I DESCRIPTION Invention Title: METHOD AND APPARATUS FOR PERFORMING INTERPOLATION BASED ON TRANSFORM AND INVERSE TRANSFORM I he present application is a divisional application iom Austaian Patent Application !No.H 239 12 the entire disclosure of which is incorporated herein by reterene Technical Field [W! Apparatuses admehods consistent with exemplary emubodimnts relate to interpolating an inage. and more particular to intepolatingbetween pixel values of integer pixel units. Background Art [21 in a related art image encodig and decoding method one pimre i divided io a plurality of mact o blocks so as to encode an imn'age. Then, each of the pluirahty of macro blocks is prediction-encoded bype fomm mier prediction or sntra prediction thereon. 3] titer predicutn is a method of compmressimg an mage bx iemox mn a temporal redundacy between pictures. A represenati e exampk of inter prediction is motion-est imatnon encoding in mot ion.-esumation encodmn, eLich block of a current picture is predicted by usmg at least one reference picture A reference block that is the most simlar to a current block is searched hi i a predetermied search range b using a predetrmed evaluation tmcton. The currem block i predicted based on the referenee blee a residual block is ohtaNed by subtracig a predicted block wbieb is the result of predictng from the current block and then the residual block is ended. in this case, in order to precisely predict the current blok sub pixels that anQ' naller than imteger pixel uits are generated by performng intrpolation in a search mnge omte reference picture, and iter pre dicton is pertorned based on the sub pixels, Disciosore 'T echnical Solu tion [15] Aspcts of one or more exemplars embodiments provide a method and apparatus for generatmig pixel vdues of fractional piel units by icrpolatig pnel values o nteger pxel unis. [6] Aspects of one m more exemplan embodiments also pro ide a computer readable recording medmm havimy recorded thereon a computer program or executing the method. Advantageous Effects [7] Aecording to the preset appleation, a fraction pixel umt can be generate more Brief Description of Drawings The waove td oiler featuren Al betane more apparent by descnbmg in detA eJenpy embodimentswith refeee to te attached Ara.nang inh Ih Home FIG. I i4 a Mck dwgran of an apparats for encoding animage cording toa exenmpiargembOdbIent [101 Flo. is akkiagran of an apparatus for decoding aniage acding to an axemrrphrv embodimnt: 1 i1 I ilhstite' hinuhical codingFGnitacardng.an exCmpkarenbuament 1121 FIG isa b o c agLam of an image encoder baed on adding i aectdito an exemplary ermbeduneht [13] FIGk. s diokagram of an image decoder based on acoding uni, according to an exempl ary emillU xdnent' 14j FiG 6 ilusrates a maxn lm xhng um a sub codig unit, and a prediction ni according to an exempry ebrodiment [1] FG i tsto a codini and a transfoGrm1 unit, according to an e.emplary nm bodiment [16] FIGS. 8A to SD nitrate division shapes of a coding unit a prediction ii and a transformn.unit according to an exempla ry etrbodiment: [# FIG. 9 is a block diamo an imag- interpola lion apparatus according to an exemplary eirnen I 8] FIG. 1 is a diagram .ilustrating a twc-dimenional (D) imerpolation method perf'ormerd by the imat ge mrerpotahon apparatus of FIG. 9. according to an exemplary emabod imem; i9] FIG. I i is a diagram illustrating an interpalation region according to an e:<emplary embodimnen t; i20] FIG is adiagam iFuara aone-lienonai ( IDinterpoauonnethoc according to an exemplary embodimem; [t] FIG. I is a diagram specifically illorating a iD interpriion method pdmem by the bige interpolation apparamsc of FIG. 9 acco ding to an exemplary enboiment.; [22U B 14 is a block diagrani of an imageainterpalatioih appraamsaccording to an exemplary embodinenr I'] )) FI illustrates 2D interpolation fibers according to an exemplary embodimen; [24 FI16 i A to 16ih ilutrate ID interpolation fiter accordting to exemiplary em-e bodenenmt [251 FIGS. Atol i liustrate optimized ID interpiation filters according to exemplary enbodimemsN 3 [26] FIGS. 18A and 18B illustrate methods of interpolating pixel values in various directions by using a ID interpolation filter, according to exemplary embodiments; [27] FIG. 19A illustrates a 2D interpolation method according to an exemplary embodiment; [28] FIG. 19B illustrates a 2D interpolation method using a ID interpolation filter, according to another exemplary embodiment; [29] FIG. 19C illustrates a 2D interpolation method using a ID interpolation filter, according to another exemplary embodiment; [30] FIG. 20 is a flowchart illustrating an image interpolation method according to an exemplary embodiment; [31] FIG. 21 is a flowchart illustrating an image interpolation method according to another exemplary embodiment; [32] FIG. 22 is a flowchart illustrating an image interpolation method according to another exemplary embodiment; and [33] FIGS. 23A to 23E illustrate methods of performing scaling and rounding off in relation to a ID interpolation filter, according to exemplary embodiments. Best Mode for Carrying out the Invention [33a] According to a first aspect, the present invention provides an apparatus of motion compensation, the apparatus comprising: a processor which is configured for extracting, from a bitstream, information about a maximum coding unit and split information indicating whether a coding block included in the maximum coding unit is split into coding blocks of a lower depth, splitting the maximum coding unit hierarchically into coding blocks of depths including at least one of a current depth and a lower depth according to the split information, determining, in a luma reference picture, a luma reference block for prediction of a current block from among the coding blocks, by using a luma motion vector; generating a luma sample of a 2/4-pixel location included in the luma reference block, by applying an 8-tap interpolation filter to luma samples of an integer pixel location of the luma reference picture; generating a luma sample of a 1/4-pixel location or a 3/4-pixel location included in the luma reference block, by applying an interpolation filter to the luma samples of the integer pixel location of the luma reference picture without using the generated luma sample of the 2/4-pixel location, determining, in a chroma reference picture, a chroma reference block for prediction of the current block, by using a chroma motion vector of the current block; and generating a chroma sample of 4/8-pixel location in the chroma reference block by applying a 4-tap interpolation filter to chroma samples of an integer pixel location of the chroma reference picture, wherein the 8-tap interpolation filter comprises eight filter coefficients for interpolation for generating the luma sample of the 2/4-pixel location and the 4-tap interpolation filter comprises four filter coefficients for interpolation for generating the chroma sample of the 4/8-pixel location. [34] There may be provided a method of interpolating an image, the method comprising: selecting, according to an interpolation location, a first filter, from among a plurality 3a of different predetermined filters, for interpolating between pixel Values of in teger pixel units of an and generaing at least one pixel value of at least one fractional pixel unit by interpolatig between the pixel values of the integer pixel units by asing the selected first filter. [35 The method may further include selecting a second filter, from among a plurality of different filters, for interpolating between the generated at least one pixel value of the at least one fractional pixel unit, according to an interpolation location; and interpolating between the generated at least one pixel value of the at least one fractional pixel unit by using the selected second filter fbr interpolating between the generated at least one pixel value of the at least one fractional pixel unit. 1361 The first filter fbr interpolating between the pixel values of the integer pixel units may be a spatial-domain filter for transfonn ing the pixel Valaes of the integer pixel its by unlg a plumlity o basis fnctions hiying different n an inerse transfoning a plurality of coefficies, ich are obtained by the transforming the piel values i the Neger Pix umits by singt r Of basis. fOnctious. phases cf which are shifted l7T The second filter for interpoating between the generated at least one pixel value of the at least one fractional pixel uni my be a spaiaaldonmin ilter fr trmsforming the generatea least one pixcl value of the at least one fractional pixe un1t by using a plurdtlitv &f [his iuictlonls IhaVIng different frCuecDies, and Iierse transi'riinga pluf3htv of 'il~ints. whiCh are obtained by the transforming te generated at ai:st one exel value of the ai leasi one fractional p>i el unit, by using the pluruahtv of basis functians phases of. which areshifted M! Th er maY be provided an apparatus for interpolat ing an %mage the apparatus eonirsn g: a iter selector wh ich selects. aeordin tOan interpoitrons location, a fihst filterifrom.anixna p1luality of different predetermined fiti or interpoiainug between pixel alues of integer pixel units; and an interpolator which generates at last one pixel alue ofat least ne fractioiu pte unmit byx itertpolatinrg between the pien values ~ ~ ~ W' ofw n thpnee ie nt yu ix tel seleted. fir "0 filterQ.a [39 Thefilter selAtor uny select a second filer. nom an a lunality of diffent filtersfora interpoiatimg between the generated at least one pnsixesam'e or the at least onefractional pixel unit according to an nterpolion loai and Te Interpoiator may interpolate between the generated at leastone pixel v rIm ofhe at least one fractional pixe unit by using the selected second filter for iuterpo'ting between the generaed at least onr pixel value of the at least one fr actional pixel unitL I4] There may be povided a computer teadabre recording medium having embodied thereon a computer program for excuting the method described above 41] There ma y be proded a method of nterpolatng an image, the method including transform pixel values in a spatial domain by using a pluralty of bass funcions having different frequencies; shifting phases of the pluralty of basis functions; and inverse transforming a plural itv of coicin obtained by h rininrmg the pixel adtes, by using the phlas-shifted pluralty of basis fur-m tins Mode forhe Invention Fill Herinate. onri 1or moexemplarebodimentsoil be desribd more fuly with referencget h accompanyig drawings. Expressions such as "at least oegof' en precdigalstpf elemnuts odtv the ntir e it of emenals but do not modtfv the indiidalelenments oftie list hithe present s pecification. an'imiace" may denote a still image for a video or a moving image that is, he vrdeo itsE i43 FW I is a block diagram of an apparatus 0 for encoding an image according to an exemplargeendi nent Referdng toiG. I the apparatus 10 o encoding an inage includesainaxirnnoding unit divider 10; an encoding deth demAiner 1,anl 5 iage data eCiZodei1 .a nd an Cnrd inrformnation enoder L40 M44 i The Amu coding n diid 10 A y idea crent frameslice bseon a Uaxinom coding uni nhat is a cxing unit of thelase size Tha is the maximnu coding unit di ider 110 maydIide the curnt frame oa dice into at leastone maimumr eoding unit 451 According to an exemplary embodiment, a codmg unt may he represented usmn a L imnonig ct and a depthk As des ribd above, the maXimum Ldg unit indicates a cding unit hav ing the laet sie from among coding ma ts of the urren frame; and the depth indicates a degrce ot'hierarchieadly decreasing the coding uit. As a depth imeases a coding uit may decrease tom a maximum coding uni to a minimumn coding unit wh erein a depth of the maximnan codimg unit is defined as a nnmmrn depth and a depth of the vninmum coding uni s defined as a max inum depth, Siee the size of a codmg unit Jereases from a maxmum codmg ut as a depth irearse a ab codig umt of a 1$ depth may include a parity of s:h codig units of a (k+nly depth w here I and ri are integers equal to or greater than [461 According to an increase of the size of a frme to be encode entoding an itr age in a greater codig Umi may cause a hig:1er image nmplession ratio. However, if a heater codig unit s tixed, an :mage may not be efficiently encoded by reflecting con tinuously changing inmage characterstie 47] For examnpie. when a smo1h area su c as the sea Or sky is encoded, the greaer a scodig unit is, the n Io a MmpIesson ratiOn may inease, likvveL when a complex area h as people or bildings is ended, the smaller a oding unit i, the more a oripemsinnation may innease Accordingly, according to an exemplary embodiment, a different maxitmur image coding unit and a ditferent maximum depth may be set for each frame or slie., Since a mniimn depth denotes the maximum number of times by whic ng unit Inay decreaCse the si7e of each minimuM coding unit lauded in a maximum image codini unit may be variably set ac(oding to a maximum depth, The maxiemm depth may he determined diferenty o each frame or sleie of lot each maximum coding uniA [A91 The encoding depth determiner20 determines adivisin shape of ha maximum coding unit Thediion shape nay be determined based n adulation of rate dist n (RDcosts Thedetermined division shape o the maximum coding unn is provided to the encoding inormatuon encoder 140. and image data accordmge to aimuncoding uniti p ided to the iage dat encder 0 5 O] A maximacoding unit may be divided into sub coding unithaving different sizes according different deps andthesub coding units hang different sizes, which are fuetided in themaximum codingunitmay beqredicted or transformed based on processing uns ain dferr sizesIn others the appartu 1XUforenceoding an imagenmay petomin aphuai ito pae5ssig operatins tor nilige endlC~ing based in proc n unqit haig aious sizesanaso shapes To enoden-naM daa processing oeations. sumb as at east oneof pirdiction.trausttrmand enpy encodinig ant performed, wherin proessing uits having the same size or dJifferent srzec mnay be usedi fo the procesong operator respectively. MI For example, the apparatus 100 for encoding an image may select a processing unit that is ditTelent front a codong unit to predict the coding unit [521 When the siZe ofa coding unit is 2Nx2N (where N is a positive integer, pmcewng units for prediction may he 2N2N, 2NxW N2'1dnd NxN. in other words, mnin prediction may be peduneJ based on a procsing unit having a shape whereby at least one of the height and width of a coding unit is equal divided by two, Hereinafter, a pkocesing wit w hich is the bas of prediction is defned as a reductionn unit [53 A prediction mode ry he at last one of a ntra mode an inter mode, an a sp mode, and a specific predieion mode may be performed f n aMprediction unit having a specific Size or shape. For example. the imra mode may be performed for only prediction units havin the sires of 2Nx2N and NxN of which the shape is a square. Purther, the skip node may be pertormed for only a predicton unit ha.'ig he size of 2Nx2N, if a plurality or prediction units exist in a coding unit. the pordiction mode with the leasr encoding errors rmay be selected afer performing prediction for everypiedition t 41 Aernatively. the appatus 100 for enceing an imae nma pefn tgr nMorm On image data., based on a proewssn unit hazving a dhi ferent size from a coding ut, For the iranfornn nthe coding un. the traashonn nay be perfored baed on a processmg unit >a ng a Sie equal to or sm i ar than that of the codong unit HCeaftr, a proenwi :nit, wh bs th bset fr-afonnm is died a a transtoti unit', The transiorm may be discrete cosine transform l'i or Kah onen isue\e trnsform dK~ or any other fl'ied point spatind traustmun. [551 The encoding depth determiner 120 may deterne sub coding units inluded in a mamum coding unmt by using RD optimization based on a L.agrangi an multiplier, in other words, the enctnudz depth determiner 120 may determine wvh ch shape a plurality oif sub coding u!nits divided from lte toaitum codingu(Un have wherein the plurahity of ub ding units have different sizes accAdingt their depths The image dat encoder 130 outpua hitstream by encodingtemaximum soding unit based on the division shapes determined by rueecoding depth detenninerd120 [561 T encodinginormation enoder40 encode infonatn about an encdinamde of the raximnum coding unit determined b the encoding depth dretmier 120 in other w ords, the encodmg information encoder 1 40 ou tputs a bitstream b'y encoding im tnnarion about a division shape of the maximum coding unit, adormation about the maxirum depth, and information bout a a encoding mode of a Nob COO ng unit tor each depth. The information about the encoding mode of the sub coding uni may include imfmaisn about a prediction un it of the sub oding unit in fofmation about a predietm mode for each pdiction unmt and inorInation abou a tranorrn un i of the sob cen. unit. J7]I The i 0rnton aoM the division shape f the rnaxinu cAin unit may he i frnatwon, e flag information, indicaing whether each coding unit is divided. For exampk, when the maximum coding unit is divided and encoded, information in dictating whether the maximum codAg unit is djded is encded,. Als, when a sub codin unit divided fron the maimu coding unit i divided and encoded, in, formiain i.iatn hther the sub coding unit is div ided is encoded. ~58] Since sub eodig units having different sizes exist for each maxum codig unit and formation about an encoding mode must be determined for each sub codung not, mtrmatson aboui at least one ec dr oJe uiay be determined t'r one maniuK coding unit. 19] The apparatus 100 for encoding an image may generate sb codin units by equally div idn both the height and width of a maximum codmg nit by two accordig to sa mncreaSe of depth. That is when the size of a coding unit of a k0' depth is 2Nx2N, the size0 of oding unit o a(ic+ Idepth iN NxN. 60] A1cidingly the apparatus Ii 0 for eucoir an 'mage may detennine a d iison shape for each omium codirg iun it, based on si;es of maintmn coding unitsandJ a. maimum depth in conderatnion of image chtteateristics. By vranaly adiuctina the size of a mnaxinum- codug unit mn considemation of imane cbanaetcrisuecs and enonng an image throughh di vision of a naximum cog unt int sub cn units of i fferent depths. images having vaious resolunons may' he more etticientixy encoded. II MG. 2 i a bkok diagram of an apparatus 2) for decoding animage according to an eempla enudr t Referg to FIG. 2, the appar 200 fo decodig an image includes aia treadata acquisition unit 210 an encoding inroration extracor 220 andan image data decoder3) 1621 The image data acquiiton Unit 2j0 acquis imagedata according t maimumn coding units by paring a btst-ant received by the apparatus 200 fAr decoding an image and outpus theimage data tothe image daa dcnd Lm er 2 The inae data aer qisidin unit210 May extract frmaton abou maximum codiag units of a current frame or sice frm a header of he curentframe or slice ther words the image data acquisin unh 2o0 dividesthe bitstnanm accondin to the maximum codieg units so that the image data decoder 230 may decode the imae data accodninto the "din""m coding ifs [63] The ncoding ifniirnation extractor 20 atacts nformationabout anwxnn coding unit maximan depth, a diviion shape of the axianm codimnitand a encoding mode af sb eno g uinits from the header of the cent fame by parningthe bitstrearn meiAed by rhe apparatus for decodng an image The informtionabout the division shape and the information about the encodin mode areprided to the anaye data decc or 230 641 The information about the division shape of the maxnin coding unit may include information about sub Coding units halving different sizes according to depths and included in the maximum coding unit, and may be infcunnation (eg tlhg irnformation) indicating whether each coding unit is dvhided. The information about the encoding mtode may include information about a predictuon unit according to sub coding uruts information about a podiction mode and information about a transform unit, 01 The image data decoder 230 restores the Curent frame by decodma inage da of each maximum coding unit, based on the infotmaton extracted by the encdin in tornation extat'o 220, 66] The image data decoder 230 ai decode the sub coding unit included in ' maximum coding unit, based on the ienfmon about the division shape of the maximum coding unit. The decoding may include intra prediction, inter prediction that inclundes~ maliOn compestion&mf, and inlverSe transootn 671 The image data decoder 230 may perorm itr prediction or intet poduition based on information abot a prediction it an inflation about . prediction odc int order to prodic a prediction unit. The image data decoder 230 may also perform inverse u-ansform for each sub coding unit based on information about a transform unit of a sub codng unit, V68] FIG 3 illustrate hierarchical coding units according to an exerplary embdi mount. Refrring to FIG . the hierarchical coding un its may include coding units whose widths and heghts ac 64x64, 3x32. 16x16, Sx8 and 4-1x4 Besdes these coding units having perfet square shapes codin units whose wIdth nd heih are 64x2,32x64, 326, I6x32 '6x8, 8 x 16, 8, and 4x8 may also exist V9] Referrn to FG. 3, tor image data 3101 whose resolution is 1920x1080S, the si ze of a maximum coding unit is set to 64x64. and a muaximum depth isnet to 2, 70 Finmae data211 whosereshitin es 1920x1080 the size of aaimnunm Moding unt isen to 64x6 iand a m in depth s set to. r image data330 whose resolution 352x28 the size of maximum codng unit is set to i6x4i6 and a maxanum depth as set to 1 U 1] When the resolution is high or he anmonni f data i great , -aximum sizea cda coding unitmay be reativevmatato increaea comipressionrartioand enacit reflect umag~e ch-a racterisrics Acoringlv for thei iunge data 3 0 .tnd 320 hang higher VCSOlttnithan thennaa ta 335646x4 may be seleted ste hsize ua fmdta coding unit. 2i W ate i a W a number of aeinhe hierircha coding [72j] nxntr ept s~tc2 W tlp~ : lay ,ers In nisSince themaximumdiepth ofthe image data 310 i 2. a coding 3 of the nuage data 310 my aideanmainium coing unitwhose lkogeaxissize is 64and ub coding unis whose longer axis-sizes are2 and 1.ading tan incse ofa depit [731 On the other hand since the maximum depth of the image data 330 is , a coding unid 335 of the image data 330 may include a. maximum coding unit whose longer axcis size is 16 and coding units whose longer axis sizes ar 8 and 4, according to an increase of a depth, 1741 However, sine the maximum depth of the image, data 320 is 3, a coding uni 323 of the image data 320 ay include a maxmn coding unit whose longer axis size s 64 and sub coding units waose longer axis sizes ar 32, 16, S and 4 according to an increase of a depth Since an imae is encoded based on a smaller sub coding unit as a depth increases, the current exemplary embodiment is suitable for encoding an image including more imu e scenes, [751 FI0. 4 is a block diagram of an image encoder 400 based on a coding unit, according to an exemplary embodin t, An it prediction unit 410 performs ima prediction on piedtiction units of the intaade in a curmat frame 40S, and a motion estimate r 420 and a motion enmpenmator 425 perform itter predicting atnd mlot iou nmpelsation on predtictinn units of the inter mode by using the current frame 405 and a reference frame 1764 Reduai v alues are generated based on the prediction unias output from the irn poretion unit 410 the moti stimaor 4'K and the monon compensuor i4 and tre then output as quantized transform coef-lcients by passing through a transformer 4M0 and a quant/er 440. [771 The quanied urnrm co316cients are restored to the residual dues by passing throtigh an inverse quantizer 460 and :a inverse tranisformer 470, are postprocessed by pasing rough a deblocsing unt 40 and a ioop fitermg unit 40 and are then output as the reference tnme 495. The quanized transform coefficents may he output as a bi nam 45 by passing through entropy encoder 450. [781 To pefon encodg baed ou an encodry methodAo ccog an eemnnphly em bod-ment components of the image encoder 45). the intu predicion unit 410, the motion etimator 420R the rotnon compensator '4. the transformer 431) the quantizer 440. the entropy encokder 4I30. the inv\erse q nanti/er 460. the Iierse transrmer 4-2l the debiocksing unit 480, and the loop tiletirmt unit 490, may perform image encoding prX(tu&Sbased oa aximnm coding uhnsb coding flos acading to deptthsa 9] FIG. 5 s a block diagramofa inge decoder500 ased onac ng unitcording o an exenphy embodiment, Refening to FIG, 5, a bituream 505 n parsed by a parser 510 in aoer to obt aI en coed image data to be decXed and encxiing information which is necessary for decodin. The encoded image data i s output as inmvre quantized data by passing through an entopy decoder 520 and an inverse qumizer 530. and is restored to residual values by passin through an inverse transformer 540. The residual values are restored according m coding units by being added to an intra prediction result of an in, prediction unit 550 or a motion comnpensaon resul or a motion compensatory 56) The restored coding units am used for predicnon of next codin unit or a next frame by passing through a debkxcking unit 70 and a loop iitenng u'R 380. 180] Is peidorm dcNding based on a decaig meh acrdm to an exemplary en bodmeto imponents of the image decoder 51 he,. the parser 5 10, te entropy Jeode 5 20 die inverse quantizer 530 the inverse transformer 54. the inta prediction unit %0, the notion compensator 561) the debiocking unit 570, and the 'oop itrigyuiti80 K ire) pt ormt image dt'codsing proceses based on a mtaxsimi m oiri unit sub coding unis acc.ording to depths. a. prediction unit, and a transform unit. In particular, ihe introa prediction unit 550 and the motion compensatory 560 determine a pied on;on unit and a 'redion mae in aub Lodo unit by onside ing a onium coding unit and a depth, and the nvur-se transtonner 54(3 perfurms imnve transform 1w considering the toe of a irnmtorrn on t [&2] FIG. 6 illustrates a maoximnum coding unit a sub coding unit and a prediction unit, according to an exemplary emxodintent The apparatus 100 forencoding an image i lustred 0n FIG I and the apparams 200 for decoding o image illustrated in FR. 2 use hierarchical coding unit to perform encoding and decoding in consideration of image characteristics A maximum coding unit and a maximum depth may be adaptively set according to the image characteristics or vaiousiv set according to xe quiremnents of a user. 1831 In IC. 6, a hieradha coding uninkucre 6011 aSa maimun coding unt 61 whlse height and widt are64 and maximumdepthis A depinreases along a vertial axi of the hierarchical coding unit structure 600. and as a depth creases the height\ am wihs of su b codine urits 620 to 650 decrease, Prediction unit; of the mniminum coing uni NO and tie sub coding units 620 to 650 are shonn along a honriolY axis of the hierarciical coding unlit strucure 600 [841 The aimim codingunit 610 ha atepth of 0 and the size Of a oeding unitie height andidtho64x64A depineaseog thevertic s and there exst a 1b cod1ag ungt 620 Whosesizig x3202ad depth is 1. a sub codingut W whos sizeis116 anddeh is 2.asuodingunit 640 whos sie is 8 and depth , asnd ub coding uni650 whose i7e is44 and deuth is 4. The suh coding nit 650 whose swie is x and depth is 4 is a mniiimum codin unit, and the minimum coding unit ,a be divide into prediction nits each of which is less than the minimnam coding unit. 851 Referring to FIG. 6, examples of a prediction unit are shown along the horizontal axis according to each depth. That is a prediction unit of the maximum coding unit 610 whose depth is 0 may be a prediction unit whose size is equal to the coding unit 610. i~e~, 64x64, or a prediction unit 612 whose size is 64x32, a prediction unit 614 whose size is 32x64, or a prediction unit 61.6 whose size is3x32, which has a size smaller than the coding unit 610 whose size is 64x64. 56I A predicton unit of the coding unt 620 whose depth s i and size is 32x32 may be a precditotn unit whose size is equal to the coding unit 620, ie, 3e x3, or a prediction uni 622 whose ,ize *K 32x16, a p1icton unit 624 whose wize is 16x32, or a prediction unit 626 whose size is I16x 16. which has a size smtalier than the coding unit 620 whose ize is I6 A predictu0n unit ot the coding unit 630 whose depth is 2 and size is 16x 16 may be a prediction unit whose size is equal to the coding unit 630, ite, i6x6, or a prediction unit 632 whose size is 16x8, a prediction unit 634 whose sire is 8xi6, or a prediction unit 636 whoe size is 8x8, which has a size smaier than the oding unit 630 whose size is 16x16. [8]A prediction unit of the coding unit 640 whose depth isi 3 and size is 8x8 may bet a prediction unit whose size is equal to the coding Unit 640, ie 8x8, or a prediction unit 642 whose sie is 8x4, a prediction unit 644 whose size is A or a predicon unit 646 whose size is 4x4, which has a size smaller thant the coding unit 640 whose size is x8, 9] Finally, the coding unit 650 whose depth is 4 and size is 4x4 is a minimum coding unit and a iouding unit of a maximum depth. and a prediction unit of the coding unit 650 may be a prediction unit 650 whose size is 4x4, a prediction unit 652 having a size of 4x2, a prediction unit 654 having a size of 2x4, or a prediction unit 656 having a size of 2x2, 0)FIG. 7 illustrates a coding unit and a unansfo unit, according to an exemplary emn bodiment, The apparatus 0t0 for encoding an image ilustrated in Fi 1 and the apparans 200 for decoding an image illustrated in FIG. 2 perform encoding and decoding with a maximum coding unit itself or witb sub coding units, which are equal to or smaller than the maximum coding nit divided from the maximum .coding unit. in the encoding and decoding process, the size of a transom unit for transform may be selected to be no larger than that of a corresponding coding unit. For exarpe, r FIng.to P 7, when a current odeg unit 710 haste sie of 6x4, !ansora mays be perftred tsing a tranmtrin W 4 ua haing~ the ie of32x32 191] FIGS. SA though RD ilustrate division shape acodingunit, apedictionunit and a tnansforn una, according to an exemplary embodime. Specifially, FICS, mk and8R iuhtraie a coding it and a prediction unit accordmg' to an exempry em bodinenit, 921 FIG. SA shows a division shape selected by the apparatus 1t0 for encoding an image illustrated in FIG. I in order to encode a maxinmm cooing unit 810. The apparatus 100 for encoding an image divides the maximum codin nit 810 into trious shapes performs encodi thereon, and selects an optimal dir iion shape by comparnn encoding results of dhe Vanous division shapes with each other based on R D costs. When i is optmal that the mIaxium codin urit 80 is encoded as t is the mainun coding unimt 10 mar be ended without diwidi theraxmum coding uit $10 as lustated n FS, 8A though 8R 1q} Referring to FIG. SB. the. ium coding unt $10 whose depth is C is encoded by dividing it iato sub codin. Inithe depths are equal to or greater than i That is the maxinumn coding unit 810 is dio ided io four .sub codui units whoedpbarL &r- .. hodets re and all or seine ol the sub codtng units whose depths are 1 are divided into sub coding units vhose depts are. 2, 94 A ub coding unit aloated m an upper-ght side and a sub coding Unit oicted in a lower-eft side among the sub odan units wh boe depths are I ame divided into sub oding umts whose depths are egal to or greater than 2i Son of the ub codog uits whose depths are equal to or greater than 2 may he di ided into sub c en unts whse depths are equai to or heater than 3 1951 FIG B show a dv on shape of a predtion unit er the mrum codmg unit 810 Refern to FIGS a piedienon unit 8t0 for the meanaum c oding umt $10 may be divded differeniy from the maxmum coding unit 810, In other words a predicton unit for each of sb cud og unts may be smaller thana corresponding sub codig unit. d] For exampe a prediction unit for a snb coding unt $4 located in a lower-ight side among the sub oding units whose depths avI ay heasaliethan thesub coding ni 854i addition predidion units for omel u Coding units 814 8 5and 852 tromt among sub coding units Xi . 16. 18 828 $50, and852 whose depths are 2 may be smaller than the sub codin units 84 816 $5and 852/respetivy [971 In addition peditoM units frsu codnd n ts221 832 and 48 whose depthsare 3 nay be smaller than the sub coding units 2 832, and $8,especvely.The prediction uitstaa' bae shape vhtxvbbyrespectivesu codingunit are qualy died by two in a direron of heigh or width or ha Mahapehereby rspentve Aoigunits are eqa diided by four in directs of he adwidth -13, [9Fj FIGS.C and d iiuatez a prdicnunit and atra ntrmy accnM Ag ia eXmrpihry ebdintenn 99] PIG. ACshos a division shapeof a predctn nt flr the rnanun dine unit k0 showin FGi. SR. and FIG.D showka divinhape of a insnfrmLnit of the omxmum coding unt 810. [100k Refening to FIG. ND, a diviion hape ofa transfmunIt870 may be set differently fron the preJction unit 80 [1!.! Fo e 0xampie, even though a prediction unit for the coding unit 854 whose depth is 1 is selected with a shape whereby the height of the coding unit 854 is equally divided by two, a I ornsfom unt may be selected with te ame size as the cudinu unit 854, ikewise, e'en though prediction unis for coding units 8 14 and 85 whose depths are 2 are selected with a shape whereby the height of each of the codina units 814 and 850 is equally divided b wo, a transform unit may be selected with the sane sie as the original ize of each of the coding units 84 and 85.0 122 A transforuntn aaybsiected withasmaerietana predicdonnit, For x eh d un or the codin ui 52 whose depth is 2 is selected wah a sApe wherb the width twe codi g uni 52 equally divided by two, a ransiorn unutnnIneselected wih a shape whereby the coding unit852is equally divided by tour in direcionsd height and width, which has a siater izeethant the shape of th Aredictin ini [103 TKG P A a block daragm of an hiage imterpolatn appamas 90 aemalng to an xemlplarmbodieInt. Imac nterpolaton may be usd to cnrt n mtage having a io resoln1ito an image havi rga high re Wion A, imae nterpIatyon may be used to convert an iterlaced image to a progresiv image or may be used :O up sampte an image havi n a tow reMlution to a higher resolution. When the niage ioder 400 of PI. 4 encodes da im., the nation estimator 420 adn the moboi compensato: 425 may perform inter prediction by using an interpolated reference frae, That is. rfeeIrinl to FMi 4. an image having a hIgI resolution may be generated by interpolating the reference frmne 495 and motion estitio and cotnpensatin oaf be errned based o thenne having the higheshltion thereby increasing the precion of inter predieton. Likewke, when the image deer 500 of HM 5 decades an image, the motion compenstor 550 nuy per ormt motin n mpensaion by usoJ an interpolated reference frame, thereby increasmg the precision of inter [00 Refening to HG , the image interpolation appartus 900includestranformer90 and an inee transomern 92 [1051 The transtrner 910 transforms pixel vaiQes by ing ahuality ofbasisutions havn different equenies Thetransforn maye one of variouserocess of 14 iradsfw i Nxehdoval en a spitti a domain ito frequee nodsinaenCO roefn ei-isngd n CkW- K ~ 1 A~l' '-en Asa ~ XitiOWe o r W E mwnpWe NCT as d rd aiM W vaue, Q an integer pse un ae ansfrndusing heoait of basis non The pin vaus may be pi vaine of h nance components or of chmoma comnonents lhe type of the piur-aliy f basis functions iS not limited, and] may be o of t rious types of functions for transformning pixel values in a Npati domain into a frequecc-deamain value(sy. for exml.the pluraity of basis functions may be cosine functions to performing DCT or i e DCT, Also, various types of bais functions suh o basis functions may be used. Emples of DOCT may include modified DOT, and modified OCT that uses windowing [106! The inverse tansfer 0 shifts the phase of the njaudty of basis funions used or performIng nsfoirm by the transfomier910. and inverse trnsforms a plurality of Oecny t e domain vlueswhi ae generated by the tranformer 910, by using the puality of basfunctions thephaseo whdheenhi01. ransto erormed by-the tansformer 0 and diverse transfrm peond by the liners transformed 920 will now bdescribedby m tw dimenional (2)DCT and one-dimensiond D1)OCT. luK <D oKo and D mirerse DCI > [10, FIG. 0 is a diagram iluaratig a 2D iaterpolation mehod performed by the image interpolation apparatus 900 uf 9I . 9, aaordng to an exemplary embodiment, Refenidg to FR I0, te -eplanion appaats 900 genei-ales piXel values (n lotions X. i, interpoaSon k tion, by tcrpolaing between pe blues of mteger pnel unts in the spatial domai, e.g, pixel values on kcanons ( in a WMock [000, The p)ixei valtes on the loations X aue pxel values of fractional pixel Es. the ternpoition locations of which are determined by to and 'of Although HG .10 i lusaaes a case wbem e block l00t has asie of4 the size o the block 1000 is not limited to 44 and it would be obvious to those of ordinary skill i the art that piel values of fractional piel units may be generated by perform ming 2D DOT and 2D imese DCT on a block that is smaller or ler than the block 1000. [109! Firt the trUMforer 910 performs 2D DCT on the pixel values of the integer piel units 2 DC F may be performed accordig to the following equation: 110 c za -)Qm ) , iarr n< ,,,i~ ii-, [lii, where 'C' denotes a block that includes trecuency-domain coefficnts dhtamed by perlfonning 1D DOT, 'REP' denotes the bkot 100 on which CT i peitormed. Dx iaamatr fo performing DCT in the Xais directon. tede horiAotd direenonJ 'Dfy' denotes a marix for performing DC in the Yaxi, direction e the vetial direction. Here x and D(y) may be defined by M follkwing egundon (2) 1r y he vein WV an MI denote ineer I gm% saisfin,, ' codo e xpressed in uon D2 C denotes a ' ro' and an P olumn ofa squa matri CN x ad 1 swhersei V and'denote intenes each sansfying the 'onuon exressed in Eg uanon (3) Db() denotes a V o and an Iolun oaqar namx fi and S 'enf e C tnkwr ejno land ertda, Ies of thes'rmit Dy 10: The vrans t er 910 peoim; 2D OCT o fd Ak MWO t valulang Equation I and the ine taformnet91 p0 erforins 2D inveseCT on the fIrequency domain coeficients genratd by the ransformer 90 by calua'nging' n t owing equnatikn 121 is"" AXKv >'A"~)Y~'F 1 1j"HATO 4 -lo. i 4) 1221 wherein 'P' denotes a flock includig pAd values Cn an teipolation loation, the location X. whih ae obtained by perfornin imerne DCT, Compared to Equation i )a Equain (4) is obt:med b" miruiping both sides of de Nock C bv 'WaY :o l \\V, respectivel, so as to perform ierse oCT on the blocd C. lere, 'Wox denotes a mxatra fot perorn'g ;me DCT m the horontal direction, and W(y denotes perorimmg inverse e DC m the vrticai diremon, 23 A described above, the invene trantormer 910 nes the phnality of basi tntions, the ph ot wMichae shited a to per'orAm 2D in verse DCT 'W' ,nd 'IWy' may be deMed by the flowing equaians ,5) nd): U141 l,<" 1 "Th where ' I and k' denote integers each sntisfyngx the Condition expressed in Equation (5h 'WA)' denotes an P row and a k& column or a square matrix Wix), and Sx denotes the horizontal and vtertical sizes of the square matix 'W(x). cx denotes a horizontal interpol avion location: a.: llustrated in FIG 10, and mayv be a fractional n umber ecg. 1/2. P 4. 3/4, 1/8.. 3/S. 5/8, 7/8, 1 /16. or ... However, the fractional numb er is no t limited thereto, and cix may be a. real number.
16 11210 (20H 1v',. 1 e (1maO{ 2 --- A-------- -- - - [1 '4 wheri F and 'denoeimegers cn sansfying hw condition n presd in E (t $ dn nUn 1 andl su lm of aVg- mr ui JW(y), and S, enoe dh hodontal and verticAizes on he quare mati Wy i denotes a vertical interp ion W01tion as hurttnedi I ad e itractionanunmbe .. 1/2, 14. 3 8/ . 3/5. 5'8, 1, or -owexet the fractional mbei is not limited thereo, and & my be a ral number. [32] (omiparedJ to Equations (2 and (3), the rhase of dhe plurity of bas functions used by the verse manormer 10, ine a plurally of cosine ihne lonS a shifted hn a and 2q respcively, i Equations ) and t. If the invene transformer 910 performs 2D inverse DCT asked on the p uaitn "f cosine functions the phaes of which are shifted, as expieed inEquaai (5) and (n then the pixe! values of the locations X are generated. 11331 FIG. I is a diarro ilustralig n mterpiation region a n) according o an exmplay eNodintm . hen Ih transtormer 910 and the invere transformer 91) f Gk 9 generate pliel on i polatione atior by erforning 2D1 and 2D inverse DCT. respecaively a' reio 1120 thantis lrger than a blockIhat is to begin terpolatede.an interpon regin 111A may h used i gener the prec n of inerjgolation my be lt eyed at the brers of the ierpLati region 1110, aridthus, nq the co values an' i ntepoitio ceaton I m i be hecorrelatiom between pixel 1ausadjacent to nnepltonlaincyb considered for ierpiation. I he tnmage interpolation apparatus 900 of HIG 9 performs 2) DCT on piel values included in the iterpolaton regIon 1 . 10 and then perrms 2D2) inverse DCT on the result of performing the 23D DCI,,hemi the correlan between the pixel ales included in the nearpalaion region I l) and piuMi Naies Sunde the intepolatir retion 1110 i no coidered. 34] flau, the in-age interpolution apparatu; 000 performs interpoaton on the region 110, which is larger thas the imerpoIaton region 11 10 and Ndudes the ierpolaion rnlg1i.n 110 and a region adjace-n io the interpolation region i1110, and uses the piAel valuer i the interpolation region 1 (10 for moan omtpenation. 135 1 iO DCT and 1 D inverse DCT> 361 PIG. e12is a diagranilustawiga interpolaion method according to a exemplairembodimtent Referr inga to FiG. 12,he image Interpolationapp'rams90101 FIC 9generaa pixel alue 1200an nzerpoaon location by iriterpaing between pixel value 1210 and a pixel value 1220 of integer piel units in a spatI V? domrauTiTe pixel vaue 120( ixa pixel vale ofa :lactiomil pma n the imer nation b~eatdon of wich isdeteunedby K he I *iterptione miend accrdingto thea curentexenplary nboadimnnl wi be described helw in detw! with reference to IG13. [137) FIG. 13 is a diagram specificaUy illustrating a iD inerpolation methxi performed by the image interpolanin apparatus 900 of FIG. 9. according to an exemplar em bodimuent Refer ins to FIG 13, a plurality of adjacent pixel values 1310 and 1320 that incde pixel values 1210 and 1220 of integer pixel units, respective, are used to generate a pixel value 1200 of a fracionol pixel unit by interpolating between ne two pixel values 1210 and 1220, In other words, ID OCT is performed on M W V to M pixel values. ie 2M pixei values, Wb inverse DCT is pernned on the result of peroming the. I D DCI. based on a plurality of basis S-unctons the phases of which are shifted, therehy interpolating between a 0N pixel and a !' pixel. FiG. 1 illustrates a case where M=6, but M'is not limited to 6 and may be any positive integer greater than 0. 08I 3 Also, FIGS. 12and 113ilstrate cases whee interpolationisaperforned between PiAl valu adcent in he horizonal direin , butitvould beobvious to thoe ot ordinary sil in the ar that e WD interpolattonmethodsof FIG i1 and 13 ay be used to interpolate between pielS values adjacent the vertical direction or a diagonal. direcic See PIGS 1NAad for more pa a iSl i3Q Te transfoner10 performs 1D DCT on pixel vines ofinteger piXi mits. Th 11 DCT ny Ve petrfmedby caKulatin g the knowing egpationt [14CY 2A / lNo((2f A~x 1 t! 1 ... 447 !421 wherein pp denotes he - 1 " to M pixel value, forexample, the -51 to E pixel values 1310 and 1320 iutrated in FIG. 13, and 'C$ denotes a plrahty of n'e ticients obtained by performing 10 DCT or the pixel values. Here 'k' denotes a positive iteger satisfying the condition expressed in Equation (7), 1143 When the transfonner 910 performs ID DCT on the pixel values 1 10 and 1 320 by calculatig Equation (7), the inverse transfonner 920 performs 1D inverse DCT on frequ- domain coefficients generated by the transformer 910 by calculating the folisrsn n quaion tS [144j' C C 2tc-I<21l&MY S) 1141 vihendn denotesain intepoadon location betwen two pixel values asdescobed aboe with reference oa FG 13. and mayn b one of various fradional nurnbers, eig 1/2. 1/4 3/4 is 3/, 5/W. 7/8. 1/16,. Th thfacsAknal numberS ar ote ain do Ami nd o 18 nay Ybe real n ber. 'Ofl C dnates the pixel vt i 200on th - erpoalon aos enerted by nerrn1ing i invsE DC ComparIed toEupatn Pq the phae f the .oi funion expressed n quaion S Whih s a basis function Used for perthming ID in erse DC7, i determined by the fRraconal number 'f other than an ieger 'and is ihus diffeni fiom the phase of tihe basis function used fur performing ID DU [1461 FIG. 1 is a block diagnua of an image mieilitpaln appatus 1i00 according to an exemplary embody menr. Refening to FIG. 14, the image interpolation apparatus 1400 im ludes a filter selector 1410 and a i nerpoator 4 The image interpolation apparatus 900 of FIG. 9 tnnAfunsm an image and in ersel tansform s the rsulk of :ransiornng baed on plurality of basis functions the phases of which are shi ed., iH o-er, if trailsform and inverse transforrm are permued whbreeer pixel \values are nput 0 tohe image inerpolaten apparatus 900, the atnou n of calculation rexibed is large, thereby decreamn the operator speed of an image processmt sstemr 11471 Tu in\umte interpolation ruy be quickly peoromed i a spatial onun without havingu i ransfonn the spatial domauin to a frequency domain byvcaiculating filter ce ef ficien s f or pertbrining transformi and ierse transform desucribed above and then tteing pixe m values m the spatial domain, which ae to be mput to the image inter poladion apparatUS 1400. by using the calculated fillier'effiIcieln [148 The filter SeleCtOr 1410 receives information regarding an inerpolaion location and selects a filter uo be used for inte; pclation, As desired abuve, the filter is used to transform pnel values based on a plurality No bais functions having diffemnt fit quencies and to inverseNi transform a plurality of coefficients which are obtained through the transfonn, bsed on the pluraity of bass funenons the phases of which are shifted, T he filter coefficients mayi vary according to an interpolati<>n locatton, and the filter is selected accorliing in th- mterpolanon loc ation [149 A descried abme with reLrye to FIG. 9 the pixel values are transformed using the pluaLty of bals funcni ns a ing Jiterent frequencies, and the phases of He piurdits of basis functions having difte em frequencies are shifted according us the in terpolauon locatin so as to petr1r inm erse transfonn. Then, the pixel values on the interpolatiton location ma's hb Miepolated by *ne ,l reason the plurality\ of coefiients by using the plurally o basis fnMtion the phases of which ue shifted., in other words if transfom is perfonued based on pielvaluedinteger pixel units and inverse tranafom is peninned ased onthe plurality of ba ficnnthe phases Mnh aeAshifted acdoring tohe nterp aton latonhen pixel values of east one f onal pixelant may degenerated for various hnterpolation location a ,e filter selctor 1410f FIG 14 preseis a plurality filters foperformig transforn and peform inverse transfonr bsed on different bais uiAms and se s one of 19 the preset fiter based on informationrgarding aninterpolation locatan. 150 The ioterpolhtar 10 pdrnis eiat poiaton by ung the fierA elected b dhe lte selentoi II43 Speedfaialy, iterpskatan is performed byfilterng a p ality of pixel ;aines of im t eger pixl n its basJ v the selected filter As te result ol i miepoltion, a pixel vl uemo on a predeermined interolanlon, me, a pixel value) of a tractiondA pixel unit isare) joined. Referring to FG. 10 if a blck that inludes a pluality o pixel wIn of integer pixel units is filered with a 2D filer. then a pluralit of' pixel valueN on interpolaion locations, eah of W hich is deternned by ' and are gene ae . Referring to HG. 13, i' a irw or cinn including a ptlralty of pixel values of itnw pixel unis is fieied wAh a ID filter, then a plurality of pixel values on intepolath are generated, Interpolation methods performed usin; the 21) fier and the ID filter. respectively, will now be described below wth reference to theaemn'ying dr'awings. [1511 Q Ihihe 11521 it Y Y. -ion trezc'r ft v; u6 s as desnried above in ehtion t Equaod This equation may also be expreed as fo=s: 153] fu (ea (9 154 wherein F x)I' denotes a filler for tr'anstorming a REF~ block in the h1orizontal direction and foir inverse transforming the result of transforming in the horizontal direction by using the plurality of basis functions, the phases of which are shifted, Fly)' denotes a filter for transforming the REF bkok in the vertical direction and for inverse trantsformning the result oft iransfotrmting in the vertical direction by uing the plurality of basi' functions, the phases of which are shifted. For eximnpe, F(x Niay denotes filter for performing DCT on the REF block in the horzonral direction, and performing inverse DCT on the result of perWonning in the horizontal direction by using a pvurahy of cosine functions the phases ot which are shifted F(y) may denote a filter for performing DC' on the REF [dock in the vertical direction, and performng inverse DCT on the result of performing in the vertical diction by using a hrality of cosine functions, the phases of which ate shifted. 155! According to Equations (2)h (3K (5n and the filers Fix) and Fy) nav be defined by the following Equations (10) and (I 1): 51'03 whee n k and 1'dente integerseach satisfying the ndrition expressed in Bquation (lt' Y deotes akP row and a IV olumn o a matrix F, and $K denotes the horizontand vertial sizes of square matrices a)and D Sineethe 20 squareiaices W@ and fls have the same siete horironmalisanered ie theeo P xalso the an Wto WodtsaI w and clnofthe are aaix\sdescred bove in relao to Equaion (5 LinK) denote an nW 5 row and an P oun o he squar matrx I) described above i relation to Equaion 06-C 13] where n and ' denote imecem each satisfy int the condrion exnvNseJ n E~qution 1 '(vV denotes a irow and a i coln of a in tris F'-" ad 8, denotes the horimnm n and vertal ires of sq r marincNes WI y) a D; y) Sir the square matnees why) and M(y; have the same size, the horinal and ertical sizes thef are also thle saeV f' dCnotes an " row aid. an.jo column of the Mquare uutrix Wa y) descr ibed above inuelatier to Equation (5 'kDyY' denotes a he iryw and a a" column of the square maix Dy) decribed. above in relation to Equation Q2). 164 If. iepoiaion is erfornued h increasing bitdepths of the fitters F) and 1y)A t:he preon of fltermn my he improve. Thus, according to an exemplary embodiment, coetiients of the filter F and H!y).are increased bn multiplying them y a prede ternined value, and an inage rnay be interpolated usig these fliers including the IthreeCd coeffi,:ients in this case, Equation ) may be changed as followN: 16. P=(E )xxRF v FM(yfl/.
2 1.2t 1661 wherein F(x'denoteS a filter scaled by multplymg the coefticient of the filter Fix by a scaling factor 'S' and rounding off de esult of mulication to an integer. and '(y 1' denotes a biter obtained by n ultiplying the coefficients of the filter ly ) by and rouindinga off the result of muitipl fiction to an integer. Since interpolation is performed using the scaled lIer, tIe pisel values on the interpolanon locations are caictaite and re then divided by '8 to compensate for the scaling effect, [671 F 2 MG. 15 illustrates 2D interpolat ion filters according to an exemplary embodiment Specifically, FIG. 15 illustrates filter coefficiems scaled accordmg to Equation (2) That is, FIG, 15 illustrates 2D interpolation filters Fix) when a is 1/4, 1/2, and 3/4, wherein the 2D interpolation filters FPx) are generated by multiplying de coefileients of the 2D interpolation filter Fs) by a scalig factor 2 A 2D interpolitri : filter Fy) when '<I 1/4. 1/2, and 3/4, may be used by transposing the filter Fi (168! Refening to FIG. 14. if the fiLer selector I410 selects one of the 2D interpolation filters ot FIG. 15 based on an itlerpolation location, the interpolator 1420 generates pixel values on the mterpolatiorn location by calculating Equation (9) or (12). 169 D ilter 21 [17(q D D C accoding t EquatiOn (I7) may be expressed as the falluwing dtetermnant: 7 C1 D xREF .. (13 1727 wherein 'C' denotes a (2Mxl) matrix for 2M coe fficients described above in elation to Equation (7y and 'R EF denotes a (2Mx 1) matrix for pixel vahves of integer pixel units described above in relation to Equation (7n ., P ,. thoMugh to P. The total number of pixel values used for interpolation, i.e, 2M, denotes the total number of taps of a I) interpolation filter, D denotes a squame matrix for ID DCT. which may be defined as folows i 171 2 4 Ti! ul KIz roi" .414),F:T [176! wherin i' nd ''denoteintegers each satisfying the cadition pressed in Equatin 1) D denotes ak wgw and a V clumn of square nmatrix I)for ID lAQh express in Egqation I 3M.and M' has been descri bed above in relation to Equation (13), [i77% 11 DI using a piuity of basis ftions the phases of which shifted, accuvring fl1qpatont mayuix be expresse s he follwing determinant 191 whein P P is the ame as PO< expressed in Equatiob i and "W(c< dentesa ) nn mat xvr e ID inveDse 11 using a rAity ofbai fnAions the phases of which au shifted WM may be dfined as IMlows: 12! wren , denotes an integer satisfying theedia expressed in Equation( nd W (e 1 dene a I column of te W ( ; ntri described above in reltion to Eq i 1 1 utrpoation flter F@ fr perfming 1D DCIand 1D inver tw uses phr of basi fmons he phaes o which are shifted, based On Equaons {1 an cm I ay be defined as follows [18 -' 187! where and dent integes each sisi W he condition expressed in Equati (17). "F: denoesan :orn of the r nF k.nd 'Two" and'' a7 e the same as W"Au and 'Y expreed in Euation K 3).
22 [I8H The precision of filtering may be improved by ineCasing the bit-depth othe 1D in terpolation filter Ft) similar to a 2D interpolaton filter. An image may be interpolated by increasing the cceffic lents of the 11D interpolation filter F(as) by multiplying them wish -a predetermined value and using the 10) inerpatan fier Fd ) inc ling the increased coefficients (189! For example. inteWolation may be performed by multiplying the ID interpolation filer ) by a scaling fator '21ri i this case, z<a RPtal x r v, expressed in Equation (17) may be changed as fhhows: 1P)1 ) ScalingBits 1 wheen F s) denotes a tier caled by mul playing the enelfciems of the II imer polation filter f0) by the scaling factor '2 and rounding off the result Wi Mui pication to an integer. 'REF denotes a t coluntm of the REF matrix expressed in Equation (17), and '2 ig5 denotes a value added to round off a filtered pixei Sace. A pixel value un an ierpulatoi location a is calculaed by mulplying sthe seated filter FLa) by a maxtax tor pixel y;-lues the result of caklulation is rounded Ol by adding the value 'Q % n ' thereto, and the resulant value is shifted by ai Seeing Bit bit so as to compensme for the scalina effect. (193! Rounding~ off used in the Equations described above s; just an example of a method of quiantizingt filter coefficients. in or der to genendaize a mejhod of quartizing filter co effkients for ease of undemanding, filter coefficients may be modified and optimized asespree nh1i following Equins (9) and C20 (19% wheir Tmi C denote, at cu ofe a (sit " a iC teai N n { qu n r; Jenotes an l cc 'ebnen: of the ilter that . quarnizd, and 'e denote aqy real nunher tha ma beselcte acori t doeme oI cpuilntzaon and my We to example', W2 FQ&) Acordmg to Eau aon' 9): when the l coefficient E) t is a rel number isa elated according to huaton f .) to 1 M tenhe ithi coefficient F/u is chamged to the I coefficient ft) satisYN g Equmi-o ( threby quanting :e h 197 When filter coefficients ar scaled by a predetermined scaling factor, quantization according to Equation (19) may be changed as follows: 9 (p Fi - -) F'c 00 whern 'p denotes a stang factor (hich may en 2 and p ' mil de note a saledfteroefficientAcordingto qu ation 2% pT'Fg)nis euo-erred toF F 10. 6.4 t NP illhtrat ID inerpolationflersacordingietemplar odients. in IGO l6A no I n the scaled lter Adescribed above in a tion to Equation Il ) allustrated acconding t the number of taps and an inte rpolation location, Specifieali FiGS i 6A to l6F illustrate a 4-ap filter, a 6 tap filer, an S-tap filter, a 10tap ilter, a 1-up filter, and a 14-tap filter, respectively. I FIGS 1 6A to 16F a saling factor 'or tuter coefficients is set to 256 ,e. a Sealingllis is set to '8. 202 i FIGS. 16A to 16F the filter cuefficiens include coeffieients fr highieqeency compOnents, whereby the precision of imerpoiation and prediction may be increased but image compression eficiency nay he degraded due to the high-frequency components. However, .mroai ispr rrJt ices mg omrpressionr ci tetency .as described aboe wAt rference to FIG. A' To sove thN problem, the filter Soefficients illistrated in FIGS. 16 A to 16F may be adiusted to inwase image can prissin effiency in this case l20l For examp., an asolu te value of each of the tiler 1cetfficent hnay be reduced, and each filuer coeicient at the nmdpoin of each biter may be mnulapled bty a target weighted value Than weighted values assigned to the other iltercoefficnts For example, trinng to FIG. 16R m the 6-ap filer for gnerating pixel values or a 1/2 mteirplation locatin, the filter coefficients, {1. -43. 160, 160 -4' 11 1 a cdjsed i such a manner that abSoWute valus of ' 11 43 and *60' ray' be teduced and only '160' at the midpoint of the 6-top tiet rn multiplied by a v eighted value. 20 FIGS. I 2A to 171 illustrate oprtmid 1M interpolation fners accord g to exempAry embAdiments. The filters ilus trated i GS. 6A to 16V may ai be adjusted to easily embody the filter by hMmhvar. \hen Fuation (17) or (1) is caulated usig a computer, filter onefficients may be optimized to minimize an arthmetical operation e.g. bit shifting of bi nary numbers and addition, OSi in FIGS. 17A and 178. the amount A calculation needed to perform filtering .1or in terplation of each filter is indicated in both "adding" and "shift' units Each of the filters of FGS 17A to iTM includes cxeficients optimized to minm e the "adding" and abift" units on a cormespondin imterpclation location. 1206 F GS, I A and 171R illustrate a -tap filter and a 12tap filter opimized to interpolate an image with the precision of 1/4 pixel sealed by an offset of bits respectveiy, FIGS. 17. 1D and NE illustrate 8-tap filters optimized to iterpolate an image with the preciion of 1/4 pixel scaled by an ofset of bth The S-tap filters of FIGS. 17 to E areassified accordingto at least oneuf whether filcrcoefficentsaretbe opdtized and a methodaof optiniing filter coefficients. FIGS. I F and 170 iuastrate 24 8-tap f~'ike optimnizedl to interpolate an im-aget with the precisioni of 1/4 pixel scaled by an offset of 6 bis. The fiers of FIGS. O7F and 170 rmy be classified according to a method of optirmizing titter coefficients, A7 FIG. 17H illustrates a 6-tap filer optimized to interpolate an image with the precision of !?8 pixel scaled by an offset of 6 bits. FG. 171 illustrates a 6-tap filter optimed to interpolate an image with the prCcision of 1/8 pixel scaled by an offset of 8 bits. 2081 FCiS, 2 17 and 17K illustate ap fliers optinized to inerpolate an imaee with tiihe precison of 1/8 oixe scaled by an offset of 5 bits. The filters of FIGS. 17J and 171K May be claWfied according to a method of optimzng filter coefficientA FIGS. I7L and 17M ilistte 4- tap filters optimized to interpolate an image with the precision of 1/S pixel scaled by an Oset of 8 bits. The fliers of IS. l''! and 17M may also be classified according to a nthod of optnimi Mg fiter coefficients !209 FIGS. 17N to 17Y illustrate a 4-tap ier, a 6 tap filter, an 8-up filter, a 1-tap fiber, and a 12-tap filter oputrized to mnterpotate am nage with the precsio of 1/8 p ci sced by an Affet of S bits respectely th filters of FIG. 17N to IY are different from the filners of FIGS. 17A to 17M inthat some of the filer coefficients are differeSt, but are the saue a the filters of FIGS, 7 to i M in that aiher coeficlent Ao itepoating a 1/$ nterpolaton location is svnunetricai wih a tilted coefic cat for interpolating a 7/; interpolaen location, a ilter coeFycjent for imerpolating a 2/8 in terp ilarion location is symmetrical w ith a filter eteffentor or interpolating a (S/ inter polation louation, and a. flter coefficisent fi: imterpotating a 3/8 inter pol..tin hxs ion is syrmmeical with a filter coeffiaiem hor iterpolating a ~/S imerpolation location. [2i0i FIGS. 23A to 23E illustrate nethos of performing sealing and roundmg off in relaon to a MD Aerpotation filter, according to exemplary enodinents [211 As descried above iterpolaton intern uses 1I~ and inverse DCT. and the ID terpolation filter thus -neldes filter coeffiets, the absolute values of which are less than ,'. Thus, as described above in relation to Eqation f12) the tiIer coe fieents arc scaled by multiply ing dhem by '2'c n&', are rounded off to Integers\. re spetivel, and are then used for interpolation. 12121 FIG. 23A illustrates filer coefficients scaled by ','ai v when ID interpolaion filters are I -tap fiers Referring to FIk 23A, the filter coefftcIents have been scaled but are not rounded off to integers U131 FIG. 23B lustrates the result of rounding off the scaled flter coeffictems of IG 23A to itegens by funding them off to the tenth decimal point Referrm to FiG. 23B. sone interpolation filters the sum of rounding oft the scaled filter coefficients of which is les than '.256' from. among the ID inerplation filters Specifically, the sum of Af ench of a filter for interpolating pixel values on an i/s in terpolation ikcaon, a filter for :nterpolating pixel values on a /8 imterpoation lOcaton asherforinterpoiat gpixel values o a 5 terpolaon lcatiand a filter6 ltnerpoang p ie ValueSona7/iterpone cen is les than 25N 'That is. the rn M itet oeffiients of a filter saled by an offetN biAhs shold '34' bt a ermr o crs during rounding off of the filter cnetTiejents [21 That the stus or filter coefficients are not the same means that pixel values may vary according to an mnterpolation location. To solve this problem, a normalized filiter may be generated by adjusting Tilter coefficients FG. 23C Rnstrates a anrualizer filer generated by the filer coefficient of the filters illustrated in FIG. 2313 2 A comparison of FIGS. 23B and 23C reveals that the sums of all the filter coef icients are normalized to '256' by adjusting some of the filter coefficiens of the filter for interpohttng the pixel values on the 1/8 interpolation location, the filter for imrer, polling thae ptxel values on th'e 3/8 'nterpolation location, the tilter for tnterpolating the pixel values on the /8 interpnlhrion location, and the filter lbr interpolating pixel vilues on the 7/8 mnterpolaior loatnon. SFIGS. 73f) ad 231 ilustate tan tfitters tha t are seaed, nand the result of no mnal ,Zing the tap liters respectiel, if the Pap filters that ae scaled by 2" are a iiiusrated in IG. 23D. then the result of rounding off filte; coefficents of the tap ler F 231 to ieger value and nonnalumn the result AM funding ot in such a manner that the surs of the filer coeffiients are ^ may be as illustrated in F1. 23F, Referrng o FG 23F, some of the filter coeficients 're difterm from the esu. of eroding off the filter coefficients o the 8tap fihlt illustrated s'1 FAD. 231. This mens that we of the ilter coeffiSients aadjted in such a mater that the sua of all the fivtei ientsare 25$ i As illustrated in FIGS, 23 and 23C. at least one of resulant filter coefficents obtained by at least one of scalmg and rounding off filter coefficients may be different from the result of normalizing the resultant filter efficients, Thus. it would be obvious to those of ordinary skill in the art that a 1 D interpolation filter, at least one trom atmog the filter coefficients of which is changed in a predetermined range of error, e.a. or +-2. rn among the llkers illustrated in FIGS. 1A to 16F or the filters illustrated in I 'IA to 'M should be understood to fall within the scope of exemplary embodiments. {2 is If the filter selector 1410 selects one of the fliers illustrated in FIGS .6A to i6F or FIGS, 'A I ?Y or FIGS, 23A to 23E. based on an interpolation location, then the in terpolator 1420 generates pixel values on the interpolation location by caiculating Equation (17) or (18. The other various factor (such as a direction of inter prediction, a type of loop-filer, a position of pixel in a block) may further be considered for the filr selector 1410 toelect one of the filers A se, ie. a tap size, f a fiker tat is to be select may be determinedby eher thesize ofa blkhat iso be itroaed o 26 adttion o f ierg Vr ierpolaTon Oexample a ge ler imy he send when a blek tht io e inrerpoated is largeard Esmall filteray be seected to minimhire mnmorygaccess wheninterpoiaiion is t be peformed in thevertical direction. Acorfin;; to an exemplary embodiment information regarding filter selection may be additionally encoded. F-o example, if an image was :nterpolated during encoding of the imae, a de din side should kowv the type of a filte sed to mieroh re te imae so as tointerpolate and decode tMe image by using the same filter used darine the enc-.ing of the image. IO this end, information spelling the filter Used to in ierpdate the inge .nay be encoded together vthme inage. However when filter selection is pertormued based (.3n the result or previous encoding aof another block, that is. context nformaon :rgrdhg ilter selection does not need to be additionally [220, i a pixel vaue generated by pertowanng nterpolatim i t than a mmimum pi i value or is greaer than a maxisrom pixel v;due then the pixel vlue is changed to the muinimtun o1 maximum pie.au.Fo xrpe if he generated pixel value is les than a mam pixel value of 0, it is changed to '0' and if the generated pixel value a greateri than natunum pixei value of 255 i i changed to '255 2i0 When interpolation is performed to poecisely perormt imeir prediction during endIng ofan image, informaon -ci an m'erpolain filtr may be encoded 10ocethem with ihe image. in other Words, aibi n rato r el'adiUg the type of the fliter selected by the falter selector 140 may he ecxlod as ar image poameter together with the image. Since a different type ot an interpolaion filter na be selected in coding unit; or m dice or pieume units inforrnaion regarding filer selection ray also be encoded in dhe coding units or the :tice or picture umit together with the image, However, if filter selections n erformed :aording to an rmplicit nlie, the information regarding filter selection may not be encoded together with the image, [22 Methods of perfonnma interpolation by the interpolator 1420 according to exemplary entbodimnemts will now be described, in detail with reference to FiG. IS8A, 18Bh arid 19. l 23 FIGS. IA and ISl illutrate meteods of inepolating pie aues in various dd rectdons by using a di) epoadnther, aegading to exnmpiry embodimnt Referring to E1iS. SA and 8 pixel ales on interpolation loatins inrvarious i og may be generated by aung Djepldnfhrta a pertoin 1D DCT on ID pixel values and petrformi ID inverse DCT on the resultof pertormiag by usang a pluality of basis unctions, the phases f wich are shifted. tY24: Refen 11 g to FI ISAa pixel value Pfct) I1Si\) n an interpolationlation a inithe vertical diection may be generated by interpolatin beween a pixe v alue N 10 and apievalue 1 1804 thatae djacen the verical directnCompaed t the ID i rolanon ntedtod oFIG. 13. terpohotisiperfornedtng vue 10ad 1820 ananaed in the vertial detOn insqtead of te pixe voas 1310 and 132C0 arranged in the horimmnal direction. hbt the imterpolation method described above in relanon to Equatons (13) to (i5: may also he applied to the methoa of IG, h8A 25r Similarly. umpared to the ID interpolation method of FIG. 13, in the method of FiG. I8B interpolation is performed using pixel values 1840 aWd 1850 araned in a diagonal direction instead of the pixel values 1310 and 1320 arranged in the horiztal direction, but a pixel viue PtP) 1830 on an inerpolation location o may be enrted by interpolating between two adjacem pixel values 1832 and 1834 as described above in relation to Equaions(lto 18 [226 FIG. 19A illustras a 2D interpohdon method according to an exemtpiary em bodmhnt. Referring to FI, i piel values 910 o 190 of fraction pixel units may be enereed based on piel values 1900 to 19% of'nteger p ael umts, 1227 Specilally fist, the fiter selector 1410 of the i inerpolarion apparamus 1400 illustrate in F11 14 selects a ID interpolaton tilte to generate pixel alues 1910. 1920. 1 930, and 1 940o fructiotal pisel units that are present between the piel vaues 1.90 to 190(6 of integer piwl uf' as desen ed acove whn reference to FiG. 14. a dif'eren flter maV be seerd recording to an 'iterpolaton loctinr. For example, different filers may be seeded for piel values 191 E 1914. arnd 1916 of a fractional pixel unit, respcively.s as to a upolate'the pixel vue 1910 between twot uper pixel values 1900 amJ 1902. For example, a fiber for generating the pievaluie 1914 of a 1!2 pixel Unit way he different fron a ilte for generting the pixel wues 1912 and 1916 of the same /i piwe unit .Also, the pixel values 1-12 and 1916 of the st:ne 1/ pi el umt may he generated usig di ferent filtes respectively As described abo e with reference to FIG 1 a degieo f shiftiNg of the phaes of hasi factions used to performD inverse II varies ac.onding to an interpolaion location, and thu a filter for performing iterpolation is selected accordmg to an interpolation lo action, 22i Similari the pixw values 12 193( and 191 of different fractional pixel units present between the pixel values .1901 to 19 of integer pixel units may be generated based on a ) interpolation filter selected according to an nterpolaton location. 12291 i the piler seller 1410 selets a filter r oeneraing te pixel values 1910, 10. 1930.ad 1940 of fractional pixel units presenbetween the pixel values 1900 to 1906 of integer pixel unit then the'it pet 42 generates the pixel values 1910, 920. 931) and 141 of fractal pixe nits on inaoaojedprspectively based on the seetedfler, According to an eseplary enbhodiment. since a filter t cnratinga pixelvalue on each of nterpolation locations has beenpreviously alculated, pixel values onll 01inheinterpoliolocations mabg enerated based on pixel vaihaes of iteger pixeI vales [23( In thewrds, siein he pkvlahs 121a0 di 16 o the /4 pel Unit ay be enerand dirctay iorn te pix values 19M( and 921 ofAan iueger pixel un there is no need o fio ilbe the piealue 1914 of a/ pixe nitand hen geneate the pixel vus1912and 1916 of e14 pixel unit baed o the pixel Aes 1900 and 1902 ofiteger pixel units amd the piel value 1914 of the 1/2 pixel unit. Since inae nterpolation doeh not need to be perfoned sequertially aconidng to the ie of a pixel unit image ime-Irpolation may be performed at high speed. ciArding No anoter exnmpary embodiment, an interpolaion method based on an mterpolaion location ac t an exemplary einbodimem. ay be combined with a ilted interpolation method. For example, a pixel value of a it2 pi:,xe unit ad a pixel value l alI pid unit may be generated d-i r pixel values 1900 and 1920 (f the oteger pixel unit by using an interpoatot filer accordmg to an exemplary em bodiment nd a px rl ue of a i/b pxel Unt may be generate tmm The pixel value of the b n pixel :it by using a related hiear interpolaion ilter iterwise, only the pixel xadue J the it/ pixel tini may be genaited directly Vam the pxel va3 e 1900 and 1920 o: the ineger pixel unit u;n the interpolation ulter accodin to an exemipl r m odiment, the piea I l ue of the 1/4 piel tn my be generated loT the pixel vai1e ot the 122 pixel nmit b\ uing the created art inear inerpolation filer,, and the pixel value o the 1 s pixel unit any he generated fm the Pxel vaie of &h 4 pixei ni bn uxing the related at lin ai inicx polation filter., 1a32] I of the pixwl s alNs 1910 I=0 1934 and 1940 of the traaicna pixel units present bet ween the pnxel values 194 to 1906 of the freger pnxei units are Lenemated hy performmg interpolation then the filter ele.:tor 1t10 elects a ID interpolation ilter again for nterpolamAg between the pixe values 1u10 1920 1930 and 1940 of flit fractional pi.el unitsn In this ase, a different filter is selected w conhin to an inter plAtion locati: similar to a manner in whih a filter is selected to interpolate between the pels aLes 19K i to 1906 of the mteger pixel traits 1233 The inter polato i 1420 generates the uel vale 1950 of a fractional nixel unit tore sponding to each of interpolation locations h using the flter selected by the filter seletor 10 l That Is the pixel value 19% ot the factinal pixel units between the pixel values 1910 192010. a nd 1940 of tie fractional pe units s generated .34 F. 191 illustrates a 2) :nterpolanon method using a ID interpo'tion ilter. according to another e emplary embodiment. Refenin to I G 19Bl a pixel vaue on :a D interpolation location may be generated by epeatediy performing interpotion in the verdial and horizontal diarectons usng te D interpolation fEler, 235% pecitialyapixel vaTerp 1 is generated hbinerpouing between a piKel value Rhw 10 and api eval REF, 1964 of integer pixel n te 29 hcwizontai direction, Alsoapixeale Temnpageisgenerated byinteipohating teen a el ae REF; 1962 an a pixel %aiue REF~ , 1966 in aie orintal direcnIhen a pixel value P, ona 2 ineipaun location is geneated by iINe>eAdadu bei ean theI pAe alue 'enp"> and the pixel value.Temt .t 41 the vertia directin. 236! The I) intepolation fiter may be a filer forperforming ID DCT and pefonning ID inerse T DC asedi on a pluraliy of basi fnctions the phaseSof wiha are shifed.lsoIIhe D terpolaion filter may be scaled lter as describedabove in rlation ato EIUMtIon (1f7 Whe interpolain i p rmed i the iontal and vetical directions base n the scaled file interpolation may be perfrnied by cda culating the foslowrIg Etuaon (.21 ~27, Temp ~ lttRFF '4 2 SagagnBst-si p Stageflits 24(l heeinFt ad F~ttf cornespond toF'O expresed in Equation I184 ihyever. ince a vertical tpolation location maey dytteint r ahorzontai rpolation locationadirernt I interpolation filter may he selected accorAng to a inter plaio ovation, [241] When thehozro linetpoiadoadtheveiclintrpiaio epermned first hA shiftng in perfoned according StaRe Bits i aer the horizontal interpolation and secondt hinug is perrnd adigtoStage Hi tsafter the vertical interpolation TotalBits St eBisI + Tageits2) If StgeBits 1 i set toeothe firsbi shiting isot peifomed. p4p TiMo, if a scaling factor for Fa0; is '21' and a scaling factor or FM) is '2" m quation (C2 x then 'TotalBits ='irl 410bir2'. 12431 FIG. 19C illustrates a 21) interpolation method using a IID interplatin filter. according to another exemplary embodiment Referrintg to FiG. 1 9C, a pixel value on a 2.D interpolation location may be generated by repeatedly performing interpolation in the vernal and horizonal directions by usng the IL) interpolation titer. 244i Specifically, a pixei value Temap , is generated by interpolating between a pixel values R EFr 1960 and a pixei value REF, 192 of an integer pixel unit in the vertiala direction. Next. a Temnp , is generated by interpolating between a pix~el value REF, , 194 and a pixe value REFO, k 1%6 in the vertical direction Then, a pixei value P on a 2 interpoianon location is generated bx interpolating between the pixel value Temps, ad the pixel value Temp . , When interpolation is perfonned in 30 the horionna aned alternterpe)uan may be prformed Mekalclainhe felloin g Equadion 22: f24$~Q Tem - =ZIM ,F(c)REF~+n -b 2s"VK Stageitsl [247i 1, 1248 FIG. 20 is a thwchanT ilustrating an image interpolanon method according to an exemplaw embodusent. Referinn to Fl 20, in operation 2010. the iage iter polation apparatus 900 of FiM 9 tansform pixel nhies in a spatial domain by uing a plurality of basis Wtachos having diferen requencies The pixel values may be a plurality of pixel values included in a predeterraned blck or may be rows or columns of paxel values arranged in the horiental o eical diremon, Here. the tratnstbn may be 2D T or ii) DL I desenbed abov i reldtion to the transformer M and Equatons Il (2), t 3 and (7. 50 In OPON0 hi inte r rs the pbses of the pluindiq of baSi fetins ued in perTai 2010. The phases of the plurlit of basis functirs nan be shifted a eurding to a 2D nterpolation kauun determined by and or cording to a 1D mnterpolunon location determined by W0 $ 1 In operation 2030 the image interpoatiov appaatu 00 inverely transforms DT coeffliits whih wete obtained by iransfonning ti-e'ixel vnlues n the sptial domain in operadon 201. by ung the pialit of basis funtctins tephasest whichweresiited in operaion20i t is pie valdes o interpolation cation ae generated by inerely trnsforning the CT coefficentsabained in operation 201.0. 2 T 1 I he nansfoM peaorm-ted in operation 2010iD DC]i then in operation 030 he imageineplaxnmapparatus 900 generate piel values on 2D interpolationloation by perfoming 2D inene DCT on the DCT oefAiens by using a plunaity ofosine vttnionshie phases of which are shifted. $253i f the transform peformed inopertion 2010 isD DCT perfrmed inrws o umns of pixel values, hen in operain2030the iWage interpdation apparatus 900 genemaes pixel ales on ID interp ton locatons by pedorming 1D ierse OCT on the OT coefficients by uing a phrainty of cosine functions the phases ofhich are shifted. , Thle phraiy of basisfunctionshe phase of whi an sifted and inverse tranform based tereon, have been describe above in relation tote invene transfer 92(1 and lqaUons 4), M) (). and 10.
21 F255 . 21 is a f lowchart iiiusfnuing: an image intferptolation methods acorw to anter eIemplauy embodimern. Referrinp to FoG, 21L in operaton 2110, the image i terplaton apparatus 1400 of loK. 14 selects a filer thy pebrmine tatsform and performing nterse' aandforn based on a pioraiy of basis foneions the phaN of which are shifted. according to an interpoition location. For example. a filter fo performing DCT anid perform invere DCI based on a plurality of cosme functions he phases of which are shifted is selected acrling to an oterpolation lation. If pixel values that are to e interpoaUted are iniuded in a predetermined block. then a fler Pit petforinri 20 DCI a d 2D invese DO is selected based on 'r and 'OW if pixel values thca are to be interpolated are mws or coluns of pixel values, then a fitei Ao performing it )O and ID invere DT is selected based on . One of the fiiTe> deuied above with eferecue to F~G 15 FIGS. 16A to 16F. and FIG. i7 may be selzected acorting to an mterpolafion locateon How ever, th~e sie of a filter may be determrrncd !w theJ 'v rious other faciors apart Irom an oIterpolanon locatin a lesrio.d above in iciation to t'e tt't selector' 1410 and with Wafrene' to FIG. 17. In operation 2120. the image inrerpolation apparun 1400 performs interpolation based on te filter selected in operation 21 10. Pixel alues on a 2D interpotaiori location o a pixel value on a 1 0 intepola locatio may be generated by filtera pixel values in a spatial domain by using the filter selected in operaon 21l, Inter polaion performed using filtering ha been described a above in fa non to Uquanons 1 9) fO k(l t. 12571 FIG. 22 is a flowchart iusaing an image iterpolianon method accrtig to another exemplary embodiment Referring to FTG. 2.1 in operation 2210, the image in erpoation apparatus 1400 of FKG 1, selects a dmferent fier for interpolating between pixel values 10, to 19%16 of integer pixel umis. accorling to an nterpolation location. n the crrnt exempawy ernhodimentm pixe vaIues lI 19 1930i ani 1941 of at least one fractional pixel unit maly be generated directly from the pixel values 1900 to 1906 of the intener pixel v alues. 'Ihus the sinage inter polation apparatus 1.40 selects iterpolation fibers correponding to the intepolaton Iocatkins respecti v el. in opentin 2210, 28] In operation 220the iage in xaion apparatus 400 generates te pixelIlues 191120. 1930 and 1940 heat least onefactanal pixel unibierpolatng betweenthe p vale 9(N) to 1906 o the Anteger pel unt, bsed on the different terlected acoing to each finterpolation l tions in operation 2210 [591 in operation 2230.e imageitepolan. appears 4 eects adifferemiter for nterpoating between the pixel vlues 910. 1920, 1930 and 1940 of the at least One tractional pixel unit generated in operations 2220. '>nim to an inter polation location. A different titter for generating the pixel values 190 of another fractional 32 pxel unit iustrated in 9, which ae resent enwen the piel valus 1910T 2093 in 1940oftheat leaston fractitialpixe nt is elected acording po a 3 nerpolt~oiocatdon. D2N, in opera lion 24( theinage h nterpok aipparans400 genera the pxel values 195 another gratin piel unit by interplAtingabe pixe alues10 1920 1930) nd194 of theat least one fractional pixeln i based on he filer seleced in operation 223& [261 While exemplry etbodAnens have been patiukliarsi hown and described aboe. it will be understood by one of ordinary skilln the art Tha vanius Change in form ani detalk may be mad e therein without departing fom the spit and supe of the iventive concept as defined by the oiowo claims and their equivalents Aso a system according to an exemplay embodimnen can be emNodied as computer readable code on a computer readable recordng medium. [2t Br example each of aappaatus for encoding an iage tn appartusfor decoding an inageanimge ecode and anmage decoder acconting to exemplary n7 bodmenta lustrated in H 2, 4, 5, 9, and 14 ma include a bus oupled to uits thereof at least one procesor connected to the busand niemory that is connected to the bus to store a onand or a reeled e r gender acted message and is coupled to the at least one procesor to execute the command 2tY The computer rdable wCording medium red he any data StOrage device That can stomI data to be mead in a computer system,. flxampies of the coaputer readable record nrg medium include rd-y memeyGRM, taudomaees ertemnoryv RM compact disc (CD>)RR, nmgrteue tapes floppy disks and optical data storage devices The computer teadable recendirig medium can also be distributed oaver netw ork-coupled computer systems so that the computer readable code may be stored and exeuted in a dlistributed abinim