New Error Model of Entropy Encoding for Image Compression

Transcription

1 Wb St: Eml: Volum, Issu, Mrch Aprl 03 ISSN Nw Error Modl of Etropy Ecodg for Img Comprsso Moht Mshr, Rjsh Tjw, Assstt Profssor Ary Collg of Egrg & IT, Jpur (RAJASTHAN Abstrct: Etropy codg provds th losslss comprsso of dt symbols d s crtcl compot sgl comprsso lgorthm. I our cs w hv dsgd w modl whch rducs th Itrpxl rdudcy d hv bttr rsults s comprd to othr modls lk losslss prdctv cod (LPC d Dffrtl puls cod modulto (DPCM. Our w proposd Schm for Huffm codg wll chv hghr comprsso s w hv lso rduc th stdrd dvto th rror mg trmdously s compr to LPC d DPCM. Kywords: Losslss comprsso, Lossy comprsso, Rdudcy, Losslss prdctv modl. INTRODUCTION Img comprsso s th pplcto of dt comprsso o dgtl mgs. I ffct, th objctv s to rduc rdudcy of th mg dt ordr to b bl to stor or trsmt dt ffct form. Img comprsso c b lossy or losslss. Losslss comprsso s somtms prfrrd for rtfcl mgs such s tchcl drwgs, cos d comcs. Ths s bcus lossy comprsso mthods. [] Espclly wh usd t low bt rts, troduc comprsso rtfcts. Losslss comprsso mthods my lso b prfrrd for hgh vlu cott, such s mdcl mgry or mg scs md for rchvl purposs. Lossy mthods r spclly sutbl for turl mgs such s photos pplctos whr mor loss of fdlty s ccptbl to chv substtl rducto bt rt.[], [3].Rdudcy d Irrlvcy Rdudcy grl trms, rfrs to th qulty or stt of bg rdudt, tht s xcdg wht s cssry or orml, or duplcto. Ths c hv gtv cootto, spclly rhtorc: suprfluous or rpttv, or postv mplcto, spclly grg: srvg s duplct for prvtg flur of tr systm. Dt comprsso s chvd wh o of th followg rdudcs r rducd or rmovd. A mportt xmpl of dt rrlvcy occurs th vsulzto of gryscl mgs of hgh dymc rg,.g. bt or mor. It s xprmtl fct tht for moochrom mgs 6 to 8 bts of dymc rg s th lmt of hum vsul sstvty; y xtr bt dos ot dd prcptul vlu d c b lmtd.[3]..typs of Rdudcs (A Codg Rdudcy A grt dl of formto bout th pprc of mg c b obtd from hstogrm of ts gry lvl. Lt us ssum tht dscrt rdom vrbl r k th trvl [0, ] rprsts th gry lvls of mg d r tht ch k occurs wth th probblty P r ( r k. Pr ( rk k k 0,,... l (. Whr L s th umbr of gry lvls, k s th umbr of tms th k th gry lvls ppr th mg. r If th umbr of bts usd to rprst ch vlu of k s L( r k th th vrg umbr of bts rqurd to rprst ch pxl s L vg L( rk. Pr ( rk (. Thus th totl umbr of bts rqurd cod M x N mg s (MNL vg (B Itr-pxl Rdudcy Th cods usd to rprst th gry lvls of mg hv othg to do wth th corrlto btw pxls. Ths corrltos rsult from th structurl or gomtrc rltoshps btw th objcts of th mgs. (C Psycho- vsul rdudcy Th brghtss of rgo, s prcvd by th y, dpds o th fctors othr th smply th lght rflctd by th rgo. For xmpl, tsty vrtos c b prcvd r of costt tsty. Such phom rsults from th fct tht th y dos ot rspod wth qul sstvty to ll vsul formto. Crt formto s sd to b psycho-vsully rdudt.[], [9]. Grl Img Comprsso Modl Trsform: Th m gol of trsform stg s to dcorrlt th orgl mg dt, so tht th orgl sgl (mg rgy s rdstrbutd mog smll st Volum, Issu Mrch Aprl 03 Pg 4

2 Wb St: Eml: Volum, Issu, Mrch Aprl 03 ISSN of trsform coffcts. Th m of d-corrlto s to rmov th tr-pxl rdudcy; thrby provdg rprstto tht c b codd mor ffctly.[] Th zroth ordr tropy of th trsformd coffct s much lowr th tht of orgl mg. Ths trsforms should b rvrsbl tur so tht th orgl mg c b rcovrd by th vrs trsform; provdd o qutzto hs b prformd o forwrd trsform oprto. Qutzto / D-qutzto Stg: Th scod stg of qutzto / D-qutzto s th procss tht lds to th lossy comprsso. I th qutzto scto psycho-vsul rdudcy th mg s rducd by rmovg uwtd bts from th trsformd coffct. Ths stg lds to hgh comprsso rto d dstorto mg fdlty. Etropy Codg / Dcodg Stg: Th thrd stg of tropy codg, dtrms th umbr of bts rqurd to rprst prtculr mg t gv qutzto lvl. Th procss of tropy codg d dcodg s losslss. It mps th qutzd trsform coffcts to bt strm usg vrbl lgth cods. Ths stg xplots th codg rdudcy.[], [5] I optmum cod, symbols tht occurs mor frqutly (hv hghr probblts of occurrc wll hv shortr cod words th symbols tht occur lss frqutly. Th two symbols tht occurs lst frqutly wll hv th sm lgth.[0]. Etropy [8] Etropy codg whch s wy of losslss comprsso tht s do o mg ftr th qutzto stg. It bls to rprst mg mor ffct wy wth smllst mmory for storg or trsmsso. Th vrg formto pr sourc symbol s kow s tropy. [8] From sourc wth symbols, th Etropy of th sourc s th mmum thortcl of th vrg umbr of bt pr symbol H P log P P (. Whr s th probblty of th -symbol. Gv comprsso schm, ts ffccy c b msurd s: Effccy = H/(vrg codword lgth..3 Dffrtl tropy Th dffrtl tropy s gv by h ( x log (. Th dffrtl tropy of Guss rdom vrbl s drctly proportol to ts vrc. Fgur. MATHEMATICAL BACKGROUND Now w dscuss bout dffrt rror-fr comprsso pproch tht dos ot rqur dcomposto of mg to collcto of bts pls. Th pproch, commoly rfrrd to losslss prdctv codg, s bsd o lmtg th tr-pxl rdudcs of closly spcd pxls by xtrctg d codg oly th w formto ch pxl.[7].huffm Codg A commoly usd mthod for dt comprsso s Huffm codg. O of th most populr tchqus for rmovg codg rdudcy s du to Huffm. Wh codg th symbols of formto sourc dvdully, Huffm codg ylds th smllst possbl umbr of cod symbols pr sourc symbol. Th dffrtl tropy for th Guss dstrbuto hs th ddd dstcto tht t s lrgr th th dffrtl tropy of y othr cotuously dstrbutd rdom vrbl wth th sm vrc. Tht s, for y rdom vrbl X, wth vrc. [], [].3. Globl thrsholdg Th globl thrsholdg s bsd o th hstogrm of mg. I ths mthod w do prtto of th mg hstogrm usg sgl globl thrshold. Th succss of ths tchqu vry strogly dpds o how wll th hstogrm c b prttod.i most pplctos; thr s usully ough vrblty btw mgs tht, v f gobl thrsholdg s sutbl pproch d lgorthm cpbl of stmtg utomtclly th thrshold vlu for ch mg s rqurd.[9].4 Vrous Prdctv Codg Modls Now w hv dscussd bout som prdctv codg modls followg subsctos:..huffm codg lgorthm.4. Losslss prdctv modl [9] Wh w tlk bout rror-fr comprsso pproch th th pproch commoly rfrrd to losslss Volum, Issu Mrch Aprl 03 Pg 4

3 Wb St: Eml: Volum, Issu, Mrch Aprl 03 ISSN prdctv codg, s bsd o lmtg th tr-pxl rdudcs of closly spcd pxls by xtrctg d codg oly th w formto ch pxl. Th w formto of pxl s dfd s th dffrc btw th ctul d prdctv vlu of tht pxl.[9] Fgur. Losslss Prdctv Ecodr Fgur. Losslss Prdctv Dcodr.5 Dffrtl puls cod modulto Th bsc dffrtl codg systm s kow s th dffrtl puls cod modulto (DPCM systm. It s most populr s spch codg systm d s wdly usd tlpho commuctos. Th DPCM systm cossts of two mjor compots th prdctor d qutzr. [0] Prdcto Error Rcostructo s sˆ (.3. s sˆ (.3. Prdctv codg s bsclly usd to lmtg th tr-pxl rdudcs of closly spcd pxls. Th dstrbuto of put dt plys vry mportt rol bfor mg trsformto stg. If th put dt s o uform tur th th trsformd dt wll b mor uform typ d most of th coffcts wll hv sm vlu of rgy d to cod ths coffcts t wll rqur mor o of bts s th tropy wll b mor (As costrd gv by Sho s frst fudmtl thorm. Whrs cs oppost to ths f yhow th dt s uform bfor mg trsform stg th vry fw o of th trsformd coffcts wll hv most prt of th rgy d mor trsform coffcts wll hv vry lss frcto of rgy. So scod cs most of th coffcts c b lmtd wth rwrd of vry lss rgy loss d crsd comprsso prformc. Our w modl s bsd o th bov fct tht. I th proposd modl w hv trd to grt rror mg bfor mg trsform stg. Th motv bhd th grto of ths rror mg s to kp th vrg pxl vlu d th stdrd dvto t mmum so s to mk th dstrbuto mor uform to gt bttr comprsso prformc. Th codr s show followg fgur-.5 grts th pxl vlu wth th hlp of pst put. Frst t subtrct d dd th pxl vlu wth th prvous vlu d th t dvds th subtrctd d ddd wth th hlp of dvdr to grt Error Img(.Th dcodr just prform th rvrs oprto to rtrv th mg. Ecodr Rcostructo rror = qutzto rror s s q (.3.3 Th codr vlu s Fgur.4 Nw Modl Ecodr (.4. Dcodr Fgur.3: Dffrtl Puls Cod Modulto Modl Fgur.5 Nw Modl Dcodr.6 Nw Proposd Error Modl Th dcodd vlu s Volum, Issu Mrch Aprl 03 Pg 43

4 Wb St: Eml: Volum, Issu, Mrch Aprl 03 ISSN (.4. Vrtcl Dgol RESULTS W wll compr our Nw Proposd modl wth th LPC.W wll prov how our modl s bttr th LPC d Orgl mg. W hv lso show vrous rsults d tbls dpcto output of our work. 3. Dtld Alyss Proposd Modl LPC Modl Fgur 3.3: Orgl Bboo Img berror Img c Dcomposd Img t lvl hr Fgur 3.:Orgl Bboo Img berror Img c Dcomposd Img t lvl hr Fgur 4.4 : dtl horzotl hstogrm of bboo b dtl dgol hstogrm of bboo c dtl vrtcl hstogrm of bboo d Approxmto Hstogrm Orgl Img Hstogrm Tbl : Dtld lyss of bboo LPC mg Fgur 3.: dtl horzotl hstogrm of bboo b dtl dgol hstogrm of bboo c dtl vrtcl hstogrm of bboo d Approxmto Hstogrm Orgl Img Hstogrm Tbl Dtld lyss of bboo rror mg Nw modl M Stdr d Dvt o mx.vl u M.vl u trop y LPC modl Bboo rror mg Approxm to Horzotl Vrtcl M Stdrd Dvto Mx. vlu M.vlu tropy Bboo rror mg Dgol Approxmt o Horzotl Comprso of Rtd Ergy d Numbr of Zros for Dffrt Prdctv Modl Proposd Modl Volum, Issu Mrch Aprl 03 Pg 44

5 Wb St: Eml: Volum, Issu, Mrch Aprl 03 ISSN Fgur 4.5 Alyzto of bboo rror Img t Lvl wth Hr Tbl 3 Dffrt st of for bboo rror Img S.o Globl Thrshold Rtd Ergy (% No of Zros (% LPC Modl Fgur 4.5 Alyzto of bboo LPC Img t Lvl Wth Hr Tbl 3. Dffrt st of for bboo Lpc Img S.o Globl Thrshold Rtd Ergy No of Zros (% (% CONCLUSION I our proposd modl th rsults gv us w st of pxl whch forms th rror mg or rsdul mg. Th rsdul dstrbuto s typclly zro m d much mor compct th th dstrbuto of orgl mg. By dcrsg th m vlu, w hv lso dcrsd th vrg codword lgth. Th w ppld stdrd wvlt trsform (Hr t lvl d clcultd th rtd rgy d th umbrs of zros d foud tht th rtd rgy d umbr of zros for our w modl hv th bttr rsults wh thrshold vlu s tk vry low. Sc th pxl r uformly dstrbutd spc dom, th th wll b o-uformly dstrbutd frqucy dom d ll th rgy wll b cotd by oly fw frqucy coffcts d t c b glctd th frqucs hvg vry low of rgy whch wll hlpd us to us lss bts of dt durg codg. So w hv cocludd tht mor compct dstrbuto (uform rsults lowr tropy whch dtrms th mmum vrg cod word lgth tht s ttbl wthout formto loss, hc chv hghr comprsso rto wthout formto loss. REFERENCES []. R.C.Gozls & R.E.Wood, Dgtl Img Procssg, d Low Prc Ed., Prtc Hll, 007. []. K Syood, Dt Comprsso, d Ed., Morg Kufm, 005. [3]. D Slomo, Dt comprsso Th Complt Rfrc, 3rd Ed., Sprgr, 004. [4]. Myug-S Sog Etropy codg Wvlt Img comprsso, 006. [5]. G.E.Bllloch Itroducto to dt comprsso, 00. [6]. W-W Lu & M.P.Gough A Fst dptv Huffm codg lgorthm IEEE Trs.Commu. vol.4, 993. [7]. K Syood, Loslss Comprsso Hdbook, d Ed., Acdmc Prss,003. [8]. D.A.Huffm, A mthod for th costructo of mmum-rdudcy cods, Proc.IRE, vol.40, pp.098-0, 95. [9]. R.C.Gozls & R.E.Wood, Dgtl Img Procssg Usg MATLAB, st Low Prc Ed., Prtc Hll, 008. [0]. Thoms Wgd, Dgtl Img Commucto, 000. AUTHOR Moht Mshr rcvd th B.Tch. Elctrocs d Commucto Eg from Jyp Isttut d Egrg d Tchology 009. Currtly h s workg Ary Collg of Eg & IT s Assstt profssor. Now h s workg o th r of mg procssg Rjsh Tjw rcvd th B.E. Computr Scc from Govrmt Eg collg Ajmr 008. Currtly h s workg Ary Collg of Eg & IT s Assstt profssor. Now h s workg o th r of mg procssg. Volum, Issu Mrch Aprl 03 Pg 45