Efficient Linear Local Features of Digital Signals and Images: Computational and Qualitative Properties

Transcription

1 WSCG 5 Confeene on Compte Gaphis Visalization and Compte Vision Effiient Linea Loal Feates of Digital Signals and Images: Comptational and Qalitative Popeties Vladislav Myasnikov Samaa State Aeospae Univesity Moskovskoye shosse 4 ssia 4486 Samaa vmyas@geosamaa ABSTACT The pape pesents the analysis of effiieny of two oiginal appoahes to the onsttion of the s of linea loal feates (LLF) whih ae sed fo digital signal and image poessing The fist appoah is based on geneating of LLF whih onsists of sepaately onstted effiient LLFs eah of whih has its own algoithm fo feate allation The seond appoah assmes the onsttion of an effiient LLF whih has a single algoithm fo joint simltaneos omptation of all feates The analysis is aied ot by seveal iatos that haateize the omptational and qalitative popeties of the onstted LLFs eywods Feates digital images and signals omptational omplexity poessing qality INTODUCTION Feate eation is one of the main stages of visal data poessing systems development and it affets the final qality of the system A loal feate of a digital signal is sally a nmeial haateisti - the eslt of a tansfomation of digital signal/image samples whih belong to a loal analysis aea [] Fo linea loal feates (LLF) this tansfomation is linea with onstant paametes Taking into aont that allation of LLF vales an be made in diffeent ways (diet algoithms o fast onvoltion esive algoithms et) a speifi LLF is haateized by two omponents a linea onvoltion kenel (we all it as LLF's kenel) and an algoithm fo allation of the onvoltion of the inpt signal/image and this kenel (we all it as LLF's algoithm o algoithm fo LLF vales allation) Moeove if LLF's kenel detemines qalitative haateistis of the speifi LLF the algoithm fo LLF vales allation haateizes omptational omplexity of the feate Sets of feates whih have not jst one bt seveal feate vales fo the same analysis aea of a digital signal ae sally sed to solve patial poblems It is essential that allation of the oesponding feate vales in a an be poded by seveal ependent algoithms as well as a geneal algoithm that Pemission to make digital o had opies of all o pat of this wok fo pesonal o lassoom se is ganted withot fee povided that opies ae not made o distibted fo pofit o ommeial advantage and that opies bea this notie and the fll itation on the fist page To opy othewise o epblish to post on seves o to edistibte to lists eqies pio speifi pemission and/o a fee exetes jointly simltaneosly allations fo all the vales of feates in a In the latte ase we speak abot a of jointly ompted feates Qalitative iatos (fo s of jointly and ependently allated LLFs) ae detemined by a of oesponding kenels The geneal fomlation of the poblem of onstting an effiient ( of) LLFs implies the onstting LLFs (o of LLFs) with the best qality iato and with speified omptational omplexity [-4] Despite the seeming simpliity of the pesented fomlation we shold aept the poblem of onstting feates and thei s extemely omplex In the atho's pape [] the fomal appoah fo effiient LLFs onsttion has been poposed and in the papes [ 4] this appoah has been extended to the ase of onstting an effiient of jointly allated LLFs These appoahes allow s to design an effiient LLF (o effiient of LLFs) fo the most applied poblems The tem effiieny of LLF efes to the satisfation of two basi eqiements: algoithm fo LLF vales allation has a pedetemined omptational omplexity vale; LLF's kenel(s) is(/ae) the best mathed to a given qality iato Unde the peeding eqiements effiient LLFs enable s to establish a easonable balane between two opposing gops of feates: feates whih ae optimal in the sense of some qality iteia and do not have sitable o fast omptation algoithm (eg feates obtained sing ahnen-loeve tansfom); feates whih ae obtained by sing fast algoithms and ae not elated to the ontent of the Shot Papes Poeedings ISBN

2 WSCG 5 Confeene on Compte Gaphis Visalization and Compte Vision poblem and elevant qality iatos (eg feates obtained sing fast Foie tansfom algoithm) Aoding to the infomation of atho the only altenative appoah of the feate onsttion that satisfies all eqiements mentioned above exists It was poposed by Pof VLabnets in and was denoted as «mltipaameti wavelet tansfoms» [] Unfotnately these papes do not povide the method of solving the effiient LLFs onsttion poblem they only show that mltipaameti (o adaptive) wavelets exist and an be onstted The main ppose of this pape is to analyze/ompae the atho's two appoahes to onstting s of LLFs The fist appoah onstts a of feates by onstting a of effiient LLFs eah of whih has its own algoithm fo feate allation The seond appoah onstts an effiient of LLFs in whih thee is a single algoithm fo ompting all feates jointly Shot desiption of these appoahes is pesented in the Setion whee the known infomation is olleted New eslts on analytial and expeimental analysis of these appoahes ae pesented in Setions and 4 SETS OF JOINTLY AND INDEPENDENTLY CALCULATED LINEA LOCAL FEATUES OF DIGITAL SIGNALS: BACGOUND This Setion pesents shot efeene infomation on the effiient linea loal feates of the digital signals: basi definitions eqations and onsttion methods Fll desiption may be fond in the papes [-4] Let N be a of natal nmbes be a N ommtative ing with nity { ( n) n= signal of length N ove the ing x be an inpt Definition A linea loal feate (LLF) of length M ( A) M ove the ing is a pai { h( m) M { ( m) m = whee h is a linea onvoltion kenel of length M whih is detemined as a finite seqene ove the ing and satisfies the onstaint h ( m) h( M ) and A is an algoithm fo allating a linea onvoltion () of an abitay inpt signal ove the ing with the kenel M { ( m) y h : M ( n) h( m) x( n m) n = M N = = m () A of ependently allated LLF of length M ove the ing is a fthe of LLFs: { ( m) h M A = Definition A of jointly allated LLFs ove the ing is a pai { h m M A whee = h M is a of kenels eah of whih { m = is detemined as a finite seqene ove the ing and satisfies the following onstaints: h ; m M h ( m) h ( M ) ; ; and A is an algoithm fo joint allation of a of linea onvoltions of an abitay inpt signal n M < N ove the ing with a of { x n= N kenels: y M ( n) h ( n) * x( n) = h ( m) x( n m) = n = M N = () To distingish the elements of s of ependently allated LLFs fom jointly allated LLFs the last will be denoted as follows: { h m M A = In atho s papes [-4] we poposed a method fo onsttion of the s of ependently and jointly allated LLFs based on designing (s of) seqenes of kenel s samples in the fom of linea (mtal) eent seqenes (LS o LMS espetively) [569] Fo these (s of) seqenes alled NMC-(s) seqenes the omptational omplexity of allating linea onvoltions () o () is minimal Fo fixed paametes of linea (mtal) eent elations (L o LM espetively) these s of NMC seqenes o NMCs of seqenes fom a olletion of seqenes M o denoted espetively ( M T Hee is an ode of L fo samples of a seqene is a nmbe of seqenes in a T is an ode of mtal eene (fo s) and a ae L s o LM s oeffiients espetively As it has been shown in papes [-4] the powes of these olletions satisfy the elations: ( a ) ( M ) M > a M > C M T ( b d )( M T C ( M ) C ( M ) () NMC - nomalized with minimal omplexity Shot Papes Poeedings 4 ISBN

3 WSCG 5 Confeene on Compte Gaphis Visalization and Compte Vision Eah seqene fom the olletion along with its paametes is also haateized by a Θ of additional ependent paametes degees of feedom The powes of degees of feedom s Θ ae detemined in the following way [-4]: Θ ( M ) = Θ ( M T = (4) A The omptational omplexity of algoithms and A fo allating elevant feates o s of feates fo all NMC seqenes o NMC s of M and seqenes fom olletions ( M T is detemined by these eqations [- 4]: ( A ) ( A ) N (5) N M N N M ( ) ( ) ξadd T (6) ( ) T The poblems of onsttion of an effiient ( of) LLF(s) ae defined as follows [-4] A patila poblem of onsttion of an effiient of LLFs is defined as a poblem of seahing in a pedefined olletion ( M T of sh a (with its oesponding algoithm of joint allation of LLFs A ) fo whih the minimm ondition fo a poblem-speifi objetive fntion Ψ : M is flfilled: Ψ ( h h ( M ) h h ( M )) { min (7) h m M M T a = Fo a patila poblem of onsttion of an effiient LLF the dafting hanges ae elated to a M and an objetive fntion olletion M Ψ : The diffeene in the soltions of these poblems lies in the fat that in the fist ase a of jointly allated LLFs is fomed { h m M A = and in the seond ase thee is only one LLF M ( A) onstted { h( m) = Note that sing a patila poblem of onstting an effiient LLF it is possible to onstt a of ependently m { allated feates ( m) h M A = fo example by thei onseqent onsttion with appopiate modifiation of objetive fntions fo eah of patila poblems The omptational omplexity of allation of the s of LLFs and the nmbe of thei degees of feedom an be sed as iatos o onstaints in the analysis of onstted s of jointly and ependently allated LLFs Additionally fo fthe analysis we an intode a fomalized notion of olletions ompaability of jointly and ependently allated LLFs as follows h Let s onside a of LMS { ( m) M ( = ) whih belongs to olletion ( M T and satisfies a LM [4]: h b m) = = T m = ( m) a h m k m min k ( k = min t = ( T ) min( m) a h k = tk t ( m k) ϕ ( m) T m (8) In ase when ϕ ( m) LMS and LM ae alled homogeneos [569] The following lemma defines haateistis of the seqenes in this Lemma (on soltion of homogeneos LM) Let T= and a homogeneos LM of ode (T) h ( m) = a k h ( m k) atk h t ( m k) k = t = k = = detemines the samples of the olletion of h = ; fo the entie domain Let seqenes { m s define matixes ( z) Q of size whee eah element qij ( z) is detemined t q ( z) q ( z) i j < min( t) with an expession: ij ij i k a k z i = j k= qij ( z) = i < j i j = i k a( i j) k z i > j k= Then evey -th seqene of the olletion fo the entie domain satisfies the following homogeneos L: ( ) = h m h m s = s s= whee the vales { matix ( z) s s= Q deteminant: ae oeffiients in the i k s det ( Q ( z) ) = a k z = s z i= k= s= Shot Papes Poeedings 5 ISBN

4 WSCG 5 Confeene on Compte Gaphis Visalization and Compte Vision It is obvios that nde the lemma s onditions the seqene of the olletion with nmbe satisfies the homogeneos L with ode not exeeding ( ) This poved onnetion allows s to give the following definition fo ompaability of jointly and ependently allated LLF olletions Definition A of olletions of LSs M and a olletion of LMs { ( ) = ( M T ae alled ompaable if these eqations ae valid: = i= k= ( ) i k s ak z = s z = s= The fat of ompatibility means that one an speify fo at least one (homogeneos) of seqenes fom M T a exatly the same of seqenes fom { ( M ) = Note also that althogh thee ae moe than one eqal s of seqenes fo ompaable olletions the fll math of s of seqenes doesn t happen The eslts of this setion allow s to make an analytial ompaison of ompaable s of olletions COMPUTATIONAL AND QUALITATIVE POPETIES: ANALYTICAL COMPAISON Compaison of Linea Loal Feates Sets fo Compaable Colletions and ( m) { h = ; A Let N M T N M is an abitay effiient of LLFs fo a olletion M T a Comptational omplexity of the algoithm of allation of the LLF oesponding to any of seqenes of this olletion satisfies the eqation (6) Fom the othe hand one an onstt ependent effiient LLFs h m A fom the ompaable {( { m ) = M = ( M T of olletions { ( M ) = Then taking into aont eqations (5) omptational omplexity of LLF h m A is allation detemined as follows: = {( { m ) = M = N ( A ) ( ) (9) N M Compaing the ight pat of this eqation with the eqation (6) one an asse of the following elation oetness: ( ) ( ) ( ) ξadd T < T Then the following statement is oet ( ) () Statement Let M T N { T s of LLFs h ( m) M A = { h ae onstted and ( m) M A = fo ompaable olletions ( M T { ( ) = and M oespondingly while elations (5) and (6) ae satisfied as eqalities Then ( A ) < ( A ) = () This statement makes it possible to onfim the potential omptationally benefits of jointly allated LLFs in ompaison with s of ependently allated effiient LLFs designed fo ompaable olletions Compaison of Linea Loal Feates Sets with Eqal Nmbe of Degees of Feedom Eqation (4) means that the nmbe of degees of feedom fo the speifi effiient of LLFs fom the olletion ( M T is eqal to Fom the othe hand one an onstt ependent M effiient LLFs fom olletions { ( ) = in sh a way that the oveall nmbe of degees of feedom beomes eqal too It is easy to pove that in this ase the following eqality is valid: = () Using () one an asse the following elation oetness ( M T N T ): ( ) ( ) ( ) ξadd T > T ( ) Statement Let M T N T jointly and ependently allated LLFs have eqal nmbe of degees of feedom (ie eqation () is oet) while elations (5) and (6) ae satisfied as eqalities Then Shot Papes Poeedings 6 ISBN

5 WSCG 5 Confeene on Compte Gaphis Visalization and Compte Vision ( A ) > ( A ) = () This statement makes it possible to onfim the potential omptational benefits of of ependently allated LLFs in ompaison with the of jointly allated effiient LLFs designed fo eqal nmbe of degees of feedom Compaison of the Comptational Complexity of Solving the Patila Poblem of Feates Consttion Let ( M T and { ( ) = M ae ompaable olletions of jointly and ependently allated LLFs To ompae the allational omplexities of the solving of the patila tasks of { h = ; A M {( { ( m) m ) = M A = LLFs ( m) and h onsttion (see Setion ) we have to ompae the nmbe of M T a and seqenes in the olletions { ( M ) = ( M T In the ase of the olletion the nmbe of seqenes is defined by eqation () When we fom the of seqenes M we an fom the olletions { ( ) = se two obvios stategies: - exhastive seah (optimal soltion): in this ase the nmbe of seqenes s takes the fom: = ( ( ) ) M ; - inemental seah (qasi-optimal soltion): in this ase we seah fo the seqene of the -th olletion when the seqene of the (-)-th olletion is fond The nmbe of possible s of seqenes has the = ( ) fom: ( ) M Taking into aont eqations () we an ompae the omptational omplexity of solving the patila poblem of LLFs onsttion by ompaing the vale C ( M ) C ( M ) with = = C ( ) M ( ) (exhastive seah ase) o C ( ) M ( ) (inemental seah ase) It may be done by analyzing the following atios: C exhastive seah: ( M ) C( M ) ( ) CM ( ) = (4) C( M ) C( M ) inemental seah: ( ) CM ( ) = Using (5) we an pove the following statement (5) Statement Let M T N T M > Then = ( ) C ( M ) C( M ) > CM ( ) This statement makes it possible to onfim that solving of the patila poblem of jointly allated LLFs onsttion is moe diffilt than the solving of the patila poblem of ependent allated LLFs Diet nmeial analysis of the atio (5) fo sefl paametes ange (M=; =4) shows that it is mh moe diffilt: vales of the atio (5) ae in the ange [ 57*^9] Unlike the sitation is onsideed with an inemental seah it the ase of exhastive seah it is not possible to make an nambigos onlsion Diet nmeial analysis of the atio (4) fo paametes anges mentioned above shows that it is in the ange [7*^-8 97] Finally we an onlde that: - qasi-optimal soltion of the patila poblem of ependently allated LLFs onsttion based on the inemental seah is less diffilt then the optimal soltion of the patila poblem of jointly allated LLFs onsttion; - optimal soltion of the patila poblem of ependently allated LLFs onsttion based on the exhastive seah may be adially diffilt then the optimal soltion of the patila poblem of jointly allated LLFs onsttion So when we ae going to f optimal soltion jointly allated LLFs ae pefeable 4 Analytial Compaison: Conlsion Analytial and nmeial eslts pesented in this Setion above make it possible to onlde that the analytial analysis annot povide the nambigos answe on the qestion what type of LLFs (s of ependently o jointly allated LLFs) is bette Theefoe we ae tying to answe this qestion sing expeiments 4 COMPUTATIONAL AND QUALITATIVE POPETIES: EXPEIMENTAL COMPAISON In ode to omplete the ompaison of the s of ependently and jointly allated LLFs and to ompae them with existent typial ways of linea loal feates allations we will onside seveal illstative tasks In evey task we will ompae Shot Papes Poeedings 7 ISBN

6 WSCG 5 Confeene on Compte Gaphis Visalization and Compte Vision omptational and qalitative popeties of the onstted LLFs Despite of the illstative haate of the hosen tasks they appea often in eal appliations in simila fomlations and expliit iteia and mathematial model of the poessing signal is neessay only to point ot the best (fom typial ways of linea loal feates allations) of feate kenels So geneal poblem statement is as follows Let we have a digital signal that may be intepeted as a ealization of the disete stationay andom poess X n with zeo mean and atooelation fntion: n ( n) = D ρ n x (6) hee D x = ρ = 95 fo definiteness We allow that the length of the poessing signal N is nlimited and to pefom the loal analysis of the signal in the speifi position n we have to se M= samples of the signal (ie «poessing wow»): X ( n ) X ( n M ) Also we allow that the qality of the loal analysis of the signal depends dietly on the qality iato that is given by the following eqation: J α E = α M = E ( ) Y h M ( m) X ( m) ( X ( m) ) h h t ( α) α [ ] { m = ; M Hee = t= t h h (7) h is a of kenels that is sed fo linea epesentation of the analyzed fagment of the signal E - the mathematial expetation opeato Obviosly the less the qality iato the bette the of feates In the eqation (7) the fist tem defines elative eo of the epesentation of the signal fagment sing weighted sm of LLF's kenels the seond tem shows the oelation ate of the kenels and the denominato of the fist tem satisfies the eqality: M E X ( m) = D M ( = ) x Let define the geneal poblem as follows: we have to h = ; and obtain the of kenels { m M algoithm(s) of allation of the of onvoltions () of the signal with these kenels whih povide minimal vale of the qality iato (7) and satisfy etain estition on the omptational omplexity of onvoltions () allation: J α min () max (8) Bellow we povide seveal ways to solve the poblem (8) Fist and seond methods (soltions that ae odinay sed in digital signal and image poessing) se "optimal" kenels that omes fom ahnen-loewe deomposition [7] of the fagment of the disete stationay andom poess (6) The only diffeene between these methods is the onvoltion algoithms Fist method (method ) ses the diet onvoltion algoithm and the seond one (method ) ses the fast onvoltion algoithm that is based on the Fast Foie Tansfom (FFT) [8] and optimal setioning of the poessing signal [] In patie the seond method is the de fato standad fo soltions of this type of poblems Method ses the of jointly allated LLF's and methods 4-7 se the s of ependently allated LLF's (desiption of these methods is given bellow) It shold be noted that the detail desiption of the poblem (8) when α= sing the of jointly allated LLF's was given in the pape [4] Some sefl eqations that ae sed hee fo allation of an eo of epesentation of the fagment of the disete stationay andom poess sing nonothogonal kenels wee given in that pape too We analyze soltions of the poblem (8) fo thee vales of paamete α namely: - gop : α= - gop : α= - gop : α=/ Soltion of the poblem (8) sing s of ependently o jointly allated LLFs (methods - 7) is pefomed by solving the patila poblem (7) of onstting an effiient of LLFs This patila poblem [-4] means that the LLF's kenels ae fom the speifi olletion and this olletion is defined both by the task estitions (the size M of the "poessing wow" and the ppe bond max of the allational omplexity of feates allation) and sbjetive hosen paametes T a and { = In o expeiments paametes ae as follows: method : olletion ( M T paametes: T = = a = a = a = a ; = methods 4-7: olletions { ( ) = paametes: - qasi-polynomial (method 4): M k k ( ) C ( = k = ( ) ) k = - qasi-exponential (method 5): k = ρ = k = ( ) ( ) k ; ; Shot Papes Poeedings 8 ISBN

7 WSCG 5 Confeene on Compte Gaphis Visalization and Compte Vision - qasi- Fibonai (method 6): = : = : = = / = = ; = / = 4 : - qasi-hamoni (method 7): = : = : = : = = os = os = / ; ( ω) = ; ( ω) = os ( ω) = os( ω) ; = 4 : Pesented olletion names ae deived fom the names of the seqenes ( ) that satisfy the homogeneos LMS (8) with the same paametes The allational omplexity of the ependently and jointly allated LLFs is defined by eqations (5)-(6) that wee sed as eqalities Figes -5 pesent the obtained eslts that show the dependene of the qality iato J α of the onstted feates on the omptational omplexity of the feates allation These eslts lead to the following onlsions Fo the fist gop of the tasks (α= Fig) qality iatos fo the s of ependently and jointly allated LLFs (methods -4) ae signifiantly less (ie the qality is signifiantly highe) then the qality iatos obtained fo «optimal» kenels (obtained sing ahnen- Loewe deomposition) and diet (method ) o fast (method ) onvoltion algoithms Patilaly when the allational omplexity of the feates allation satisfies max = 4 the qality of the of jointly allated LLFs is six time highe (vs method )! Fo this patila ase Fig shows fo onstted kenels fo the jointly allated LLFs It is easy to see that these kenels ae simila to the «optimal» kenels (sinsoids of diffeent phases and feqenies) that may be obtained sing ahnen-loewe tansfom Fo the fist gop of the tasks (α= Fig) qality iatos fo the of jointly allated LLFs is less (ie the qality is highe) then the qality iatos fo the s of ependently allated LLFs Fo the nd and d gops of the tasks (α< Figs4-5) qality iato fo all types of LLFs depends signifiantly on the olletion paametes Theefoe hanging these paametes we an obtain diffeent answes whih type of feate s ( of jointly o of ependently allated LLFs) is bette In patie the best type of LLFs may be fond sing global optimization methods: geneti algoithms simlated annealing et The obtained expeimental eslts allow s to make two onlsions: - the poposed effiient LLFs have advantage in ompaison with the taditional way of solving sh a type of poblems even when the optimal kenels/bases exist; - jointly and ependently allated effiient LLFs have ompaable effiieny ie neithe of two appoahes has lea advantages J method method method method 4 Fige Compaison of the poposed effiient LLFs (methods -4) with taditional way of feates onsttion (methods -); task gop : α= h (m) = = = =4 Fige Fist fo onstted kenels fo the jointly allated LLFs (fo onveniene we pt kenels to the ange [-]) J method method 4 method 5 method 6 method 7 Fige Analysis of the poposed effiient LLFs: ompaison of the s of jointly (method ) and ependently (methods 4-7) allated LLFs; task gop : α= m Shot Papes Poeedings 9 ISBN

8 WSCG 5 Confeene on Compte Gaphis Visalization and Compte Vision J method method 4 m method 6 method 7 Fige 4 Analysis of the poposed effiient LLFs: ompaison of the s of jointly (method ) and ependently (methods 4-7) allated LLFs; task gop : α= J / 4 method method 4 method 6 method 7 Fige 5 Analysis of the poposed effiient LLFs: ompaison of the s of jointly (method ) and ependently (methods 4-7) allated LLFs; task gop : α=/ 5 CONCLUSIONS In this pape two appoahes to the onsttion of a of linea loal feates fo digital signals ae analyzed It is shown that depending on the ompaison iteia the poposed appoahes an have advantages and disadvantages In the geneal ase it an be onlded that these appoahes ae ompaable by effiieny vale (in tems of paametes pai - qality and omptational omplexity) This fat allows the develope of a patila signal o image poessing system to hoose the appoah that is onvenient and/o familia to him Condted in the pape expeiments show that the poposed appoahes have onvining advantages ove a typial "best" way to solve the model digital image analysis/epesentation poblem (in tems of paametes pai - qality and omptational omplexity) Fthe eseah will be elated to the following: - development of altenative ways to intode effiient linea loal feates; - development of nmeial methods and algoithms fo a qik soltion of the patila (and extended patila) poblem of onstting an effiient of jointly allated LLFs and of ependently allated effiient LLFs 6 ACNOWLEDGMENTS This wok was finanially sppoted by the ssian Fondation of Basi eseah (pojets: -7- -ofi a) 7 EFEENCES [] Fosyth DA Pone J Compte Vision: A Moden Appoah Pentie Hall Uppe Saddle ive New Jesey [] Myasnikov VV Effiient Loal Linea Feates fo Digital Signals and Images Compte Optis (4) pp [] Myasnikov VV Constting effiient linea loal feates in image poessing and analysis poblems Atomation and emote Contol 7() pp54-57 [4] Myasnikov VV Effiient mtally-allated feates fo linea loal desiption of signals and images In poeedings of the IASTED Intenational Confeene on Atomation Contol and Infomation Tehnology - Infomation and Commniation Tehnology pp9-4 [5] Agawal P Diffeene Eqations and Ineqality: Theoy Methods and Appliations Mael Dekke New Yok [6] Lidl Niedeeite H Finite Fields Seond edition Cambidge Univesity Pess 997 [7] Gigoi M Stohasti Calls: Appliations in Siene and Engineeing Bikhase Boston [8] Nssbame HJ Fast Foie Tansfom and Convoltion Algoithms Seond edition Spinge- Velag New Yok 98 [9] Andeson JA Disete Mathematis with Combinatois Pentie Hall Uppe Saddle ive New Jesey [] Gold B ade CM Digital Poessing of Signals MGaw-Hill Book Company New Yok 969 [] Labnets V Gainanov D Beenov D Mltipaameti wavelet tansfoms and pakets In Poeedings of the -th Intenational Confeene on Patten eognition And Image Analysis Vol pp 5-55 [] Labnets V Gainanov D Beenov D The best mltipaameti wavelet tansfoms In Poeedings of the -th Intenational Confeene on Patten eognition And Image Analysis Vol pp Shot Papes Poeedings 4 ISBN