ieee transactions on fuzzy systems, vol. 22, no. 6...

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 6, DECEMBER 2014 1515

Edge-Detection Method for Image Processing Basedon Generalized Type-2 Fuzzy Logic

Patricia Melin, Senior Member, IEEE, Claudia I. Gonzalez, Juan R. Castro, Olivia Mendoza,and Oscar Castillo, Senior Member, IEEE

Abstract—This paper presents an edge-detection method that isbased on the morphological gradient technique and generalizedtype-2 fuzzy logic. The theory of alpha planes is used to implementgeneralized type-2 fuzzy logic for edge detection. For the defuzzi-fication process, the heights and approximation methods are used.Simulation results with a type-1 fuzzy inference system, an inter-val type-2 fuzzy inference system, and with a generalized type-2fuzzy inference system for edge detection are presented. The pro-posed generalized type-2 fuzzy edge-detection method was testedwith benchmark images and synthetic images. We used the meritof Pratt measure to illustrate the advantages of using generalizedtype-2 fuzzy logic.

Index Terms—Alpha planes representation, edge detection, gen-eralized type-2 fuzzy logic, image processing.

I. INTRODUCTION

AN edge may be the result of changes in light absorption,color, shade, and texture, and these changes can be used

to determine the depth, size, orientation, and surface propertiesof a digital image [1]. In analyzing the image digitally, edgedetection involves filtering irrelevant information to select theedge points. The detection of subtle changes may be mixedup by noise and this depends on the pixel threshold of changethat defines an edge. Detection of these continuous edges isvery difficult and time consuming especially when an image iscorrupted by noise [2].

Edge detectors have been an essential part of many computervision systems. The edge-detection process is useful for simpli-fying the analysis of images by drastically reducing the amountof data to be processed [3]. The main application areas of edgedetectors include: geography, military, medicine, robotics, me-teorology, and pattern recognition systems [4]–[8].

In the area of image processing, there exist some edge-detection methods that make use of type-1 fuzzy systems [9],[10], [11], neural networks [12], genetic algorithms with particle

Manuscript received May 17, 2013; revised July 30, 2013 and September 23,2013; accepted November 26, 2013. Date of publication January 2, 2014; dateof current version November 25, 2014. This work was supported by CONACYTcontract Grant 44524.

P. Melin and O. Castillo are with the Division of Graduate Studies, Tijuana In-stitute of Technology, Tijuana 22500, Mexico (e-mail: [email protected];[email protected]).

C. I. Gonzalez is with the School of Engineering, University of BajaCalifornia, Tijuana 22379, Mexico, and also with the Tijuana Institute of Tech-nology, Tijuana 22500, Mexico

J. R. Castro and O. Mendoza are with the School of Engineering, Universityof Baja California, Tijuana 22379, Mexico (e-mail: [email protected];[email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TFUZZ.2013.2297159

TABLE ISOME EDGE DETECTION METHODS

swarm optimization (PSO) [2], ant colony optimization (ACO)[13], [14], interval-valued fuzzy operators [15], interval type-2 fuzzy systems combined with the Sobel operator [16], [17],interval type-2 fuzzy systems, and the morphological gradient(MG) [18] and an improved Canny method that is based on in-terval type-2 fuzzy logic [19]. Of course, we can also find thetraditional methods for image processing, like the Canny [3],MG, Sobel [20], Roberts [21], and Kirsch methods [22].

The main goal of the vision systems that are based on com-putational intelligence techniques is to achieve better edge de-tection when image processing is performed under high noiselevels [2]. In Table I, a summary of these methods is presented.

In [18], an improved method for edge detection that is basedon interval type-2 fuzzy logic is proposed. This paper applies theMG technique that is combined with a type-1 fuzzy inferencesystem (T1FIS) and with an interval type-2 fuzzy inferencesystem (IT2FIS), where the authors conclude that the IT2FISis better than T1FIS. The IT2FIS achieves better control of thedetected edges in the image.

Recently, there has been a significant increase in the researchon higher order forms of fuzzy logic, in particular, the use ofinterval type-2 fuzzy logic [23]–[26] and more recently gen-eralized type-2 fuzzy logic. Of course, the idea of going intohigher orders or types of fuzzy logic is to construct better mod-els of uncertainty. In this sense, it is theoretically expected thatgeneralized type-2 fuzzy logic will allow better management ofuncertainty [27]. However, generalized type 2 requires a highercomputational overhead and several efforts have been made inorder to limit the complexity of general type-2 fuzzy logic;for example, Wagner and Hagras [27], [28] have introducedthe zSlices-based representation, and Mendel and Liu [29]–[31]have put forward a representation that is based on alpha-planes,which both enable the representation of, and computation with,general type-2 fuzzy sets.

1063-6706 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications standards/publications/rights/index.html for more information.

1516 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 6, DECEMBER 2014

The main contribution of this paper is the proposed newmethod for edge detection that is based on generalized type-2 fuzzy logic and the MG, which allows for better modeling ofthe uncertainty that exists in processing digital images, as wellas to obtain a comparative study of T1FIS, IT2FIS, and gener-alized type-2 fuzzy inference system (GT2FIS) fuzzy inferencesystems as tools for enhancing edge detection in digital imageswhen used in conjunction with the MG.

The rest of this paper is organized as follows. Section IIdescribes the MG technique. In Section III, some basic conceptsof generalized type-2 fuzzy logic and the theory of alpha planesare presented, which are used for the implementation of theproposed method. Section IV describes the methodology that isused to develop the proposed edge-detection method. Section Vexplains the technique for evaluating the quality of the detectededges. Section VI presents simulation results with benchmarkimages to illustrate the advantages of the proposed generalizedtype-2 fuzzy edge-detection method. Finally, Section VII offerssome conclusions of the proposed method.

II. EDGE DETECTION USING THE MORPHOLOGICAL GRADIENT

The MG of a gray-scale image can be defined as the differ-ence between the intensity values of two neighboring pixels thatbelong to a given structural element. The core of gradient edgedetection is, of course, the gradient operator ∇. In continuousform and applied to a continuous space image, the gradient,fc (x, y), is defined by

∇fc (x, y) =∂fc (x, y)

∂xix +

∂fc (x, y)∂y

iy (1)

where ix and iy are the unit vectors in the x− and y−directions,respectively. Notice that the gradient is a vector, having boththe magnitude and direction. Its magnitude, |∇fc (x0,y0)|, mea-sures the maximum rate of change in the intensity at the location(x0,y0). Its direction is that of the greatest increase in intensity.To produce an edge detector, we consider the effect of findingthe local extrema of ∇fc (x, y) or the local maxima of

|∇fc (x, y)| =

√(∂fc(x, y)

∂x

)2

+(

∂fc(x, y)∂y

)2

. (2)

The precise meaning of “local” is very important here, if themaxima of (2) are found over a 2-D neighborhood, the result isa set of isolated points rather the desired edge contours [32].

In this paper, we are going to use Di instead of∇fc (x, y); weapply (2) for a matrix of 3 × 3 as shown in Fig. 1; therefore, wecan obtain the coefficients zi with (3), and the possible directionof the edge Di with (4). The edges S can be calculated with(5) [18], [33]

z1 = f (x − 1, y − 1) z2 = f (x, y − 1)

z3 = f (x + 1, y − 1) z4 = f (x − 1, y)

z5 = f (x, y) , z6 = f (x + 1, y)

z7 = f (x − 1, y + 1) z8 = f (x, y + 1)

z9 = f (x + 1, y + 1) (3)

Fig. 1. Matrix of 3 × 3 indicating the coefficients Zi and the edge directionDi .

D1 =√

(z5 − z2)2 + (z5 − z8)

2

D2 =√

(z5 − z4)2 + (z5 − z6)

2

D3 =√

(z5 − z1)2 + (z5 − z9)

2

D4 =√

(z5 − z3)2 + (z5 − z7)

2 (4)

S = D1 + D2 + D3 + D4. (5)

III. GENERALIZED TYPE-2 FUZZY LOGIC

A. Definition of Type-2 Fuzzy Sets

A generalized type-2 fuzzy set (T2 FS), which is denoted byA, is characterized by a type-2 membership function μA (x, u),where xεX, uεJx ⊆ [0, 1], and 0 ≤ μA (x, u) ≤ 1, and can berepresented by (6) [29], [31], [34]–[36]

A = {((x, u) , μA (x, u)) |∀x ∈ X,∀u ∈ Jx ⊆ [0, 1]} . (6)

If A is continuous, it can be denoted by the followingequation:

A ={∫

x∈X

μA (x)/x

}

=

{∫x∈X

∫u∈J u

x ⊆[ 0 , 1 ]

μA (x, u)/(x, u)

}

=

{∫x∈X

[∫u∈J u

x ⊆[ 0 , 1 ]

fx(u)/u

]/x

}(7)

where∫ ∫

denotes the union for x and u. Here, Jx is called theprimary membership of x in A. At each value of x say x = x′,the 2-D plane, whose axes are u and μA (x′, u), is called avertical slice of A [34]. A secondary membership function is avertical slice of μA (x, u). It is μA (x = x′, u), for x′ ∈ X and∀u ∈ Jx ′ ⊆ [0, 1], and it is described as

μA (x = x′, u) ≡ μA (x′u)

=∫

u∈J x ′fx ′(u)/u Jx ′ ⊆ [0, 1] (8)

in which 0 ≤ fx ′ (u) ≤ 1.

MELIN et al.: EDGE-DETECTION METHOD FOR IMAGE PROCESSING BASED ON GENERALIZED TYPE-2 FUZZY LOGIC 1517

Fig. 2. Generalized type-2 membership function.

Fig. 3. FOU of the generalized type-2 membership function.

Fig. 4. Example of the associated T2 FS for the alpha-plane.

In Fig. 2, we can find a representation of a generalized type-2membership function, and in Fig. 3, the footprint of uncertainty(FOU) is illustrated, which is associated with the third dimen-sion and allows a better modeling of real-world uncertainty.

B. α-Planes Representation

An α-plane for a generalized T2 FS, in this case A, is denotedby Aα , and it is the union of all primary membership functionsof A, for which secondary membership degrees are higher orequal to α (0 ≤ α ≤ 1) [29], [31]. The equation of the alphaplane is represented by (9). In Fig. 4, the representation of an

Fig. 5. Proposed Model for the GT2FIS.

alpha plane is shown

Aα = {(x, u) , μA (x, u) ≥ α|∀x ∈ X,∀u ∈ JX ⊆ [0, 1]}

=∫∀x∈X

∫∀u∈Jx

{(x, u) |fx (u) ≥ α} . (9)

IV. EDGE DETECTOR USING A GENERALIZED TYPE-2FUZZY SYSTEM

In this section, the proposed model for edge detection that isbased on a GT2FIS is described. In Fig. 5, the block diagram ofthe GT2FIS for edge detection is presented.

A. Input Image

The first step in the whole process is reading an input imageto apply the edge-detection method. In this case, we are onlyconsidering images with a gray scale.

B. Obtaining the Image Gradients

In this step, the MG technique that has been described inSection I is applied to obtain the gradients in the four direc-tions using (3) and (4), and then they are used as inputs for theproposed generalized type-2 fuzzy inference system.

C. Fuzzification

The fuzzifier maps crisp inputs into generalized type-2 fuzzysets to process within the FLS. In this paper, we will focus onthe type-2 singleton fuzzifier as it is fast to compute and, thus,suitable for the generalized type-2 FLS real-time operation. Sin-gleton fuzzification maps the crisp input into a fuzzy set, whichhas a single point of nonzero membership. Hence, the singletonfuzzifier maps the crisp input x′

p into a type-2 fuzzy singleton,whose MF is μAp

(xp) = 1/1 for xp = x′p , and μAp

(xp) = 0for all xp = x′

p for all p = 1, 2, ..., P, where P is the number ofFLS inputs [27], [37].

For this case study, the inputs are represented by the gradientsDi of the original image, and each of them will be an input to


Fig. 6. Generalized type-2 fuzzy inference system.

the fuzzy system. In the following lines, we define each of theinput and output linguistic variables.

1) Input Linguistic Variables: Four inputs are defined, inwhich each one has three Gaussian membership functions withuncertain mean. The linguistic variables that are used for thefour inputs are: low, medium, and high. In order to adapt themembership functions to the range of gray tones dependingon the image, we obtain the maximum, minimum, and middlevalues of Di with (10)–(12), and we use these values to calculatethe mean of the membership functions, but adding different sizesof the FOU. For this task, we made tests using different sizes ofthe FOU for the Di input variables.

The Gaussian membership functions for each D input areobtained with (21)–(25), and the means of each function areobtained with (14)–(15). For example, for the high membershipfunctions, the first mean was obtained with (14), the secondmean was calculated with (15), and the σ value was obtainedwith (13).

The inference system has one output S (the edge), the linguis-tic values that are used for the output are: edge and no_edge, andwe selected the range [0, 1], since the input image was normal-ized in this range, where the minimum value for the output isrepresented by (16) and maximum by (17). The Gaussian mem-bership functions for the output are obtained with (21)–(25),the means of each function are obtained with (19)–(20), andthe σ value with (18). The FOU for the output variable S wascalculated in a similar way to the inputs variables. This is themethod that we propose to adapt the parameters of the member-ship functions depending on the contrast level of each image. InFig. 6, we show the linguistic variables with generalized type-2membership functions, where the value for the FOU is 0.2 onthe inputs and 0.25 for the outputs. We have to note that weare using a number between 0 and 1 to represent the size of theFOU, which of course is only a crisp value that represents theaverage size of the footprint to model the particular problem

lowi = min(Di) (10)

highi = max(Di) (11)

mediumi = lowi + (highi − lowi)/2 (12)

σi = highi/5 (13)

m1 = highi (14)

m2 = m1 + (m1 × FOU), where FOU is in (0, 1) (15)

no edgei = 0 (16)

edgei = 1 (17)

σi = edge/4 (18)

m1 = edgei (19)

m2 = m1 + (m1 × FOU), where FOU is in (0, 1) (20)

μ (x, u) = gausmgausstype2 (x, u, [σx,m1 ,m2 ]) (21)

where “gausmgausstype2” stands for the Gaussian generalizedtype-2 membership function with uncertain mean

μ (x) =[μ (x) , μ (x)

]= igausmtype2 (x, [σx,m1 ,m2 ]) (22)

where “igausmtype2” stands for the Gaussian interval type-2membership function with uncertain mean

mx =m1 + m2

2

mx =m1 + m2

2

σu =δ

2√

6+ ε (23)

px = gaussmf (x, [σx,mx ]) = exp

[−1

2

(x − mx

σx

)2]

(24)

μ (x, u) = gaussmf (u, [σu , px ])

= exp

[−1

2

(x − px

σu

)2]

. (25)


D. Inference

Once the input and output variables are defined, with theirrespective membership functions, the inference process is per-formed in the system, and for this the following steps are needed.

1) Define the Fuzzy rules:The structure of the rules in the generalized type-2 FLS is

the standard Mamdani-type FLS rule structure used in the type-1 FLS and an interval type-2 FLS; however, in this paper, weassume that the antecedent and the consequent sets are repre-sented by generalized type-2 fuzzy sets. Therefore, for a type-2FLS with p inputs x1εX1 , . . . , xp ∈ XP and one output y ∈ Y ,multiple input single output (MISO), if we assume that thereare M rules, the kth rule in the generalized type-2 FLS can bewritten as follows [38]:

Rk : IF x1 is F k1 and . . . and xp is F k

p , THEN y is Gk . (26)

To model the process with the fuzzy system, we considerthree rules that help describe the existing relationship betweenthe image gradients. The fuzzy rules are the following.

a) If (D1 is HIGH) or (D2 is HIGH) or (D3 is HIGH) or(D4 is HIGH), then (S is EDGE).

b) If (D1 is MEDIUM) or (D2 is MEDIUM) or (D3 isMEDIUM) or (D4 is MEDIUM), then (S is EDGE).

c) If (D1 is LOW) and (D2 is LOW) and (D3 is LOW) and(D4 is LOW), then (S is NO_EDGE).

2) Performing inference with the alpha planes:To perform the inference in the fuzzy system, the alpha planes

representation was used (see Section III). In this case, the alphaplanes are obtained in the secondary membership functions ofthe antecedents F k

i and consequents Gki of the ith input and kth

rule. The alpha planes create an interval type-2 fuzzy set, [29],[31], [35], which is defined by the following equations:(

F ki

)∝

=

⎧⎨⎩∫

x ′i ∈Xi

⎡⎣∫

μ∝F k

i

(x ′i )∈

[μ∝

F ki

(x ′i ),μ∝

F ki

(x ′i )

] 1/μ∝F k

i(x′

i)

⎤⎦/

x

⎫⎬⎭

(F k

i

)∝

=

{∫x ′

i ∈Xi

[μ∝

F ki

(x′i) , μ∝

F ki

(x′i)]/

x′i

}(F k

i

)∝

={[

μ∝F k

i

(x′i) , μ∝

F ki

(x′i)]/x′

i

}(27)

(Gki )∝

=

⎧⎨⎩∫

yi ∈Yi

⎡⎣∫

μ∝G k

i

(yi )∈[μ∝G k

i

(yj ),μ∝G k

i

(yj )]1/μ∝

Gkj(yi)

⎤⎦/

yj

⎫⎬⎭

(Gki )∝ =

{∫x ′

i ∈Xi

[μ∝Gk

i

(yj ), μ∝Gk

i(yj )]/yj

}. (28)

3) Firing strengths:The firing strengths of the rules are calculated, where the

firing sets μαF k

i

(x′i) for each alpha plane α, of the ith input and

kth rule of a singleton type-2 FLS are represented as

Ωk∝ (x′) = n

i=1

{μα

F ki

(x′i)}

Ωk∝ (x′) = n

i=1

{μα

F ki

(x′)}

.(29)

4) In a multiple input single output FLS:The inferred output μα

Bj(yj ) and μα

Bj(yj ) of each rule k are

represented by

μαBj

(yj ) = Ωk∝(x′) μ∝

Gkj

(yj )

μαBj

(yj ) = Ωk∝(x′) μ∝

Gkj(yj ) (30)

where μαGk

j

is the type-2 fuzzy MF that represents the αth alpha

plane, kth rule, and jth input of the consequents.5) The outputs of the fired rules (M) are combined using the

join operation to produce the overall output set, which can bewritten as follows:

μαBj

(yj ) = �rk=1{μ∝

B kj

(yj )}

μαBj

(yj ) = �rk=1{μ∝

B kj(yj )}. (31)

E. Type Reduction

To perform the defuzzification process, the heights and ap-proximation methods are used.

1) Heights Method: This method replaces each intervaltype-2 output by a T2 FS whose y-domain consists of a sin-gle point (y), the secondary membership function of which isa type-1 fuzzy set. The lth output set is replaced by a singletonthat is situated at yl , where yl can be chosen to be the pointhaving the highest primary membership in the principal mem-bership function of the output set Gk [38]–[40]. In this case,we have an interval type-2 output, which was created by the

alpha plane(Gk

j

)∝

, the output Gaussian membership function

was generated with uncertain mean (see Section IV); therefore,the output set is replaced by two points yl and yr , which aregiven by

ml = min(

μ∝Gk

j

(yj ))

, mr = max(

μ∝Gk

j

(yj ))

(32)

ml = min(μ∝

Gkj(yj )

), mr = min

(μ∝

Gkj(yj )

)(33)

yl = (ml + ml) /2 (34)

yr = (mr + mr ) /2. (35)

Type reduction is performed by applying the type reductionalgorithm of Mendel and Wu [41], [42], and this reduction isgiven by

yl∝ (x′) =

∑Lk=1 Ω

k∝ (x′) yi

l +∑M

j=L+1 Ωj∝ (x′) yj

l∑Lk=1 Ω

k∝ (x′) +

∑Mj=L+1 Ωj

∝ (x′)(36)

yr∝ (x′) =

∑Rk=1 Ωk

∝ (x′) ykr +

∑Mk=R+1 Ω

k∝ (x′) yk

r∑Ri=1 Ωi

∝ (x′) +∑M

i=R+1 Ωi∝ (x′)

. (37)


2) Approximation Method: This method is defined by thefollowing theorem. If each Zl ∈

[Z l , Zl

]is an interval

type-1 set, having center cl and spreads sl , and if each

Wl

[μ

l(y) , μl (y)

]is also an interval type-1 set with center hl

and spreads Δ1 , then Y is approximately an interval type-1 set,with center C and spreads S [30], [31], and can be representedby the following equations:

[yleft, yright ] = Y (Z1 , . . . , ZM,W1 , . . . ,WM )

=∫

Z1

. . .

∫ZM

∫W 1

. . .

∫WM

1/∑M

l = 1 Wl Zl∑Ml = 1 Wl

(38)

C =∑M

l=1 hlcl∑Ml=1 hl

(39)

and

S =∑M

l=1 [hlsl + |cl − C|Δl ]∑Ml=1 hl

(40)

provided that ∑Ml=1 Δl∑Ml=1 hl

� 1 (41)

where

hl =μα

Bj(yj ) + μα

Bj(yj )

2(42)

Δ1 =μα

Bj(yj ) − μα

Bj(yj )

2(43)

Cl =Zl + Zl

2(44)

and

Sl =Zl − Zl

2(45)

ylj (x′) = C − S (46)

yrj (x′) = C + S. (47)

3) Alpha Plane Integration: The results of the alpha planesare integrated by [38]

ylj (x′) =

∑Ni=1 ∝i

∝i ylj (x′)∑N

i=1 ∝i

(48)

yrj (x′) =

∑Ni=1 ∝i

∝i yrj (x′)∑N

i=1 ∝i

. (49)

F. Defuzzification

After realizing the type reduction and integrating the resultsof all the alpha planes, defuzzification is performed by usingthe average of yl and yr to obtain the defuzzified output of ageneralized singleton type-2 FLS [30], [31], [38]

yj (x′) =yl

j (x′) + yrj (x′)

2. (50)

Fig. 7. Images used for the simulation results.

Fig. 8. Synthetic image.

V. EDGE-DETECTION METRICS

There are different types of methods to evaluate the detectededge of an image, which usually apply different parameters toassess the abrupt change of color in the pixels. One of the mostfrequently used techniques is the figure of merit (FOM) of Pratt.This measure represents the deviation of an actual (calculated)edge point from the ideal edge and it is defined as

FOM =1

max (II , IA )

IA∑i=1

11+ ∝ d2

i

(51)

where IA is the actual number of detected edge points, II isthe number of edge points on the ideal edge, d(i) is the distancebetween the edge of the current pixel and its correct position inthe reference image, and α is a scaling constant (usually 1/9).To implement this metric, a test image is needed, such as the onein Fig. 7 and the reference image that represents II ; in this case,the reference image (with the ideal edges) of Fig. 8 is Fig. 9.Then, we apply any edge detector to obtain the value of IA thatrepresents the number of detected edge points. Now, if the resultof (51) is 1 or very close to 1, this means that the detected edgeIA is the same or very similar to the ideal edge II . Otherwise,the more the value is closer to 0, this means that there is a highdifference between the edge detected and ideal edge [43], [44].


Fig. 9. Reference image of Fig. 8.

TABLE IISIMULATION RESULTS WITH VARIATION IN FOU

VI. SIMULATION RESULTS

The images for testing were obtained from the database ofUSC-SIPI [24], [45], and we also used synthetic images; someof the images that are used for the tests can be seen in Figs. 7and 8. To test the proposed edge-detection method, the necessarycomputer programs were developed in MATLAB to detect theedges in real images. For all the images, the MG was obtainedwith (3) and (4). For the generalized type-2 system (GT2FIS),the gradients that are obtained with (4) are used as inputs, andthe fuzzy system was build using the membership functions thatare presented in Fig. 6.

Tests were performed using images with noise as well aswithout noise. The applied noise was of the Gaussian type withlevels of 0.001 and 0.002, and 30 runs were executed to calculatethe average values that are reported in Tables III, IV, and VIII.

In the first test, a comparative analysis was performed varyingthe σ parameter of the membership functions, which are usedin the generalized type-2 system. The type of the membershipfunctions that are used in this paper is described in SectionIV, where σ represents the deviation in the mean in (24) and(25), and the FOU of the membership functions. To measure thequality of detected edges with variation in the FOU, the FOMwas used (described in Section V); the results of these tests areshown in Table II. It can be observed that the measurements

TABLE IIISIMULATION RESULTS WITH VARIATIONS IN ALPHA PLANES

TABLE IVSIMULATION RESULTS APPLYING DEFUZZIFICATION METHOD BY

HEIGHT AND APPROXIMATION

that are obtained with the FOM were better when using an FOUwith a value of 0.2.

For the proposed method, the theory of alpha planes was used(described in Section III), for this reason, another test that wasperformed to make a comparative study varying the number ofalpha planes necessary to approximate the output. For this test,the number of alphas planes with values of 5, 10, 50, 100, 150,200, and 1000 were applied. The results of this comparison areshown in Table III, in the same manner the FOM was usedto measure the edge detector quality. In Table III, it can benoted that in images without noise, the metric value obtainedwas 0.9554 for all numbers of alpha planes, but in images withnoise, a variation in the metric values was observed, for imageswith a noise level of 0.001, the metric value was 0.9382 with50 alpha planes, and for images with a noise level of 0.002, thebest value obtained was 0.9319 with 100 alpha planes.

Another comparative study was performed to show the ad-vantages of using the heights or approximation defuzzifica-tion methods, and the results of this comparison are shown inTable IV. In this case, in the image without noise, the defuzzi-fication method by heights was better, where the highest valuewas 0.9554, but in the simulations with Gaussian noise, the per-formance of the approximation method was better, obtainingvalues of 0.9528 and 0.9533, for images with noise level of0.001 and 0.002, respectively. In Fig. 10, we can note that themethod by approximation was better (than the heights method)when noise was added to the test image, because it remainscloser to the ideal value of 1.

Finally, other simulations were performed, with the goal ofmaking a comparative study between the MG edge detectors,MG edge detectors that are based on type-1 [46], [47], inter-val type-2 [48], [49], [50], [51], and generalized type-2. Wehave to mention that all the fuzzy systems were implemented in


Fig. 10. Simulation results applying the defuzzification method by height andapproximation.

TABLE VSIMULATION RESULTS USING MORPHOLOGICAL GRADIENT, TYPE-1, INTERVAL

TYPE-2, AND GENERALIZED FUZZY SYSTEMS

MATLAB for making the same tests, using the same number ofinputs and outputs, membership functions and fuzzy rules.

The first simulation was performed with the database of testimages and the four edge detectors were applied; the obtainedresults are shown in Table V. In this table, it can be noted thatthe edge detector that is based on generalized type-2 fuzzy logicachieves better detection of the edges than the other methods.

Two other experiments were performed applying Gaussiannoise with levels of 0.001 and 0.002 and executing 30 runs foreach case, and we are only showing representative results of

TABLE VISIMULATION RESULTS APPLYING GAUSSIAN NOISE WITH A LEVEL OF 0.001

TABLE VIISIMULATION RESULTS APPLYING GAUSSIAN NOISE WITH A LEVEL OF 0.002


TABLE VIIIFIGURE OF MERIT OF PRATT (FOM)

Fig. 11. Figure of merit of Pratt (FOM) of Table VII.

these tests in Tables VI and VII. In both tests, it can be notedthat the generalized type-2 fuzzy logic achieves better detectionof the edges and has a better management of the uncertainty inthe images.

To measure the quality of the detected edges with the proposedmethod and comparing with the results that are obtained byT1FIS, IT2FIS and MG edge detectors, the FOM was used.

The results of these tests are shown in Table VIII. It canbe observed that the measurements that are obtained with theFOM were better when using the edge detection that is basedon generalized type-2 fuzzy inference systems. In this case, forthe image without noise a metric of 0.9543 was obtained, in theimage with noise of 0.001 and 0.002, the FOM was 0.9528 and0.9533, respectively. The second best method was the IT2FIS,which obtained the values of 0.9482, 0.9484, and 0.9332; fol-lowed by T1FIS with FOMs 0.8660, 0.8769, and 0.9051; andfinally the lowest values were obtained by the traditional MG,this is because the traditional technique does not consider anyparameters to prevent noise or regions with very low or high con-trast. In Fig. 11, the obtained results can be better appreciated.In Fig. 11, we can note that when the noise level is increased andMG is applied the value of FOM decreases, this means that thedifferences between the ideal edge and detected edge is high,this is because this technique has no additional parameters tomodel uncertainty. Otherwise, when fuzzy techniques are used,the value of the FOM increases, which means that the differencebetween the ideal edge and detected edge decreases; therefore,we have better control of the uncertainty.

VII. CONCLUSION

In this paper, an edge-detection method that is based on gen-eralized type-2 fuzzy logic has been proposed. As it can be notedin Table IV, when comparing the defuzzification methods usedin the generalized type-2 fuzzy inference systems, the heightsmethod achieved better results in image without noise, and themethod of approximations in images with noise.

In the results presented in Fig. 11 and Table VIII, when theproposed method is applied, better results are obtained, and themain reason is that the uncertainty in edge detection is modeledmore closely with generalized type 2 fuzzy logic. This is incontrast with the traditional MG method, or even its type-1 andinterval type-2 versions that are not able to cope with higherdegrees of uncertainty. This study leads to the conclusion thatthe use of generalized type-2 fuzzy systems can be a good choicewhen there is a high level of uncertainty in the problem. In otherwords, general type-2 fuzzy logic allows for better modelingof uncertainty, because it gives more degrees of freedom incomparison to interval type-2 and type-1 fuzzy logic.

The complex nature of the uncertainty that is encounteredin the real world indicates that generalized type-2 is needed inreal-world devices and applications, in particular in the imageprocessing area that is the case study in this paper, becausethe devices that capture digital images are always exposed toexternal interference adding high noise levels or uncertainty tothe images.

In the future, we plan to implement other defuzzificationmethods, since in this paper only the heights and approxima-tion methods were used. Additionally, we envision using opti-mization techniques that would help find the optimal parametervalues of the membership functions and the optimal number ofalpha planes for the automatic implementation of the proposedmethod. Finally, we look forward to improving the generalizedtype-2 fuzzy logic algorithms to consider other areas of appli-cation.

ACKNOWLEDGMENT

The authors would like to thank the MyDCI program at theUniversity of Baja California and the Division of Graduate Stud-ies and Research, Tijuana Institute of Technology.

REFERENCES

[1] V. Torre and T. A. Poggio, “On edge detection,” IEEE Trans. Pattern Anal.Mach. Intell., vol. PAMI-8, no. 6, pp. 147–163, Mar. 1986.

[2] M. Setayesh, M. Zhang, and M. Johnston, “Effects of static and dynamictopologies in particle swarm optimisation for edge detection in noisyimages,” in Proc. IEEE Congr. Evol. Comput., 2012, pp. 1–8.

[3] J. Canny, “A computational approach to edge detection,” IEEE Trans.Pattern Anal. Mach. Intell., vol. PAMI-8, no. 6, pp. 679–698, Nov. 1986.

[4] T. Shimada, F. Sakaida, H. Kawamura, and T. Okumura, “Application ofan edge detection method to satellite images for distinguishing sea sur-face temperature fronts near the Japanese coast,” Remote Sens. Environ.,vol. 98, no. 1, pp. 21–34, 2005.

[5] A. A. Goshtasby, 2-D and 3-D Image Registration: For Medical, RemoteSensing, and Industrial Applications. Hoboken, NJ, USA: Wiley, 2005,pp. 34–39.

[6] A. Kazerooni, A. Ahmadian, N. D. Serej, H. S. Rad, H. Saberi, H. Yousefi,and P. Farnia, “Segmentation of brain tumors in MRI images using multi-scale gradient vector flow,” in Proc. IEEE Annu. Int. Conf. Eng. Med.Biol. Soc., 2011, pp. 7973–7976.


[7] B. Siliciano, L. Sciavicco, L. Villani, and G. Oriolo, Robotics: Modelling,Planning and Control. New York, NY, USA: Springer, 2010, pp. 415–418.

[8] Z. Hocenski, S. Vasilic, and V. Hocenski, “Improved canny edge detectorin ceramic tiles defect detection,” in Proc. IEEE 32nd Annu. Conf. Ind.Electron., 2006, pp. 3328–3331.

[9] C. Tao, W. Thompson, and J. Taur, “A fuzzy if-then approach to edgedetection,” in Proc. IEEE 2nd Int. Conf. Fuzzy Syst., 1993, vol. 2, pp. 1356–1360.

[10] Z. Talai and A. Talai, “A fast edge detection using fuzzy rules,” in Proc.Int. Conf. Commun., Comput. Control Appl., Mar. 2011, pp. 1–5.

[11] L. Hu, H. D. Cheng, and M. Zhang, “A high performance edge detectorbased on fuzzy inference rules,” Inf. Sci., vol. 177, no. 21, pp. 4768–4784,Nov. 2007.

[12] T. Pham and L. Van Vliet, “Blocking artifacts removal by a hybrid filtermethod,” in Proc. 11th Annu. Conf. Adv. School Comput. Imag., 2005,pp. 372–377.

[13] O. P. Verma and R. Sharma, “An optimal edge detection using universallaw of gravity and ant colony algorithm,” in Proc. World Congr. Inf.Commun. Technol., Dec. 2011, pp. 507–511.

[14] P. Agrawal, S. Kaur, H. Kaur, and A. Dhiman, “Analysis and synthesis ofan ant colony optimization technique for image edge detection,” in Proc.Int. Conf. Comput. Sci., Sep. 2012, pp. 127–131.

[15] H. Bustince, E. Barrenechea, M. Pagola, and J. Fernandez, “Interval-valued fuzzy sets constructed from matrices: Application to edge detec-tion,” Fuzzy Sets Syst., vol. 160, no. 13, pp. 1819–1840, Jul. 2009.

[16] O. Mendoza, P. Melin, and G. Licea, “A new method for edge detectionin image processing using interval type-2 fuzzy logic,” in Proc. IEEE Int.Conf. Granular Comput., Nov. 2007, pp. 151–151.

[17] O. Mendoza, P. Melin, and G. Licea, “Interval type-2 fuzzy logic foredges detection in digital images,” Int. J. Intell. Syst., vol. 24, no. 11,pp. 1115–1133, 2009.

[18] P. Melin, O. Mendoza, and O. Castillo, “An improved method for edgedetection based on interval type-2 fuzzy logic,” Expert Syst. Appl., vol. 37,no. 12, pp. 8527–8535, Dec. 2010.

[19] R. Biswas and J. Sil, “An improved canny edge detection algorithm basedon type-2 fuzzy sets,” Procedia Technol., vol. 4, pp. 820–824, Jan. 2012.

[20] I. Sobel, “Camera models and perception,” Ph.D. dissertation, Dept. Com-put. Sci., Artif. Intell. Lab, Stanford University, Stanford, CA, USA, 1970.

[21] J. M. S. Prewitt, “Object enhancement and extraction,” in Picture Analysisand Psychopictorics, B. S. Lipkin and A. Rosenfeld, Eds. New York,NY, USA: Academic, 1970, pp. 75–149.

[22] R. Kirsch, “Computer determination of the constituent structure of bio-logical images,” Comput. Biomed. Res., vol. 4, pp. 315–328, 1971.

[23] S. Chen, Y. Chang, and J. Pan, “Fuzzy rules interpolation for sparsefuzzy rule-based systems based on interval type-2 Gaussian fuzzy setsand genetic algorithms,” IEEE Trans. Fuzzy Syst., vol. 21, no. 3, pp. 412–425, Jun. 2013.

[24] C. Hsu and C. Juang, “Evolutionary robot wall-following control us-ing type-2 fuzzy controller with species-DE-activated continuous ACO,”IEEE Trans. Fuzzy Syst., vol. 21, no. 1, pp. 100–112, Feb. 2013.

[25] H. Zhou and H. Ying, “A method for deriving the analytical structure of abroad class of typical interval type-2 Mamdani fuzzy controllers,” IEEETrans. Fuzzy Syst., vol. 21, no. 3, pp. 447–458, Jun. 2013.

[26] X.-J. Li and G.-H. Yang, “Switching-type H∞ filter design for T–S fuzzysystems with unknown or partially unknown membership functions,”IEEE Trans. Fuzzy Syst., vol. 21, no. 2, pp. 385–392, Apr. 2013.

[27] C. Wagner and H. Hagras, “Toward general type-2 fuzzy logic systemsbased on zSlices,” IEEE Trans. Fuzzy Syst., vol. 18, no. 4, pp. 637–660,Aug. 2010.

[28] C. Wagner and H. Hagras, “Employing zSlices based general type-2 fuzzysets to model multi level agreement,” in Proc. IEEE Symp. Adv. Type-2Fuzzy Logic Syst., Apr. 2011, pp. 50–57.

[29] J. M. Mendel, F. Liu, and D. Zhai, “α -Plane representation for type-2fuzzy sets: Theory and applications,” IEEE Trans. Fuzzy Syst., vol. 17,no. 5, pp. 1189–1207, Oct. 2009.

[30] J. M. Mendel, “Comments on alpha-plane representation for type-2 fuzzysets: Theory and applications,” IEEE Trans. Fuzzy Syst., vol. 18, no. 1,pp. 229–230, Feb. 2010.

[31] F. Liu, “An efficient centroid type-reduction strategy for general type-2fuzzy logic system,” Inf. Sci., vol. 178, no. 9, pp. 2224–2236, May 2008.

[32] A. C. Bovik, The Essential Guide to Image Processing. New York, NY,USA: Academic, 2009, pp. 498–500.

[33] Y. Becerikli and T. M. Karan, “A new fuzzy approach for edge detection 2detection of image edges,” in Computational Intelligence and BioinspiredSystems. Berlin, Germany: Springer Verlag, 2005, pp. 943–951.

[34] J. M. Mendel and R. I. B. John, “Type-2 fuzzy sets made simple,” IEEETrans. Fuzzy Syst., vol. 10, no. 2, pp. 117–127, Apr. 2002.

[35] D. Zhai and J. M. Mendel, “Uncertainty measures for general Type-2fuzzy sets,” Inf. Sci., vol. 181, no. 3, pp. 503–518, Feb. 2011.

[36] D. Zhai and J. Mendel, “Centroid of a general type-2 fuzzy set computedby means of the centroid-flow algorithm,” in Proc. IEEE Int. Conf. FuzzySyst., 2010, pp. 1–8.

[37] A. Bel, C. Wagner, and H. Hagras, “Multiobjective optimization andcomparison of nonsingleton type-1 and singleton interval type-2 fuzzylogic systems,” IEEE Trans. Fuzzy Syst., vol. 21, no. 3, pp. 459–476, Jun.2013.

[38] J. Mendel, Uncertain, Rule-Based Fuzzy Logic Systems: Introduction andNew Directions. Englewood Cliffs, NJ, USA: Prentice-Hall, 2001.

[39] N. N. Karnik, J. Mendel, and Q. Liang, “Type-2 fuzzy logic systems,”IEEE Trans. Fuzzy Syst., vol. 7, no. 6, pp. 643–658, Dec. 1999.

[40] L. A. Lucas, T. Centeno, and M. Delgado, “General type-2 fuzzy inferencesystems: Analysis, design and computational aspects,” in Proc. IEEE Int.Conf. Fuzzy Syst., London, U.K., 2007, pp. 1107–1112.

[41] J. M. Mendel, “On KM algorithms for solving type-2 fuzzy set problems,”IEEE Trans. Fuzzy Syst., vol. 21, no. 3, pp. 426–446, Jun. 2013.

[42] D. Wu, “Approaches for reducing the computational cost of interval type-2 fuzzy logic systems: Overview and comparisons,” IEEE Trans. FuzzySyst., vol. 21, no. 1, pp. 80–99, Feb. 2013.

[43] F. Perez-Ornelas, O. Mendoza, P. Melin, and J. R. Castro, “Interval type-2fuzzy logic for image edge detection quality evaluation,” in Proc. Annu.Meeting North Amer. Fuzzy Inf. Process. Soc., 2012, no. 1, pp. 1–6.

[44] I. Abdou and W. Pratt, “Quantitative design and evaluation of enhance-ment/thresholding edge detectors,” Proc. IEEE, vol. 67, no. 5, pp. 753–763, May 1979.

[45] (1977). The USC-SIPI image database. [Online]. Available: http://sipi.usc.edu/database/.

[46] J. Mendel, “A quantitative comparison of interval type-2 and type1fuzzylogic systems: First results,” in Proc. IEEE Int. Conf. Fuzzy Syst., 2010,pp. 1–8.

[47] O. Castillo and P. Melin, Type-2 Fuzzy Logic Theory and Applications.Berlin, Germany: Springer-Verlag, 2008.

[48] D. Zhai and J. Mendel, “Computing the centroid of a general type-2 fuzzyset by means of the centroid-flow algorithm,” IEEE Trans. Fuzzy Syst.,vol. 19, no. 3, pp. 401–422, Jun. 2011.

[49] J. R. Castro, O. Castillo, and P. Melin, “An interval type-2 fuzzy logictoolbox for control applications,” in Proc. IEEE Int. Fuzzy Syst. Conf.,2007, pp. 1–6.

[50] M. Pagola, C. Lopez-Molina, J. Fernandez, E. Barrenechea, andH. Bustince, “Interval type-2 fuzzy sets constructed from several mem-bership functions: Application to the fuzzy thresholding algorithm,” IEEETrans. Fuzzy Syst., vol. 21, no. 2, pp. 230–244, Apr. 2013.

[51] C. Juang, S. Member, and K. Juang, “Reduced interval type-2 neural fuzzysystem using weighted bound-set boundary operation for computationspeedup and chip implementation,” IEEE Trans. Fuzzy Syst., vol. 21,no. 3, pp. 477–491, Jun. 2013.

Patricia Melin (M’98–SM’04) received the Doctorin Science degree (Doctor Habilitatus D.Sc.) in com-puter science from the Polish Academy of Sciences,Warsaw, Poland, with the dissertation “Hybrid In-telligent Systems for Pattern Recognition using SoftComputing.”

She has been a Professor of computer science withthe Division of Graduate Studies, Tijuana Institute ofTechnology, Tijuana, Mexico, since 1998. In addi-tion, she is serving as the Director of graduate studiesin computer science and the Head of the research

group on computational intelligence (2000–present). Her research interests in-clude type-2 fuzzy logic, modular neural networks, pattern recognition, andneuro-fuzzy and genetic-fuzzy hybrid approaches. She has published more than90 journal papers, five authored books, 15 edited books, and 200 papers inconference proceedings.

Dr. Melin is currently the President of the Hispanic American Fuzzy Sys-tems Association and is the Founding Chair of the Mexican Chapter of theIEEE Computational Intelligence Society. She is the Chair of the Task Forceon Hybrid Intelligent Systems of Neural Networks Technical Committee of theIEEE Computational Intelligence Society. She is member of the North Ameri-can Fuzzy Information Processing Society and the International Fuzzy SystemsAssociation. She belongs to the Mexican Research System (SNI) with level III.


Claudia I. Gonzalez received the Master’s degree incomputer science from the Tijuana Institute of Tech-nology, Tijuana, Mexico, in 2004. She is currentlyworking toward the Ph.D. degree in computer sci-ence with the University of Baja California (UABCUniversity), Tijuana.

She is also an Assistant Professor of computerscience with the Division of Graduate Studies andResearch, Tijuana Institute of Technology, where shelectures at the undergraduate and graduate levels. Herresearch interests include interval type-2 fuzzy logic,

generalized type-2 fuzzy logic, digital image processing, artificial vision, mod-ular neural networks, and pattern recognition.

Juan R. Castro received the Master’s degree in com-puter science from the Tijuana Institute of Technol-ogy, Tijuana, Mexico, in 2005 and the Ph.D. degree incomputer science from University of Baja California(UABC University), Tijuana, in 2009.

He is currently a Professor of computer sciencewith the School of Engineering, UABC University,where he lectures at the undergraduate and gradu-ate levels. His research interests include type-2 fuzzylogic, genetic fuzzy systems, type-2 neuro-fuzzy neu-ral networks, and new techniques for time series pre-

diction and intelligent control.

Olivia Mendoza received the Master’s degree incomputer science from the Tijuana Institute of Tech-nology, Tijuana, Mexico, in 2004 and the Ph.D. de-gree in computer science from the University of BajaCalifornia (UABC University), Tijuana, in 2009.

She is currently a Professor of computer sciencewith the School of Engineering, UABC University,where she lectures at the undergraduate and graduatelevels. Her research interests include fuzzy logic ag-gregation operators, modular neural networks, hybridneuro-fuzzy systems, and new techniques for image

processing and pattern recognition.

Oscar Castillo (M’98–SM’04) received the Doctorin Science (Doctor Habilitatus) degree in computerscience from the Polish Academy of Sciences, War-saw, Poland, with the dissertation “Soft Computingand Fractal Theory for Intelligent Manufacturing.”

He is currently a Professor of computer sciencewith the Division of Graduate Studies, Tijuana In-stitute of Technology, Tijuana, Mexico. In addition,he is serving as the Research Director of computerscience and the Head of the research group on fuzzylogic and genetic algorithms. His research interests

include type-2 fuzzy logic, fuzzy control, and neuro-fuzzy and genetic-fuzzyhybrid approaches. He has published more than 90 journal papers, five authoredbooks, 20 edited books, and 200 papers in conference proceedings.

Dr. Castillo is the Vice President of the Hispanic American Fuzzy Sys-tems Association and the Past President of the International Fuzzy SystemsAssociation (IFSA). He is also the Chair of the Mexican Chapter of the IEEEComputational Intelligence Society. He is a member of the Mexican ResearchSystem (SNI) with level III and the IEEE Technical Committee on Fuzzy Sys-tems, where he belongs to the Task Force on “Extensions to Type-1 FuzzySystems.” He is also a member of the North American Fuzzy Information Pro-cessing Society and IFSA.

ieee transactions on fuzzy systems, vol. 22, no. 6...

Documents