An Efficient Lloyd-Max Quantizer for Matching Pursuit Decompositions

Transcription

1 BRAZILVI ITERATIOAL TELECOMMUICATIOS SYMPOSIUM (ITS6), SEPTEMBER 3-6, 6, FORTALEZA-CE, BRAZIL A Efficiet Lloyd-Max Quatizer for Matchig Pursuit Decompositios Lisadro Lovisolo, Eduardo A B da Silva ad Paulo S R Diiz Abstract Several applicatios are usig the Matchig Pursuit algorithm for sigal ad video compressio. The Matchig Pursuit approximates sigals iteratively usig liear combiatios of pre-defied atoms of a dictioary. I compressio applicatios Matchig Pursuits coefficiets, which multiply the atoms i the liear combiatio, eed to be quatized. The Lloyd- Max quatizer is kow to be the best quatizer for a give source. However, to desig a Lloyd-Max quatizer the statistics of the source eed to be kow. The statistics of Matchig Pursuit coefficiets are difficult to model. I this paper, startig from the observatio that the statistics of the agles betwee the residues ad the atoms preset little variatio alog Matchig Pursuit iteratios, we propose to use these statistics to model the oes of Matchig Pursuit coefficiets. This permits the desig of Lloyd-Max quatizers for Matchig Pursuit coefficiets. The Lloyd-Max quatize is compared to a state-of-the-art off-loop Matchig Pursuit quatizatio scheme. Results show that the proposed scheme has good rate-distortio performace, specially at low rates. Idex Terms Matchig Pursuit, Quatizatio, Compressio. After M iteratios, the MP obtais the M-term approximatio ˆx, or simply M-term, give by ˆx = γ g i(), (3) = which has distortio x ˆx = r M x,themth residue. The performace of the MP for sigal compressio applicatios depeds heavily o two aspects: i) Dictioary it should iclude atoms that are good matches to the possible compoets of the sigals to compress. I additio, the dictioary cardiality affects the data rate. ii) Quatizatio for compressio the coefficiets γ must be quatized. Usig a coefficiet quatizatio rule Q[ the compressed sigal is retrieved by the quatized M-term I. Itroductio The Matchig Pursuit algorithm [ (MP) approximates sigals iteratively. The approximatio is obtaied usig atoms (pre-defied sigals) g k from a dictioary D (the collectio of possible atoms). The MP is beig used i compressio schemes for -dimesioal [, [3 ad - dimesioal sigals [, [4. The MP works as follows. Let D = {g k } with k [,...,C(D), such that g k = k, C(D) is the dictioary cardiality (umber of elemets i D). At each iteratio, the MP searches for the atom g i(), i() [...C(D), with largest ier product with the residual sigal r x [, [5, which is give by γ = r x, g i(). () Observe that the first residue is equal to the sigal to be decomposed, that is r x = x. The ier product γ,which is a measure of how much of g i() is icluded i r x,is used to compute the ext residue r x = r x γ g i(). () Lisadro Lovisolo (lisadro@uerj.br) is with DETEL-FE-UERJ, Eduardo A B da Silva ad Paulo S R Diiz (eduardo, diiz@lps.ufrj.br) are with POLI/COPPE-UFRJ. ˆx q = Q[γ g i(). (4) = I order to desig efficiet quatizers Q[, oe eeds a statistical model for MP coefficiets. However, MP coefficiets are difficult to model. For example, i [6 it has bee observed that MP residues have a chaotic behavior. Here, istead of searchig for a good model for MP coefficiets, our approach is based o a statistical model for the MP agles. It relies o the observatio that the agles betwee the atom chose ad the residue beig decomposed alog MP iteratios have well behaved statistics. Defie, at each MP iteratio, the agle betwee the residue r x ad the selected atom g i() as ( r θ = arccos x, g i() r x ). (5) We have verified, experimetally (see sectio II), that the statistics of θ are approximately idepedet of. Here, it is cojectured that the agles betwee the residues ad the atoms i MP iteratios ca be statistically modeled as idepedet ad idetically distributed. SBrT 97

2 BRAZILVI ITERATIOAL TELECOMMUICATIOS SYMPOSIUM (ITS6), SEPTEMBER 3-6, 6, FORTALEZA-CE, BRAZIL II. Agles i Matchig Pursuit Iteratios Usig the defiitio of θ, eq. (5), the MP algorithm is such that γ = x cos (θ ), (6) γ = x si (θ )cos(θ ), (7). γ = x si (θ i )cos(θ ). (8) i= However, if D icludes the elemets g k ad g k for ay k, forcig the ier products to be positive, the γ = x si (θ i )cos(θ ). (9) i= This work assumes that g k ad g k belog to D obtaiig always positive coefficiets, ad all subsequet refereces to dictioaries cosider that. That is, if D does ot iclude g k, the g k is icluded i D ad C(D) is updated accordigly. A. Statistics of the Agles i MP Iteratios I [6, it has bee observed that MP residues have a chaotic behavior, thus it would be reasoable to assume that, after some MP iteratios, the residues ca have ay orietatio. I additio, oe may assume that these orietatios may have uiform probability desity fuctio o the uit-ball. This is equivalet to assume that the residues are realizatios from a memoryless idepedet ad idetically distributed (iid) Gaussia source. That is, the sigal source is such that if a outcome is x = [x(),x(),...,x(), the the x(j) have the same Gaussia distributio (,σ ).This source does ot privilege ay sigal orietatio. Therefore, we use this source to ivestigate MP agle statistics. Fig. shows the relative frequecy histograms of the RVs Θ, which correspod to the agles θ that result from decompositios of realizatios of a Gaussia source usig a dictioary composed of 6 ormalized radom atoms also draw from a Gaussia source i R 4.These histograms were obtaied usig a esemble of 5, MP decompositios of radom sigals from a Gaussia source. Such dictioaries, composed of C(D) ormalized sigals draw from a -dimesioal Gaussia source, are referred here as GS D(C(D),) ad thus the former dictioary is deoted by GSD(6, 4). It should be oted that this dictioary has its cardiality doubled i order to obtai just positive coefficiets. I Fig. oe otes that the histograms of the RVs Θ have very similar shape for all. This leads to the cojecture that the pdfs f Θ (θ )are idepedet ad idetically distributed at ay iteratio. The results preseted i [6 corroborate this assumptio, where it is show that, uder specific coditios, MP residues have chaotic behavior. Fig. shows the mea ad the variace of Θ for several. Fig. depicts also the covariaces betwee MP agles i.5..5 ormalized histogram of Θ θ.5..5 ormalized histogram of Θ θ ormalized histogram of Θ θ ormalized histogram of Θ θ.5..5 ormalized histogram of Θ θ ormalized histogram of Θ θ Fig.. Relative frequecy histograms of Θ for a Gaussia source i R 4 usig the GSD(6, 4). 4 Expected value of Θ Cov[Θ Θ 5 5 Cov[Θ 5 Θ Variace of of Θ Cov[Θ Θ 5 5 Cov[Θ Θ 5 5 Fig.. Mea, variace ad covariace of Θ for a Gaussia source i R 4 usig the GSD(6, 4). differet steps. From it oe observes that Cov[Θ i Θ k =, i k, that is, the agles are ucorrelated. I this work we assume that the agles are idepedet RVs. This is ot a ureasoable assumptio, sice it does ot cotradicts the behavior observed i Fig.. The results preseted so far make use of a dictioary of relatively low dimesio ad cardiality C(D). I practice, the MP is commoly used i large dimesioal spaces with dictioaries such as the Gabor oe [5. The elemets of this dictioary are defied by traslatios, modulatios ad dilatios of a prototype sigal. The most commo choice for the prototype f() is the Gaussia widow. We aalyze here a real Gabor dictioary composed of atoms π with predefied phases i multiples of V.Eachatomis the give by [5 8 δ(), j = >< «p j g() = K (j,p,v) f cos kπ j + πv, j (,L), >: j V, j = L () where f() = 4 e π, is the sample, K (j,p,v) provides uit-orm atoms, ad v [,...,V. Above, j defies the atom scale, p defies the time shift, ad k defies the atom modulatio, ad for L =log () scales their rages are [ j [,L, p [, j ),k [, j ), ad v [,V. SBrT 98

3 BRAZILVI ITERATIOAL TELECOMMUICATIOS SYMPOSIUM (ITS6), SEPTEMBER 3-6, 6, FORTALEZA-CE, BRAZIL 3 Fig. 3 shows f Θ (θ ), for some, obtaied for a esemble of 8, decompositios of Gaussia sigals i R 64 usig the Gabor dictioary with four phases. Fig. 4showsf Θ (θ ), for some, obtaied for a esemble of 8, decompositios of sigals drive from a source that has gamma distributed coordiates i R 64 for the same dictioary. ote that the agle statistics show for each sigal source differ strogly oly at the first MP iteratio, beig visually very similar for the other iteratios. It ca be oted that for this dictioary the agles i differet MP steps have similar statistics eve for very differet sources. ote that, although the statistics of Θ are ot exactly equal to the statistics obtaied i the first MP iteratio for a Gaussia source, they are reasoably close to these; therefore f Θ (θ ), for >, ca be reasoably approximated by the f Θ (θ ) obtaied for a memoryless Gaussia source. This is a reasoable assumptio, sice the memoryless Gaussia source does ot privilege ay orietatio, what seem to be the case for the residues r x for >. B. Discussio of Results It was verified that the statistics of the agles, after the first MP iteratio, ca be cosidered to be ivariat with respect to the step umber. For some dictioaries, for example the Gabor oe, the agle pdf, after some steps, is slightly differet from the agle pdf obtaied for a memoryless Gaussia source. However, oe ca still use a Gaussia source to obtai good estimates of the pdf of the agle i MP iteratios. The differece i the statistics meas that, although the residues of MP iteratios have similar orietatios, these orietatios are ot uiformly distributed. evertheless, the iid statistical model is ot too bad a assumptio. I the sequel, this model is used to desig Lloyd-Max quatizers of MP coefficiets. This permits to verify the validity of the statistical model for the agles i MP iteratios. It should be oted that the statistics of the first MP agle, ad therefore of the first coefficiet, are much more source depedet tha they are at further MP steps. Therefore, i order, to make a appropriate use of the preseted agle model, the first coefficiet, γ, will be quatized with egligible error ad ecoded as side iformatio. ote that this value replaces x, the value of the sigal orm which is usually trasmitted as side iformatio i practical MP based compressio schemes [, [4. III. Quatized Matchig Pursuit Decompositios I compressio applicatios, oe ecodes the quatized versios Q[γ of the coefficiets γ of M-terms. Oe way to obtai the Q[γ is to quatize the coefficiets offloop, i.e. first the whole decompositio is obtaied ad the the coefficiets are quatized. Aother strategy is iloop quatizatio [4, i which the quatized coefficiet is used to compute the residue, ad due to that i-loop quatizatio might result i atoms i M-term that deped o the quatizer used. 3 x 6 ormalized histogram of Θ x 6 ormalized histogram of Θ x 6 ormalized histogram of Θ x 6 ormalized histogram of Θ x 6 ormalized histogram of Θ x 6 ormalized histogram of Θ Fig. 3. ormalized histograms of MP agles for a Gaussia source i R 64, usig bi, at = {, 8, 6, 3, 64, 7}, for the 4-phase Gabor dictioary i R x 6 ormalized histogram of Θ x 6 ormalized histogram of Θ x 6 ormalized histogram of Θ x 6 ormalized histogram of Θ x 6 ormalized histogram of Θ x 6 ormalized histogram of Θ Fig. 4. ormalized histograms of MP agles for a source with coordiates drive from a gamma distributio, usig bi, at = {, 8, 6, 3, 64, 7}, for the 4-phase Gabor dictioary i R 64. We choose off-loop quatizatio sice it allows a simple rate distortio (RD) optimizatio procedure, i which differet quatizers are tried out i order to fid oe that meets a prescribed RD criterio. I cotrast, i-loop quatizatio would require, for RD optimizatio, several MP sigal decompositios, what would greatly icrease the computatioal demads. Sice i off-loop quatizatio the residues are computed without kowledge of the quatizer, the quatizatio error i oe step caot be compesated i subsequet steps. However, these quatizatio errors ca be miimized by desigig appropriate quatizers [. I this work, MP coefficiet quatizers desig is achieved by employig the iid statistical model of MP agles to desig Lloyd-Max quatizers for the MP coefficiets. The quatizers desiged make use of dead-zoes because some advatage i RD performace ca be gaied by quatizig small coefficiets to zero [, [4: either the coefficiets quatized to zero or their atoms idices are SBrT 99

4 BRAZILVI ITERATIOAL TELECOMMUICATIOS SYMPOSIUM (ITS6), SEPTEMBER 3-6, 6, FORTALEZA-CE, BRAZIL 4 set, reducig the data rate. For trasmissio, it is importat that if coefficiets ad/or idices are lost, the decoder ca just successfully igore the lost terms whe recostructig the sigal. Therefore, it would be highly desirable for the quatizer for a give γ to be idepedet of the quatized values of other γ m (m ). A. Distortio Due to Off-Loop Quatizatio Whe M-terms are quatized off-loop, each coefficiet γ is replaced by its quatized versio Q[γ ad the sigal approximatio is retrieved usig eq. (4). The distortio criterio employed for MP quatizer desig ca be: i) the eergy of the error relative to the actual sigal d = x ˆx q ; () ii) the eergy of the error relative to the M-term d M = ˆx ˆx q = (γ Q[γ )g i(). () = I this work, the secod oe is used, sice we desig oe Lloyd-Max quatizer for each umber of coded terms M. The distortio per sample of the quatized M-term is give by d M = d M,thatis, d M = (γ Q[γ )(γ m Q[γ m ) g i(), g i(m), = m= (3) where is the sigal legth. D is composed of uit orm vectors ( g i() = ). Thus, defiig the quatizatio error e q (γ )=γ Q[γ, (4) it follows that [ d M = M e q (γ )+ (5) = e q (γ )e q (γ m ) g i(), g i(m). = m=, For a M-term ˆx, the miimizatio of eq. (5) implies also the miimizatio of the distortio per sample (see eq. ()) give by d = d = x ˆx q. (6) ote that if d M is equal to zero the d = r M x /. ) Distortio for a Give Source: For a give sigal source X, oe may cosider the expected value of d M { E[d M = M E [ e q(γ ) + (7) = E [ e q (Γ )e q (Γ m ) g i(), g i(m). = m=,m Each Γ is a radom variable (RV) that correspods to γ,for M, for sigals draw from X. I eq. (7), E [ e q(γ ) stads for the expected value of the squared quatizatio errors of the RV Γ. B. MP Coefficiet Quatizatio Usig MP Agles I eq. (9) the value of x is required to compute the MP coefficiets from the MP agles. Accordig to thediscussioattheedofsectioiiweusethefirst coefficiet γ istead of x, the eq. (9) ca be rewritte as γ = γ δ, δ =ta(θ ) si (θ i )cos(θ ),. (8) i= Thus, the pdfs of the coefficiets γ ca be computed from the pdfs of the agles Θ, assumig that the Θ are idepedet. The pdfs of the θ, >, ca be obtaied as explaied i sectio II. For a kow γ, the pdf of the RV Γ,for, is give by f (γ γ )=f (γ δ γ ), where is the RV whose outcome is δ, see eq. (8). If a optimal quatizer Q is desiged for the RV Y, the the optimal quatizer for Z = cy (c is a costat) is simply a scaled versio of Q. Thus,forγ kow, the quatizatio of δ istead of γ,fromeq.(7),gives { E[d M γ = γ M E [ e q ( ) + (9) = E [ e q ( )e q ( m ) g i(), g i(m). = m=,m Oce the pdfs of the RVs are kow E[d M γ cabe computed for ay quatizatio rule applied to.sice the quatizatio is applied to δ istead of γ, the value of γ is required at the decoder. I this work we use γ to defie γ,for. The use of γ to compute γ ( ) guaratees its correct value at the decoder. For quatizer desig usig eq. (9) the pdfs of are eeded which ca be computed from the RVs Θ i, i [,...,, correspodig to the agles betwee the residues ad the atoms selected i MP steps, as i eq. (8). IV. Lloyd-Max Quatizers for MP Coefficiets For a give first coefficiet γ the quatizatio of MP coefficiets should aim to miimize the distortio i eq. (9). I this case, oe should desig quatizers for,. Therefore, if just oe quatizer is to be used the it shall be desiged for = M =,thervgiveby the uio of the ( ). Sice MP iteratios are disjoit evets the pdf of = M = is give by f (δ) = M M = f (δ ). A. Distortio for a Optimal Quatizer The distortio per sample i eq. (9) has two terms. The first term is the sum of the squared quatizatio errors of, whereas the secod cotais a sum of ier products betwee dictioary atoms weighted by the products of the quatizatio errors of the atoms ivolved i the ier products. As verified, i sectio II, the RVs of the agles Θ ca be assumed to be ucorrelated. Although ad m may be correlated, whe desigig a quatizer for = M = the assumptio that the quatizatio errors e q ( ) are ucorrelated is reasoable. It is also SBrT 9

5 BRAZILVI ITERATIOAL TELECOMMUICATIOS SYMPOSIUM (ITS6), SEPTEMBER 3-6, 6, FORTALEZA-CE, BRAZIL 5 reasoable to assume that the quatizatio errors products e q ( )e q ( m ) are ot correlated to the ier products g i(), g i(m). Usig the assumptios above, the secod term i eq. (9) becomes E [e q ( ) E [e q ( m ) E [ g i(), g i(m). = m=,m () The atoms selected at differet MP steps may be correlated. But due to the ivariat ature of the agles statistics i differet MP steps, oe ca cosider the expected value of the ier product betwee the atoms selected i ay two differet MP steps ad m to be also ivariat. That is E [ g i(), g i(m) = c. Therefore, otig that E [e q ( ) = eq. () yields E [e q( ) = (M )E [e q( ) M m=,m E [e q ( ), () = = m=,m E [e q( m) c = () E [e q( m) c. The optimal quatizatio of leads to E [e q ( ) =, so that the expressio above vaishes. As a result eq. (9) becomes E[d M γ = γ E[( Q[ ), (3) = This result is a sum of terms E[( Q[ ), thus E[d M γ = γ (δ Q[δ ) f (δ )dδ. (4) = Sice the same quatizer Q[ is applied to all, M, oe has that E[d M γ =γ M (δ Q[δ) f (δ)dδ, (5) with f (δ) = f (δ ) (6) M = If = M = ad defiig eq. (5) becomes MSE( ) = E [ ( Q[ ) (7) E[d M γ = γ (M ) MSE( ). (8) The desig of a optimal quatizer for = M =,that miimizes eq. (8), is accomplished by Lloyd-Max quatizers [7 (). ote that sice Lloyd-Max quatizers are ubiased estimates of the iput the E[e q ( ) =, ad thus the term i eq. () vaishes validatig eq. (3). B. Lloyd-Max Quatizer Desig The desig of Lloyd-Max quatizers requires f (δ ), for M; these are estimated from f Θ (θ ). I tur, f Θ (θ ) is estimated by applyig oe MP decompositio step to a large set of sigals draw from a memoryless Gaussia source. As discussed i sectio II, this is a acceptable procedure sice all Θ,, have statistics that are similar to the oes obtaied for a memoryless Gaussia source. The estimated f (δ ) are the used to calculate f (δ), ad f (δ) istheusedtoobtaithe of b coef bits (L = b coef levels). The quatizer thresholds ad recostructio levels are calculated usig a iterative algorithm [7. The f (δ ), that are used to compute f (δ), are obtaied recursively from a estimate of f Θ (θ ) that is i tur obtaied usig the agle i the first MP step for a esemble of C(D) realizatios from a Gaussia source. The same quatizer law is used for all coefficiets γ. ote that f (δ) varies with M, therefore each M leads to a differet quatizer. The quatizer desig is idepedet of γ ad it suffices to desig quatizers for γ =, storig copies of the quatizers i both ecoder ad decoder. The ecoder seds γ, the umber of bits of the quatizer, ad the umber of terms of the decompositio (M), i a header, to the decoder. The parameter γ is used to scale the quatizer i both coder ad decoder, a simple strategy that makes good use of resources. V. Lloyd-Max Quatizers Performace The state-of-the-art for off-loop quatizatio of MP coefficiets is preseted i [. There, uiform quatizers whose umber of quatizatio levels ad rage adapt accordig to the coefficiets of the M-term are used. Here, this quatizatio scheme is referred to as adaptive bouded uiform quatizer (). The implemets a bit-allocatio per coefficiet that relies o the kow result that MP coefficiets magitudes decrease o average at each MP iteratio. I the, previous to quatizatio all the coefficiets eed to be sorted i decreasig magitude ad the is fed with the coefficiets i this order. For each coefficiet, the employs a uiform quatizer of differet rage ad umber of levels; the quatizer rage for the l th coefficiet depeds othequatizedvalueofthe(l ) th coefficiet, ad the umber of levels of each coefficiet quatizer is decided usig a criterio based o a Lagragia multiplier a bitallocatio procedure. For decodig the quatized decompositio the umber of bits used to quatize the secod coefficiet as well as the larger coefficiet are set as side iformatio. The total rate of MP quatized decompositios is give by R = S [log (C(D)) + r coef, (9) where S is the umber of terms that remai after quatizatio, ad r coef is the rate icurred i codig the quatized coefficiets. Therefore, the rate i bits per sample is give by R/, where is the sigal space dimesio. The SBrT 9

6 BRAZILVI ITERATIOAL TELECOMMUICATIOS SYMPOSIUM (ITS6), SEPTEMBER 3-6, 6, FORTALEZA-CE, BRAZIL 6 strategy employed to geerate the coded bitstream is to etropy code the differeces betwee the quatizatio idices of successive coefficiets. I the comparisos betwee Lloyd-Max quatizatio ad preseted here, this strategy is also employed to code the Lloyd-Max quatized M-terms. Fig. 5 shows the RD curves of quatized MP expasios arrivig from three differet radom sources i R (a memoryless Gaussia, a memoryless uiform ad a memoryless Gamma distributed source) usig both the Lloyd- Max quatizatio ad the ad a GSD(8, ). For each distict source the results are averages over a esemble of quatized MP decompositios of sigals from each source. For this experimet the s were desiged with bit-depth ragig from to 8. It ca be see i Fig. 5 that both quatizers have similar performace for the three sigal sources; however the teds to be slightly better at low rates (below 8 ). Fig. 6 shows the RD curves of quatized MP expasios arrivig from three differet radom sources i R 64 for the ad the for the Gabor dictioary of 4 phases i R 64, see eq. (). For that purpose the s were desiged with bit-depth ragig from to 6. The decompositios to be coded allowed a maximum of 56 terms. For each distict source the results are averages over a esemble of quatized MP decompositios of sigals from each source. The decompositios quatized with the require a larger bitrate because its bitallocatio scheme does ot maages to cotrol the rate as the shape of f Θ (θ ) for this dictioary implies a low decay rate of the coefficiets ivolved i M-terms obtaied usig this dictioary. VI. Coclusio The agles betwee the residues ad the selected atoms i Matchig Pursuit iteratios ca be statistically modeled as idepedet ad idetically distributed (iid) at each iteratio. That is, the agles i Matchig Pursuit steps ca be cosidered statistically ivariat with respect to the decompositio step. As a result, the statistics of Matchig Pursuit agles ca be obtaied from the statistics of the first Matchig Pursuit agle for sigals draw from a memoryless Gaussia source. Based o the iid statistical model for Matchig Pursuit agles, Lloyd-Max quatizers for Matchig Pursuit coefficiets were desiged. The Lloyd-Max quatizer is desiged to be the same for all the coefficiets i a M-term, ad its desig requires both the umber of terms to be coded ad the probability desity fuctio of the first Matchig Pursuit agle for a memoryless Gaussia source, resultig i dictioary depedet quatizers. It is importat to poit out that if the source is ot Gaussia the first MP agle has differet statistics tha the MP agles i other steps. However, eve i this case, the Lloyd-Max quatizers obtaied for the Gaussia source ca still be employed sice the first coefficiet is ot quatized but trasmitted with egligible error as side iformatio. E[d E[d E[d Memoryless Gaussia Source Memoryless Uiform Source Memoryless Gamma Distributed Source Fig. 5. ad RDs for three differet radom sources usig the GSD(8, ), the distortio is defied i eq. (6). E[d E[d E[d Memoryless Gaussia Source Memoryless Uiform Source Memoryless Gamma Distributed Source Fig. 6. ad RDs for three differet radom sources usig the 4-phase Gabor dictioary i R 64, the distortio is defied i eq. (6). The Lloyd-Max quatizatio preseted was compared to the state-of-the-art off-loop quatizatio scheme i [ ad both quatizatio schemes showed similar rate distortio performace. Refereces [ S. Mallat ad Z. Zhag, Matchig pursuits with time-frequecy dictioaries, IEEE Tras. o Sigal Processig, vol. 4, o., pp , December 993. [ P.Frossard,P.Vadergheyst,R.M.FiguerasIVetura,ad M. Kut, A posteriori quatizatio of progressive matchig pursuit streams, IEEE Tras. o Sigal Processig, vol. 5, o., pp , February 4. [3 L.Lovisolo,M.A.M.Rodrigues,E.A.B.daSilva,adP.S.R. Diiz, Efficiet coheret decompositios of power systems sigals usig damped siusoids, IEEE Tras. o Sigal Processig, vol. 53, pp , 5. [4 R. eff ad A. Zakhor, Modulus quatizatio for matchigpursuit video codig, IEEE Tras. o Circuits ad Systems for Video Techology, vol., pp ,. [5 S. Mallat, A Wavelet Tour of Sigal Processig, sted. Sa Diego, Califoria, USA: Academic Press, 998. [6 G. Davis, S. Mallat, ad M. Avellaeda, Adaptive greedy approximatios, Joural of Costructive Approximatio, vol. 3, pp , 997. [7 A. K. Jai, Fudametals of Digital Image Processig, 3rded., ser. Pretice Hall Iformatio Ad System Scieces Series. Eglewood Cliffs, J 763: Pretice Hall, 989. SBrT 9