Image Segmentation using Tissue-like P Systems with Multiple Auxiliary Cells

Image Segmen a ion using Tissue-like P Sys ems wi h Mul iple Auxilia y Cells Ra´ul Reina-Molina, Ja ie Ca ne o, Daniel D´ıaz-Pe nil 1Resea ch G oup on Compu a ional Topology and Applied Ma hema ics Depa men o Applied Ma hema ics I Uni e si y o Se ille [email p o ec ed], [email p o ec ed], [email p o ec ed] Abs ac . We p esen a solu ion o he segmen a ion p oblem using a dis ibu ed, non de e minis ic and pa allel compu a ional model known as issue-like P sys ems. We p esen a new echnique o segmen images wi h espec o he algo i hm appea ed in [1], whe e we use mul iple auxilia y cells and no only one. 1 In oduc ion Memb ane sys ems a e dis ibu ed and pa allel compu ing de ices p ocessing mul ise s o objec s in compa men s delimi ed by memb anes. Compu a ion is ca ied ou by applying gi en ules o e e y memb ane con en , usually in a maximal non-de e minis ic way, al hough o he seman ics a e being explo ed. In ou wo k, we p esen a amily o issue-like P sys ems ( lP sys ems) which sol es he Segmen a ion P oblem in Digi al Image y using mul iple cells. Seg- men a ion in compu e ision (see [4]) e e s o he p ocess o pa i ioning a digi al image in o mul iple segmen s (se s o pixels). Image segmen a ion is yp- ically used o loca e objec s and bounda ies (lines, cu es, e c.). Mo e p ecisely, image segmen a ion is he p ocess o assigning a label o e e y pixel in an image such ha pixels wi h he same label sha e ce ain isual cha ac e is ics. In he li e a u e, one can ind se e al a emp s o b idging p oblems om Digi al Image y wi h Memb ane Compu ing. We can ci e he wo ks by K.G. Sub amanian e al. [?] o ecen ly some p oblems om Digi al Image y ha e been sol ed in he amewo k o Memb ane Compu ing (see [3]). 2 Image P ocessing: Segmen a ion P oblem The m-D Segmen a ion P oblem wi h k auxilia y cells (mDSP-kC) can be se led as ollows: gi en an m-D digi al image, Io size nm, o de e mine he edge pixels o his image using kauxilia y cells. The usual de ini ion o edge pixel p esen s p oblems om a p ac ical poin o iew wi h he noise and he deg ada ion o colou s, because we ake as bo de poin s a lo o pixels ha a e no edge pixels om a p ac ical poin o iew. 25 (a) Ini ial con en (b) Con en o second cell (c) Con en o second cell a e cleaning s age (d) Con en o second cell be o e edge de ec ion s age (e) Con en o second cell a e edge de ec ion ( ) Con en o i s cell once hal ing condi ion is eached Fig. 1: Full segmen a ion p ocess zoomed. Nex , we will show ha 2DSP-kC can be sol ed in linea ime (in he numbe o pixels o he image) by a amily o lP sys ems (see Fig.1). To his aim, le us cons uc a amily Π={Π(n, k):n, k ∈N}whe e each sys em o he amily will p ocess e e y ins ance uo he p oblem (a 2D image Iwi h n2pixels) and using kauxilia y cells. Mo e o mally, we de ine he size o he ins ance as s(u)="n, k#, whe e "x, y#=(x+y)(x+y+ 1)/2+xis he G¨odel mapping. In o de o p o ide a sui able encoding o his ins ances in o he sys ems, we will use he objec s I(ij)ij , wi h 1 ≤i, j ≤n, o ep esen he pixels o he g aph, and we will p o ide cod(u) as he ini ial mul ise o he sys em, whe e cod(u) is he mul ise o objec s I(ij)! ij o 1 ≤i, j ≤n. Then, gi en an ins ance uo he 2DSP-kC p oblem, he sys em Π(s(u)) wi h inpu cod(u) gi e a solu ion o his p oblem, implemen ed in he ollowing s ages: –Cleaning noise. –Homogenize colou s using a gene al h esholding in colou space. –Segmen ing image p ocess. The amily Π={Π(n, k):n, k ∈N}o lP sys ems o deg ee k+1 is de ined as ollows: o each n, k ∈N, Π(n, k) = (Γ,Σ,E,w 1, . . . , wk+1,R,i Π,o Π), de ined as ollows: –Γ=Σ∪{aija!! ij,¯aij,A ij,A ! ij,A !! ij,¯ Aij :1≤i, j ≤n, a ∈C} ∪{∗ij,∗ji :i= 0,n+1,0≤j≤n+1},Σ={a! ij :1≤i, j ≤n, a ∈C},E=Γ−Σ, 26 –w1=∗ij,∗ji wi h i=0,n+1,0≤j≤n+ 1, w2=· · · =wk+2 =T!n2/k", –Ris he ollowing se o communica ion ules: •(1,a # ij/a8 ijAij,0) o 0 ≤i, j ≤n+ 1 and a∈C∪{∗} • 1, ci−1j−1di−1jei−1j+1 bij−1Aij ij+1 oi+1j−1hi+1jgi+1j+1 / T,   o 1 ≤i, j ≤n,a, b, c, d, e, , g, h, o ∈C∪{∗}and 2 ≤ ≤k+1 indica ing an auxilia y wo king cell. These ules a e used o gene a e new elemen s. The P sys em uses hese elemen s o wo k wi h he noise o ou image. •  , ci−1j−1di−1jei−1j+1 bij−1Aij ij+1 oi+1j−1hi+1jgi+1j+1 /z " ij ,0 ,  , ci−1j−1di−1jei−1j+1 bij−1Aij ij+1 oi+1j−1hi+1jgi+1j+1 /a " ij ,0  o 1 ≤i, j ≤n,a, b, c, d, e, , g, h, o ∈C∪{∗}. We ake µas he numbe o pixels adyacen s o he ij possi ion wi h colou s in Cand ∗= 0. Then, a =(b+c+d+e+ +g+h+o)/µ and z= max{s∈C:s≤a }and |a−a |≤ρ1, whe e ρ1∈(0,+∞). This se o ules is used o de ec he noise and co ec i wi h he a e age o colou s o i s adjacen pixels. We ind he e a local h esholding (wi h espec o he colou s) wi h p ede ined h eshold ρ1. The use o ∗g an s a simple wo king o pixels in he bo de o I. •( , b# ij/A# ij,0) o 1 ≤i, j ≤n,ν=(|C|/ρ2), l=0,1,2,...,ρ2. I b∈C hen a∈C(a<b≤a+(ν−1) and a=ν·l) o (b=a=ν·l) and, i b=∗ hen A=∗. These ules a e used o disc e ize he colou s di iding he se o colou s in ρ2subse s o leng h ν. We ind he e a gene al h esholding (wi h espec o he colou s) wi h p ede ined h eshold ν. •( , A# ij/T, 1) o a∈C,0≤i, j ≤n+ 1 and 2 ≤ ≤k+ 1. This se o ules a e used o send ou ans o med image o he cell 1. Now, he objec s A# ij codi y he pixels o ou image. •(1,A # ij/A## ija8 ij,0) o a∈C∪{∗}and 0 ≤i, j ≤n+ 1. The P sys em uses hese ules o gene a e an numbe o copies o ou image o do he segmen a ion p ocess in he cells 2, . . . , k and k+ 1. • 0, ci−1j−1di−1jei−1j+1 bij−1A## ij ij+1 ii+1j−1hi+1jgi+1j+1 / T,   o 1 ≤i, j ≤nand a, b, c, d, e, , g, h, i ∈C∪{∗}. These ules a e de ined o send o he emainde o cells he new objec s. •( , A## ijbkl/Aij,0), o 1 ≤i, j, k, l ≤n,(i, j),(k, l) adjacen pixels, a, b ∈C and a<b. These ules a e used o ma k edge pixels. When we ha e wo adjacen pixels wi h di e en colou , we ake as edge pixel he pixel wi h less associa ed colou . In ac , he P sys em b ings om he en i onmen an objec Aij (in his case). •( , Aij/T, 1) o a∈Cand 1 ≤i, j ≤n. These ules send o he cell 1 he ou pu objec s. 27 –iΠ=oΠ= 1. A li le s udy o he complexi y aspec s o his solu ion is gi en by he Fig.2 showing his is an e icien algo i hm om a heo e ical poin o iew. mDSP-kC P oblem Complexi y Numbe o s eps o compu a ion 9 Resou ces needed Size o he alphabe 8n2+4n+5 Ini ial numbe o cells k+1 Ini ial numbe o objec s (n+ 2)2 Numbe o ules O(n2 ·h9 ·k) Uppe bound o he leng h o he ules 10 Fig. 2: Complexi y aspec s, whe e he size o he inpu da a is O(n2), |C| =his he numbe o colou s o he image and kis he numbe o wo king cells. 3 Final Rema ks We ha e designed a amily o issue-like P sys ems o do a segmen a ion o a digi al image. Wi h his algo i hm we wo k wi h mul iple cells, so we can ob ain a bigge pa alleliza ion wi h espec o he design p esen ed in [1]. Acknowledgemen DDP acknowledges he suppo o he p ojec s TIN2008-04487-E and TIN-2009- 13192 o he Minis e io de Ciencia e Inno aci´on o Spain and he suppo o he P ojec o Excellence o he Jun a de Andaluc´ıa, g an P08-TIC-04200. Re e ences 1. 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