A new method for constructing kernel vectors in morphological associative memories of binary patterns

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1 DOI:.2298/CSIS B A new method fo constuctng kenel vectos n mophologcal assocatve memoes of bnay pattens Yanns.S. Boutals 1 1 Depatment of Electcal and Compute Engneeng, Democtus Unvesty of Thace, GR-670 Xanth, Geece ybout@ee.duth.g Abstact. : Kenel vectos epesent an elegant epesentaton fo the eteval of patten assocatons, whee the nput pattens ae coupted by both eosve and dlatve nose. Howeve, the acton completely fals when a patcula knd of eosve nose, even of vey low pecentage, coupts the nput patten. In ths pape, a theoetcal ustfcaton of ths fact s gven and a new method s poposed fo the constucton of kenel vectos fo bnay pattens assocatons. The new kenels ae not bnay but gay, because they contan elements wth values n the nteval [0, 1]. It s shown, both theoetcally and expementally that the new kenel vectos cay the good popetes of conventonal kenel vectos and, at the same tme, they can be easly computed. Moeove, they do not suffe fom the patcula nose defcency of the conventonal kenel vectos. The ecallng esult s n geneal a gay patten, whch n the sequel undegoes a smple thesholdng acton and passes though a smple Hammng netwok to poduce hgh ecall ates, even n heavly coupted pattens. Reteval of patten assocatons s vey sgnfcant fo a vaety of scentfc dscplnes ncludng data analyss, sgnal and mage undestandng and ntellgent contol. Keywods: Neual netwoks, Assocatve memoy, Kenel vectos, Nose Robustness. 1. Intoducton Mophologcal Neual Netwoks (MNNs) epesent atfcal neual netwoks whose neuons pefom an elementay opeaton of mathematcal mophology [1], [2]. Unlke Hopfeld netwok [3,4], MNNs povde the esult n one pass though the netwok, wthout any sgnfcant amount of tanng. A numbe of eseaches devsed MNNs fo a ange of applcatons lke those appeang, fo example, n [5-13].

2 Yanns.S. Boutals Atfcal neual netwok models ae specfed by the netwok topology, unt chaactestcs, and tanng o leanng ules. The undelyng algebac system used n these models s the set of eal numbes R togethe wth the opeatons of addton and multplcaton and the laws govenng these opeatons. Ths algebac system, known as ng, s commonly denoted by (R,+,x). The basc computatons occung n mophologcal netwoks ae based on the algebac lattce stuctue ( R,,, ), whee, denote the bnay opeatons fo mnmum and maxmum, espectvely. The basc axomatc opeatons of lattce algeba [14] ae: ( ) ( R R 0 R 0 R An assocatve memoy (AM) s an nput-output system that descbes a elaton R R m R n. If (x, y)r,.e. f the nput x poduces the output y, then the assocatve memoy s sad to stoe o ecod the memoy assocaton (x, y). NN models sevng as assocatve memoes ae geneally capable of etevng a complete output patten y even f the nput patten x s coupted o ncomplete. The pupose of auto-assocatve memoes s the eteval of x fom coupted o ncomplete vesons x. If an atfcal assocatve memoy stoes assocatons (x, y), whee x cannot be vewed as a coupted o ncomplete veson of y, then we speak about heteo-assocatve memoy. Reseach on neual assocatve memoes goes back n the 1950s [15-17]. Lnea assocatve memoy o coelaton memoy [17-19] s one well known neual assocatve memoy. Assocaton of pattens x R n and y R m s acheved by means of a matx-vecto poduct y Wx. If we suppose that the goal s to stoe k vecto pas (x 1, y 1 ),...,(x k, y k ), whee x R n and y R m fo all = 1,...,k, then W s an m n matx gven by the followng oute poduct ule: k 1 (1.1) W y ( x ) (1.2) If X denotes the matx whose column ae the nput pattens x 1,...,x k and Y denotes the matx whose columns ae the output pattens y 1,...,y k, then ths equaton can be wtten n the smple fom Y X'. If the nput pattens x 1,...,x k ae othonomal, then 1 1 k k Wx (( y x )... ( y x ) ) x y (1.3) Thus, we have pefect ecall of the output pattens y 1,...,y k. In case the pattens ae not othonomal the capacty of the memoy s extemely lmted and ts ablty to eteve assocatons s futhe educed when nput pattens ae nosy []. Mophologcal assocatve memoes (MAMs) ae based on the algebac lattce stuctue ( R,,, ). They have been ntally poposed n [14], [21], [22] fo assocatng bnay patten vectos and ae fa moe obust to nose 142 ComSIS Vol. 8, No. 1, Januay 11

3 A new method fo constuctng kenel vectos n mophologcal assocatve memoes of bnay than the conventonal lnea assocatve memoes. Moeove, the memoy capacty n stoed assocatons s sgnfcantly lage than the capacty of conventonal lnea assocatve memoes. These popetes and the fact that a numbe of notable featues such as optmal absolute stoage capacty and one-step convegence have been shown to hold n the geneal case fo ealvalued pattens [14], motvated fo the extenson to gay-coded vecto assocatons [23] o ecently n geneal gay-scale MAMs and fuzzy oented teatments [24], [25],[26-28]. Recent woks epot also on ecall and stoage pefomance of vaous types of MAMs n colo pattens unde vaous types and pecentages of nose [29], [], whle othe woks pesent new dynamc MAMs and test the pefomance n ecallng gay and colo patten assocatons [31], [32], [33]. Accodng to [14], thee ae two altenatve appoaches to constuct MAMs. In the fst appoach the constucted memoy s vey obust n the pesence of nput pattens contanng eosve nose. In the dual appoach the constucted memoy s vey obust n the pesence of nput pattens contanng dlatve nose. In case the nput patten x s coupted by both eosve and dlatve nose none of the memoes s able to ecall y. To ovecome ths poblem Rtte and hs cowokes [34-36] poposed the dea of two-step MAMs and the poducton of the so-called kenel vectos, as an elegant epesentaton of the assocatons (x, y ). These vectos, n conuncton wth appopate mophologcal opeatons, ae sutable to ecall y when an abtaly coupted (both eoded and dlated) nput patten x appeas at the nput of the MAM. A bnay vecto z s sad to be a kenel of the assocaton (x, y) between the bnay vectos x and y f t s a subset of x and satsfes some condtons gven n [34], [35]. Moe ecent developments on econstuctng pattens fom nosy nputs extend the defnton of kenels and assocate them wth the popety of stong mophologcal ndependence whch should be pesent n the stoed pattens [37-]. Thee ae two mao ssues elated to the kenel method. The fst s the vey selecton of epesentatve kenel vectos. The condtons fo a vecto z to be a kenel vecto, gven n [34], [35], they do not povde wth a fast method fo selectng such vectos. As t wll be ponted out n secton 3 the pocedue of selectng kenel vectos s qute tme consumng, especally when the patten vectos ae of lage dmenson. Addtonal condtons fo kenels and selecton pocedues, whch allevate n some extent ths poblem, ae poposed n [35],[41-43] whle n [37-] the pocedue of selectng the kenels emeges though the steps of the poof of the elevant theoems, povded that the pattens ae ntally modfed to meet the popety of stong mophologcal ndependence. The second mao ssue, whch s of ths pape concen, s elated to the behavo of kenel method n the pesence of nosy nput pattens. Although ths ssue has eceved only mno attenton n the elevant lteatue [43], the kenel method s not entely obust to nose. It can be expementally vefed that, although kenel vectos ae qute satsfactoy n etevng assocatons based on nosy (both dlated and eoded) data, n some cases, they entely fal even n the pesence of vey small nose pecentages. Ths ComSIS Vol. 8, No. 1, Januay

4 Yanns.S. Boutals happens when one o moe nonzeo element of the nput patten, whch coespond to espectve nonzeo elements of the kenel vecto ae ht by eosve nose, whch convets them to zeo-valued elements. In ths case, the eteval pocedue ecalls nothng, even f the amount of nose s qute lmted. In ths pape, a theoetcal poof of ths obsevaton s establshed and the mechansm of poducng ths falue s explaned. Next, a new kenel defnton method s poposed whch ovecomes ths poblem and s moe obust to such knd of nose than the tadtonal kenel vectos. The new kenels ae not anymoe bnay but they contan elements of vaable values n the ange [0, 1]. The values of the elements of each selected kenel ae elated to the fequency of appeaance of the coespondng elements of the nput tanng pattens. Dffeent altenatves of selectng kenels based on fequences ae poposed and ae compaed n espect to the obustness n nose. All fequency based kenel vectos no longe suffe fom the patcula nose defcency of conventonal kenels. Moeove, the pocedue of selectng the kenel vectos s qute smple and well defned; theefoe t s not tme consumng. The development s gven fo an auto-assocatve scheme, but heteo-assocaton can be tackled wth the same pocedue. The patten assocaton ecall s pefomed n thee steps. The fst step s the (nput patten -> kenel) and (kenel -> output patten) ecall. The esult s an output patten, whch n geneal s not bnay but gay and keeps n a lage extend the shape of the output patten to be ecalled. Theefoe a smple thesholdng acton s the second step and may poduce the coespondng bnay patten. The thd step nceases the pefomance of the poposed method by passng the ecalled bnay patten though a smple Hammng netwok equpped wth ndex selecton. The pefomance of the new kenel n bnay patten ecall s demonstated by usng extensve chaacte ecognton expements unde vaous types and pecentages of nose. The pape s oganzed as follows. Fst, n secton 2, a bef but complete ntoducton to the mophologcal assocatve memoes s gven. The kenel method s also epoted n the same secton and ts weaknesses ae demonstated. In the sequence, n secton 3, the theoetcal ustfcaton of the falue of the kenel method n the pesence of the patcula knd of eosve nose s gven. In secton 4, the new kenel defnton method s ntoduced and ts valdty and nose obustness s theoetcally establshed. Secton 5 povdes expemental esults egadng the capacty and nose obustness of the poposed kenels, pesentng also compasons between the altenatve vesons of the fequency based kenels. Conclusons and dscusson ae fnally gven n secton ComSIS Vol. 8, No. 1, Januay 11

5 A new method fo constuctng kenel vectos n mophologcal assocatve memoes of bnay 2. Mophologcal assocatve memoes and the kenel method Thee ae two basc appoaches to ecod k vecto pas (x 1, y 1 ),...,(x k, y k ) usng a mophologcal assocatve memoy (MAM) [14]. The fst appoach conssts of stuctung an m n matx W XY wth elements computed by k 1 w ( y x ), 1,..., m 1,..., n (2.1) The dual appoach conssts of constuctng an m n matx M XY wth elements computed by k 1 m ( y x ), 1,..., m 1,..., n (2.2) If matx W XY eceves a vecto x as nput, the poduct y WXY x s fomed. The poduct s called max poduct and each element of the esultng vecto y s computed by the fomula n ( y ), 1,..., w x m (2.3) 1 Lkewse, f matx M XY eceves x as nput the so-called mn poduct y MXY x s fomed, whee each element of the esultng vecto y s computed by the fomula. n ( y ), 1,..., m x m (2.4) 1 Matces W XY and M XY, computed by (2.1) and (2.2) espectvely, consttute the memoy of the MAM. The only equed tanng of the netwok s smply the computaton of ethe of the two matces W o M. The dffeence between the two memoes ases when nosy pattens appea at the nput. Memoy M XY s able to eteve y n case a nosy vecto x coupted by dlatve nose appeas at ts nput, whle memoy W XY s able to eteve y n case an eoded veson of the nput patten appeas at ts nput. The natue of dlatve and eosve nose n bnay pattens, as well as the eteval esults of M XY and W XY espectvely s depcted n Fg. 2.1 and 2.2. In case the nput patten x s coupted by both eosve and dlatve nose none of the memoes s able to ecall y. Othe types of memoes have also been poposed n the ecent yeas. The medan opeato (nstead of mn o max) s ntally poposed n [31] to poduce a memoy that s obust n stctly mxed type of nose, whch pesents pefect ecall f ths nose s of medan zeo and f the fundamental (uncoupted) patten set fulflls some stct condtons. To ovecome these estctons the authos n [31] popose the constucton of the memoy usng specally constucted suogates of the fundamental patten set and an algothm fo the tanng and ecall phase usng these suogates. The dea of usng suogates appeas also n [32], [33], whee a dynamc type of memoy s pesented and the md opeato s employed. Although successful n stong and ecallng geneal patten assocatons, dynamc assocatve memoes ae ComSIS Vol. 8, No. 1, Januay

6 Yanns.S. Boutals not appopate fo bnay patten assocatons, because an essental pat of the algothm, namely the detemnaton of actve egon [32], fals when bnay pattens ae nvolved. M XY W XY Fg Patten ecall of a dlated nput usng the mn poduct and memoy matx M. Fg Patten ecall of an eoded nput usng the max poduct and memoy matx W. Anothe type of memoes fo bnay patten assocatons follows fom fuzzy oented teatments [25-28]. A new set theoetc ntepetaton of ecodng and ecall of bnay AMM s gven n [25-27] and a genealzaton s povded usng fuzzy set theoy. Ths esults n the defnton of the so-called fuzzy mophologcal assocatve memoes and the use of the fuzzy max poduct of W and x and the fuzzy mn poduct of M and x. When a coupted bnay patten appeas as nput to memoy W o M the outcome s a gaylevel patten. The espectve bnay ecall s obtaned by usng appopate nomalzed theshold, whch howeve has to be dffeent dependng on the type of memoy used. An teatve algothm s also poposed n [25] fo the detemnaton of the appopate theshold. The ecall scoes ae futhe mpoved n [28], whee an enhanced fuzzy autoassocatve mophologcal memoy s pesented and s combned wth a dscete Hammng neual netwok that nceases the accuacy of the ecall. Expements wth pattens coupted wth vaous nose levels ae pesented, howeve the patten set used s vey lmted and s not suffcent fo dawng clea conclusons. A well establshed appoach to ovecome the poblem of the complete falue of memoes W (eq. 2.1) and M (eq. 2.2) n dlatve and eosve nose espectvely s the kenel method. To ovecome the poblem Sussne, Rtte and cowokes [34], [35] poposed the dea of two-step MAMs and the poducton of the so-called kenel vectos, as an elegant epesentaton of the assocatons (x, y ). These vectos, n conuncton wth matces M and W, ae sutable to ecall y when an abtaly coupted (both eoded and dlated) nput patten x appeas at the nput of the MAM. A bnay vecto z s sad to be a kenel of the assocaton (x, y) between the vectos x and y f t s a subset of x and satsfes the followng condtons. M x z zz (2.5) W z y, ZY (2.6) whee matces M ZZ and W ZY ae computed accodng to (2.1), (2.2). In case of autoassocaton (2.5) and (2.6) ae wtten as M x z ZZ (2.7) and W z x (2.8) ZX 146 ComSIS Vol. 8, No. 1, Januay 11

7 A new method fo constuctng kenel vectos n mophologcal assocatve memoes of bnay espectvely. Fg. 2.3 shows a sample of bnay pattens (lettes) and the coespondng kenel pattens, whch satsfy equatons (2.7) and (2.8). When an abtaly coupted nput patten x appeas at the nput of a mophologcal autoassocatve netwok, the ognal uncoupted nput patten x s ecalled by the followng equaton: WZX MZZ x x (2.9) Moe ecent developments on econstuctng pattens fom nosy nputs extend the defnton of kenels and assocate them wth the popety of stong mophologcal ndependence whch should be pesent n the stoed pattens [37-]. A pocedue fo selectng such kenels s poposed n [37], whch also nvolves a mechansm of alteng the ntal pattens so that they fulfll the popety of stong mophologcal ndependence. Fg Input pattens and undeneath the espectve kenels Thee ae two mao ssues elated to the tadtonal kenel method. The fst s the vey selecton of epesentatve kenel vectos. Equatons (2.5), (2.6) (o (2.7), (2.8) fo autoassocaton) povde the necessay and suffcent condtons fo a vecto z to be a kenel vecto but they do not povde wth a fast method fo selectng such vectos. An appaent pocedue s the andom selecton of kenel vectos and the acceptance o eecton on the bass of (2.5) and (2.6). Ths pocedue s qute tme consumng, especally when the patten vectos ae of lage dmenson. Addtonal condtons fo kenels and selecton pocedues ae poposed n [35],[41-43]. The second mao ssue, whch s of ths pape concen, s elated to the behavo of kenel method n the pesence of nosy nput pattens. Although ths ssue has eceved only mno attenton n the elevant lteatue [43], the kenel method s not entely obust to nose. It can be expementally vefed that, although kenel vectos ae qute satsfactoy n etevng bnay pattens assocatons based on nosy (both dlated and eoded) data, n some cases of eosve nose, they entely fal even n the pesence of vey small nose pecentages. Ths happens when one o moe nonzeo elements of the nput patten, whch coespond to espectve nonzeo elements of the kenel vecto ae ht by nose, whch convets them to zeo-valued elements. In ths case, the eteval equaton (2.9) ecalls nothng. A theoetcal explanaton of ths fact s gven n the followng secton. Closng ths secton, t s woth notng that even n the dynamc MAM pesented n [32], [33], thee s a pat of the memoy, namely a one column vecto of the espectve matx of the memoy, that the authos call kenel of the assocatve memoy. The elements of such a kenel play an mpotant ComSIS Vol. 8, No. 1, Januay

8 Yanns.S. Boutals ole n ecallng pattens alteed by some knd of nose. The ecall pocedue wll fal f kenel changes, leadng to loss of memoy [32], [33]. Howeve, n ths pape, we wll not futhe efe to ths type of memoes, because dynamc MAM s not appopate fo stong and ecallng bnay patten assocatons. Ths s due to the falue of detemnng the so-called actve egon [32] when bnay pattens ae nvolved. Anothe eason s that, wth bnay pattens, the low accuacy condton (see [32], secton 4.2) s vey often actvated. 3. Nose defcency of the kenel method The basc dea of kenel method s to constuct a vecto (the kenel vecto), whch s a subset, o othewse a spase veson, of the bnay nput vecto t epesents. Actually, the stcte the subsethood t s the bette the pefomance of the method s. Two vey useful condtons fo bnay kenels ae poposed n [34] and n [35]. They ae descbed by the followng equatons z x (3.1) z z 0 (3.2) Equaton (3.1) expesses the condton of subsethood descbed above, whle equaton (3.2) expesses the demand that dffeent kenels should not have common nonzeo ponts. A pctoal ntepetaton of the subsethood condton can be dawn fom Fg Each bnay lette of dmenson (x) can be scanned ow by ow and be epesented by a (0x1) vecto contanng only the values 0 and 1. Hee, 1 epesents the black pxel and 0 epesents the whte pxel. Smlaly, the coespondng kenel pattens can be epesented by the (0x1) kenel vectos. The 1s n the kenel vectos ae much fewe than the 1s of the coespondng pattens and ths s a dect ntepetaton of (3.1). Moeove, the kenel pattens do not have common non-zeo elements and theefoe condton (3.2) s fulflled. Snce each kenel vecto satsfes (3.1), each nput vecto x can be consdeed as a dlated veson of ts coespondng kenel vecto z. Theefoe, snce memoy M ZZ s vey capable n etevng pattens coupted by dlatve nose t can ecall z by usng equaton (2.5) (o (2.7)) and the vecto x as ts nput. In the sequel, each ecalled kenel vecto z can be used as the nput to the memoy W ZX o W ZY to eteve x o y by usng equaton (2.6) o (2.8) espectvely. The combnaton of these two steps s expessed n equaton (2.9), In case an abtaly coupted veson x of the nput patten s consdeed, equaton (2.9) can stll povde easonably good esults, povded that x can be consdeed as a dlated veson of z. If, howeve, x can be consdeed as a coupted veson z, whch contans even the slghtest amount of eosve nose, the ecall pocedue entely fals and equaton (2.5) and consequently (2.9) ecalls nothng. The theoems that follow theoetcally explan ths stuaton. A pctoal epesentaton of the 148 ComSIS Vol. 8, No. 1, Januay 11

9 A new method fo constuctng kenel vectos n mophologcal assocatve memoes of bnay stuaton s also gven n Fg It can be obseved that f the nput patten s ht by eosve nose, whch affects even one pxel that coesponds to a nonzeo element of the coespondng kenel patten, then the ecall pocedue completely fals. Fo notatonal convenence, we call as kenel pont any element of the kenel vecto whch has nonzeo value (value 1). We also ecall that, kenel vectos satsfy (3.1) and (3.2). Theoem 3.1. The ows of Mzz ae vectos wth values 1. All zeos ( m 0, ) f the ow ndex () does not coespond to any kenel pont of any kenel vecto. 2. All ones ( m 1, ) f the ow ndex () coesponds to a kenel pont belongng to any kenel vecto z, except when the column ndex () coesponds to anothe kenel pont of the same kenel vecto z. In ths case m 0 The poof of Theoem 3.1 s gven n the Appendx. A pctoal epesentaton of Mzz fo the kenels of Fg. 2.3 s gven n Fg. 3.2, whee the values of Mzz ae dsplayed n the fom of a bnay mage (0 = black, 1= whte). In ths example, Mzz s of dmenson (0x0). Theoem 3.2. If thee s at least one ndex, such that x 0 and z 1, then equaton (2.5) ecalls nothng. That s M x 0 ZZ (3.3) and consequently WZX MZZ x 0 (3.4) The poof of Theoem 3.2 s gven n the Appendx. It explans why thee s such a complete falue n the pesence of ths patcula knd of eosve nose, even f the nose pecentage s vey low. 4. The new kenel defnton method In the sequel we popose a new method fo ecallng assocatons between bnay pattens, by constuctng kenel vectos whch does not pesent the nose defcency of the conventonal kenel vectos descbed n the pevous secton. Moeove, the pocedue of constuctng the kenels eques only modeate pocessng tme and theefoe s computatonally much moe effcent than the conventonal kenel method. In ou appoach kenel vectos ae not bnay but the elements ae allowed to take values n the nteval [0, 1]. Each kenel vecto z s constucted to be a subset of an nput vecto x (see equaton (3.1)), but the subsethood s defned n a fuzzy-lke manne. ComSIS Vol. 8, No. 1, Januay

10 Yanns.S. Boutals Ognal Kenel Coupted Recall Fg Falue of the ecall pocedue when even a slght amount of eosve nose affects pxels, whch coespond to nonzeo kenel elements. Fg Pctoal epe-sentaton of the values of Mzz. A black pxel co-esponds to m 0, whle a whte pxel coesponds to m 1. Defnton 4.1. The new kenel vectos ae fomed based on the followng condtons. 1. Kenel vectos contan values that ae n the nteval [0,1]. 2. The values of the elements of each kenel vecto ae zeo when the coespondng values of the nput vecto ae zeo and nonzeo when the coespondng nput vecto values ae nonzeo. 3. The nonzeo values of each kenel vecto ae fomed based on the fequency, f, of appeaance of the coespondng nonzeo nput vecto element n all the nput vectos accodng to the followng method 1: 4. z 1 f f 1, z 0.8 f f 2, z 0.7 f f 3, z 0.4 f f 4 We call the non-zeo elements of a kenel vecto kenel ponts. Each kenel pont s assocated wth ts stength. The kenel ponts wth z 1 ae the stongest kenel ponts, whle kenel ponts wth z 0.8, 0.7, 0.4 ae weake kenel ponts. In usng the fequences to constuct kenels thee ae moe than one altenatve. Two such altenatves, temed method 2 and method 3, ae pesented late on n ths secton. Howeve, method 1 s enough to demonstate the dea and daw the elevant conclusons. Fg. 4.1 shows examples of kenel vectos constucted accodng to method 1 and assocated wth the nput pattens (lettes) of Fg Dffeent gay levels epesent the kenel ponts of dffeent stength wth the stonge kenel ponts coespondng to the bghte ntensty. The algothm fo constuctng kenel vectos s qute smple and obvous and theefoe ts computatonal buden s qute low n compason wth the conventonal kenel method. Indeed, to constuct a kenel vecto z that coesponds to an nput vecto x we poceed as follows: If the value of the element of the nput vecto s 0 the coespondng element of the kenel vecto s set to 0. Fo each nonzeo element of the nput vecto we count the nonzeo occuences of the same element n the othe nput vecto. We call ths fequency of 150 ComSIS Vol. 8, No. 1, Januay 11

11 A new method fo constuctng kenel vectos n mophologcal assocatve memoes of bnay appeaance. Next we assgn a nonzeo value to the coespondng element of the kenel vecto accodng to the counted fequency and accodng to method 1of defnton 4.1. The same pocedue s appled fo the constucton of all the kenel vectos Patten Kenel 50 Patten Kenel Patten Kenel Patten Kenel Fg Pctoal epesentaton of pattens and the coespondng kenel pattens constucted usng the poposed new method. The bghte pxels coespond to stonge kenel ponts. Each new kenel vecto s a subset of ts coespondng nput vecto accodng to equaton (3.1), but the subsethood s now defned n a fuzzy lke manne snce fo each element of the kenel vecto the followng elaton holds z x, (4.1) The new kenel defnton comes as an ntutve extenson of conventonal bnay kenels. In meetng (3.1) and (3.2), denotng subsethood and unqueness, the esultng bnay kenels ae spase vectos, cayng all nfomaton n the few nonzeo values. All these nonzeo values ae cucal and, as ponted out n secton 3, even f one of them s mssng (has zeo value) n the nput patten the whole ecallng pocedue collapses. In the new defnton, the gay kenel vectos ae allowed to nclude also othe suppotng non-zeo values, assocated wth non-zeo patten elements appeang n moe than one patten. Due to the defnton of the new kenel vectos, whch s based on the fequences of the elements of the coespondng nput vectos, the fom of memoy matx Mzz s now completely dffeent than the coespondng matx poduced by conventonal kenels. The new fom of Mzz allows the ecallng pocedue to be successful even f the nput vecto s coupted by the specal eosve nose descbed n secton 3. The patcula method 1 fo selectng the gay values {1, 0.8, 0.7, 0.4} to be attbuted n kenel ponts assocated wth the espectve patten non-zeo element fequences {1, 2, 3, 4}, came out of autho s obsevatons and s suffcent fo demonstatng the dea and dawng the conclusons. As t s poved n the next theoems, the poposed method 1 seves the pupose of emovng the patcula nose defcency of tadtonal kenels. Othe fequency based methods may also be vald. Two such altenatve methods (method 2 and method 3) ae pesented at the end of ths secton, whch cay the good popety of method 1, and pefom bette n lage patten sets and ae moe obust n mxed nose. Theoem 4.1. The ows of Mzz ae vectos wth values All zeos ( m 0, ) f the ow ndex () does not coespond to any kenel pont of any kenel vecto (fequency 0). ComSIS Vol. 8, No. 1, Januay

12 Yanns.S. Boutals m I 1 1, 0.6, 0.3, 0.2, 0 f the ow ndex coesponds to kenel value z 1 (fequency 1). m I 2 0.8, 0.4, 0.1, 0 f the ow ndex coesponds to kenel value z 0.8 (fequency 2). m I 3 0.7, 0.3, 0 f the ow ndex coesponds to kenel value z 0.7 (fequency 3) m I 4 0.4, 0 f the ow ndex coesponds to kenel value z 0.4 (fequency 4 ). The poof of Theoem 4.1 s gven n the Appendx. Theoem 4.2. If thee ae one o moe ndces, such that x 0 and z 0, then equaton (2.5) ecalls the kenel vecto o an eoded veson of t. That s M x z ZZ (4.2) and consequently WZX MZZ x x (4.3) The poof of Theoem 4.2 s gven n the Appendx. Fg. 4.2 shows examples of pefect ecall usng the new kenel vectos constucted by method 1 and as nput vectos uncoupted pattens. Fgue 4.3 shows examples of pefect ecall usng the new kenel vectos and as nput vectos pattens coupted by the eosve nose descbed n secton 3. (a) (b) (a) (b) (c) (d) (c) (d) Fg Examples of pefect ecall usng the new kenel vectos and nput vectos the uncoupted pattens: (a) the nput, uncoupted vecto (b) the ognal kenel vecto (c) pefect ecall of the kenel vecto by (2.5) (d) pefect ecall of the patten by (2.9) (a) (b) (a) (b) (c) (d) (c) (d) Fg Examples of pefect ecall usng the new kenel vectos and nput vectos coupted by the eosve nose descbed n secton 3. (a) the coupted vecto used as nput (b) the ognal kenel vecto (c) non-pefect ecall of the kenel vecto by (2.5) (d) pefect ecall of the patten by (2.9) 152 ComSIS Vol. 8, No. 1, Januay 11

13 A new method fo constuctng kenel vectos n mophologcal assocatve memoes of bnay It has to be noted that tem M x n (4.3) wll poduce z, whch can ZZ be consdeed as an eoded veson of x. Memoy WZX s obust n eosve nose but ts obustness depends also on the amount of the eoson. Theefoe, wth the excepton of expements nvolvng small patten sets wth pobably small pecentages of mxed nose, eq. (4.3) wll not poduce a bnay patten but a gay suogate of t, havng a shape qute close to t. A bnay veson of the ecall can be obtaned afte a smple thesholdng acton. Smla to [28], the ecallng esults can be futhe mpoved f the bnay ecall passes though a smple Hammng netwok to poduce an ndex pontng to a patten of the ntal uncoupted patten base. The oveall ecallng scheme s shown n Fg. 4.4 and ts pefomance s demonstated by the expements appeang n the next secton. k x ZX W M x ZZ k x k gay Smple Thesholdng k x bnay Hammng & Index ecall k Fg The thee step ecall pocedue. The coupted bnay patten x passes k though eq. (4.3) to poduce the gay ecall x, whch afte smple thesholdng wll poduce a pobably coupted bnay patten ecall. The thd step wll poduce an ndex to be used fo etevng the uncoupted bnay patten x y. If k we have pefect ecall. In defnton 4.1 the values of the elements of kenel vectos ae detemned by the fequency of appeaance of the coespondng element n the pattens. Howeve, n method 1 only 4 dffeent values ae employed, as all elements havng fequency geate than 4 eceve the same value z 0.4. Two altenatve fequency based methods ae pesented below. Method 2: A plausble extenson of method 1 s to allow fo a fne detemnaton of kenel pont values takng nto account all avalable fequences of appeaance. Snce the maxmum fequency of a non-zeo pxel cannot be geate than the numbe of avalable pattens (nop) the kenel elements eceve the followng gadng f z 1 f f 1, z 1 f f 2 nop Regadng the specal nose defcency of kenels, method 2 caes the same good popetes of method 1, because theoems 4.1 and 4.2 can be easly extended to cove ths method by smply nceasng the numbe of dffeent cases examned to at most nop cases. In ths appoach the sets I n Theoem 4.1 now contan moe and dffeent values than those of method 1, but the atonale of the development and the usage n Theoem 4.2 eman the same. Regadng nose obustness, n the expements caed out n the next secton, method 2 poves to be moe obust than method 1. gay x ComSIS Vol. 8, No. 1, Januay

14 Yanns.S. Boutals Method 3: A slghtly dffeent appoach fo constuctng the gay kenels s the fequency dependent two-step kenel constucton method, whch s descbed hee n detal. In the fst step fequency pattens ae ceated. Then the kenel vectos ae detemned by gvng values to the kenel ponts accodng to the stength of common fequences n the fequency pattens. Moe specfcally, Step 1. Fo each ntal patten x a coespondng fequency patten p s constucted. The elements of the fequency patten ae constucted accodng to the followng ule p 0 f x 0, p f f x 0 Whee, as usual, f s the fequency of appeaance of the non-zeo th element n all pattens. Appaently the mnmum value f can take s 1. Step 2. Based on the set of fequency pattens, the set of common fequences F s detemned. That s, F contans all the dstnct fequences that appea n all the fequency pattens. Appaently, the cadnalty cf of F s at most equal to the numbe of pattens nop; usually t s smalle. Let also fmn mn f F and fmax max f F denote the mnmum and maxmum of the fequences patcpatng n F. Next, fo each patten, ts kenel vecto z s constucted accodng to the followng ule. z 0 f p 0, o z 1 f p f, mn p z F p 1 f max f f mn mn That s, the nonzeo elements of each kenel vecto coespond to elements of the fequency pattens havng fequences that appea n all othe fequency pattens. The concept of stong and weake kenel ponts eman, because the values of the kenel ponts ae detemned by the coespondng fequences, wth the lagest value (=1) taken by elements that coespond to the mnmum common fequency and the othe values (<1) ae gadually educng accodng to the ncease of the coespondng common fequency. Regadng the specal nose defcency of kenels, method 3 caes the same good popetes of method 1 and method 2, because the constucton of Mzz s analogous to that of the othe methods. Theoems 4.1 and 4.2 can be easly extended to cove ths method by smply consdeng dffeent numbe of cases examned (the numbe of cases s cf 1, countng also the m 0 case). The sets I n Theoem 4.1 now contan dffeent values than those of the othe methods, but the atonale of the development and the usage n Theoem 4.2 eman the same. Regadng nose obustness, n the expements caed out n the next secton, method 3 poves to be much moe obust than the othe two methods. f p f mn 154 ComSIS Vol. 8, No. 1, Januay 11

15 A new method fo constuctng kenel vectos n mophologcal assocatve memoes of bnay 5. Expemental esults To test the pefomance of the poposed method, a numbe of expements wee caed out. In ode to have a meanngfully szed data set, 26 uncoupted bnay pattens wee ntally ceated. The set contans the 35x34 bnay mages of the captal lettes of the Latn alphabet and s shown n Fg Some examples of gay kenels poduced by method 1, 2 and 3 espectvely ae shown n Fg Fg The complete patten set consstng of 35x34 bnay mages of the 26 captal lettes of Englsh alphabet. Fg Sample of kenel vectos poduced by the thee fequency based methods. (a) Kenels of method 1 (b) Kenels of method 2 (c) Kenels of method 3. All methods (1, 2 &3) of constuctng kenels wee ntally tested on the ecall pefomance of uncoupted pattens, when the thee step pocedue of Fg. 4.4 was appled. All methods had a 0% success n ecallng the uncoupted patten. Fg. 5.3 shows thee examples (one fo each method) of the oveall pocedue. The patten nteacts wth memoy M accodng to (4.2) to poduce z. Then, z nteacts wth memoy W accodng to (4.3) to poduce a gay patten x havng the shape of x. Next, the gay outcome passes fom a smple thesholdng scheme to poduce a bnay patten. In ths smple thesholdng scheme evey gay value geate than zeo eceves value 1. In the sequel, the bnay ecall may pass though a Hammng netwok to ecall an ndex pontng to a membe of the ntal patten set. Fnally, the uncoupted patten assocated wth ths ndex appeas as the fnal ecall of ths pocedue. It has to be noted that, n the expements caed out, we dd not use a Hammng netwok. Instead the Hammng dstances n 1 H x x 1,, k (5.1) ComSIS Vol. 8, No. 1, Januay

16 Yanns.S. Boutals between the bnay ecall and each one of the pattens of the ntal patten space wee computed and the ndex of the patten gvng the smallest dstance was selected. The patten ecalled s x, whee ag mn H, 1,, k (5.2) In (5.1), n denotes the sze (n pxels) of a patten, denotes the th element of the patten vecto and k s the numbe of pattens n the patten space. Fg Steps of uncoupted patten ecall accodng to the pocedue of Fg. 4.4 usng the thee fequency based kenel methods. (a) ecall usng method 1, (b) ecall usng method 2, (c) ecall usng method 3. Fom left to ght: 1 st column - nput patten, 2 nd column - kenel ecall usng (4.2), 3 d column - patten ecall usng (4.3), 4 th column bnazaton of column 3 ecall usng smple thesholdng, 5 th column fnal ecall afte computng the mnmum Hammng dstance. The use of (5.1) and (5.2) povdes also a means fo ndectly evaluatng the stoage capacty of the oveall scheme. Statng fom a small szed patten space (k = 5) and gadually nceasng the sze by 2 untl k = 26, the aveage mnmum dstance of all patten ecalls at each patten space fo each method s ecoded. Fgue 5.4 shows the plot of the aveage mnmum dstance (AMD) of ecalls n espect to the sze of the patten space. It s evdent that the AMD nceases n the begnnng but t appoaches an uppe bound as the numbe of pattens nceases. That s, afte a cetan numbe of pattens the AMD and ndectly the stoage capacty of the scheme s not affected by the sze of the patten space. Moeove, Fg. 5.4 demonstates the supeoty of method 3, snce t povdes always moe confdent esults (smalle AMD). The next set of expements s concened wth the obustness of all methods unde the pesence of vaous pecentages of mxed type (both dlatve and eosve) nose. Mxed nose was andomly appled on each bnay patten. The nose pecentage s computed by countng the numbe of the alteed (nosy) pxels and dvde them wth the total numbe of mage pxels. Fgue 5.5 shows thee examples (one fo each method) of the oveall ecall pocedue, when a coupted by % mxed nose patten appeas n the nput of the poposed 3-step ecall scheme. 156 ComSIS Vol. 8, No. 1, Januay 11

17 Aveage mnmum Hammng dstance A new method fo constuctng kenel vectos n mophologcal assocatve memoes of bnay method 1 method 2 method Numbe of pattens Fg Aveage mnmum Hammng dstance fo the thee fequency based kenel methods n espect to the sze of the patten space. Fg Steps of successful patten ecall of a coupted patten accodng to the pocedue of Fg. 4.4 usng the thee fequency based kenel methods. (a) ecall usng method 1, (b) ecall usng method 2, (c) ecall usng method 3. Fom left to ght: 1 st column coupted nput patten, 2 nd column - kenel ecall usng (4.2), 3 d column - patten ecall usng (4.3), 4 th column bnazaton of column 3 ecall usng smple thesholdng, 5 th column fnal ecall afte computng the mnmum Hammng dstance. The vaous ecall stages ae smla wth those appeang n Fg. 5.3, howeve the effect of nose s now evdent n the ntemedate esults. It s also clea that method 2 s moe obust to nose than method 1, because the ntemedate esults ae close to the actual ones. Smlaly, method 3 pefoms much bette than the othe two. Fnally, Fg. 5.6 depcts the obustness of each method n mxed nose. The falue ecall ate of each method s dsplayed n espect to the pecentage of the mxed nose. Snce the nose s andomly appled, n ode to have statstcally elable esults, each expement coespondng to a dffeent nose pecentage was caed out 50 tmes and the aveage falue ecall ate s actually ecoded and dsplayed. It s evdent that method 3 s fa moe obust than the othe two methods and method 2 s sgnfcantly moe obust than method 1. Theefoe, although all fequency based kenel methods do not pesent the patcula nose defcency of tadtonal kenel methods, they pesent sgnfcant dffeences egadng the nose obustness. Ths n tun may pompt fo futue seekng of altenatve ComSIS Vol. 8, No. 1, Januay

18 Falue ate (%) Yanns.S. Boutals fequency based kenel constucton methods wth bette pefomance n ecall of nosy pattens. 90 method 1 80 method 2 method Mxed nose (%) Fg Falue ecall ates of the thee fequency based kenel vecto methods n espect to mxed nose pecentage. 6. Concluson A new method fo constuctng kenel vectos was poposed n ths pape to be used fo ecallng assocatons between bnay pattens. The need fo a new kenel defnton aose fom a specal nose defcency of the conventonal bnay kenel vectos of the elevant lteatue. The new kenels ae not bnay but gay, because they contan elements wth values n the nteval [0, 1]. These values ae detemned by the fequency of appeaance of nonzeo elements of the nput patten vectos. Altenatve schemes of poducng kenels based on fequency wee pesented and t was shown, both theoetcally and by chaacte ecall examples that the new kenel vectos cay the good popetes of conventonal kenel vectos and, at the same tme, they can be easly computed. Moeove, they do not suffe fom the patcula nose defcency of the conventonal kenel vectos. Expements, pefomed on lage set of bnay pattens coupted by vaous degees of mxed nose demonstate the obustness and the lmts of the poposed appoach. Futue extenson of ths wok mght extend the poposed appoach to cove moe fequency dependent kenel altenatves as well as an nvestgaton egadng the applcablty of the poposed concepts n gay and possbly colo patten assocatons. Acknowledgement.The autho wants to acknowledge the contbuton of M. S. Chatzchstofs n pefomng the expements caed out. 158 ComSIS Vol. 8, No. 1, Januay 11

19 Refeences A new method fo constuctng kenel vectos n mophologcal assocatve memoes of bnay 1. Pete Sussne and Macos Eduado Valle, Mophologcal and cetan fuzzy mophologcal assocatve memoes fo classfcaton and pedcton. In Vassls Kabulasos and Gehad Rtte, (eds), Computatonal Intellgence Based on Lattce Theoy, chapte 8, Spnge-Velag, Hedelbeg, Gemany, , (07). 2. J.L.Davdson, G.X.Rtte, A theoy of mophologcal neual netwoks, n: Dgtal Optcal Computng II, Vol. 1215, Poc. SPIE, , (July 1990). 3. J.J.Hopfeld, Neual netwoks and physcal systems wth emegent collectve computatonal abltes, Vol. 79, Poc. of the Natonal Academy of Scences, U.S.A., , (Apl 1982). 4. J.J.Hopfeld, Neuons wth gaded esponse have collectve computatonal popetes lke those of two-state neuons, Vol. 81, Poc. of the Natonal Academy of Scences, U.S.A., 88 92, (May 1984). 5. J.L.Davdson, Smulated annealng and mophologcal neual netwoks, n: Image Algeba and Mophologcal Image Pocessng III, Vol. 1769, Poc. SPIE, San Dego, , (1992). 6. J.L.Davdson, F.Humme, Mophology neual netwoks: an ntoducton wth applcatons, Ccuts, Systems and Sgnal Pocessng, Vol. 12, No. 2, 177 2, (1993). 7. C.P.Suaez Aauo, Novel neual netwok models fo computng homothetc nvaances: an mage algeba notaton, Jounal of Mathematcal Imagng and Vson, Vol. 7, No. 1, 69 83, (1997). 8. Y. Won, P. D. Gade, Mophologcal shaed weght neual netwok fo patten classfcaton and automatc taget detecton, In Poc. of the IEEE Intenatonal Confeence on Neual Netwoks, Peth, Austala, , (Novembe 1995). 9. P. D. Gade, M. Khabou, and A. Koldobsky, Mophologcal egulazaton neual netwoks, Patten Recognton., Specal Issue on Mathematcal Mophology and Its Applcatons, Vol. 33, No. 6, , (00).. L. Pessoa and P. Maagos, Neual netwoks wth hybd mophologcal/ ank/lnea nodes: A unfyng famewok wth applcatons to handwtten chaacte ecognton, Patten Recognton, Vol. 33, , (00). 11. Handbook of Neual Computaton, E. Fesle and R. Beale, (Eds)., IOP Publshng and Oxfod Unvesty Pess, pp. C1.8:1 C1.8:14. W. Amstong, M. Thomas, Adaptve logc netwoks, (1997). 12. B. Raducanu, M. Gaña, and X. F. Albzu, Mophologcal scale spaces and assocatve mophologcal memoes: Results on obustness and pactcal applcatons, J. Math. Imagng and Vson, Vol. 19, No. 2, , (03). 13. M. Gaña, J. Gallego, F. J. Toealdea, and A. D Anou, On the applcaton of assocatve mophologcal memoes to hypespectal mage analyss, Lectue Notes n Compute Scence, Vol. 2687, , (03). 14. G.X.Rtte, P.Sussne, J.L.Daz de Leon, Mophologcal assocatve memoes, IEEE Tans. Neual Netwoks, Vol. 9, No. 2,, , (1998). 15. W.K. Taylo. Eletcal smulaton of some nevous system functonal actvtes, Infomaton Theoy, Vol. 3, , ( 1956). 16. K. Stenbuch. De lenmatx. Kybenetck, Vol. 1, 36 45, (1961). 17. J. A. Andeson and E. Rosenfeld, (Eds), Neuocomputng: Foundatons of Reseach, Vol. 1, MIT Pess, Cambdge, MA, (1989). 18. T.Kohonen, Coelaton matx memoy, IEEE Tans. Comput. Vol. 21, , (1972). ComSIS Vol. 8, No. 1, Januay

20 Yanns.S. Boutals 19. James A. Andeson, Andas Pellonsz, and Edwad Rosenfeld, (Eds). Neuocomputng: Dectons fo Reseach, Vol. 2, MIT Pess, Cambdge, MA, (1990).. Toshyuk Tanaka, Shnsuke Kakya, and Yoshyuk Kabashma, Capacty Analyss of Bdectonal Assocatve Memoy, In Poc. Seventh Int. Conf. Neual Infomaton Pocessng, Taeon, Koea, Vol. 2,, , (Nov. 00). 21. G. X. Rtte and P. Sussne. An ntoducton to mophologcal neual netwoks, In Poceedngs of the 13th Intenatonal Confeence on Patten Recognton, Venna, Austa, , (1996). 22. G. X. Rtte and P. Sussne, Mophologcal neual netwoks In Intellgent Systems: A Semotc Pespectve, In Poceedngs of the 1996 Intenatonal Multdscplnay Confeence, Gathesbug, Mayland, , G. Costantn, D. Casal, and R. Pefett, Neual assocatve memoy stong gay-coded gay-scale mages, IEEE Tans. Neual Netwoks, Vol. 14, No. 3, , (03). 24. P. Sussne and M. E. Valle, Gay-Scale Mophologcal Assocatve Memoes, IEEE Tans. Neual Netwoks, Vol. 17, No. 3, , (06). 25. P. Sussne, Genealzng opeatons of bnay mophologcal autoassocatve memoes usng fuzzy set theoy, J. Math. Imagng and Vson, Specal Issue on Mophologcal Neual Netwoks, Vol. 9, No. 2, 81 93, (03). 26. P. Sussne, A elatonshp between bnay autoassocatve mophologcal memoes and fuzzy set theoy, In Poceedngs of the Int. Jont Conf. on Neual Netwoks, Washngton DC, (01). 27. P. Sussne, A Fuzzy autoassocatve mophologcal memoy, In Poceedngs of the Int. Jont Conf. on Neual Netwoks, Potland OR, (03). 28. G. Ucd, J.A. Neves-V., and C.A. Reyes-G, Enhanced fuzzy autoassocatve mophologcal memoy fo bnay patten ecall, In Poceedngs 4th IASTED Int. Conf. on Sgnal Pocessng, Patten Recognton, and Applcatons, Innsbuck, Austa, (07). 29. Robeto A. Vazquez and H. Sossa, Behavo of mophologcal assocatve memoes wth tue-colo mage pattens, Neuocomputng, Vol. 73, N0. 1-3, , (09).. M.E. Valle: A Class of Spasely Connected Autoassocatve Mophologcal Memoes fo Lage Colo Images, IEEE Tansactons on Neual Netwoks, Vol., No. 6, 45 50, (09). 31. H. Sossa, R. Baon and Robeto A. Vazquez, New assocatve memoes to ecall eal-valued pattens, Lectue Notes n Compute Scence, Spnge, Vol. 3287, 195-2, (04). 32. Robeto A. Vazquez and H. Sossa, A new assocatve model wth dynamcal synapses, Neual Pocessng Lettes, Vol. 28, No. 3, 189-7, (08). 33. Robeto A. Vazquez and H. Sossa, A bdectonal heteoassocatve memoy fo tue-colo pattens, Neual Pocessng Lettes, Vol. 28, No. 3, , (08). 34. G.X.Rtte, J.L.Daz de Leon and P.Sussne, Mophologcal bdectonal assocatve memoes, Neual Netwoks, Vol. 12, , (1999). 35. P. Sussne, Obsevatons on mophologcal assocatve memoes and the kenel method, Neuocomputng, Vol. 31, , (00). 36. P. Sussne, Assocatve mophologcal memoes based on vaatons of the kenel and dual kenel methods, Neual Netwoks, Vol. 16, No. 5, , (03). 37. G.X. Rtte, G.Ucd, and L. Iancu, Reconstucton of nosy pattens usng mophologcal assocatve memoes, Jounal of Mathematcal Imagng and Vson, Vol. 19, , (03). 160 ComSIS Vol. 8, No. 1, Januay 11

21 A new method fo constuctng kenel vectos n mophologcal assocatve memoes of bnay 38. [G. Ucd and G.X. Rtte, Kenel computaton n mophologcal assocatve memoes fo gayscale mage ecollecton, In Poceedngs of 5th IASTED Int. Conf. on Sgnal and Image Pocessng, Honolulu, Hawa, (03). 39. G.X. Rtte and P. Gade, Fxed Ponts of Lattce Tansfoms and Lattce Assocatve Memoes, n P. Hawkes (Ed.) Advances n Imagng and Electon Physcs Elseve, Vol. 144, San Dego, Calfona, 06.. G. Ucd, G.X. Rtte, L. Iancu, Kenel computaton n mophologcal bdectonal assocatve memoes, Pogess n Patten Recognton, Speech, and Image Analyss, Spnge LNCS 2905 Poceedngs, , (03). 41. T. Ida, S. Ueda, M. Kashma, T. Fuchda and S. Muashma, On a method to decde kenel pattens of mophologcal assocatve memoy, Systems and Computes n Japan, Vol. 33, No. 7, 85-94, (02). 42. M. Hatto, A. Fuku and H. Ito, A fast method of constuctng kenel pattens fo mophologcal assocatve memoy, In Poceedngs of 9th Intenatonal Confeence on Neual Infomaton Pocessng, 58-63, (Sngapoe 02). 43. H. Hashguch and M. Hatto, Robust mophologcal assocatve memoy usng multple kenel pattens, In Poceedngs th Intenatonal Confeence on Neual Infomaton Pocessng, (03). 7. Appendx (Poof of Theoems) Notaton: In the followng theoems an element of a conventonal bnay kenel vecto s sad to be a kenel pont f ts value s 1. An element of a new, gay kenel vecto s sad to be a kenel pont f ts value s > 0. Poof of Theoem 3.1. To pove the theoem we stat fom the fomula of computng the elements of Mzz. Accodng to equaton (2.2) the elements m of Mzz ae computed by p 1 m ( z z ), 1,..., n 1,..., n (A.1) whee n s the length of kenel vecto z. In geneal the ntenal tem ( z z ) can take the followng values (a) z z 0. Ths happens when fo a specfc both kenel ponts o both ae not kenel ponts (b) z z 1 Ths happens when fo a specfc z s not a kenel ponts and (c) z z 1 Ths happens when fo a specfc z s a kenel pont z and z ae z s a kenel ponts and z s not a kenel pont Takng now nto account that (A.1) s computed by takng the maxmum of all the ntenal tems ( z z ) fo 1 p, we conclude the followng. ComSIS Vol. 8, No. 1, Januay

22 Yanns.S. Boutals (1) If ndex does not coespond to any kenel pont of any kenel vecto then z 0,. Consequently, fo each ( z z ), 1 p takes the values 0 and 1. Theefoe, the maxmum of these values s 0 and theefoe m 0,. (2) In case ndex equals the ndex of a kenel pont n any kenel vecto z, then takng nto account stuaton (b) above, m 1, except when z 1 (that s stuaton (a) above holds) and smultaneously z z 0,, whee m 0. Ths means that when the kenel ponts of a kenel vecto do not concde wth the kenel ponts of anothe kenel vecto (o othewse kenel ponts appea only once) (whch holds tue due to condton (3.2)), then m 0 when both and coespond to kenel ponts. Poof of Theoem 3.2. We take nto account that n computng equaton (2.5), that s, M zz x z, the th element of z s computed usng the elements of the th ow of Mzz accodng to equaton (2.4) as follows n ( z ), 1,..., m x n 1 We dstngush the followng cases 1. Index of lne,, of Mzz does not coespond to any kenel pont of any kenel vecto. In ths case accodng to Theoem 3.1 m 0,, and theefoe n n m x x 1 1 ( ) 0. The last equaton holds because thee s the plausble assumpton that at least one element of the nput vecto x s of zeo value. 2. Index of lne,, of Mzz coesponds to a kenel pont of kenel vecto z. In ths case, accodng to Theoem 3.1, ( m 1, ), except when the column ndex () coesponds to a kenel pont of the same kenel vecto z. In ths case m 0.Theefoe, n ( m ) x 1 0 f x 0 : ke npont s ( (1 x), (0 x) ) ke npo nt s ke npo nt s 1 othewse In the above notaton an ndex,, s sad to belong to the kenel ponts f the element of the kenel vecto havng the same ndex,, s a kenel pont. The above equaton poves that a kenel vecto z can be ecalled by equaton (2.5) only f all the elements of the nput vecto x, whch coespond to kenel ponts, have the value 1. In othe wods, f even one element of the nput vecto, whch coesponds to a kenel pont, s ht by eosve nose then equaton (2.5) ecalls nothng and so does equaton (2.9). 162 ComSIS Vol. 8, No. 1, Januay 11

23 A new method fo constuctng kenel vectos n mophologcal assocatve memoes of bnay Poof of Theoem 4.1. To pove the theoem we stat fom the fomula (A.1), whch computes the elements of Mzz. Now, the new kenel vectos ae used. In geneal the ntenal tem ( z z ) can take values fom the set {-1, - 0.8, -0.7, -0.6, -0.4, -0.2, -0.1, 0,, 0.1, 0.2, 0.3, 0.4, 0.6, 0.8, 1} wth the exteme values to appea as follows (a) z z 0. Ths happens when fo a specfc both z and z ae kenel ponts of the same value (stength). (b) z z 1 Ths happens when fo a specfc z s one of the stongest kenel ponts (value 1) and and z s not a kenel pont (value 0) (c) z z 1 Ths happens when fo a specfc z s one of the stongest kenel ponts z s not a kenel ponts Takng now nto account that (A.1) s computed by takng the maxmum of all the ntenal tems ( z z ) fo 1 pwe conclude the followng. (1) If ndex does not coespond to any kenel pont of any kenel vecto then z 0,. Consequently, fo each ( z z ), 1 p takes the values 0, -0.4, -0.7, -0.8, 1. Theefoe, the maxmum of these values s 0 and theefoe m 0,. (2) If ndex coesponds to a kenel pont wth value z 1 (fequency 1), then m I 1 1, 0.6, 0.3, 0.2, 0 wth the value m 1 beng the most fequent. It has to be noted that the values of m ae fomed by ( z z ) of kenel only. The values ( z z ), ae always smalle snce z 0 (Fequency 1 means only z 1) The values of m ae esolved as follows: m 1, f z 0, m 0.6, f z 0.4, m 0.3, f z 0.7, m 0.2, f z 0.8, m 0, f z 1 (3) If ndex coesponds to a kenel pont wth value z 0.8 (fequency 2), then m I 2 0.8, 0.4, 0.1, 0 wth the value m 0.8 beng the most fequent. Fo the fomaton of m only two kenel vectos ae actually esponsble, whch have z 0.8. We call ths set of (two) kenels as fomng se and denote t as F. The values of ( z z ), F ae always smalle snce z 0 F (Fequency 2 means only two z 0.8, F ) The values of m ae esolved as follows: m 0.8 f z 0,at least fo one F m 0.4 f z 0.4, at least fo one F and fo the emanng z 0.4 ComSIS Vol. 8, No. 1, Januay