Available online Journal of Scientific and Engineering Research, 2018, 5(4): Research Article

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1 Availabl onlin Journal of Scintific and Enginring Rarch, 018, 5(4): Rarch Articl SSN: CODEN(USA): JSERBR Analyi of Rat-Ditortion function for Multi-Hypothi Motion Compnation (MMC) in Vido Comprion Sytm K.A. Buari 1, T.A. Buari, M.A. Afolabi 3,.D. Adamu 4 1 Dpartmnt of Phyic, Fdral Univrity, Gahua, Yob Stat, Nigria Nigrian Tulip ntrnational Collg, Kano, Nigria 3 Th light Collg, Kano, Nigria 4 Dpartmnt of Phyic, Bayro Univrity, Kano, Nigria Abtract Thi work prnt a computational analyi of ditortion function for multi-hypothi fram at thir rpctiv bit-rat in vido comprion ytm. Thortical bit aignmnt for multi-hypothi fram i achivd with Lagrang multiplir and it cot function for raonabl low cot comprion ytm at minimum bit- rat. Vido ignal ar comprd at a minimum tranmiion bit-rat to achiv accurat dcoding or rcontruction dpnding upon th numbr of hypothi ud in coding th ignal. Thi ha mad multi-hypothi motion compnation (MMC) a grat proc in th prdiction of th actual ignal through th combination of mor than on motion compnatd prdiction (MCP). Th ditortion function wr analyzd at bit-rat rang 0.0 [bpp] to 1.0 [bpp] uing MATLAB in ordr to optimiz th numbr of hypothi to b ud in dcoding for a raonabl ignal comprion. Rult how that a minimum ditortion of th rang wa ralizd with ight numbr of hypoth fram. Thi impli that varying numbr of hypoth actualiz minimum ditortion of ignal du to optimization. Thi alo addr a bttr prdiction du to gratr tim corrlation btwn hypoth which conquntly hlp in th improvmnt of data comprion, coding and rcontruction quality of vido ignal. Kyword Bit-rat, Motion Compnation, Rat-Ditortion Function, Vido Comprion 1. ntroduction n rcnt yar advanc in vido ignal comprion ha mad it vidnt for th dvlopmnt of hybrid codr lik MPEG, H.63, H64 tc toward noi and rdundancy rduction through motion compnation. Thrfor, digital vido bing immun to noi, i air to tranmit and alo abl to provid a mor intractiv intrfac to ur. n th vido cn, th data rdundancy ari du to chang in patial, tmporal and tatitical corrlation btwn th fram [1]. Vido comprion, motion compnation, Tranform coding, ntropy coding, ar ud to rduc th tmporal, patial and tatitical rdundancy btwn th ucciv fram in thir rpctiv domain [1]. Th ntropy coding tchniqu (lol comprion tchniqu) i alo commonly ud in fil comprion. Thrfor fficint comprion i achivd, at rducd bit-rat for a crtain minimum ditortion of vido ignal. Digital rprntation of imag i important for digital tranmiion and torag on diffrnt mdia uch a magntic or lar dik []. Howvr, pictorial matrial rquir vat amount of bit if rprntd through dirct quantization, thi mad comprion (coding) of vido data vidnt [3]. Thrfor, for ay tranmiion of vido ignal, vido data ar comprd (rducd in iz) through vral coding algorithm that mploy diffrnt coding and comprion tchniqu through motion compnation for an accptabl ound and pictur quality at th rcivr. Thi alo giv room for ignal to b prad acro many frqunci for many bnfit, including ritanc to jamming and intrfrnc, allowing multipl ur to nd Journal of Scintific and Enginring Rarch 448

2 Buari KA t al Journal of Scintific and Enginring Rarch, 018, 5(4): data imultanouly ovr th am frquncy rang and alo nabling ncryption through rror dtction and corrction. Multi-hypothi motion compnation ha found many application in vido coding uch that codr mploy motion-compnatd prdiction ignal that ar uprimpod to prdict th original fram for ay rcontruction at th dcodr. Thi trm wa firt ud to provid a fram work for ovrlappd block motion compnation (OBMC). OBMC wa introducd to rduc blocking artifact in motion-compnatd prdiction [4]. Concatnatd block cod hav bn conidrd for mbddd bit tram tranmiion with minimum ditortion varianc and ovr rror-pron mmoryl channl [5]. Th thortical motivation for multihypothi motion compnation hav alo bn prntd in [6], whil th mathmatical framwork on tmporal and patial prdictiv procing in motion-compnatd vido comprion hav alo bn prntd in [7]. A mathmatical modl wa alo introducd for rat-ditortion function and coding gain of multihypothi motion compnatd vido ignal [].. Thory Figur.1: Hybrid vido comprion ytm Figur 1.: Group of pictur (GOP) tructur Virtually all vido comprion ytm idntify and rduc four baic typ of vido data rdundancy: intrfram (tmporal) rdundancy, intr-pixl rdundancy, pycho-viual rdundancy, and coding rdundancy [3]. Figur.1 how a typical diagram of a hybrid vido comprion ytm. Firtly th currnt fram i prdictd from prviouly dcodd fram by timating th motion of block or objct, thu rducing th intr-fram rdundancy. Aftrward to rduc th intr-pixl rdundancy, th ridual rror aftr fram prdiction i tranformd to anothr format or domain uch that th nrgy of th nw ignal i concntratd in fw componnt and th componnt ar a uncorrlatd a poibl. Th tranformd ignal i thn quantizd according to th dird comprion prformanc (ubjctiv or objctiv). Th quantizd tranform cofficint ar thn mappd to cod word that rduc th coding rdundancy [3]. A hown in figur (.), all tandard hybrid vido ytm lik MPEG-1, MPEG-, MPEG-4, H.63, H.64 ncod t of imag ignal to rduc patial and tmporal rdundancy by dividing th fram of vido ignal into group of pictur (GOP). Thraftr, claifying ach fram into, P or B fram (intra, intr and Bidirctional fram). mag data in ach fram ar plit into rgion (block) for th prdiction of a block in th currnt fram by uing rfrnc fram a th prviouly dcodd fram (lat or P-fram). n figur.1, th diplacmnt d of th block i not fixd and mut b ncodd a id information (S) in th bit-tram. Thi giv room for motion compnation and timation to b block-bad in ordr to minimiz th S and implify Journal of Scintific and Enginring Rarch 449

3 Buari KA t al Journal of Scintific and Enginring Rarch, 018, 5(4): th ncoding proc. Thrfor, by th proc of multi-hypothi motion compnation and timation dicud, ach t of block availabl for prdiction (multi-hypothi) ar aignd motion vctor which ar tranmittd with th prdiction rror p btwn th original and rfrnc fram (ignal rror) for th final rcontruction of th imag at th dcodr aftr undrgoing tranformation and quantization. M and N rprnt th ditanc btwn cor pictur and th numbr of cor pictur rpctivly. Thrfor, M N=Q which man th total numbr of pictur in a group of pictur..1. Quantization Quantization i th mapping of vctor or calar of an information ourc into a finit collction of codword for torag or tranmiion. Thi involv two proc, th ncoding and dcoding. t can furthr b imply dfind a man of providing approximation to ignal and ignal paramtr by a finit numbr of rprntation lvl (mapping a larg t of input valu to a countabl mallr t). Thi proc i irrvribl and thu alway introduc quantization noi (rror) which form part of loy comprion algorithm (ratditortion) in vido qunc. Thi rror i a a rult of th diffrnc in th input and quantizd ignal of th ourc ncodr. A dvic or algorithmic function which prform quantization i calld a quantizr. Simultanou quantization of vral ampl i calld vctor quantization (VQ), which i a gnralization of calar quantization (SQ). t involv th application of multi-dimnional (vctor valud) input ignal [8]. Thrfor, w map a continuou N-dimnional vctor x to a dicrt-valud N-dimnional vctor according to th rul. X C Q X i Y i whr C i i an N-dimnional cll. Th, i 1,... L poibl cll ar non-ovrlapping and contiguou to fill th ntir gomtric pac. Th vctor {y i } corrpond to th rprntation lvl in a calar quantizr. n a VQ tting, th collction of rprntation lvl i th codbook. Th cll C i, alo calld Voronoi rgion, corrpond to th dciion rgion. n VQ an indirct approach i utilizd via a ditortion maur d (x, y) to tt for th intrval which a ignal ampl blong. Q X Y d x, y d x, y, i for j 0... L 1 i j Aftr obtaining th bt match y i, th indx i idntifi th vctor and it thrfor codd a an fficint rprntation of th vctor. Th rcivr (dcodr) can thn rcontruct th vctor y i by looking up th contnt of cll numbr i in a copy of th codbook. Thu, th bit rat in bit pr ampl in thi chm i log L=N whn uing traight forward bit rprntation for i... Tranform Coding (TC) Tranform coding contitut an intgral componnt of contmporary imag/vido procing application. Mot imag and vido comprion chm apply a tranformation to th raw pixl or to th ridual rror rulting from motion compnation bfor quantizing and coding th rulting cofficint[3]. Tranform coding rli on th prmi that pixl in an imag xhibit a crtain lvl of corrlation with thir nighbouring pixl. Similarly in a vido tranmiion ytm, adjacnt pixl in concutiv fram how vry high corrlation. Conquntly, th corrlation can b xploitd to prdict th valu of a pixl from it rpctiv nighbour. A tranformation i thrfor, dfind to map thi patial (corrlatd) data into tranformd (Uncorrlatd) cofficint. Th tranformation ub-block d-corrlat th imag data thrby rducing or rmoving intr-pixl rdundancy [9]..3. Bit-Rat Lagrang Multiplir (λ) By minimizing th avrag rcontruction rror varianc [6] a a ditortion w allocat bit-rat utilizing th Lagrang multiplir (λ) mthod[10], [11]with a Lagrang cot function givn by j D R c (3) (1) () Journal of Scintific and Enginring Rarch 450

4 Buari KA t al Journal of Scintific and Enginring Rarch, 018, 5(4): min r 1 Q Q 1 (4) r, taking into account th contant bit-rat contraint R th targt function bcom c Q R QR givn by R 1, (5) Q 1 R Q 1 min j min r, 1 R QR. (6) Q By rlating quation (3) and quation (6) th drivativ of th function j i givn by dj dd 0 dr dr Sinc j R, R, Q 1 j R R Q 1 r, 1 R QR[10], Q 1Q Q Thn log Q 1 0, 1 log, Q (8) Sinc R R (9) ubtituting qn (8) into qn (9) w hav 1 R R log. (10) Q Hnc, whr whr and ar th quantization prformanc factor and bit rat for th multi-hypothi fram rpctivly and R i th ovrall bit rat. w now alo obtain that for th ca of P-fram,, 3, 4 and 8 in []: 1 r t RPfram R log t, Q (11) 1 3 r t 1 h R R log, t Q (1) 1 4 r t log r R3 R, t Q 3 3 (13) 1 5 r t 1 r 1 log r R4 R, t Q 4 4 (14) 1 9 r t 1 r 1 r 1 r 1 log r R8 R, (15) t Q hypoth rpctivly givn.4. Rat-Ditortion Function Rat-ditortion function i a function that allow u to calculat prformanc bound without conidration of a pcific coding mthod. Rat ditortion function dcrib th trad-off btwn loy comprion rat and th corrponding ditortion. t provid a minimum tranmiion bit-rat, if a ditortion D btwn th original (7) Journal of Scintific and Enginring Rarch 451

5 Buari KA t al Journal of Scintific and Enginring Rarch, 018, 5(4): imag at th tranmittr and th rcontruction imag at th rcivr will not xcd a maximum accptabl ditortion []. Th rat ditortion function for Gauian ourc with a quard rror ditortion i givn by [1]. Hnc thi can b rwrittn in trm of rat a, D( R) R (17) Suppoing th ovrall numbr of fram tnd to infinity and taking into account th optimal bit allocation, th ditortion rat function D(R) can b obtaind in th clod form [13]. D D ( R ) Pfram R R (18) R r t (19) t R 3 r t 1 r D R t (0) R 4 r t 3r D3 R t 3 3 (1) R 5 r t 1 r 1 r D4 R t 4 4 () R 9 r t 1 r 1 r 1 r 1 r D8 R t (3) Alo rprnt th radial diplacmnt of ach hypothi. t man th Euclidan ditanc to zro diplacmnt rror vctor i.. whil ( man th tim ditanc btwn hypoth. Thrfor, and ar th patial and tmporal corrlation cofficint of a pictur lmnt rpctivly. =0.93 i th avrag corrlation btwn horizontally or vrtically adjacnt pictur lmnt in a typical vido ignal [14] 3. Mthodology n thi work, w analyzd th rat-ditortion function which ar all polynomial function (quation 19-3) uing MATLAB in ordr to optimiz th numbr of hypothi to b ud in dcoding ignal. W ud ditor cript and command window of MATLAB to run th analyi with th following imulation procdur W utilizd output function (rat-ditortion function of multi-hypothi fram) uch a DP, D, D3, D4, D8 and input paramtr uch a. W gnratd cript for th output function in th ditor cript nvironmnt of th MATLAB by dfining a poly-function for th input paramtr a [DP, D, D3, D4, D8]= poly3 ( n th firt ca, w t at 0.8, 0.93, 1.0, 0.54, 1.0 and rpctivly. A cript i thn ud to run a poly function dfind on th command window for a raonabl optimization. n th cond ca, w t at 0.8, 0.93, 0.3, 0.64, 1.0 and rpctivly. A cript i thn ud to run a poly function dfind on th command window for a raonabl optimization. n th third ca, w t at 0.95, 0.93, 1.0, 0.50, 1.0 and rpctivly. A cript i thn ud to run a poly function dfind on th command window for a raonabl optimization. n th lat ca, w t at 0.95, 0.93, 0.3, 0.95, 1.0 and rpctivly. A cript i thn ud to run a poly function dfind on th command window for a raonabl optimization. W obtaind th output valu of th ditortion function for multi-hypothi fram with th givn valu of th input paramtr on th command window du to th gnratd cript in ach ca. (16) Journal of Scintific and Enginring Rarch 45

6 Ditortion D for Fram Ditortion D for Fram Buari KA t al Journal of Scintific and Enginring Rarch, 018, 5(4): W plottd a two dimnional (-D) graph of ditortion function of multi-hypothi fram againt thir bit-rat for ach ca uing plot command which contain grid, lgnd of tring and color with x and y labl for R and th output function rpctivly a, plot(r,dp,'m.',r,d,'b--',r,d3,'k-.',r,d4,'k-',r,d8,'r:'); Finally, th diplay of th -D graphic plot how th prformanc limit for ditortion of multi-hypothi fram in ignal comprion. 4. Rult and Dicuion 4.1. Rult Stting th valu of paramtr a givn in figur 4.1a, ditortion of 8 fram i minimal at whil that of P fram i minimal at Thrfor, a ditortion of i collctivly obtaind for 8 fram ovr P fram Pfram fram 3fram 4fram 8fram Bit-rat R[bpp] Figur 4.1a: Ditortion function for multi-hypothi with rpctivly. Stting th valu of paramtr a givn in figur 4.1b, ditortion of 3 fram i minimal at whil that of P fram i minimal at Thrfor, ditortion of i collctivly obtaind for 3 fram ovr P fram Pfram fram 3fram 4fram 8fram Bit-rat R[bpp] Figur 4.1b: Ditortion function for multi-hypothi with rpctivly. Journal of Scintific and Enginring Rarch 453

7 Ditortion D for Fram Ditortion D for Fram Buari KA t al Journal of Scintific and Enginring Rarch, 018, 5(4): Stting th valu of paramtr a givn in figur 4.1c, ditortion of 8 fram i minimal at 0.05 whil that of P fram i minimal at Thrfor, a ditortion of i collctivly obtaind for 8 fram ovr P fram Pfram fram 3fram 4fram 8fram Bit-rat R[bpp] Figur 4.1c: Ditortion function for multi-hypothi with rpctivly. Stting th valu of paramtr a givn in figur 4.1d, ditortion of 3 fram i minimal at 0.09 whil that of P fram i minimal at Thrfor, a ditortion of i collctivly obtaind for 3 fram ovr P fram Pfram fram 3fram 4fram 8fram Bit-rat R[bpp] Figur 4.1d: Ditortion function for multi-hypothi with rpctivly. 4.. Dicuion Th Figur abov (4.1a, 4.1b, 4.1c and 4.1d) how th ffct of ditortion for multi-hypothi fram (fram) uing intr-fram vido coding at minimum bit-rat with th valu of th paramtr uch a tim corrlation cofficint, radial diplacmnt tim ditanc btwn pictur ( ) and patial corrlation cofficint uggtd by vido qunc [15]. A MATLAB program i ud for th imulation of th paramtric quation of rat-ditortion function of th multi-hypothi fram (19 to 3). Th ditortion of th multi-hypothi fram ar varid ovr bit-rat with th radial diplacmnt ranging from 0.3 to 1.0. Th lowt rat-ditortion valu i obtaind with thr hypoth from th figur (4.1a, 4.a, 4.3a and 4.4a) which Journal of Scintific and Enginring Rarch 454

8 Buari KA t al Journal of Scintific and Enginring Rarch, 018, 5(4): ignify a bttr prdiction and rcontruction of fram uing up to ight hypoth in ordr to achiv improvd comprion of vido ignal tranmiion. n Figur 4.1a and 4.1b, a collctiv ditortion valu of and i obtaind for 8 fram and 3 fram at a minimum ditortion valu of and rpctivly ovr P fram all at th am bit-rat [0.0bpp-1.0bpp]. Thrfor th ditortion dcra for about whn tim corrlation cofficint and radial diplacmnt rang from 1.0 to 0.3 at a gratr tim intrval ( ) of 0.54 to n Figur 4.1c and 4.1d, a collctiv ditortion valu of and i obtaind for 8 fram and 3 fram at a minimum ditortion valu of 0.05 and 0.09 rpctivly ovr P fram all at th am bit-rat [0.0bpp-1.0bpp]. Thrfor th ditortion dcra for about whn tim corrlation cofficint and radial diplacmnt rang from 1.0 to 0.3 at a gratr tim intrval ( ) of 0.5 to W can alo obrv that a gratr tim corrlation cofficint ( ) of th rang btwn pictur lmnt and low ditortion rat of rang impli a bttr vido comprion. 5. Concluion Rat-ditortion function for multi-hypothi fram ha bn thortically and computationally analyzd in ordr to dduc th prformanc bound without conidration of a pcific coding mthod. Alo, th trad-off btwn loy comprion rat and th corrponding ditortion ha bn dcribd to provid a minimum tranmiion bit-rat, for an accptabl non-xcding maximum ditortion btwn th original imag at th tranmittr (ncodr) and th rcontruction imag at th rcivr (dcodr). A low ditortion rat of rang i achivd at minimum tranmiion bit-rat of th rang 0.0bpp to 1.0bpp with ight hypoth du to optimization. Thi how a gratr prformanc of a codc for motion compnatd vido comprion with application in both digital torag and communication rvic Rfrnc [1]. J.Ratnottar, R. Johi, and M. Shrivatav, (01). Comparativ tudy of motion timation and motion compnation for vido comprion, nt. J. of Emrging Trnd and Tcchnology in Computr Scinc. ( JETTCS ), vol.1 no.1, pp ,. []. A. Sam, (015). Mathmatical modling of coding gain and rat-ditortion function in multihypothi motion compnation for vido ignal, nt. J. Elctron. Commun. ( AEÜ ), vol. 69, pp ,. [3]. V. K. Maditti, (010). Vido, Spch, and Audio Signal Procing and Aociatd Standard, in Th Digital Signal Procing Handbook, nd Editio., V. K. Maditti, Ed. Boca Raton London Nw York: CRC Pr Taylor & Franci Group, pp [4]. S. Nogaki and M. Ohta, (199). An ovrlappd block motion compnation for high quality motion pictur coding, Proc. EEE nt. Symp. Circuit Syt., pp [5]. S. S. Arlan, (014). Minimum Ditortion Varianc Concatnatd Block Cod for Embddd Sourc Tranmiion, Advancd Dvlopmnt Laboratory Quantum Corporation rivin CA 9617, pp [6]. M. T. Orchard and G. J. Sullivan, (1994). Ovrlappd Block Motion Compnation: An Etimation- Thortic Approach, EEE Tranaction on mag Procing, vol. 3, no. 5, pp [7]. A. Sam, (01). ANALYSS OF CODNG GAN AND OPTMAL BT ALLOCATON N MOTON COMPENSATED VDEO COMPRESSON, Journal of Elctrical Enginring, vol. 63, no., pp [8]. A. Grho, and R.M. Gray, (199). VECTOR QUANTZATON AND SGNAL COMPRESSON. Kluwr Acadmic Publihr, Boton,MA,. [9]. S. A. Khayam, (003). "Th Dicrt Coin Tranform (DCT): Thory and Application. Dpartmnt of Elctrical & Computr Enginring Michigan Stat Univrity Michigan Stat Univrity, [10]. X. Li, N. Ortl, and A. Huttr, (009). Laplac Ditribution Bad Lagrangian Rat-Ditortion Optimization for Hybrid Vido Coding, EEE Tranaction on Circuit Sytm for Vido Tchnology, vol. 19, no., pp [11]. M. Flirl, T. Wigand, and B. Girod, (1998). A Locally Optimal Dign Algorithm for Motion- Compnatd Prdiction, Proc. Data Comprion Conf., pp Journal of Scintific and Enginring Rarch 455

9 Buari KA t al Journal of Scintific and Enginring Rarch, 018, 5(4): [1]. M. C. Thoma, and J.A. Thoma, (1991). Elmnt of nformation Thory. John Wily & Son, nc. Print SBN Onlin SBN , [13]. N. S. Jayant, and P. Noll, (1984). Digital coding of wavform: principl and application to pch and vido. Englwood Cliff, NJ: Prntic-Hall vol 75, no 4 pp [14]. B. Girod, (000). Efficincy Analyi of Multihypothi Motion-Compnatd Prdiction for Vido Coding, EEE Tranaction on mag Procing vol. 9, no., pp ,. [15]. M. Ohta, and J.Katto, (1995). Mathmatical Analyi of MPEG Comprion Capability and it Application to Rat Control, nt. Conf. mag Proc., vol, pp Journal of Scintific and Enginring Rarch 456

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