Procdings of IC-IDC0 ( and (, ( ( and (, and (f ( and (, rspctivly. If two input signals ar compltly qual, phas spctra of two signals ar qual. That is

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1 Procdings of IC-IDC0 EFFECTS OF STOCHASTIC PHASE SPECTRUM DIFFERECES O PHASE-OLY CORRELATIO FUCTIOS PART I: STATISTICALLY COSTAT PHASE SPECTRUM DIFFERECES FOR FREQUECY IDICES Shunsu Yamai, Jun Odagiri, Masahid Ab, Masayui Kawamata Dpartmnt of Elctronic Engring, Graduatd School of Enginring, Tohou Univrsity , Aoba, Aramai, Aoba-u, Sndai, , Japan {yamai, odagiri, masahid, awamata}@m.ci.tohou.ac.jp Abstract: This papr analyzs ffcts of stochastic phas spctrum diffrncs on phas-only corrlation (POC functions. assum phas spctrum diffrncs btwn two signals ar statistically constant for frquncy indics. That is, thy hav idntical probability dnsity function for all frquncy indics. driv th gnral xprssions of th xpctation and varianc of th POC functions. Rlationships btwn th POC functions and th phas spctrum diffrncs ar formulatd. This rsult mathmatically guarants th validity of th POC functions usd for similarity masur in matching tchniqus. Kywords: Phas-spctrum diffrncs; Phas-only corrlation functions; Expctation; varianc Introduction Phas-only corrlation (POC functions hav bn widly usd for valuating similarity btwn two signals. Thy hav bn applid in many filds, such as imag rgistration [-4], pattrn rcognition [5]-[7], th fram displacmnt for old films [8]-[9], motion stimation [0], matching priodic DA squncs [], optics [], and so on. It has bn nown that th POC function is th dlta function if th phas spctra of two signals givn ar compltly qual. This proprty has bn xploitd in many matching tchniqus. Howvr, in practical signal procssing scn, it is almost impossibl that th phas spctra of two signals ar compltly qual. In actual cas of imag rgistration, for xampl, a rfrnc imag and an obsrvd imag ar hardly idntical. In such cass, th proprty of th POC function to b th dlta function with two qual phas spctra cannot b xploitd in matching tchniqus sinc two phas spctra ar not qual. Thrfor, w hav to clarify th ffcts of nonzro phas spctrum diffrncs on th POC functions. In this papr, w analyz ffcts of stochastic phas spctrum diffrncs on th POC functions. assum phas spctrum diffrncs btwn two signals ar statistically constant for frquncy indics. That is, phas spctrum diffrncs hav idntical probability //$ IEEE dnsity function for all frquncy indics. driv th xpctation and varianc of th POC functions with stochastic phas-spctrum diffrncs. Phas-only corrlation functions. Dfinition Considr complx discrt-tim signals ( and ( of lngth. Th discrt Fourir transforms of ( and ( ar givn by X ( DFT[ x( Y( DFT[ y( n0 n0 x( n y( n X ( j ( j Y( ( rspctivly, whr =xp ( /is th twiddl factor, and ar phas spctra of ( and (, rspctivly. Th phas-only corrlation (POC function ( btwn two signals ( and ( is dfind by th invrs discrt Fourir transform of normalizd cross-powr spctrum btwn two signals ( and ( as follows: ( ( X Y ( IDFT j n r n X ( Y( 0 ( n 0,,, (3 whr = ar phas spctrum diffrncs. ot that th POC function ( can b considrd as th invrs discrt Fourir transform of a signal, calld phas factors.. Proprtis Figur shows xampls for calculating POC functions. Figur (a, (b, and (c illustrat th original signal (, its circular shiftd vrsion ( =(, whr ( dnots mod, and nois-corruptd vrsion ( =( +(, whr ( is an additiv whit Gaussian nois, rspctivly. Figur (d, (, and (f illustrat th POC functions btwn (d 360

2 Procdings of IC-IDC0 ( and (, ( ( and (, and (f ( and (, rspctivly. If two input signals ar compltly qual, phas spctra of two signals ar qual. That is, = =0. In such cas, POC function ( is th dlta function ( sinc j0 n rxx( 0 ( n 0 (. 0 ( n 0 0 n This rsult is shown in Figur (d. If two input signals ar rlatd by circular shift, that is, on is ( and th othr is ( =(, POC function ( is th dlta function shiftd by sampls sinc r xxs n ( n ( 0 0 ( n mod ( n 0 ( n mod. (4 (5 Thrfor, th location of th pa corrsponds to th shift amount. This rsult is shown in Figur (. Th proprty in Eq. (4 holds only in cas that phas spctra ar compltly qual, and th proprty in Eq. (5 holds only in cas that two signals ar rlatd by circular shift. In most ralistic cas, input signals ar corruptd by nois, which causs corruptd phas spctra. As a rsult, POC function cannot b th dlta function. Suppos that w hav original signal ( and its noiscorruptd vrsion ( =( +(. Ths two signals ar similar but not qual. Th rsult of calculation of th POC function btwn ths two signals is shown in Figur (f. can obsrv in this figur that (0 dcrass and ( 0 incrass, compard with th dlta function in Figur (d. Th additiv nois causs such ffcts, which ma wav form matching by POC functions difficult. Thrfor, it is important to valuat variation of th POC functions undr th nois, assumd to b random variabls. hav to clarify ffcts of stochastic phas spctrum diffrncs on th POC functions. 3 Phas-only corrlation functions with stochastic Phas-Spctrum diffrncs In this sction, w analyz ffcts of stochastic phas spctrum diffrncs on th POC functions. (a original signal x( (b circular shiftd signal x s ( (c nois-corruptd signal x n ( (d POC function r xx ( ( POC function r xxs ( (f POC function r xxn ( Figur Exampls for calculating POC functions 3. Stochastic assumptions forphas-spctrum diffrncs In ordr to analyz ffcts of stochastic phas-spctrum diffrncs on th POC functions, w giv som stochastic assumptions for phas spctrum diffrncs s. assum th phas spctrum diffrncs s to b random variabls. Sinc phas factors s ar also random variabls, w can valuat th xpctation and varianc of thm. dnot th xpctation and varianc of phas factors as =[ ] and = Var[ ], whr [ ] and Var[ ] dnots th xpctation and varianc oprators, rspctivly. Ths xprssions ar drivd as 36

3 Procdings of IC-IDC0 A j E j B Var E E E j j E A A j j E A A AA AA. (6 j j j j (7 Th valus and ar dtrmind by giving a probability dnsity function of phas-spctrum diffrncs s. Th valus and ar constant which do not dpnd on frquncy indics = 0,, sinc w assum th phas spctrum diffrncs s hav idntical probability dnsity function for all frquncy indics =0,,. Furthrmor, w assum that phas spctrum diffrncs s ar statistically indpndnt for frquncy indics = 0,, as follows: E E E (8 l which yilds statistical indpndncy of phas factors s such as E j jl j jl E E AA ( l. (9 For xcption, whn =, it is obvious that E j jl ( l. l (0 3. Gnral xprssions for xpctation and varianc of POC function ( driv th xpctation and varianc of POC function (. first driv th xpctation [(] of POC function ( in Eq. (3 as follows: j n j Ern ( E E 0 0 n An ( 0 A A (. 0 0( n 0 n ( Eq. ( shows that [(] has only non-zro valu at =0. On th othr hand, [(] =0 for 0. nxt driv th varianc Var[(] of POC function ( as follows: Var rn ( Ernrn ( ( Ern ( Ern ( Er( r( AA (. ( From Eq. (, it is found to b ncssary to driv th man squard POC function [(( ]. thus driv th man squard POC function [(( ] as follows: j jl n( l E (3 0 l0 j n jl nl E r( r( E 0 l0 0 l0 E j jl n( l Substituting Eqs. (9 and (0 into Eq. (3 yilds Er( r( n( l n0 ( AA AA 0 l0 0 AA AA n nl 0 l0 AA ( AA. Substituting Eq. (4 into Eq. (, w hav Var rn ( AA B ( n 0,,,. It is rmarabl that Var[(] dos not dpnd on indx, which mans that th POC function ( has constant varianc for all. 4 Exprssions of th xpctation and varianc of POC functions basd on charactristic functions. (4 (5 Givn a probability distribution for phas spctrum diffrncs s, w can xprss th xpctation and varianc of th POC functions by using charactristic function. Th charactristic function ( of a probability dnsity function ( dfind by ( j t t p( d. (6 It is rmarabl that =[ ] can b xprssd by charactristic function as follows: A E j j p d (. ( (7 Thrfor, w can driv th xpctation and varianc of POC functions by using th charactristic functions. 4. Uniform distribution Lt th phas spctrum diffrncs s b probability variabls uniformly distributd in rang of [, ]. Probability dnsity function of is givn by a p( a (8 0 a of which charactristic function is nown to b. ( t sinc at (9 36

4 Procdings of IC-IDC0 whr sinc( is dfind by ( x 0 sinc( x sin( x (0 ( x 0. x Thrfor, w driv =[ ] as follows: a A ( sinc. ( Substituting Eq. ( into Eqs. ( and (5, w hav th xpctation and varianc of th POC function ( as follows: a Ern [(] sinc ( n ( a Var[ rn ( ] sinc. (3 Th varianc of is givn by = /3. can rwrit Eqs. ( and (3 as functions of th varianc as follows: 3 Ern [(] sinc ( n (4 3 Var[ rn ( ] sinc. (5 As incrass from 0 to /3, incrass from 0 to, [(0] monotonically dcrass from to 0, and Var[(] monotonically incrass from 0 to /, rspctivly. Figur Expctation and varianc of POC function r( in cas of uniformly distributd phas spctrum diffrncs shows th xpctation [(0]and varianc Var[(] vrsus with signals of lngth. In Figur Expctation and varianc of POC function r( in cas of uniformly distributd phas spctrum diffrncs, w st th rang of th varianc to b [0, /3] sinc th rang of is [, ] whn = /3. (7 Var[ rn ( ] sinc (. /3 4. Gaussian distribution Lt th phas spctrum diffrncs s b probability variabls following Gaussian distribution (0,. Probability dnsity function of is givn by p (8 ( of which charactristic function is nown to b t ( t. (9 Thrfor, w driv =[ ] as follows: Substituting Eq. (30 into Eqs. ( and (3, w hav th xpctation and varianc of th POC function ( as follows: As incrass, [(0] monotonically dcrass from to 0, and Var[(] monotonically incrass from 0 to /, rspctivly. Figur 3 shows th xpctation [(0] and varianc Var[(] vrsus varianc with signals of lngth. In Figur 3, w st th rang of th varianc to b [0,] sinc th rang of can b approximatly considrd as [, ] whn =. / A (. (30 / E[ r( ( (3 Var[ r (. (3 (a Expctation [(0] (a Expctation [(0] (b Varianc Var[(] Figur Expctation and varianc of POC function r( in cas of uniformly distributd phas spctrum diffrncs Th xpctation [(0] and varianc Var[(] for = /3 ar drivd as follows: Er /3 [(0] sinc( 0 (6 (b Varianc Var[(] Figur 3 Expctation and varianc of POC function r( in cas of Gaussian distributd phas spctrum diffrncs Th xpctation [(0] and varianc Var[(] for = ar calculatd as follows: / E[ r(0] (33 Var[ r( (34 363

5 Procdings of IC-IDC0 5 umrical xampls Figur 4 shows th POC functions for stochastic phas spctrum diffrncs. st lngth of signals to b =6. assum phas spctrum diffrncs s follow Gaussian distribution (0,, and calculat POC functions in Eq. (3 for = 0, 0.5, 0.5,. can obsrv (0 dcrass as th varianc incrass. On th othr hand, ( 0 tnds to incras as th varianc incrass. Ths numrical rsults agr with th thortical rsults prsntd in th prvious sction (a =0 (b =0.5 (c =0.5 (d = Figur 4 POC functions r( for various variancs of phas spctrum diffrncs 6 Conclusions In this papr, w hav analyzd th ffcts of stochastic phas spctrum diffrncs on th POC functions. hav assumd phas spctrum diffrncs btwn two signals to b random variabls. hav drivd th xpctation and varianc of th POC functions with stochastic phas spctrum diffrncs. As phas diffrnc btwn two signals incrass, th xpctation [(0] dcrass and th varianc Var[(] incrass, rspctivly. This rsult mathmatically guarants th validity of th POC functions usd for similarity masur in matching tchniqus. Acnowldgmnt A part of rsults prsntd in this papr was achivd by carrying out an MIC program ''Rsarch and dvlopmnt of tchnologis for ralizing disastr-rsilint ntwors'' (th o. 3 supplmntary budgt in 0 gnral account. Rfrncs [] C. D. Kuglin and D. C. Hins, Th phas corrlation imag alignmnt mthod, in Proc. Int. Conf. Cybrntics and Socity, 975, pp [] Q. Chn, M. Dfris, and F. Dconinc, Symmtric phas-only matchd filtring of fourir-mllin transforms for imag rgistration and rcognition, IEEE Trans. Pattrn Anal. Mach.. Intll., vol. 6, no., pp , Dc [3] B. S. Rddy and B.. Chattrji, An fft-basd tchniqu for translation, rotation, and scal-invariant imag rgistration, IEEE Trans. Imag Procss., vol. 5, no. 8, pp. 66 7, Aug [4] Foroosh, J. Zrubia, and M. Brthod, Extnsion of phas corrlation to subpixl rgistration, IEEE Trans. Imag Procss., vol., no. 3, pp , Mar. 00. [5] J. L. Hornr and P. D. Gianino, Pattrn rcognition with binary phasonly filtrs, Applid Optics, vol. 4, pp , 985. [6] B. V. K. V. Kumar, A. Mahalanobis, and R. D. Juday, Corrlation Pattrn Rcognition. Cambrig Univrsity Prss, 005. [7] K. Miyazawa, K. Ito, T. Aoi, K. Kobayashi, and H. aajima, An ffctiv approach for iris rcognition using phas-basd imag matching, IEEE Trans. Pattrn Anal. Mach.. Intll., vol. 30, no. 0, pp , Oct [8] M. Hagiwara, M. Ab, and M. Kawamata, Estimation mthod of fram displacmnt for old films using phasonly corrlation, Journal of Signal Procssing, vol. 8, no. 5, pp. 4 49, Spt [9] X. Zhang, M. Ab, and M. Kawamata, Rduction of computational cost of POC-basd mthods for displacmnt stimation in old film squncs, IEICE Trans. Fundamntals of Elctronics, Communications and Computr Scincs, vol. E94-A, no. 7, pp , July 0. [0] J. Rn, J. Jiang, and T. Vlachos, High-accuracy subpixl motion stimation from noisy imags in fourir domain, IEEE Trans. Imag Procss., vol. 9, no. 5, pp , May 00. [] A. K. Brodzi, Phas-only filtring for th masss(of DA data: A nw approach to squnc alignmnt, IEEE Trans. Signal Procssing, vol. 54, no. 6, pp , Jun 006. [] J. L. Hornr and P. D. Gianino, Phas-only matchd filtring, Applid Optics, vol. 3, no. 6, pp. 8 86, Mar

10. The Discrete-Time Fourier Transform (DTFT)

10. The Discrete-Time Fourier Transform (DTFT) Th Discrt-Tim Fourir Transform (DTFT Dfinition of th discrt-tim Fourir transform Th Fourir rprsntation of signals plays an important rol in both continuous and discrt signal procssing In this sction w