Image Processing Using Cellular Neural Networks Based on Multi-Valued and Universal Binary Neurons

Transcription

1 Image Processig Usig Cellular Neural Networks Based o Multi-Valued ad Uiversal Biary Neuros Igor Aizeberg ad Costatie Butakoff Neural Networks Techologies Ltd., 55, Bialik str., Ramat-Ga, 553, Israel igora@etvisio.et.il igora@t-group.com (IA); cbutakoff@yahoo.com cbutakoff@t-group.com (CB) Abstract Multi-valued ad uiversal biary euros (MVN ad UBN) are the eural processig elemets with the comple-valued weights ad high fuctioality. It is possible to implemet a arbitrary mappig described by partially defied multiplevalued fuctio o the sigle MVN. A arbitrary mappig described by partially defied or fully defied Boolea fuctio, which ca be o-threshold, may be implemeted o the sigle UBN. The quickly covergig learig algorithms eist for both types of euros. Such features of the MVN ad UBN may be used for solvig the differet problems. Oe of the most successful applicatios of the MVN ad UBN is their usage as basic euros i the Cellular Neural Networks (CNN). It opes the ew effective opportuities i oliear image filterig ad its applicatios to oise reductio, edge detectio ad solvig of the super resolutio problem. A umber of eperimetal results are preseted to illustrate the performace of the proposed algorithms. Keywords: eural etworks, oliear filterig, image processig. Itroductio The itesive developmet of the eural etworks durig last years makes their applicatio to image processig, aalysis ad recogitio very attractive. E.g., may image filterig algorithms i spatial domai may be reduced to the same operatio, which has to be performed over all the piels of the processed image. For eample, may of the algorithms of liear ad oliear filterig i spatial domai are reduced to the processig withi a local widow aroud each piel of the image. We will cocetrate here o the oliear filterig algorithms that are reduced to the local processig withi a widow aroud each piel ad based o the oliear trasformatio of the result of liear covolutio with the weightig kerel (template). The mai operatios i this case are: oise reductio, frequecy correctio (etractio of image details) ad edge detectio. A prelimiary report o the followig aspects has bee preseted at the 000 IEEE Iteratioal Workshop o Neural Networks for Sigal Processig []. Sice processig withi a local widow aroud the piel is ot recursive, it may be orgaized simultaeously for all the piels, idepedetly o each other. So it is atural to orgaize this process usig some appropriate kid of eural etwork. The most appropriate eural etwork for solvig of these problems is the Cellular Neural Network (CNN). CNN has bee itroduced i [] as a special high-speed parallel eural structure for image processig ad recogitio. CNN cocept has bee itesively developed. May results from simple filters for biary images [-3] to algorithms for color image processig [4] were carried out. The appearace of CNN progressed i three fields: o-liear dyamic systems, eural etworks ad image processig. We omit here the first field, ad cosider the secod ad the third oes. A revolutioary role of the CNN i eural etworks is first of all i its local coectivity. The eural etworks cosidered before the paper [] usually were the fully coected etworks (e.g., Hopfield etwork), or multi-layered eural etworks with full coectios betwee euros of eighborig layers (e.g., multi-layered perceptro) [5]. A CNN cocept supposes a cellular structure of the etwork: each euro is coected oly with the euros from its earest eighborhood (see Fig. ). It meas that the correspodig iputs of each euro are coected with outputs of euros from the earest rr eighborhood, ad o the other had, the output of each euro is coected with the iputs of euros from the same eighborhood. The euros of such a etwork are also ofte called cells. Depedig o the type of euros that are basic elemets of the etwork, it is possible to distiguish cotiuous-time CNN (CTCNN) [], discrete-time CNN (DTCNN) [6] (orieted especially o biary image processig), CNN based o multi-valued euros (CNN-MVN) [7, 8] ad CNN based o uiversal biary euros (CNN-UBN) [8, 9, 0]. CNN-MVN makes possible processig, which is defied by some multiple-valued threshold fuctios, ad CNN-UBN allows processig defied ot oly by threshold, but also by arbitrary Boolea fuctio. These properties of the CNN-MVN ad CNN-UBN will be used here to develop a cocept of the oliear cellular eural filterig (NCNF) [, 0].

2 (i, j) Fig.. CNN of a dimesio 35 with the local coectios i a 33 eighborhood: each euro is coected with 8 euros aroud it ad with itself We would like to cosider below some effective image processig algorithms that should be implemeted usig CNN- MVN ad CNN-UBN. There are the algorithms that belog to the family of oliear cellular eural filters. The first group of algorithms icludes multi-valued filters (MVF) for oise reductio ad frequecy correctio. The secod oe icludes filters for the precise edge detectio. We will also cosider how MVF should be effectively used for solvig the super resolutio problem. All the problems to be cosidered here are very importat for differet tasks of applied image processig: medical imagig, satellite imagig, graphic art, etc. Additioally to the refereces metioed above, the followig CNN applicatios to image processig should be oted: implemetatio of order statistic oliear filterig [], morphological operatios [], image segmetatio [3, 4] ad image halftoig [5]. Sice the fial purpose of this work is to preset some ew effective solutios i oliear image filterig we would like to make here a brief ad cocetrated observatio of the cosidered problem. We will ot preset here a comprehesive observatio of the oliear filterig. The reader, who is iterested, should address to the latest moographs devoted to this problem [6-9]. We wat to metio here the most importat ad key issues of the oliear cellular eural filterig (NCNF). The purpose of the filterig operatio is assumed to be a effective elimiatio or atteuatio of the oise that is corruptig the sigal. We will cosider here D sigals (images) ad therefore D filters. Let us cosider N M image corrupted by oise. The simplest spatial domai filterig operatio is the mea filter ad it is defied as B$ = B kl, ( m+ )( + ) i k i+ j m l j+ m () where B kl are the values of the iput sigal, B $ is the corrected value of the th piel. The rage of idices k ad l defies the filter widow. It is evidet, but should be metioed that mea filter is a liear operatio. A more effective liear filter is a simple low-pass filter, which may be cosidered as a geeralizatio of the mea filter (): w0 0, wkl > 0 w 0 = 0 w where. If, ad $B = w+ w B kl = 0 kl kl, i k i+ j m l j+ m ( + )( m+ ) the filter () is trasformed to the filter (). Computig simplicity of the liear filters () ad () makes them very attractive. But ofte it is impossible to obtai ice results with them. They reduce a oise, but ot completely, ad simultaeously they smooth a sigal. May be the simplest oliear filter, which is more complicated from the computatioal poit of view, but usually more effective for the oise reductio is a simple media filter, which is defied as (3) $B MED B, = i k i + j m l j+ m kl where MED defies a media value withi the m filter widow. A fudametal differece betwee the behavior of liear ad oliear filters is the followig. The impulse respose (that is, the output sigal whe the iput sigal is equal to at time istat 0 ad zero otherwise) of a otrivial timeivariat liear filter clearly caot be a zero-valued sequece [6]. It meas that a liear filter always will smooth the ()

3 sigal. A oliear filter preserves sigal carefully, ad usually is more effective for oise removal. Differet types of oliear filters were elaborated ad ivestigated, e.g., L-filters [0], rak-order filters [0]. The reader who is iterested may address also to the weighted media filters [], stack filters [6, 7], etc. It is easy to see from eq. () - (3) that a problem of the image filterig i the spatial domai is reduced to the ) replacemet of the curret value of brightess i -th piel by the value B, which is better from some poit of view. B Let us cosider the liear filters () ad (). The arithmetic mea is a most popular, but ot a sigle mea. There is a umber of other widely used types of mea, e.g., geometric, harmoic,. But calculatio of these types of mea is coected with oliear operatios. A geeral form of oliear mea filters has bee cosidered recetly [6, ]. It is (for D filter for simplicit: where g() is the followig fuctio g B i = g : R R : k k= wgb ( k) wk k= the arithmetic mea / the harmoic mea g ( ) =. l the geometric mea p, p R\ {, 0, } the L p mea Sice fuctio g() is used for averagig of the sigal, a followig questio is atural: is it possible to defie other oliear meas, ad therefore, other types of averagig, which may be used for the filterig? The mai part of the paper gives a positive aswer to this questio. The issue of the approach, which is developed here, is the applicatio of the o-liearity of the euros activatio fuctios. A coveiece of the spatial liear filters implemetatio usig cellular eural etworks is a additioal stimulus for the research i this directio. Itroductio of the CNN based o MVN (CNN-MVN) [8, 3] ad CNN based o UBN (CNN-UBN) [8, 9] opeed a possibility for the eural implemetatio of the origial oliear spatial domai filters, amely multi-valued filters (MVF) ad cellular eural Boolea filters (CNBF). They should be cosidered as parts of the ew oliear filters family. This family, which we would like to call as oliear cellular eural filters (NCNF), will be cosidered below.. Multi-Valued ad Uiversal Biary Neuros Uiversal Biary Neuro (UBN) performs mappigs described by arbitrary Boolea fuctio of variables. Multi-Valued Neuro (MVN) performs mappigs described by full-defied threshold or partial-defied k-valued fuctio (fuctio of k- valued logic), where k is i geeral a arbitrary positive iteger. Commo basic mathematical backgroud of the UBN ad MVN is the followig. A arbitrary Boolea fuctio of variables or k-valued fuctio of variables is represeted by + comple-valued weights [8]: where,...,, L p w 0,w,...,w f,..., ) = P( w +w w ), (4) ( 0 are the variables, which the performed fuctio depeds o (euro iputs) ad P is the activatio fuctio, which is defied i the followig way. ) For Multi-Valued Neuro: P ( z) = ep( iπj/k), if πj/k arg( z) < π ( j+ ) /k, (5a) or with iteger-valued output: P ( z) =j, if πj/k arg( z) < π ( j+ ) /k, (5b) where j=0,,..., k- are the values of the k-valued logic, i is the imagiary uity, z = w0 + w+ w is the weighted sum, arg(z) is the argumet of the comple umber z (values of the fuctio ad of the variables are coded by the comple umbers, which are k th j roots of uity: ε = ep( iπ j/k ), j { 0,,..., k- }, i is a imagiary uity; i other words values j ε of the k-valued logic are represeted by k th roots of uity: ); 3 j

4 ) For Uiversal Biary Neuro j P ( z) = ( ) if (πj/m) arg( z) (π ( j+ ) /m) B < z w + w + w = where m is a positive iteger, j is a o-egative iteger 0 j<m, 0 is the weighted sum, arg(z) is the argumet of the comple umber z. Estimatio for m is m ad it is cosidered i [8]. Fig. illustrates defiitio of the activatio fuctios (5) ad (6). (6) i 0 0 k- k- Z ε i - ε 0-0 m- ε m m- Z (a) MVN activatio fuctio P( z) = ε k- (b) UBN activatio fuctio P (z) = Fig. Applicatio of the MVN ad UBN activatio fuctios to the weighted sum Z. Evidetly, fuctios (5) ad (6) separate the comple plae ito k sectors ad m sectors, respectively. There are at least two quickly covergig learig algorithms for MVN [8]. Oe of them is based o the error-correctio learig rule: Wl Wl+ q s Wl+ = W l+ Cl ( ε -ε ) X ( + ) where ad are curret ad et weightig vectors, X is the vector of the euro's iput values (complecojugated), ε is the primitive k-th root of uity (k is chose from (5ab)), (7) is the scalig coefficiet, q is a umber of the desired sector o the comple plae( correct sector), s is a umber of the sector, where the actual value of the weighted sum fell to( icorrect sector), is a umber of euro s iputs. Learig for the UBN may also be reduced to the rule (7). For UBN q has to be chose o each step based o the followig rule: if the weighted sum Z gets ito icorrect sector umber s the both of the eighborig sectors are correct [8]: C l q = s- (mod m), if Z is closer to (s- )-th sector (8a) q = s+ (mod m), if Z is closer to (s+)-th sector. (8b) 3. MVN ad UBN based Cellular Neural Network Cellular eural etworks (CNN) were itroduced by Chua ad Yag i [] ad became a very appropriate computig model for the solutio of differet kids of image processig problems. CNN local coectivity (see Fig. ) is a key poit for the implemetatio of the differet spatial domai filterig algorithms. The dyamics of Cotiuous time CNN (CTCNN) euro (cell) is described by the followig equatio []: r r r r v( i, j) = F z&( i, j) + ( A( k, l) v( i+k, j+l )) + ( B( k,l) u( i+k,j+l ) + I. k=-r l=-r k=-r l=-r The dyamics of discrete time CNN (DTCNN) cell is described by the followig equatio [6]: (9) 4

5 r r r r v( i, j, t ) =Fd ( A( k, l) v( i+k, j+l, t- )) + ( B( k, l) u( i+k, j+l, t- )) + I. (0) k=-r l=-r k=-r l=-r Here (i, j) are the coordiates (umber) of -th euro, v is the euro s output, u is the euro s iput, z is the euro s state, r is the size of a euro s earest eighborhood (r r), t, t+ are the time slots t ad t+, respectively, A, B are the r r matrices, which defie syaptic weights (feedback ad cotrol templates respectivel, I is a bias, F ad F d are the activatio fuctios of a CTCNN ad DTCNN euro, respectively. For a CTCNN the activatio fuctio is the followig: Fz () = For the DTCNN the activatio fuctio is the followig: z+ - z-. () Fd = sg( z). () The followig remark should me made. A oliear D-template for the implemetatio of oliear filters (e.g., media ad rak-order oes) has bee itroduced i [, 4]). The equatio (9) is trasformed i this case as it is show below: r r r r r r v( i, j) = F z& ( i, j) + ( A( k,l) v( i+k, j+l)) + ( B( k,l) u( i+k,j+l) + ( Dˆ ( k,l) Vuzv ( t) ) + I, k=-r l=-r k=-r l=-r k=-r l=-r where Dˆ ( k, l) is the geeralized oliear term applied to V, the voltage differece of either the iput, state or output values. The activatio fuctio () is a piecewise liear fuctio. The activatio fuctio () is a simple sig fuctio, ad therefore DTCNN is based o the usual threshold euros. Evidetly, the liear part of the fuctio () esures the implemetatio of ay algorithm of liear D filterig i spatial domai, which is reduced to the liear covolutio with the weightig widow, o the CTCNN cell. DTCNN esures the implemetatio of biary image processig algorithms, which may be described by threshold Boolea fuctios. Of course, the CTCNN ad DTCNN ope may possibilities to solve the differet problems of image processig. Ideed, they eable the implemetatio of liear ad oliear filters i spatial domai. Moreover it is a good way to implemet the image processig algorithms described by threshold Boolea fuctios. O the other had two problems should be metioed here. The first oe is that a derivatio of correspodig templates is ecessary for the implemetatio of ay algorithm. Such a derivatio may be achieved by several ways [6]: learig (usig some learig algorithm to fid the weightig templates, which perform the desired global mappig from iput state to the correspodig desired output); desig (e.g., programmig the etwork to have some desired fied poits); mappig (e.g., simulatio of biological pheomeo based o a set of equatios). A problem is that may learig ad desig algorithms for CNN (see e.g., [6] for observatio) ofte are reduced to the solutio of differetial equatios, which may be a specific computig problem. The secod problem is that the DTCNN ca t be used aywhere for the implemetatio of the processig algorithms described by o-threshold Boolea fuctios. The o-threshold fuctio ca t be implemeted with ay learig algorithm o the threshold eural elemet. But the threshold euro itself is a basic elemet of DTCNN. These two problems should ot be cosidered as a disadvatage of CNN. Ideed, may iterestig applied problems were solved withi CTCNN ad DTCNN traditioal cocept. But the followig ways for developmet of CNN paradigm are atural amog other: ) icreasig of CNN fuctioality by usig of MVN ad UBN as basic euros i the cellular etwork with the architecture preseted i Fig.. This brigs us to the CNN based o MVN ad UBN (CNN-MVN ad CNN-UBN); ) a search for the ew problems, which could ot be solved withi traditioal CNN cocept, but may be solved o CNN-MVN ad CNN-UBN. The efficiecy of CNN-MVN ad CNN-UBN is based o the followig observatios: ) high fuctioality (because of uiversal fuctioality of UBN, ad practically uiversal fuctioality of MVN); ) quickly covergig learig algorithms; 3) use of the o-liearity of MVN activatio fuctio (5) for the developmet of oliear filters. Let us ow defie CNN-MVN ad CNN-UBN. Cosider a CNN with the obvious local coectio structure (see Fig. ) of a dimesio N M. Cosider MVN or UBN as a basic euro of such a etwork. Evidetly, each cell of CNN-MVN (or CNN-UBN) will perform the followig correspodece betwee euro s iputs ad output: r r i j i j Y i j ( t+ ) = F w0 + wkl kl ( t) k= rl= r where: are the two-dimesioal coordiates (umber) of a euro; Y is the euro's output; (3) is the syaptic weight correspodig to the iput of th euro, to which the sigal from the output of kl th euro is trasmitted (oe may make a compariso to the elemets of B-template i the traditioal CNN (see (9) ad (0) ); 5 kl w kl is the iput of th euro, to which

6 the sigal from the output of kl th euro is trasmitted; F is the activatio fuctio of the euro, which is defied by the equatio (5) for MVN, ad by the equatio (6) for UBN; r m= r meas that each euro is oly coected with the euros from its earest r-eighborhood, as usual for CNN. Sigle weightig template of CNN-UBN ad CNN-MVN may be writte i the followig form: wi r j r... w w ;,, 0 W =... w... wi r, j r... w, i r j+ r + i+ r j+ r where W is a rr matri. What is commo ad what is differet with the traditioal CNN? First of all the cellular architecture of etwork is a importat commo poit. It is evidet, ad it will be show below o may eamples, that CNN-MVN ad CNN-UBN are orieted towards the solutio of the same problems of image processig ad recogitio, as traditioal CNN. The mai differece betwee the traditioal CNN, ad CNN-MVN, CNN-UBN is the use of comple-valued weights i the last oes. At first sight it is a disadvatage. But at the same time CNN-UBN ad CNN-MVN use a sigle comple-valued weightig template, ad the traditioal CNN uses two real-valued weightig templates. So, from the poit of view of the weights compleity the traditioal CNN, ad CNN-UBN, CNN-MVN are equivalet. To coclude this observatio of CNN, it should be metioed that CNN-UBN ad CNN-MVN are ot alterative to the traditioal CNN. They may be cosidered as a ice supplemet, which gives high efficiecy i applicatios (to be cosidered below). O the other had the elaboratio of the ew image processig algorithms has bee iitiated by the cocept of CNN-UBN ad CNN-MVN. 4. Noliear Cellular Neural Filterig, Two-dimesioal oliear cellular eural filter (NCNF) may be defied i the followig way: ) B = F( w 0 + w Y kl ) kl i k i+ j m l j+ m (4) Here m is a filter widow, Y kl w, w kl are the comple-valued weightig coefficiets, 0 B ) is a sigal value i the th piel after filterig, defiitio of ad F depeds o a type of the two NCNF subfamilies multi-valued filters (MVF) ad cellular eural Boolea filters (CNBF). Let us cosider both subfamilies. 4.. Multi-valued oliear filterig Multi-valued oliear filters (MVF) have bee itroduced recetly [, 8, 9]. A specific comple o-liearity of the multivalued euro activatio fuctio [8] is a key for the developmet of a ew filters family. Let 0 B k be a dyamic rage of D sigal. Let cosider a set of the k th roots of uity. We ca put a root of uity to the correspodece with the real value B: e B ep( iπb/k) =Y =. (5) B ε B Thus, we have the uivalet mappig, where ε is a primitive k th root of uity. A two-dimesioal Multi-Valued Filter (MVF) is a filter, which is defied by the followig formula [0]: ) B = P( w + w 0 kl Ykl ) i k i+ j m l j+ m (6) where P is the activatio fuctio (5b) of the multi-valued euro, Y is obtaied from by the equatio (5), i,j are the coordiates of the filtered piel, m is a filter widow, w kl are the weightig coefficiets (comple-valued i geeral). Evidetly, the equality (6) defies a class of filters. Each of them is defied by the correspodig set of the weightig coefficiets (weightig template). kl B kl 6

7 Let us compare the filter (6) to the liear filters () ad (). We ca make the followig coclusio: without the oliear fuctio P ad comple additio (if Y will be replaced by B), ad with w kl > 0 filter (6) is trasformed to the simple low-pass spatial domai liear filter. But the filter (6) is pricipally oliear. First of all the fuctio P is oliear, but additioally the comple additio together with the trasformatio (5) defies a specific oliear averagig of the sigal. A detailed mathematical ivestigatio of the properties of o-liearity carried by comple additio ad fuctio P may be a subject of a special work. It should be metioed here that impulse respose of MVF is a zero-valued sequece, ad this feature brigs MVF together with other oliear filters. Oe remark should be made before we will move to the applicatios of MVF. The itegratio of MVF with the differet well-kow oliear spatial domai filters leads to their promisig modificatios. Let us cosider, e.g., a filter, which is defied by equatio (6), but simultaeously coected with the weighted rak-order filter. The order-statistic filters ad the weighted rak-order filters are based o the compromise betwee a pure oliear operatio (orderig) ad a pure liear operatio (weightig). Actually e.g., a weighted rak-order filter may be defied by the same equatio () as the low-pass filter, with a additioal coditio that addeds uder a sig Σ have to satisfy the rak-order criteria (the order-statistic of addeds has to be i the limited eighborhood of the order-statistic of the filtered piel). If we will implemet a weighted rak-order filter, but together with the trasformatio (5), comple additio, ad the fuctio P applied to the weighted sum, we will obtai a rak-order multi-valued filter (ROMVF). MVF may be also coected with other differet oliear filters i the same way. Evidetly, the most appropriate structure for the MVF implemetatio is CNN-MVN. The most simple, but very effective template for the reductio of Gaussia or uiform oise is the followig: W = G (7) where G is a parameter. The template (7) is desiged from the followig cosideratios. If G = 0 ad all other weights are equal to the a maimal sigal averagig will be achieved. If G > the the smoothig of the sigal will be more light ad the image boudaries will be preserved with more accuracy. At the same time for G > 3 the filter will be degeerated because it will ot chage the sigal. Thus, to remove a oise with the impulsive compoet we have to recommed 0 G. To remove the Gaussia or uiform oise, which is ot mied with the impulsive oe our recommedatio for G is the followig: G 3. Let us cosider a eample. It shows a reductio of zero-mea Gaussia oise with a dispersio equal to 0.3σ, ad it is preseted i Fig. 3. The results of rak-order filterig of the same images are give for compariso. The ehaced differeces betwee oisy image ad the resultig images are also show. Table cotais the objective statistical characteristics of the origial, corrupted, ad filtered images. Table. Statistical characteristics for the eample give i Fig. 3 Image Rose Noisy Rose MVF, G=8 MVF, G=6, MVF, G=6, ROMVF ROF iteratio iteratios ND (σ) PSNR SD ND (σ) estimatio of white oise dispersio, PSNR estimatio of peak sigal to oise ratio, SD stadard deviatio from the origial image. The eample shows advatages of MVF. The oise is reduced effectively by simple MVF, image boudaries are preserved effectively by rak-order MVF (ROMVF). All implemetatios of MFV give better results tha the usual rak-order filter. 4.. Multi-valued frequecy correctio Multi-valued filters are also a highly effective mea for the frequecy correctio. A frequecy correctio problem may be cosidered as a specific ati-filterig. Ideed, the filterig is a reductio of the high frequecy. But it is atural to cosider the iverse problem: amplificatio of the high ad medium frequecy. It is clear that by fidig a appropriate solutio of such a problem it is possible to solve the problems of image sharpeig, etractio of details agaist a prevetig backgroud, local cotrast ehacemet, etc. A atural solutio to the frequecy correctio problem is a filterig i the frequecy domai. It is clear that this approach is complicate from the computig poit of view. Direct ad iverse Fourier trasformatios, search for the appropriate filter mask, ad multiplicatio of the spectrum by this mask require may operatios ad therefore it will take a log time. A very iterestig idea [5] is the approimatio of the correspodig filters i frequecy domai by simple liear filters i spatial domai. 7

8 (a) the origial image (b) MVF, 33 G=8 (c) MVF, 33, G=6, iteratio (d) oisy image (Gaussia oise, 0.3σ) (e) differece betwee images (d) ad (b) (f) differece betwee images (d) ad (c) (g) MVF, 33, G=6, iteratios (h) ROMVF, 33, G=, rak=9 (i) simple ROF, 33, rak=9 (j) differece betwee images (d) ad (g) (k) differece betwee images (d) ad (h) (l) differece betwee images (d) ad (i) Fig. 3. Noise reductio usig multi-valued filterig 8

9 The etractio, localizatio, ad detectio of importat features o the comple image backgroud were proposed to cosider [5] as a problem of localizatio of objects or details of a give type o the image. Such a type has bee defied as sizes of the objects. The this idea has bee developed i [9], where the solutio of the problem usig traditioal CNN also has bee cosidered. The followig liear filters have bee proposed for the solutio of the global [9, 5] ad high [9, 6] frequecy correctio problems, respectively ) B =G B +G ( B -B ) +G B +c m 3 m Bˆ = ( G ) where B m is a local mea value i a widow aroud the piel B ; B, ad (8) + G B G Bm+c, (9) ) B are the sigal values i th piel before ad after the processig, respectively; G a d G 3 defie the correctio of the low spatial frequecies, G defies the correctio of the high spatial frequecies, c is a mea value of the backgroud after processig. It should be oted that filter (9) could be obtaied from the filter (8). By the way, if G = the filter (9) coicides with the filter called usharp maskig i differet issues (see, e.g., [6]). Although the filters (8) ad (9) give good results, they have some sigificat disadvatages. They are very sesitive to the choice of the weightig coefficiets. Eve miimal chage of ay of them may move a geeral dyamic rage of the image to white or black side. At the same time the global image histogram may dramatically chage. It meas, that some iformatio will be lost, because some piels will be replaced by truly white or truly black oes. O the other had eve a small icremet of the weightig coefficiets ofte leads to the etremely high amplificatio of the high frequecy ad a oise may appear o the image as a result. To break these disadvatages, liear filters (8) ad (9) should be trasformed to the oliear oes. Sice filters (8) ad (9) operate with the arithmetic mea, they should be geeralized usig oliear meas. Oe of the atural solutios is implicated from the followig cosideratios. Multi-valued filter (6) is a geeralizatio of the liear filters ()-(). What will be a multi-valued geeralizatio of the filters (8)-(9)? Takig ito accout (5), it is easy to obtai the followig multi-valued geeralizatio for the filter (8): B$ = P c+ ( G + G ) Y + ( G G ) Y Y( k, l) R 3 kl, where Y kl is obtaied from B kl by equatio (5), R is a local widow aroud piel Y ; B (Y ) ad B $ are the sigal values i th piel before ad after processig, respectively, G, G 3 (0) are the coefficiets, which defie the correctio of the low frequecies, G defies the correctio of the high frequecies, c is a costat. Filter (0) solves a global frequecy correctio problem. Its applicatio to the image leads to the etractio of details whose sizes are approimately equal to the sizes of the filter widow. MVF for the high frequecy correctio may be obtaied from (0) by puttig G : 3 = 0 ˆ =P c+ ( G +G ) Y -G Ykl, Y kl R B () where Y kl is obtaied from B kl by equatio (5), R is a local widow aroud piel Y ; B (Y ) ad B $ are sigal values i th piel before ad after processig respectively, G ad G are the correctio coefficiets, c is a costat. Filters (0) ad () are ot so sesitive to the choice of the correctio coefficiets i compariso to the filters (8) ad (9). They always preserve a global dyamic rage of the image ad its global histogram. Such a tolerace may be used for the processig of the images that cotai may details of differet sizes ad differet cofiguratio. The followig weightig templates implemet filters (0) ad () o the CNN-MVN, respectively: G3 G G3 G G+ G w=c; 0 W=.. G G... G G ()

10 (a) The origial X-ray image (tumor of lug) (b) Histogram of the origial image (a) (d) Histogram of the image (c) after MVF-global frequecy correctio. This histogram does ot cotai ay peak (c) MVF-global frequecy correctio, eq. (0), (e) Liear global frequecy correctio, eq. (8), 3535 (may piels became truly white or truly black) (f) Histogram of the image (e) after liear global frequecy correctio. High peaks i 0 ad 55 are clearly visible. It meas that may piels have become truly white or truly black. As a result, iformatio i the correspodig piels is lost Fig. 4. MVF-global frequecy correctio ad its compariso to liear global frequecy correctio 0

11 (a) The origial image Sydey (b) Histogram of the origial image (a) (c) MVF-high frequecy correctio, eq. (), 33 (d) Histogram of the image (c) after MVF-high frequecy correctio. This histogram does ot cotai ay peak (e) Liear high frequecy correctio, eq. (9), 33 (may piels became truly white or truly black) (f) Histogram of the image (e) after liear high frequecy correctio. High peaks i 0 ad 55 are clearly visible. It meas that may piels have become truly white or truly black. As a result, iformatio i the correspodig piels is lost Fig. 5. MVF-high frequecy correctio ad its compariso to liear high frequecy correctio

12 G.. w 0=c; W=. G Filter (0) etracts image details with the sizes approimately equal to the sizes of the filter widow. Filter () etracts the smallest image details. Thus, the most appropriate size for the widow of the filter () is 33. Estimatios for the values of the weightig coefficiets i (0) ad () are the followig: m G < m, G < G3 < G, m G3 < G < m, where m is a filter widow. Estimatios for the values of the weightig coefficiets i () ad (3) are the followig: m G m <, 0 < G <. Let us cosider some eamples of image processig usig filters (0) ad () ad their compariso to the results obtaied by filters (8) ad (9) (see Fig 4 ad Fig. 5). The preseted eamples show high efficiecy of the MVF frequecy correctio for image sharpeig ad etractio of image details. They also show advatages of the MVF method i compariso to the liear oe. The liear frequecy correctio leads to etremely high amplificatio of the high frequecy. As a result, may truly white ad black piels appear at the image. Some details become sharper, but some other disappear, due to dramatic chage of the global image histogram. The MVF frequecy correctio is free of these disadvatages. It is very importat for the applicatios, for eample, i medical imagig, satellite imagig, digital photography, graphic arts, etc. G G... G Cellular Neural Boolea Filterig Idea of cellular eural Boolea filterig (CNBF) has bee proposed i [9]. This idea is based o a separate processig of the image biary plaes usig special spatial-domai filter defied by some Boolea fuctio ad further itegratio of the resultig biary plaes ito the resultig image. This method has bee called a cellular eural Boolea filterig i [8, 0]. A key issue for the method was ability of CNN-UBN to implemet spatial domai filter described by arbitrary Boolea fuctio [8, 9] (ot oly threshold Boolea fuctio as for DTCNN [6]). Let B be a sigal value i the th k piel ad B { 0,,..., }. Let us split the image oto k biary plaes kl directly, without ay thresholdig. Let B be a biary value of the sigal correspodig to s th s biary plae, is the same s value trasformed to the Boolea alphabet {, -}. Let F i (4) be the UBN activatio fuctio (6). Takig ito accout (6), (4) is trasformed to the followig form: ) B s s = P ( w 0 + w Y ) B kl i k i+ j m l j+ m The equatio (4) establishes CNBF for a biary plae. CNBF for the gray-scale image or for the color chael of the color image should be writte i the followig form: k s= 0 kl, k s B ) = PB ( w0 + wklykl ), (5) s= 0 i k i+ j m l j+ m where deotes the itegratio of k biary plaes from 0 th util (k-) th ito the gray-scale image (color chael). Equatios (4) ad (5) may be also writte i the followig form, respectively: Y (3) (4) ) s s s s B = f B k l, Bk l... B,, k l m i k i+ j m l j+ m (6)

13 k ) s s s B = f B k l Bk l...,, B k = l s 0, m i k i+ j m l j+ m where f is a Boolea fuctio, which defies spatial domai filter with m widow for the biary image (biary plae). Evidetly, (4) describes the implemetatio of the Boolea fuctio (6) usig UBN, or CNN-UBN cell (oe may compare the equatio (4) to the equatio (3) ). The most impressive applicatio of CNBF is a precise edge detectio, which has bee proposed i [9] ad the developed (see, e.g., [, 8, 0]). May of the differet edge detectio algorithms are well kow. The most kow group of edge detectors is based o the covolutio with zero-sum kerels [6, 7]. Some of these algorithms are eve classical (e.g., Laplacia, Sobel, Prewitt). These algorithms are also classified as covolutios with the st order derivative kerels (e.g. Previtt) ad d order derivative kerels (e.g. Laplacia). Aother group of edge detectors is based o a local statistical aalysis [9, 8]. A problem is that all of the metioed algorithms do have a commo ad sigificat disadvatage: they are usually very sesitive to the sigificat brightess jumps ad uable to detect the edges correspodig to the small brightess jumps. It meas that all these algorithms give good results while applied to the images with a good cotrast. The applicatio of the same algorithms to the images with a poor cotrast or to the images with may brightess jumps of a differet level is possible, but the results ofte are ot satisfactory. Other problems are the detectio of the edges correspodig to the details of a complicated form ad edge detectio o the oisy images. Oe of the good alterative solutios is the edge detectio usig CNBF. Let us cosider a backgroud for the desig of the Boolea fuctios, which implemet the correspodig CNBF. The followig key items should be take ito accout: ) whe we say that the edge i some piel of biary image is detected, it meas that the brightess value i such a piel differs from the brightess value of at least oe of the other piels from the widow aroud it. If the isolated piels should be removed from the cosideratio, the the followig supplemet to this coditio should be added: the brightess value i such a piel must be equal to the brightess value of at least oe of the other piels from the widow aroud it. We will ot cosider this case further; ) the upward brightess jumps (from 0 to, if is the value of brightess i the aalyzed piel), ad the dowward oes (from to 0, if 0 is the brightess value i the aalyzed piel) are ot equivalet i geeral; 3) it follows from the item that the et three cases of edge detectio should be cosidered to obtai the most geeral picture of the jumps of brightess: a) separate detectio of edges correspodig to the upward brightess jumps; b) separate detectio of edges correspodig to the dowward brightess jumps; c) joit detectio of edges correspodig to the upward ad dowward brightess jumps. The Boolea fuctios used for the CNBF edge detectio with a widow 33, are preseted i [9, 0]. For eample, the edge detectio correspodig to the upward brightess jumps detectio withi a 33 widow is implemeted usig the followig Boolea fuctio: 3 f = 5 & ( ) This fuctio may be easily implemeted o the UBN. The learig algorithm (7)-(8) requires a few iteratios to coverge, startig from ay radom vector. The eample of precise edge detectio usig the filter defied by (7)-(8) ad the compariso to other edge detectors are preseted i Fig. 6. I [8] CNBF with a 33 widow for the precise edge detectio i a particular directio has bee also cosidered. At the same time it is very attractive to cosider the precise edge detectio usig CNBF with a 55 widow. It is importat for the detectio of edges correspodig to the details of a complicated form. Evidetly, it is possible to desig may fuctios for the edge detectio with a 55 widow i the differet particular directios. For eample, the followig fuctio may be cosidered: (7) (8) Upward jumps, South-East f = ( 3 & 9 & 5) & ( 4 8) & ( 0 4) 3

14 (a) Iput image from electroic microscope (b) Edge detectio usig (c) Image (b) with the CNBF (7)-(8) ehaced cotrast (d) Edge detectio usig Sobel operator: edges are smoothed (e) Edge detectio usig the algorithm based o a local statistics aalysis [9]: edges are smoothed Fig. 6. Precise edge detectio usig CNBF with a 33 widow (f) Edge detectio usig the variace ( d order) ehacemet based o [8]: edges are smoothed The fuctio, which implemets CNBF for complete precise edge detectio correspodig to the upward brightess jumps: f up = [( [( [( [( [( [( [( [( & & & & & & & & & ) & ( & ) & ( 3 & ) & ( 5 & & & & & ) & ( ) & ( ) & ( ) & ( ) & ( ) & ( ) & ( ) & ( 9 7 ) & ( ) & ( ) & ( ) & ( ) & ( a )] a )] a 0 a )] 6 5 a a )] 4 4 )]. )] )] )] (9) Evidetly, it cosists of the disjuctio of the fuctios that implemet edge detectio i all the directios. Fig. 7 presets the results of precise edge detectio usig CNBF with a 55 widow. This eample is a very ice illustratio of how it is possible to detect the sea streams i the differet directios usig the preseted techique. It is clear from the eample that CNBF with a 55 widow is more effective tha CNBF with a 33 widow for detectio of the edges correspodig to the details of a complicated form (oe may compare images i Fig. 7e ad 7f). It is also more effective tha a classical solutio for the edge detectio i a particular directio (oe may compare images i Fig. 7c ad 7d). Here ad further the edged images are give with the iverted palette ( black o white ) 4

15 (a) The origial image: sea surface (b) North-East+North-West+ South-West+South-East (c) North-East+South-West (d) For compariso: North-East+South-West usig the zero-kerel covolutio preseted i [7] (e) All directios (filter (7)+(9) ) (f) Edge detectio usig CNBF with a 33 widow (filter (7)-(8) ) Fig. 7. Precise edge detectio usig CNBF with a 55 widow 5. Applicatio of the Multi-Valued Filters to the Solutio of the Super-Resolutio Problem The idea of the Fourier spectrum etrapolatio to reach a super-resolutio effect has bee proposed i [9, 30]. The approach to solutio of the super-resolutio problem, which will be developed here, has bee proposed i [7] ad the cosidered i [8]. This approach is based o the iterative approimatio of the Fourier, Cosie or Walsh image spectra i the highest frequecy domai. We will develop this approach here. Let f (, be a two-dimesioal sigal or discrete image of a sizes, without loss of geerality. Let A be a spatial domai, o which the sigal is defied, A A be its subdomai. The fuctio F ( u,v) = Φ[ f (, ]; u,v {0,,..., }, where Φ is Fourier, Cosie, or Walsh (ordered by Walsh) trasform, is a spectrum of the sigal f (,. The superresolutio problem ca be separated ito two problems: etrapolatio of spectrum to the domai 5

16 u, v {0,,...,,,..., } ad iterpolatio of sigal o the whole domai A. To solve these two problems, we will cosider the followig iterative process [8]. To iitiate the iterative process, we eed to eted the sigal from domai A to domai A. So let us build the first approimatio to the resultig sigal: f (,, if (, A g (, = s(,, if (, A \ A (30), where s(, ca be the uiform oise with a small dispersio ad mea equal to the mea of f (,. s(, ca also be the result of iterpolatio of the sigal f (, to the whole domai A. It should be oted that it was proposed i [9, 30] to use a zero-costat as s(,. But this could ot be recogized as a good solutio because the results of the followig Fourier spectrum etrapolatio leads to the resultig image, which is very close to the result of iterpolatio. For simplicity ad without loss of geerality g (, may be cosidered as f (,, corrupted by additive uiform oise, ad furthermore we kow, that f (, is corrupted oly i the domai A \ A. This assumptio is true also, whe s (, is the result of iterpolatio. Sice f (, ad s(, were obtaied separately i this case, they have a differet statistics i geeral. This feature leads to the oisy visual effect. So we ca reformulate our problem of the superresolutio as a problem of oise reductio ad correctio of the highest frequecies. But if we take a closer look, we ca see that the spectrum of origial sigal f (, is also kow. It ca be used for particular recostructio of the spectrum of g(,. Thus let us assume that f (, = g(, ). The th approimatio f (, for f (, ca be obtaied i the followig way: y F F ( u,v); u,v {0,,..., -} (3) ( u, v) = Φ[ f - (,]; u,v {,..., - } - g (, = Φ [ F ( u, v)] (3) f (,, if (, A f (, = g (,, if (, A \ A (33) Epressios (3) - (33) defie the iterative process, which allows to obtai the best approimatio for the sigal f(, ad its spectrum F(u,v). It has bee prove [9] for cotiuous sigals that This iequality leads us to the followig coclusio It meas that lim F F( u, v) F ( u, v) dudv < F( u, v) F ( u, v) dudv. lim F( u, v) F ( u, v) dudv = 0. ( u, v) = F( u, v). I this case the followig is also true: lim f (, = f (,. The last coclusio meas that the iterative process defied by (3)-(33) is fiite, ad it meas that this process is covergig. The practice shows that this iterative process always coverges quickly. Usually the process requires o more tha 7-8 iteratios for the stabilizatio. As a result, we obtai the followig: f f (,, if (, A (, = f (, + s, ), if ( ) \ ( y,y A A, (34) where s (, y ) is a oisy compoet ad 6

17 f = 0, if (, A, f (, ε, if (, A \ A, ( where ε is close to zero. Accordig to the equatio (34) f (, = f (, cotais a additive oise i A \ A subdomai. It meas that to complete a process of spectrum ad sigal restoratio, we have to remove the oise ad to correct the highest frequecies, which may be distorted durig the oise removal. There are may differet filters, which may be used to solve this problem. The best filter to apply, both from the poit of view of oise reductio, quality ad preservatio of the image boudaries, is MVF defied by (6). So, if MVF [ h(, ] is the filter (6) applied to the fuctio h(,, a d MVF fc [ h(, ] is the filter (9) applied to h(,, we obtai the followig formulas, which defie the fial correctio of the super-res oluted image: f (,, if (, A S (, = MVF [ f (, ], if (, A / A f (,, if (, A S (, = MVFfc[ S(, ], if (, A / A A ccordig to the equatios (36) ad (37), the sigal is preserved i the subdomai, where it is origially defied. It is also possible to etrapolate the image spectrum without the use of the origial oe i (3). To go i this wa y, the followig correctio has to be made i (3) - (33): A (35) (36) (37) F Φ[ f - (, ]; u, v {0,,..., -} ( u, v) = 0; u, v {,..., -} (38) This techique requires a few iteratios more for the covergece of the iterative process (3) (33), but at the same time a quality of the resultig image ofte ca be better, especially for the Cosie spectrum. Aother importat improvemet to the proposed super-resolutio algorithm is the followig. It is well kow that spatial limitatio of the discrete images ad therefore limitatio of their discrete spectral represetatio leads to the appearace of so-called boudary effects at the results of frequecy domai filterig. These boudary effects become apparet as series of stripes, which are ofte clearly visible at the regios of image boudaries. To elimiate this effect, it is possible to use a symmetrical etesio of the origial image i all directios (eve etesio of the image). Of course, this approach requires more resources ad leads to the icrease of the processig time. At the same time a quality improvemet of the resultig image will be achieved. Let us illustrate the proposed algorithm by the followig eample. The origial 0404 image (see Fig. 8a) has bee subsampled four times usig the etractio of the low-frequecy part from its Fourier spectrum. The the obtaied 5656 image (Fig. 8b) has bee iterpolated usig several algorithms (the best result of iterpolatio is achieved usig Laczos iterpolatig filter, see Fig. 8c) ad upsampled usig several variatios of the proposed super-resolutio algorithm (the best result is preseted i Fig. 8d). It is clearly visible that a visual quality of the image obtaied usig the super-resolutio techique is better tha a quality of the image obtaied by iterpolatio. The image i Fig. 8d is sharper tha oe i Fig. 8c, ad it is very difficult to detect some differece betwee the result of the super-resolutio procedure ad the origial image. Of course, these images do ot coicide with each other, but the stadard deviatio betwee them is miimal. The stadard deviatios betwee the origial image ad images obtaied by differet iterpolatio algorithms are the followig: Bicubic iterpolatio 3.98 Sic (Laczos) iterpolatio (Fig. 8c) 3.86 Super-resolutio (algorithm [30] with Fourier trasform) 3.93 Super-resolutio, algorithm described here, Cosie trasform, MVH, without eve etesio 3.45 fc Super-resolutio, algorithm described here, Cosie trasform, MVH fc, with eve etesio (Fig. 8d) 3.0. The obtaied results show efficiecy of the proposed algorithm. Fig. 9 presets a compariso betwee average values of the Cosie spectrum amplitudes correspodig to the origial (Fig. 8a), iterpolated (Fig. 8c) ad supersampled (Fig. 8d) images. It is clear from this compariso, why the super-resolutio algorithm described here is better tha the iterpolatio. We see that the preseted algorithm restores a higher frequecy part of the image spectra more effectively tha the iterpolatio. The amplitude of the supersampled image spectrum is very close to the amplitude of the origial image 7

18 spectrum especially i the highest frequecy domai, while the amplitude of the iterpolated image spectrum is close to zero. (a) The origial image (b) 4-times origial image dowsamplig (c) Sic (Laczos) iterpolatio of the image (b) (d) Super-resolutio of the image (b) (4-times upsamplig): sic iterpolatio as startig approimatio; Cosie trasform; iput image eve etesio; model (38) for the iput image spectrum (without its preservatio); 8 iteratios for the process (3)-(35); MVF high frequecy correctio (37) Fig. 8. Super-resolutio: dowsamplig ad upsamplig, compariso with the iterpolatio 8

19 Fig. 9 Compariso betwee average values of the Cosie spectrum amplitude correspodig to the origial (Fig. 8a), iterpolated (Fig. 8b) ad super-resoluted images (Fig. 8c) 6. Coclusios We have preseted the cellular eural etworks based o the multi-valued ad uiversal biary euros (CNN-MVN ad CNN-UBN). The results preseted i this paper show high efficiecy of CNN-MVN ad CNN-UBN i image processig applicatios. O the other had, cosideratio of these eural etworks ivolved developmet of a ew family of oliear filters oliear cellular eural filters. As it was show, these filters are very effective ad useful for the solutio of such problems as oise reductio, frequecy correctio ad edge detectio. Multi-valued filter (MVF) may be successfully used for the oise reductio ad for the solutio of the frequecy correctio problem (etractio of image details). MVF comparisos to the order-statistic filters show its advatages for oise reductio. At the same time it was show that MVF could be effectively combied with other oliear filters (for eample, with the same order-statistic filters) to obtai better results. The comparisos of MVF frequecy correctio algorithms to usharp maskig also show advatages of MVF for etractio of image details ad image sharpeig. It is show that cellular eural Boolea filters (CNBF) are a very good mea for the solutio of edge detectio problem. The precise edge detectio algorithm based o CNBF with a 55 widow is preseted. Advatages of precise edge detectio i compariso to the classical edge detectors are cosidered. It is also show that the described filters may be used for the solutio of the super-resolutio problem, to overcome the results obtaied by iterpolatio algorithms ad the results obtaied without the use of multi-valued filters. ACKNOWLEDGEMENTS The work was completely supported by Neural Networks Techologies Ltd. (Israel). Special thaks to Mr. David Halfo, the Presidet of this compay. REFERENCES. I. Aizeberg, N.Aizeberg, T.Bregi., C.Butakoff. ad E.Farberov Image Processig Usig Cellular Neural Networks Based o Multi-Valued ad Uiversal Biary Neuros, Proceedigs of IEEE 000 Iteratioal Workshop o Neural Networks for Sigal Processig, Sydey, Australia, December 3-5, 000, IEEE Computer Society Press, 000, pp

20 . L.O.Chua ad L.Yag, "Cellular eural etworks: Theory", IEEE Trasactios o Circuits ad Systems. Vol. CAS-35, No 0, 988, pp T.Roska ad J.Vadewalle (ed.) Cellular Neural Networks, New York: Joh Wiley & Sos, C.-C.Lee ad.j.pieda de Gyvez Color Image Processig i a Cellular Neural -Network Eviromet, IEEE Tras. O Neural Networks, vol. NN-7, No 5, 996, pp S.Hayki Neural Networks. A Comprehesive Foudatio. Secod Editio. New York: Macmilla College Publishig Compay, H.Harrer ad J.A.Nossek Discrete-time cellular eural etworks, Iteratioal Joural of Circuit Theory ad Applicatios, vol. CTA-0, 99, pp I.Aizeberg ad N.Aizeberg Applicatio of the eural etworks based o multi-valued euros i image processig ad recogitio, SPIE Proceedigs, Vol. 3307, 998, pp I.Aizeberg, N.Aizeberg ad J.Vadewalle Multi-valueapplicatios. Bosto/Dordrecht/Lodo: Kluwer Academic Publishers, 000. ad uiversal biary euros: theory, learig, 9. I.Aizeberg Processig of oisy ad small-detailed gray-scale images usig cellular eural etworks Joural of Electroic Imagig, vol. EI-6, No 3, 997, pp I.Aizeberg, N.Aizeberg, S.Agaia, J.Astola ad K.Egiazaria Noliear cellular eural filterig for oise reductio ad etractio of image details, SPIE Proceedigs, Vol. 3646, 999, pp C.Rekeczky, T.Roska ad A.Ushida CNN Based Self-Adjustig Noliear Filters Proc. of the Fourth IEEE Iteratioal Workshop o Cellular Neural Networks ad their Applicatios, Seville, Spai, Jue 996, pp A.Zarady,. A.Stoffels, T.Roska, ad L.O.Chua Morphological Operatios o the CNN Uiversal Machie, Proceedigs of the Fourth Iteratioal Workshop o Cellular Neural Networks ad their Applicatios, Seville, Spai, Jue 4-6, 996, pp T.Sziray, J.Zerubia ad D.Geldreich Cellular Neural Network for Markov Radom Field Image Segmetatio, Proceedigs of the Fourth Iteratioal Workshop o Cellular Neural Networks ad their Applicatios, Seville, Spai, Jue 4-6, 996, pp D.L.Vilario, D.Cabello, M.Balsi ad V.M.Brea Image Segmetatio Based o Active Cotours usig Descrite-Time Cellular Neural Networks, Proceedigs of the Fifth IEEE Iteratioal Workshop o Cellular Neural Networks ad their Applicatios, Lodo, UK, April 4-7, 998, pp K.R.Crouse, T.Roska, ad L.O.Chua Practical Halftoig o the CNN Uiversal Machie, Proceedigs of the Fifth IEEE Iteratioal Workshop o Cellular Neural Networks ad their Applicatios, Lodo, UK, April 4-7, 998, pp J.Astola, ad P.Kuosmae Fudametals of o-liear digital filterig. New York: CRC Press, Boca Rato, S.Agaia, J.Astola ad K.Egiazaria Biary polyomial trasforms ad oliear digital filters. New York/Bosto/Hog-Kog: Marcel Dekker, Ic., I.Pitas ad A.N.Veetsaopoulos Noliear digital filters: Priciples ad Applicatios. Bosto: Kluwer Academic Publishers, X.Z.Su, ad A.N.Veetsaopoulos Adaptive Schemes for Noise Filterig ad Edge Detectio by use of Local Statistics, IEEE Trasactios o Circuits ad Systems, vol. CAS-35, 988, pp A.C.Bovik, T.S.Huag, ad D.C.Mudso A Geeralizatio of Media Filterig Usig Liear Combiatios of Order Statistics, IEEE Trasactios o Acoustic, Speech, Sigal Processig vol. ASSP-3, pp , B.I.Justusso Media Filterig: Statistical Properties. Topics i applied Physics - 43, Two-dimesioal Digital Sigal Processig II, (Ed. T.S.Huag), Berli: Spriger-Verlag, 98, pp I.Pitas, ad A.N.Veetsaopoulos Noliear Mea Filters i Image Processig. IEEE Trasactios o Acoustic, Speech, Sigal Processig, vol. ASSP-34, 996, pp N.Aizeberg ad I.Aizeberg CNN Based o multi-valued euro as a model of associative memory for gray-scale images", Proceedigs of the Secod IEEE Iteratioal Workshop o Cellular Neural Networks ad their Applicatios (CNNA-9), Techical Uiversity Muich, Germay, October 4-6, 99, pp B.E.Shi Order Statistic Filterig with Cellular Neural Networks, Proceedigs of the Third IEEE Iteratioal Workshop o Cellular Neural Networks ad their Applicatios, Rome, Italy, December 994, pp T.P.Belikova Simulatio of the liear filters i the problems of medical diagostics", Digital Image ad Fields Processig i the Eperimetal Researches, (I. Ovseevich - Ed.), Moscow: Nauka Publisher House, 990, pp W.KPratt "Digital Image Processig. Secod Editio" New York: Joh Wiley & Sos, J.R.Jese Itroductory Digital Image Processig: A Remote Sesig Perspective Eglewood Cliffs, New Jersey: Pretice-Hall, R.M.Haralick Statistics ad Structural Approach to Teture, Proceedigs IEEE, vol. 67, No 5, 979, pp M.Shaker Sabri ad W.Steeart A approach to bad-limited sigal etrapolatio: the etrapolatio matri, IEEE Tras.Circuits Syst., vol. CAS-5, No, 978, pp H. Stark Applicatios of optical Fourier trasforms, New York: Academic Press, 98. 0

21 AUTHORS: IGOR N. AIZENBERG Dr. Igor Aizeberg graduated from the State Uiversity of Uzhgorod (USSR) with M.D. i 98, received Ph.D. i the Computig Ceter of the Academy of Scieces of the USSR (Moscow, USSR) i 986. I he was with the Istitute for Iformatio Trasmissio Problems of the Academy of Scieces of the USSR (Moscow) as a researcher. I ad he was with the Departmet of Cyberetics i the State Uiversity of Uzhgorod (USSR-Ukraie) as Associate Professor (Docet). I he was with the Departmet of Electrical Egieerig (ESAT) of the Catholic Uiversity of Leuve (K.U. LEUVEN, Belgium) as a visitig researcher. Sice 999 he is a head of the research departmet of the compay Neural Networks Techologies Ltd. (Israel). He is the author of about 80 papers ad oe book i the field of eural etworks ad their applicatios i image processig ad patter recogitio. He also holds 4 patets. CONSTANTINE V.BUTAKOFF Costatie Butakoff graduated from the State Uiversity of Uzhgorod (Ukraie) with M.D. i 999. I he was with the compay Neurotechologies (Ukraie) as a researcherprogrammer. Sice 999 he is with the compay Neural Networks Techologies Ltd. (Israel) as a researcher. He is the author of 7 papers i the field of eural etworks ad their applicatios i image processig ad patter recogitio.