ORTHOGONAL RADIAL MOMENTS, TRANSFORMS AND THEIR APPLICATIONS

Transcription

1 Intnational Confnc of Tchnology Managmnt and Social Scincs (ICTMS-5) ORTHOGONAL RADIAL MOMENTS TRANSFORMS AND THEIR APPLICATIONS AJAY PRASHAR HARINDER KAUR RAHUL UPNEJA Si Guu Ganth Sahib Wold Univsity Fathgah Sahib Punjab India Abstact: aious adial momnts i.. Znik momnts Psudo-Znik momnts Othogonal Foui Mllin momnts Chbyshv Hamonic momnts Radial Hamonic Foui momnts and Pola Hamonic Tansfoms such as ola comlx xonntial tansfoms ola cosin tansfoms and ola sin tansfoms satisfy th condition of othogonality. Thfo ths momnts and tansfoms ossss minimum infomation dundancy and having good chaactistic of imag sntation. In this a w discuss about th vaious othogonal adial momnts tansfoms and thi alications in imag ocssing. Indx tms: Radial momnts Znik momnts Psudo-Znik momnts Othogonal Foui Mllin momnts Radial Hamonic Foui momnts Chbyshv s Hamonic Foui momnts Pola Hamonic Tansfoms othogonal momnts.. INTRODUCTION Momnts a th scala quantitis usd to chaactiz a function and to catu its significant fatu. In th mathmatical fom momnts a ojction of a function onto a olynomial basis function. Momnts comutd fom a digital imag givs infomation about gomtical fatus of th imag. Momnts a of two tys othogonal and non othogonal. Othogonal momnts scify indndnt fatus of th imag thfo thi infomation dundancy is minimum. Othogonal momnts hav diffnt alications in imag analysis and attn cognition du to thi ability to snt imag fatus. Ths momnts includ Znik momnts (ZMs) Psudo-Znik momnts (PZMs) Othogonal Foui Mllin momnts (OFMMs) Radial Hamonic Foui momnts (RHFMs) and Chbyshv s Hamonic Foui momnts (CHFMs). Tagu[] sntd a st of othogonal Znik momnts which a lss snsitiv to nois and has otation invaianc oty in continuous domain. ZMs a widly usd in attn cognition alications lik imag constuction imag sgmntation dg dtction fac cognition tc. Bhatia and Wolf [] intoducd anoth class of othogonal adial momnts calld PZMs. ZMs and PZMs hav simila chaactistics of thi minimum infomation dundancy and immunity to nois. Shng and Shn [3] intoducd anoth st of otational invaiant momnts calld OFMMs. OFMMs hav btt fomanc than ZMs and PZMs tms of nois snsitivity fo small imags. OFMMs giv mo numb of momnts as comad to ZMs and PZMs fo th sam maximum momnt od. Bcaus th tition aamt q in Znik and sudo Znik olynomials is not indndnt. Othogonal adial momnts a comutationally intnsiv and numically instabl at high od of momnts. In od to minimiz ths oblms Radial Hamonic Foui momnts (RHFMs) [4] a intoducd which a also othogonal. Ths momnts fac th oblm of numical stability whn th high od momnts a concnd i.. at high od of momnts o tansfom th quality of constuctd imags dtioats significantly which is du to numical instability in momnts and tansfoms. Th oblm of ths momnts is latd to th fact that many factoial tms a involvd in th ocss of calculating th momnt knls. In od to ovcom this oblm Y t al. [5] intoducd vaious Pola Rsach Cll : An Intnational Jounal of Engining Scincs Scial Issu ICTMS Dcmb 05 ISSN: (Pint) ISSN: (Onlin) 05 MID Publications in association with idya Publications. Authos a sonsibl fo any lagiaism issus. 47

2 Intnational Confnc of Tchnology Managmnt and Social Scincs (ICTMS-5) Hamonic Tansfoms (PHTs) such as Pola Comlx Exonntial Tansfoms (PCETs) Pola Cosin Tansfoms (PCTs) and Pola Sin Tansfoms (PSTs). Comutation of Pola Hamonic Tansfom (PHT) knls is siml as coma to ZMs and PZMs hnc can b fomd at a high sd and th is no numical instability issu. Ths momnts and tansfoms suff fom vaious os and numical instability fo high od of momnts. Th common os ffct on th imag constuction. Th adial momnts a comutd by maing a ctangula o squa imag gomtically insid a unit cicl. Gomtic o and numical intgation o occus by using this maing. In this maing only thos ixls a involvd in momnt comutation whos cnts li insid th unit disc whil discading oth ixls. To liminat th gomtic o Chong t.al intoducd an altnativ aoach in which th comlt imag is containd insid th unit disc. All ixls a includd in th comutation of adial momnts. Numical intgation o occus whn th doubl intgation is includd in th momnt comutation is aoximatd by summation. Numical instability is anoth majo o in calculation of adial momnt which aiss du to high ods of momnts. Numical instability is aiss du to th high od factoial tms snt in adial olynomials. At high ods of momnts imags constuctd using PZMs OFMMs CHFMs RHFMs oduc o at th cnt of th constuctd imag. But ZMs hav bst imag constuction caability thn PZMs OFMMs CHFMs RHFMs.. MATHEMATICAL FORMULATION OF MOMENTS AND TRANSFORMS. Znik Momnts Znik intoducs a st of comlx olynomials which fom a comlt othogonal st ov th intio of th unit cicl i.. x + y.th two dimnsional Znik momnt of od with tition q ov a unit disc is givn by []: + ( ) Z q = f x y ( x dxdy. q x + y () wh ositiv intg o zo q ositiv and ngativ intgs subjct to constaints q = vn q q Th functions ( x a th comlx conjugat of Znik olynomial q ( x which is othogonal and comlt. It is dfind as θ ( x = R Z θ ( x jθθ () θ = tan ( y / x) θ [ 0π ] j = Radial olynomial of ZMs is givn by s ( q ) s ( ) ( )! ( ) Z s + y R ( ) = x q x y s= 0 + q q s!! s! s (3) Znik momnts with ngativ valus of tition q a obtaind dictly by making us of th comlx conjugat of Znik momnts fo ositiv valus.. PSEUDO ZERNIKE MOMENTS (PZMS) Psudo Znik momnts a also dfind insid th unit cicl. Th two dimnsional PZMs od and tition q of an imag function f ( x ov a unit disk a dfind by [6]: Rsach Cll : An Intnational Jounal of Engining Scincs Scial Issu ICTMS Dcmb 05 ISSN: (Pint) ISSN: (Onlin) 05 MID Publications in association with idya Publications. Authos a sonsibl fo any lagiaism issus. 48

3 Intnational Confnc of Tchnology Managmnt and Social Scincs (ICTMS-5) A q + = x + y f ( x ( x dxdy. q (4) Th function q ( x is th comlx conjugat of Psudo Znik olynomial q ( x which is othogonal and comlt. It is dfind as jθθ P θ ( x = R θ ( x (5) wh is a non-ngativ intg and q is an intg q θ = tan ( y / x) θ [ 0π ] j = and s s ) ( + s)!( x + y ) ( + q + s)!( q s) ( ( ) = q R q x y. s= 0 s!! (6).3 Othogonal Foui- Mllin Momnts (OFMMs) Othogonal Foui Mllin momnts hav btt fomanc than ZM s and PZM s in tms of imag dscition and nois snsitivity in cas of small imags. Fo an imag function f(x th two dimnsional OFMMs of od with tition q is givn by [3]: + O q = f ( x q ( x dxdy x + y (7) wh is a non-ngativ intg q is an intg (ngativ o ositiv) and q ( x is th comlx conjugat of th momnt basis function q ( x and th basis function q ( x is othogonal to ach oth. It is dfind as wh x = + y θ = tan ( y / x) j = and th adial olynomial R O () is givn by + s s O ( ) ( + s + )! q = R ( ) s= 0 s! ( s + )!( s)! (9) Othogonal Foui Mllin Polynomials (OFMPs) a indndnt of q th is no stiction on q whil comuting OFMMs unlik ZMs and PZMs. In fact th PZPs bcom OFMPs if q=0 is substitutd in PZPs..4 Radial Hamonic Foui Momnts (RHFMs) Th othogonal adial momnts a comutational intnsiv and numically instabl at high ods of momnts. Thfo to ovcom o minimiz ths oblms Radial Hamonic Foui Momnts (RHFMs) a us which a also othogonal. Th RHFMs of od and tition q with 0 and q 0 ov a unit disk a dfind as [4]: M π πθ πθ ) = f ( θ ) π (0) ( θ ddθ wh ( θ θ ) is th comlx conjugat of θ ( θ ) of od and tition q and q ( q ) = R ( ) () Th adial knl functions a dfind by q (8) O q ( x = R ( ) Rsach Cll : An Intnational Jounal of Engining Scincs Scial Issu ICTMS Dcmb 05 ISSN: (Pint) ISSN: (Onlin) 05 MID Publications in association with idya Publications. Authos a sonsibl fo any lagiaism issus. 49

4 Intnational Confnc of Tchnology Managmnt and Social Scincs (ICTMS-5) R () ( ) = cos( ) sin( ( + ) ) = 0 vn odd Th othogonal oty fo adial knl is givn as: R ( ) Rk ( δ = δ k ) 0 (3) Th othogonality of basis function is givn as: π πθ ( θ ) π' θ' ( θ ) δδθ = π δ ππ' δ θθ' (4).5 Chbyshv-Foui Momnts (CHFMs) Chbyshv-Foui momnts (CHFMs) xhibiting almost simila fomanc as OFMMs fo dscibing th imags [7]. CHFMs of od and tition q with 0 and q 0 a dfind in ola fom as: π πθ πθ ) Ψ = f ( θ ) π (5) ( θ ddθ wh ( θ θ ) is th comlx conjugat of th basis function θ ( θ ) dfind by: q ( q ) = R ( ) (6) wh th adial knl function is givn as: (7).6 Pola Hamonic Tansfoms Pola hamonic tansfoms (PHTs) a othogonal otation invaiant tansfoms that ovid many numically stabl fatus as th knl comutation of PHTs is vy siml and is numically stabl. Th diffnt tansfoms namly Pola Comlx Exonntial Tansfom (PCET) Pola Cosin Tansfom (PCT) and Pola Sin Tansfom (PST) a goud und PHTs. Ths tansfoms hav also low tim comlxity and numical stability. Fo an imag function f ( θ ) th two dimnsional PHTs of od with tition q a dfind as π µ M πθ = f ( θ ) H πθ ( θ ) dd θ (8) wh 0 q 0 and µ = 0 q 0 and µ = if q 0 and µ = = 0 PST PCET µ = and H θ ( θ ) is th comlx conjugat of th basis function H θ ( θ ). Fo PCET H θ ( θ ) is dfind as H q ( q ) = R ( ) (9) if 0 P R ( ) = 8 / 4 / k ( k)! ( ) k! ( k)! ( ( ) ) k k = 0 Rsach Cll : An Intnational Jounal of Engining Scincs Scial Issu ICTMS Dcmb 05 ISSN: (Pint) ISSN: (Onlin) 05 MID Publications in association with idya Publications. Authos a sonsibl fo any lagiaism issus. 50

5 Intnational Confnc of Tchnology Managmnt and Social Scincs (ICTMS-5) wh R () is adial knl tatmnt of diffnt disass lik canc j = θ = tan bain tumo tc. Th aim of mdical imag ( y x). Simila to PCET analysis is to find out th lvant th basis functions of PCT and PST a infomation o knowldg fom mdical dfind sctivly as imags. Th diffnt mthods can b goud into sval boad catgois: imag C H q ( q ) = R ( ) sgmntation imag gistation dtction (0) nois duction and oths. Th main challng of mdical imag analysis is to S H q ( q ) = R ( ) ocss and analyz ffctivly th mdical () imags to xtact quantify and intt this infomation to undstand and insight into and thi adial knls a th stuctu and function of th ogans bing imagd and ut it to fo actical us. i ( ) R = R ( ) = cos( ) R ( ) = sin( ) Mdical imags ovid imotant C 3. APPLICATIONS OF ORTHOGONAL RADIAL MOMENTS AND TRANSFORMS Th a vaious alications of othogonal adial momnts and tansfoms viz. to xtact th fatus and th contnts (colo txtu sha tc.) of mdical imaging (tumo dtction canc dtction) fingint cognition fac cognition data comssion tc. 3. Mdical Imag Analysis Mdical imag analysis is th tchniqu and ocss of solving o analyzing mdical oblms and cating a ictu of insid (intio) of a body fo clinical uos and mdical action. Mdical Imags a basd on diffnt imaging mods and digital imag analysis tchniqus. Mdical imaging finds to mak known th stuctus which a not visibl by th skin and bons and also to idntify and cu th disas. Th most imotant fild in mdical imag ocssing is mdical imag visualization and advancs in analysis of vaious mthods fo xaml an ssntial at of th aly dtction diagnosis and S infomation about th disas which is basd on th macoscoic micoscoic biochmical and molcula xamination of ogans and tissus. Th gowth of Mdical imags in databas is vy high in th ast fw yas whn th mdical digital imag quimnts such as CT (Comutd tomogah MRI (Magntic sonanc imaging) a usd in th clinic woks. Iscan [8] t. al. oosd D continuous wavlt tansfom(cwt) fo fatu xtaction dtmination of asymmty by using momnts and dtction of th tumo. Tansfoms has gat otntial in mdical imag comssion and dg dtction. A.S.Tolba [9] discovs th bst dsign aamts fo a data comssion schm alid to mdical imags of diffnt imaging modalitis. This tchniqu ducs th tansmission cost whil sving th diagnostic intgity. Psudo-Znik momnts [0] of an imag a usd fo sha dscito which hav btt fatus sntation caabilitis and a mo obust to nois than oth momnt sntations. 3. Fingint Rcognition Fingints cognition fs to th automatd mthod of vifying a match Rsach Cll : An Intnational Jounal of Engining Scincs Scial Issu ICTMS Dcmb 05 ISSN: (Pint) ISSN: (Onlin) 05 MID Publications in association with idya Publications. Authos a sonsibl fo any lagiaism issus. 5

6 Intnational Confnc of Tchnology Managmnt and Social Scincs (ICTMS-5) btwn two fingints of th sam son. A fingint cognition systm can b usd fo both vification and idntification. Th ang of alications also includs Banking Scuity (ATM Scuit ison visitos idntification of missing childn tc. Th sts fo fingint cognition includ imag acquisition ocssing fatu xtaction and matching. Znik momnts lays imotant ol in fingint cognition. ZMs a usd fo fatu xtacto du to its obustnss to nois and othogonal otis. ZMs a lss snsitiv to nois and with suio imag sntation caability. ZMs a vy usful fo undstanding th imag. ZMs a usd in attn cognition alications and imag constuction. PZMs a usd to cat invaiant fatu vctos fo th fingint biomtic. It has otation invaiant otis. PZMs a usd to xtact global fatus of th imag. PZMs xtact imag fatus indndntly with lss infomation dundancy. Ths asons mak PZMs mo dsiabl fo imag cognition. PZMs hav imotant otis such as obust to noisy imags and good imag constuction caability. Othogonal Foui Mllin momnts hav btt fomanc than ZM s and PZM s in tms of imag dscition and nois snsitivity in cas of small imags. Pola hamonic tansfoms (PHTs) hav bn usd in imag ocssing alications. PHTs hav vaious alications such as fingint classification attn cognition chaact cognition imag constuction tc. PHTs a comutationally fast. Ths tansfoms hav otis such as otation invaiant fatus and numical stability. PHTs a usd wh maximal disciminant infomation is ndd. Th comutation of PHTs knl is siml as comad to Znik momnts (ZMs) and sudo Znik momnts (PZMs). 3.3 Fac Rcognition Fac Rcognition is a attn cognition task fomd scifically on facs and it can b dscibd as classifying a fac ith "known" o "unknown" aft comaing it with stod known individuals. Fac cognition is an imotant and activ aa of sach in digital imag ocssing du to th comlxity of fac cognition oblm and its wid alications in many filds. Ths alications ang fom static matching of contolld fomat hotogahs such as assots cdit cads hoto IDs div's licnss tc. Th ang of alications also includs building o offic scuity ciminal idntification and authntication in scu systms lik comuts o bank tll machins. Th a th stags in fac cognition oblms namly fac sgmntation fom an imag fatu xtaction and classification. Th stag of fatu xtaction is vy imotant as it ultimatly dtmins th fomanc of a cognition systm. Although th a standad mthods fo classification and many sachs us thm to tst th fficincy of imag (fac) fatus divd by thm. Thus th stag of fatu xtaction bcoms vy imotant. Th xisting fatu xtaction o fac sntation tchniqus fall into two catgois local basd and global basd. Th fist on is basd on xtaction gomtical and stuctual facial fatus such as shas of ys nos mouth and distanc btwn ths. Ths mthods a not affctd by ilvant infomation in th imag but a snsitiv to undictability of fac aaanc and nvionmntal conditions. In th scond aoach which is statistical basd mthods fatus fom whol imag a xtactd. Sinc th fatus so divd snt th whol imag thy a lss snsitiv to imag nois and gomtical distotions. Howv thi Rsach Cll : An Intnational Jounal of Engining Scincs Scial Issu ICTMS Dcmb 05 ISSN: (Pint) ISSN: (Onlin) 05 MID Publications in association with idya Publications. Authos a sonsibl fo any lagiaism issus. 5

7 Intnational Confnc of Tchnology Managmnt and Social Scincs (ICTMS-5) fomanc is imaid by ilvant lmnts lik backgound ffcts. Sinc facs may undgo gomtic tansfomations such as otation scaling tanslation th othogonal adial momnts ovid th dsid chaactistics of bing invaiant to such changs. Among th widly usd adial othogonal momnts includ th Znik momnts sudo Znik momnts and othogonal Foui Mllin momnts. In th cnt ast wavlt momnts and wavlt tansfom hav also bn usd fo fatu xtaction and hybid aoachs combining fatus fom diffnt aoachs a also in us. 4. CONCLUSION Th a numb of othogonal adial momnts and hamonic tansfoms that a otation invaiant. Ths momnts and tansfoms a mad scal invaiant whn thy a comutd in a unit disc. Du to thi uniqu chaactistic of bing otation and scal invaiant ths momnts and tansfoms hav bn usd vaious imag ocssing alications. With th futh dvlomnt of othogonal adial momnts and tansfoms thy will b widly alid to th domain of imag ocssing. Int. Conf. on Fuzzy Systms and Knowldg Discovy (FSKD07) (Haikou 007) ol P.T.Ya X.Jiang and A.C.K.Fllow Two Dimnsional Pola Hamonic Tansfoms fo Invaiant Imag Rsntation IEEE Tans. Pattn Anal. Mach. Intll. 3(7) (00). 6. C.H.Th and R.T.Chin On Imag Analysis by th Mthods of Momnts IEEE Tans. Pat. Anal. Mach. Intll. 0(4) (988). 7. Z.L.Ping and Y.L.Shng Imag Dscition with Chbyshv-Foui Momnts Acta Ot. Sin. 9(9) (00). 8. Zaf Iscan Zumay Doku and Tam Olmz Tumo dtction by using Znik momnts on sgmntd magntic sonanc bain imags Ext Systm with Alications 37(00) A.S. Tolba Wavlt ackt comssion of Mdical imags Digital signal ocssing 00 vol. No. 4 : B.Jyothi Y.Madhav Latha P.G.Kishna Mohan.S.K.Rddy Mdical Imag Rtival Using Momnts IJAIEM ol. Issu (03). Rfncs:. M.R.Tagu Imag Analysis via th Gnal Thoy of Momnts J.Ot. Soc. Am (980).. A.B. Bhatia and E.Wolf On th Cicl Polynomials of Znik and Rlatd Othogonal Sts Poc. Camb. Phil. Soc (954). 3. Y.Shng and L.Shn Othogonal Foui Mllin Momnts fo Invaiant Pattn Rcognition IEEE Tans. J. Ot. Soc. Am. (6) (994). 4. H.Rn A.Liu J.Zou D.Bai and Z.Ping Chaact Rconstuction with Radial Hamonic Foui Momnts in Poc. 4 th Rsach Cll : An Intnational Jounal of Engining Scincs Scial Issu ICTMS Dcmb 05 ISSN: (Pint) ISSN: (Onlin) 05 MID Publications in association with idya Publications. Authos a sonsibl fo any lagiaism issus. 53