Scalar-Vector Quantization of Medical Images

Nat1t.r Molxienian* aiitl FTo1~l;tpu11 SIiahrit * lBhf ('orporatiorr, 1701 North Street, Dept. V58, Bldg. 32-4, Fnclicott, NY 13760

ATk'l' Bell Laboratoriw, Room 5D102, 600 Mountain A v e , Murray Ifill, N J 07974

Ab s t r a(: t

A new cocliiig scheme bastd on the scalar-vector quail- tizer (SVQ) i s developrd for coniprewon of meclical i i i i -

ages SVQ i s a fixcd-ratc rncotler and its rate-distortion performance is close to that of optirnal entropy-constrained scalar quantizers (E( 'SQ's) for memoi ylrss sources For a set of riiagriet i c rfwmaiicc ( M R ) iiriagcs, totling results oh- taincd from SVQ a n d E('SQ a t low hit-rates are indistin- guishahle Furtherinore, our encotlrd ii1iagc.s arc perceptu- ally intlist itigiiis1inl)le froiti I tie original, wlicn c1iy)layed on a monitor This iriakes our SVQ Ix~scd code i a i i attractibe compressiori sclitwie for picture arcliiv iiig ant1 coiiiinuiiica- tion systems (PAOS), cnrrcwtl~ i i i iclci coi~sideratiori for an all digital radiology enviroiiiiit~iif 111 hospitals, w l i r ~ e IPI I - able transmissioii, storage, and high fidelity recoiistruct ion of images are tltwred

I. INTRODUCTION

'rhc popii1a.rit.y of tligit,al tlia.gnost,ic imaging systseiiis is v&ly incrca.siiig in radiology departments of various 1iealt.h care cenkr s . A typical hospital caii genera.t.e mul- tiple terabyt>es of iiic-.tlicnl iiiiagc tiatma per year. This forniidahle amount, of data. 1ia.s t,o I)c ma.iiagrvl efIiciciit,ly in t8he hospit,al environrticot, iisiug PACS. Iniagcy cmiiprr's- sion is ca.pahle of bot.h rrduciiig t.he storage cost, and the amount, of t<iinr> req U i red t,o t, ra.nsi1i i t. a. r d i ogra phi c i I ri age, and hence, make t,he image tla.t,a iiiore inanageahlt~.

Most) popular medical image couipressioii sclieines are either iinplenientcd i n t,ransforiii domain [2] or a.pplied in spa.tia1 domain usirig vt,ct,or quant,ization (VQ) and its variant [I]. In t1ot.h tloiiia.iris, entropy-coiist,raint, scalar/vector quant8izers, comhiiictl wit81i variable lciigt,h coding (VI,(:?), producc 1 r coniprvssioii result,s than level-constraint, scalarlvec pant.izers at a cost, of hav- ing encoder/decoder buffers for c,onversion bdween t,he va.ria.hle\-rat8e out,put,s and fix(:d-ra.tte hit-streams. This buffer increases the pr g delay. Also, a sophisticat,ed buffer maiiagcmcnt~ is regulat,t, t ,he generat,ed I)it-st,rca.iri. T h ~ us(' o f VJL: conipresscd tla.ta t,o lie inore susceptible to t,raiisiiii rrors sincc a single bit, error inay result in t.he loss of cotlc syiicliroiiizat,ion. On tmhe contrary, t,lic coding t , c w for a fist+rat,(, qua.nt,izcr is less complex hut, as mcnt~ionecl r.ar1ic.r it,s perforinaiict, is not as gooti a.s va.riahle-rat,e (ent,ropy coded) ciuaiit8izers.

In this paper w t present, a new fix(-d-rate coding schemc, which does not, suffer from t,lie usual tlrawhacks of t,lie vari-

able lrrigth coding schemes It is based on scalar-vector quantizer (SVQ) which was first introduced i n [4] The SVQ scheme IS a viable compression solution for storage necds of hospitals and clinics as well as delivering medical images over digital networks

11. THE SCALAR,-VECTOR QUANTIZER.

The eiit,iol)y-coiist,raiiied sca.1a.r qua.nt,ieers a.re known t,o perforin close t80 the rate distorttion bound (within 0.25 hits/sa,mple) for meiiioryless sources. However, some t,ype of i~tr icrble- lcngih coding is needcd t,o irnplernent such qmiit,izcrs. As tlisciissrtl allove, t,he use of variahle-lengtmh codes coulcl rtwlt, ill loss of syiiclironizat~iori a.iid coiisc- qiient,ly loss of clat#a. An out,put, buffer is also needed t,o convert, t,he va,ri;thlc-ra.t,e out,piit, of the quant,izer t,o a fixed-rate hit, st,reani. Alt,liough near opt,imal, ECSQ is of 1imit)ed usc' for applica.tions involving tra.nsmission over noisy c1ia.iiiicls. On t,lie ot>lier hand Lloyed-Max quant,izers (LMQ's) miniiiiize t,he tlist,ortion and utilize j i m d - l e n g t h cotlewords. Alt,lioiigli inimitne to error propagation LMQ's ptdorniaiice is w o r s ~ t,lian t,liat, of t.he opt,ima.l ECSQ. The pcrformarrce gap could be large for cert,a.in sonrces of in- terest,. To ltritlgc the ga.p het,wecii t8he perforimnce of LMQ and ECSQ one coiiltl use fixed-ra.t.e ve&r quant'izers. In fact, vect,or quarit,izers of sufficient,ly large dimensions ca.n pc,rforni arliit,rarily close to the ratre-distortion bound. Rowever, vect.or quaiit,izers require training, and t,he en- coding coinpl(-.xitmy grows exponentially wit#h the size of t'he cotlebook. The iriotivat.ioii liehind the sca1a.r-vector quan- t,izer (SVQ) [4] is t,o dcsign a fixed-ratme VQ sclieine such that it perfornis close t,o variable-rate ECSQ and does not. suffer froiii the aforeriieiit,ioiied t h w b a c k s of veclor quail- t,izat,ion.

Assume an 11-level EClSQ scheme with a set, of quati- tization levels Q = {y;}y=l and the corresponding set. of lengths C = where t i ' s , are t.he 1engt.h of t.he 1)ina.ry codeword corresponding t,o y i . Let, ~ 1 x 1 = ( ~ 1 , ~2~ . . . , x,,,,} be an n-dimcnsioiial block of samples ta.ken from a st,a.- t-ionary niemorylcss source. Ea.cli sa.niple of vect,or xixi is t,Iieii encoded using t,he ECSQ system represenkd by ( Q , L) and a, clist,ortioii ineastire bet5ween the input' and the quaiit,izer rcprotluction Iwels. T h e result,ing yimntized vwt80r is i n l = { X I , .r2, ~ .r,?&} E Q"", and Q'" is an m- tliiiiensioiial qiiant,izer having nm tnemhcrs. If t,he size m of the inputj vect,or is sufEcicntly large, t,licii sun1 of the lengt,lis of reprotluct,ion 1c:vels approach a value given by

0-7803-3 180-X/96$5.0001996 1425

4

I Figurr 2: C'la.ssificat,ion of AC coefEcient,s in an 8 x 8 DCT hlock.

Figure 1: 't'liresliolds arid quaiitizatioii Ievc3ls of the urii- form threshold quant.izer.

put . levels y I ' s can he coiiiput,cd from

mh where h is thc entropy of the H ' S Q system \.Ye us(' a threshold L to dtwotr the lmgtli c o r r ~ ~ p o ~ i d i i i g to %Ill

where p( s) is t.he proha.bilit,y densihy furict,ion (pdf) of iii- pu t source s E R. Thus qi defines t.he center of the proh- abilit,y mass for t#he range t i p l t,o t i .

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Where t,hc index fuiict,ioii f : Q J,, is defined as: f ( q i ) = i wit,h i E J,, = { 1 , 2, . . . , 7 1 1 . From t h s oh- servation i t can t i c , conclrrtled taliat, if a11 rri-tlii~ierisional reprodiict8iou veckors t,lia,t, arc' i n Q" such that t,lieir to- tal lengt31i a.re close t,o I,, a.re placed iii tht, coc le l~~ok Z , the vect,or qua.ntjizt~r defined hy this codrbook will perform close t,o ECSQ. This special t8ype of \'& is called SVQ ancl is f o r n d l y represent8ed hy (Q, C, L ) wit.11 a. codebook 2 of size 2mr where 2 = {z E ( i l l , 2 2 , . . , s,,,) E Q n ' } with

C, f i z , ) 5 L , arid is t,lie Iiit,-rate at, wIiicli the fixed- rate SVQ scheme operates. In t.lie nest suhsect,ioiis, we explain the SVQ design procedure.

A. Design of the SVd)

B. Deteriiiiiiatioii of C

The set, of leiigtlis C is det,ermined such that, the SVQ cotlevrctors correspond to the 27nT most, proba.hle grid points in t.he 771 diiiiensi0na.l space. 'This eiisiires tha t with highest probability t,hr input vector is quaiitmized to t>hc niiniiinim (list o h o n point, 011 t8he ?n-dimensional grid.

Let p L . i E J,, he the proba.hility tha t t,lie input sa.mple is quaiit~izcd t,o q i . Tlieii t.he probability of an arbitrary input vector v = ( ( $ 1 , 0 2 , . . . , z ) , ~ ) is

1=1

If the length C, is defiiied as !, = log, ( l / p t ) , we have - _ ~ ,,/

There exist several approaches to SVQ design [4]. The SVQ schenie enir)loyed in t.liis paper is based 011 synimet- ric iiniforni t.liresholt1 quantizer (IJTQ) of rate (ent,ropy) T . UTQ's a.re known t.o perforin as good as optiinuni scalar qua.ntizers, i.e., ECSQ for a. large i i u m l w of i n e n - oryless sources at. high ra.t,es [5]. The UTQ's can also be

p ( v ) z 2- L,=, ' f ( ~ ) . This iixiplies tha t the highest, proh- ability grid poillt,s are the o~ics wit,li smallest t,ot,al lengths.

For t,lie special ca.se of a source with zero mean Gener- alized C;aussiall (C,G) proha.hilit,y densit,y, defined as

(YC .fr. ( ( ' 1 = ~ F S ~ { - ~ C V ~ ~ ~ } = C:G(7n, a , U ) (4) very effectjive a.t, lower rat~cs arid are very simple t,o de- 'mi) sign. Each element, 2 , of codevect,or x is a quantization level of an [JTQ, descr ihd by ( Q . L ) . ThrPsholtl L i s se- lected such t1ia.t the SVQ codebook 2 has 2""' c o d ~ v ~ c - tors. Thresholds and quant,izatioii levels of the IJTQ are shown in Fig. 1. The t,Iiresliold t i is chosen such t.hat. t i+l - ti = 60, i = 0, 1 , 2 , . . . , n - 1 where 6 is t,lie nornial- ized st,cp size mid (T is t81ic, st,andartl deviation of t,he input source. By varying st,ep size 6 , we caii ot>t.ain a u n i f o r m sca,la.r qiiaiit,izer t,ha.t operaks at, a desired rat.e T . For a. fixed va,lue of 6 and a. set, of t81iresholds. t.he opt,iiiiuni out-

where 7 7 1 , n, aiid N are t,lie mean, st>andard deviat,ion aiid t,hc shape para.met.er of t,he soiirce, respectively, and e = ( r ( ~ / ~ ) ~ - 2 r - 1 ( l / ~ ) ) 1 ' 2 . It. can be shown from t,he Asyrnptot.ic Equipart.it,ion Propertmy (AEP) [3] t,lia.t. the input, vect,ors lie 011 aiid nea.r a surface defined by Cif, / I . , / ~ = I<, wherc I< is a const,aiit t1ia.t. depends on t8he parainet.ers of the GC; tlist4ributjion. Hence, it, can be seen t ,hat for this specia.1 ca.se an equivderit measure of lengt.11 is given by Ci = / i i i l n . Moreover, it, can be concluded t,lia.t t,he higlipst proba.bi1it.y grid p o i i h correspond to the input,

For a givtw C, the dctrmiiiiation of I , is as follows Wt star t wit11 codevectors of siiialle5t Ic~ngth (Iiiglicst pro1)a- bility) z E Q”’ FVe thcn tncrcaw the lcngtli and collect all the vectors with this 1 1 w lcmgtli The vectors are then added to the cotlehook ’Tills p r o c w contluues untll 2?11“‘ vectors of sinallcst lcngtlis (liigli~st probability) are cho- sc’n We thcn fii i t l the cotlmwtol that has tlie maxiiiium leiigth 111 the codebook ‘I‘lit~ Imgt 11 of that vector rcpre- srmts the tltwrrtl t hrcslioltl L T h e following representq an algoritli~ii for clr.t~r111111111g tI i f1 tI~resI~oItI L. Let A’: repre- s m t the iiiiiiilwr of vectors ( ~ 1 , 0 2 , , Q ) E Q‘ such that their total lerigth E, f f ( ? , , ) = J All N;1’s can be com- putet1 recursivc>ly front N: = E;=, NJ:;~ wllcrr N;‘ = o for j < 0 and N:; = 1 The thrc~lioltl I, cat1 tlierefore he obtained as

D. Codebook Search

There exist, various i l i e t l l ~ ~ d ~ to find the v codebook 2 of tot,al leiigt,li less t,han or equal to L whicli is closest, to t,he given input vcc,t,or. The codebook search can be done efficientmly usiiig dynariiic progra.tiiiiiing [4]. However, since we take aelva.nt,age of AEP to shape our codebook boundary, we can use a sub-opt3iiiia.l codebook searc,h scheme which is fa.st. a.iid very effective. We start, by quantizing the given input, vrct,or (notme t,liat, our quan- tizer is t<he direct, product, of 11 niforiii sca,lar quant>izers) and cornputme its t,otal length. If tlie lcwgth of tthe vector is less t,han or equal t80 L , t,lie vect>or is insidc t,he coclehook. If t,he length of t,he vect,or is la,rgcr than 1, the vect,or lies outside the codebook and a, codehook search must be performed t80 find the closest cotlebook vector to t,he given input, vector.

Our codebook sc:arcli is a.s follows: If t,he optiinum hountla.ry siirfacc of t,lie cotlchook can be identified, we bring the input, vect,or closer to t,Iw surfa.ce along t,he per- pendicular linr, from t,he inpiit vector t o t,lie surface un- tail its Icngt,Ii is less t,lian or tyua.1 to L . For exainple for a Gaussian sourcc whose optiriiuin boundary i s a mult,i- dinietisiona,l spherc, we simply scale t,he vector by a siiiall constant,. We t8hen quant~ize t8he vector antl ineasurc its new lengt,li. If tjhc leiigt81i is I( t,lian or equal t,o L t,he vector is in t,he codehook antl t>lie smrch t,twniiiatcs, else the vector is scaled again. Tlie process continues uiit,il t,he lengt,h const,raint, is sat,isfied.

E. Encoding and Decodiiig of tlie Codevectors

Since there a.re 2mr cotlevect80rs in the SVQ codebook, the encoder is a biject,ive map from t,lie set of Ynr code- vectors onto mr bit binary cotirwords (or indices). One such mapping is preseiit,ed below. Every codevector z = ( 2 1 , 2 2 , . . . , zm) E 2 ca.n be represented by an m-digit ba.se- n number N ( z ) = ( f ( z 1 ) - l , f ( z ~ ) - l ; . . , . f ( z T n ) - l) , where j”(z1) is the most significant digit. Obviously, N(Z~) = N(z2) ++ z1 = 2 2 . The encoder funct,ion

7 2 --+ {O}UJamr-1 , where y(z) IS the number of vec- tor5 in 2 that are “sniallcr” than R in the sense that N ( z ) > ,v(w) for R , W E 2. Let C“: denote the nuin- hr r of distinct ?-vectors (z1,fz,. , z , ) E Q2 such that fj(,l + r f ( i 2 ) + + Tfc. , I j Also, let

0, j < O (3; = { 1, j 2 0.

It can lie shown that t,hc index of the codeword associated with codevector z is:

k = l 3 = 1

where Lo = 0 and L, = Note that (7f can he rxpresscd as C,” = E”,=, N: For fast encod- ing. tlie C:’s can Ise computed once and stored in inem- or) Decoding is the mapping froiii the set of codewords to the codevectors It is the inverse of tlie above algontlini Namely, given the index I = y(z) we niu51 determine z Define I , = I and 1, = 1 k - l - - y k - = , ( z ) wltere -yo(z) = 0 Then i t can be shown that

P f c , , ) , i E J,,

j = l

where $J, - 1 IS the largest integer sucli that the ~nequal i ty in ( 7 ) Iiolds We tlirii have .f( z k ) = gik Thus, one compo- nent of z IS dcteriiiiued at each step of the iteration. For fast decoding tlie C,”’s inay be coiriputed once and stored in memory

111. DISTRIBUTION OF DCT COEFFICIENTS

There exist,s a g r m t a.niount of correlation bet>ween neighboring image samples of a medical ima.ge. Discrete cosine transforinat,ion (DCT) is used to exploit t>his corre- 1at.ion. The DCT transform coefficierit,s are more likely to have ind~peiideiit,ly distributed comporieiits compared t80 the original iina.ge sa.niples. In order tto closely nia.tdi the quant,izers t o the dist,ribution of the transform coefficient,s we need t,o obt,a.iii a.n accurat,e model for t,he st,at,istical di~t~rihut~ion of tslie DCT coefficient,s.

The input, iinage is suhdivided into non-overlapping blocks of A4 x A’s a.nd a11 A4 x A4 DCT is performed on each block. We classify the transform coefficicnt,s int,o sev- eral groups defined by t,lie set S = {Si; i = 1,2, . . ., IC} as illust.rat8etl in Fig. 2. Each class ,Si could ha.ve a differ- twt. prohahilitmy cleiisity (derived from t81ie histograin of t,be d a h ) whicli can be inoclcled by the Generalized Gaussian pdf GCr((rn, U, w ) defined in (4). m, D and cr are the niea.n, standartl deviation and t,lie sliape paramrt.er of the GG dist,ribution of source 5; = (st, s:, . . . , ss’) . gi is t,he t,ota.l number of elements of the class i . The distribution of the DCT coefficient,s in each class are fitted t80 t8he G G distri- bution using the E;olmob.orov-Sinirnov (KS) goodness of fit test [ G I . The KS t,estt compares ttlie cumulative probability

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Figure 3 : T h e goodness of t,lic fit, t,est, for t,he sample d a t a of class 1 A(': coefficieiit,s using t'he IiS st,at,ist,ic,

distjrihution function (C'UE') of each class i with a given dist8ribut8ioii furict#ion F'( .) derived froin ( 4 ) . using tlie f q norm of tlie of t,he diffrrciicr of tlie two (-'DF's as the dis- tar i ce iiieasur(3. Tlic sa iii plc ti is t rib II t, ion fu ii c t, ioii is clef i IF d bY

Wlien t,txt.ing t.he tlist,rihut ioii of sa.iii111e da.t a S, against t,hc GG tlistmributtioii, t,he tlistrihutioii t,liat yields the siiiall- est lis distance iii( lire is t , l i c ~ 1)est~ fit, for t l i ~ saiiiplc, d a t a . Figure 3 illustra.t.rs t8hc I m t , fit. for the liistograin of tlir sample dat,a of c h s s 1 pc~for i i i rc l oil 5 AIR iiiiagcs. It, is observed from the lis t,c.st, t.hat. t 1ic c1istril)iitioii of the l ~ s s significant, A ( ~ ' coeficie1it.s a.pproac1i t,he Laplaciaii tlistri- butiori while t.lie dist,ril~ut,ioii of t h e more significant A C ' co- efficients have sharper p e a . 1 ~ aiid Iiroatlcr tails. The value N of GG prohahilitmy clist~rihiit~ion that, best, fit,s t,lie saiiiple d a t a distrihut,ioii t8heii gives t'lie appropriat,r pdf for each S i . The o p h i u n ~ reprotlrrct'ioii lwels q i ' s are coinpntcd according t,o (2). 'This ensures t,lia.t t.lie hcxt UTQ's are designed for the SVQ coding schrnie.

IV. CODING SIMULATIONS

In our simulat,ions t,he tlcscrilwd SVQ schcmr is applied t o a set of MR images of 1iunia.n l irart o f sizcs 256 x 256 and 12-bit gra,y scale resoliit,ioii. A 1)loclc diagram of t.he SL'Q coding scheme is shown in Fig. 4. Each sliccl of the medical data is tra.iisforrrlrd using 8 x 8 ll(-'T blocks. iVe used a 12

E'igurc 4: Block iliagraiii of the SVQ coder/decoder.

hit, iiiiiforiii qiiaiit8ize>r t,o rrprwtmt, t,he original DC conipo- iit'iits. This is liiglily tlt~sirahle siiicc our S V Q - h s r d coding scheme is also fixed-rate. Note that, one could d c $ g a. 12 hit SVQ to quantize t,he different,ia.l DC coinponeiit,s, re- s ul t,i ng in slight iin provoii ieii t, in t, he overa.11 per form a ncc of oiir coder. I Iomr~cr , , since t,ransniission errors coiiltl have catast.rophic dfcct,s 011 cliff(,rent8ial da.t~a, resiilt,iiig in loss of synchronization and possibly loss of da ta , we did not' at- tcinpt to (lo so. Iiist.ead we quaritizc t8he UC componc~it~s directl~-.

T l i ~ remaining A(I1 coefEcieiit,s are divided into 7 classes as i l lus t ra td hy Fig. 2. Wc desigii 7 different3 SVQ banks defined hy

a i= { ( $ ' 7 1 ~ ~ 7 , ~ , ~ 7 ) ; = o . a s j , j = 1 , 2 ; . . , 2 3 } , i = 1 , 2 , . " , 7

(91 .ic-lierc. for class i , 9; defines a set. of SVQ-based eiicodcrs a t various fract,ional rat,c:s rj ' s in bit,s per pixrl. Since ea.cli class i has a different, mean i n aiid varia.nce d, we iiorina.1- ize racli source to ze ro 1nea.n ancl unit, variance prior to the qiiantizcr tlc-.sigii of sectlion T1.A. Values m and U are r e p rcseiittd 11y 32 bit, floa.t,ing point, numbers in the overhead iriforination aiid t,raiismit,ted over t.lie channel. In order to miiiiinize the overall dist,ort,ioii in the DCT block, we eiiiploy a near optimal priiiiirig algorit,hni [7] to distribute hits aiiiong t he AC coefficierit,s. The bit-dlocation scheme in [ i ] is lmsed oil t,hc sterpest,-clescent, algorit,hm. In order to fwsiirc: that, no c.orrelat,ioii I t s among t,lic coniponcnt,s of the SJ'Q vect,ors, t,lit= coiiiponeiitjs of t,he vectors are cho- sei i 13y scaiiiiirig a given class in a reversible prp-dct,t.riiiiii~?[l i i o i i ~ s i ~ q u e i i t i a l ratitloin iiiaiiiier.

'llie S\,'Q sclipiiie is i~setl l,o encode t,he MR da.t.a sct at, different, I,it-rat,cs. 'Ihc result is compared a,gaiiist, variahlc- leiigt81i (Huffnian) coded ECSQ and displayed in Fig. 5. This figure demonstrates the excellent, rate-distortion (R- D ) perforiiia~ice of the proposed SVQ t,echiiique. Clodiiig perforinmice is obt>aiiietl in terms of Sigiial-t,o-Nois~~-rR.ntio ( S V R )

n hcrc yK aiid y, are original ancl reconstructed pixels and 'S = 2.56 x 2.56 i s the 5ize of tlie image E(y2) is tlic average

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28 , I

22 -1 I 1 1 1 I S 1 7 1 9 2 1

Aveiage hl mle (bpl

E’igurc 5. (’ornp”oii of tlic coding rcsults for SVQ aiid VLC-FCSC) schetnes for a hrar t wqutwce

energy of random ficld y rcpiesciii iiig tlic original i m a g e

closely wliile the SVQ rcsiilt 19 oiily 0 tj d H worse than the VLC-ECSQ at ratrs close to 2 0 h p p For coniparison purposc5, we show an arliitrary sltcc~ taken from tlie M R data in Fig 6(a) , aiid the eiicodcd version at 1 4 2 1 bpp with SNR = 24 18 dZ1 in Fig It can be seen fiom Fig 6 that tlie t w o iiii<i\gcs ai(’ irithstiiiguisliahlc

Ai low hit-ratcs ( 5 1 45 h p p ) tlir two encot

G ( h )

V. CONCLUSION

We have prcwiitt>tl a n m cotling scheme hased on scalar- vector quantization (SVQ) for coniprcssioii of incdical 1111-

ages Results preseiiietl i n t h i s paper dcmoiistrate that the R-11 performance of t hc SVQ Ii d cotler. approach t h a t of thc VT,C‘-E( ‘ S Q 111 tlie rrgioii of high to iiiodPrate coin- pression ratios wh lc thP output cotlewords are kept at a fixed-rate The SVQ scheme c a n he further improved 112

different dircctioris Since etitIopy-coiist raiiit vector quan- tization (ECVQ) 1s known to oiitpcrform ECSQ, we may replace tlic scalar coiiiporieiits of SVQ hy vectors and cle- sign the quantizers hascd on the EVVQ algortthni rather than the variable-lriigth scalar quantizer. Another possible approach is to apply trellis-hased SVQ (TB-SVQ) liased on Ungcrboek’s four-state trellis code Using TB-SVQ is ex- pected to result in an iiiiprovement of about one dB in the signal to i i o i ’ ~ ~ ratio

References E. A. Klskln, ‘r. Lookahaiigh, P A (‘11011, and R M. Gray, “Variahle rate kector quaiit izatiori for medical image coiiiprcmioii,” IEEC Tr n n 7 M c d l i n n g , vol. MI-9, no. 3 , pp. 290-208, Sept. 1990.

I<. E; ( ‘hati, S-I, Lou, aiicl I f . E; Huang, “E’ull-frame trarisforiii coiiipression of (*’I arid MR images,” H a d ? - ology, vol 171, no 3 , pp. 847- 851, 1989.

(h)

Figure G : (a) Original lieart, slice, (b) encoded lieart at, 1.421 hpp wit,li SNR = 24.18 dB using the SVQ scheme.

A . Gersho arid H. M. Gray, Vector &u,nn.tizatioi, nn.d Signal Conzpression. Boston: Kluwer Academic Pub- lishers, 1992.

FZ. Laroia a n d N . l?a.rva.rdin, “A shuctured fixed-mte vect,or quaiit,izer derived from a va.ria.ble-length scalar qi~arit~izer: part, I-memoryless sources.” IEEE Trans. 17rf01-1t2. Th,tory, vol. IT-39, no. 3 , pp. 851-867, May 1993.

N . Fa.rva,rtlin a n d .J. W. M o d t h n o , “Opt~iiiiurri qiim- tizer perforinancc for a. class of non-gaussian meinory- less sources,” IEEE Trans. Iiiforni. Theory, vol. IT-30, no. 3 , pp. 485-497, May 1984.

R. C . Rciiiinger a.ud J . D. Gibson, “Dist.ribut,ions of the two- dirriensioiial DClT coefficieiits for images,” IEEE ‘7’7~7n.s. C’omrrr., vol. COM-31, no. 6, pp. 835- 839, .June 1983.

E. A. Itiskin, “Opt,imal bit allocat,ion via tlne general- ized BFOS algorit,hiii,” IEEE Trans. Inform. Theory, vol. IT-37, pp. 400-402, Mar. 1991.

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