A New Algorithm for Training Multi-layered Morphological Networks

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1 A New Algorthm for Tranng Mult-layered Morphologcal Networs Rcardo Barrón, Humberto Sossa, and Benamín Cruz Centro de Investgacón en Computacón-IPN Av. Juan de Dos Bátz esquna con Mguel Othón de Mendzábal Mexco Cty, 07738, Mexco Abstract. In ths wor we present an algorthm for tranng an assocatve memory based on the so-called mult-layered morphologcal perceptron wth maxmal support neghborhoods. We compare the proposal wth the orgnal one by performng some experments wth real mages. We show the superorty of the new one. We also gve formal condtons for correct classfcaton. We show that the proposal can be appled to the case of gray-level mages and not only bnary mages. Keywords: Assocatve memores, Morphologcal neural networs, maxmal support neghborhoods. 1 Introducton Neural networs have shown to be an excellent alternatve to face problems where t s dffcult to fnd no algorthmc soluton. Based on the functonng of the human nervous system, lots of researchers have proposed dfferent neural processng models. Probably the best now model s the Bac-propagaton Neural Networ [1], [2] and [3]. The study of the nternal structure of neural cells has revealed that all cells have the same smple structure, ndependently of ther sze and shape. Informaton from a cell voyages through the sgnals that neurons send to other neurons through ther dendrtes. It s beleved that the cellular body adds up the receved sgnals; when enough nputs are avalable a dscharge s produced. Intally, ths dscharge occurs n cellular body, t then propagates between the axon untl the synapses that sends a new sgnal to the other neurons. An artfcal neural networ can be seen as a non-lnear mappng between two pattern spaces: the nput pattern set and the output pattern set. Normally the nternal parameters of ths mappng are determned by means of a tranng process and are denoted as synaptc weghts. In the decade of the 50 s, Rosenblatt ([2], [3]) ntroduces the well-nown perceptron, whch s the classcal model that has been used for most of the actual developments. However, n the 90 s, Rtter et al., ([4], [5], [6]) and Sussner [12] presented a new nd of neural networ model, the so-called morphologcal neural L. Rueda, D. Mery, and J. Kttler (Eds.): CIARP 2007, LNCS 4756, pp , Sprnger-Verlag Berln Hedelberg 2007

2 A New Algorthm for Tranng Mult-layered Morphologcal Networs 547 networ. Here, the classcal operatons of multplcaton and addton are replaced by summatons and max (or mn), respectvely. One dfference between ths model and the classcal models s the computatonal cost when computng the value of -th neuron at tme t + 1 that n ths model s less. In the last years t has been found that, apparently the processng of nformaton not only occurs at the cellular body but also at the dendrtes [7]. Ths affrmaton could gve an explanaton of the great effcency of our nervous system, snce processng of the nformaton happens practcally along the communcaton channel. The materal ust presented along wth the morphologcal paradgm s the departure pont of ths paper. 2 Assocatve Memory Based on the Morphologcal Perceptron In [7] t s presented how a morphologcal perceptron allows classfyng any compact set n the pattern s doman, whch can be used to buld an assocatve memory able to recall patterns affected by mxed nose. The dea s to buld a three layer assocatve memory (an of nput, one hdden, and of output) as can be apprecated n Fgure 1. Fg. 1. Assocatve memory of three layers based on the morphologcal perceptron Frst layer wors as regster of the number of elements of the nput pattern. Second one (the hdden layer) s composed of morphologcal perceptrons [7], one for each class, whle the output layer s formed by perceptrons based on the max operator wth a lnear gatng functon, one perceptron for each component of the output patterns. The perceptrons of the hdden layer are morphologcal perceptrons wth one nput dendrte. The classfed regon s thus a hyper-rectangle n R n, whose goal s to classfy the correspondng pattern nsde the on-regon. For the better functonng of ths model, the perceptron outputs a zero n ts on-regon and n ts complements. The output of a perceptron at the output layer s gven as K ϕ ( ) (1) z = y + x = 1 where ϕ x s the output of the -th perceptron of the hdden layer. Thus, when tang the max of the y s the -th component of the -th output pattern, and ( )

3 548 R. Barrón, H. Sossa, and B. Cruz ϕ that output a unqualfy pattern outputs of the perceptrons y as possble output of the assocatve memory. To tran ths assocatve memory, we have smply to defne the supports of each ey pattern. In ths case the on regons of the correspondng patterns represent these supports. From [3], durng tranng, all supports of the memory are bult as hyper-cubes of sde equal to α, where α s obtaned as where (, ) d x x s defned as and ( x ) 1,, α 1 mn d x, x 2 < ( ) < (2) (, ) max{ } = l l d x x x x = = m s the set of ey patterns. l 1, n To avod collsons at the moment of classfcaton t s necessary that the supports are dsont two by two. Ths s not demonstrated n [7]. Next a bref proof that ths happens s gven. Proposton 1. Let M a mult-layered assocatve memory. If each ey pattern a support S { x: d( x, x ) α} (3) x has = <, wth α as defned n equaton (2), then t hold S S =. that, Proof. Let us suppose that such that S S, ths means that there s a x such that d( x, x ) < α and d( x, x ) α nequalty we get: d( x, x ) d( x, x ) d( x, x ) ( α α 2α) d( x, x ) d( x, x ) < 2α, or n other words α <. When summng and usng trangles + + =, then 2 <!!, whch s a contradcton, thus the proposton hold. One of the drawbacs of ths method to construct the supports s that f two of them are of them too close, the (radus) of the neghborhood s supports reduce drastcally. In [7] t s proposed another method to ncrease the neghborhoods of the supports by means of the ernels method. Accordng to [7], wth ths the range of permssble nose s ncreased. However, ths method maes expensve the computatonal cost and besdes t mposes restrctons over the patterns very dffcult to get. 3 Proposed Tranng Algorthm In the content of ths wor, let us suppose that a pattern s represented n terms of n obect features; then at each coordnated axs we can compute the varaton obtaned

4 A New Algorthm for Tranng Mult-layered Morphologcal Networs 549 per feature by orderng all components. By computng the average of varablty per pattern t s possble to defne a threshold. For practcal purposes, ths threshold allows to consder f two patterns can be consdered to be the same from the pont of vew of one of ther components. Ths way we avod havng very tny supports wth respect to the coordnated axs. Ths way the drawbac of the algorthm descrbed n last secton s surpassed. Fg. 2. Flowchart to get average varaton threshold The algorthm to fnd the average varaton threshold by axs s observed n Fg. 2. In ths case, X s a matrx of n m. At -th column t s -th pattern, whle at - th lne are the m features of the -th axs of each pattern. In U t s the average varaton threshold for the -th axs. The ey pont for the tranng of the multlayered morphologcal perceptron conssts on constructng the supports for each pattern. In general these supports must be dsont two by two to avod that two patterns be assgned to the same output. Ths s fulflled n the proposed algorthm thans to the followng: Proposton 2. Let Ω and Ω two arbtrary supports correspondng to dfferent patterns n R n, bult accordng to average varaton threshold, then t holds that Ω=Ω or Ω Ω =. [13]. Proof. Let x and x the correspondng patterns to supports Ω and Ω, respectvely, then we have two cases: a. If t holds that x x < U, = 1, n, wth U s the average varaton. If ths holds then Ω=Ω. such that x x > U would mply that at th b. If = 1,, n coordnate, the support does not concde and on the axs they are dsont, thus they are dsont n R n.

5 550 R. Barrón, H. Sossa, and B. Cruz Evdently, when the patterns are too close, the neghborhoods would present occlusons. In these cases Ω=Ω for, one varant of ths algorthm would be to consder that Ω Ω, but that each neghborhood s centered at ts respectve ey pattern. The advantage of ths enhancement s that neghborhoods allowng more nose to be added to the patterns wll be enlarged. 4 Numercal Examples Example No. 1. Let the followng set of ey patterns n R n : x =, x =, x =, x =, x =, x = a) Soluton obtaned by means of algorthm proposed n [7]: By applyng the algorthm proposed n [7], accordng to equaton (2) α = 0.5. Fgure 3 shows the neghborhoods obtaned when usng ths value of α. (4) Fg. 3. Neghborhoods obtaned when algorthm proposed n [7] s used b) Soluton obtaned by means of algorthm proposed n ths paper (frst varant): When usng the algorthm to get the maxmal support neghborhoods, by applyng the flowchart shown n Fgure 3, we get the threshold value U as: U = 1.5 The neghborhoods obtaned by usng ths threshold value are shown n Fgure 4. The dfferences can be mmedately apprecated. In ths second case the range on nose for each pattern, as can be apprecated, s bgger. (5)

6 A New Algorthm for Tranng Mult-layered Morphologcal Networs 551 Fg. 4. Neghborhoods obtaned when usng frst varant of the proposed algorthm c) Soluton obtaned by means of algorthm proposed n ths paper (second varant): We apply the same procedure used by the frst varant, but n ths case the neghborhoods are centered at the correspondng ey patterns. The threshold s the same gven by equaton (5). Fg. 5. Neghborhoods obtaned when usng second varant of the proposed algorthm

7 552 R. Barrón, H. Sossa, and B. Cruz The neghborhoods obtaned are shown n Fg. 5. Due to the neghborhoods are now centered at ther respectve ey patterns, n ths case the support for nose s bgger. Note also the occlusons between classes that do not occur n the frst varant. In the followng secton we show how the proposal descrbed n ths paper can be used not only to recall bnary patterns but also gray-level patterns such mages. 5 Experments wth Real Patterns For ths experment we used the mages shown n Fg. 6. These mages are gray-level of elements. They were perturbed wth nose from 5% to 15%, n steps of 5%. Fg. 6. Orgnal mages of experments The nput ey patterns where formed by descrbng each mage by means of nown Hu nvarants ([10], [11]). Fgures 7 to 9 show graphcally the results obtaned when usng the algorthm proposed n [7], and the algorthm proposed n ths paper to the mages s shown n Fgs. 10 to 12. From all of these fgures we can observe that when addng, even small quanttes of nose to the patterns, the orgnal algorthm proposed n [7] fals to recall practcally of the patterns, whle the proposal, although wth modest percentage, several of the desred patterns are correctly recalled. Fg. 7. Images altered wt 5% of nose. Images recalled usng [7].

8 A New Algorthm for Tranng Mult-layered Morphologcal Networs Fg. 8. Images altered wt 10% of nose. Images recalled usng [7]. Fg. 9. Images altered wt 15% of nose. Images recalled usng [7]. Fg. 10. Images altered wt 5% of nose. Images recalled usng the proposal. 553

9 554 R. Barrón, H. Sossa, and B. Cruz Fg. 11. Images altered wt 10% of nose. Images recalled usng the proposal. Fg. 12. Images altered wt 15% of nose. Images recalled usng the proposal 6 Conclusons In ths paper we have presented an algorthm to tran the mult-layered morphologcal perceptron that allows to buld more effcent support neghborhoods, snce the pont of vew of pattern recall, from the set of ey pattern patterns of the tranng set of an assocatve memory n both ts auto-assocatve or hetero-assocatve way of operaton. By several experments wth real patterns, we have shown that the proposal can be used to recall gray-level mages and not only bnary mages. We show the superorty of the new one. Acnowledgements. Ths wor was economcally supported by CIC-IPN, COFFAA- IPN and CONACYT under grant and SIP under grants and , respectvely.

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