Fuzzy B-spline Modeling of Uncertainty Data - HIKARI Ltd

Applied Mathematical Sciences, Vol. 6, 2012, no. 140, 6971 - 6991

Fuzzy B-spline Modeling of Uncertainty Data Rozaimi Zakaria Department of Mathematics, Faculty of Science and Technology Universiti Malaysia Terengganu, Malaysia [email protected] Abd Fatah Wahab Department of Mathematics, Faculty of Science and Technology Universiti Malaysia Terengganu, Malaysia [email protected]

Abstract In this paper, we purposed a new method in modeling the uncertainty data based on the theory of fuzzy sets and the interpolation B-spline model curve. The uncertainty data firstly defined by using fuzzy number concept and fuzzy relation by defined them in real data form. The existed coefficient control of interpolation B-spline basis function is substituted with the fuzzy data points(FDPs) which will introduce fuzzy interpolation B-spline curve model. The fuzzification method is applied in order to obtain a fuzzy interval of crisp fuzzy solution curve based on the alpha-cut values. After the fuzzification method was applied, then we used defuzzification method to achieve the final solution which is the crisp fuzzy solution curve. This purposed model is applied in verification of offline handwriting signature(OHS) as the hypothetical example. Keywords: B-spline curve, fuzzy data points, fuzzy B-spline curve, fuzzification, defuzzification.

1 Introduction One of the important role of mathematical models is how they were used in real world phenomena to describe the criteria of their behavior. Generally, in modeling the real world phenomena, the set collection of data are necessary needed as the representative of the real world phenomena. Basically, in modeling the set collection of data through mathematical model, what is important is the needed

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mathematical function of these collected data can be visualized in curve and surface forms. There are many mathematical functions of curve and surface which become the main fundamental in modeling field such geometric modeling [9,10,23]. In geometrical modeling field, curve and surface become the main factors in designing curve and surface [14] in order to representative the real data form. These forms of designing curve and surface become the sources for users to understand the designing curve and surface of representative real data form before the users making the analysis, decision and conclusion [6]. This shape form can be modeled if the representative data have the complete data set which can be used to reconstruct the required shape. However, there is major problem in shape reconstruction due to uncertainty, imprecise and vague of the real data forms which are representative the real phenomena. This situation make us unable to model the uncertainty data through existed functions of curves and surface. In addition, the uncertainty data making us fail to understand the nature of this kind of data. Thus, one of methods which is used to define the uncertainty issues is fuzzy set theory which was introduced by [24]. This theory become basic theory in order to handle the uncertainty matters. Then, this theory was expanded along the arising the uncertainty issues. Therefore, for defining the uncertainty data in geometrical modeling, the issues of uncertainty data can be solved by using the definitions of fuzzy number concepts which was discuss by [4]. When the uncertainty data issues had been defined which become fuzzy data, then there are several studies were extended due to the issues modeling the uncertainty data in geometrical modeling field such fuzzy Bezier curve and surface [2,4,18], fuzzy interpolation Bezier curve [18,20], fuzzy interpolation rational Bezier curve [1,3,19,21] and fuzzy B-spline curve and surface [5,11,12,13]. Our approach to solve the problem is based on the integration between interpolation B-spline curve and fuzzy set theory especially fuzzy number in handling the uncertainty data. The result is fuzzy interpolation B-spline curve(FIBsC) which give user to modeling the fuzzy data after been defined through fuzzy number in curve form where its function is B-spline function. Then, fuzzfication and defuzzification methods of FIBsC are introduced where the fuzzification method is the α -cut operations which used to reduce the fuzzy data interval and defuzzification method is used to obtain crisp fuzzy solution curve of FIBsC. The remainder of the paper is organized as follows: first, the discussing about how to define uncertainty data through fuzzy set theory which includes fuzzy number concept and fuzzy relation. Then, a brief description on B-spline curve and interpolation B-spline curve are given together with the parametrization methods of interpolation B-spline curve. In the fourth section, we develop the

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method of FIBsC based on the theories and definitions which had been given before. Example is given through verification of OHS and finally, discussion and some concluding remarks are given and also future developments will be outlined.

2 Preliminaries Since the theory of fuzzy sets was introduced by Zadeh [24], there were existed others theories were introduced due to requirement in handling the uncertainty problem such as fuzzy number which dealing with the uncertainty in numbers form [15,16,25]. Therefore, in this section, we discussed the fuzzy number concept in defining the uncertainty real data.

Definition 1. Let R be a universal set which R is a real number and A is subset to t R. Fuzzy set, A in R(number around A in R) called fuzzy number which explained through the α -level set(strong and normal α -cut) that is if for every t α ∈ (0,1] , there exist set Aα in R until Aα = { x ∈ R : μ Aα ( x) > α } and t Aα = { x ∈ R : μ Aα ( x ) ≥ α } .

t t Definition 2. If triangular fuzzy number represent as A = ( a, d , c ) and Aα be a α -cut operation of triangular fuzzy number, then crisp interval by α -cut t operation is obtained as Aα = [aα , cα ] = [ (d − a)α + a, −(c − d )α + c ] with

α ∈ (0,1] where the membership function, μ At ( x ) given by ⎧0 ⎪x−a ⎪ ⎪ μ At ( x) = ⎨ d − a ⎪c − x ⎪c − d ⎪0 ⎩

for x < a for a ≤ x ≤ d (1) for d ≤ x ≤ c for x > c

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μ b

1

α

x

0

a

a

α

d

c

α

c

t Figure 1. Triangular fuzzy number, A = ( a, d , c ) .

Fig. 1 shows the triangular fuzzy number where point b is a crisp point and a and c left fuzzy point and right fuzzy point. The α symbol means the α values of triangular fuzzy α -cut. In order to define the uncertainty real data, we used the definition of fuzzy relation to convert the definition of fuzzy number to become FDPs. The definition of fuzzy relation and then the definition of FDPs are given as follow. Definition 3. Let X , Y ⊆ R be universal sets then t R = {(( x, y ), μ Rt ( x, y )) | ( x, y ) ⊆ X × Y } is called a fuzzy relation on X × Y [25]. t Definition 4. Let X , Y ⊆ R and A = {( x, μ At ( x) | x ∈ X } and t B = {( y, μBt ( y) | y ∈ Y } are two fuzzy sets. Then t t t R = ⎡⎣ ( x, y ), μ Rt ( x, y ) ⎤⎦ , ( x, y ) ∈ X × Y is a fuzzy relation on A and B if μ Rt ( x, y ) ≤ μ At ( x ), ∀( x, y ) ∈ X × Y and μ Rt ( x, y ) ≤ μ Bt ( y ), ∀( x, y ) ∈ X × Y [25].

{

}

t X ,Y ⊆ R Definition 5. Let and and M = {( x, μ Mt ( x ) | x ∈ X } t N = {( y , μ Nt ( y ) | y ∈ Y } are two fuzzy data. Then, the fuzzy relation between both t fuzzy data is given by P = ⎡⎣ ( x, y ), μ Pt ( x, y ) ⎤⎦ , ( x, y ) ∈ X × Y .

{

}

D = {( x, y ), x ∈ X , y ∈ Y | x and y are fuzzy data} Definition 6. Let and t D = { Pi | P is data point} are the set of FDPs which is Di ∈ D ⊂ X × Y ⊆ R with R is universal set and μ P ( Di ) : D → [0,1] is membership function defined as

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t

μ P ( Di ) = 1 where can be formulated as D = {( Di , μ D ( Di ) ) | Di ∈ R} . Therefore, ⎧ 0 ⎪ μ P ( Di ) = ⎨c ∈ (0,1) ⎪ ⎩ 1

if

Di ∉ R t Di ∈ R

if

Di ∈ R

if

with μ D ( Di ) = μ P ( Di← ), μ P ( Di ), μ P ( Di→ )

(2)

where μ D ( Di← ) and μ D ( Di→ ) are

left-grade and right-grade membership points values respectively. This can be overwritten as t t D = Di = ( xi , yi ) | i = 0,1,.., n (3) t t t t t for all i, Di = Di← , Di , Di→ with Di← , Di and Di→ are left FDP, crisp data

{

}

point and right FDP respectively. The defining process and the form of FDP after defined using type-1 fuzzy relation can be shown as Fig. 2 and Fig. 3 respectively. μ A ( x)

μ A ( x)

1

1

0.8

0.8

0.6

0.6

0.4

0.4 0.2

0.2 0

x 0

1

2

3

4

5

6

0

0

1

2

3

4

5

x

6

Fuzzy point data

Ordinary point data μ A ( x)

1 0.8 0.6 0.4 0.2 0

0

1

2

3

4

6x

5

Crisp point data

Figure 2. Process of defining fuzzy data. Fig. 2 shows that the process of defining fuzzy data from ordinary point to fuzzy point. The crisp point is the ordinary point which its membership function is equal to 1. μ ( x, y ) 1 0. 0. 0. 0. 0

y

6 4

x

2 0

0

1

2

3

4

5

6

7

8

Figure 3. FDP form after defined by fuzzy relation.

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Fig. 3 shows that the FDP formed by using the definitions fuzzy relation and fuzzy number. This FDP was constructed with the combination of xy-axis through fuzzy relation. Fig. 2 shows that the process of how FDP was define which it is defined a x-axis. In Fig. 3, the construction FDP in xy-coordinate which represent the real FDP form. This real form FDP was obtained through the definition fuzzy number and fuzzy relation which being used in define the real uncertainty data point.

3 α -Cut Operation and Defuzzification Method of Fuzzy Data Point The next proses in getting crisp fuzzy solution of FDP is the α -cut operation(fuzzification process) which is applied to obtain a new FDP( α -FDP) with a new fuzzy interval of α -FDP. Therefore, this following definition is the fuzzification process for FDP.

t t t t Definition 7. Let D be the set of FDPs with Di ∈ D . Then Dα is the α -cut operation of FDPs which is given by t t t→ Diα = Di← , D , D (4) i iα α t t where Di← , Di and Di→ are α -cut of left FDPs, crisp data points and α -cut α α of right FDPs with i = 0,1, 2,..., n . Uncertainty data

t t t t Di = D0 , D1,..., Dn

{

}

Uncertainty data

Uncertainty data

αi

∗ t ∗ D0

Di

Figure 4. FDPs and the interval

∗ t ∗

D2

D1

t Dα

∗ ∗ t ∗

...

∗ t ∗

i

∗

α i − level set

t t Dα←i , Pα i , Dα→i

∗ t← Dα Dα i

i

∗ t→ Dα i

at α i -level set in (0,1] .

Fig. 4 shows that the α -cut operation has been applied against FDPs which t t gives the new fuzzy interval, Dα←i , Pα i , Dα→i at α i -level set values in (0,1] . When the new fuzzy interval has been obtained, then the crisp fuzzy solution is within in it.

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The Def. 7 is extended to the certain cases included the α -cut operation of fuzzy diagonal points beside the α -cut operation of FDPs at x- and y- axis. Therefore, the definition of this cases can be given as follow which is defined based on the α -cut of triangular fuzzy number. Definition 8. From Fig. 1, let say there exists two FDPs in triangular forms which t t is constructed by Dx = ax , bx , cx , and Dy = a y , by , c y where bx and by are crisps FDP at x- and y- axis respectively which is viewed by Fig. 5. Then, α -FDP is obtained based on the α -cut operations which can be defined through the Def. 2 such as: 1) α -FDP at x-axis t t Dx←α = (bx − ax )α + ax , Dx→α = −(cx − bx )α + cx (5)

(

2)

)

α -FDP at y-axis t t ( Dy←α = (by − ay )α + a y , Dy→α = −(cy − by )α + cy ) cy

nR

αy

nL ay

by

. . .

∗ ∗ ∗

. . . ∗

∗

∗ ∗ ∗

(6)

. . .

bx

αx ax

mL

mR

cx

Figure 5. Two FDP with their intersection points after fuzzification process. Fig. 5 shows that the intersection of two FDP which involves two different situation. The two different situation are the FDP at x-axis intersection with the FDP at y-axis. From the illustration based on the Fig. 5, there exists a situation where the FDPs intersect at diagonal which produced fuzzy diagonal data points among two different axis. Therefore, in order to obtain α -FDP in diagonal form, then the fuzzification process need to be extended from the previous methods which are given as follow. 3) α -cut of FDP diagonal at (mL , nL )

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Rozaimi Zakaria and Abd Fatah Wahab t←

(D

x

4)

t (mL ) = (bx − ax )α + ax , Dy← (nL ) = (by − a y )α + a y

)

α -cut of FDP diagonal at (mR , nL ) t t Dx→ (mR ) = −(cx − bx )α + cx , Dy← (nL ) = (by − a y )α + a y

(

5)

(7)

)

(8)

α -cut of FDP diagonal at (mL , nR ) t t Dx← (mL ) = (bx − ax )α + ax , Dy→ (nR ) = −(c y − by )α + c y

)

(9)

α -cut of FDP diagonal at (mR , nR ) t t Dx→ (mR ) = −(cx − bx )α + cx , Dy→ (nR ) = −(c y − by )α + c y

(10)

(

6)

(

)

From Def. 8, the formula in getting the α -FDPs at diagonal had been constructed by using the definition of α -cut operation. Also, this definition can be applied for FDP in 3D form(surface data). After we obtained the α -FDPs, the final steps in getting the crisp fuzzy data solution is the defuzzification method. Defuzzification method is used as the final solution to obtain crisp fuzzy solution which will give the single value output. This method is being used after the fuzzification method has been applied. Therefore, the definition of defuzzification for FDP is given as follow.

t Definition 9. Let α i be the α -cut for every FDPs, Di with i = 0,1,..., n . Then t t t t Di named as defuzzification FDPs for Di if for every Di ∈ D ,

D = { Di } for i = 0,1,..., n t t Di← + Di + Di→ ∑ where for every Di = i =0 which 3 defuzzification process can be illustrated by Fig 6. μ A ( x)

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

The

0.2

x

0

1

2

3

4

5

6

0

0

1

Ordinary data point

2

3

4

5

x 6

Defuzzify data point

μ A ( x)

μ A ( x)

1

1

0.8

0.8

0.6

0.6

0.4

0.4 0.2

0.2 0

i = 0,1,..., n .

μ A ( x)

1

0

(11)

x 0

1

2

3

4

Crisp data point

5

6

0

x 0

1

2

3

4

5

6

Fuzzy data point

Figure 6. Defuzzification process of FDP. Fig. 6 shows how to obtain crisp fuzzy solution of FDP through defuzzification method. This method used after α -FDP was obtained.

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4 Interpolation B-spline Curve Model As we know, the interpolation curve means that all data points are interpolated by a single curve which the curve must goes through all the data points. Therefore, interpolation B-spline curve can be defined via Def. 10. Definition 10. Given a list of data points di ∈ m , 0 ≤ i ≤ n , the B-spline interpolation problem of order k is to find: (1) the knot vector T = (T0 , T1 ,..., Tn + k −1 , Tn + k ) ; (2) the parameter value ti for each di , 0 ≤ i ≤ n , and ; (3) the control points such that the resulting B-spline curve n

Bs(t ) = ∑ PN i i , k (t )

(12)

i =0

has the property Bs (ti ) = di , 0 ≤ i ≤ n [7,8,22]. The computation of B-spline interpolation is straightforward. First, we choose knot vector T = (T0 , T1 ,..., Tn + k −1 , Tn + k ) so that the B-splines N i ,k (t ) can be defined. In open curve cases, the first k Ti ’s are equal and the last k Ti ’s equal, such that, T0 = T1 = ... = Tk −1 and Tn +1 = Tn + 2 = ... = Tn + k because we want the interpolating curve Bs(t ) to have the property Bs (tstart ) = P0 and Bs (tend ) = Pn [14]. For the other knot values, equally-spaced values can be assigns, such that, Ti +1 − Ti = constant , for example. Then, the parametrization method of choice assigns an appropriate parameter value ti to each data point Pi . Once we have done the parametrization, the following equation holds. (13) P0 N 0,k (ti ) + P1 N1,k (ti ) + ... + Pn N n ,k (ti ) = d i , 0 ≤ i ≤ n . In the matrix form, the equation becomes

AP = d , where

⎛ N 0,k (t0 ) N1,k (t0 ) L N n,k (t0 ) ⎞ ⎜ ⎟ N 0,k (t1 ) N1,k (t1 ) L N n,k (t1 ) ⎟ ⎜ A= , ⎜ M M M M ⎟ ⎜⎜ ⎟⎟ ⎝ N 0,k (tn ) N1,k (tn ) L N n ,k (tn ) ⎠ P = ( P0 , P1 ,..., Pn )T and d = ( d 0 , d1 ,..., d n )T [8]. Then, the interpolation B-spline curve can be illustrated in Fig. 7.

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D2

D1

D5

D7

D8

D4 D3

D6

D0

Figure 7. Interpolation B-spline curve of nine points data.

Fig. 7 shows that the interpolation B-spline curve which this curve perform in cubic form where interpolated nine data points. 4.1. Parametrization Method of Interpolation B-spline Curve

The form and quality of interpolating B-spline curve, Bs(t ) is depend to the parametric values, ti for every given data points, di . There are several parametrization techniques which commonly used are the uniformly spaced method, the chord length method and the centripetal method [8,22]. We define the chord length method: t0 = 0, tk = tk −1 + distk −1 , for k = 1,..., n,

(14)

distk = (∇x)2 + (∇y )2 , for k = 0,..., n − 1. where ∇xk = xk −1 − xk , ∇yk = yk −1 − yk . D2

D1

D5

D7 D8

D4 D3

D6

D0

Figure 8. Interpolation B-spline curve using chord length parametrization.

Fig. 8 shows that the interpolation B-spline curve where its parametrization is based on the Eq. 14 which also known as chord length parametrization.

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The uniformly spaced method is defined by dividing the domain [ x0 , xn ] into equal subintervals, t0 = x0 , x −x (15) tk = k n 0 , for 1 ≤ k ≤ n − 1, n tn = xn .

D2 D1

D5

D7

D8

D4 D6

D3

D0

Figure 9. Interpolation B-spline curve using uniformly spaced parametrization.

Fig. 9 shows that the interpolation of cubic B-spline curve using uniformly spaced method parametrization which mentioned by Eq. 15. Then, the centripetal method is given by t0 = 0, tk = tk −1 + (distk −1 ) 2 , for k = 1, 2,..., n, 1

(16)

distk = (∇x)2 + (∇y )2 , for k = 0,1,..., n. where ∇xk = xk −1 − xk , ∇yk = yk −1 − yk D2

D1

D5

D7

D8

D4

D3

D6

D0

Figure 10. Interpolation B-spline curve using centripetal parametrization.

Fig. 10 shows that the interpolation of nine points data formed by cubic B-spline curve using centripetal parametrization.

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5 Fuzzy Interpolation B-spline Curve In this section, we discuss about the construction of fuzzy Interpolation B-spline curve in order modeling the uncertainty points data. Therefore, the uncertainty points data is defined through fuzzy number concepts and fuzzy relation as in Def. 6. Before we define the fuzzy Interpolation B-spline curve, we would like to define fuzzy B-spline curve first. 5.1. Fuzzy B-spline Curve

Fuzzy B-spline curve which had been discuss in [2,4,17]. Therefore, fuzzy B-spline can be defined as in Def. 11. t Definition 11. Let Pi

{ }

n

be a set of fuzzy control points relative to crisps knot sur sequences t1 , t2 ,..., tm = k + 2( n −1) . A fuzzy B-spline curve is a function Bs (t ) from a i =0

real line to the set of real fuzzy number and it defined by k + h −1 t sur Bs(t ) = ∑ PB i i , k (t )

(17)

i =1

t where Pi are fuzzy control points and Bi ,k (t ) ’s are crisp B-spline basic function [2].

t P1

t P3

t P0

t P2

Figure 11. Fuzzy B-spline curve with four fuzzy control points.

Fig. 11 shows that the fuzzy B-spline curve which is interpolate first and last fuzzy control points(fuzzy data) is obtained. The fuzzy control points is defined at x-axis which means the uncertainty data points at x-axis are defined using fuzzy number and fuzzy relation concepts. 5.2. Fuzzy Interpolation B-spline Curve

In this part, we discussed the construction of fuzzy interpolation B-spline curve based on the definition of interpolation B-spline curve, fuzzy control points and

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fuzzy B-spline curve. Then, we applied fuzzification process of triangular α -cut against the fuzzy interpolation B-spline curves which gives us the fuzzy interval of fuzzy interpolation B-spline curve. We used the centripetal parametrization to model the fuzzy interpolation B-spline curve because it has less of bending curve from the other paramterization methods as in shown before. The definition of fuzzy interpolation B-spline curve can be given in Def. 12 by using Def. 10.

t Definition 12: Let di ∈ m be a list of fuzzy data points with 0 ≤ i ≤ n , the fuzzy interpolation B-spline curve can be defined as k + h −1 t sur sur t Bs(t ) = ∑ PB ( t ) Bs ( t ) = d which (18) i i ,k i i, i =0

t where t is crisp knot sequences t1 , t2 ,..., tm = k + 2( n −1) , Pi are fuzzy control points and Bi ,k (t ) is basic function of B-spline. Therefore, Fig. 10 shows that the illustration

of fuzzy interpolation B-spline curve. t D2

t D1 t D0

t D5

t D7

t D8

t D4 t D3

t D6

Figure 12. Fuzzy Intrpolation B-spline curve of nine fuzzy data points.

Fig. 12 shows that the modeling of fuzzy interpolation B-spline curve which interpolated nine fuzzy data points. Also, this modeling fuzzy interpolation B-spline curve of non-symmetric fuzzy data points means that the fuzzy interval values between crisp points and left-right fuzzy points are not equally same. Then, fuzzification method of triangular α -cut is applied against the fuzzy data points to give a new fuzzy data points. Thus, the definition of triangular α -cut is given as follow. Definition 13. Let us assume that membership function of triangular fuzzy number t A denoted by A = (c, δ , β ) is defined as

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c−x ⎧ ⎪ 1 − δ if c - δ ≤ x ≤ c ⎪ x−c ⎪ if c ≤ x ≤ c + β μ A ( x) = ⎨ 1 − β ⎪ ⎪ 0 otherwise ⎪ ⎩ is the centre and δ > 0 is the left spread, β > 0 is the right spread where c ∈ of A. If δ = β , then the triangular fuzzy number is called symmetric fuzzy number and denoted by (c, δ ) . The triangular fuzzy number can be shown in Fig. 13 as the illustration.

1

c −δ

c

c+β

Figure 13. Triangular fuzzy number.

Fig. 13 shows that the triangular fuzzy number which was mentioned in Def. 13. This triangular fuzzy number is used to define the fuzzy control point as mentioned before. Then, for the α -cut concepts of triangular fuzzy number can be defined in Def. 14. Definition 14. Let ∀α ∈ [0,1] , then from

( c − δ ) (α ) − ( c − δ ) ( c + β ) − ( c + β ) (α ) = α, =α c − (c − δ ) (c + β ) − c we obtained (c − δ )(α ) = (c − (c − δ ))α + (c − δ ) (c + β )(α ) = −((c + β ) − c )α + (c + β ) therefore t Aα = [(c − δ ) (α ) , c, (c + β )(α ) ] = [(c − (c − δ ))α + (c − δ ), c, −((c + β ) − c)α + (c + β )].

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Thus, the α -cut of triangular fuzzy number can be applied in the process fuzzification of fuzzy control points which its result can be shown in Fig. 14. t D2

t D1

t D5

t D7

t D8

t D4

t D6

t D3

t D0

(a)

t D2

t D1

t D5

t D7

t D8

t D4

t D3

t D0

t D6

(b)

t D2

t D1

t D5

t D7

t D8

t D4 t D3

t D0

t D6

(c)

Figure 14. New fuzzy interpolation B-spline curve after α -cut of triangular fuzzy number operation with the value of α are (a) 0.3, (b) 0.6 and (c) 0.9 respectively.

From Fig. 14 (a), (b), and (c) shows that the operation of α -cut triangular fuzzy number against the fuzzy control points and shows them in curve form. When the fuzzification process is done, then in order to find the fuzzy crisp solution of fuzzy interpolation B-spline curve, we used defuzzification method as the fuzzy crisp solution. Based on Def. 9 which are applied to Fig. 14 (a), (b) and (c), we obtained these result and then modeling them in Fig. 15.

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( D2 , D2 )

( D5 , D5 )

( D7 , D7 )

( D1 , D1 )

( D8 , D8 )

( D4 , D4 ) ( D6 , D6 )

( D3 , D3 )

( D0 , D0 )

(a)

(a) ( D2 , D2 ) ( D7 , D7 ) ( D1 , D1 )

( D8 , D8 )

( D5 , D5 )

( D4 , D4 ) (D6 , D6 )

( D3 , D3 ) ( D0 , D0 ) (b)

( D2 , D2 )

( D5 , D5 )

( D7 , D7 )

( D1 , D1 )

( D8 , D8 )

( D4 , D4 ) ( D3 , D3 )

(D6 , D6 )

( D0 , D0 ) (c) (Pink=Defuzzification, Blue=Crisp)

Figure 15. Defuzzification process against fuzzy interpolation B-spline curve.

Fig. 15 shows that the defuzzification process of fuzzy interpolation B-spline curve has been applied based on Fig. 14 (a), (b) and (c). The defuzzify curve is marked by green color and the crisps curve is marked by red color. If the α values tend to one of fuzzification curve, then defuzzification curve is tend to crisp curve. This is the relation between fuzzification and defuzzification.

6 Example of Fuzzy Interpolation B-spline Curve Application After the construction of fuzzy interpolation B-spline curve was obtained, then we need some example of application to apply in fuzzy interpolation B-spline curve model. Therefore, we choose verification offline handwriting signature (OHS) model as the basic and simple example.

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6.1. Verification of Offline Handwriting Signature

The verification of offline handwriting signature using fuzzy geometric curve had been discussed in [1,9] using fuzzy interpolation rational Bezier curve model. This model had been developed in piecewise rational curve. Therefore, we demonstrated the fuzzy interpolation B-spline curve in modeling OHS. The crisp OHS can be given in Fig. 16.

D1 D2

D10 D6

D8

D3

D0

D5 D7

D9

D11

D4 D12

Figure 16. Crisp OHS model.

Fig. 16 shows that the crisp OHS model which is model by using interpolation B-spline curve with centripetal parametrization. This OHS model is the signature model which doesn’t use global interpolation B-spline curve. The signature model is modeling by piece of interpolation B-spline curve such that the C0 -continuity are at D6 , D8 and D10 . If there exists uncertainty of OHS, then fuzzy set theory especially fuzzy number concept is used to define the uncertainty data of OHS which called fuzzy data. By FDPs definition, then the modeling of fuzzy OHS model through fuzzy interpolation B-spline curve can be pictured as Fig. 17.

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t D1

t D2

t D0

t D6 t D3

t D10 t D8

t D5 t D7 t D4

t D9

t D11

t D12

Figure 17. Fuzzy OHS which modeled through fuzzy interpolation B-spline curve.

Fig. 17 shows that the fuzzy OHS which is modeled using fuzzy interpolation B-spline curve with twelve fuzzy data points. The fuzzy data points of fuzzy OHS had different situation of FDPs which has FDPs at x-axis, y-axis and diagonal. Therefore, for fuzzification of all these kind of fuzzy data, we applied the α -cut triangular fuzzy number which has mentioned before in order to obtain a new fuzzy data points of fuzzy OHS after α -cut operation. Thus, we obtained the new fuzzy data which is modeled by fuzzy interpolation B-spline curve of fuzzy OHS by Fig. 18 with the values of α is 0.5. t D 1α = 0 .5

t D 2α = 0 .5

t D 0α = 0 .5

t D 5α = 0 .5

t D 6α = 0 .5 t D 8α = 0 .5 t D 3α = 0 .5 t t D 9α = 0 .5 D 7α = 0 .5

t D 4α = 0 .5

t D 1α0 = 0 .5

t D 1α1 = 0 .5

t D 1α2 = 0 .5

Figure 18. The new fuzzy OHS( α -fuzzy OHS) of fuzzy interpolation B-spline curve.

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The α -fuzzy OHS is obtained by Fig. 18 based on the operation of α -cut against FDPs in Fig. 15. When the α -fuzzy OHS is achieved, then the next step is to find crisp fuzzy solution of the fuzzy OHS. Based on Def. 9, the crisp fuzzy solution of fuzzy OHS can be given through Fig. 19.

( D1 , D1 )

( D2 , D2 )

(D10 , D10 )

( D6 , D6 ) ( D8 , D8 ) ( D0 , D0 ) ( D5 , D5 )

( D3 , D3 )

( D7 , D7 )

( D9 , D9 )

(D11, D11)

( D4 , D4 ) ( D12 , D12 )

Green=Defuzzification, Red=Crisp

Figure 19. Crisp fuzzy solution(defuzzification) of α -fuzzy OHS.

Fig. 19 shows that the defuzzification of α -fuzzy OHS(green curve) is plotted together with crisp OHS(red curve). Based on that figure, the crisp fuzzy solution of fuzzy OHS which is modeled by fuzzy interpolation B-spline curve have a little error with the crisp OHS. This is happen because that the certain FDPs of fuzzy OHS had non-symmetric FDPs and the others are symmetric FDPs.

7 Discussion and Conclusion The construction of fuzzy B-spline interpolation curve had been discovered based on its required in modeling of the uncertainty data via geometric modeling. The fuzzy set theory once more is used to develop a new model based on the expended of fuzzy geometric field. The fuzzy B-spline interpolation curve which is used in modeling of fuzzy OHS in verification of the signature is one of the application of this fuzzy model. This fuzzy B-spline interpolation curve also can be extended to variety of FDPs forms and also can be applied in many field in order to model the FDPs which defined by fuzzy number and fuzzy relation concepts.

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Acknowledgement The first authors would like to acknowledge Research Management and Innovation Centre (RMIC) Universiti Malaysia Terengganu and Ministry of Higher Education (MOHE) Malaysia for their funding(FRGS, vot59244) and providing the facilities for doing this research.

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Received: September, 2012