1 Proprtis o Hxgonl Til lol n XYZ-lol Sris Jy Arhm 1, Anith P. 2, Drsnmik K. S. 3 1 Dprtmnt o Bsi Sin n Humnitis, Rjgiri Shool o Enginring n, Thnology, Kkkn, Ernkulm, Krl, Ini. jyjos1977@gmil.om 2 Dprtmnt o Mthmtis B.J.M. Govrnmnt Collg, Chvr, Kollm, Krl, Ini. nitnson123@ymil.om 3 Dprtmnt o Mthmtis, Fthim Mt Ntionl Collg, Kollm, Krl, Ini. rsnp@yhoo.om Astrt In this ppr w will hrtriz th mily o pitur sris, rogniz y WHPA (wight hxpoli pitur utomton n lso, y using wight hxgonl tiling n hxgonl omino systms. W will prov tht hxgonl pitur sris is th projtion o hxgonl til lol sris n lso o xyz-lol sris. Thus w otin roust inition o lss o rognizl hxgonl pitur sris. Kywors: Wight hxgonl tiling systm, hxgonl omino systm, wight hxpoli pitur utomton Mthmtis Sujt Clssiition. Appli Mthmtis I INTRODUCTION Gnrlly whn ompr with othr mthos pitur is us or unrstning things in ttr wy. A lot o thnologis r thr to omput piturs with th hlp o omputrs. This hs rsult in th introution o pitur gnrting vis. Wight init utomt, two imnsionl utomt, th onpt o gnrliztion lol lngugs rom wors n irnt typs o grmmrs r th sis on whih ths vis vlop. Jurgn Dssow [1] hs isuss irnt typs o grmmrs in his ppr. In 1991, Gimmrsi n Rstivo propos th mily o rognizl lngugs (REC [2]. Tiling systms r us to in th lngugs in REC. K.S. Drsnmik t l. [3] hv in tht hxgonl piturs r us prtiulrly in pitur prossing n lso in img nlysis. In 2003, K.S. Drsnmik t l. us irnt ormlism or hxgonl pitur lngugs. Th ormlisms r hxgonl tiling systm, lol hxgonl pitur lngugs, rognizl hxgonl pitur lngugs, lll hxgonl Wng tils n inlly Wng systms. Introution out xyzomino systms wr m y thm n prov tht it is quivlnt to hxgonl tiling systms. In this ppr w will ssign wights to ths lol n xyz-lol hxgonl pitur vis using hxgonl tils or hxgonl ominos. By projtion vry simpl lol hrtriztions o WHPArognizl hxgonl pitur sris n on. II PRELIMINARIES Hr w rll th inition, strutur, xis n 2 2 2 Hxgonl tils n orrsponing ominos rom K.S. Drsnmik t l. [3] Fig. 1. II.1 x Hxgonl Piturs [3] Hr w rll th notions o hxgonl piturs n th hxgonl pitur lngug [4]. Lt Σ init lpht o symols. A hxgonl pitur p ovr Σ is hxgonl rry y z 40
2 Fig. 2. g or g o symols o Σ n th st o ll hxgonl piturs ovr Σ is not y Σ H A hxgonl pitur ovr th lpht,,, is shown in igur 1 Th st o ll hxgonl rrys ovr th lpht Γ is not y Γ H. Dinition II.1. [3] I x Γ H, thn ˆx is th hxgonl rry otin y surrouning x with spil ounry symol Γ. A hxgonl pitur ovr th lpht,,,,, surroun y is shown in ig. 1. Dinition II.4. [3] Lt L hxgonl pitur lngug. Th lngug L is xyz-lol i thr xists st o ominos s in ov ovr th lpht Σ suh tht L = ω Σ H /ll omino tils rlting to ω W writ L = L( i L is xyz-lol n is st o ominos ovr Γ. A hxgonl pitur lngug L is xyz-omino rognizl i thr xists omino systm D suh tht L = Π(L(. Th lss o hxgonl lngugs rognizl y omino systms is not y L(HDS Thorm II.5. [3] I HREC is th mily o ll rognizl hxgonl pitur lngugs n L(HDS is th lss o hxgonl lngugs rognizl y omino systms, thn w hv HREC = L(HDS II.2 Hxgonl Tils n Dominos [3] A hxgonl pitur o th orm shown in ig. 2 is ll hxgonl til ovr n lpht,,,,,,g. Givn hxgonl pitur p o siz (l,m,n, or g l,h m,k n w not B g,h,k (p, th st o ll hxgonl loks (or hxgonl su-piturs o p o siz (g,h,k. B 2 2 2 is in t hxgonl til. Dinition II.2. [3] Lt Σ init lpht. A hxgonl pitur lngug L Γ H is ll rognizl i thr xists hxgonl lol pitur lngug L (givn y st o hxgonl tils ovr n lpht Γ n mpping Π : Γ Σ suh tht L = Π(L. Th mily o ll rognizl hxgonl pitur lngugs will not y HREC. Exmpl II.3. Th st o ll hxgons ovr Σ = is rognizl y hxgonl tiling systm. St L = L n π(1 = π(0 = III TILE LOCAL AND xyz-local SERIES Hxgonl tils r piturs o siz (2,2,2 n hxgonl ominos hv siz (2,1,1 or (1,2,1 or (1,1,2. For pitur p w not T(p (rsptivly D(p th st o ll suhxgonl til (rsptivly su-hxgonl ominos o p. A lngug L Γ H is lol (rsptivly xyz-lol i thr xists st o hxgonl tils (rsptivly hxgonl ominos ovr Γ, suh tht L = p Γ H T(ˆp rsptivly L = p Γ H D(ˆp Thn (Γ, hrtrizs L. W writ L = L(. For pitur p Σ l m n, w will onsir su-hxgonl tils (su-hxgonl ominos t rtin positions o ˆp. In s o hxgonl tils, w n in: 1 0 0 1 0 0 1 L π For ll 1 i l+1, 1 j m+1, 1 k n+1 : t(ˆp i,j,k = ˆp i,j,k ˆp i,j,k+1 Now w onsir nothr ormlism to rogniz hxgonl piturs whih is s on omino systms introu y Lttux, t l. [5]. Hr w onsir ominos o th ollowing typs. ˆp i+1,j,k ˆp i,j+1,k ˆp i,j+1,k+1 ˆp i+1,j+1,k ˆp i+1,j+1,k+1 41
3 Also w onsir th su-hxgonl ominos in z, y, x- irtion istinguish y thir positions in ˆp. For ll 1 i l+2, or ll 1 j m+2, 1 k n+1 : z (ˆp i,j,k = ˆp i,j,k ˆp i,j,k+1 or ll 1 i l+2, or ll 1 j m+1, 1 k n+2 : y (ˆp i,j,k = ˆp i,j,k ˆp i,j+1,k or ll 1 i l+1, or ll 1 j m+2, 1 k n+2 : x (ˆp i,j,k = W gt th ollowing inition. ˆp i+1,j,k ˆp i,j,k Dinition III.1. A (wight hxgonl til systm is in y T 1 = (Σ,T whr T : (Σ 2 2 2 K is untion mpping hxgonl tils ovr Σ to K. It omputs th hxgonl pitur sris T 1 : Σ H K, in y or ll p Σ H ; T 1 (p = T(t(ˆp i,j,k 1 i I x(p+1 1 j I y(p+1 1 k I z(p+1 W ll S : Σ H K, hxgonl til-lol i thr xists hxgonl til systm T 1 stisying T 1 = S. Similrly or hxgonl ominos w hv th ollowing inition. Dinition III.2. A (wight hxgonl omino systm is in y D 1 = (Σ,D, whr D : (Σ (2,1,1,(1,2,1,(1,1,2 K mps hxgonl ominos ovr Σ to K. It omputs th sris D 1 : Σ H K in y, or ll p Σ H : D 1 (p = D( x (ˆp i,j,k 1 i I x(p+2 1 j I y(p+1 1 k I z(p+2 1 i I x(p+1 1 j I y(p+2 1 k I z(p+2 D( y (ˆp i,j,k 1 i I x(p+2 1 j I y(p+2 1 k I z(p+1 D( z (ˆp i,j,k. A hxgonl pitur sris S : Σ H K is ll xyzlol i thr xists omino-systm D 1 stisying D 1 = S. Th milis o hxgonl til lol sris n xyz-lol sris n not y K lo Σ H n K xyz Σ H rsptivly. Now, T 1 (rsptivly D 1 r ll hxgonl til (rsptivly hxgonl omino untion. Th milis o sris tht r projtions o hxgonl tillol (rsptivly hxgonl xyz-lol sris n not y K plo Σ H (rsptivly K pxyz Σ H. Proposition III.3. A hxgonl pitur lngug L Γ H is lol (xyz-lol rsptivly i n only i its hrtristi sris I L B Γ H is hxgonl til lol (xyz-lol rsptivly. In orr to prov prposition III.7, w n th ollowing inition III.4 n propositions III.5 n III.6, tht w hv isuss in th prvious ppr [7]. Dinition III.4. A wight hxpoli pitur utomton (WHPA is n 8 tupl B = (Q,R,F N, F S, F NW, F SW,F NE,F SE onsisting o init st Q o stts, init st o rulsr Σ K Q 6, s wll s six pols o ptn F N,F S,F NW,F SW,F NE,F SE Q [N-north, S-south, NW -north wst, SW -south wst, NEnorth st, SE-south st] Proposition III.5. Lt S K r Γ H sris omput y rul trministi WHPA. Thn S is rtionl Hxgonl pitur sris. Proposition III.6. Lt B WHPA ovr Σ. Thr xists rul trministi WHPA ovr n lpht Γ n projtion Π : Γ Σ stisying B = Π( B Proposition III.7. W hv K r Σ H K pxyz Σ H Proo. Using th projtion (III.6 w now onsir th rul trministi utomt s ollows. Lt B = (Q,R,F N,F S,F NW,F SE,F NE,F SW rul trministi omputing B = S. W my us th nottions vlop or th proo o prposition III.5 n inition III.4. For, Σ, in s th ourring ruls xist, w in hxgonl omino-untion D : (Σ 2 1 1,1 2 1,1 1 2 K s ollows: 1 i σ SW (r( F SW 1 i σnw (r( F NW 1 i σ NE (r( F NE 1 i σse (r( F SE 1 i σ NE (r( = σ SW (r( 1 i σ SE (r( = σ NW (r( 42
4 1 1 1 wight (r( i σn (r( F N 1 i σs (r( F S wight (r(, i σs (r( = σ N (r( D mps ll othr hxgonl ominos to 0. Thn D 1 = (Σ, D is hxgonl omino-systm. For hxgonl pitur p with (uniqu sussul omputtion R H in B, th prout vlus od(tkn ovr th nonil omino ovring o ˆp oinis with wight (. This is us or vry position p i,j,k o p, thr xists prisly on tor wight (r(p i,j,k in th prout or D 1 (p. On th othrhn, ip hs no sussul omputtion in B, thn lrly th inition o D givs D 1 (p = 0. Thus D 1 = S Proposition III.8. Evry xyz-lol sris is hxgonl tillol. Proo. Lt S : Γ H K n xyz-lol sris ovr n lpht Γ, omput y D 1 = (Γ,D. W in T 1 = (Γ,T s hxgonl til-systm omputing S suh tht T : (Γ 2 2 2 K nots th hxgonl til untion. For ritrry Γ n,,,, Γ, w put (hr nwnorth wst ornr, n- north st ornr, n- north ornr, n- north st g, w- wst g, m- mil ( nw : T = D( D( ( D( D( n : T ( = D( D( D( D( D( D( ( n : T = D( D( D( D( n : T w : T ( = D( D( D( D( D( D( ( = D( D( D( D( D( D( ( m : T g = D( g D( D( Furthrmor w st ( T = 0. Th vlus o D ovr th hxgonl omino-ovring o hxgonl pitur p r istriut with T ovr th hxgonl til-ovring or p. Evry x (ˆp i,j,k, y (ˆp i,j,k n z (ˆp i,j,k hppns prisly on in th prout or T 1 (p. Thn or p Γ H, w gt T1 (p = D 1 (p = (S,p. Proposition III.9. W hv K plo Σ H K r Σ H Proo. It suis to prov th rsult or hxgonl til lol sris. Lt S : Σ H K hxgonl til lol, omput y T1 = (Σ, T with hxgonl til untion T : (Σ 2 2 2 K. W in B = (Q,R,F N,F S,F NW,F SW,F NE,F SE s WHPA ovr Σ omputing S y putting Q = (Σ 2 2 2 n F N = Σ,,, Σ F S = Σ,,, Σ F NW = Σ,,,, Σ F SW = Σ,,,, Σ F NE = Σ,,,, Σ 43
5 F SE = Σ,,,, Σ W st R = R nw R n R n R n R w R m Σ K Q 6 with ritrry,,,,,, g,h,t, x,y,z Σ. R nw =,ω (,, g,, g, g, g,/ Σ ( ( whr ω ( = T ( ( g ( g R n =,ω (,,,, g g, z, g Σ x y ( ( whr ω ( = T ( ( g g R n =,ω (,, g, g,, y x Σ g whr ω ( = T ( x ( ( ( g R n =,ω (, g,, g, y z g,, x Σ x y ( ( whr ω ( = T g g ( ( g R w =,ω (,, g, g, g g, x, Σ ω ( = T ( ( g ( g ( x x y R m =,ω (,, g z, g, z x,, y Σ g y y g ( whr ω ( = T To prov B = S, w osrv th ollowing. Th ov onstrution with its pting onition givs wight 44
6 hxpoli pitur utomton tht hs prisly on sussul run or vry lmnt in Σ H. This run simults th istriution o wights long th nonil hxgonl tilovring o th hxgonl pitur. Suppos tht B Σ H with sussul omputtion R H in B is givn wight ( = i,j,k wight ( i,j,k. In th multiplition, th img o th hxgonl til-untion o th rsptiv hxgonl til t vry position o th nonil hxgonl til ovring o ˆp ours xtly on. Agin, th inition o th vlus o ω ( simult th ovring in suh wy tht no othr wights our. For p Σ H, w hv B (p = T(t(ˆp i,j,k = T 1 (p 1 i I x(p+1 1 j I y(p+1 1 k I z(p+1 = (S,p IV CONCLUSION In this ppr w introu irnt vis to hrtriz rognizl hxgonl pitur sris. W tri to prov tht hxgonl pitur sris is th projtion o hxgonl til lol sris n lso o xyz-lol sris. In similr mnnr, w n xtn vrious othr proprtis o rognizl rtngulr pitur sris to hxgonl rognizl pitur sris. REFERENCES [1] Jurgn Dssow (2006 Grmmtil Pitur Gnrtion, Otto-vongurik-univrsitt Mgurg, Fkultt Fr Inormtik. [2] Gimmrrsi D. & Rstivo A., Rognizl pitur lngugs. In Prlll img prossing, 31 46 (Worl Sintii [3] K.S. Drsnmik, K. Krithivsn, C. Mrtin-Vi & K.G. Surmnin, Lol n Rognizl hxgonl pitur lngugs. Int. journl pttrn rognition Arti-Intllign 19, 853 871 [4] Bozpliis S. & Grmmtopoulou, A Rognizl Pitur Sris. J. Autom-Lng. Com. 10, 159 183 (2005 [5] M. Lttun n D. Simplot, Rognizl pitur lngugs n omino tiling, Thortil Computr Sin, 178, (1997, (275 283 [6] K.S. Drsnmik, Kml Krithivsn, K.G. surmnin, (2003 P systm gnrting Hxgonl pitur lngugs, Ltur nots in Computr Sin, Springr Brlin Hilrg, Volum 2933 pp. 168 180. [7] Jy Arhm & Drsnmik K.S., Chrtriztions o Hxgonl rognizl Pitur Sris, in Ntionl Conrn on Emrging Trns in Mthmtis n Applitions in Enginring n Thnology 2018 45