OPTIMAL PIECEWISE UNIFORM VECTOR QUANTIZATION OF THE MEMORYLESS LAPLACIAN SOURCE
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1 Joural of ELECTRICAL EGIEERIG, VOL. 56, O. 7-8, 2005, OPTIMAL PIECEWISE UIFORM VECTOR QUATIZATIO OF THE MEMORYLESS LAPLACIA SOURCE Zora H. Perić Veljo Lj. Staović Alesadra Z. Jovaović Srdja M. Bogosavljević I this paper we will preset a desig procedure of a optimal piecewise uiform vector quatizer of a memoryless Laplacia source. We will derive expressios for the graular distortio ad the optimum umber of the output poits i each subregio. We will also compare the obtaied results with the results preseted i referece [7] ad o the basis of that we will derive some coclusios about the proposed models. K e y w o r d s: piecewise uiform vector quatizer, Laplacia source, optimal asymptotic aalysis 1 ITRODUCTIO Quatizers play a importat role i the theory ad practice of moder day sigal processig. The asymptotic optimal quatizatio problem, eve for the simplest case uiform scalar quatizatio, is very topical owadays [1, 2]. It was show i the literature that vector or multidimesioal quatizatio ca yield a smaller average mea squared error per dimesio tha scalar quatizatio for the case of fie quatizatio. Desigig a optimal vector quatizer is equivalet to fidig a partitio of the vector space ad assigig a represetative poit to each partitio such that a predefied distortio measure betwee the iput ad output is miimized. Ufortuately, the optimal partitios i higher dimesioal spaces are uow for eve the simplest source distributios ad the most commo distortio measures. Extesive results have bee developed o fidig the optimal output poits distributio i multidimesioal space for a specific probability distributio. The first approximatio to the log-time-averaged probability desity fuctio (pdf) of amplitudes is provided by Laplacia model [3]. I a umber of papers the vector quatizatio of the memoryless Laplacia source was aalysed sice the pdf of the differece sigal for a image waveform follows the Laplacia fuctio [3]. Also, the Laplace source is a model for the speech [4]. The aalysis of a vector quatizer for a arbitrary distributio of the source sigal was give i referece [5]. The authors derived a expressio for the optimum graular distortio ad optimum umber of output poits. However, they did ot prove the optimality of the proposed solutios. Also, they did ot defie the partitio of the multidimesioal space ito subregios. I referece [6] they derived expressios for the optimum umber of output poits, however, the proposed partitioig of the multidimesioal space for memoryless Laplacia source does ot tae ito cosideratio the geometry of the multidimesioal source. I referece [7], vector quatizers of Laplacia ad Gaussia sources were aalysed. The proposed solutio for the quatizatio of memoryless Laplacia source, ulie i [6], taes ito cosideratio the geometry of the source, however, the proposed vector quatizer desig procedure is complicated ad gives results that are quite differet from optimal for low bit rates. We preseted a improved variat of the method i [7] ad also the optimal solutio for piecewise uiform vector quatizatio. I this paper we will give a exact aalysis of the proposed quatizer ad preset our systematic aalysis of a piecewise uiform vector quatizer of the Laplacia memoryless source. We will give a geeral ad simple way for the desig of the piecewise uiform vector quatizer. We will derive the optimum umber of output poits ad prove the optimality of the proposed solutios. Also we will give a short discussio of the vector quatizatio solutio proposed i referece [7] by Jeog ad Gibso. 2 OPTIMAL ASYMPTOTIC AALYSIS Let us cosider a multidimesioal piecewise uiform quatizer, where the iput space is partitioed ito subregios. Each subregio is divided usig cubic cells with differet sizes. The -dimesioal vector of Laplacia radom variables is the iput to the -dimesioal quatizer. For -dimesioal vector x = [x 1 x 2... x ] cosistig of idepedet ad idetically distributed (i.i.d.) Laplacia variables x i with zero mea ad uit variace, the joit pdf of x is f(x) = f ( ) x i = 2 2 exp ( ) 2 x i. (1) Faculty of Electroic Egieerig, Uiversity of is, Serbia; peric@elfa.i.ac.yu Uiversity of Techology Ilmeau, Germay, Telecom Serbia, is, Serbia ISS c 2005 FEI STU
2 Joural of ELECTRICAL EGIEERIG 56, O. 7 8, The cotour of costat probability desity fuctio (pdf) is give by x i = l( 2 fc 2 ) r0, r 0 > 0, (2) where f c is the value of pdf. This is a expressio for the -dimesioal hyperpyramid with radius r 0, where we defie the radius as r = x i. (3) Sice each x i has a expoetial distributio with mea 1/ 2 ad variace 1/2, the radom variable r has a gamma pdf give as where f (r) = K r 1 e 2r, r 0, (4) K = 2 2 Γ(). (5) The probability that the vector x = [x 1 x 2... x ] is i the th cell deoted by P is equal to P = f(x)dx = R r+1 r f (r)dr, (6) where r ad r +1 deote the radius of the th ad ( + 1)th regio R ad R +1, respectively. Graular distortio for piecewise vector quatizatio of the sigal geerated by the Laplacia source ca be writte as follows D g = 1 2 P = 1 ( V ) 2 P, (7) where the volume V of the th subregio is give by the followig equatio V = 2 r+1 Γ( + 1) 2 r Γ( + 1). (8) ow, we will derive the optimum solutio for the piecewise vector quatizatio of the memoryless Laplacia source. We ca obtai the optimum umber of cells i oe subregio by usig the Lagragia multipliers o equatio (7). I order to do that we cosider the expaded fuctio J J = D g + λ. (9) After differetiatig J with respect to, = 0, q 1, ad equalizig with zero, with some mathematical maipulatio, we obtai the umber of poits i the th regio as = +2 P +2 i=0 +2 i P +2 i, (10) with the respect to the coditio that the total umber of output poits is equal to =. (11) If we substitute from equatio (10) i equatio (7), we have D g = ( 1 2/ P ) +2. () I the literature [8] we foud that this form of equatio (equatio ()) is Zador Gersh formula for the piecewise uiform vector quatizer. The total distortio ca be calculated usig the followig expressio D = ( 1 2/ + 1 r q P ) +2 [ (r rq ) 2+ ( 1) 2 q ] f (r)dr, (13) where r q deotes the radius of the iput space. From the equatio (13) we ca see that the total distortio depeds oly o the total umber of the output poits, dimesio of the quatizer, the umber of the subregios ad the radius of the iput space. If we assume that the total umber of the output poits ad the dimesio of the quatizer are ow, miimizig the total distortio give by the equatio (13) we ca obtai the optimum radius of the iput space ad the optimum umber of the subregios. It is importat to emphasize that this solutio is valid for all ids of space partitio ito subregios. Oe example of partitio is preseted with the followig equatio for radius of subregio r = r q q, = 0,... q, (14) where r q ad q deote optimum radius of the iput space ad the umber of the subregios, respectively. ow, we perform asymptotic aalysis, ie we assume the ifiite umber of output poits. For this special case the costat pdf f(r) over the volume of the th subregio V ca be supposed. Tha we ca substitute P with P = V f(r ) ad after that the optimum umber of the output poits is give with = V f +2 (r ) i=0 V i f +2 (ri ), (15)
3 202 Z.H.Perić V.Lj.Staović A.Z.Jovaović S.M.Bogosavljević: OPTIMAL PIECEWISE UIFORM VECTOR QUATIZATIO... while the graular distortio is D g ( 1 ) +2 f 2/ +2 (r )V. (16) The assumptio that P = V f(r ) has to be very carefully applied. The problem is that the former assumptio is ot valid for the small bit rates per dimesio, ie for small codeboo. The equatios (15) ad (16) are suitable for high-resolutio case, ie i practice for desig of the high bit rate quatizer or the large dimesio quatizer. 3 JEOG AD GIBSO ASYMPTOTIC AALYSIS I [7] Jeog ad Gibso proposed a codeboo dilutio where a codeboo is costructed by a series of successive subregios filled with differet sets of cubic cells of legth c. It was show that for the miimum distortio the followig equatio must be satisfied [ 2(r B r A ) ] c B = exp c A, (17) + 2 where c A, c B, r A ad r B represet the legths of cells ad radii of two successive subregios. Also, they supposed the costat pdf f(r) o matter which bit rate it is ad from this they derived that the legths of cells i two successive subregios satisfy the followig costrait c +1 = s c, (18) where s represets the scale factor ratio. After some mathematical maipulatio with equatios (17) ad (18) they obtai ( 2r ) exp = s. (19) + 2 From equatio (19), they cocluded that the radial parameters have to be equidistat r = ( + 2)l s 2, = 0, q 1. (20) The umber of the output poits i each subregio was calculated i [7] as = V (s c 0 ), (21) where c 0 is the legth of cells i the first subregio. Tha the total umber of output poits i [7] is equal to the ifiite summatio of the output poits i each subregio, which is a defectiveess of this model. From equatios (21), (8) ad (20) the total umber of output poits ca be calculated as = = c 0! [ ] 2( + 2)l s s [ ( + 1) ]. (22) Sice the sum i the above equatio coverges to some costat S whe s > 1 ad = 2 R, the legth of the cells i the first subregio ca be calculated from (22) as [ (s c 0 = 2 R+0.5 1)S ] 1 ( + 2)l s. (23)! The way to determie the desired value of the scale factor ratio s is to calculate the miimum value of the distortio. I order to do that, we eed to assume a value for scale factor ratio, ad calculate the legth of the cells i the first subregio usig (23). After that we calculate radial parameter from equatio (20), the volume of each subregio usig (8) ad the umber of output poits usig equatio (21). With this parameters we ca tha calculate the total distortio. We have to repeat this procedure i order to obtai the miimum distortio ad the optimum value for the scale factor ratio s. 4 PERFORMACE COMPARISO I this sectio we will compare performace of the quatizers proposed i [7] ad i this paper. We will do that o the example of the two dimesioal uiform piecewise quatizer described i [7, Fig. 5]. Total umber of output poits is = 165 which reflect o a bit rate per dimesio ( R = bits/dimesio) ad the umber of subregios ( q = 3). Scale factor ratio is s = 1.5, ie the radius of iput space r q = First we will derive oe improvemet of the model proposed i [7]. We will start from the equatios (21) ad (22), ad replace the ifiite summatios with the summatios over q subregios = = 1 c 0 V. (24) s The legth of the first cell c 0 ca tha be obtaied from the previous equatio as c 0 = 1 ( ) 1/ V 1/ s. (25) Usig equatio (1) ad equatio (19) we ca express s as s = 2 1 2(+2) f +2 (r ). (26) If we substitute s from equatio (26) i equatio (25) we ca write c 0 = 2 ( 2(+2) 1/ V 1/ f +2 (r )). (27)
4 Joural of ELECTRICAL EGIEERIG 56, O. 7 8, The, if we substitute (26) ad (27) i (21) we obtai the elegat way of calculatig the umber of output poits i each subregio = V f +2 (r ) i=0 V i f +2 (ri ). (28) The umber of output poits i subregios obtaied o this way (equatio (28)) is equivalet to our optimal solutio for the high-resolutio case (equatio (15)). The distortios i these cases are calculated by equatio (16). ow, we will compare the umbers of output poits i subregios as they are give i [7, Fig. 5], equatio (21), with our results obtaied usig equatios (10) ad (15), ie (28) (Table 1.). If the umber of the output poits i subregio is calculated by meas of equatio (28), tha the miimizig of the graular distortio (equatio (16), = 165 ad q = 3) gives the scale factor ratio optimum value s = Table 1. [7, Fig. 5] eq. (10) eq. (28) s = The values of the total distortio calculated usig the values give i Table 1 are give i Table 2. Table 2. ext, i Table 4 we will show the values for the optimum radii of the two-dimesioal quatizer, umber of subregios ad distortio for various values of the bit rate per dimesio R. Distortio as a fuctio of the iput load r max ad the dimesio of the vector quatizer is plotted i Fig. 1. The bit rate per dimesio is R = 4 bit/dimesio, umber of subregios q = 5. As we ca see the distortio decreases ad the optimum iput load icreases as the dimesio of the vector quatizer icreases. Table 3. R = 1 R = 2 P f(r )V P f(r )V E E0 3.64E E E E E E E E E E E E E E E E E E E E E E E E E E E E-4 Table 4. R r opt q,opt D SRQ (db) E E E [7, Fig. 5] eq. (10) eq. (28) s = Distortio r opt distortio = 2, 4, 6 It is obvious from Table 2 that distortio calculated as i [7] is more tha two times larger tha our optimal distortio, ie our distortio is about 3.6 db lesser tha distortio i [7]. This meas that the uiform piecewise vector quatizatio with our space partitio ito subregios has better performace tha the quatizatio proposed i [7]. Also for give bit rate i this example, ie for give codeboo dimesio, the assumptio that the pdf f(x) is costat iside subregio is ot suitable sice the distortio i this case (Table 2, colum 3) is very differet from optimal (Table 2, colum 2). I order to show oes more that the assumptio of the costat pdf fuctio over a subregio is ot suitable for the low bit rate, i the followig Table 3 we compare the values for the probability that the iput vector is i the th subregio ad the product of the pdf fuctio ad the volume of that subregio. The differece betwee these two is obvious r max Fig. 1. Ifluece of the dimesio of the vector quatizer o the distortio 5 COCLUSIO I this paper we have preseted a desig procedure of a optimal piecewise uiform vector quatizer of memoryless Laplacia source. The preseted desig procedure
5 204 Z.H.Perić V.Lj.Staović A.Z.Jovaović S.M.Bogosavljević: OPTIMAL PIECEWISE UIFORM VECTOR QUATIZATIO... of piecewise uiform quatizer is based o miimum distortio criterio. By optimisig the total distortio, that icludes graular ad overload distortio, we ca exactly determie all the parameters of the quatizer that are eeded. We derived the expressios for the graular distortio ad the optimum umber of the output poits i each subregio. We use closed forms to calculate all parameters of the quatizer. We proved the optimality of the proposed solutios. Also, the preseted method is simpler ad more practical for applicatio tha the method i referece [7]. Refereces [1] HUI, D. EUHOFF, D. L.: Asymptotic Aalysis of Optimal Fixed-Rate Uiform Scalar Quatizatio, IEEE Tras. 47 o. 3 (2001), [2] A, S. EUHOFF, D. L.: O the Support of MSE-Optimal, Fixed-Rate Scalar Quatizers, IEEE Tras. o Iformatio Theory 47 o. 6 (2001), [3] JAYAT,. S. OLL, P.: Digital Codig of Waveforms Priciples ad Applicatios to Speech ad Video, Pretice-Hall, ew Jersey, [4] GERSHO, A. GRAY, R. M.: Vector Quatizatio ad Sigal Compressio, Kluwer Academ. Pub, [5] KUHLMA, F. BUCKLEW, J. A.: Piecewise Uiform Vector Quatizers, IEEE Tras. o Iformatio Theory 34 o. 5 (1988), [6] SWASZEK, P. F.: Urestricted Multistage Vector Quatizers, IEEE Tras. o Iformatio Theory 38 o. 3 (1992), [7] JEOG, D. G. GIBSO, J. D.: Uiform ad Piecewise Uiform Lattice Vector Quatizatio for Memoryless Gaussia ad Laplacia Sources, IEEE Tras. o Iformatio Theory 39 o. 3 (1993), [8] GRAY, R. M. EUHOFF, D. L.: Quatizatio, IEEE Tras. o Iformatio Theory 44 o. 6 (1998), Received 1 October 2004 Zora H. Perić was bor i is, Serbia, i He received the BSc degree i electroics ad telecommuicatios from the Faculty of Electroic Sciece, is, Serbia, Yugoslavia, i 1989, ad MSc degree i telecommuicatio from the Uiversity of is, i He received the PhD degree from the Uiversity of is, also, i He is curretly Professor at the Departmet of Telecommuicatios, Uiversity of is, Yugoslavia. His curret research iterests iclude the iformatio theory, source ad chael codig ad sigal processig. He is particulary worig o scalar ad vector quatizatio techiques i compressio of images. He has authored ad coauthored over 70 scietific papers. Dr Zora Peric has be a Reviewer for IEEE Trasactios o Iformatio Theory. Veljo Lj. Staović was bor i is, Serbia, i He received the BSc Degree i electroics ad telecommuicatios from the Faculty of Electroic Egieerig, is, Serbia, i 2000, ad MSc Degree i telecommuicatios from the Uiveristy of is, i He has authored ad coauthored 10 scietific papers. His curret iterests iclude the commuicatio theory, source codig, quatizatio. Alesadra Z. Jovaović was bor i is, Serbia, i She received the BSc Degree i electroics ad telecommuicatios from the Faculty of Electroic Egieerig, is, Serbia, i 1995, ad MSc Degree i telecommuicatios from the Uiveristy of is, i She has authored ad coauthored 30 scietific papers. Her curret iterests iclude the commuicatio theory, source codig, sigal detectio. Srdja M. Bogosavljević was bor i is, Serbia, i He received the BSc Degree i electroics ad telecommuicatios from the Faculty of Electroic Egieerig, is, Serbia, i 1992, ad MSc Degree i telecommuicatios from the Uiveristy of is, i He has authored ad coauthored 30 scietific papers. His curret iterests iclude the iformatio theory, source codig, polar quatizatio. SLOVART G.T.G. s.r.o. GmbH of periodicals ad of o-periodically prited matters, boo s ad CD - ROM s Krupisá 4 PO BOX 152, Bratislava 5,Slovaia tel.: , fax.: gtg@iteret.s, SLOVART G.T.G. s.r.o. GmbH
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