Quaternion Fourier Transform for Colour Images

Transcription

1 Vikas R. Duby / (IJCSIT) Intrnational Journal of Computr Scinc and Information Tchnologis, Vol. 5 (3), 4, Quatrnion Fourir Transform for Colour Imags Vikas R. Duby Elctronics, Mumbai Univrsity VJTI, Matunga, Mumbai, India Abstract Th Fourir transforms plays a critical rol in broad rang of imag procssing applications, including nhancmnt, rstoration, analysis and comprssion. For filtring of gray scal imags D Fourir transform is an important tool which convrts th imag from spatial domain to fruncy domain and thn by applying filtring mask filtring is don. To filtr color imags, a nw approach is implmntd which uss hypr complx numbrs (calld as Quatrnion) to rprsnt color imags and uss Quatrnion Fourir transform for filtring. This transform allows color imags to b transformd as whol, rathr than as color sparatd componnts. This papr is concrnd with fruncy domain nois rduction of colour imags using Quatrnion Fourir transform. Th approach is basd on obtaining uatrnion Fourir transform of colour imag and applying th Gaussian filtr to it in th fruncy domain. Th filtrd imag is thn obtaind by calculating th invrs uatrnion fourir transform, which xplains th accuracy of this mthod. Kywords Hyprcomplx numbrs, Quatrnion Fourir Transform(QFT), Gaussian LPF and HPF. I. INTRODUCTION Fourir transform hav bn widly usd in signal and imag procssing vr sinc th discovry of Fast Fourir transform in 965 which mad th computation of discrt Fourir transform fasibl using a computr[]. Fourir transform finds major application in imag procssing, such as imag filtring, imag analysis, imag rconstruction and imag comprssion. In gnral, thr ar two main approachs for filtring an imag. Th first is th convolution of an imag and krnl in th spatial domain. Th scond is th multiplication of an imag s Fourir transform with a filtr in th fruncy domain. Until rcntly, thr was no dfinition of Fourir transform applicabl to colour imags in holistic mannr. Th ida of computing th Fourir transform of a colour imags has only rcntly bn ralizd. It is possibl to sparat a colour imag into thr scalar imags and comput th Fourir transforms of ths imags sparatly but this mthod is infficint. In this papr w ar concrnd with th computation of a singl, holistic, Fourir transform which trats a colour imag as a vctor fild and application of this transform to colour imag filtring such as low pass filtring for smoothing and high pass filtring for sharpning. Th application of a uatrnion Fourir transform to colour imags is basd on rprsnting colour imag pixls using uatrnion discovrd by Hamilton in 843 []. Th first dfinition of a uatrnion Fourir transform was that of Ell [3],[4],[5] and th first application of a uatrnion Fourir transform of colour imags was rportd in 996 [4] using a discrt vrsion of Ell transform. II. QUATERNION Th concpt of th uatrnion was introducd by Sir. William Hamilton in 843 []. It is th gnralization of a complx numbr. A complx numbr has two componnts: th ral and th imaginary part. Howvr, th uatrnion has four componnts i.. on ral part and thr imaginary parts and can b rprsntd in Cartsian form as: w xi yj zk () Whr w, x, y, and z ar ral numbrs and i, j and k ar complx oprators which oby th following ruls; i* j k, j * k i, k * i j j * i k, k * j i, i k j i j k i* j * k * () From ths ruls, it is clar that multiplication is not commutativ. Th uatrnion conjugat is, w xi yj zk (3) and th modulus of a uatrnion is givn by, w x y z (4) A uatrnion with zro ral part is calld a pur uatrnion and a uatrnion with unit modulus is calld a unit uatrnion. i j k 3 (5) Th imaginary part of a uatrnion has thr componnts and may b associatd with a 3-spac vctor. For this rason, it is somtims usful to considr th uatrnion as composd of a vctor part and a scalar part.thus can b xprssd as; S( ) V ( ) (6) whr S() is th ral or scalar part i. S() = w and V() is th vctor part which is a composition of thr imaginary componnts; V ( ) xi yj zk 44

2 Vikas R. Duby / (IJCSIT) Intrnational Journal of Computr Scinc and Information Tchnologis, Vol. 5 (3), 4, A. Rprsntation of colour pixl as Quatrnion Colour imag pixl has thr componnts viz, Rd, Grn and Bl and thy can b rprsntd in uatrnion form using pur uatrnion [6]. For imags in RGB colour spac, th thr imaginary parts of a pur uatrnion can b usd to rprsnt th rd, grn and blu colour componnts. A pixl at imag coordinats ( x, in an RGB imag can b rprsntd as, r( x, i g( x, j b( x, k (7) whr r(x,, g(x, and b(x, ar rd, grn and blu componnts of a colour imag pixl rspctivly. Using uatrnion to rprsnt th RGB colour spac, th thr colour channls ar procssd ually in oprations such as multiplication. Th advantag of using uatrnion basd oprations to manipulat colour information in an imag is that w do not hav to procss ach colour channl indpndntly, but rathr, trat ach colour tripl as a whol unit. W bliv, by using uatrnion oprations, highr colour information accuracy can b achivd bcaus a colour is tratd as an ntity. III. QUATERNION FOURIER TRANSFORM Basd on th concpt of uatrnion multiplication and xponntial, th Discrt Quatrnion Fourir Transform (DQFT) has bn introducd[7]. Du to th noncommutativ proprty of th uatrnion, thr ar thr diffrnt typs of DQFT dfind: th lft sid DQFT, th right sid DQFT and th two sids DQFT rprsntd as, Two-sidd DQFT (Typ-) F LR M N x y Lft-sidd DQFT (Typ-) F L M N x y Right-sidd DQFT (Typ-3) F R M N x y xu M xu yv M N xu yv M N vy N (8) (9) () Similarly, th Invrs Discrt Quatrnion Fourir Transforms (IDQFT) can b dfind for th thr typs of QFT rspctivly as, Lft-sidd IDQFT: M N xu yv M N FL () x y Right-sidd IDQFT: Two-sidd IDQFT: M N x y M N x y F R xu M F xu yv M N LR vy N () (3) μ is any unit pur uatrnion. μ dtrmins a dirction in color spac and an obvious choic for color imags is th dirction corrsponding to th luminanc axis which conncts all th points r=g=b. In RGB color spac this is th gray lin. IV. QUATERNION FILTERING Th fruncy domain is th spac dfind by valus of th Fourir Transform and its fruncy variabls (. In this sction, w dfin diffrnt typs of uatrnion smoothing and sharpning filtrs. In th cas whr h ( x, (impuls rspons of uatrnion filtr) has th vn symmtry rlation, h( x, h( x, (4) QFT of h ( x, dnotd by H also has th sam symmtry rlation and th rlation btwn th uatrnion convolution and th QFT can b simplifid to, G H F( (5) th uatrnion convolution opration in th spatial domain corrsponds to th product opration in th fruncy domain [8]. This is th sam as th cas of th convntional convolution. whr F is th uatrnion Fourir transform of th color imag to b filtrd and G is th uatrnion Fourir transform of th filtrd output imag. Th objctiv is to slct a uatrnion filtr transfr function H that yilds G Th filtrd imag is obtaind simply by taking th invrs uatrnion Fourir transform of G. A. Quatrnion Low pass filtring Th dgs and othr sharp transitions in th pixls of an imag contribut significantly to th high fruncy contnt of its Fourir transform [6]. Hnc, smoothing of a colour imag is achivd in th fruncy domain by attnuating th spcifid rang of high fruncy componnts in th uatrnion Fourir transform of th imag. Now w propos two typs of uatrnion low pass filtrs whos impuls rsponss satisfy E. (4): idal and Gaussian [9]. 44

3 Vikas R. Duby / (IJCSIT) Intrnational Journal of Computr Scinc and Information Tchnologis, Vol. 5 (3), 4, ) Idal uatrnion Low pass filtrs: Th simplst low pass uatrnion filtr is a filtr that cuts off all high fruncy componnts of th uatrnion Fourir Transform that ar at a distanc gratr that a spcifid distanc D from th origin of th transform. Such a filtr has th transfr function, H,, D D (6) D is a non-ngativ uantity, and ) whr D v is th distanc from point ( u, to th origin of th fruncy rctangl ( M /, N / ), givn by M N D u v (7) Fig. shows th rsults of idal low pass filtring in th hypr complx spctral domain. In th first column of Fig., ar th original imag and its spctral modulus. Th scond, third and fourth columns, in th sam Fig., show th rsults of idal low pass filtring th imag by masks with radii, and 3. As xpctd th rsulting imags ar blurrd consistnt with this opration. As th filtr radius incrass, lss and lss powr is rmovd, rsulting in lss blurring. Column- Column- ) Gaussian Quatrnion low pass filtrs: Th two dimnsional Gaussian low pass filtr transfr function with th cut-off fruncy at a distanc D is givn by, H D (8) whr σ is a masur of Gaussian sprad and is ual to D and D as in E.(7) is th distanc from th origin of th Fourir transform. Fig. shows th rsults of Gaussian uatrnion low pass filtring th Lna imag in th hypr complx spctral domain. As in th cas of th Idal low pass filtr, w not a smooth transition in blurring as a function of incrasing cutoff fruncy. Th Gaussian low pass filtr did not achiv as much smoothing as th Buttrworth low pass filtr of ordr for th sam valu of cutoff fruncy. Fig. : Th rsults of idal low pass filtring basd on typ QFT, first column shows th original noisy colour imag and its spctral modulus. Th low pass filtring of th imag by th masks with radii, and 3 and th corrsponding typ two invrs QFT(Lft-Sid) imags ar likwis shown in th scond,third and fourth columns, rspctivly

4 Vikas R. Duby / (IJCSIT) Intrnational Journal of Computr Scinc and Information Tchnologis, Vol. 5 (3), 4, high pass filtring procss, which attnuats th low fruncy componnts without disturbing high fruncy information in th uatrnion Fourir transform. Bcaus th intndd function of th uatrnion high pass filtr is to prform th rvrs opration of th uatrnion low pass filtr, th transfr function of th high pass filtrs can b obtaind using th rlation: H hpf H (9) lpf Whr H lpf and H hpf is th transfr function of th corrsponding low pass uatrnion filtr. In this sction also, w propos two typs of uatrnion high pass filtrs that satisfy E. (4): idal and Gaussian [9]. ) Idal Quatrnion High pass filtrs: A two dimnsional uatrnion high pass filtr is dfind as, Column- Column- H,, D D () D is a non-ngativ uantity, and ) Whr D v is th distanc from point ( u, to th origin of th fruncy rctangl, as givn by E(7). Fig.3 shows th rsults of idal high pass filtring in th hypr complx spctral domain. In th first column, ar th original imag and its spctral modulus. Th scond, third and fourth columns, in th sam Fig. show th rsults of high pass filtring th imag by masks with radii, and 3. Again, as xpctd, th imag contains contnt whr thr is rapid luminanc and chrominanc variation. From this Fig. It can b sn that th sharpning of th imag incrass as radius of mask incrass. ) Gaussian Quatrnion High pass filtrs: Th transfr function of th Gaussian high pass uatrnion filtr with cut-off fruncy locus at a distanc D from th origin is givn by, Fig. : Th rsults of Gaussian low pass filtring basd on typ QFT, first column shows th original noisy colour imag and its spctrum modulus. Th low pass filtring of th imag by th masks with radii, and 3 and th corrsponding typ two invrs QFT(Lft-Sid) imags ar likwis shown in th scond,third and fourth columns, rspctivly. B. Quatrnion High pass filtring Th dgs and othr abrupt changs in pixls ar associatd with high fruncy componnts; imag sharpning can b achivd in th fruncy domain by a D () H whr σ is a masur of Gaussian sprad and is ual to D and D as in E.(7) Fig.4 shows th rsults of Gaussian high pass filtring th imag in th hypr complx spctral domain. In th first column, ar th original imag and its spctral modulus. Th scond, third, and fourth columns, in th sam Fig. show th rsults of Gaussian high pass filtring th imag by masks with radii, and 3. Th rsults obtaind with Gaussian high pass filtring ar smoothr than with idal high pass filtrs

5 Vikas R. Duby / (IJCSIT) Intrnational Journal of Computr Scinc and Information Tchnologis, Vol. 5 (3), 4, Column- Column- Column- Column- Fig. 3: Th rsults of idal high pass filtring basd on typ QFT, first column shows th original colour imag and its spctrum modulus. Th high pass filtring of th imag by th masks with radii 5, and and th corrsponding typ two invrs QFT(Lft-Sid) imags ar likwis shown in th scond,third and fourth columns, rspctivly. Fig. 4: Th rsults of Gaussian high pass filtring basd on typ QFT, first column shows th original colour imag and its spctrum modulus. Th high pass filtring of th imag by th masks with radii 5, and and th corrsponding typ two invrs QFT(Lft-Sid) imags ar likwis shown in th scond,third and fourth columns, rspctivly

6 Vikas R. Duby / (IJCSIT) Intrnational Journal of Computr Scinc and Information Tchnologis, Vol. 5 (3), 4, V. CONCLUSIONS This work dmonstrats that th Quatrnion Fourir Transform is wll suitd for dscribing th spctral contnt of colour imags. Similar to gray scal imags, colour imags rprsntd as uatrnion valud imags can also b transformd into th Fruncy domain and can b rprsntd as uatrnion fruncy signals, basd on which diffrnt imag procssing tchnius such as filtring can b prformd fficintly. Filtring in uatrnion fruncy domain has th advantag that th colour tripls ar procssd as a whol unit rathr than daling with RGB channls sparatly. W bliv mor accurat colour information can b prsrvd this way, sinc all colour channls ar procssd as a singl unit. And w would lik to point out that, th Quatrnion Fourir transform is not limitd to this, but also can b applid to othr colour imag procssing filds, such as imag rgistration, dg dtction, and data comprssion. REFERENCES [] Cooly. Jams W. and John W. Tuky, An Algorithm for th machin calculation of complx Fourir sris, Math. Comput, Vol pp [] W. R. Hamilton, Elmnts of Quatrnions, London, U.K. Longmans Grn, 866. [3] Ell T.A., Quatrnion-Fourir transforms for analysis of twodimnsional linar tim-invariant partial diffrntial systms, in Proc. 3nd Con. Dcision Contr., Dc. 993, pp [4] S. J. Sangwin, Fourir transforms of colour imags using uatrnion, or hyprcomplx, numbrs, Elctron. Ltt. vol. 3, no., pp , Oct [5] Transforms of Color Imags", IEEE transaction Todd A. Ell and S. J. Sangwin, "Hyprcomplx Fourir on Imag procssing, VOL. 6, NO., JANUARY 7. [6] M. I. Khalil, Applying Quatrnion Fourir Transform for Enhancing colour imags, I.J. Imag, Graphics and Signal Procssing,,, 9-5. [7] S. J. Sangwin and T. A. Ell, Th discrt Fourir transform of a colour imag, in Proc. Imag Procssing II Mathmatical Mthods, Algorithms and Applications, J.M. Blackdg and M.J. Turnr, Eds., Chichstr, U.K.,, pp [8] S. C. Pi, J.J. Ding, and J. -H. Chang, Efficint implmntation of uatrnion Fourir transform, convolution, and corrlation by -D complx FFT, IEEE Trans. Signal Procss., vol. 49, no., pp , Nov.. [9] B.D. Vnkatramana Rddy and Dr. Jayachandra Prasad, "Fruncy Domain Filtring of Colour Imags using Quatrnion Fourir Transform", IJCST Vol., Issu, Dcmbr. [] Rafal C. Gonzalz and Richard E. Woods, Digital imag procssing Parson Education, 7. [] Bihan, N.L., Sangwin, S.J., "Quatrnion principal componnt analysis of color imags", IEEE Intrnational Confrnc on Imag Procssing, 3, Vol,, pp [] S. J. Sangwin and N. L Bihan, Quatrnion Toolbox for Matlab, Softwar Library [Onlin]. Availabl: sourcforg.nt. [3] Ja-Han Chang, S.C.P., Ding, J.J., "d uatrnion fourir spctral analysis and its applications", IEEE Intrnational Symposium on Circuits and Systms, 4, Vol. 3, pp