Orthogonal and Biorthogonal FIR Hexagonal Filter Banks with Sixfold Symmetry

Transcription

1 IEEE TRANS. SIGNAL PROC. VOL. 56 NO DEC. 008 Orthogonl n Biorthogonl FIR Hxgonl Filtr Bns with Sixol Symmtry Qingtng Jing Astrt Rntly hxgonl img prossing hs ttrt ttntion. Th hxgonl ltti hs svrl vntgs in omprison with th rtngulr ltti th onvntionlly us ltti or img smpling n prossing. For xmpl hxgonl ltti ns wr smpling points; it hs ttr onsistnt onntivity; it hs highr symmtry; its strutur is plusil to humn vision systms. Th multirsolution nlysis mtho hs n us or hxgonl img prossing. Sin th hxgonl ltti hs high gr o symmtry it is sirl tht th hxgonl ltr ns sign or multirsolution hxgonl img prossing lso hv high orr o symmtry whih is prtinnt to th symmtry strutur o th hxgonl ltti. Th orthogonl or prt ronstrution (PR) hxgonl ltr ns whih r vill in th litrtur hv only -ol symmtry. In this ppr w invstigt th onstrution o orthogonl n PR FIR hxgonl ltr ns with 6-ol symmtry. W otin lo struturs o -siz rnmnt (-hnnl -D) orthogonl n PR FIR hxgonl ltr ns with 6-ol rottionl symmtry. -rnmnt orthogonl n iorthogonl wvlts s on ths lo struturs r onstrut. In this ppr w lso onsir FIR hxgonl ltr ns with xil (lin) symmtry n w prsnt lo strutur o FIR hxgonl ltr ns with psuo 6-ol xil symmtry. Inx Trms Hxgonl ltti hxgonl img ltr n with 6-ol symmtry orthogonl hxgonl ltr n iorthogonl hxgonl ltr n -rnmnt wvlt -siz rnmnt multirsolution omposition/ronstrution. EDICS Ctgory: MRP-FBNK I. INTRODUCTION Img prossing is ommonly rri out on th rtngulr ltti sin imgs r onvntionlly smpl on suh ltti. Img prossing on th hxgonl ltti hs ttrt ttntion rntly. S squr n hxgonl lttis in th lt n right prts o Fig.. Th hxgonl ltti hs svrl vntgs in omprison with th rtngulr ltti. For xmpl hxgonl ltti ns wr smpling points; it hs ttr onsistnt onntivity; it hs highr symmtry; its strutur is plusil to humn vision systms ]-8]. Thus th hxgonl ltti hs n us in mny rs suh s g ttion 9] 0] n pttrn rognition ]-5]. It hs lso n us in gosin 6]-9]. Whn hxgonl ltti L is us or img smpling h no (sit) on L rprsnts smll hxgonl ll ll Mnusript riv Mrh 008; rvis Jun This wor ws support y UM Rsrh Bor 0/05 n UMSL Rsrh Awr 0/06. Th ssoit itor oorinting th rviw o this mnusript n pproving it or pulition ws Dr. Alpr Erogn. Th uthor is with th Dprtmnt o Mthmtis n Computr Sin Univrsity o Missouri St. Louis St. Louis MO 6 USA -mil: jingq@umsl.u w: jing. Fig.. Squr ltti (lt) n hxgonl ltti (right) th lmntry ll. A no n th hxgonl lmntry ll (show) it rprsnts r shown in Fig.. All th hxgonl lmntry lls orm hxgonl tsslltion o th pln (s Fig. ). W ll this tsslltion to th hxgonl tsslltion ssoit with L. Fig.. Hxgonl ltti n ssoit hxgonl tsslltion It ws shown in 0] ] tht hxgonl ltti llows thr intrsting rnmnts (suivisions): 4-siz (4-rnh) -siz (-rnh) n -siz (-rnh) rnmnts. As n xmpl w sri -siz rnmnt ( -rnmnt) low. In th lt prt o Fig. th nos o th unit rgulr hxgonl ltti G r not y l ots n th nos with irls orm nw ltti whih is ll th -siz (-rnh) sultti o G hr n it is not y G. G is lso rgulr hxgonl ltti. From G to G th nos r ru y tor. So G is ors ltti o G n G is rnmnt o G. Sin G is lso rgulr hxgonl ltti w n rpt th sm prour to G n w thn hv highorr (ors) rgulr hxgonl ltti with wr nos thn G. Rpting this prour w hv st o lttis with wr n wr nos. This st o lttis orm pyrmi or tr whr high-orr ltti hs wr nos thn its prssor y tor o. Th hxgonl tsslltion (with thi hxgon gs) ssoit with G is shown in th right prt o Fig. whr th hxgonl tsslltion (with thin hxgon gs) ssoit with G is lso provi to giv us pitur on how ths two tsslltions r rlt to h othr. Th suivisions ssoit with 4-siz -siz n - siz rnmnts r ll rsp. th yi (-to-4 split) n rnmnts in th r o Computr Ai Gomtry Dsign ]-9] whil thy r ll prtur 4 prtur

2 IEEE TRANS. SIGNAL PROC. VOL. 56 NO DEC. 008 Fig.. Hxgonl ltti G n its -siz sultti G (lt) n hxgonl tsslltions ssoit with G n G (right) n prtur (rnmnts) in isrt glol gri systms in 9]. Th rnmnts o th hxgonl ltti llow th multirsolution (multisl) nlysis mtho to us to pross hxgonlly smpl t. Th 4-siz rnmnt is th most ommonly us rnmnt or multirsolution img prossing s.g. ] ] 0] or multirsolution hxgonl img prossing pplitions n 0]-6] or th onstrution o hxgonl ltr ns. Th -siz rnmnt is most ppling mong ths rnmnts sin th -siz rnmnt gnrts mor rsolutions n hn givs pplitions mor rsolutions rom whih to hoos. Compr with 4- siz n -siz rnmnts th -siz rnmnt is not so ppling or multirsolution prossing sin this rnmnt rsults in rution in rsolution y tor whih is ors. Howvr osrv rom th right pitur o Fig. tht th hxgonl lmntry lls (with thin hxgon gs) ssoit with G n ggrgt in groups o svn to orm ojts whih r lmost th ors-rsolution hxgonl lls (with thi hxgon gs) ssoit with G whil nithr G 4 nor G hs suh proprty. Thror th - siz rnmnt ws wily us in plnr multirsolution n hxgon-s gri s 9]. On th othr hn though th - siz rnmnt multirsolution img prossing is onsir (s.g. ]) th -tp orthogonl ltr ns in 8] r th only -siz rnmnt orthogonl/iorthogonl ltr ns vill in th litrtur. Ths ltrs wr onstrut or th purpos o img oing n thir ssoit wvlts r not ontinuous whih mns tht thos ltrs r not suitl or multirsolution img prossing sin rtin smoothnss o wvlts is rquir whn th ltrs r us or multirsolution img prossing pplitions. Thror th onstrution/sign o -siz rnmnt hxgonl ltr ns srvs our invstigtion. A rgulr hxgonl ltti hs 6-ol symmtry whil squr ltti hs 4-ol symmtry. Th tur o highr symmtry or th hxgonl ltti ms img prossing mor urt s ]. Th symmtry o hxgonl ltr ns whih is losly rlt to th symmtry strutur o th hxgonl ltti is lso importnt or img prossing. For xmpl th symmtry o th hxgonl ltr ns in 0] ls to simplr lgorithms n int omputtions s 0]. Thror or th hxgonl ltti it is sirl tht th ltr ns long it lso hv 6-ol symmtry. Th lowpss ltrs onsir in 0]-] o hv 6-ol symmtry ut th nit impuls rspons (FIR) ltr ns onstrut in ths pprs r not prt ronstrution ltr ns. Th ltr ns onstrut in ]-6] r orthogonl or prt ronstrution ltr ns ut thy hv only -ol symmtry. In our stuy o -siz rnmnt ltr ns w n tht it is possil to onstrut -siz rnmnt orthogonl FIR hxgonl ltr ns with 6-ol symmtry whil on n h irtly tht it is impossil to onstrut 6-ol symmtri 4-siz or -siz rnmnt orthogonl FIR ltr ns with rsonl lrg ltr lngth. Th min ojtiv o this ppr is to onstrut -siz rnmnt orthogonl n iorthogonl FIR hxgonl ltr ns with 6-ol symmtry. This ppr is orgniz s ollows. In Stion II w provi -siz rnmnt multirsolution omposition n ronstrution lgorithms n som si rsults on th orthogonlity/iorthogonlity o -siz rnmnt ltr ns. In Stion III w prsnt lo strutur o FIR ltr ns with 6- ol rottionl symmtry. Th onstrution o orthogonl n iorthogonl FIR ltr ns o 6-ol rottionl symmtry r isuss in Stions IV n V rsp. Finlly in Stion VI w onsir FIR ltr ns with 6-ol xil symmtry. W provi onition or ltr n to hv 6-ol xil symmtry. W n it is hr to otin lo strutur o orthogonl or iorthogonl ltr ns with 6-ol xil symmtry. Bus o this w onsir in Stion VI nothr typ o symmtry ll psuo 6-ol xil symmtry n otin lo strutur o FIR ltr ns with suh symmtry. In this ppr w us th ollowing nottions. For positiv intgr n I n nots th n n intity mtrix. For mtrix M w us M to not its onjugt trnspos M T n or nonsingulr mtrix M M T nots (M ) T. W us ol- lttrs suh s x ω to not lmnts o IR. x y nots th ot (innr) prout x T y o x = x x ] T y = y y ] T IR. For x = x x ] T IR not x = x x n or untion on IR ˆ nots its Fourir trnsorm: ˆ(ω) = (x) IR ix ω x. II. MULTIRESOLUTION IMAGE PROCESSING WITH -SIZE REFINEMENT FILTER BANKS In this stion w provi -siz rnmnt multirsolution omposition n ronstrution lgorithms n prsnt som si rsults on th orthogonlity/iorthogonlity o -siz rnmnt ltr ns. A. -siz rnmnt multirsolution lgorithms Rll tht G is th ors ltti o G tr on -siz rnmnt itrtion whr G is th unit rgulr hxgonl ltti givn y G = {n v + n v : (n n ) Z } whr v = 0] T v = ]T. In gnrl lt G n not th ors ltti tr n stps o -siz rnmnt itrtions. For n (input) img smpl on G th nos o G n n onsir s th smpling points o th susmpl img whn th multirsolution omposition lgorithm is ppli n tims to th input img. To provi th multirsolution img omposition n ronstrution

3 Q. JIANG: ORTHOGONAL AND BIORTHOGONAL FIR HEXAGONAL FILTER BANKS WITH SIXFOLD SYMMETRY lgorithms w n to hoos mtrix N ll th iltion mtrix suh tht it mps th hxgonl ltti G j onto its ors ltti G j nmly NG j = G j whr NG j = {Ng : g G j }. On my hoos N to mtrix tht mps A = {v v v +v v v v v } onto B = {V V V +V V V V V } whr V = v v V = v + v. (A n B r susts o G n G rsp. with thir lmnts orming hxgons.) Thr r svrl hois or suh mtrix N. For xmpl w my hoos N to on o th mtris: ] ] N = 5 5 N =. () As sts oth N G j n N G j r th ors ltti G j. But onsiring th lmnts o N G j n N G j on osrvs tht N G j ps th orinttion ut is rott lowis out 9. with rspt to th xs o G j (s th lt prt o Fig. 4 whr r th imgs o with N ) whil th xs o N G j r rott n rt rom thos o G j (s th right prt o Fig. 4 whr r th imgs o with N ). On lso osrvs tht th xs o (N ) G j r th sm s thos o G j sin (N ) is I. In 9] th susmpling with N (N rsp.) is ll th spirling (toggling rsp.) susmpling. For -siz rnmnt multirsolution hxgonl img prossing on hooss on o suh iltion mtris (n thn ps using this mtrix uring th prour o multirsolution prossing). It shoul up to on's spi pplition to us N N or nothr iltion mtrix. Fig. 4. Hxgonl ltti with spirling rnmnt (lt) n hxgonl ltti with toggling rnmnt (right) For squn {H g } g G o rl numrs ssoit with G lt H(ω) not th ltr with its impuls rspons oints ing H g (hr tor / is or onvnin): H(ω) = (/) g G H g ig ω. Suh ltr is ll hxgonl ltr. For -siz rnmnt multirsolution img prossing lowpss hxgonl ltr P (ω) hs 6 ssoit highpss ltrs Q () (ω) Q (6) (ω). Lt N iltion mtrix whih mps G j onto G j. I w us on ltr n {P Q () Q (6) } s th nlysis ltr n n us nothr ltr n { P Q () Q (6) } s th synthsis ltr n thn th multirsolution (sun) omposition lgorithm with iltion mtrix N or n input img C g0 g G smpl on G is { Chj+ = (/ ) g G P g NhC gj D (l) hj+ = (/ ) g G Q(l) g Nh C gj with l = 6 h G or j = 0 J whr J is positiv intgr; n th ronstrution lgorithm is Ĉ gj = ( P g Nh Ĉ hj+ + Q (l) g Nh D(l) hj+ ) h G l 6 h G or j = J J 0 whr ĈhJ = C hj. I Ĉ gj = C gj 0 j J or ny input img C g0 g G thn w sy tht {P Q () Q (6) } n { P Q () Q (6) } r prt ronstrution ltr ns (PR) or tht thy r iorthogonl ltr ns. W sy ltr n {P Q () Q (6) } to orthogonl i it is iorthogonl to itsl. Assum tht {P Q () Q (6) } n { P Q () Q (6) } r two FIR hxgonl ltr ns. Lt Φ n Φ th sling untions (with iltion mtrix N) ssoit with lowpss ltrs P (ω) n P (ω) rsp. nmly Φ Φ stisy th rnmnt (two-sl) qutions: Φ(x) = g G P g Φ(Nx g) Φ(x) = g G P g Φ(Nx g). () Lt Ψ (l) Ψ (l) l 6 th untions n y Ψ (l) (x) = g G Q(l) g Φ(Nx g) Ψ (l) (x) = (l) g G Q g Φ(Nx g). I {P Q () Q (6) } n { P Q () Q (6) } r iorthogonl to h othr thn unr rtin mil onitions (s th nxt sustion) Φ n Φ r iorthogonl uls: Φ(x) Φ(x g) x = δ IR g g G whr δ g is th ronr-lt squn: δ g = i g = (0 0) n δ g = 0 i g (0 0). In this s Ψ (l) Ψ (l) l 6 r iorthogonl wvlts nmly {Ψ (l) jg : l 6 j Z g G} n (l) { Ψ jg : l 6 j Z g G} r Risz ss o L (IR ) n thy r iorthogonl to h othr: Ψ (l) jg (x) Ψ (l ) j g (x)x = δ j j δ l l δ g g IR () or j j Z l l 6 g g G whr Ψ (l) jg (x) = j Ψ (l) (N j x g) Ψ (l) jg (x) = j Ψ(l) (N j x g). For th -siz rnmnt on n otin iorthogonlity onition or th sling untions Φ Φ n orthogonlity/iorthogonlity onitions or hxgonl ltr ns. Th rr rrs to 0] 40] or th iorthogonlity onitions or 4-siz rnmnt hxgonl ltr ns. Consiring tht most multirsolution nlysis thory n lgorithms or img prossing r vlop long th squr ltti Z n tht most popl r mor milir with ltr ns on th squr ltti in th nxt sustion w giv iorthogonlity onitions or pir o Φ n Φ n or pir o -siz rnmnt hxgonl ltr ns tr w trnsorm thm into sling untions φ φ n ltr ns long Z. In th rst o this sustion w giv th nitions o th symmtris o ltr ns onsir in this ppr. Dnition : A hxgonl ltr n {P Q () Q (6) } is si to hv 6-ol rottionl symmtry i its lowpss ltr P (ω) is invrint unr π π π 4π 5π rottions n its

4 4 IEEE TRANS. SIGNAL PROC. VOL. 56 NO DEC. 008 v highpss ltrs Q () Q (6) r π π π 4π n 5π (ntilowis) rottions o th highpss ltr Q () rsp. Som untions suh s -irtion ox-splins B v v v v v v v v ] long th hxgonl ltti G hv 6-ol xil (lin) symmtry nmly thy r symmtri roun xs S 0 S 5 in G shown in Fig. 5 (s th symmtry o -irtion ox-splins in 4] t p.4-6 whr 6-ol xil symmtry is ll th ull st o symmtris). It is nturl tht on onsirs th onstrution o sling untions with suh symmtry. To this rgr w onsir hxgonl ltr ns with 6-ol xil symmtry. Fig. 5. v S 5 6 xs (lins) o symmtry in rgulr hxgonl ltti Dnition : A hxgonl ltr n {P Q () Q (6) } is si to hv 6-ol xil symmtry or 6-ol lin symmtry i its lowpss ltr P (ω) is symmtri roun S 0 S 5 n its highpss ltr Q () is symmtri roun th xis S 0 n othr v highpss ltrs Q () Q (6) r π π π 4π n 5π (ntilowis) rottions o highpss ltr Q() rsp. W n tht i {P Q () Q (6) } hs 6-ol xil symmtry thn it hs 6-ol rottionl symmtry s Stion VI. W lso n tht it is hr to otin mily o orthogonl or iorthogonl ltr ns with 6-ol xil symmtry. Bus o this w onsir in Stion VI nothr typ o symmtry ll psuo 6-ol xil symmtry n provi mily o FIR ltr ns with suh symmtry. B. Trnsorming th hxgonl ltti to th squr ltti Z Lt U th mtrix n y U = v S 0 S 4 0 ] S S S. (4) Thn U trnsorms th rgulr unit hxgonl ltti G into th squr ltti Z. For hxgonl ltr H(ω) = g G H g ig ω tr th trnsormtion with th mtrix U w hv orrsponing ltr h(ω) = Z h i ω or squrly smpl t with its impuls rspons h = H U. Convrsly or squr ltr (ltr on th squr ltti) h(ω) = Z h i ω w hv orrsponing hxgonl ltr H(ω) = g G H g ig ω with H g = h Ug. Th mtrix U lso trnsorms th sling untions n wvlts long th hxgonl ltti to thos long th squr ltti Z. Mor prisly suppos {P Q () Q (6) } n { P Q () Q (6) } r pir o FIR hxgonl ltr ns. Lt Φ n Φ th sling untions (with iltion mtrix N) ssoit with P (ω) n P (ω) rsp. n Ψ (l) n Ψ (l) l 6 r th untions n y (). Lt {p q () q (6) } n { p q () q (6) } th orrsponing squr ltr ns. Dn φ(x) = Φ(U x) ψ (l) (x) = Ψ (l) (U x) φ(x) = Φ(U x) ψ(l) (x) = Ψ (l) (U x). Thn φ n φ r sling untions stisying th ollowing rnmnt qutions: φ(x) = p φ(mx ) φ(x) = p φ(mx ) (6) Z Z n ψ (l) ψ (l) l 6 r givn y ψ (l) (x) = Z q(l) φ(mx ) ψ (l) (x) = Z q(l) φ(mx ) whr p p q (l) q(l) r xtly th impuls rspons o- ints o p(ω) p(ω) q (l) (ω) q (l) (ω) rsp. n M = UNU. Osrv tht or iltion mtris N n N n y () th iltion mtris M = UN U n M = UN U or φ n φ r intgr mtris: ] ] M = M =. (8) Equtions (6) r th stnr (tritionl) rnmnt qutions or sling untions long th ommonly us intgr ltti Z with (gnrl) intgr iltion mtrix M n () r th stnr ormuls to n th orrsponing wvlts. For pir o squr ltr ns {p q () q (6) } n { p q () q (6) } th tritionl multirsolution omposition n ronstrution lgorithms with iltion mtrix M or n input squrly smpl img 0 r (rr to 4]) { nj+ = (/ ) Z p Mn j (l) nj+ = (/ ) Z q(l) Mn j with l = 6 n Z or j = 0 J n ĉ j = ( p Mn ĉ nj+ + n Z l 6 n Z q (l) Mn (l) (5) () nj+ ) with Z or j = J J 0 whr ĉ nj = nj. W sy squr ltr ns {p q () q (6) } n { p q () q (6) } to PR ltr ns i ĉ j = j 0 j J or ny input squrly smpl img 0. Clrly {P Q () Q (6) } n { P Q () Q (6) } r PR ltr ns (with N) i n only i thir ountrprts {p q () q (6) } n { p q () q (6) } r PR ltr ns (with M = UNU ). From th intgr-shit invrint multirsolution nlysis thory w n otin tht {p q () q (6) } n { p q () q (6) } r PR ltr ns i n only i p(ω + πm T η ) p(ω + πm T η ) = (9) p(ω + πm T η ) q (l) (ω + πm T η ) = 0 (0) q (l ) (ω + πm T η ) q (l) (ω + πm T η ) = δ l l ()

5 Q. JIANG: ORTHOGONAL AND BIORTHOGONAL FIR HEXAGONAL FILTER BANKS WITH SIXFOLD SYMMETRY 5 or l l 6 ω IR whr η j 0 j 6 r th rprsnttivs o th group Z /(M T Z ) with η 0 = 0 0] T. Whn M is th iltion mtrix M or M in (8) η j j 6 r η = ] T η = 0 ] T η = 0] T η 4 = ] T η 5 = 0 ] T η 6 = 0] T. () Squr ltr ns {p q () q (6) } n { p q (6) } r ommonly si to iorthogonl i thy stisy (9)-(); n ltr n {p q () q (6) } is ommonly rrr to orthogonl i it stiss (9)-() with p = p q (l) = q (l) l 6. Z p i ω lt For n FIR lowpss ltr p(ω) = T p not its trnsition oprtor mtrix (with iltion mtrix M): T p = A M j ] j KK] () whr A j = (/) n Z p n jp n n K is suitl positiv intgr pning on th ltr lngth o p n th iltion mtrix M. W sy tht T p stiss Conition E i is its simpl ignvlu n ll othr ignvlus λ o T p stisy λ <. From th intgr-shit invrint multirsolution nlysis thory (s.g. 4] 44] 4]) i {p q () q (6) } n { p q () q (6) } r iorthogonl to h othr n th trnsition oprtor mtris T p n Tp ssoit with p n p stisy Conition E thn φ n φ r iorthogonl uls: φ(x) φ(x ) x = δ IR Z ; n ψ (l) ψ (l) l 6 n y () r iorthogonl wvlts nmly {ψ (l) j : l 6 j Z Z (l) } n { ψ j : l 6 j Z Z } r iorthogonl ss o L (IR ) whr ψ (l) j (x) = j ψ (l) (M j x ) ψ(l) j (x) = j ψ(l) (M j x ). From th ov isussion w now tht to sign orthogonl/iorthogonl hxgonl ltr ns w n only to onstrut squr ltr ns {p q () q (6) } n { p q () q (6) } suh tht thy stisy (9)-(). As onsqun {P Q () Q (6) } n { P Q () Q (6) } with impuls rsponss P g = p Ug Q (l) g = q (l) Ug P (l) g = p Ug Q g = r orthogonl/iorthogonl hxgonl ltr ns. Furthrmor i th trnsition oprtor mtris T p n Tp ssoit with th squr lowpss ltrs p n p stisy Conition E (n hn sling untions φ φ ssoit with p(ω) p(ω) orm iorthogonl uls n ψ (l) ψ (l) l 6 n y () gnrt iorthogonl wvlts systms) thn th hxgonl sling untions Φ Φ ssoit with P (ω) P (ω) orm iorthogonl uls n Ψ (l) Ψ (l) l 6 n y () gnrt iorthogonl wvlt systms. In ition i on wnts to sign hxgonl ltrs suh tht th orrsponing (hxgonl) sling untions Φ Φ n wvlts Ψ (l) Ψ (l) l 6 hv sirl rtin proprtis suh s pproximtion powr smoothnss n ni tim-rquny loliztions thn on ns only to onsir suh proprtis or th sling untions φ φ n wvlts ψ (l) ψ (l) l 6 ssoit with squr ltrs sin thy r rlt s in (5). Rmr : Sin Z /M T Z n Z /M T Z hv th sm rprsnttivs η j 0 j 6 i {p q () q (6) } n { p q () q (6) } r iorthogonl with on o M M sy M thn thy r lso iorthogonl to h othr with th q (l) Ug othr iltion mtrix M. Consquntly i pir o hxgonl ltr ns r iorthogonl with on o iltion mtris N N thn thy r lso iorthogonl to h othr with th othr iltion mtrix. In th ollowing stions w onsir onstrution o orthogonl/iorthogonl ltr ns with 6-ol symmtry. From Rmr w n only to onsir on o th iltion mtris M M. In th rst o this ppr without loss o gnrlity w hoos M to M. III. FILTER BANKS WITH 6-FOLD ROTATIONAL SYMMETRY In this stion w onsir ltr ns with ] 6-ol rottionl os θ sin θ symmtry. Lt R(θ) = not th rottion sin θ os θ mtrix. Dnot R = R( π ) Rj = ( R ) j j 5. (4) In othr wors R R 5 r th (lowis) rottion mtris o π π π 4π 5π rsp. Thn 6-ol rottionl symmtry o ltr n {P Q () Q (6) } mns tht P = P Rjg g Q (j+) g = Q () j 5 g G. (5) R j g For hxgonl ltr n {P Q () Q (6) } lt {p q () q (6) } its orrsponing squr ltr n y th trnsormtion with th mtrix U in (4). Nmly th impuls rsponss p q (l) o p(ω) q (l) (ω) l 6 r P U Q (l) U rsp. Lt R R R 5 not th mtris U R U U R U U R 5 U rsp. Mor prisly ] ] 0 R = R = 0 (6) R = I R 4 = R R 5 = R. Thn on n show tht P Q () Q (6) stisy (5) i n only i p q () q (6) stisy p Rj = p q (j+) = q () R j j 5 Z. () To summriz w hv th ollowing proposition. Proposition : Lt {P Q () Q (6) } hxgonl ltr n n {p q () q (6) } its orrsponing squr ltr n. Thn {P Q () Q (6) } hs 6-ol rottionl symmtry i n only i {p q () q (6) } stiss (). In th ollowing or onvnin w sy squr ltr n {p q () q (6) } hs 6-ol rottionl symmtry i it stiss (). Clrly () is quivlnt to p(r T j ω) = p(ω) q (j+) (ω) = q () (R T j ω) j 5. This n th ts tht R j = R j j 5 n R6 = I imply th ollowing proposition. Proposition : A ltr n {p q () q (6) } hs 6-ol rottionl symmtry i n only i it stiss p q () q (6)] T (R T ω) = M 0 p(ω) q () (ω) q (6) (ω) ] T (8)

6 6 IEEE TRANS. SIGNAL PROC. VOL. 56 NO DEC. 008 whr M 0 = (9) Nxt w onstrut th ltr n {p q () q (6) } s th prout o pproprit lo mtris. Assum tht w n writ p(ω) q () (ω) q (6) (ω)] T s B(M T ω)p s (ω) q s () (ω) q s (6) (ω)] T whr M is M n in (8) B(ω) is mtrix whos ntris r trigonomtri polynomils n {p s q s () q s (6) } is nothr FIR ltr n with shortr ltr lngths. I oth {p q () q (6) } n {p s q s () q s (6) } hv 6-ol rottionl symmtry thn Proposition implis tht B(ω) stiss B(M T R T ω) = M 0 B(M T ω)m 0 (0) whr M 0 is th mtrix n y (9). Dnot I 0 (ω) = i(ω +ω ) iω iω i(ω +ω ) iω iω ] T. () Thn rom R T ω = ω +ω ω ] T w hv I 0 (R T ω) = iω iω i(ω +ω ) iω iω i(ω +ω ) ] T. Thus I 0 (ω) stiss (8). Thror -tp ltr n { i(ω+ω) iω iω i(ω+ω) iω iω } hs 6- ol rottionl symmtry n it oul us s th initil ltr n with 6-ol rottionl symmtry. Dnot D(ω) = ig( i(ω +ω ) iω iω i(ω +ω ) iω iω ). () Thn D(M T ω) = ig( i(ω+ω) i(ω+ω) i(ω ω) i(ω+ω) i(ω+ω) i(ω ω) ). On th othr hn ] with M T R T = on hs tht D(M T R T ω) = ig( i(ω +ω ) i(ω ω ) i(ω +ω ) i(ω +ω ) i(ω ω) i(ω+ω) ). Thror D(ω) stiss (0) n it oul us to uil th lo mtris. Nxt w us B(ω) = BD(ω) s th lo mtrix whr B is (rl) onstnt mtrix. Bs on th ov isussion w now tht B(ω) = BD(ω) stiss (0) i n only i B stiss M 0 BM0 = B whih is quivlnt to th t tht B hs th orm: B = () Bs on th ov isussion w rh th ollowing rsult on th ltr ns with 6-ol rottionl symmtry. Thorm : I {p q () q (6) } is givn y p(ω) q () (ω) q (6) (ω)] T = (4) B n D(M T ω)b n D(M T ω) B D(M T ω)b 0 I 0 (ω) or som n Z + whr I 0 (ω) is n y () n B 0 n r onstnt mtris o th orm () thn {p q () q (6) } is n FIR ltr n with 6-ol rottionl symmtry. In th nxt two stions w show tht th lo strutur in (4) yils orthogonl n iorthogonl FIR ltr ns with 6-ol rottionl symmtry. IV. ORTHOGONAL FILTER BANKS WITH 6-FOLD ROTATIONAL SYMMETRY In this stion w stuy th onstrution o orthogonl ltr ns with 6-ol rottionl symmtry. For n FIR ltr n {p q () q (6) } not q (0) (ω) = p(ω). Lt U(ω) mtrix n y U(ω) = q (l) (ω + η j ) ] 0 lj 6 whr η 0 η η 6 r givn in (). Thn {p q () q (6) } is orthogonl i U(ω) is unitry or ll ω IR tht is it stiss U(ω)U(ω) = I ω IR. (5) Nxt w writ q (l) (ω) 0 l 6 s q (l) (ω) = ( q (l) 0 (M T ω) + q (l) (M T ω) i(ω +ω ) +q (l) (M T ω) iω + q (l) +q (l) 5 (M T ω) iω + q (l) 6 (M T ω) iω (M T ω) iω + q (l) 4 (M T ω) i(ω +ω ) ) whr q (l) (ω) r trigonomtri polynomils. Lt V (ω) not th polyphs mtrix (with iltion mtrix M) o {p(ω) q () (ω) q (6) (ω)}: V (ω) = q (l) ]0 l 6 (ω). (6) Clrly p(ω) q () (ω) q (6) (ω)] T = V (M T ω)i 0 (ω) whr I 0 (ω) is n y (). On n vriy tht th mtrix I 0 (ω + πm T η 0 ) I 0 (ω + πm T η ) I 0 (ω + πm T η 6 )] is unitry or ll ω IR whih implis tht (5) hols i n only i V (ω) is unitry or ll ω IR nmly V (ω) stiss V (ω)v (ω) = I ω IR. () Thror to onstrut n orthogonl ltr n {p q () q (6) } w n only to onstrut trigonomtri polynomil mtrix V (ω) whih stiss (). I {p q () q (6) } is givn y (4) thn its polyphs mtrix V (ω) is V (ω) = B n D(ω)B n D(ω) B D(ω)B 0. Sin D(ω) = ig( i(ω +ω ) iω iω i(ω +ω ) iω iω ) is unitry w now tht i onstnt mtris

7 Q. JIANG: ORTHOGONAL AND BIORTHOGONAL FIR HEXAGONAL FILTER BANKS WITH SIXFOLD SYMMETRY B 0 n r orthogonl thn V (ω) stiss (). Nxt w onsir th orthogonlity o mtrix B o th rom (). To this rgr lt W not th unitry mtrix: W = ( 6/6) w j] πi whr w = 6 = 0 j 5 + i. With W = ig( W ) on n gt tht whr W BW = = = 5 + ( ) i( ) 44 = + 5 ( ) i( ) 55 = = 44 =. Thus B is orthogonl i n only i ] 6 6 orthogonl n jj = j whih implis tht n jj j = 4 5 n writtn s = s 0 os θ 6 = sin θ 6 = s 0 sin θ = os θ = os γ i sin γ 44 = os ζ i sin ζ 55 = s whr s 0 = ± s = ± θ γ ζ IR. From ths qutions w hv = s 0 os θ = 6 6 sin θ = s sin θ = 6 (s os θ + os γ + os ζ) = 6 ( s os θ + os γ + sin γ os ζ + sin ζ) 4 = 6 (s os θ os γ + sin γ os ζ sin ζ) 5 = 6 ( s os θ os γ + os ζ) 6 = 6 (s os θ os γ sin γ os ζ + sin ζ) = 6 ( s os θ + os γ sin γ os ζ sin ζ). (8) Thus n orthogonl mtrix B o th orm () hs thr prmtrs. W hv thror th ollowing thorm. Thorm : Suppos {p q () q (6) } is givn y (4). I h B 0 n is o th orm () n its ntris ij r givn y (8) or som θ γ ζ thn {p q () q (6) } is n orthogonl FIR ltr n with 6-ol rottionl symmtry. Trnsorming {p q () q (6) } givn in Thorm with th mtrix U to th ltr ns long th hxgonl ltti w hv mily o orthogonl FIR hxgonl ltr ns with 6-ol rottionl symmtry n lo strutur. With suh mily o orthogonl ltr ns y slting th r prmtrs on n sign th ltrs with sirl proprtis or on's spi pplitions. For xmpl on my onsir to sign ltrs with optimum tim-rquny loliztion (rr to 45]-4] or th sign o -D mtrixvlu ltrs with optimum tim-rquny loliztion). Hr is w onsir th ltrs s on th Soolv smoothnss o th ssoit sling untions φ. For s 0 lt W s not th Soolv sp onsisting o untions (x) on IR with IR ( + ω ) s ˆ(ω) ω <. Th Soolv smoothnss o sling untion φ n givn y th ignvlus o th trnsition oprtor mtrix T p n in () whr p(ω) is th lowpss ltr ssoit with φ. Mor prisly ssum tht n FIR lowpss ltr p(ω) hs sum rul orr m (with iltion mtrix M): p(0 0) = D α Dα p(πm T η j ) = 0 j 6 (9) or ll (α α ) Z + with α +α < m whr η j j 6 r n y () D n D not th prtil rivtivs with th rst n son vrils o p(ω) rsp. Unr som onition sum rul orr is quivlnt to th pproximtion orr o φ s 48]. Lt σ σ th ignvlus o th iltion mtrix M. (Whn M is M in (8) σ = 5 + i σ = 5 i.) Dnot S m := {σ α σ α : α α Z + α + α < m} n S m := sp(t p )\ S m. Thn φ is in Soolv sp W log ρ 0 whr ρ 0 = mx{ λ : λ S m }. (0) S 49] 50] or th tils n rr to 4] or lgorithms n Mtl routins to n th Soolv smoothnss orr. Rmr : Osrv tht sum rul orr o lowpss ltr p(ω) with iltion mtrix M is hrtriz y th vlus o p(0 0) n D α Dα p(ω) t πm T η j j 6. Osrv tht {M T η j : j 6} = {M T η j : j 6}. Thus i p(ω) hs sum rul orr m with iltion mtrix M thn it lso hs sum rul orr m with iltion mtrix M. Exmpl : Lt {p q () q (6) } th orthogonl ltr n with 6-ol rottionl symmtry givn y (4) with n = 0: B 0 I 0 (ω). This is -tp ltr n. Th lowpss ltr p(ω) pns on on r prmtr θ 0. W n hoos this prmtr suh tht th rsulting p(ω) hs sum rul orr. Th nonzro oints p o th rsulting p(ω) r p 00 = p = p 0 = p 0 = p = p 0 = p 0 =. W n tht th ling ignvlus o T p (with M = M ) r σ σ. Thus ρ 0 =. With log ( ) = w now tht th orrsponing sling untion φ is in W W n lso hoos othr prmtrs γ 0 η 0 or highpss ltrs suh tht th nonzro oints q () o q () (ω) r q () 00 = q() = q () 0 = q() 0 = q() = q() 0 = q() 0 =. 6 Th rsulting sling untion φ is hrtrsiti untion o rgion with rtl ounry. φ n pproximt y φ n n y φ n (x) = Z p φ n (Mx ) n =

8 8 IEEE TRANS. SIGNAL PROC. VOL. 56 NO DEC. 008 whr φ 0 is suitl omptly support untion with Z φ 0(x ) =. W show th support o φ on th lt o Fig. 6 with φ 0 (x) = χ )(x). From Rmrs n this ltr n is orthogonl with iltion mtrix M n th rsulting p(ω) hs sum rul orr with M. W h tht th ssoit sling untion not y ϕ is lso in W ϕ is hrtristi untion o rgion with rtl ounry. Th support o ϕ (x) is shown on th right o Fig.6 whr ϕ n (x) = Z p ϕ n (M x ) n = n ϕ 0 (x) = χ )(x). S 5] 5] or th rgions with rtl ounry ll rgon n twin rgon sts gnrt y th quinunx mtris. or θ 0 = tn ( ) γ 0 = π tn ( 5 ) θ = sin ( 6 ). () Thus w hv lowpss ltr p(ω) pning on on r prmtr η 0. Thn w slt this r prmtr suh tht ρ 0 in (0) s smll s possil. With η 0 = π tn ( 0 ) n θ 0 γ 0 θ givn in () w hv th (numrilly) most smooth sling untion φ whih is in W Thr r two r prmtrs γ η lt or th highpss ltrs o this orthogonl ltr n. Hr w simply st γ = η = 0. In Fig. 8 w show th piturs o th rsulting φ n ψ () Fig. 8. φ (lt) n ψ () (right) Fig. 6. Approximt support o φ with iltion M (lt) n tht o ϕ with iltion M (right) Fig.. Impuls rsponss P g (lt) n Q () g +5 (right) whil Q (j+) 6 g r πj with = = 6 rottions o Q() g j 5 Lt {P Q () Q (6) } th orrsponing orthogonl hxgonl ltr n. W pply this ltr n to hxgonlly smpl img in Fig. 9. This is prt o th hxgonl img r-smpl rom 5 5 squrly smpl img Ln y th ilinr intrpoltion in 6]. Th ompos imgs with th lowpss ltr n highpss ltrs r shown on th lt o Fig. 0 n in Fig. rsp. Ths imgs r in rott out 9. with rspt to th originl img. Th ompos img with th lowpss ltr ppli twi is shown on th right o Fig. 0. Lt {P 0 Q () 0 Q(6) 0 } th orrsponing orthogonl hxgonl ltr n. Th nonzro impuls rspons oints P g Q () g o P 0 (ω) Q () 0 (ω) r isply in Fig. whil Q (j+) g r πj rottions o Q() g j 5. Noti tht th lowpss ltr P 0 (ω) hs 6-ol xil symmtry nmly it is symmtri roun xs S 0 S 5 shown in Fig. 5 n tht Q () 0 (ω) is symmtri roun xis S 0. Thror this orthogonl ltr n hs 6-ol xil symmtry. In 8] thr -tp orthogonl hxgonl ltr ns wr onstrut or img oing with ll ltr ns hving th sm lowpss ltr whih is xtly P 0 (ω). Th highpss ltrs in 8] r irnt rom Q (l) 0 (ω) l 6. Thr o thir highpss ltrs r symmtri roun th xis S n th othr thr r nti-symmtri roun S. Exmpl : Lt {p q () q (6) } th orthogonl ltr n with 6-ol rottionl symmtry givn y (4) or n = : B D(M T ω)b 0 I 0 (ω) with s 0 s in (8) ing. Th lowpss ltr p(ω) pns on our prmtrs θ 0 γ 0 η 0 θ. By solving th qutions or sum rul orr w gt θ 0 = tn ( 6 58) γ 0 = tn ( 5 ) θ = sin ( 6 ); () Fig. 9. Originl (hxgonl) img Fig. 0. Dompos imgs with lowpss ltr P with on (lt) n two (right) itrtions W lso onsir orthogonl ltr ns givn (4) with w mor los B D(M T ω). W n tht it is hr to onstrut orthogonl sling untions

9 Q. JIANG: ORTHOGONAL AND BIORTHOGONAL FIR HEXAGONAL FILTER BANKS WITH SIXFOLD SYMMETRY 9 Fig.. Dompos imgs with highpss ltrs Q () Q () Q () (top row rom lt) n Q (4) Q (5) Q (6) (ottom row rom lt) with high smoothnss orr. For xmpl it sms tht B D(M T ω)b D(M T ω)b 0 I 0 (ω) nnot l to n orthogonl sling untion in W. Thus to onstrut orthogonl sling n wvlts with ni smoothnss orr w n to us mor los B D(M T ω). In th nxt stion w onsir 6-ol rottionl symmtri iorthogonl ltr ns whih giv us som xility or th onstrution o PR ltr ns. V. BIORTHOGONAL FILTER BANKS WITH 6-FOLD ROTATIONAL SYMMETRY In this stion w onsir iorthogonl ltr ns with 6-ol rottionl symmtry. Suppos {p q () q (6) } n { p q () q (6) } r pir o ltr ns. Lt V (ω) n Ṽ (ω) thir polyphs mtris n s in (6). Thn on n otin s in th ov stion tht {p q () q (6) } n { p q () q (6) } r iorthogonl to h othr i n only i V (ω) n Ṽ (ω) stisy V (ω)ṽ (ω) = I ω IR. I {p q () q (6) } is th FIR ltr n givn y (4) or som rl mtris B thn V (ω) = B n D(ω)B n D(ω) B D(ω)B 0. I th onstnt mtris B 0 n r nonsingulr thn rom th t tht (D(ω) ) = D(ω) w now (V (ω) ) = Bn T D(ω)Bn T D(ω) B T D(ω)B0 T. Clrly i B hs th orm o () nmly M 0 B M0 = B thn M 0 B T M0 = B T sin M0 T = M0. Thus B T lso hs th orm o (). Hn y Proposition { p q () q (6) } givn y p(ω) q () (ω) q (6) (ω)] T = () Bn T D(M T ω) B T D(M T ω)b0 T I 0 (ω) with I 0 (ω) n y () hs 6-ol rottionl symmtry. For this ltr n { p q () q (6) } sin its polyphs mtrix Ṽ (ω) is (V (ω) ) w now it is iorthogonl to {p q () q (6) }. To summriz w hv th ollowing thorm. Thorm : Suppos tht {p q () q (6) } r th FIR ltr n givn y (4) whr B 0 n r nonsingulr onstnt mtris o th orm (). Lt { p q () q (6) } th FIR ltr n givn y (). Thn { p q () q (6) } is n FIR ltr n iorthogonl to {p q () q (6) } n it hs 6-ol rottionl symmtry. Thorm provis mily o iorthogonl FIR ltr ns with 6-ol rottionl symmtry. Compr with th orthogonl ltr ns in Thorm this mily o iorthogonl ltr ns llows or som xiility in ltr sign. Exmpl : Lt {p q () q (6) } n { p q () q (6) } th iorthogonl ltr ns with 6-ol rottionl symmtry givn y Thorm with n =. W n hoos th r prmtrs or B 0 n B suh tht th rsulting sling untions φ W.80 φ W 0.68 with th orrsponing lowpss ltrs p(ω) n p(ω) hving sum rul orrs n rsp. Th slt prmtrs r provi in Appnix A. Exmpl 4: Lt {p q () q (6) } n { p q () q (6) } th iorthogonl ltr ns with 6-ol rottionl symmtry givn y Thorm with n =. In this s w n hoos th r prmtrs or B 0 B n B suh tht th rsulting sling untions φ W.98 φ W W n lso slt othr prmtrs suh tht φ W.809 φ W 0.4 n w us Bio to not th rsulting pir o iorthogonl ltr ns. In ithr s th orrsponing lowpss ltrs p(ω) n p(ω) hving sum rul orrs n rsp. Th slt othr prmtrs or Bio r provi in Appnix B. VI. ORTHOGONAL/BIORTHOGONAL FILTER BANKS WITH PSEUDO 6-FOLD AXIAL SYMMETRY In Stions III-V w isuss th onstrution o ltr ns with 6-ol rottionl symmtry. In this stion w onsir ltr ns with 6-ol xil symmtry. Dnot ] L 0 = L = ] 0 L = 0 L = L 0 L 4 = L L 5 = L. Lt Rj j 5 th rottionl mtris n in (4). Thn 6-ol xil symmtry o {P Q () Q (6) } mns tht or 0 5 j 5 P L g = P g Q () L = Q() g 0g ] Q (j+) g = Q () g G. (4) R jg On n vriy tht R = L L0. This togthr with th t R j = ( R ) j j 5 implis tht i {P Q () Q (6) } hs 6-ol xil symmtry thn it hs 6-ol rottionl symmtry. On th othr hn rom th t tht L = R L0 L = L L0 L L = L L0 L L 4 = L 0 L L0 L 5 = L 0 L L0 w now tht th onition P L0g = P g n P Rg = P g or P (ω) is quivlnt to th onition P L g = P g 0 5. Thus w hv simplr onitions or th 6-ol xil symmtry o ltr n s summriz in th nxt proposition. Proposition : A hxgonl ltr n {P Q () Q (6) } hs 6-ol xil symmtry i n only i or j 5 g G P L0g = P Rg = P g Q () L = Q() g 0g Q (j+) g = Q () R. (5) jg

10 0 IEEE TRANS. SIGNAL PROC. VOL. 56 NO DEC. 008 For hxgonl ltr n {P Q () Q (6) } lt {p q () q (6) } th orrsponing squr ltr n y th trnsormtion with th mtrix U in (4). Lt L 0 not th mtrix U L 0 U nmly L 0 = 0 0 Thn P Q () Q (6) stisy (5) i n only i p q () q (6) stisy p L0 = p R = p q () ]. L = 0 q() or j 5 Z whih is quivlnt to q(j+) = q () R j p(l 0 ω) = p(r T ω) = p(ω) q () (L 0 ω) = q () (ω) q (j+) (ω) = q () (R T j ω) (6) or j 5. From Proposition (6) n th rltionship mong L 0 R j j 5 on n otin th ollowing proposition (th til o rivtion is omitt hr). Proposition 4: A hxgonl ltr n {P Q () Q (6) } hs 6-ol xil symmtry i n only i its orrsponing squr ltr n {p q () q (6) } stiss (8) n p q () q (6)] T (L0 ω) = () whr N 0 p(ω) q () (ω) q (6) (ω)] T N 0 = Suppos {p q () q (6) } n writtn p(ω) q () (ω) q (6) (ω)] T = C(ω)p s (ω) q () s (ω) q (6) s (ω)] T. (8) whr C(ω) is mtrix with trigonomtri polynomil ntris n {p s q s () q s (6) } is nothr FIR ltr n. I oth {p q () q (6) } n {p s q s () q s (6) } stisy (6) thn Proposition implis tht C(ω) stiss C(R T ω) = M 0 C(ω)M 0 C(L 0ω) = N 0 C(ω)N 0 (9) whr M 0 n N 0 r th mtris n y (9) n (8) rsp. For I 0 (ω) n y () w show in Stion III tht I 0 (ω) stiss (8). On th othr hn I 0 (L 0 ω) = i(ω +ω ) iω iω i(ω +ω ) iω iω ] T Thus I 0 (ω) lso stiss (). Tht is ltr n { i(ω +ω ) iω iω i(ω +ω ) iω iω } hs 6-ol xil symmtry n hn this -tp ltr n shoul us s th initil symmtri ltr n. Lt D(ω) th igonl mtrix n y (). Thn on n vriy tht D(ω) stiss (9). Thror i {p q () q (6) } is givn y p(ω) q () (ω) q (6) (ω)] T = (40) C n D(ω)C n D(ω) C D(ω)C 0 I 0 (ω) whr n Z + n h C 0 n stisy (9) or quivlntly it hs th orm o C = (4) thn {p q () q (6) } stiss (). Thror th orrsponing hxgonl ltr n hs 6-ol xil symmtry. Th prolm is tht th strutur in (40) hrly yils orthogonl or iorthogonl ltr ns. To otin i(orthogonl) ltr ns w shoul toriz p(ω) q () (ω) q (6) (ω)] T s C(M T ω)p s (ω) q s () (ω) q s (6) (ω)] T. Howvr i th si lo mtrix is C(M T ω) or som trigonomtri polynomil mtrix C(ω) thn it will hr to n simpl C(ω) suh tht C(M T ω) stiss (9). This is u to th t tht thr is 9. rottion o th xs o sultti G with rspt to th xs o G. Bus o ths iultis to onstrut lo strutur o ltr ns with 6-ol xil symmtry in th ollowing w onsir nothr typ o symmtry. Dnition : A hxgonl ltr n {P Q () Q (6) } is si to hv psuo 6-ol xil symmtry i th polyphs mtrix V (ω) o its orrsponing squr ltr n {p q () q (6) } stiss (9). Th lt prt o Fig. shows 49-tp lowpss ltr with psuo 6-ol xil symmtry whil th right prt o Fig. shows -tp lowpss ltr with 6-ol xil symmtry. Noti tht th symmtry o th 49-tp lowpss ltr is not only losly rlt to th symmtry strutur o hxgonl ltti G ut lso rlt to th strutur o sultti G. I th polyphs mtrix V (ω) o {p q () q (6) } stiss th rst qution in (9) nmly V (R T ω) = M 0 V (ω)m0 thn y th ts R T M T R T = M T n I 0 (R T ω) = M 0 I 0 (ω) w hv p q () q (6) ](R T ω) = V (M T R T ω)i 0 (R T ω) = (M 0 V (R T M T R T ω)m 0 )M 0I 0 (ω) = M 0 V (M T ω)i 0 (ω) = M 0 p q () q (6) ](ω). This togthr with Proposition implis tht {p q () q (6) } hs 6-ol rottionl symmtry. Thror w lso now tht i hxgonl ltr n hs psuo 6-ol xil symmtry thn it hs 6-ol rottionl symmtry. I th polyphs mtrix V (ω) o {p q () q (6) } is givn y th prout o C D(ω) or som C o th orm (4) thn V (ω) = V n (ω) V (ω)v 0 stiss (9). Hn th

11 Q. JIANG: ORTHOGONAL AND BIORTHOGONAL FIR HEXAGONAL FILTER BANKS WITH SIXFOLD SYMMETRY g g g g g g Fig.. Impuls rspons o lowpss ltr with psuo 6-ol xil symmtry (lt) n impuls rspons o lowpss ltr with 6-ol xil symmtry (right) n = : C D(M T ω)c 0 I 0 (ω) whr C 0 C r orthogonl mtris o th orm (4) with thir ntris givn y (4) or som ξ 0 ξ IR. With s j = 0 j or C 0 C y slting ξ 0 = ξ = w hv th (numrilly) most Soolv smooth sling untion φ whih is in W In this s th ssoit lowpss ltr p(ω) hs sum rul orr. Th ontours o th piturs o φ n ψ () r shown in Fig. whih my givn us som i out th psuo 6-ol xil symmtry. orrsponing hxgonl ltr n {P Q () Q (6) } hs psuo 6-ol xil symmtry. In ition i C 0 n r orthogonl thn {P Q () Q (6) } is orthogonl. To summriz w hv th ollowing rsult. Thorm 4: I {p q () q (6) } is givn y p(ω) q () (ω) q (6) (ω)] T = C n D(M T ω) C D(M T ω)c 0 I 0 (ω) (4) whr n Z + I 0 (ω) is n y () n C 0 n r onstnt mtris o th orm (4) thn th FIR hxgonl ltr n {P Q () Q (6) } orrsponing to {p q () q (6) } hs psuo 6-ol xil symmtry. In ition i C 0 n r orthogonl thn {P Q () Q (6) } is n orthogonl ltr n. For mtrix C o th orm (4) i it is orthogonl thn on n show irtly or rom (8) or th xprssion o orthogonl B o th orm () tht th ntris ij o C n writtn s = s 0 os ξ = 6 6 sin ξ = s sin ξ = 6 (s os ξ + s + s ) = 6 ( s os ξ + s s ) 4 = 6 (s os ξ s s ) 5 = 6 ( s os ξ s + s ) (4) whr s j = ± j = 0 n ξ IR. Th lo strutur in (4) lso yils iorthogonl FIR ltr ns with psuo 6-ol xil symmtry s shown in th nxt thorm. Thorm 5: I {p q () q (6) } n { p q () q (6) } r givn y p(ω) q () (ω) q (6) (ω)] T = C n D(M T ω) C D(M T ω)c 0 I 0 (ω) p(ω) q () (ω) q (6) (ω)] T = Cn T D(M T ω) C T D(M T ω)c0 T I 0 (ω) whr n Z + I 0 (ω) is n y () n C 0 n r nonsingulr onstnt mtris o th orm (4) thn th FIR hxgonl ltr ns {P Q () Q (6) } n { P Q () Q (6) } orrsponing to {p q () q (6) } n { p q () q (6) } rsp. r iorthogonl to h othr n oth o thm hv psuo 6-ol xil symmtry. Exmpl 5: Lt {p q () q (6) } th orthogonl ltr ns with psuo 6-ol xil symmtry givn y (4) with 0 Fig.. 0 Contours o φ (lt) n ψ () (right) 0 0 Compr with ltr ns with 6-ol rottionl symmtry ltr ns with psuo 6-ol xil symmtry hv highr symmtry n thus thy hv wr r prmtrs. Thror th ov lo strutur o orthogonl/iorthogonl ltr ns with psuo 6-ol xil symmtry ls to sling untions with lss smoothnss. Bus o this in this ppr w woul not provi mor xmpls on th sign o suh ltr ns s on th smoothnss o sling untions. APPENDIX A Slt prmtrs in Exmpl : th slt ij or B 0 r = = = = = = = = =.69455; n th slt ij or B r = = = = = = = = = APPENDIX B Slt prmtrs or Bio in Exmpl 4: th slt ij or B 0 r = = = = = = = = =.450;

12 IEEE TRANS. SIGNAL PROC. VOL. 56 NO DEC. 008 th slt ij or B r = =.9590 = = = = = = =.58090; n th slt ij or B r = =.0958 = = = = = = = Anowlgmnts. Th uthor thns Dl K. Pouns or his in hlp to rt Figs 9-. Th uthor lso thns th nonymous rviwrs or ming vlul suggstions tht signintly improv th prsnttion o th ppr. REFERENCES ] D.P. Ptrsn n D. Milton Smpling n ronstrution o wv-numr-limit untions in N-imnsionl Eulin sps Inormtion n Control vol. 5 no. 4 pp. 9- D. 96. ] R.M. Mrsru Th prossing o hxgonl smpl twoimnsionl signls Pro. IEEE vol. 6 no. 6 pp Jun. 99. ] R.C. Stunton n N. Story A omprison twn squr n hxgonl smpling mthos or piplin img prossing In Pro. o SPIE Vol. 94 Optis Illumintion n Img Snsing or Mhin Vision IV 990 pp ] G. Tirunlvli R. Goron n S. Pistorius Comprison o squr-pixl n hxgonl-pixl rsolution in img prossing in Proings o th 00 IEEE Cnin Conrn on Eltril n Computr Enginring vol. My 00 pp ] D. Vn D Vill T. Blu M. Unsr W. Philips I. Lmhiu n R. Vn Wll Hx-splins: novl splin mily or hxgonl lttis IEEE Trn. Img Pro. vol. no. 6 pp. 58 Jun ] L. Milton n J. Sivswrmy Hxgonl Img Prossing: A Prtil Approh Springr 005. ] X.J. H n W.J. Ji Hxgonl strutur or intllignt vision in Proings o th 005 First Intrntionl Conrn on Inormtion n Communition Thnologis Aug. 005 pp ] X.Q. Zhng Eint Fourir Trnsorms on Hxgonl Arrys Ph.D. Dissrttion Univrsity o Flori 00. 9] R.C. Stunton Th sign o hxgonl smpling struturs or img igitiztion n thir us with lol oprtors Img n Vision Computing vol. no. pp Aug ] L. Milton n J. Sivswrmy Eg ttion in hxgonl-img prossing rmwor Img n Vision Computing vol. 9 no. 4 pp D. 00. ] A.F. Lin S. Shulr J. Fn n W. Hu Mmmogrphi tur nhnmnt y multisl nlysis IEEE Trns. M. Imging vol. no. 4 pp D ] A.F. Lin n S. Shulr Hxgonl wvlt rprsnttions or rognizing omplx nnottions in Proings o th IEEE Conrn on Computr Vision n Pttrn Rognition Sttl WA Jun. 994 pp ] A.P. Fitz n R. Grn Fingrprint lssition using hxgonl st Fourir trnsorm Pttrn Rognition vol. 9 no. 0 pp ] S. Priswmy Dttion o mirolition in mmmogrms using hxgonl wvlts M.S. thsis Dpt. o Computr Sin Univ. o South Crolin Columi SC ] R.C. Stunton On-pss prlll hxgonl thinning lgorithm IEE Proings on Vision Img n Signl Prossing vol. 48 no. pp F ] A. Cmps J. Br I.C. Snhuj n F. Torrs Th prossing o hxgonlly smpl signls with stnr rtngulr thniqus: pplition to -D lrg prtur synthsis intrromtri riomtrs IEEE Trns. Gosin n Rmot Snsing vol. 5 no. pp Jn. 99. ] E. Antrriu P. Wltul n A. Lnns Apoiztion untions or -D hxgonlly smpl synthti prtur imging riomtrs IEEE Trns. Gosin n Rmot Snsing vol. 40 no. pp D ] D. Whit A.J. Kimrling n W.S. Ovrton Crtogrphi n gomtri omponnts o glol smpling sign or nvironmntl monitoring Crtogrphy n Gogrphi Inormtion Systms vol. 9 no. pp ] K. Shr D. Whit n A.J. Kimrling Gosi isrt glol gri systms Crtogrphy n Gogrphi Inormtion Sin vol. 0 no. pp. 4 Apr ] M.J.E. Goly Hxgonl prlll pttrn trnsormtions IEEE Trns. Computrs vol. 8 no. 8 pp. 40 Aug ] P.J. Burt Tr n pyrmi struturs or oing hxgonlly smpl inry imgs Computr Grphis n Img Pro. vol. 4 no. pp ] L. Kolt -suivision in SIGGRAPH Computr Grphis Proings pp ] U. Lsi n G. Grinr Intrpoltory -suivision Computr Grphis Forum vol. 9 no. pp. 8 Sp ] Q.T. Jing n P. Oswl Tringulr -suivision shms: th rgulr s J. Comput. Appl. Mth. vol. 56 no. pp. 4 5 Jul ] Q.T. Jing P. Oswl n S.D. Rimnshnir -suivision shms: mximl sum ruls orrs Constr. Approx. vol. 9 no. pp ] P. Oswl n P. Shrör Composit priml/ul -suivision shms Comput. Ai Gom. Dsign vol. 0 no. pp Jun. 00. ] C.K. Chui n Q.T. Jing Mtrix-vlu symmtri tmplts or intrpoltory sur suivisions I: rgulr vrtis Appl. Comput. Hrmoni Anl. vol. 9 no. pp. 0 9 Nov ] P. Oswl Dsigning omposit tringulr suivision shms Comput. Ai Gom. Dsign vol. no. pp Ot ] C.K. Chui Q.T. Jing n R.N. No Tringulr n quriltrl 5 suivision shms: rgulr s J. Mth. Anl. Appl. vol. 8 no. pp. 04 F ] E. Simonlli n E. Alson Non-sprl xtnsions o qurtur mirror ltrs to multipl imnsions Proings o th IEEE vol. 8 no. 4 pp Apr ] E.A. Alson E. Simonlli n R. Hingorni Orthogonl pyrmi trnsorms or img oing In SPIE Visul Communitions n Img Prossing II (98) vol pp ] H. Xu W.-S. Lu n A. Antoniou A nw sign o -D non-sprl hxgonl qurtur-mirror-ltr ns in Pro. CCECE Vnouvr Sp. 99 pp ] A. Cohn n J.-M. Shlnr Comptly support iimnsionl wvlts ss with hxgonl symmtry Constr. Approx. vol. 9 no. / pp Jun ] J.D. Alln Coing trnsorms or th hxgon gri Rioh Cli. Rsrh Ctr. Thnil Rport CRC-TR-985 Aug ] J.D. Alln Prt ronstrution ltr ns or th hxgonl gri In Fith Intrntionl Conrn on Inormtion Communitions n Signl Prossing 005 D. 005 pp. 6. 6] Q.T. Jing FIR ltr ns or hxgonl t prossing IEEE Trns. Img Pro. in prss. ] L. Milton n J. Sivswrmy Frmwor or prtil hxgonlimg prossing J. Eltroni Imging vol. no. pp Jn ] A.B. Wtson n A. Ahum Jr. A hxgonl orthogonl-orint pyrmi s mol o img prsnttion in visul ortx IEEE Trns. Biom. Eng. vol. 6 no. pp Jn ] A. Lunmr N. Wströmr n H. Li Hirrhil susmpl giving rtl rgion IEEE Trns. Img Pro. vol. 0 no. pp. 6 Jn ] S. Shulr n A.F. Lin Hxgonl QMF ns n wvlts hptr in Tim Frquny n Wvlts in Biomil Signl Prossing M. Ay (E.) IEEE Prss 99. 4] C. Boor K. Höllig n S. Rimnshnir Box splins Springr- Vrlg Nw Yor 99.

13 Q. JIANG: ORTHOGONAL AND BIORTHOGONAL FIR HEXAGONAL FILTER BANKS WITH SIXFOLD SYMMETRY 4] C.K. Chui n Q.T. Jing Multivrit ln vtor-vlu rnl untions in Morrn Dvlopmnt in Multivrit Approximtion ISNM 45 Birhhäusr Vrlg Bsl 00 pp. 0. 4] A. Cohn n I. Duhis A stility ritrion or iorthogonl wvlt ss n thir rlt sun oing shm Du Mth. J. vol. 68 no. pp ] R.Q. Ji Convrgn o vtor suivision shms n onstrution o iorthogonl multipl wvlts In Avns in Wvlts Springr- Vrlg Singpor 999 pp ] Q.T. Jing Orthogonl multiwvlts with optimum tim rquny rsolution IEEE Trns. Signl Pro. vol. 46 no. 4 pp Apr ] Q.T. Jing On th sign o multiltr ns n orthonorml multiwvlt ss IEEE Trns. Signl Pro. vol. 46 no. pp. 9 0 D ] T. Xi n Q.T. Jing Optiml multiltr ns: sign rlt symmtri xtnsion trnsorm n pplition to img omprssion IEEE Trns. Signl Pro. vol. 4 no Jul ] R.Q. Ji Approximtion proprtis o multivrit wvlts Mth. Comp. vol. 6 no. pp Apr ] R.Q. Ji n S.R. Zhng Sptrl proprtis o th trnsition oprtor ssoit to multivrit rnmnt qution Linr Algr Appl. vol. 9 no. pp My ] R.Q. Ji n Q.T. Jing Sptrl nlysis o trnsition oprtors n its pplitions to smoothnss nlysis o wvlts SIAM J. Mtrix Anl. Appl. vol. 4 no. 4 pp ] J. Kovvić n M. Vttrli Nonspprl multiimnsionl prt ronstrution ltr ns n wvlt ss or IR n IEEE Trns. Inorm. Thory vol. 8 no. pp ] K. Gröhnig n W.R. Myh Multirsolution nlysis Hr ss n sl-similr tilings o IR n IEEE Trns. Inorm. Thory vol. 8 no. pp PLACE PHOTO HERE Qingtng Jing riv th B.S. n M.S. grs rom Hngzhou Univrsity Hngzhou Chin in 986 n 989 rsptivly n th Ph.D. gr rom Ping Univrsity Bijing Chin in 99 ll in mthmtis. H ws with Ping Univrsity rom 99 to 995. H ws n NSTB postotorl llow n thn rsrh llow t th Ntionl Univrsity o Singpor rom 995 to 999. Bor h join th Univrsity o Missouri-St. Louis in 00 h hl visiting positions t Univrsity o Alrt Cn n Wst Virgini Univrsity Morgntown. H is now Prossor in th Dprtmnt o Mth n Computr Sin Univrsity o Missouri-St. Louis. His urrnt rsrh intrsts inlu timrquny nlysis wvlt thory n its pplitions ltr n sign signl lssition img prossing n sur suivision.