Exponential intuitionistic fuzzy entropy measure based image edge detection.

Transcription

1 Expoetial ituitioistic fuzzy etropy measure based image edge detectio. Abdulmajeed Alsufyai 1, Hassa Badry M. El-Owy 2 1 Departmet of Computer Sciece, College of Computers ad Iformatio Techology, Taif Uiversity, Taif, KSA; 2 Aswa uiversity, Aswa, Egypt; 2 Departmet of Computer Sciece, Taif Uiversity, Taif, KSA; Abstract. This paper address a ew algorithm for image edge detectio usig expoetial etropy with Attaassov's ituitioistic fuzzy set theory. Attaassov's Ituitioistic fuzzy set takes ito accout the ucertaity i determiig the degree of membership kow as hesitatio degree. A distace measure, called Expoetial Ituitioistic Fuzzy Divergece have bee itroduced. Edge detectio is carried out usig the suggested distace measure. Experimetal results reveal that the proposed method exhibits better performace ad may efficietly be used for the detectio of edges i images. Keywords: Image edge detectio, Attaassov's ituitioistic fuzzy set, expoetial etropy, fuzzy divergece. INTRODUCTION The cocept of fuzzy set proposed by Zadeh [1, 2]. I recet times, may applicatios of this theory ca be foud, such that i sigal ad image processig, patter recogitio, operatio research, artificial itelligece, expert systems, ad robotics [3-7]. I fuzzy set theory, each elemets has a degree of membership μ ragig from zero to oe, where the degree of o-membership γ is just automatically equal to oe mius the degree of membership (γ = 1 μ). The cocept of ituitioistic fuzzy set itroduced by Ataassov [8-11], which is geeralizatio the cocept of fuzzy set. I Ataassov's ituitioistic fuzzy set the degree of omembership is ot always equal to (1 μ), but there may be some hesitatio degree π. The distace measures betwee ituitioistic fuzzy sets itroduced by Szmidt ad Kacpryzk [12, 13], which are geeralizatio of the Euclidea distace ad the Hammig distace. Grzegorzewski [14] proposed Euclidea distace ad Hammig distace usig ituitioistic fuzzy sets based o Hausdorff metric. The similarity measures betwee two ituitioistic fuzzy sets ad its applicatios i patter recogitio suggested by Degfeg ad Chutia [7]. Etropy is a ucertaity measure first itroduced by Shao [15, 16] ito iformatio theory to describe how much iformatio is cotaied i a source govered by a probability law. Pal ad Pal [17] proposed aother measure called expoetial etropy. They idicated to some advatages for cosiderig expoetial etropy over Shao's etropy. I this paper, a ew attempt to use expoetial etropy with the ituitioistic fuzzy set theory to detect edges i gray level images. Due to fuzzy divergece takes ito accout oly the membership degree, a defiitio of distace measure, called Expoetial Ituitioistic Fuzzy Divergece (EIFD), is also itroduced. EIFD takes ito accout the degrees of membership ad o-membership, ad also the hesitatio degree. To describe the applicability of EIFD i practical applicatios, it has bee used i this paper for edge detectio. Experimetal results reveal that the proposed measure exhibits better performace ad may efficietly be used for edge detectio comparig with the previous approaches. The rest of the paper is orgaized as follows. A overview about edge detectio ad its related work is give i the ext sectio. A brief itroductio of expoetial etropy is, fuzzy sets ad Attaassov's ituitioistic fuzzy sets, ad few related distace measures are itroduced i Sectio 3. Sectio 4 cotais the suggested distace measure. Sectio 5 describes the proposed methedology. The experimetal results for edge detectio are displayed ad discussed i Sectio 6. Fially, coclusio is icluded i Sectio 7. RELATED WORK A edge i a image is defied a cotour or boudary where a abrupt chage occurs i some physical aspect such as gray level value of a image [18]. Edge detectio is oe of the most importat tasks i image processig. Especially registratio, segmetatio, idetificatio ad recogitio are based o edge detectio algorithm. May edge detectio methods are available i the literature. Classical edge detectio techiques, such as Sobel, Prewitt ad Cay detectors were based o the cocept of spatial derivative filterig [19, 20], where local gradiet operators are used to detect edges of certai orietatios oly. Derivative filter suffer whe the edges are oisy ad blurred, ad are ot flexible. May fuzzy ad crisp edge detectio methods have bee proposed by researchers [20-26]. Some of the related work i fuzzy set theory is metioed. Alsheawy et al. [20] suggested a edge detectio approach based o fuzzy logic without determiig the threshold value. They segmeted the images ito regios usig 3 3 biary matrix. A fuzzy system mapped a rage of values distict from each other i the matrix to detect the edge. Lopera ad Ilhami [27] used Jese-Shao divergece of gray level histogram obtaied by slidig a double widow over a image for edge detectio. Kha ad Thakur [24] proposed fuzzy logic based edge detectio algorithm for gray scale images. Tao ad Thomso [26] used gradiet approximatios as iput variables. They used two fuzzy sets, small ad large, as liguistic variables ad 16 fuzzy rules from 16 cotour structures were obtaied. Ho ad Ohishi [22] used fuzzy edge detector for edge detectio. They used several fuzzy templates ad the edge detect a image usig a similarity measure. I all fuzzy edge detectio methods, the pixel degree of membership is equal to 1 mius the o-membership degree. However, it does ot represet a 8518

2 eutral of evidece which resulted i the real world there are may iformatio caot be represeted ad processed usig fuzzy set theory. Ituitioistic fuzzy set has bee prove to be highly useful to deal with ucertaity ad vagueess. Recetly, ituitioistic fuzzy set theory was used to improve accuracy i detectio [6, 23, 28-30]. Chaira [28] itroduced ituitioistic fuzzy etropy i objective fuctio of covetioal clusterig algorithm ad applied to medical images. Hu ad Li [23] itroduced a ovel method to detect hardwood leaves edges, which clusters, thresholds, ad the detects edges of hardwood seedligs leaves usig ituitioistic fuzzy set theory. PLIMINARIES I this sectio we preset some basic cocepts related to expoetial etropy, fuzzy set theory ad Attaassov's ituitioistic fuzzy set, also we itroduce some of the related distace measures. Expoetial Etropy. Etropy is a ucertaity measure first itroduced by Shao [15, 16] ito iformatio theory to describe how much iformatio is cotaied i a source govered by a probability law. Let P = {p 1, p 2,, p } be the probability distributio of a discrete source. Therefore, 0 p i 1, i = 1,2,, ad p i = 1, where is the total umber of states ad P is called the etropy of the distributio. The etropy of a discrete source is ofte obtaied from the probability distributio. The Shao Etropy ca be defied as: H(P) = p i l(p i ) (1) This formalism has bee show to be restricted to the domai of validity of the Boltzma Gibbs Shao (BGS) statistics. Geerally, systems that obey BGS statistics are called extesive systems. If we deem that a physical system ca be decomposed ito two statistical idepedet subsystems O ad B, the probability of the composite system is P O+B = P O P B, it has bee verified that the Shao etropy has the extesive property (additive) H(O + B) = H(O) + H(B). It is to be oted from the logarithmic etropic measure (1) that as p i 0, its correspodig self iformatio of this evet, I(p i ) l(p i ) but I(p i 1) l(1) = 0 ad I(p i 0) l(0) is udefied. Pal ad Pal [17] proposed aother measure called expoetial etropy give by E(p) = p i e (1 p i ) 1 (2) Expoetial etropy (2) [31] has some advatages over Shao's etropy, which is widely acclaimed, we fid that the measure of self iformatio of a evet with probability p i is take as l( 1 p i ) = l(p i ), a decreasig fuctio of p i. The same decreasig character alteratively may be maitaied by cosiderig it as a fuctio of 1 p i rather tha of (1/p i ). Also for example, for the uiform probability distributio P = ( 1, 1,., 1 ) expoetial etropy has a fixed upper boud lim H (1, 1,., 1 ) = e 1 which is ot the case for Shao's etropy. Basic Cocept of Attaassov's Ituitioistic fuzzy sets. Defiitio 1:[2] A fuzzy set A i a fiite set X = {x 1, x 2,, x } may be give mathematically as A = {(x, μ A (x)) x X} where the fuctio μ A (x): X [0,1] is the membership fuctio of a elemet x i the fiite set X; μ A (x) [0,1] is the degree of membership of x X. Expoetial etropy for fuzzy set A correspodig to (2) has bee itroduced by Pal ad Pal as: 1 E(A ) ( e 1) [μ A (x i )e 1 μ A (x i ) (1 μ A (x i ))e μ A (x i ) 1] (3) 1 = ( e 1) [1 μ A (x i )e 1 μ A (x i ) (1 μ A (x i ))e μ A (x i ) ] Further, Ataassov [8-11] suggested a geeralizatio of fuzzy sets, called Ituitioistic Fuzzy Sets (IFS). Defiitio 2: [11] A ituitioistic fuzzy set A i a fiite set X = {x 1, x 2,, x } may be mathematically represeted as A = {(x, μ A (x), γ A (x)) x X} where the fuctios μ A (x), γ A (x): X [0,1] are, respectively, the membership degree ad the o-membership degree of a elemet x i a fiite set X with the coditio 0 μ A (x) + γ A (x) 1 Obviously, each fuzzy set A is a particular case of ituitioistic fuzzy set A, ad may be represeted as A = {(x, μ A (x), 1 μ A (x)) x X} I [12, 13] Szmidt ad Kacpryzk stressed the ecessity of takig ito cosideratio a parameter π A (x), kow as the ituitioistic idex of x i A or degree of hesitatio, which appears due to the lack of kowledge durig calculate the distaces betwee two fuzzy sets. Defiitio 3: [9, 32] For each ituitioistic fuzzy set A i a fiite set X, which may be represeted as if A = {(x, μ A (x), γ A (x), π A (x)) x X}, π A (x) = 1 μ A (x) γ A (x), (4) the π A (x) is called the Attaassov's ituitioistic idex (or hesitatio degree) of a elemet x i a fiite set X I real world, the sum of membership ad o-membership values may ot be equal to 1. This is due to the occurrece of ucertaity i defiig the membership fuctio. This ucertaity is amed as hesitatio degree. Therefore the summatio of membership, o-membership, ad hesitatio degree is equal to 1. It is obvious that 0 π A (x) 1, for each x X. I may theoretical ad practical problems, i order to determie the differece betwee two objects, it is ecessary to kow the distace betwee two fuzzy sets. Szmidt ad Kacprzyk [12, 13] preseted some distace measures betwee two Ituitioistic Fuzzy Sets (IFS) A ad B that take ito accout the degree of membership μ, the degree of o- 8519

3 membership γ, ad the hesitatio degree π i X = {x 1, x 2,, x }. EXPONENTIAL INTUITIONISTIC FUZZY ENTROPY DIVERGENCE Assume two ituitioistic fuzzy sets A ad B are expressed as A = {(x, μ A (x), γ A (x), π A (x)) x X} ad B = {(x, μ B (x), γ B (x), π B (x)) x X}, respectively. Let the rage of the membership degree of the two ituitioistic fuzzy sets A ad B may be, respectively, itroduced as {μ A (x), (1 π A (x))} ad {μ B (x), (1 π B (x))} where μ A (x), μ B (x) are the degrees of membership ad π A (x), π B (x) are the degrees of hesitatio i the particular sets, with π A (x) = 1 μ A (x) γ A (x), ad π B (x) = 1 μ B (x) γ B (x). This rage due to the lack of kowledge i determiig the membership values. The proposed distace measure takes ito accout the hesitatio degrees. Fa ad Xie [33] itroduced fuzzy divergece from fuzzy expoetial etropy usig a row-vector. I this paper the divergece cocept of Fa ad Xie is exteded to a image, represeted by a matrix. Let p 1, p 2,, p k 1 be the probability distributio for a image of size M M with K gray-levels, accordig to (2) the expoetial etropy for a image A is K 1 defied as E = p i e 1 p i i=0. Accordig to (3), the expoetial fuzzy etropy of a image A of size M M is defied as [17, 31]: 1 M 1 M 1 H(A) = [μ ( e 1) A(a ij )e 1 μ A(a ij ) + (1 μ A (a ij )) e μ A(a ij ) 1] (5) where = M 2 ad μ A (a ij ) is the membership values of the pixels i the image ad a ij is the (i, j)th pixel of the image A. I order to determie the differece betwee two images A ad B (i.e., at pixels a ij ad b ij ), it is ecessary to kow the amout of iformatio betwee the membership degrees of images A ad B which is give by: Due to the membership values of the (i, j)th pixels of images A ad B (i.e., μ A (a ij ) ad μ B (b ij ) of the (i, j)th pixels, respectively): e μ A(a ij ) μ B (b ij ). Correspodig to the expoetial fuzzy etropy, the expoetial fuzzy etropy divergece betwee images A ad B may be give as D E (A, B) [1 (1 μ A (a ij )) e μ A(a ij ) μ B (b ij ) μ A (a ij )e μ B(b ij ) μ A (a ij ) ] Similarly, the expoetial fuzzy etropy divergece of B agaist A is: D E (B, A) [1 (1 μ B (b ij )) e μ B(b ij ) μ A (a ij ) μ B (b ij )e μ A(a ij ) μ B (b ij ) ] So, the total divergece betwee the pixels a ij ad b ij of the images A ad B is D T = D E (A, B) + D E (B, A) = [ 2 (1 μ A(a ij ) μ B (b ij )) e μ A(a ij) μ B (b ij) (1 μ B (b ij ) μ A (a ij )) e ] μ B(b ij ) μ A (a ij ) Accordig to 1 π A (a ij ) ad 1 π B (b ij ) of the (i, j)th pixels of A ad B, this distace measure take ito accout the hesitatio degrees: e (1 π A(a ij )) e (1 π B(b ij )) or e (π B(b ij ) π A (a ij )) I similar maer, the total Ituitioistic Expoetial Fuzzy Divergece betwee the pixels a ij ad b ij of the images A ad B is: D π [ 2 (1 π B(b ij ) π A (a ij )) e π B(b ij) π A (a ij) (1 π A (a ij ) + π B (b ij )) e ] (7) π A(a ij ) π B (b ij ) Cosequetly, the overall Ituitioistic Expoetial Fuzzy Divergece (IEFD), betwee the images A ad B by addig equatios (6) ad (7), is give by : IEFD(A, B) = 4 (1 μ A (a ij ) + μ B (b ij )) e μ A(a ij ) μ B (b ij ) i j (1 μ B (b ij ) + μ A (a ij )) e μ B(b ij ) μ A (a ij ) (1 π B (b ij ) + π A (a ij )) e π B(b ij ) π A (a ij ) (1 π A (a ij ) + π B (b ij )) e π A(a ij ) π B (b ij) (8) PROPOSED METHODOOLOGY The process of spatial filterig cosists simply of movig a filter mask w of order m from poit to poit i a image. At each poit (i, j), the respose of the filter at that poit is calculated a predefied relatioship. We will use the usual masks for detectio the edges. For this purpose, smallest meaigful size of 8 fuzzy mask templates each of size 3 3, as show i Figure 1. Where the a ad b are template parameters, a, b [0,1], which are chaged with the differet traied parameters. a a a a 0 b 0 b b a a a 0 b a 0 b a 0 b b b b a 0 b a a 0 0 b b a a a 0 a b a b 0 a a a b b b 0 a b a b 0 b b b a b a b 0 Figure 1: Set of eight 3 3 template Move each fuzzy template o the whole biary image ad fid The IEFD measure at each cetral pixel positio (i, j) of the image uder the fuzzy template. IEFD(i, j), is calculated betwee the image widow (same size as the template) ad the template, as give below: IEFD(i, j) = max N (6) [mi (IEFD(A, B))] (9) r where N= umber of templates, r= umber of elemets i the square template. It is oted that, IEFD(i, j) is calculated for all pixel positios of the image. IEFD(A, B), is calculated by determiig the IEFD betwee each of the elemets a ij ad b ij of image widow A ad of template B usig Equatio (8). The IEFD matrix, which has the same size as that of image, is created with values of IEFD(i, j) at each poit of the matrix. This IEFD matrix is thresholded ad thied to get a edge- 8520

4 detected image. The selectio of threshold value is maually curbed for gettig the fial edge-detected result. I our experimet, For the two images A ad B, the membership degree, o-membership degree ad the ituitioistic fuzzy idex (hesitatio degree) of images A ad B are, respectively, give by: The membership degrees μ A (a ij ) are the ormalized values of the (i, j)th pixel of image A, where the image A represets the chose widow i the test image. The membership degrees of the template pixels, μ B (b ij ) are the values of the template, where the image B is the fuzzy template. Due to the ituitioistic characteristics, the membership degrees of images A ad B may take the values i the rage {μ A (x), (1 π A (x))} ad {μ B (x), (1 π B (x))}, respectively. From (4) we kow that γ A (a ij ) = 1 μ A (a ij ) π A (a ij ). So, the o-membership degree= 1 membership degree hesitatio degree. For calculatig the hesitatio degree or ituitioistic fuzzy idex, we have assumed Hesitatio degree = h c (1 membership) (10) where 0 < h c 1 is a hesitatio costat. The value of h c should be such that the (4) holds. The followig Algorithm summarize the proposed techique for carryig out the edge detector. Algorithm : Edge Detectio 1. Iput: A digital grayscale image A of size M N. 2. Create 8 fuzzy templates with values a ad b. 3. Apply the fuzzy templates over the digital image by placig the ceter of each template at each poit (i, j) over the digital grayscale image, 4. Apply equatio (8) to calculate the IEFD betwee each elemets of each template ad the image widow (same size as that of template) ad choose the miimum IEFD value. 5. For all the 8 miimum IEFD values, that determied i step 4. Choose the maximum value betwee them by usig equatio (9). 6. Put the maximum value at the poit where the template was cetered over the image. 7. For all the pixel positios the max mi value has bee determied ad positioed. 8. A ew ituitioistic expoetial divergece matrix (IED- Matrix) has bee formed. 9. Create a biary image ad thi: For all i, j if IED-Matrix(i, j) th the f(i, j) = 0 else f(i, j) = 255, where th is the threshold value. 10. Output: The edge detectio image f of A. RESULTS AND DISCUSSION To demostrate the efficiecy of the proposed approach, the algorithm is tested over a umber of differet Gray scale images ad compared with classical operators. Results with differet combiatio of a ad b are show for each images, but the edge-detected results are ot at all good as show i the Figures 2-6. The best combiatio is a = 0.4, b = 0.6 ad with these values, the edge-detected results are foud better. Also, it is observed that, o decreasig the umber of templates, may edges ca be missig ad whe icreasig the umber of templates, there is o chage i the edge detectio results. Results with differet values of hesitatio costat (hesitatio degree) h c have bee show i Figures 2-6. To demostrate the efficiecy of our proposed algorithm, the proposed approach is tested over a umber of differet grayscale ormally medical images or atural images ad compared with Cay's method [19], Laplacia's method [34] ad wavelet method [35] as show i Figures 6 ad 7. I each of the images i Figures 2-7, th ad h c deote, the threshold value ad the edge template hesitatio costat, respectively. a) "Blood" image. CT scaed Blood image of size cotaiig the gray level is show i Figure 2a. The IEFD matrix has bee thresholded ad the thied. The results with differet values of h c, are show. The results with h c = 0.3 ad h c = 0.4 have bee foud better, as show i Figures 2c ad 2e. The edges are clearly extracted usig our proposed approach, with proper h c, implies that the edge detectio is completely depedet o the choice of h c. Figure 2f, where the edges are ot properly detected, is the result with a = 0.9, b = 0.8. b) "Camerama" image. Camerama image of size is show i Figure 3a. The result of edge detectio with varyig values of h c are show. Better results are obtaied i Figures 3b ad 3c. a) Origial image d) a=0.4, b=0.6, hc=0.2, th=0.11 b) a=0.2, b=0.7, e) a=0.2, b=0.7, hc=0.4, th=0.15 c) a=0.9, b=0.5, hc=0.3, th=0.10 f) a=0.9, b=0.8, hc=0.3, th=0.12 Figure 2: The results of the edge detectio for blood image usig IEFD a) Origial image d) a=0.5, b=0.3, hc=0.6, th=0.13 b) a=0.6, b=0.4, e) a=0.8, b=0.5, hc=0.2, th=0.1 c) a=0.8, b=0.4, hc=0.5, th=0.12 f) a=0.9, b=0.8, hc=0.4, th=0.11 Figure 3: The results of the edge detectio for Camerama image usig IEFD 8521

5 a) Origial image b) a=0.75, b=0.35, hc=0.5, th=0.12 c) a=0.8, b=0.4, CONCLUSION I this paper, the ucovetioality lies i usig Attaassov's ituitioistic fuzzy set theory i image edge detectio. A ew distace measure, called expoetial ituitioistic fuzzy divergece has bee itroduced. Edge detectio was carried out by usig the suggested distace measure. The result of edge detectio is completely depedet o the selectio of hesitatio costat ad thus be the hesitatio degree. The domiat edges is clearly detected by usig the proposed method, whereas removig the uwated edges. d) a=0.7, b=0.7, hc=0.4, th=0.13 e) a=0.7, b=0.65, hc=0.4, th=0.11 f) a=0.9, b=0.8, hc=0.5, th=0.10 Figure 4: The results of the edge detectio for Brai image usig IEFD c) "Brai" image of size is show i Figure 4a. Edge-detected results with varyig values of h c are show. Figures 4b ad 4c give a better result. Figure 4f, where the edges are ot properly detected, is the result with differet values of a = 0.9, b = 0.8. d) "MRI" image is of size is show i Figure 5a. The edge detectio results with differet values of h c are show i Figures 5b-5f. But the result i Figure 5b is better. The results with a = 0.4, b = 0.65 is show i Figure 5f, where the edges are ot properly detected. For compariso with other methods, two results are show. e) "Lea" image is of size is show i Figure 6a. The results of our approach have bee compared with a Cay's method, Laplacia's method ad wavelet method. The edge detectio results with differet values of hesitatio costats, h c, are show i Figures 6b ad 6c. I Figure 6b, with h c = 0.5, better result is obtaied. Figure 6d is the result usig Wavelet method. Figure 6e is the result usig Cay's method. Figure 6f is the result usig Laplacia's method f) "Rice" image is of size is show i Figure 7a. Figure 7b-7d are the result usig proposed approach with h c = 0.4. Figure 7e is the result usig Cay's method. Figure 7f is the result usig Laplacia's method. a) Origial image b) a=0.7, b=0.4, hc=0.5, th=0.1 c) a=0.4, b=0.6, d) Wavelet method e) Cay's method f) Laplacia method Figure 6: The results of the edge detectio for Lea image usig IEFD ad other methods a) Origial image d) a=0.8, b=0.8, hc=0.4, th=0.1 b) a=0.9, b=0.8, hc=0.4, th=0.1 c) a=0.2, b=0.5, hc=0.4, th=0.1 e) Cay's method f) Laplacia method Figure 7: The results of the edge detectio for Seeds image usig IEFD ad other methods. a) Origial image b) a=0.2, b=0.8, c) a=0.3, b=0.7, hc=0.5, th=0.10 REFERENCES [1]. Zadeh, L. A., (1965). "Fuzzy Sets". Iformatio ad Cotrol, 8(3), pp [2]. Zadeh, L. A. (1975). "Cocept of a liguistic variable ad its applicatio to approximate reasoig-i". Iformatio Scieces, 8(3), pp [3]. Ataassov, K. T. (1989). "More o Ituitioistic fuzzy set". Fuzzy sets ad systems. 33(1), pp d) a=0.8, b=0.4, hc=0.5, th=0.11 e) a=0.7, b=0.5, f) a=0.4, b=0.65, Figure 5: The results of the edge detectio for MRI image usig IEFD [4]. Bustice, H., ad Burillo P. (1996). "Vague sets are ituitioistic fuzzy sets". Fuzzy sets ad systems. 79(3), pp

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