Gabor window grid (900 samples) dual window (grid) dual window (quincunx) quincunx (800 samples)

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1 alclating th dal Gabor window for gnral sampling sts Ptr Prinz Univrsitat Win, Institt f Mathmatik, U H A G Strdlhofgass 4 tl: ++43 / / fax: ++43 / / mail: prinztychmatniviacat Abstract Mthods for th calclation of th so-calld dal Gabor window for gnral sampling sts ar dvlopd First w considr th standard sitation whr th Tim- Frqncy plan (TF-plan) is (ovr)sampld on a grid, i a rctanglar lattic is considrd with mor points than th dimnsion of th signal spac Scondly w show that th proposd mthoan b asily xtndd to gnral sbgrops of th TF-plan On typical xampl of sch a sampling st is th so calld qincnx-lattic Kywords Gabor transform, Psdo{invrs, dal Gabor fram I Th discrt Gabor rprsntation Lt x (x[n]) n dnot a discrt complx vald signal, i, x is an lmnt of th -dimnsional complx vctor spac Th signal x is considrd to b - priodic, i, if th indx xcds, th vctor is dnd as: x[n] x[n + ] for all n Z: Th tim shift oprator T k ; k Z and th frqncy shift oprator M l ; l Z ar dnd as T k (x) T k (x[]; : : : ; x[ ]) : (x[k]; x[k + ]; : : : ; x[k ]) () M l (x) : (x[]; il x[]; : : : ; i( )l x[ ]): () A st (g i ) of tim-and frqncy shiftopis of a discrt fnction g is calld a Gabor family, if it is of th following form: g i : (M li T ki g); for all (l i ; k i ) H Z Z : (3) If H < Z Z (i H is a sbgrop of Z Z ) thn th fnction g is calld th Gabor window, th family g i ; if;:::;s g is calld Gabor family and th (s)-matrix (s : jhj) G g (g i ) M l T k g M l T k g M ls T ks g A ; (l i; k i ) H; i f; :::; s g (4) is calld th Gabor asic Matrix In or convntion, th rows of G g ar in an incrasing ordr of rst tim shifts and thn frqncy shifts jhj dnots th nmbr of lmnts of H If th rank of G g qals, i, G g sn as a linar mapping from th signal spac into th cocint spac s has fll rank (i rang(g g ) ), thn th st (g i ) is calld a Gabor fram (or complt Gabor family) In this cas vry signal x can b xprssd as a linar combination of (g i ), i x sx i a i g i a i ;, x ag g (5) Formla (5) is calld th Gabor xpansion of x On possibility to obtain th cocint vctor a is []: a x(g g G g) G g xg + g ; (6) whr G + g is calld th psdo-invrs of G g Thorm : (S [], [3]) Lt G g b a Gabor fram d- nd on H < Z Z as in (3)and (4), thn thr is a so calld dal Gabor fram G ~g satisfying x xg ~gg g for all x ; (7) which is of th sam strctr as G g, i G ~g (M li T ki ~g) i,(l i ; k i ) H, whr ~g is calld a dal Gabor window Dal Gabor windows ar not niq []If ~g is obtaind by G ~g G+ g thn ~g has minimal L -norm [4], [5] and is th most orthogonal-lik window [6] II Th calclation of th dal Gabor window In th following sctions w discss dirnt mthods of dtrmining th dal Gabor window ~g W show that th comptational ort of sch algorithms mainly dpnd on th strctr and siz of th sampling st H W always assm that G g has fll rank A Ovrsampling on th grid In th cas of ovrsampling on th grid, th nmbr of sampling points s ab (a; b ar dnd blow) is gratr than th lngth of th Gabor-window Th sflnss of ovrsampling is a consqnc of th alian-low- thorm, which shows that in th cas of critical sampling \nic" Gabor-windows do not hav \nic" dals (g [7]) Lt a b th tim gap and b th frqncy gap of th sampling st H, i l f; b; b; :::; bg; (g i ) ii (M l T k g); (8) k f; a; a; :::; ag:

2 and F b th Forir-Matrix of siz b b (i F kl ilk b i ) Th (s s) matrix F is dnd as F F : : : F A : (9) : : : F Lt th matrix b dnd as F G g, and G b g dnots a b matrix containing th rst b rows of G g Sinc th matrix F is block-diagonal (with blocksiz b ), w xamin th rst b rows of sparatly: kl (FG b g ) kl b X j X b j b g[l] ikj b ilj b g[l] ij b (l k) g[l] for k f; :::; b g if (l k) is a divisor of b othrwis () (any nmbr is considrd to b a divisor of zro) Th nxt b rows of dir from th rst ons in th way, that only th nonvanishing vals ar dirnt Hr th l-th ntry of kl is not jst b g l, bt th indx l is dcrasd by a, th tim-shift gap This lads to th gnral rslt for th matrix : kl b g[l a kb ] if (l k) is a divisor of b othrwis () kb whr dnots th gratst intgr not gratr than kb Mor prcisly looks as in qation () (with b : b ) To mak s of th spars strctr of w nd two prmtation matrics: P r is a (s s) matrix which rarrangs th rows of a (s ) matrix in th following ordr: P r f; ; : : : ; s g! ; a ; a ; : : : ; b b a ; ; a +; ; ab ; (3) g [] : : : g [b ] : : : g [( b )] g [] g[(b +)] : : : g [(b )] g [( a)] : : : g [(b a)] g [(b a+)] Th maning of (3) is th following: row # bcoms row #, row # bcoms row #, row # bcoms row #, tc a a and P c is a ( ) matrix which rarrangs th colmns of a ( ) matrix in th following ordr: P c f; ; : : : ; g! ; b ; b ; ; (b ) b ; ; b +; ; b ; : (4) Mltiplying th matrix from th lft and th right sid by P r and P c, rarrangs th rows anolmns and rslts in a block-diagonal matrix W : W P r P c P r F G g P c W i g [a] g [(a b )] b W : : : W A ; : : : W (b ) (5) g [i] g [(i+b )] : : : g [(i+ b )] g [(i a)] g [(i+b a)] : : : g [(i+ b a)] g [(i ( a))] g [(i ( a)+b )] : : : g [i+a b ] (6) Lmma : If Q is an nitary matrix, thn th following formla holds tr for vry matrix A: A A : (AQ) + Q A + (7) Lmma : If W is a block-diagonal matrix, thn th psdo-invrs W + is also block diagonal in th following way: W + W : : : W : : : W n A + (W ) + : : : (W ) + A : (8) : : : (W n ) + For a proof of ths two lmmata s g [8] ()

3 3 Sinc th matrics P r ; P c and F ar nitary and W is block diagonal, th psdo-invrs of G g is asily calclatd: W P r F G g P c, G g F P r W P c, G g + P c W + F P r : (9) ths colmns (or rows) ar pointwis dirnt, i W i (n ; m ) 6 W i (n ; m ) for all n ; n f; ; :::; a g if jm m j a c ; W i (n ; m ) 6 W i (n ; m ) for all m ; m f; ; :::; bg if jn n j b c : Sbstitting th window fnction g in qation (9) by th dal window fnction ~g givs G ~g F P r f W P c ; () If th siz of W i is gratr or qal ( b c a c ) i a b c and b a c (3) whr W f is similar to W in (5) and (6) rplacing th gi by ~g i In Thorm it is shown that G + g G ~g, which implis that G ~g P c f W P r F P c W + P r F G + g ) f W W + : () Rmark: Th calclation of th dal Gabor window is ths rdcd to th calclation of th psdo-invrs of th blockdiagonal matrix W This is, according to Lmma th calclation of th psdo-invrss of ( b ) matrics of siz ( a b) Eqation (6) shows that vry lmnt of th matrix W i niqly dtrmins on lmnt of th vctor g Thorm and qation () lad to th fact that vry lmnt of ~g is niqly rprsntd by a matrix W + i (for som i) Thrfor th nmbr of matrics W i for which w hav to calclat th psdoinvrs dpnds on th nmbr of dirnt matrics W i, ncssary to rprsnt ach lmnt of th window fnction g at last onc Th following proposition shows that this nmbr is givn by th gratst common divisor of (a; b ) Proposition : Lt fw ; W ; :::; W (b ) g b th family of matrics as dnd in qation (6) Th minimal nmbr of W i rqird to rcovr th window g is givn by th gratst common divisor (gcd) of (a; b ) Proof: According to qation (6) ach lmnt of th matrix W i can b rprsntd in th following way: W i (n; m) g[(i + mb na)]; () whr (n; m) dnots th row-colmn position W show how many dirnt indics (i + mb na) of g[(i + mb na)] in th W i ar possibl for givn ; a; b Sinc g is -priodic (or nit), any indx i qals an indx j if j i + d for som d Z In or notation w will writ: j i Lt c b th gratst common divisor of (a; b ), thn g[i + m b n a] g[i + m b n a];, i + m b n a i + m b n a, (n n ) (m m ) b c a c : This lads to th fact that (m m ) is a mltipl of a c and (n n ) is a mltipl of b c Hnc thr ar m a c colmns (or n b c rows), whr ach two lmnts ot of a vry matrix W i contains a c b b c c mtal dirnt lmnts Hnc w n matrics to rprsnt an - dimnsional vctor Finally w show that inqality (3) always holds tr: b a b c b a c ) c b a ) c b a ) b a: (4) c Q If jj p j is th prim factorization of, than a and b c can b niqly rprsntd by a prodct of ths p j's Sinc a and b c ar rlativ prim, no prim-nmbr p j can b a part of th prim factorization of a and b c This provs inqality (4) Smming p, th calclation of th dal Gabor window can b obtaind by calclating th psdo-invrss of gcd(a; b ) matrics of siz ( a b) A ritical sampling and rdndancy of intgr vals Th cas of critical sampling, i ab, was vry wll dscribd in g [9] Th main ida is, that in th critical cas th matrics W i ar sqar anyclic Hnc th W i ar compltly rprsntd by th rst row and invrtd qickly sing fast Forir transform Th cas of intgr rdndancy, i abd for d I is vry similar to th critical cas In this cas th prodct of W W is a cyclic matrix, anan also asily invrtd by FFT-mthods Sampling on an arbitrary sbgrop of th TF-plan In this sction w calclat th dal Gabor window for mor gnral sampling sts H < Z Z than jst a grid Th ida is to rdc this problm to that of th prvios sction, i H is a grid Sinc a grid is niqly dtrmind by th tim/frqncy gap w nd th following lmma for gnral sbgrops: Lmma 3: If H is a sbgrop of Z Z thn vry row (colmn) of H ithr contains a constant nmbr of lmnts or it contains no lmnt Th narst nighbors x ; x - in th sam row (colmn) - of an lmnt x hav constant distanc d r ( ) to this lmnt Proof: Assm thr ar for lmnts (x; y ); (x; y ); (x; y ); (x; y ) satisfying

4 4 y y > y y d ) (; d) H This lads to th fact, that (x; y ); (x; y ) ar not narst nighbors, bcas (x; y + d) is closr to (x; y ) than (x; y ) Sinc th distanc btwn two narst nighbors is constant, th nmbr of ths lmnts mst also b constant Lmma 3 and Figr show that any H < Z Z may b rprsntd by th smallst row anolmn distanc (d r ; ) and by vry lmnt (y; x) 6 (; ) which satiss y < d r and x < (Th top lft dot of ach box in Figr rprsnts th lmnt (,) of H) In othr words, givn th top lft box of Figr th sampling st H < Z Z is niqly dtrmind Sinc th row-coordinats of H rprsnt th modlation of th signal, w jst "shift" spcic colmns of H in sch a way, that H bcoms a grid, which w call K (S Figr ) W now writ G H g and G K g for th Gabor asic Matrics according to th sampling sts H and K rsp G K g dirs from G H g jst in modlations of svral rows Ths modlations may b obtaind by pointwis mltiplication with a matrix M (dnotd by a dot): G H g M G K g ; (5) According to Figr th matrix M has th following form: Th calclation of th dal Gabor window may now b prformd as dscribd in th prvios sction Th additional ort is jst th modlation of svral vals of g corrsponding to th vals of Wi H bfor and aftr th calclation of th psdoinvrs of W i Figr 3 shows a typical Gabor window and its dals according to a grid and a qincnx lattic Thogh th qincnx lattic has lss lmnts than th grid, th corrsponding dal window is bttr locatd in th sns of scond ordr momnts III onclsion A gnral mthod for th calclation of th dal Gabor window has bn introdcd First w considrd th standard sitation whr th TF-plan is (ovr)sampld on a grid Scondly, w gnralizd this mthod to calclat th dal window for Gabor frams gnratd by mans of an arbitrary sbgrop of th Z Z This covrs th most gnral sampling strctr of th TF-plan, for which it is still tr that thr is jst on singl dal Gabor atom dtrmining th dal fram An algorithm has bn givn Th propsd aproach can b xtndd to highr dimnsions W ar crrntly working on dimnsional and 3 dimnsionsl imag transformations M T ; ; ; M (); M (); M (); M 4 (); M 4 (); M 4 (); ; ; ; M (); M (); M (); M 4 (); M 4 (); M 4 () (6) whr (; : : : ; ) T 8 and M ; M 4 ar as dnd in () Lt m b th nmbr of lmnts (y; x) H satisfying y < d r and x < Th tim gap of K is dc m and th frqncy gap is d r According to (5) thr is a blockdiagonal matrix W K with: W K P r F G K g P c : (7) Sinc th nmbr of blocks of W in (5) qals th nmbr of frqncy shifts pr tim shift, and sinc qation ()-() jst handl with b ( d r ) modlatd vrsions of a signal, thr is a blockdiagonal matrix W H as follows: W H P r F G H g P c: (8) Lt Wi K (Wi H ) b th block matrics of W K (W H ) Whil th matrics Wi K ar as in (6) th matrics Wi H ar dirnt: Th rows of ach W i do not contain vals of g, bt vals of modlatd vrsions of g According to Figr th Wi H hav th following form: W H i d r Acknowldgmnts Th athor wold lik to thank H G Fichtingr T Strohmr and W Kozk for instrctiv discssions and gnral hlp dring his work Th athor acknowldgs partial spport from th Astrian Scinc Fonds \FWF", projct nmbr \S7-MAT" Rfrncs [] J Wxlr and S Raz Discrt Gabor Expansions Signal Procssing, (3):7{, ovmbr 99 [] O hristnsn Atomic dcomposition via projctiv grop rprsntations Rocky Montain J Math, 996 to appar [3] HG Fichtingr, O hristnsn, and T Strohmr A gropthortical approach to Gabor analysis Optical Enginring, 34:697{74, 995 [4] I Dabchis, H Landa, and Z Landa Gabor tim-frqncy lattics and th Wxlr-Raz idntity J For Anal Appl, (4):437{478, 995 [5] AJEM Janssn On rationally ovrsampld Wyl-Hisnbrg frams sbmittd, 995 [6] S Qian and D hn Discrt Gabor Transform IEEE Trans SP, 4(7):49{438, 993 [7] JJ ndtto and DF Walnt Gabor frams for L and rlatd spacs In ndtto J and Frazir M, ditors, Wavlts: Mathmatics and Applications, pags 97{6 R Prss, 993 [8] SL ampbll and D Myr Jr Gnralizd Invrss of Linar Transforms Dovr, 99 [9] DF Stwart, L Pottr, and S Ahalt omptationally attractiv ral Gabor transforms IEEE Trans SSP, pags 77{84, Jan 995 g[i] g[i + d r ] : : : g[i + d r ] (M g)[i dc m ] (M g)[i + d r dc m ] : : : (M g)[(i + d r dc (M 4 g)[i dc m ] (M 4g)[i + d r dc m ] : : : (M 4g)[(i + d r dc g[i 3 dc m ] g[i + d r 3 dc m ] : : : g[(i + d r 3 dc (M g)[i 4 dc m ] (M g)[i + d r 4 dc m ] : : : (M g)[(i + d r 4 dc (M 4 g)[i 5 dc m ] (M 4g)[i + d r 5 dc m ] : : : (M 4g)[(i + d r 5 dc A (9)

5 5 6 d r? - - Fig Sbgrop of Z 8 Z 8 - Fig Gnral sbgrop and lattic H : ) K : Gabor window dal window (grid) grid (9 sampls) dal window (qincnx) qincnx (8 sampls) Fig 3 dal Gabor window according to grid and qincnx lattic (Signal lngth 6, Grionstants a b ) 9 sampls, Qincnx row/colmn distanc d r 3 ) 8 sampls)

6 6 Fig 4 Algorithm to calclat th dal Gabor window according to a qincnx lattic Gabor Dal for Qincnx Lattic Inpt: g; ; d r Otpt: ~g Df: lngth of g, dim(w ) ( )(d r ) h : M dr g, (M as dnd in () ) for k to gcd( dc ; d r ) for i to stp for j to d r stp d r W (i; j) g[k i dc d r ] W (i + ; j + ) h[k (i + ) dc + (j + ) d r ] fw dr (W + ) for i to stp for j to d r stp d r ~g [k i dc + j d r ] f W (i; j) ~g [k (i + ) dc + (j + ) d r ] f W (i + ; j + ) ~g ~g + M dr ~g (Rmark M dr M dr )