AN edge may be the result of changes in light absorption,

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1 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL., NO. 6, DECEMBER Edge-Detecton Method for Image Processng Based on Generalzed Type- Fuzzy Logc Patrca Meln, Senor Member, IEEE, Clauda I. Gonzalez, Juan R. Castro, Olva Mendoza, and Oscar Castllo, Senor Member, IEEE Abstract Ths paper presents an edge-detecton method that s based on the morphologcal gradent technque and generalzed type- fuzzy logc. The theory of alpha planes s used to mplement generalzed type- fuzzy logc for edge detecton. For the defuzzfcaton process, the heghts and approxmaton methods are used. Smulaton results wth a type-1 fuzzy nference system, an nterval type- fuzzy nference system, and wth a generalzed type- fuzzy nference system for edge detecton are presented. The proposed generalzed type- fuzzy edge-detecton method was tested wth benchmark mages and synthetc mages. We used the mert of Pratt measure to llustrate the advantages of usng generalzed type- fuzzy logc. Index Terms Alpha planes representaton, edge detecton, generalzed type- fuzzy logc, mage processng. TABLE I SOME EDGE DETECTION METHODS I. INTRODUCTION AN edge may be the result of changes n lght absorpton, color, shade, and texture, and these changes can be used to determne the depth, sze, orentaton, and surface propertes of a dgtal mage [1]. In analyzng the mage dgtally, edge detecton nvolves flterng rrelevant nformaton to select the edge ponts. The detecton of subtle changes may be mxed up by nose and ths depends on the pxel threshold of change that defnes an edge. Detecton of these contnuous edges s very dffcult and tme consumng especally when an mage s corrupted by nose []. Edge detectors have been an essental part of many computer vson systems. The edge-detecton process s useful for smplfyng the analyss of mages by drastcally reducng the amount of data to be processed [3]. The man applcaton areas of edge detectors nclude: geography, mltary, medcne, robotcs, meteorology, and pattern recognton systems [4] [8]. In the area of mage processng, there exst some edgedetecton methods that make use of type-1 fuzzy systems [9], [10], [11], neural networks [1], genetc algorthms wth partcle Manuscrpt receved May 17, 013; revsed July 30, 013 and September 3, 013; accepted November 6, 013. Date of publcaton January, 014; date of current verson November 5, 014. Ths work was supported by CONACYT contract Grant P. Meln and O. Castllo are wth the Dvson of Graduate Studes, Tuana Insttute of Technology, Tuana 500, Mexco (e-mal: epmeln@hafsamx.org; ocastllo@tectuana.mx). C. I. Gonzalez s wth the School of Engneerng, Unversty of Baa Calforna, Tuana 379, Mexco, and also wth the Tuana Insttute of Technology, Tuana 500, Mexco J. R. Castro and O. Mendoza are wth the School of Engneerng, Unversty of Baa Calforna, Tuana 379, Mexco (e-mal: cgonzalez@tectuana.mx; rcastror@uabc.edu.mx; omendoza@uabc.edu.mx). Color versons of one or more of the fgures n ths paper are avalable onlne at Dgtal Obect Identfer /TFUZZ swarm optmzaton (PSO) [], ant colony optmzaton (ACO) [13], [14], nterval-valued fuzzy operators [15], nterval type- fuzzy systems combned wth the Sobel operator [16], [17], nterval type- fuzzy systems, and the morphologcal gradent (MG) [18] and an mproved Canny method that s based on nterval type- fuzzy logc [19]. Of course, we can also fnd the tradtonal methods for mage processng, lke the Canny [3], MG, Sobel [0], Roberts [1], and Krsch methods []. The man goal of the vson systems that are based on computatonal ntellgence technques s to acheve better edge detecton when mage processng s performed under hgh nose levels []. In Table I, a summary of these methods s presented. In [18], an mproved method for edge detecton that s based on nterval type- fuzzy logc s proposed. Ths paper apples the MG technque that s combned wth a type-1 fuzzy nference system (T1FIS) and wth an nterval type- fuzzy nference system (ITFIS), where the authors conclude that the ITFIS s better than T1FIS. The ITFIS acheves better control of the detected edges n the mage. Recently, there has been a sgnfcant ncrease n the research on hgher order forms of fuzzy logc, n partcular, the use of nterval type- fuzzy logc [3] [6] and more recently generalzed type- fuzzy logc. Of course, the dea of gong nto hgher orders or types of fuzzy logc s to construct better models of uncertanty. In ths sense, t s theoretcally expected that generalzed type- fuzzy logc wll allow better management of uncertanty [7]. However, generalzed type requres a hgher computatonal overhead and several efforts have been made n order to lmt the complexty of general type- fuzzy logc; for example, Wagner and Hagras [7], [8] have ntroduced the zslces-based representaton, and Mendel and Lu [9] [31] have put forward a representaton that s based on alpha-planes, whch both enable the representaton of, and computaton wth, general type- fuzzy sets IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See standards/publcatons/rghts/ndex.html for more nformaton.

2 1516 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL., NO. 6, DECEMBER 014 The man contrbuton of ths paper s the proposed new method for edge detecton that s based on generalzed type- fuzzy logc and the MG, whch allows for better modelng of the uncertanty that exsts n processng dgtal mages, as well as to obtan a comparatve study of T1FIS, ITFIS, and generalzed type- fuzzy nference system (GTFIS) fuzzy nference systems as tools for enhancng edge detecton n dgtal mages when used n conuncton wth the MG. The rest of ths paper s organzed as follows. Secton II descrbes the MG technque. In Secton III, some basc concepts of generalzed type- fuzzy logc and the theory of alpha planes are presented, whch are used for the mplementaton of the proposed method. Secton IV descrbes the methodology that s used to develop the proposed edge-detecton method. Secton V explans the technque for evaluatng the qualty of the detected edges. Secton VI presents smulaton results wth benchmark mages to llustrate the advantages of the proposed generalzed type- fuzzy edge-detecton method. Fnally, Secton VII offers some conclusons of the proposed method. II. EDGE DETECTION USING THE MORPHOLOGICAL GRADIENT The MG of a gray-scale mage can be defned as the dfference between the ntensty values of two neghborng pxels that belong to a gven structural element. The core of gradent edge detecton s, of course, the gradent operator. In contnuous form and appled to a contnuous space mage, the gradent, f c (x, y), s defned by f c (x, y) = f c (x, y) x x + f c (x, y) y (1) y where x and y are the unt vectors n the x and y drectons, respectvely. Notce that the gradent s a vector, havng both the magntude and drecton. Its magntude, f c (x 0, y 0 ), measures the maxmum rate of change n the ntensty at the locaton (x 0, y 0 ). Its drecton s that of the greatest ncrease n ntensty. To produce an edge detector, we consder the effect of fndng the local extrema of f c (x, y) or the local maxma of ( fc ) ( ) (x, y) fc (x, y) f c (x, y) = +. () x y The precse meanng of local s very mportant here, f the maxma of () are found over a -D neghborhood, the result s a set of solated ponts rather the desred edge contours [3]. In ths paper, we are gong to use D nstead of f c (x, y);we apply () for a matrx of 3 3 as shown n Fg. 1; therefore, we can obtan the coeffcents z wth (3), and the possble drecton of the edge D wth (4). The edges S can be calculated wth (5) [18], [33] z 1 = f (x 1,y 1) z = f (x, y 1) z 3 = f (x +1,y 1) z 4 = f (x 1,y) z 5 = f (x, y), z 6 = f (x +1,y) z 7 = f (x 1,y+1) z 8 = f (x, y +1) z 9 = f (x +1,y+1) (3) Fg. 1. D. Matrx of 3 3 ndcatng the coeffcents Z and the edge drecton D1 = (z 5 z ) +(z 5 z 8 ) D = (z 5 z 4 ) +(z 5 z 6 ) D3 = (z 5 z 1 ) +(z 5 z 9 ) D4 = (z 5 z 3 ) +(z 5 z 7 ) (4) S = D1+D+D3+D4. (5) III. GENERALIZED TYPE- FUZZY LOGIC A. Defnton of Type- Fuzzy Sets A generalzed type- fuzzy set (T FS), whch s denoted by Ã, s characterzed by a type- membershp functon μã (x, u), where xɛx, uɛj x [0, 1], and 0 μã (x, u) 1, and can be represented by (6) [9], [31], [34] [36] à = {((x, u),μã (x, u)) x X, u J x [0, 1]}. (6) If à s contnuous, t can be denoted by the followng equaton: { } à = μã (x)/x = = { { x X x X x X u J u x [0, 1] [ u J u x [0, 1] μã (x, u)/(x, u) ] } f x (u)/u /x where denotes the unon for x and u. Here, J x s called the prmary membershp of x n Ã. At each value of x say x = x, the -D plane, whose axes are u and μã (x,u), s called a vertcal slce of à [34]. A secondary membershp functon s a vertcal slce of μã (x, u). Itsμà (x = x,u), forx X and u J x [0, 1], and t s descrbed as μã (x = x,u) μã (x u) = f x (u)/u J x [0, 1] (8) u J x n whch 0 f x (u) 1. } (7)

3 MELIN et al.: EDGE-DETECTION METHOD FOR IMAGE PROCESSING BASED ON GENERALIZED TYPE- FUZZY LOGIC 1517 Fg.. Generalzed type- membershp functon. Fg. 5. Proposed Model for the GTFIS. Fg. 3. FOU of the generalzed type- membershp functon. alpha plane s shown à α = {(x, u),μã (x, u) α x X, u J X [0, 1]} = {(x, u) f x (u) α}. (9) x X u J x IV. EDGE DETECTOR USING A GENERALIZED TYPE- FUZZY SYSTEM In ths secton, the proposed model for edge detecton that s based on a GTFIS s descrbed. In Fg. 5, the block dagram of the GTFIS for edge detecton s presented. A. Input Image The frst step n the whole process s readng an nput mage to apply the edge-detecton method. In ths case, we are only consderng mages wth a gray scale. Fg. 4. Example of the assocated T FS for the alpha-plane. B. Obtanng the Image Gradents In ths step, the MG technque that has been descrbed n Secton I s appled to obtan the gradents n the four drectons usng (3) and (4), and then they are used as nputs for the proposed generalzed type- fuzzy nference system. In Fg., we can fnd a representaton of a generalzed type- membershp functon, and n Fg. 3, the footprnt of uncertanty (FOU) s llustrated, whch s assocated wth the thrd dmenson and allows a better modelng of real-world uncertanty. B. α-planes Representaton An α-plane for a generalzed T FS, n ths case Ã, s denoted by Ãα, and t s the unon of all prmary membershp functons of Ã, for whch secondary membershp degrees are hgher or equal to α (0 α 1) [9], [31]. The equaton of the alpha plane s represented by (9). In Fg. 4, the representaton of an C. Fuzzfcaton The fuzzfer maps crsp nputs nto generalzed type- fuzzy sets to process wthn the FLS. In ths paper, we wll focus on the type- sngleton fuzzfer as t s fast to compute and, thus, sutable for the generalzed type- FLS real-tme operaton. Sngleton fuzzfcaton maps the crsp nput nto a fuzzy set, whch has a sngle pont of nonzero membershp. Hence, the sngleton fuzzfer maps the crsp nput x p nto a type- fuzzy sngleton, whose MF s μã p (x p )=1/1 for x p = x p, and μã p (x p )=0 for all x p x p for all p =1,,..., P, where P s the number of FLS nputs [7], [37]. For ths case study, the nputs are represented by the gradents D of the orgnal mage, and each of them wll be an nput to

4 1518 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL., NO. 6, DECEMBER 014 Fg. 6. Generalzed type- fuzzy nference system. the fuzzy system. In the followng lnes, we defne each of the nput and output lngustc varables. 1) Input Lngustc Varables: Four nputs are defned, n whch each one has three Gaussan membershp functons wth uncertan mean. The lngustc varables that are used for the four nputs are: low, medum, and hgh. In order to adapt the membershp functons to the range of gray tones dependng on the mage, we obtan the maxmum, mnmum, and mddle values of D wth (10) (1), and we use these values to calculate the mean of the membershp functons, but addng dfferent szes of the FOU. For ths task, we made tests usng dfferent szes of the FOU for the D nput varables. The Gaussan membershp functons for each D nput are obtaned wth (1) (5), and the means of each functon are obtaned wth (14) (15). For example, for the hgh membershp functons, the frst mean was obtaned wth (14), the second mean was calculated wth (15), and the σ value was obtaned wth (13). The nference system has one output S (the edge), the lngustc values that are used for the output are: edge and no_edge, and we selected the range [0, 1], snce the nput mage was normalzed n ths range, where the mnmum value for the output s represented by (16) and maxmum by (17). The Gaussan membershp functons for the output are obtaned wth (1) (5), the means of each functon are obtaned wth (19) (0), and the σ value wth (18). The FOU for the output varable S was calculated n a smlar way to the nputs varables. Ths s the method that we propose to adapt the parameters of the membershp functons dependng on the contrast level of each mage. In Fg. 6, we show the lngustc varables wth generalzed type- membershp functons, where the value for the FOU s 0. on the nputs and 0.5 for the outputs. We have to note that we are usng a number between 0 and 1 to represent the sze of the FOU, whch of course s only a crsp value that represents the average sze of the footprnt to model the partcular problem low = mn(d ) (10) hgh = max(d ) (11) medum = low +(hgh low )/ (1) σ = hgh /5 (13) m 1 = hgh (14) m = m 1 +(m 1 FOU), where FOU s n (0, 1) (15) no edge =0 (16) edge =1 (17) σ = edge/4 (18) m 1 = edge (19) m = m 1 +(m 1 FOU), where FOU s n (0, 1) (0) μ (x, u) =gausmgausstype(x, u, [σ x,m 1,m ]) (1) where gausmgausstype stands for the Gaussan generalzed type- membershp functon wth uncertan mean μ (x) = [ μ (x), μ (x) ] = gausmtype(x, [σ x,m 1,m ]) () where gausmtype stands for the Gaussan nterval type- membershp functon wth uncertan mean m x = m 1 + m m x = m 1 + m σ u = δ 6 + ε (3) [ p x = gaussmf (x, [σ x,m x ]) = exp 1 ( ) ] x mx (4) μ (x, u) =gaussmf (u, [σ u,p x ]) [ =exp 1 ( ) ] x px. (5) σ u σ x

5 MELIN et al.: EDGE-DETECTION METHOD FOR IMAGE PROCESSING BASED ON GENERALIZED TYPE- FUZZY LOGIC 1519 D. Inference Once the nput and output varables are defned, wth ther respectve membershp functons, the nference process s performed n the system, and for ths the followng steps are needed. 1) Defne the Fuzzy rules: The structure of the rules n the generalzed type- FLS s the standard Mamdan-type FLS rule structure used n the type- 1 FLS and an nterval type- FLS; however, n ths paper, we assume that the antecedent and the consequent sets are represented by generalzed type- fuzzy sets. Therefore, for a type- FLS wth p nputs x 1 ɛx 1,...,x p X P and one output y Y, multple nput sngle output (MISO), f we assume that there are M rules, the kth rule n the generalzed type- FLS can be wrtten as follows [38]: R k : IF x 1 s F k 1 and... and x p s F k p, THEN y s G k. (6) To model the process wth the fuzzy system, we consder three rules that help descrbe the exstng relatonshp between the mage gradents. The fuzzy rules are the followng. a) If (D1 s HIGH) or (D s HIGH) or (D3 s HIGH) or (D4 s HIGH), then (S s EDGE). b) If (D1 smedium)or(d smedium)or(d3 s MEDIUM) or (D4 s MEDIUM), then (S s EDGE). c) If (D1 s LOW) and (D slow)and(d3 s LOW) and (D4 s LOW), then (S s NO_EDGE). ) Performng nference wth the alpha planes: To perform the nference n the fuzzy system, the alpha planes representaton was used (see Secton III). In ths case, the alpha planes are obtaned n the secondary membershp functons of the antecedents F k and consequents G k of the th nput and kth rule. The alpha planes create an nterval type- fuzzy set, [9], [31], [35], whch s defned by the followng equatons: ( ) F k / = x X { ( F k ) = [ μ F k (x ) x X ] 1/μ F k μ F k (x ), μ F k (x ) } [ μ F k (x ), μ F k (x )] / x ( ) {[ ] F k = μ F k (x ), μ F k (x ) ( G k ) = y Y ( G k ) = { /x } μ G k (y ) [μ G k (y ), μ G k (y )] x X (x ) / 1/μ G k (y ) y x (7) } [μ G k (y ), μ G k (y )]/y. (8) 3) Frng strengths: The frng strengths of the rules are calculated, where the frng sets μ α F k (x ) for each alpha plane α, oftheth nput and kth rule of a sngleton type- FLS are represented as { } Ω k (x )= n =1 μ α F k (x { ) } Ω k (x )= n =1 μ α F k (x ). (9) 4) In a multple nput sngle output FLS: The nferred output μ α B (y ) and μ α B (y ) of each rule k are represented by μ α B (y )=Ω k (x ) μ G k (y ) μ α B (y )=Ω k (x ) μ G k (y ) (30) where μ α G k s the type- fuzzy MF that represents the αth alpha plane, kth rule, and th nput of the consequents. 5) The outputs of the fred rules (M) are combned usng the on operaton to produce the overall output set, whch can be wrtten as follows: μ α B (y )= r k=1{μ B k (y )} μ α B (y )= r k=1{ μ B k (y )}. (31) E. Type Reducton To perform the defuzzfcaton process, the heghts and approxmaton methods are used. 1) Heghts Method: Ths method replaces each nterval type- output by a T FS whose y-doman conssts of a sngle pont (ȳ), the secondary membershp functon of whch s a type-1 fuzzy set. The lth output set s replaced by a sngleton that s stuated at ȳ l, where ȳ l can be chosen to be the pont havng the hghest prmary membershp n the prncpal membershp functon of the output set Gk [38] [40]. In ths case, we have an( nterval ) type- output, whch was created by the alpha plane Gk, the output Gaussan membershp functon was generated wth uncertan mean (see Secton IV); therefore, the output set s replaced by two ponts ȳ l and ȳ r, whch are gven by ( ) ( ) m l = mn μ G k (y ),m r = max μ G k (y ) (3) ( ) ( ) m l = mn μ G k (y ), m r = mn μ G k (y ) (33) ȳ l =( m l + m l ) / (34) ȳ r =( m r + m r ) /. (35) Type reducton s performed by applyng the type reducton algorthm of Mendel and Wu [41], [4], and ths reducton s gven by L y l (x k=1 )= Ωk (x )ȳl + =L+1 Ω (x )ȳ l L k=1 Ωk (x )+ (36) M =L+1 Ω (x ) R y r (x k=1 )= Ωk (x )ȳr k + k=r+1 Ωk (x )ȳr k R =1 Ω (x )+. (37) M =R+1 Ω (x )

6 150 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL., NO. 6, DECEMBER 014 ) Approxmaton Method: Ths method s defned by the followng theorem. If each Z l [ Z l, Z ] l s an nterval type-1 set, havng ] center c l and spreads s l, and f each W l [μ l (y), μ l (y) s also an nterval type-1 set wth center h l and spreads Δ 1, then Y s approxmately an nterval type-1set, wth center C and spreads S [30], [31], and can be represented by the followng equatons: [ŷ left, ŷ rght ]=Y(Z 1,...,Z M, W 1,...,W M ) = Z 1 W 1 W M Z M / l =1 W l Z l l =1 W l (38) l=1 C = h lc l l=1 h l (39) and S = l=1 [h ls l + c l C Δ l ] l=1 h l (40) Fg. 7. Images used for the smulaton results. provded that l=1 Δ l l=1 h l 1 (41) where and μ α B (y )+μ α B (y ) h l = μ α B (y ) μ α B (y ) Δ 1 = C l = Z l + Z l S l = Z l Z l (4) (43) (44) (45) ŷ l (x )=C S (46) ŷ r (x )=C + S. (47) 3) Alpha Plane Integraton: The results of the alpha planes are ntegrated by [38] N ŷ l (x =1 )= y l (x ) N =1 (48) N ŷ r (x =1 )= y r (x ) N =1. (49) F. Defuzzfcaton After realzng the type reducton and ntegratng the results of all the alpha planes, defuzzfcaton s performed by usng the average of y l and y r to obtan the defuzzfed output of a generalzed sngleton type- FLS [30], [31], [38] )=ŷl ŷ (x (x )+ŷ r (x ). (50) Fg. 8. Synthetc mage. V. EDGE-DETECTION METRICS There are dfferent types of methods to evaluate the detected edge of an mage, whch usually apply dfferent parameters to assess the abrupt change of color n the pxels. One of the most frequently used technques s the fgure of mert (FOM) of Pratt. Ths measure represents the devaton of an actual (calculated) edge pont from the deal edge and t s defned as FOM = 1 max (I I,I A ) I A = d (51) where I A s the actual number of detected edge ponts, I I s the number of edge ponts on the deal edge, d() s the dstance between the edge of the current pxel and ts correct poston n the reference mage, and α s a scalng constant (usually 1/9). To mplement ths metrc, a test mage s needed, such as the one n Fg. 7 and the reference mage that represents I I ; n ths case, the reference mage (wth the deal edges) of Fg. 8 s Fg. 9. Then, we apply any edge detector to obtan the value of I A that represents the number of detected edge ponts. Now, f the result of (51) s 1 or very close to 1, ths means that the detected edge I A s the same or very smlar to the deal edge I I. Otherwse, the more the value s closer to 0, ths means that there s a hgh dfference between the edge detected and deal edge [43], [44].

7 MELIN et al.: EDGE-DETECTION METHOD FOR IMAGE PROCESSING BASED ON GENERALIZED TYPE- FUZZY LOGIC 151 TABLE III SIMULATION RESULTS WITH VARIATIONS IN ALPHA PLANES Fg. 9. Reference mage of Fg. 8. TABLE II SIMULATIONRESULTSWITH VARIATIONIN FOU TABLE IV SIMULATION RESULTS APPLYING DEFUZZIFICATION METHOD BY HEIGHT AND APPROXIMATION VI. SIMULATION RESULTS The mages for testng were obtaned from the database of USC-SIPI [4], [45], and we also used synthetc mages; some of the mages that are used for the tests can be seen n Fgs. 7 and 8. To test the proposed edge-detecton method, the necessary computer programs were developed n MATLAB to detect the edges n real mages. For all the mages, the MG was obtaned wth (3) and (4). For the generalzed type- system (GTFIS), the gradents that are obtaned wth (4) are used as nputs, and the fuzzy system was buld usng the membershp functons that are presented n Fg. 6. Tests were performed usng mages wth nose as well as wthout nose. The appled nose was of the Gaussan type wth levels of and 0.00, and 30 runs were executed to calculate the average values that are reported n Tables III, IV, and VIII. In the frst test, a comparatve analyss was performed varyng the σ parameter of the membershp functons, whch are used n the generalzed type- system. The type of the membershp functons that are used n ths paper s descrbed n Secton IV, where σ represents the devaton n the mean n (4) and (5), and the FOU of the membershp functons. To measure the qualty of detected edges wth varaton n the FOU, the FOM was used (descrbed n Secton V); the results of these tests are shown n Table II. It can be observed that the measurements that are obtaned wth the FOM were better when usng an FOU wth a value of 0.. For the proposed method, the theory of alpha planes was used (descrbed n Secton III), for ths reason, another test that was performed to make a comparatve study varyng the number of alpha planes necessary to approxmate the output. For ths test, the number of alphas planes wth values of 5, 10, 50, 100, 150, 00, and 1000 were appled. The results of ths comparson are shown n Table III, n the same manner the FOM was used to measure the edge detector qualty. In Table III, t can be noted that n mages wthout nose, the metrc value obtaned was for all numbers of alpha planes, but n mages wth nose, a varaton n the metrc values was observed, for mages wth a nose level of 0.001, the metrc value was wth 50 alpha planes, and for mages wth a nose level of 0.00, the best value obtaned was wth 100 alpha planes. Another comparatve study was performed to show the advantages of usng the heghts or approxmaton defuzzfcaton methods, and the results of ths comparson are shown n Table IV. In ths case, n the mage wthout nose, the defuzzfcaton method by heghts was better, where the hghest value was , but n the smulatons wth Gaussan nose, the performance of the approxmaton method was better, obtanng values of and , for mages wth nose level of and 0.00, respectvely. In Fg. 10, we can note that the method by approxmaton was better (than the heghts method) when nose was added to the test mage, because t remans closer to the deal value of 1. Fnally, other smulatons were performed, wth the goal of makng a comparatve study between the MG edge detectors, MG edge detectors that are based on type-1 [46], [47], nterval type- [48], [49], [50], [51], and generalzed type-. We have to menton that all the fuzzy systems were mplemented n

8 15 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL., NO. 6, DECEMBER 014 TABLE VI SIMULATION RESULTS APPLYING GAUSSIAN NOISE WITH A LEVEL OF Fg. 10. Smulaton results applyng the defuzzfcaton method by heght and approxmaton. TABLE V SIMULATION RESULTS USING MORPHOLOGICAL GRADIENT,TYPE-1, INTERVAL TYPE-, AND GENERALIZED FUZZY SYSTEMS TABLE VII SIMULATION RESULTS APPLYING GAUSSIAN NOISE WITH A LEVEL OF 0.00 MATLAB for makng the same tests, usng the same number of nputs and outputs, membershp functons and fuzzy rules. The frst smulaton was performed wth the database of test mages and the four edge detectors were appled; the obtaned results are shown n Table V. In ths table, t can be noted that the edge detector that s based on generalzed type- fuzzy logc acheves better detecton of the edges than the other methods. Two other experments were performed applyng Gaussan nose wth levels of and 0.00 and executng 30 runs for each case, and we are only showng representatve results of

9 MELIN et al.: EDGE-DETECTION METHOD FOR IMAGE PROCESSING BASED ON GENERALIZED TYPE- FUZZY LOGIC 153 Fg. 11. TABLE VIII FIGURE OF MERIT OF PRATT (FOM) Fgure of mert of Pratt (FOM) of Table VII. these tests n Tables VI and VII. In both tests, t can be noted that the generalzed type- fuzzy logc acheves better detecton of the edges and has a better management of the uncertanty n the mages. To measure the qualty of the detected edges wth the proposed method and comparng wth the results that are obtaned by T1FIS, ITFIS and MG edge detectors, the FOM was used. The results of these tests are shown n Table VIII. It can be observed that the measurements that are obtaned wth the FOM were better when usng the edge detecton that s based on generalzed type- fuzzy nference systems. In ths case, for the mage wthout nose a metrc of was obtaned, n the mage wth nose of and 0.00, the FOM was and , respectvely. The second best method was the ITFIS, whch obtaned the values of 0.948, , and 0.933; followed by T1FIS wth FOMs , , and ; and fnally the lowest values were obtaned by the tradtonal MG, ths s because the tradtonal technque does not consder any parameters to prevent nose or regons wth very low or hgh contrast. In Fg. 11, the obtaned results can be better apprecated. In Fg. 11, we can note that when the nose level s ncreased and MG s appled the value of FOM decreases, ths means that the dfferences between the deal edge and detected edge s hgh, ths s because ths technque has no addtonal parameters to model uncertanty. Otherwse, when fuzzy technques are used, the value of the FOM ncreases, whch means that the dfference between the deal edge and detected edge decreases; therefore, we have better control of the uncertanty. VII. CONCLUSION In ths paper, an edge-detecton method that s based on generalzed type- fuzzy logc has been proposed. As t can be noted n Table IV, when comparng the defuzzfcaton methods used n the generalzed type- fuzzy nference systems, the heghts method acheved better results n mage wthout nose, and the method of approxmatons n mages wth nose. In the results presented n Fg. 11 and Table VIII, when the proposed method s appled, better results are obtaned, and the man reason s that the uncertanty n edge detecton s modeled more closely wth generalzed type fuzzy logc. Ths s n contrast wth the tradtonal MG method, or even ts type-1 and nterval type- versons that are not able to cope wth hgher degrees of uncertanty. Ths study leads to the concluson that the use of generalzed type- fuzzy systems can be a good choce when there s a hgh level of uncertanty n the problem. In other words, general type- fuzzy logc allows for better modelng of uncertanty, because t gves more degrees of freedom n comparson to nterval type- and type-1 fuzzy logc. The complex nature of the uncertanty that s encountered n the real world ndcates that generalzed type- s needed n real-world devces and applcatons, n partcular n the mage processng area that s the case study n ths paper, because the devces that capture dgtal mages are always exposed to external nterference addng hgh nose levels or uncertanty to the mages. In the future, we plan to mplement other defuzzfcaton methods, snce n ths paper only the heghts and approxmaton methods were used. Addtonally, we envson usng optmzaton technques that would help fnd the optmal parameter values of the membershp functons and the optmal number of alpha planes for the automatc mplementaton of the proposed method. Fnally, we look forward to mprovng the generalzed type- fuzzy logc algorthms to consder other areas of applcaton. ACKNOWLEDGMENT The authors would lke to thank the MyDCI program at the Unversty of Baa Calforna and the Dvson of Graduate Studes and Research, Tuana Insttute of Technology. REFERENCES [1] V. Torre and T. A. Poggo, On edge detecton, IEEE Trans. Pattern Anal. Mach. Intell., vol. PAMI-8, no. 6, pp , Mar [] M. Setayesh, M. Zhang, and M. Johnston, Effects of statc and dynamc topologes n partcle swarm optmsaton for edge detecton n nosy mages, n Proc. IEEE Congr. Evol. Comput., 01, pp [3] J. Canny, A computatonal approach to edge detecton, IEEE Trans. Pattern Anal. Mach. Intell., vol. PAMI-8, no. 6, pp , Nov [4] T. Shmada, F. Sakada, H. Kawamura, and T. Okumura, Applcaton of an edge detecton method to satellte mages for dstngushng sea surface temperature fronts near the Japanese coast, Remote Sens. Envron., vol. 98, no. 1, pp. 1 34, 005. [5] A. A. Goshtasby, -D and 3-D Image Regstraton: For Medcal, Remote Sensng, and Industral Applcatons. Hoboken, NJ, USA: Wley, 005, pp [6] A. Kazeroon, A. Ahmadan, N. D. Sere, H. S. Rad, H. Saber, H. Yousef, and P. Farna, Segmentaton of bran tumors n MRI mages usng multscale gradent vector flow, n Proc. IEEE Annu. Int. Conf. Eng. Med. Bol. Soc., 011, pp

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Pratt, Quanttatve desgn and evaluaton of enhancement/thresholdng edge detectors, Proc. IEEE, vol. 67, no. 5, pp , May [45] (1977). The USC-SIPI mage database. [Onlne]. Avalable: usc.edu/database/. [46] J. Mendel, A quanttatve comparson of nterval type- and type1fuzzy logc systems: Frst results, n Proc. IEEE Int. Conf. Fuzzy Syst., 010, pp [47] O. Castllo and P. Meln, Type- Fuzzy Logc Theory and Applcatons. Berln, Germany: Sprnger-Verlag, 008. [48] D. Zha and J. Mendel, Computng the centrod of a general type- fuzzy set by means of the centrod-flow algorthm, IEEE Trans. Fuzzy Syst., vol. 19, no. 3, pp , Jun [49] J. R. Castro, O. Castllo, and P. Meln, An nterval type- fuzzy logc toolbox for control applcatons, n Proc. IEEE Int. Fuzzy Syst. Conf., 007, pp [50] M. Pagola, C. Lopez-Molna, J. Fernandez, E. Barrenechea, and H. Bustnce, Interval type- fuzzy sets constructed from several membershp functons: Applcaton to the fuzzy thresholdng algorthm, IEEE Trans. Fuzzy Syst., vol. 1, no., pp , Apr [51] C. Juang, S. Member, and K. Juang, Reduced nterval type- neural fuzzy system usng weghted bound-set boundary operaton for computaton speedup and chp mplementaton, IEEE Trans. Fuzzy Syst., vol. 1, no. 3, pp , Jun Patrca Meln (M 98 SM 04) receved the Doctor n Scence degree (Doctor Habltatus D.Sc.) n computer scence from the Polsh Academy of Scences, Warsaw, Poland, wth the dssertaton Hybrd Intellgent Systems for Pattern Recognton usng Soft Computng. She has been a Professor of computer scence wth the Dvson of Graduate Studes, Tuana Insttute of Technology, Tuana, Mexco, snce In addton, she s servng as the Drector of graduate studes n computer scence and the Head of the research group on computatonal ntellgence (000 present). Her research nterests nclude type- fuzzy logc, modular neural networks, pattern recognton, and neuro-fuzzy and genetc-fuzzy hybrd approaches. She has publshed more than 90 ournal papers, fve authored books, 15 edted books, and 00 papers n conference proceedngs. Dr. Meln s currently the Presdent of the Hspanc Amercan Fuzzy Systems Assocaton and s the Foundng Char of the Mexcan Chapter of the IEEE Computatonal Intellgence Socety. She s the Char of the Task Force on Hybrd Intellgent Systems of Neural Networks Techncal Commttee of the IEEE Computatonal Intellgence Socety. She s member of the North Amercan Fuzzy Informaton Processng Socety and the Internatonal Fuzzy Systems Assocaton. She belongs to the Mexcan Research System (SNI) wth level III.

11 MELIN et al.: EDGE-DETECTION METHOD FOR IMAGE PROCESSING BASED ON GENERALIZED TYPE- FUZZY LOGIC 155 Clauda I. Gonzalez receved the Master s degree n computer scence from the Tuana Insttute of Technology, Tuana, Mexco, n 004. She s currently workng toward the Ph.D. degree n computer scence wth the Unversty of Baa Calforna (UABC Unversty), Tuana. She s also an Assstant Professor of computer scence wth the Dvson of Graduate Studes and Research, Tuana Insttute of Technology, where she lectures at the undergraduate and graduate levels. Her research nterests nclude nterval type- fuzzy logc, generalzed type- fuzzy logc, dgtal mage processng, artfcal vson, modular neural networks, and pattern recognton. Olva Mendoza receved the Master s degree n computer scence from the Tuana Insttute of Technology, Tuana, Mexco, n 004 and the Ph.D. degree n computer scence from the Unversty of Baa Calforna (UABC Unversty), Tuana, n 009. She s currently a Professor of computer scence wth the School of Engneerng, UABC Unversty, where she lectures at the undergraduate and graduate levels. Her research nterests nclude fuzzy logc aggregaton operators, modular neural networks, hybrd neuro-fuzzy systems, and new technques for mage processng and pattern recognton. Juan R. Castro receved the Master s degree n computer scence from the Tuana Insttute of Technology, Tuana, Mexco, n 005 and the Ph.D. degree n computer scence from Unversty of Baa Calforna (UABC Unversty), Tuana, n 009. He s currently a Professor of computer scence wth the School of Engneerng, UABC Unversty, where he lectures at the undergraduate and graduate levels. Hs research nterests nclude type- fuzzy logc, genetc fuzzy systems, type- neuro-fuzzy neural networks, and new technques for tme seres predcton and ntellgent control. Oscar Castllo (M 98 SM 04) receved the Doctor n Scence (Doctor Habltatus) degree n computer scence from the Polsh Academy of Scences, Warsaw, Poland, wth the dssertaton Soft Computng and Fractal Theory for Intellgent Manufacturng. He s currently a Professor of computer scence wth the Dvson of Graduate Studes, Tuana Insttute of Technology, Tuana, Mexco. In addton, he s servng as the Research Drector of computer scence and the Head of the research group on fuzzy logc and genetc algorthms. Hs research nterests nclude type- fuzzy logc, fuzzy control, and neuro-fuzzy and genetc-fuzzy hybrd approaches. He has publshed more than 90 ournal papers, fve authored books, 0 edted books, and 00 papers n conference proceedngs. Dr. Castllo s the Vce Presdent of the Hspanc Amercan Fuzzy Systems Assocaton and the Past Presdent of the Internatonal Fuzzy Systems Assocaton (IFSA). He s also the Char of the Mexcan Chapter of the IEEE Computatonal Intellgence Socety. He s a member of the Mexcan Research System (SNI) wth level III and the IEEE Techncal Commttee on Fuzzy Systems, where he belongs to the Task Force on Extensons to Type-1 Fuzzy Systems. He s also a member of the North Amercan Fuzzy Informaton Processng Socety and IFSA.