Image Denoising Using Adaptive Neuro-Fuzzy System

Transcription

1 IAENG Intenational Jounal of Applied Mathematics, 36:1, IJAM_36_1_11 Image Denoising Using Adaptive Neuo-Fuzz Sstem Nguen Minh Thanh and Mu-Song Chen Abstact In this pape, we popose a genealized fuzz infeence sstem (GFIS) in noise image pocessing. The GFIS is a multi-lae neuo-fuzz stuctue which combines both Mamdani model and TS fuzz model to fom a hbid fuzz sstem. The GFIS can not onl peseve the intepetabilit popet of the Mamdani model but also eep the obust local stabilit citeia of the TS model. Simulation esults indicate that the poposed model shows a high-qualit estoation of filteed images fo the noise model than those using median filtes o wiene filtes, in tems of pea signal-to-noise atio (PSNR). Index Tems Genealized Fuzz Infeence Sstem, Mamdani model, TS model, PSNR I. INTRODUCTION The image coupted b diffeent inds of noises is a fequentl encounteed poblem in image acquisition and tansmission [1]. The noise comes fom nois sensos o channel tansmission eos. Seveal inds of noises ae discussed hee. The impulse noise (o salt and peppe noise) is caused b shap, sudden distubances in the image signal; its appeaance is andoml scatteed white o blac (o both) pixels ove the image. Gaussian noise is an idealized fom of white noise, which is caused b andom fluctuations in the signal. Specle noise (o moe simpl just specle) can be modeled b andom values multiplied b pixel values, hence it is also called multiplicative noise. If the image signal is subject to a peiodic, athe than a andom distubance, we might obtain an image coupted b peiodic noise. Usuall, peiodic noise equies the use of fequenc domain filteing. This is because wheeas the othe foms of noise can be modeled as local degadations, peiodic noise is a global effect. Howeve, impulse noise, gaussian noise and specle noise can all be cleaned b using spatial filteing techniques, such as Ode Statistic Filte (OSF). Ode statistic filtes have been applied to image pocessing poblems []. Given N obsevations X 1, X,,X N of a andom vaiable X, the ode statistics ae obtained b soting the {X (i) } in ascending ode. This poduces {X (i) } satisfing X (1) X () X (N). The {X (i) } ae the ode statistics of the N obsevations. The ode statistic filte is an estimato F(X 1,X,,X N ) of the mean of X which uses a linea combination of ode statistics F( X, X,, X ) = α X + α X + + α X 1 N 1 (1) () N ( N) Some common filtes which fit the ode statistic filte famewo ae the linea aveage filte, the median filte, and timmed mean file. Among them, the median filte sots the suounding pixels values in the window to an odel set and eplaces the cente pixel within the define window with the middle value in the set. That means the coefficients α i s in ae defined as 1, i = ( N + 1) / αi = () 0, othewise In (), N is an odd numbe. The median filteing is a non-linea filteing technique that wos best with impulse noise whilst etaining shap edges in the image. Although the median filte can achieve easonabl good pefomance fo low coupted images, it will not wo efficientl when the noise ate is too high. Anothe disadvantage of the median filte is the exta computation time needed to sot the intensit value of each set. Man othe impoved algoithms, such as weighted [3] o cente-weighted median filte [4], have been poposed to impove thei pefomances. Recentl, application of fuzz techniques in image noise eduction pocessing is a pomising eseach field [5]. Fuzz techniques have alead been applied in seveal domains of image pocessing, e.g. filteing, intepolation, and mopholog, and have numeous pactical applications in industial and medical image pocessing. These fuzz filtes, including FIRE-filte [6], the weighted fuzz mean filte [7], and the iteative fuzz contol based filte [8], ae able to outpefom an-ode filte schemes (such as the median filte). Nevetheless, most fuzz techniques ae not specificall designed fo gaussian(-lie) noise o do not poduce convincing esults when applied to handle this tpe of noise. In this pape we poposed a genealized fuzz infeence mode, which is a hbidizations of the Mamdani and TS models. The GFIS can be chaacteized b the neuo-fuzz spectum, in light of linguistic tanspaenc and input-output mapping accuac. In section II, the concepts of fuzz basis function (FBF) ae intoduced. The fuzz basis functions expansion can be egaded as a geneal fom of fuzz sstem. Two famous fuzz infeence modes ae also intoduced. The similaities between them ae also discussed to povide the foundation of the GFIS. In section III, the GFIS and its epesentation in neuo-fuzz (1) (Advance online publication: 1 Febua 007)

2 achitectues ae demonstated. Section IV shows the simulation esults and ou conclusions ae given in the last section. II. FUZZY BASIS FUNCTIONS AND FUZZY INFERENCE MODELS Fuzz sstems with singleton fuzzification, poduct as the fuzz conjunction opeato, addition fo fuzz ule aggegation, and cente of aveage defuzzification, can be expessed as a linea combination of fuzz basis functions (FBFs) [9] in the following = φ ( ) = 1 x whee is the cente of gavit of the output fuzz set and φ (x) ae called fuzz basis functions and ae given b n µ ( ) 1 j x j= j φ ( x) = n (4) µ ( x ) µ j (x j ) denotes the membeship function of input x j belonging to the th ule, µ j : R [0,1]. Note that (4) is not well-defined if n µ ( ) 0 m 1 j 1 mj x = = j = fo some x, which could happen if the input space is not wholl coveed b fuzz ule patches. Howeve, thee ae seveal eas fixes fo this poblem. Fo example, we can eithe foce the output to some constant when the denominato is zeo, o add a fuzz ule so that the denominato is alwas geate than zeo fo all x. On the basis of the stone-weiestass theoem [10], the univesal appoximation theoem was demonstated that an FBF netwo can pefom appoximation of continuous function to abita accuac. Consequentl, the expession of fuzz models b fuzz basis functions maes the models moe easil in solving model s paametes. The univesal function appoximation popet gives a stong mathematical gound when appling fuzz sstem in citical applications, anging fom contol, to time seies pediction, to pattens ecognition as well as image pocessing. In this pape, we stud the fuzz sstem based on a multiple-input/multiple-output (MIMO) of the FBF netwo. Speciall, if thee ae n-input, fuzz ules and m-output, the th ule s activation can be witten as In fact, is the degee (o the fiing stength) the input x matches ule computed b the T-nom o poduct opeato. The membeship function can have diffeent shapes. Fo an extensive oveview of othe membeship functions, the eade is efeed to [11]. In this stud, we choose Gaussian function as the membeship function The output value of the ith output is = 1 j= 1 j n j j = µ ( x ) j = 1 1 xj cj µ j ( x j ) = exp σ j j (3) (5) (6) i = 1 i = 1 iφ = = = 1 fo i = 1,,,m and i is the cente of gavit of the th aggegated output fuzz membeship function associated with the output i. In geneal fuzz sstems epesented b (7) can be boadl categoized into two families, depending on the THEN-pat of fuzz ules and the wa to combine fuzz ules. The fist one includes linguistic models based on collections of IF THEN ules, whose antecedents and consequents utilize fuzz sets. The Mamdani model [1] falls in this goup. The Mamdani model uses fuzz easoning and the sstem behavio can be descibed in natual was. A Mamdani model is pesented as a collection of fuzz ules in the following fom Rule : If x1 is A1 and x is A (8) then is B whee x = [x 1,x,,x n ] T R n and scala ae the input and output linguistic vectos, espectivel. A 1, A,, A n and B ae the coesponding fuzz sets fo x and in ule. It can be found fom (8) that ule maps a fuzz egion in the input space to fuzz egions in the output space b infeence mechanisms, such as max-min o max-poduct infeences. In Mamdani model, b using max-poduct infeence mechanism and cente of aveage defuzzification, the output i can be expessed as We can eplace the integal in (9) with small discete sums indexed onl b the numbe of fuzz sets that quantize the fuzz vaiables. This eliminates both the need to appoximate the centoid and its computational buden. Theefoe, (7) (9) (10) whee is the same as in (5), v is the volume of the output fuzz set B and b is the coesponding centoid of ' B which is the scaled output set of B. In the applications of Mamdani model, such classification o appoximation of static functions, the linguistic ules usuall can be undestandable to the use who does not have to be an expet in the consideed poblem domain. Without aiming at its accuac, fuzz sstems constucted b Mamdani models ae compehensible and intepetable. This is a ve impotant popet, since it allows the tansfomation of numeical data o vague nowledge into fuzz ules. The second catego, based on Taagi-Sugeno (TS) model sstems [13], uses a ule stuctue that has fuzz antecedent and functional consequent pats. Fo TS model, the fuzz nowledge is epesented as Rule : If x1 is A1 and x is A (11) then is f ( x) i i = B ( ) d B ( ) d = = vb = 1 v 1 ( x)

3 Depending on the fom of f (x), a singleton o a linea combination of input vecto, equation (11) can descibe eithe a zeo-ode o fist-ode TS model. Fo a fist-ode TS model, f (x) can be a linea function of x, i,e., f ( x) = p ix = p + p x + + p x n n (1) In (1) X is the augmented vecto of x, X = [1,x 1,x,,x n ] T and p = [p 0,p 1,,p n ] T is the coefficient vecto. The dot in (1) denotes the inne poduct of two vectos. In the fist-ode TS model, the ule s output detemines a local input-output elation b means of the eal-valued coefficient vecto p and theefoe the coesponding hpe-planes chaacteizing the local stuctue of the function to be appoximated. The featue of localit of the TS model allows us to epesent even quite complicate input-output elationships though a collection of local models. Fo an input x, the sstem output i is computed b centoid defuzzification f( ) 1 x = i (13) = = 1 Because the outputs ae calculated as a weighted aveage of the individual outputs, a sot of smooth tansition of the individual models should be guaanteed. Theefoe, TS models ae capable of appoximating an continuous eal-valued function on a compact set to an degee of accuac [14]. The obtained ule set fom the fist-ode TS model can be used to povide sstem output values fo an given input values though an intepolation of all the elevant individual ules. Such models ae capable of epesenting both qualitative and quantitative infomation and allow elativel easie application of poweful leaning techniques fo thei identification fom data. This tpe of nowledge epesentation, howeve, does not allow the output vaiables to be descibed in linguistic tems and the paamete optimization is caied out iteativel using a nonlinea optimization method. Thee is a tadeoff between eadabilit and pecision. If one is inteested in a moe pecise solution, then one is usuall not so botheed about its linguistic intepetabilit. Sugeno-tpe sstems ae moe suitable in such cases. Othewise, the choice is fo Mamdani-tpe sstems. Theefoe, the combination of both the Mamdani and TS model as neuo-fuzz models appea as an attempt to mege the advantages of both sstems in tems of tanspaenc with the advantages of leaning capabilities. Fom (8) and (11), the Mamdani and TS models ae diffeent in THEN-pat onl, theefoe both models can be expessed in a moe compact fom as Rule : If x1 is A1 and x is A (14) then is F whee F can be eithe B fo the Mamdani model o f fo the TS model. In this wa, the fuzz sstem epesentation in (14) is consistent with that of the fuzz basis functions expansion in (7) and the Mamdani model is expessed as v i b = 1 (15) v = = 1 while the TS model is i = 1 = x = 1 ( ) (16) It is ve obvious that (15) and (16) shae the similait b using a linea combination of FBFs. With the conditions of v 1 = v = = v, whee the Mamdani model owns the same size of output membeship functions, and f (x) = b, whee the TS model is zeo-ode, (15) and (16) become equivalent. Consequentl, the geneal fom of (15) and (16) is v i = f ( ) = 1 x (17) Equation (17) is the hbidizations of both models and a genealized fuzz infeence model (GFIS), which is an integation of the meits of Mamdani and TS models, an be designed to build a moe eadable model while maintains its pecision. In the next section, the GFIS model is pesented b the neuo -fuzz netwo achitectue [15][16]. The neuo-fuzz netwo, which is an integation of the meits of neual and fuzz appoaches, enables one to build moe intelligent decision-maing sstems. This incopoates the geneic advantages of atificial neual netwos lie massive paallelism, obustness, and leaning in data-ich envionments into the sstem. The modeling of impecise and qualitative nowledge as well as the tansmission of uncetaint ae possible though the use of fuzz logic. Besides these geneic advantages, the neuo fuzz appoach also povides the coesponding application specific meits. Speciall, the GFIS is designed fo multiple outputs in each ule such that the model can be extended to moe eal-wold applications. Besides, fo the applications of patten ecognition, the obtained model can be consideed as an extension of the quadatic Baes classifie that utilizes mixtue of models fo estimating the class conditional densities. III. GENERALIZED FUZZY INFERENCE SYSTEM, GFIS The poposed GFIS is assumed to have n inputs, ules, and m outputs in each ule. Fig. 1 shows the GFIS with two inputs, fou ules, and thee outputs. Each ule in the GFIS is witten as Rule : If x1 is A1 and x is A (18) then is B ( v, f ( x))...and is B ( v, f ( x)) In (18), the output membeship function B j (.) is chaacteized b two pavements, v and f j (x). v, which is common to ule, epesents the aea of B j (.) and f j (x) is the centoid of B j (.). The essential featue of (18), when compaed with (8) o (11), is the output epesentation of the consequent pats. The GFIS can not onl peseve the intepetable capabilit of the Mamdani model but also maintain the accuac of the TS model. To optimize the sstem pefomance, a two-phased hbid paamete leaning algoithm is applied with a given netwo stuctue. In hbid leaning, each iteation is composed of a f = m m m

4 fowad pass and a bacwad one. In the fowad pass, afte the input vecto is pesented, we calculate the node outputs in the netwo laes and on the basis of this the linea paametes ae adjusted using pseudo invese based on ecusive least-squaed technique. Afte the linea paametes ae identified we can compute the eo fo taining data pais. In the bacwad pass the eo signals popagate fom the output end towad the input nodes; the gadient vecto is calculated and the nonlinea paametes updated b steepest descent method. The leaning step of the nonlinea paametes update is adjusted using adaptive appoach. This pocess is epeated man times until the sstem conveges o eo is below some pedefined theshold. Detail desciptions of the paamete-tuning method can be found in [19]. IV. SIMULATION RESULTS The poposed model is tested fo emoving salt-and-peppe impulse noise and Gaussian noise. The nois image is divided b seveal pxp bloc. Each image bloc is fed into the GFIS model. The infeed output fom the GFIS is compaed with the noise-fee image bloc. Fig. shows the schematic diagam fo nois image pocessing. The poposed filte is applied to gascale test images, afte adding eithe Gaussian noise o salt-and-peppe noise of diffeent levels. Such a pocedue allows us to compae and evaluate the filteed images against the oiginal images. In the case of salt and peppe, images will be coupted b salt (with value 55) and peppe (with value 0) with equal pobabilit. A wide ange of salt-and-peppe noise levels vaied fom 10 % to 90 % with incements of 5 % will be tested. Also the additive Gaussian white noise is zeo mean with vaied vaiance fom 0.1 to 0.9. Restoation pefomances ae quantitativel measued b the pea signal-to-noise atio (PSNR) and the mean absolute eo (MAE), and mean squaed eo (MSE) [18] 55 PSNR = 10log 10 1 (, ij x ) i j ij MN (19) 1 1 MAE =, MSE ( ) i, j ij xij = i, j ij xij MN MN whee NxM is the numbe of pocessed pixels and ij and x ij denote the pixel values of the estoed image and the oiginal image, espectivel. Fo compaison pupose, the 3x3 median filte and the 3x3 wiene filte ae tested to compae with ou poposed GFIS. In the fist test, a 8-bit gascale image of Lena is coupted b salt-and-peppe noises. We summaize the estoed images of median filte, wiene filte, and the GFIS in Fig. 3. Among the estoations, ou poposed GFIS gives the best pefomance in tems of noise suppession and detail pesevation. The median filte and the wiene filte can achieve easonabl good pefomance fo low coupted images, but the will not wo efficientl when the noise level is above 50%. Fom visual diffeences, salt-and-peppe noise with noise atio as high as 90% can be cleaned quite efficientl. In addition, the GFIS equies onl ules and 36 fee paametes to be tuned. The compact stuctue maes the model lean ve fast. The second simulation is conducted b adding zeo mean Gaussian noise in tests image with diffeent vaiance. The vaiance stats fom 0.1 to 0.9 with incements of 0.1. Fig. 4 shows plots of PSNR, MAE, and MSE vesus vaiance of Gaussian noise. Fom the plots, we have found that all the methods have simila pefomance when the noise level is low. This is because the median and wiene filtes focus on the noise detection. Howeve, when the noise level inceases, noise patches will be fomed and the ma be consideed as noise fee pixels. This causes difficulties in the noise detection and emoval and the esults of filteed images ae not ecognizable. On the othe hand, ou denoising method achieves a high PSNR and low MAE even when the noise level is high. This is mainl based on the accuate noise detection b the adaptive neuo-fuzz model. Expeimental esults show that the GFIS impessivel outpefoms othe techniques. V. CONCLUSION This pape poposed an adaptive neuo-fuzz appoach fo additive noise eduction. The main featue of the GFIS is the hbidizations of the Mamdani and TS models. Expeimental esults ae obtained to show the feasibilit and obustness of the poposed appoach. These esults ae also compaed to othe filtes b numeical measues and visual inspection. In the nea futue, we will extend GFIS to pocess colo images. Moeove, the unifom distibution impulsive noise model will be futhe studied. ACNOWLEDGMENT This eseach was suppoted b the National Science Council unde contact numbe NSC94-13-E REFERENCES [1] R. C. Gonzalez and R. E. Woods, Digital Image Pocessing, nd ed. Englewood Cliffs, NJ: Pentice-Hall, 001. [] A. C. Bovi, T. S. Huang, and D. C. Munson. A genealization of median filteing using linea combinations of ode statistics. IEEE Tansactions on Acoustics, Speech, and Signal Pocessing, 31(6): , Decembe [3] D. Bownigg, The weighted median filte, Commun. Assoc. Comput., pp , Ma [4] S. J. o and S. J. Lee, Cente weighted median filtes and thei applications to image enhancement, IEEE Tans. Cicuits Sst., vol. 15, no. 9, pp , Sep [5] E. ee and M. Nachtegael, Eds., Fuzz Techniques in Image Pocessing. New Yo: Spinge-Velag, 000, vol. 5, Studies in Fuzziness and Soft Computing. [6] F. Russo and G. Ramponi, A fuzz opeato fo the enhancement of blued and nois images, IEEE Tans. Image Pocessing, vol. 4, pp , Aug [7] C.-S. Lee, Y.-H. uo, and P.-T. Yu, Weighted fuzz mean filtes fo image pocessing, Fuzz Sets Sst., no. 89, pp , 1997.

5 [8] F. Fabiz and M. B. Menhaj, Fuzz Techniques in Image Pocessing. New Yo: Spinge-Velag, 000, vol. 5, Studies in Fuzziness and Soft Computing, ch. A fuzz logic contol based appoach fo image filteing, pp [9] im, H.M., and Mendel, J.M.. Fuzz basis functions: Compaisons with othe basis functions. IEEE Tans. on Fuzz Sstems, 3, 1995, pp [10] Rudin, W. Pinciples of Mathematical Analsis. McGaw-Hill, Inc., [11] Yage, R. and Filev, D., Appoximate clusteing via the mountain method, IEEE Tansactions on Sstems, Man and Cbenetics. 4(8): , [1] E. H. Mamdani and S. Assilian, An expeiment in linguistic snthesis with a fuzz logic contolle, Intenational Jounal of Man-Machine Studies, 7(1):1-13, [13] T. Taagi and M. Sugeno, Fuzz identification of sstems and its applications to modeling and contol, IEEE Tans. Sst., Man, Cben., vol. 15, pp , Jan [14] J. J. Bucle and T. Feuing, Fuzz and Neual: Inteactions and Applications, se. Studies in Fuzziness and Soft Computing. Heidelbeg, Geman: Phsica-Velag, [15] C. T. Lin and C. S. Geoge Lee, Neual Fuzz Sstems--A Neuo-Fuzz Snegism to Intelligent Sstems. Englewood Cliffs, NJ:Pentice-Hall, [16] J. S. R. Jang, C. T. Sun, and E. Mizutani, Neuo-Fuzz and Soft Computing. Englewood Cliffs, NJ: Pentice-Hall, [17] A. Papoulis, Pobabilit, Random Vaiables, and Stochastic Pocesses, McGaw Hill, Inc., [18] A. Bovi, Handboo of Image and Video Pocessing. New Yo: Academic, 000. [19] Nguen Minh Thanh, Stuctue leaning and Paamete Leaning fo neuo-fuzz model, Ms Thesis, June,

6 Nois imag Fig. 1. Schematic diagam of the genealized fuzz infeence model. Restoe d Noise-fe e Fig.. Schematic diagam fo nois image pocessing. (a) (b) (c) (d) (e) Fig. 3. Restoation esults. (a). Oiginal image. (b). Coupted Lena image with 90% salt-and-peppe noise (53.7 db). (c). median file (54.45 db). (d). wiene file (59.35 db). (e). GFIS (66.06 db).

7 70 clown, Gau noise, mean PSNR = clown, Gau noise, mean MAE = PSNR (db) MAE va va 0.5 clown, Gau noise, mean MSE = MSE noise median filte wiene filte GFIS va Fig. 4. Results in PSNR, MAE, and MSE fo the Clown image at vaious vaiance levels fo median filte, wiene filte, and GFIS.