Fuzzy Filtering Algorithms for Image Processing: Performance Evaluation of Various Approaches
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1 Proceedngs of the Internatonal Conference on Cognton and Recognton Fuzzy Flterng Algorthms for Image Processng: Performance Evaluaton of Varous Approaches Rajoo Pandey and Umesh Ghanekar Department of Electroncs and Communcaton Engneerng, atonal Insttute of Technology, Kurukshetra (Haryana) Abstract onlnear technques have recently assumed sgnfcance as they are able to suppress non-gaussan and sgnal dependent nose to preserve mportant sgnal elements such as edges and fne detals and elmnate degradatons occurrng durng sgnal formaton or transmsson through nonlnear channels. Among nonlnear technques, the fuzzy logc based approaches are mportant as they are capable of reasonng wth vague and uncertan nformaton. In ths paper varous fuzzy flterng schemes are mplemented and ther performance s evaluated for dfferent test mages and nose models. The mean square error () s used as performance measure to examne the effcacy of these methods.. ITRODUCTIO Flterng s an essental part of any sgnal processng system. Ths nvolves estmaton of a sgnal degraded, n most cases, by addtve random nose. Several flterng technques have been proposed over the years. Among them the lnear processng technques have been the method of choce for many years because of ther smplcty. Most of these technques, however, assume a Gaussan model for the statstcal characterstcs of the underlyng process and try to optmze the parameters of a system for ths model []. In mage processng problems, however, the conventonal lnear technques are proved nadequate as they cannot cope wth the nonlneartes of the mage formaton model and do not take nto account the nonlnear nature of the human vsual system. Flters havng good edge and mage detal preservaton propertes are hghly desrable for mage flterng. Lnear flterng technques tend to blur edges and degrade other fne mage detals. ose smoothng and edge enhancement are nherently conflctng processes. Therefore, a sngle flter may not be sutable for flterng of dfferent parts of an mage. The flterng algorthm should, deally, vary from pxel to pxel dependng upon the local context. Snce n some portons of an mage the local condtons can be evaluated only vaguely, t s extremely dffcult to set the condtons under whch a certan flter should be selected. Therefore, a flterng system should possess the capablty of reasonng wth vague and uncertan nformaton; ths suggests the use of fuzzy logc for mage flterng. Most of the avalable fuzzy flterng technques can be dvded n two broad categores: fuzzy-rule -based technques and adaptve fuzzy technques. In fuzzy-rule based methods, one may use human knowledge expressed n lngustc terms. Russo employed heurstc knowledge to bu ld the rules for operatons such as smoothng, sharpenng and edge detecton [,3]. An edge preservng smoothng flter, based on weghted mean flter s porposed n [4]. A robust approach for mage enhancement based on fuzzy logc can be found n [5]. Ths approach combnes the outputs of three dfferent flters based on the local context. Another fuzzy-logc-control-based flter s proposed n [6] for mage enhancement. Some methods utlze fuzzy rule based systems to extend the classcal structure of a weghted lnear flter. The fuzzy weghts are evaluated by fuzzy rules whose nputs are local features that extract nformaton from the vcnty of the sgnal value to be processed. Hybrd fuzzy flters combnng the nonlnear flters and lnear fuzzy weghted flters are proposed n [7]. In the second category, a good account of adaptve fuzzy systems s avalable n [8]. In ths approach the system s presented as a fuzzy weghted average of the pxel values nsde the flter wndow. The weghtng coeffcents are determned adaptvely usng functons of a dstance crteron between the nput pxels. Ths approach has been used for multchannel sgnal processng. In ths paper some recent contrbutons, for nstance, robust fuzzy flters of [5], fuzzy control flters of [6] and adaptve fuzzy flterng technques of [8] along wth the conventonal medan and averagng flters have been consdered for the study. Secton 95
2 Fuzzy Flterng Algorthms for Image Processng: Performance Evaluaton of Varous Approaches brefly ntroduces varous fuzzy flterng approaches consdered for the present study. Smulaton results are presented n secton 3 whereas secton 4 summarzes the overall fndngs of the present study.. FUZZY APPROACHES. Robust Fuzzy Flters Let I( X),..., I( X ) be the gray levels of pxels X,..., X, respectvely, n a gven wndow wth pxels n t. In weghted average flterng, the gray level of the center pxel X s replaced by I( X ) = wix ( ) j j = j = w j j () where w j s the weght assocated wth a neghborng pxel X j. In ths approach of [5] three flters are used as follows. Flter A: Ths flter s obtaned by mnmzng an objectve functon that depends on a membershp functon and the varatons n the pxel ntenstes n a gven spatal wndow wth respect to the center pxel. Ths objectve functon s based on the assumpton that a gven center pxel s a prototype of ts neghborng pxel. Flter B: If the gven center pxel s a nose pxel, then the objectve functon used for flter A s not vald and another objectve functon s needed. In flter B the objectve functon s chosen so that the gray level of the center pxel s updated n such a way that the new value maxmzes the degree of membershp to whch ts neghbors represent the center pxel. Flter C: The objectve funct on for ths flter s smlar to that of flter B n the sense that ths also maxmzes the degree of membershp to whch ts neghbors represent the center pxel. However, n ths case the membershp functons representng the degree of compatblty between two pxels are combned usng averagng operator n the objectve functon, whereas flter B uses the multplcaton operator. The fltres A, B and C are used based on the total compatblty defned by TC = µ K where j=, j j K = w. j=, j j and µ represents the degree of compatblty of a neghborng pxel j X wth respect to j spatal dstance between the two pxels. The followng flterng systems are consdered n the present study. Flter-: For ths flter, followng rules are used. Rule : If total compatblty s small Then y = output of flter A Rule : Else y = I( X ). Flter : Ths flterng system nvolves flters B and C. Rule : If total compatblty s small Then y = output of flter B. Rule : Else, y = output of flter C. Flter 3: For ths flter system, rules nvolvng flters A, B and C are gven as: Rule : If total compatblty s small Then y = output of flter A. Rule : If total compatblty s small Then y = output of flter B. Rule 3: Else y 3 = output of flter C. X. The parameter w depends on the j 96
3 Proceedngs of the Internatonal Conference on Cognton and Recognton. Fuzzy-logc Control Based Flters In ths approach as gven n [6], the lumnance dfference between pxels and the central pxel, n a wndow, are used. I ( X ) = I( X ) I( X) where X represents the central pxel. dff Then followng fuzzy rules are used to form the flterng system flter 4. Rule : If (more of I ( X ) are B) Then y s B. dff Rule : If (more of I ( X ) are M) Then y s M. dff Rule 3: If (more of I ( X ) are S) Then y s S. dff Rule 4: If (more of I ( X ) are PS) Then y s PS. dff Rule 5: If (more of I ( X ) are PM) Then y s PM. dff Rule 6: If (more of I ( X ) are PB) Then y s PB. Rule 0: Else y s Z. dff The membershp functons are trangular coverng the range of lumnance dfference from 55 to 55 nto sets of B, M, S, Z, PS, PM and PB. The functon more represents a fuzzy functon gven by µ more () z = + exp( αz + β) () where α and β are constants. The actvty degree of rule s computed by number of I ( X ) support( B) dff λ = mn{ µ B( Idff ( X)): Idff( X) support( B)} µ more total number of Idff ( X) (3) The actvty degrees of other rules are computed smlarly. For the rule 0, the degree of actvty s 6 λ0 = max 0, λ (4) = The output s nferred from the correlaton-product nference mechansm as 6 y = c λ (5) = where c s the center pont of the membershp functon used n the th rule. Flter 5: Another flterng system s obtaned by replacng the Rule 0 of Flter 4 by Rule 0: Else f (more of I ( X ) are Z) Then y s ave( I ( X )) dff dff Where ave(.) represents the average n the support of set Z, and by modfyng (5) as y = kavei. ( ( X )) + ( k). c λ dff = 6 (6) 97
4 Fuzzy Flterng Algorthms for Image Processng: Performance Evaluaton of Varous Approaches where number of I ( X ) support(z) dff k = µ more ' total number of Idff( X) wth more denotng a fuzzy functon smlar to that of (). The operator mn n (3) s replaced by medan..3 Fuzzy Adaptve Flters Flter 6: In ths approach, the weghts of () are consdered as membershp functons wth respect to the specfc wndow component [8]. The adaptve algorthm evaluates a membershp functon based on a gven pxel and then uses ths value to calculate the fltered output. The membershp functon s a functon of the dstance between pxel I(X ) and the reference I(X r ). For example, t can be defned as λ d(( I X), I( Xr)) µ (( I X )) = exp β (7) where d(.) represents the dstance metrc, λ s a postve constant and β s a dstance threshold. In ths flter, the dstance metrc of (7) s taken as the aggregated dstance from all other pxels nsde the wndow d( I( X ), I( X )) d(( I X ), I( X )) = (8) r j j= Flter 7: In ths flter, a robust estmate of the pxel, evaluated n a smaller subset of the nput pxels, s utlzed as the reference pxel. Usually the medan s the preferred choce for ths estmator. Ths flter can be vewed as a double-wndow, two-stage estmator n whch the orgnal sgnal s fltered by a medan flter n a small processng wndow n order to reject possble outlers, then an adaptve fuzzy flter wth data dependent weghts s appled to provde the fnal estmates. 3. SIMULATIO RESULTS To examne the effectveness of the flterng systems of secton, two dfferent test mages: lenna and fruts are consdered. Frst of all, these mages are corrupted by three dfferent types of nose vz. 0% salt and pepper, Gaussan wth varance 0.0 and a mxture of these two. Fg. (a) and Fg. (a) show the orgnal mages, whle mages corrupted by Gaussan and salt and pepper nose are gven n Fg. (b) and Fg. (b), respectvely. The lenna mage fltered by the medan flter and flter5 s shown n Fg. (c d), respectvely, for Gaussan nose. In case of the salt and pepper nose, the fruts mage fltered by the medan flter and flter 7 s gven n Fg. (c-d). The resultng for varous flters s gven n Table (a). for the lenna mage and n Table (a) for the fruts mage. Table : (a) Performance of Varous Flters on the lenna Image Flter Type () Salt & pepper nose (0%) () Gaussan nose σ = 0.0 Mxture of () and () Flter Flter Flter Flter Flter Flter Flter Medan Averagng
5 Proceedngs of the Internatonal Conference on Cognton and Recognton Table : (b) Performance of Varous Flters on the lenna Image Flter Type () Salt & pepper nose (5%) () Gaussan nose σ = 0.05 Mxture of () and () Flter Flter Flter Flter Flter Flter Flter Medan Averagng Flter Type Table : (a) Performance of Varous Flters on the Fruts Image () Salt & pepper no se (0%) () Gaussan nose σ = 0.0 Mxture of () and () Flter Flter Flter Flter Flter Flter Flter Medan Averagng Table : (b) Performance of Varous Flters on the Fruts Image Flter Type () Salt & pepper nose (5%) () Gaussan nose σ = 0.05 Mxture of () and () Flter Flter Flter Flter Flter Flter Flter Medan Averagng These experments are then repeated for hgher nose level. ow, 5% salt and pepper nose, Gaussan nose wth varance 0.05 and a mxture of these two are consdered. The for varous flters s gven n Table (b) and (b) for lenna and fruts mages, respectvely. From Tables (a) and (a), t can be observed that, n general, fuzzy flters outperform the conventonal flters. However, flter 4 requres more than one teraton for flterng. The results gven n Table and are obtaned after fve teratons for ths 99
6 Fuzzy Flterng Algorthms for Image Processng: Performance Evaluaton of Varous Approaches flter. The of ths flter s stll larger than most of other flters consdered n the study. When the mages are corrupted by salt and pepper nose, flter 7 gves mnmum, whereas, n case of Gaussan nose, flter 5 appears most effectve among the flters consdered n ths study. However, as Tables (b) and (b) reveal, at hgher nose evel, flter7 s no longer effectve for salt and pepper nose. The obtaned for flt er 7 s hgher than the medan flter. In ths case the mnmum s obtaned by usng flter 6. The neffectveness of flter 7 s attrbuted to the smaller wndow sze for obtanng the reference pxel, as compared to the medan flter. For the Gaussan nose, on the other hand, flter 5 s stll most effectve. It s also observed that at hgher nose level the of varous fuzzy flters s not sgnfcantly lower than that of the medan flter. (a) (b) (c) (d) 4. COCLUSIO Fg. : (a) The orgnal le nna mage, (b) Image corrupted wth Gaussan nose (varance = 0.0), (c) Image fltered by medan flter, (d) Image fltered by flter 5 Ths paper has presented varous fuzzy flters based on dfferent approaches. The performance of the flters s examned under three dfferent nose models. The results n terms of, reveal that, n general, the fuzzy flters outperform the conventonal flters. Flter 5 based on fuzzy logc control s found most effectve when the nose s Gaussan. In case of salt and pepper nose, fuzzy adaptve flters are better than the other flters. Flter 7 and flter 6 are found to gve mnmum at 0% and 5% nose, respectvely. The ncrease n of flter 7, wth hgher nose level, s due to the smaller sze of the wndow used for obtanng the reference pxel. 00
7 Proceedngs of the Internatonal Conference on Cognton and Recognton (a) (b) (c) (d) Fg. : (a) The orgnal fruts mage, (b) Image corrupted wth salt & pepper nose (0%), (c) Image fltered by medan flter, (d) Image fltered by flter 7 REFERECE [] R. Chelappa et al. (998), The past, present and future of mage and multdmensonal sgnal processng, IEEE Sgnal Processng Magazne. [] F. Russo and G. Rampon (99) Fuzzy operators for sharpenng of nosy mages, IEEE Electron Lett., vol. 8, pp [3] F. Russo and G. Rampon (996) A fuzzy flter for mage corrupted by mpulse nose, IEEE Sgnal Processng Lett., vol. 3, pp [4] M. Muneyasu,Y. wada and T. hnamoto (996) Edge preservng smoothng by adaptve nonlnear flters based on fuzzy logc laws, Proc. 3rd IEEE Int. Conf. Image Processng, vol., pp [5] Y. Cho and R. Krshnapuram (997) A robust approach to mage enhancement based on fuzzy logc, IEEE Trans. Image Processng, vol. 6(6), pp [6] F. Farbz et al. (000) A new fuzzy logc flter for mage enhancement, IEEE Trans. Systems Man and Cybernetcs-Part B, vol. 30(), pp [7] S. Peng and L. Lucke, (996) A hybrd flter for mage enhancement, Proc. nd IEEE Int. Conf. Image Processng, vol., pp [8] K.. Platanots, D. Androutsos and A.. Venetsanopoulos (999) Adaptve fuzzy systems for multchannel sgnal processng, Proceedngs of the IEEE, vol. 87 (9), pp
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