38]. In mathematics, Green’s function is a function used toolve inhomogeneous differential equations subject to specific ini-ial conditions or boundary conditions. It is also used in physics,pecifically in quantum field theory, electrodynamics, and statisti-al field theory. The proposed Green’s function based differentialperator, according to Canny’s criteria which is mentioned in [39],s an efficient edge detector [37,38]. In order to design this edgeetector for single-valued images, I′(x) begins from definition oferivative of I(x) as

′(x) = limu→0

I(x + u) − I(x − u)2u

(1)

here I is the signal level (intensity of image) and u is the scal-ng factor [38]. Then I(x) is represented using signal I− such that−(x + u) = I(x). I− may result in a differential equation by using aaylor-series expansion as follows:

u3

6I′′′− (x) + u2

2I′′−(x) + uI′− + I−(x) = I(x) (2)

A gray-level edge detection operator can be obtained by solving2) using Green’s function approach [37].

The paper is organized as follows. In Section 2, the proposedethod is described and a suggested recursive version of the pro-

osed method is introduced for better estimation of color edges.n Section 3, we use some criteria to find the optimal values ofree parameters. Experimental results of the proposed method andlso comparison between this method and other edge detectionethods are presented in Section 4. We describe our conclusions

n Section 5.

. The proposed method

In Section 2.1, the Green’s function approach is used to find alosed form relationship for color edge detection operator and inection 2.2, this new operator is implemented recursively to reachptimal edge values.

In [40] it has been tried to find edges through relationship

etween R, G, and B components of color images. Finally, the ampli-ude of horizontal and vertical edges achieved as:

x =√

R2x + G2

x + B2x (3)

24 (2013) 4847– 4854

Ly =√

R2y + G2

y + B2y (4)

where Rx, Gx, Bx, Ry, Gy and By are the partial differential in x and ydirections.

2.1. Design of differential operator

In order to obtain differential operator for color image, by usingR-(x + u) = R(x), Eq. (2) can be written for R which is the first compo-nent of the color image. The final result can simply be written forthe other components of the color image i.e. G and B [40].

u3

6R

′′′−(x) + u2

2R′′

−(x) + uR′−(x) + R−(x) = R(x) (5)

By transporting R− to the right hand side of (5), and by assumingR′(x) = [R(x) − R−(x)]/u as an alternative of derivative, the followingequation can be derived, using (3)

u2

6R′′′

−(x) + u

2R′′

−(x) + R′−(x) =

√L2

x − G2x − B2

x (6)

In this equation√

L2x − G2

x − B2x is the input function and R− is

the output function. Therefore (6) can be written as:

T(R−) =√

L2x − G2

x − B2x (7)

where T is an operator as:

T = u2

6∂3

∂x3+ u

2∂2

∂x2+ ∂

∂x(8)

If (8) is supposed as a system, system output equals to convolv-ing system input with the impulse response or Green’s functionof output. Therefore to solve this inhomogeneous equation underthe boundary conditions limx→±∞R(x) = 0, we must firstly solve thefollowing equation using the same boundary conditions,

u2

6Gr

′′′u−(x) + u

2Gr′′

u−(x) + Gr′u−(x) = ı(x) (9)

where Gru−(x) is Green’s function of R−.

Gru(x) ={

Gr1u−(x) x < 0 ( lim

x→−∞Gr1

u−(x) = 0)

Gr2u−(x) x ≥ 0 ( lim

x→+∞Gr2

u−(x) = 0)(10)

Public general solution for Gr1u−(x) and Gr2

u− is similar except forcoefficients.

Gru−(x) =

⎧⎪⎨⎪⎩

A1 + e−3x/2u

(B1 sin

(√15

2ux

)+ C1 cos

(√15

2ux

))x < 0

A2 + e−3x/2u

(B2 sin

(√15

2ux

)+ C2 cos

(√15

2ux

))x ≥ 0

(11)

By applying the boundary conditions on Gru−(x)(limx→−∞Gr1

u−(x) = 0 and limx→+∞Gr2u−(x) = 0) and also by

using continuity of Green’s function, Gru−(x) is:

Gru−(x) =

⎧⎨⎩

0 x < 0

B2 e−3x/2u sin

(√15

2ux

)x ≥ 0

(12)

We integrate (9) on the interval (−ε, +ε) to obtain B2 (See Appen-dices (A.1) and (A.2)). So Green’s function is found as

Gru−(x) =

⎧⎨⎩

0 x < 0

−6√15

e−3x/2u sin

(√15

2ux

)x ≥ 0

(13)

So, R− or the system output is

R−(x) = Gru−(x) ∗√

L2x − G2

x − B2x (14)

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here ‘*’ denotes convolution operation.Similarly, if we define R+(x − u) = R(x), Green’s function for R+,

ru+, is obtained as (See Appendices from (A.3) to (A.5)),

ru+(x) =

⎧⎨⎩

0 x < 0

6√15

e3x/2u sin

(−

√15

2ux

)x ≥ 0

(15)

According to (13) and (15), Gru+(x) = Gru−(− x). Therefore R+ is

+(x) = Gru−(−x) ∗√

L2x − G2

x − B2x (16)

A more general form can be reached by includ-ng a free parameter � into derivative definition as�x(x) = limx→u(R(x + �u) − R(x − �u))/2u. Therefore an optimalstimation of derivative can be achieved by a combination of Rx

nd R�x using weight coefficient a. Therefore the final operator iss follows (See Appendices from (A.6) to (A.10)),

(x) = 12u

(F(−x) − F(x)) (17)

here F(x), for x > 0, is given by

(x) = A

[e−3x/2�u sin

(√15x

2�u

)+ Ke−3x/2u sin

(√15x

2u

)](18)

with A = −6a/[√

15(1 + a)(�u)] and K = �/a. R− and R+ are

btained by convolving input (√

L2x − G2

x − B2x ) with F(x) and F(–x),

espectively.The amplitude of the horizontal edge of the R plane can be

btained by convolving D with (√

L2x − G2

x − B2x ). This edge detector

imply can be extended for vertical edges of R plane and also for thether components (B and G) in horizontal and vertical directions.

.2. Recursive implementation

The major problem in this method is that edge in each plane (e.g.) depends on the edge of the other components (e.g. G and B), sodge detection seems impossible. In this section, a recursive form ofhe proposed method is implemented for removing this ambiguity.

To obtain recursive form, the discrete version of function F(x)an be written as [38]

(n) = A

[e−3n/2�u sin

(√15n

2�u

)+ Ke−3n/2u sin

(√15n

2u

)](19)

It is assumed that F(n) is a linear combination of two parts asollows,

(n) = h˛2 (n) + Kh˛1 (n) (20)

here h˛i(n) = Ae−3˛in/2 sin(

√15˛in/2) for ˛1 = 1/u and ˛2 = 1/�u.

he Z-transform of h˛ican be found as

˛i(z) = OUTi(z)

IN(z)= diz

−1

1 − biz−1 − ciz−2(21)

here IN(z) and OUTi(z) are the Z-transforms of the input (in(n))nd output (outi(n)) sequences, di = Ae−3˛i/2 sin(

√15˛i/2), bi =

e−3˛i/2 cos(√

15˛i/2), and ci = 2e−3˛i . For horizontal edge of the

olor image in the R plane, in(n) is√

L2x (n) − G2

x (n) − B2x (n) and

uti−(n) is a part of R-.The filter h˛i

can then be expressed in its recursive form throughhe relation

uti(n) = bi · outi(n − 1) − ci · outi(n − 2) + di · in(n − 1) (22)

Outputs of (22) for ˛1 and ˛2 are combined according to (20), tobtain the sequence R- from in(n).

Then, we similarly compute the output {R+(n)} by means of theiscrete version of function F(−x) as F(−n) = h˛2 (−n) + Kh˛1 (−n).

24 (2013) 4847– 4854 4849

By using Z-transform and inverse of it, outi+(n) obtain as a part ofR+

outi(n) = bi · outi(n + 1) − ci · outi(n + 2)

+ di ·[√

L2x (n + 1) − G2

x (n + 1) − B2x (n + 1)

](23)

According to (23) the value of R+ in each pixel depends on theGx and Bx values in one next pixel that have not been calculated.So instead of computing R+, (22) is used for the sequence inr(n)to Rr+ is calculated, where inr(n) and Rr+ are the spatially reversedsequence of in(n) and R+. Finally, we combine both outputs into thefinal response as

Rx(n) = R+(n) − R−(n) (24)

Similarly, we compute Gx and Bx and finally combine resultsaccording to (3) to achieve the horizontal edge of the color image.Then we repeat this procedure in the y direction to arrive at thevertical edge of the color image.

3. Analytical results

In this section, we evaluate the ability of edge detection ofthe proposed method using Canny’s edge detection criteria [39].Canny’s edge detection criteria contain three components definedas: (1) good detection, such that the probability of finding falseedges or missing real ones is minimized which can be expressedas the ratio of signal-to-noise, (2) good localization, such thatthe detected edges are close to the real ones. The localizationmeasure can be expressed as the reciprocal of root-mean-squaredistance between the detected edge and the true edge, and (3) sin-gle response, such that multiple detections due to noise are notassociated with a single true edge. This criterion can be based onthe distance between adjacent maxima of the filter output. For anedge detector operator, f(x), assuming a step edge located at x = 0,each of the above criteria has been mathematically represented.The detection and localization criteria are respectively defined as

˙(f ) =∫ 0

−wf (x) dx√∫ w

−wf 2(x) dx

, �(f ′) =∣∣f ′(0)

∣∣√∫f ′2(x) dx

(25)

and usually the product of these two criteria (˙�) is used. Also,the single response measure is defined as follows,

SRC = xmax

W(26)

where

xmax = 2�

√√√√ ∫ w

−wf ′2(x) dx∫ w

−wf ′′2(x) dx

(27)

is the mean distance between maxima of the filter output due tonoise, and W is the filter width.

There are two free parameters (namely a and �) in the pro-posed operator which can affect the performance of detection.These parameters should be selected such that the filter D(x) yieldsmaximum overall performance as measured by its (��)SRC index.

Fig. 1 shows the curves of the performance indices,��, SRC, and(��)SRC. As shown in Fig. 1 and Table 1, the power of edge detec-tion of proposed filter is high ((�� ≥ 1.225), while single-responsecriterion is eligible as max(SRC) = 2.472 in � = 1 and 0 < a ≤ 3. For

a = 2.85 and � = 0.35, the proposed filter yields maximum overallperformance. Also, as shown in Table 1, in gray-scale Green functionoperator [32] the best value of (��)SRC is occurred in a = 1.1 and� = 0.28. Fig. 2 shows the curve of D(x) for optimal values of � and a.

4850 Z. Zareizadeh et al. / Optik 124 (2013) 4847– 4854

Fig. 1. (a–c) Performance of Canny’s criteria for filter D(x), as functions of � and a.(d) Performance of ˙�SRC criteria for filter D(x). The various curves correspond todifferent values of a.

Table 1The values of Canny’s criteria for edge detection operators.

Operator ˙� SRC ˙�SRC � a

Proposed method 1.5 2.33 3.477 0.35 2.85Gray-scale Green [32] 1.673 2.121 3.547 0.28 1.1Canny [39] 0.92 3.245 2.985 – –

Ai

4

dd

etf

tomi

Deriche [41] 1.644 2.028 3.335 – –Sarkar and Boyer [42] 1.21 2.8 3.388 – –

lso in Table 1, the power and performance of proposed operators compared with other edge detection operators [38,39,41,42].

. Experimental results

In this section, the experimental results of our proposed edgeetector are presented and then compared with other color edgeetector.

Here both quantitative and qualitative measures are used tovaluate the performance of our edge detector in comparison withhe monochromatic based method based on the gray-scale Green’sunction operator and some of vector-valued techniques.

The quantitative performance measures can be grouped intowo types: probabilistic measure which is based on the statistics

f corrected edge detection and false edge rejection, and distanceeasure which is based on edge deviation or error distance which

s the minimum distance between the detected and the true edges.

Fig. 2. The curve of D(x) for a = 2.8 and � = 0.35.

Fig. 3. Comparison of FOM performance for color edge detectors as function ofsignal-to-noise ratio for (a) Gaussian noise and (b) impulsive noise.

A distance measure that is often used in edge detector’s evaluationsis Pratt’s figure of merit (FOM) [43]. It is defined as

FOM = 1max{ID, II}

ID∑n=1

1

1 + ˛(dn)2(28)

where ID and II are the number of detected and ideal edge points,dn is the distance from the nth detected edge point to the nearesttrue edge point and ̨ (>0) is a scaling constant. FOM takes valuesin (0, 1], and being equal to 1 if detected edges coincides with idealedges and while the detected edges deviate from the ideal edges,FOM approaches to zero. The scale parameter ̨ controls the sensi-tivity of FOM to the differences between the detected and the idealedges: for small values of ˛, FOM is close to 1 only if the detectededges are very similar to ideal edges, while for large values of ˛,larger differences between the detected and the ideal edges can beneglected.

FOM has advantages over the probabilistic measures that rendera more realistic assessment of the detected edges [44]. If we con-sider, for example, the case where all the edges are 1 pixel shiftedfrom the ground edge map, a probabilistic measure would give avery poor rating but FOM still gives a performance measure veryclose to unity (say 0.9). Hence FOM is used in this work. We adoptthe scaling constant ̨ = 1/9 that is proposed in [43].

Here, the proposed method is compared with a monochromatic-based color edge detection based on gray-scale Green functionapproach. For applying gray-scale Green function operator in coloredge detection, first this operator is applied to each R, G, and Bplanes separately, and finally combined edge image obtained byaveraging the three edge images. Also, the proposed method iscompared with some of vector valued techniques such as the PCA(principle component analysis) method [28], similarity relationmatrix-based color edge detector [33] and the extended version

of Canny filter [7] as a vector gradient-based method.

An artifact image is used for performance evaluation of the pro-posed method and comparison with other methods. Gaussian noisecontaminates the artifact image at different levels.

Z. Zareizadeh et al. / Optik 124 (2013) 4847– 4854 4851

F onocs filter

pFns(oa

s

ig. 4. Results of edge detectors applied to color images: (a) original images, (b) mimilarity relation matrix-based color edge detector, (e) extended version of Canny

Also, several different color natural images such as “Lena,” “Air-lane” and “Peppers” have been used for assessing our method byOM. These images were corrupted by Gaussian and impulsiveoise at various noise levels. In each case, FOM has been mea-ured and used as the performance criterion. The ground truthreal edges) that is needed for the computation of FOM is trivially

btained for the noise free (original) image with the application ofny edge detector.

Fig. 3 presents the FOM results for Gaussian and impul-ive noise in differential edge detectors. For Gaussian noise, in

hromatic-based method based on gray-scale Green function operator, (c) PCA, (d)method and (f) proposed method.

comparison with the proposed method, the FOM level of similaritymatrix method is near particularly in 15 dB and 30 dB but in colorCanny, PCA and monochromatic-based techniques the FOM valuesare lower. For impulsive noise, the proposed method has a FOM thatis higher than those of other techniques.

Subjectively, the performance of the proposed method has

been assessed using real color images. Results for the edgedetection algorithm for original color images are shown inFig. 4 and are compared with the other color edge detectionmethods. As shown, the proposed vector-valued method has

4852 Z. Zareizadeh et al. / Optik 124 (2013) 4847– 4854

Fig. 5. Noise behavior of color edge detectors: (a) color images corrupted with Gaussian noise wit SNR = 10 dB, (b) monochromatic-based method based on gray-scale Greenfunction operator, (c) PCA, (d) similarity relation matrix-based color edge detector, (e) extended version of Canny filter method and (f) proposed method.

F lsive nf (e) ex

fd

FrTa

in

5

Gibs

′ = 13

u2

2Gr′′

′+ u

2Gr′′ + Gr′ = u2

6Gr′′

′+ 1

6

(uGr′′ + 6Gr′)

( )

ig. 6. Noise behavior of color edge detectors: (a) color images corrupted with impuunction operator, (c) PCA, (d) similarity relation matrix-based color edge detector,

easible edge preservation in comparison to other color edgeetectors.

The noise behavior of the color edge detectors is illustrated inigs. 5 and 6. In these figures, color images are respectively cor-upted by Guassian noise and impulsive noise with SNR = 10 dBhese figures show the edge maps producted by four selected oper-tors and the proposed mehod.

As can be verified the performance of the proposed methods superior to other edge detectors mainly in the case of impulseoise.

. Conclusion

In this paper, a color edge detection operator based on thereen’s function approach was proposed by assuming that the color

mages have relationship between theirs components. To find theest color edge estimation, the proposed operator is applied recur-ively to color images.

u2

6Gr′′′(x) + u

2Gr′′(x) + Gr′(x) = 1

3

(u2

2Gr′′′ + 3u

2Gr′′

)+ Gr

( )

= u2

6Gr

′′′+ 1

6

[(uGr′)′ + 5Gr

oise wit SNR = 10 dB, (b) monochromatic-based method based on gray-scale Greentended version of Canny filter method and (f) proposed method.

The results of the numerical and subjective evaluations demon-strate a superiority of the proposed method.

As experimental results show, the proposed method detectsmajority of variations of images. This ability enables ones to selectarbitrary edges from a wide range of estimated edges throughthresholding methods. Also, robustness of the proposed methodwas shown for different types of noise and even for very low SNR.

Before edge detection often a smoothing operator is applied todecrease noise. In this method, by integrating introduced operatorwe can simply achieve a smoothing operator which is applied inthe direction parallel to the edge to obtain the best result.

In design of the operator, if one-dimensional derivativedefinition is replaced by two-dimensional one as I′ = [I(x + u,y + u) − I(x − u, y − u)]/2u and then by using two-dimensionalGreen’s function approach, an extended version of the proposedoperator was obtained.

Appendix A.

Finding B2:

[( ) ] ( )

′] = u2

6Gr′′

′+

(u

6Gr′

)′+ 5

6Gr′ (A.1)

ptik 124 (2013) 4847– 4854 4853

∫(x)dx ⇒ u2

6Gr′′(+ε) + u

6Gr′(+ε) + 5

6Gr(+ε)

u2

6

(−3

√15

2u2B2

)+ u

6

(√15

2uB2

)+ 0 = 1 ⇒ B2 = −6√

15(A.2)

R R+(x) = R(x) ⇒ u2

6R′′′

−(x) − u

2R′′

−(x) + R′−(x) = R+(x) − R(x)

u

(A.3)

= 0

= 32u

± j

√15

2u

⇒ Gr+ ={

Gr1+ x < 0 limx→−∞

Gr1+ = 0

Gr2+ x ≥ 0 limx→+∞

Gr2+ = 0⇒ Gr

+ C1 cos

(√15

2ux

))x < 0

+ C2 cos

(√15

2ux

))x ≥ 0

⇒

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

A1 = 0

C1 = 0

A2 = 0

B2 = 0

C2 = 0

(A.4)

G r′′+(x) + Gr′

+(x)

)dx =

∫ +ε

−ε

ı(x) dx ⇒ B1 = 6√15

⇒ Gr+

(A.5)

R + �u) − R(x − u�)

2�u

)u))

](A.6)

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

(A.7)

L2x − G2

x − B2x ∗

(Gr�u(x) + �

aGru(x)

)](A.8)

F (A.9)

F

Z. Zareizadeh et al. / O

+ε

−ε−→∫ +ε

−ε

(u2

6Gr′′

)′dx +

∫ +ε

−ε

(u

6Gr′

)′dx +

∫ +ε

−ε

56

Gr′ dx =∫ +ε

−ε

ı

= 1 ⇒

Finding Gru+:

+(x − u) = R(x)Taylor−series−Extension−→ − u3

6R′′′

+(x) + u2

2R′′

+(x) − uR′+(x) +

= Rx =√

L2x − G2

x − B2x

This equation solve by Green function method:

u2

6Gr′′′

+ (x) − u

2Gr′′

+(x) + Gr′+(x) = ı(x) ⇒ u2

6r3 − u

2r2 + r = 0 ⇒

{r

r

=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

A1 + e3x/2u

(B1 sin

(√15

2ux

)

A2 + e3x/2u

(B2 sin

(√15

2ux

)

r+ =

⎧⎨⎩ B1 e3x/2u sin

(−

√15

2ux

)x < 0

0 x ≥ 0

∫ +ε

−ε

(u2

6Gr

′′′+(x) − u

2G

=

⎧⎨⎩

6√15

e3x/2u sin

(−

√15

2ux

)x < 0

0 x ≥ 0

Finding D(x):

′(x) = R1x

1 + a+ a · R�x

1 + a= 1

1 + a

(R(x + u) − R(x − u)

2u

)+ a

1 + a

(R(x

= a

2(1 + a)�u

[�

a(R(x + u) − R(x − u)) + (R(x + �u) − R(x − �

R(x − u) = Gru−(x) ∗√

L2x − G2

x − B2x

R(x − �u) = Gr�u−(x) ∗√

L2x − G2

x − B2x

R(x + u) = Gru−(−x) ∗√

L2x − G2

x − B2x

R(x + �u) = Gr�u−(−x) ∗√

L2x − G2

x − B2x

R′(x) = a

2(1 + a)�u×

[√L2

x − G2x − B2

x ∗(

Gr�u(−x) + �

uGru(−x)

)−

√R′(x) =

[√L2

x − G2x − B2

x ∗ F(−x) −√

L2x − G2

x − B2x ∗ F(x)

]

(x) = a

2(1 + a)�u×

(Gr�u(x) + �

aGru(x)

)

(x) = −6a

2√

15(1 + a)�u

[e

−3x2�u sin

(√15

2�u

)+ �

ae

−3x2u sin

(√15

2u

)](A.10)

4 ptik 1

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854 Z. Zareizadeh et al. / O

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