Simple Compression Algorithm for Memoryless Laplacian Source Based on the Optimal Companding Technique

Transcription

1 Smple Compresson Algorhm for Memoryless Laplacan Source Based on he Opmal Compandng Technque Zoran H Perć Marko D Pekovć Mlan R Dnčć ) Faculy of Elecronc Engneerng Aleksandra Medvedeva 4 perc@elfaknacyu ) Faculy of Scences and Mahemacs Všegradska dexerofns@gmalcom Absrac Ths paper has wo achevemens The frs am of hs paper s opmzaon of he lossy resson coder realzed as andng quanzer wh opmal resson law Ths opmzaon s acheved by opmzng mal amplude for ha opmal andng quanzer for Laplacan source Approxmae expresson n closed form for opmal mal amplude s found Alhough hs expresson s very smple and suable for praccal mplemenaon sasfy opmaly creron for Lloyd-Max quanzer (for R 6 bs/sample) In he second par of hs paper novel smple lossless resson mehod s presened Ths mehod s much smpler han Huffman mehod bu gves beer resuls Fnally a he end of he paper general resson mehod s obaned by onng opmal andng quanzer and lossless codng mehod Ths mehod s appled on he concree sll mage and good resuls are obaned Besdes sll mages hs mehod also could be used for resson speech and bomedcal sgnals Keywords: smple lossless resson algorhm andng quanzaon opmal mal amplude Inroducon Quanzers play an mporan role n every A/D converson sysem They are appled for he purpose of sorage and ransmsson of connual sgnals A vas of research was done n hs opc An effcen algorhm for desgn of he opmal quanzer for he source wh known dsrbuon was developed by Lloyd and Max (Max 960) Ths s an erave mehod whch gves he sequence of quanzers convergng o he opmal quanzer Lloyd-Max algorhm s effcen for he small number of quanzaon pons (usually for ) However for he large number of quanzaon pons s me consumng Also he realzaon lexy of he opmal quanzers wh many quanzaon levels s very hgh One soluon whch overcomes hese dffcules s he andng model (Jayan and oll 984; Gray 004) Quanzers based on he andng model known as andng quanzers have smple realzaon srucure and performances close o he opmal Also he desgn of such quanzers s more effcen han Lloyd-Max algorhm snce does no requre he erave mehod Ths dfference s very noable for some commonly used sources ncludng he Laplacan source We wll consder andng quanzer because of less lexy n desgn and mplemenaon Memoryless Laplacan source s commonly used model n many applcaons due o s smplcy and fac ha many parameers and characerscs can be found as he closed form relaons (ncludng he opmal resson funcon) Examples are speech sgnal mages vdeo sgnal bo-medcal mages (for example ued omography scanner) ec For ransmsson processng and sorng ha sgnals smple and fas resson algorhms are

2 desrable One soluon s gven n (Sarosolsky 007) for unform quanzer In hs paper we gve smpler soluon for non-unform quanzer In paper (Ramalho 00) lossless resson algorhm was gven ha provded only he addonal resson of he dgzed sgnal (PCM) bu whou provdng a qualy mprovemen Our algorhm gves qualy mprovemen as well as furher resson In paper (kolc and Perc 008) lossy resson and Lloyd-Max quanzer are appled In our paper smple algorhm for lossy and lossless resson s gven For lossy resson smpler quanzer s used whou deeroraon of performances Usng hs algorhm for speech sgnal resson mprovemen of 0 bs/sample s acheved n relaon o algorhm used n (kolc and Perc 008) In hs paper we wan o aclsh wo goals Our frs goal s o opmze performances of he andng quanzer for memoryless Laplacan source by opmzng mal amplude In he paper (a and euhoff 00) s gven one heursc soluon for he suppor regon of he opmal quanzer Also paper (a 004) deals wh he upper bound of hs suppor regon The heursc soluon for he mal sgnal amplude n he adng model s gven n (kolc and Perc 008) An nenson of hs paper s o generalze hese resuls by gvng several approxmae analycal expressons for he mal sgnal amplude and s numercal uaon As s shown one of he analycal expressons almos concdes wh he opmal value The second goal of hs paper s he consrucon of he lossless coder for Laplacan source wh smple realzaon srucure The well-known effcen algorhm for lossless codng of he nformaon sources wh known symbol probably s Huffman algorhm (Hankerson e al 004; Sayood 006) I requres very lex realzaon srucure and also s me consumng Hence we gve he smple codng algorhm for M -ary exended nformaon source and gve he opmzaon of he exenson order M Our algorhm has beer performances (smaller b-rae) and drascally smpler realzaon lexy han Huffman algorhm Our algorhm s also smpler han algorhm proposed n (Sarosolsky 007) because we have b n prefx and ( r ) bs or r bs n suffx ha s we have consan number of bs n prefx On he oher hand algorhm n (Sarosolsky 007) uses varable number of bs n prefx whch requres much lex coder and especally decoder Prefx defnes regon where symbol of M-ary source exenson falls ( M s a number of samples n frame and we can consder frame as M -ary source exenson) Suffx represens codeword for symbol of M -ary source exenson Fnally we have oned opmal andng quanzer and smple lossless codng n a generalzed resson mehod Ths generalzed mehod s appled on a sll mage and good resuls are obaned as shown n he paper Besde sll mages hs resson mehod can be appled on speech and bo-medcal sgnals Ths paper s organzed as follows Secon recalls some basc heory of quanzers and andng model In secon opmzaon of andng quanzer was done by fndng opmal value for he mal sgnal amplude We gave hree approxmae expressons n closed form for and also one numercal mehod for fndng whch s based on mnmzaon of approxmae expresson for dsoron In secon 4 we are he approxmaons wh numercally obaned mal sgnal amplude Secon 5 deals wh he consrucon and performance analyss of he smple lossless codng algorhm In Secon 6 we gave generalzed resson algorhm whch represens combnaon of lossy and lossless resson In hs secon applcaon of hs generalzed mehod on a sll mage s presened Secon 7 concludes he paper

3 Fxed rae scalar quanzers and andng echnque An -pon fxed rae scalar quanzer s characerzed by he se of real numbers called decson hresholds whch sasfy = < < L < < = + 0 and se y y called represenaon levels whch sasfy y α = ( ] for = Ses α α α form he paron of he se of real numbers R and are called quanzaon cells The quanzer s defned as many-o-one mappng Q : R R defned by Q( x) = y where x α In pracce npu sgnal value x s dscrezed (quanzed) o he value y Cells α α are called nner cells (or granular cells) whle α and α are ouer cells (or overload cells) In such way cells α α form granular whle cells α and α form an overload regon As fxed-rae and scalar are only ypes of quanzers consdered n he paper for brevy we wll ordnarly om hese adecves Suppose ha an npu sgnal s characerzed by connuous random varable X wh probably densy funcon (pdf) p( x ) In he res of he paper we wll suppose ha nformaon source s Laplacan source wh memoryless propery and zero mean value The pdf of hs source s gven by x p( x) = e σ σ The sources wh exponenal and Laplacan pdf are commonly encounered and he mehods for desgnng quanzers for hese sources are very smlar Whou loosng of generaly we can suppose ha σ = and las expresson becomes x p( x) = e Snce p( x ) s an even funcon an opmal quanzer Q mus sasfy = and y = y for arbrary = The qualy of he quanzer s measured by dsoron of resulng reproducon n arson o he orgnal Mosly used measure of dsoron s mean-squared error I s defned as ( ) ( ( )) ( ) ( ) = D Q = E X Q x = x y p x dx The -pon quanzer Q s opmal for he source X f here s no oher -pon quanzer Q such ha D( Q ) < D( Q) We also defne granular Dg ( Q ) and overload Dol ( Q ) dsoron by g ( ) = ( ) ( ) = D Q x y p x dx + ol ( ) = ( ) ( ) + ( ) ( ) g ol D D Q x y p x dx x y p x dx Obvously holds D( Q) = D ( Q) + D ( Q) Denoe by dsoron of an opmal -pon quanzer Consderable amoun of work has been focused on he desgn of opmal quanzers for resson sources n mage speech and oher applcaons As frs dscovered by Paner and De (Paner and De 95) for large holds D = c / Here c s he Paner-De consan + ( ) / c = p ( x) dx ()

4 The general mehod for desgn of an opmal -pon quanzer for he gven source X s Lloyd-Max algorhm (Max 960; Jayan and oll 984) Ths s erave process sars from (0) ( n) some nal quanzer Q and consrucs he sequence Q n = of quanzers whch converge o he opmal quanzer Q (values of hresholds ( n ) converge o he hresholds he opmal quanzer Q ) Due o he uaonal lexy of hs mehod s no suable for he desgn of opmal quanzers wh more han 8 levels Hence here are developed oher mehods for consrucon of nearly opmal quanzers for large number of quanzaon levels One of he such echnques whch s commonly used s andng echnque (Judell and Scharf 986) I forms he core of he ITU-T G7 PCM sandard recommended for codng speech sgnals Compandng echnque consss of he followng seps: Compress he npu sgnal x by applyng he ressor funcon c( x ) Apply he unform quanzer Q u on he ressed sgnal c( x ) Expand he quanzed verson of he ressed sgnal usng an nverse ressor funcon c ( x) The correspondng srucure of a nonunform quanzer conssng of a ressor a unform quanzer and expandor n cascade s called andor Opmal resson law s used n hs paper Compresson funcon for ha law n dependence of pdf s known bu opmal mal amplude for ha resson law has no deermned up o now In hs paper hs opmal mal amplude for opmal resson law wll be deermned There are several ways o defne he ressor funcon for opmal resson law Orgnally n (Judell and Scharf 986) and also n (a 004) s he funcon c : R ( ) defned by We wll use he smlar defnon as above x / p ( x) dx c0 ( x) = + / p ( x) dx x < x / p ( x) dx / p ( x) dx x > c( x; ) = + x [ ] Value s called mal sgnal amplude Snce an mage of he funcon c( x; ) s ( ) decson hresholds u and represenaon levels y u of he unform quanzer are defned by u = + ( = 0 ) yu = + = Hence decson hresholds and represenaon levels of equvalen non-unform quanzer Q( x) = c ( Q ( c( x))) can be deermned as he soluons of he followng equaons u c( ; ) = c( y ; ) = y u u By solvng he las equaons we oban he values and amplude of () y as he funcons of he mal 4

5 + ( )exp( ) = / log 0 = log / < + ( )exp( ) + ( + ) exp( ) y = / log y = log / < + + ( )exp( ) where log x represens naural logarhm of x Expressons for and y wll be used n furher consderaon and hence we wll gve hem explcly = log (5) + ( ) exp ( ) y = log (6) + ( ) exp ( ) For he ease of noaon we wll denoe he dsoron granular dsoron and overload dsoron by D D and D respecvely (we wll om he quanzer Q ) Usng () we oban ha g ol dsoron D s funcon of one parameer Same holds also for D g and fnd he opmal value (Perc e al 007) (4) D ol Our goal s o whch mnmzes dsoron D One approxmae soluon s gven n Esmaon of he opmal mal sgnal amplude Closed-form expresson for he exac dsoron D = D( ) as he funcon of he mal sgnal amplude D = D( ) can be obaned by combnng relaons () and (4) Obaned funconal dependence s very lcaed snce appears n bounds of negrals and also n funcons under negraons Hence he exac mnmzaon of D( ) s very lex (see secon 4) Our frs goal n hs paper s o fnd some approxmae closed-form expressons for opmal mal amplude Accordng o he (Perc and kolc 007) granular dsoron D g can be approxmaed by he value D % as follows g 9 / D g ( p ( x) dx) exp = = ( ) ( ) On he oher sde we can explcly ue overload dsoron by % (7) + (8) D = ( x y ) p( x) dx ol = e ( + + y y + y ) 5

6 Hence we wll consder an approxmae dsoron D% = D% g + Dol I s worh menonng ha bounds of he negral (7) are usually se o and respecvely and herefore mum sgnal amplude s equal o he suppor regon Maxmal error n hs approxmaon s obaned on he nerval ( ) Therefore n order o mprove he approxmaon we se he correspondng bound o and herefore lower bound of negral n (8) s also Dsorson 8x0-4 7x0-4 7x0-4 7x0-4 Exac value Approxmaon Relave error of dsorson 0% 08% 06% 04% 0% Maxmal sgnal amplude 00% Maxmal sgnal amplude Fgure a Graphs of D and D % (lef) and relave error (rgh) as he funcon of for = 8 Dsorson x0 - x0 - x0 - x0 - x0 - x0 - Exac value Approxmaon Relave error of dsorson 5% 4% % % % 0x Maxmal sgnal amplude 0% Maxmal sgnal amplude Fgure b Graphs of D and D % (lef) and relave error (rgh) as he funcon of for = 64 Dsorson 46x0-45x0-44x0-4x0-4x0-4x0 - Exac value Approxmaon Maxmal sgnal amplude Relave error of dsorson 5% 4% % % % 0% Maxmal sgnal amplude Fgure c Graphs of D and D % (lef) and relave error (rgh) as he funcon of for = Value D % s he funcon of and y and snce hese values are boh funcons of (expressons (5) and (6)) also D % can be expressed as he funcon of oe ha approxmaon of D by D % s very accurae especally for he large values of 6

7 On Fgure (a b c) we show graphs of D and D % for = 8 = 64 and = as he funcons of and graph of relave error D D% D We gve hree esmae expressons for he opmal mal sgnal amplude The frs esmaon s based on he fac ha he value of he las represenaon level deermned by (6) s no opmal n general For fxed an opmal value of he followng expresson y op + xp( x) dx = = + + p( x) dx y as y s gven by On he oher hand for he fxed y value of mnmzng D % s obaned as he soluon of he followng equaon D% = 0 (0) The soluon of he las equaon s gven by + ˆ = log () Value s deermned such ha nex condon holds ˆ + = y () Usng () and (6) we oban he followng equaon n erms of + log log + = + ( ) exp ( ) whose soluon s gven by () = log () + exp( / ) In he second approach we consder he smlar condon as () amely nsead of ˆ we wll pu he value log = ( ) gven by a (a 004) as he upper bound of he suppor regon In such way we oban he followng esmaon of he mal sgnal amplude () = log (4) exp( / ) Fnally n he hrd approach we wll search for he value of such ha ˆ = By solvng hs equaon we oban = log( + ) (5) I can be seen n he las secon ha quanzers correspondng o he mal sgnal ampludes () () and has almos opmal dsoron especally for he large values of ow we wll descrbe he erave mehod for he mnmzaon of he approxmae dsoron D % As we have already seen oal dsoron D % s he funcon of and y (accordng o he (9) 7

8 (7) and (8)) Moreover relaons (5) and (6) explcly deermne hese wo quanes as he funcon of he opmal mal sgnal amplude Hence we oban closed form analycal expresson for oal dsoron D % as he funcon of number of quanzaon levels and opmal mal sgnal amplude e D% = D% ( ) An opmal value % can be obaned by applyng ewon erave mehod ddd % ( ) + = (6) d Dd % ( ) () () Esmaes and can be used as he sarng pons We can oban D% ( ) as he closed-form expresson and hen perform he symbolc dfferenaon o oban closed-form expressons for he frs and second dervave Smlar procedure (ewon mehod) can be repeaed for he numercal uaon of mnmum of he exac dsoron D( ) denoed by Snce n (6) we need second dervave of D( ) produced expressons wll be oo large Insead of ha s more suable o use some non-graden opmzaon mehod for mnmzaon of D( ) A good choce s Smplex mehod (Chong and Zak 00) due o s smplcy and ease of mplemenaon Recall ha exac expresson for D( ) s obaned by combnng equaons (4) wh equaon () The procedure for uaon of can be descrbed as follows: frs we fnd he exac expresson for D( ) subsung (4) n () and hen apply Smplex opmzaon mehod From he fgure we can see ha boh D and D % has only one global mnmum and hence boh ewon erave mehod and non-graden opmzaon mehods wll converge o % and respecvely A he end of hs secon we recall ha s he exac opmal mal amplude obaned by mnmzaon of he exac dsoron D bu ha mnmzaon procedure s very lcaed Because of ha we found four approxmae values for he opmal mal amplude: % () () % was obaned by mnmzaon of approxmae expresson for dsoron D % bu () ha mnmzaon procedure s also very lcaed On he oher hand expressons for are very smple and n closed-form and because of ha hey are very useful n praccal () applcaons As would be seen from he nex secon especally () and are very closed o 4 umercal examples In hs secon we wll are he approxmae values opmal value () respecvely whle % and () () () and % wh he exac s calculaed usng expressons () (4) and (5) s found by mnmzaon of D % and D as was descrbed n prevous secon Mnmzaon procedures for D and D % are mplemened n he symbolc programmng package MATHEMATICA (see for example (Wolfram 00)) The performance of a quanzer s ofen specfed n erms of sgnal-o-nose rao ( SRQ ) whch s drecly obaned from he dsoron D usng he followng relaon 8

9 σ SRQ = 0 log 0 D oe ha SRQ s descendng funcon of D and hence he quanzer s beer when SRQ has hgher value In Table we show he mal sgnal amplude approxmae values () () and % and exac value for dfferen values of number of quanzaon levels I can be seen ha values and are very close () () % Table : () () % and for dfferen values of Table gves he opmal SRQ of andng based quanzer ( SRQ o he ) as well as he dfferences beween SRQ value correspondng o he () () and % ( SRQ SRQ SRQ and SRQ respecvely) and SRQ ( ) ( ) SRQ = SRQ SRQ for = and SRQ = SRQ SRQ correspondng () () We denoed Table : SRQ () SRQ () SRQ SRQ SRQ * * * * * * * * * *0-6 99* SRQ () SRQ () SRQ SRQ and SRQ for dfferen values of 9

10 I s evden from Table ha andng quanzer wh mal sgnal amplude has SRQ almos equal o he mum possble value SRQ of he quanzer based on he andng model (dfference s arable wh he machne precson) Ths esmaon s even beer han % obaned by mnmzaon of he approxmae dsoron D % Oher wo esmaes have also close values of SRQ o he opmal Therefore we can conclude ha esmaon s very accurae approxmaon of he opmal mal sgnal amplude and can be effcenly used n he praccal applcaons oe ha for 64 all consdered andng models (wh mal sgnal amplude () equal o % and sasfes he soppng creron of he Lloyd-Max algorhm Accordng o he (Gray 004) hs soppng creron can be expressed as follows D D δ = < D where D s he dsoron of he opmal -level quanzer Comparng SRQ values soppng creron can be expressed as SRQ SRQ < 0 0 SRQ where SRQ = 0 log 0( / D ) Ths approves ha quanzers based on he andng model wh opmzed mal sgnal amplude are close o he opmal quanzers On he oher hand le us remember ha n pracce andng quanzers has much smpler srucure han opmal quanzers especally for he large number of quanzaon levels Hence hey are very suable for he praccal applcaons Opmzaon of he lossy algorhm based on opmal resson law was done n hs and prevous secon Also we gave some smple approxmae expressons n closed form for he mal amplude whch sasfed sop creron of Max-Lloyd algorhm The bes approxmae expresson s whch s very smple and gves neglgble error The am was o choose approxmae expresson wh mnmal error n arson o opmal soluon Imporance of he good choce of can be seen from Fgure : for small values for (eg = ) error of dsoron can be greaer han 5% Opmal andng quanzer presened n hs and prevous secon wll be used as a par of generalzed resson algorhm whch wll be presened n secon 6 5 ovel smple lossless resson mehod In hs secon novel smple lossless resson mehod wll be presened Ths s our second goal n hs paper Suppose ha he oupu of he -level quanzer are he ndces = 0 In he res of hs secon we wll suppose ha = r where r s posve neger Hence all ndces = 0 can be coded by he r -b codeword and average b-rae (average number of bs per sample) s equal o R = r We wll develop he smple resson algorhm whch mproves ha b-rae bu akng no accoun he properes of he source X Snce X has Laplacan dsrbuon mos probable ndces are mddle ones e ndces from he segmen I = / 4 / 4 The correspondng range for he source sgnal s [ x x ] where 0

11 x log exp = + Snce I consss of / ndces requred number of bs for he represenaon of each ndex I s r ow we wll descrbe he srucure of he coder Frs ndces are grouped no he frames conssng of M ndces Frs b of each frame s conrol b If all ndces n he group belongs o I hey are all coded wh r bs each and he frs b s se o Oherwse frs b s se o 0 and all ndces are coded wh r bs each Denoe by p he probably ha oupu ndex belongs o I (e ha sample of he sgnal belongs o ( x x ) ) I s equal o p = exp( x ) ow he probably ha M consecuve M ndces belongs o I s equal o p Hence we can ue average b-rae as follows M M R = p ( r ) + ( p ) r + (7) M where he las erm corresponds o he frs b of he frame For example by akng M = = 8 and = we oban x = 4540 p = and R = We wll are hs resul wh he enropy H of he source and average b-rae R h obaned by usng Huffman algorhm (Hankerson e al 004) Le us remember ha las wo quanes are gven by H = P log Rh = P log = P = P The lengh of source exenson M can be opmzed usng relaon (6) For = 8 can be shown ha opmal value of M s M = Also holds H = 6 88 and R = 6847 Le us noce ha H < R < H + and R < Rh Smlarly holds for he oher values of Hence we can conclude ha our lossless coder gves beer performances han Huffman algorhm I s worh menonng ha our coder s drascally smpler for he realzaon han Huffman code Hence s suable for mplemenaons where smplcy of he coder srucure s mporan h 6 Generalzed resson algorhm combnaon of lossy and lossless resson Generalzed resson algorhm consss of lossy and lossless resson algorhm Opmzaon of lossy algorhm wh opmal resson funcon was done by opmzaon of n secons and 4 An smple lossless resson algorhm was proposed n secon 5 where also opmzaon of exended nformaon source was done by opmzng M where M s he order of he source exenson In hs secon we are gvng he algorhms descrbng he lee srucure of our coder conssng of lossy coder (andng model based quanzer) only wh he presened lossless coder I s assumed ha sack S has lengh M and s empy on algorhm sar Also assume ha s equal o and hs value s preued Algorhm (Coder) Inpu Sgnal sample x

12 Sep Compue y = c( x ) Sep Apply unform quanzer on y and fnd he ndex of he cell where y belongs Sep Pu on he op of he sack S Sep 4 If S s no full go o he Sep and connue wh he nex sample Oherwse connue Sep 5 Take M ndces from he sack S Sep 6 If I for every = M hen code every ndex wh r bs and produce he bnary codeword C = ( c c( r ) M of he lengh ( r ) M Transm C Oherwse code every ndex wh r bs and produce he bnary codeword C = ( c c rm ) of he lengh rm Transm 0C quanzer Reurn A he end of hs secon we presen praccal example of usage of generalzed resson mehod for resson of a sll mage Fgure (lef) presens orgnal mage and Fgure (rgh) presens mage afer resson We can see ha ressed mage s vsually very close o he orgnal mage Frs we found dfferences of he consecuve mage samples because hey have Laplacan dsrbuon and hen appled resson mehod on hose dfferences To preven error propagaon besde dfferences we also ransm one orgnal mage sample on every 8 samples We use andng quanzer wh = 64 levels o quanzng dfferences of he samples and for hs quanzer SRQ = 808dB We use lossless algorhm wh M = Average b-rae for ressed mage s R = 5470 bs / pxel and for orgnal mage s R = 8 bs / pxel so he resson rao s R / R = 46 org M Algorhm (Decoder) Sep Take one b c from he npu Sep If c = ake (r )M bs from he npu and decode ndces M each from r bs Oherwse ake rm bs from he npu and decode ndces M each from r bs Sep For each = o M perform: Sep Se y o he value of represenaon level of -h cell of unform x = c ( y) r org r

13 7 Concluson Fgure : Orgnal mage (lef) and ressed mage (rgh) Ths paper provdes he smple srucure coder for memoryless Laplacan source We used andng model based quanzer and perform he opmzaon of he mal sgnal amplude There are derved analycal esmaes of he opmal mal sgnal amplude as well as one numercal esmae (based on he mnmzaon of approxmae dsoron) and exac numercal mehod for s uaon I s shown ha one of he esmaes has almos opmal dsoron Due o he exac smple analycal expresson hs esmae s very suable for he praccal applcaons Also we develop smple lossless coder algorhm for Laplacan source and ared wh he Huffman code and source enropy Man advanage of our mehod s smple realzaon srucure and less b-rae ared wh he Huffman mehod Generally our coder gves he very smple realzaon srucure and performances close o opmal and hence s very useful n praccal applcaons such as speech sgnals mages bomedcal mages ec References Chong EKP SH Zak (00) An Inroducon o Opmzaon John Wley & Sons nc ew York Chcheser Wenhem Brsbane Sngapore Torono Gray R (004) Quanzaon and Daa Compresson Lecure oes Sanford Unversy Hankerson D GA Harrs PD Johnson Jr (004) Inroducon nformaon heory and daa resson nd ed CHAPMA & HALL/CRC Jayan S P oll (984) Dgal Codng of Waveforms Prence-Hall ew Jersey Chaper 4 pp 9 9 Judell L Scharf (986) A smple dervaon of Lloyd s classcal resul for he opmum scalar quanzer IEEE Transacons on Informaon Theory () 6 8 Max J (960) Quanzng for mnmum dsoron IRE Transacons on Informaon Theory Vol IT-6 7- a S DL euhoff (ov 00) On he Suppor of MSE-Opmal Fxed-Rae Scalar Quanzers IEEE Transacons on Informaon Theory 47(7) a S (May 004) On he Suppor of Fxed-Rae Mnmum Mean-Squared Error Scalar Quanzers for Laplacan Source IEEE Transacons on Informaon Theory 50(5) kolc J Z Perc (008) Lloyd-Max s algorhm mplemenaon n speech codng algorhm based on forward adapve echnque Informaca 9() Paner PF W De (Jan 95) Quanzaon dsoron n pulse coun modulaon wh nonunform spacng of levels Proc IRE Perć ZH JR kolć DM Pokraac (007) Esmaon of he suppor regon for Laplacan source scalar quanzers Journal of Elecrcal Engneerng 58() 47 5 Perć ZH JR kolć (007) An effecve mehod for nalzaon of Lloyd-Max s algorhm of opmal scalar quanzaon for Laplacan source Informaca 8() -0 Ramalho M (00) Ramalho G7 Lossless (RGL) Codec Whepaper Cysco Sysems Inc Sayood K (006) Inroducon o daa Compresson rd ed Elsever Inc Sarosolsky R (007) Smple fas and adapve lossless mage resson algorhm Sofware-Pracce and Experence Wolfram S (00 ) Mahemaca Book 5h ed Wolfram Meda

14 Zoran H Perć was born n s Serba n 964 He receved he B Sc degree n elecroncs and elecommuncaons from he Faculy of Elecronc Engneerng š Serba Yugoslava n 989 and M Sc degree n elecommuncaons from he Unversy of š n 994 He receved he Ph D degree from he Unversy of š also n 999 He s currenly Professor a he Deparmen of Telecommuncaons and vcedean of he Faculy of Elecronc Engneerng Unversy of š Serba Hs curren research neress nclude he nformaon heory source and channel codng and sgnal processng He s parculary workng on scalar and vecor quanzaon echnques n speech and mage codng He was auhor and coauhor n over 00 papers n dgal communcaons Dr Perc has been a Revewer for IEEE Transacons on Informaon Theory He s member Edoral Board of Journal Elecroncs and Elecrcal Engneerng Marko D Pekovć was born n š Serba n 984 He graduaed mahemacs and uer scence a he Faculy of Scences and Mahemacs š Serba n 006 and elecommuncaons a Faculy of Elecronc Engneerng š Serba n 007 He receved PhD degree n uer scence from Unversy of š n 008 Currenly he s he research asssan a he Faculy of Scences and Mahemacs š Serba Hs research neress nclude he source and channel codng generalzed nverses of marces Hankel deermnans and opmzaon mehods He s he auhor of abou 0 papers ( of hem n peer-revewed nernaonal ournals) Dr Pekovć has been Revewer for Journal of Compuaonal and Appled Mahemacs Compuers and Mahemacs Wh Applcaons and Inernaonal Journal of Compuer Mahemacs He s suppored by Mnsry of Scence Republc of Serba Gran o 440 Mlan R Dnčć was born n š Serba n 98 He receved he B Sc degree n elecommuncaons from he Faculy of Elecronc Engneerng s n 007 He s currenly on docoral sudes on he same faculy and he s scholar of Mnsry of Scence Republc of Serba Hs curren research neress nclude source codng and quanzaon of speech sgnals and mages 4