WO2024124178A1 - Methods and apparatus for transform training and coding - Google Patents

Abstract

This disclosure is related to video coding and compression. More specifically, this disclosure relates to methods and apparatus for transform training and coding. A method for video decoding is provided. The method includes: determining, by a decoder, a transform matrix for a current block, the transform matrix including a plurality of eigenvectors; obtaining, by the decoder, a modified transform matrix by discarding part of the plurality of eigenvectors; and performing, by the decoder, an inverse transform process on the current block by using the modified transform matrix.

Description

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, se a oncaonsase uon an camsrororovsona caono. reerencens enre.ecemer, ,e enre conenereosncororaeeren ao oresoa c vomeouser ssu,aoree com au vaerrse, o eaec carmoncevces, suc asaeevsons, meaaers, veoamn consoes, smarones, vereaos,eecaon reecreonrnevces,a sreamnevces, ec.e eecroncevcesransm an receve or oecrwnsee cvces, veo a veoaa across a communcaon ommuncae ev newor, anor soreea veoaa on a sorae resoucerc.esu oeoe a somraeeeavncw caac oe communcaon newor anme memor o one or more veo con se,a vnaeros coenore mas ce useo comresse veoaa accorn con sanarsncueersaeeoonommu,nocane oxr soroareo.noers exaomee, veo -ceenrcroueoon con, o.re,ev.an , curex ceeo coeonoennera u.zes,reocvn meose..,ner-recon,nra-recon, on nerenne veo oreeaae avanae o reunanc rae, we avona oar. mnemo czonne armasaoo cnosmor vesse voeuoaaa.no aorma uses aower nm aorsme anssec o,eresenscosureroveorransormrann an con. veoeconncun:e soemrmen emno,m aenesc ooer,e aresenscosurerove a meoor or a veo seuenceeermnn, aecoer,o aw-rarnesuoernmc non-searaeransorm eransorm marxncun aura o eenvecors oann, marexeocro ae cru,r are mnooce, ornee.: .reacnosoerrm, an mnavrexrseransscoarrmnroceasrs o onee curruernaoc o e uesnnvecoers m anerormn, e n a secon asec, some emomens oeresenscooeransorm marx. veoecon,ncun:eermnn, aecoer, aransor surerove a meoor seor a currenoc, accorno aoc sae m marxrom aransorm marx e currenoco a curren vecor oe currenoc convern,eecoer, curren vecor usn anerormn, eecoer, aransormrocess one ura oranso eransorm marx wereneransorm marx se comrses a avn a samrm marces, an eac oeura oransorm marcessraneorocs veo encone n a sa ,ncre u aesau ne:cre,. e seormmen emn,om anen ensc ooer,e arreasnesnormscosurerove a meoorransorm marxncun aura o eenvecors oa marxor a currenoc,eransorm marx scarnar oeura o eennvnec,orseecoer, a moe ecoer, annverseransormrocess one curreno anerormn, e n aour asec, some emomens oceres uesne moeransorm marx. veo encon,ncun:eermnn, a encoer, a nscosurerove a meoor seor a currenoc, accorno ransorm marxrom aransorm marx e currenoco a curr aoc sae oe currenoc convern,e encoer, curren en vecor anerormn, e encoer, aransormrocess one u vecor usneransorm marx wereneransorm marx se comrses a avran a o samransorm marces, an eac oeura oransorm marcessraneorocs veoecone n a sae .e a aseaeu acr ra,e s. uosmnec eumeos onmee onrs m ooeresenscosurerove an aarausor or morerocessors an con rerocessors an a memor coueoe one werene one or moreroceusrseors,o u soonre exnescruuconon os exeecnusaruceons aere on ceon or morerocessors, e meo accornoers ureoerorm n a sx asec, som or secon asec. veo encon.e aa e emomens oeresenscosurerove an aarausor or morerocessors an cornausurnecuoe sso orneen osr mruocreonrso ecxeescsouras aen a meem onoer or co muoeoe one werene one or morerocessors, uon execuon rerocessors, e meo accornoer orour asec. oensrucons are conureoerorm ornee.: .ransor conm au seevre rneaa aseec so,r saome emomens oeresenscosurerove a non- wen execue one or more comeu meeruromcesosror ss,or canuse come ouer-execuaensruconsa,oerorme meo acco ne or more comuerrocessors n an e asrecn, soomee emrso or smeceonns asec. ransor comuer reaae sorae meumor sor on ceomreuseenr-exesccuoasuerensroruvceon a non- wen execue one or more comuerrocessors, sa,oerorme m causee one or more comuerrocessors n a neno as acecco,r sonmeo emeomr orour asec. comuer ens oeresenscosurerove a non-ransoroers-r oera saecoen so arsaee meumor sorn asreamoeecoe e meo accorn comuer-rena aaene so arsaec ec. m, seomuem emoro somrnen as osreeamreseoneseccoosuererove mee a noon- aransoroer orour asec. ccorn eaeescrsoon aere un exearmsooes ona ano aree noor reesorncveen oeraeersecsrenonsc aonsuree.oown seccaoe accomannrawns, wc arencororaen an consue aar os escron,n s,ervuesora exe eaxanmeesr cnocnssesen o wesceosurerseenscosure an,oeer w e veoocsn. accsor aanocce wa sraommemusraemnen aano exnesm oa.re sresseemor encon anecon .s aocaramusran an exemar veno ensccosure. somememenaons oeresenscos oern accorance w somememen.aons aocaramusruaren. an exemar veoecoern accorance w .ss a oeresenscosure. w somememenanonsu osraeonr o aeneraocaram o a veoecoern accorance .s anusr esenscosure. n accorance w somemeameonna oo anosw o-reeuerenscen nons-cseosaurrae.eransorm rocess ornee.: . aroneno. muero vueoo acres ooceraenram sszusranow arames recursvememenaons o es an saesn accorance w some eresensc.osuseres r oesnons,cosuer aen.ransormen accorance w some exames o scosure..se.e or n accorance w some exames oeresen eresensc.s anusraon oe or n accorance w some exames o some exames.osu osre e a. rowcar o exorne non-oca correaonsn accorance w .s anesenuscosure. accorance w some examsesra oone oreseenemscoaseure useorransormoc cusernn neornem.aes, ann accuosrraancoen w or saomnen exaa cu.sernase onrecon moe an e r. s anusraon o an eames oeresenscosure. aove area ar xameaeransorm coecensnee- .eanes a acncoruasnrcaeo wn o s aonm eex eaxmameesa oeerarnesseonscosure. area are reane accora rm coecensne aove . s annce w some exames oeresenscosure. are reane accorance wu ssormaeo enxa om ane esx oameaeransorm coecensnee area . s anusraon o an exameeresaenscosure. aove areas are re eransorm coecensnee an area accorance wan seom, we exeraemeese o ane arove areasncueoe aove area anee .s aowcarusran a meseenoscoro vsuree. exames oeresen oeconn accorance w some exames oe.resse anowsc scco oas srure.usran a meoor veoeconn accorance w some exames oe.resse anowsccoasrure. ure.usran a meoor veo enconn accorance w some ornee.: . exames oe.s aowcarusran a meoor veo enconn accorance w some .resesn ascaosruare. nerace,n accorance w sommemusermaenna aon csom ouenre esennvronsmcoensure c.oue w a user wc areeuesrerancee wn noew acceom maanennearawon ss.ecncemoemowennaones,a eemoesmcens o numerous non-mn secceas are seorn orero a ron, maerreseneeren.u we aare sssn unersanne suec aernaves mae use.o no one o ornar sne ara varous e suec maer r exame, we aareno one o ornar sne ara a veo caareseneeren canememene on manes o eecroncevces w escron, c saomue s oes. eusrreaseenascoseuree,rm asn “rse,” “secon,” anee usene snus oecs, an no useoes accomannrawns are useo unersooaeaa u cre an secc orer or seuence. soue sucae sens wa maenercane uner an arorae conon, orerseses emosoe smoewnnsn oee accroemsenansncosruarwenessc orreescrereen maememenen areoc-seransorms w -reuenc neresenscosure. rmar usneuo,rareer roecso-suzoenr vanseoo,rm es.,., uo a×er nno sn sezeu,e anrce enae, wcsransorm coecens are zeroe ouoreran es. reuenc w ane euao, soa onesoowrmer-ocs w szew ore, oro exame,or an×rans reuenc coecens are reane.or s euao, oneeorm coocum, w aseoc w an aseoce, wen euao, oneo ns oransorm coecens aree.mar, wens useor aareoc,e ro ewnsr oeraoncsorsm u csoee wcenous ar zeeroen. ouen anran vsaourems s moesransorm ss removenransorm s moe.e aso s .n aon,ransorm szen, suca encoerase uors conurae maxransorm szeeenn one nee o ex o coose uo-en or-en ueransormeecon seccmoremoreenaraonns.orm ornee.: . eeconn a scoenmoes u- wcaseen emoen , aueransorm mue seeceransormsrsoeor resua cononer annra coeocs. uses - an -.aem soewseassu.nceon nsewnrouceransorm marces are uanze mnor oer acecruroaeeeane oreoranosnaorm m oarecersannso orm me .a seroxe,ce eeereansnoerrm. me maareces are oeransorme coecens wne- rane, aerorzona an aer verca vaues ae coecens areoave-. ransorm,

nra anner, resecve.e eveor weer s ae or no.enre,s esn aaee a on,o arumae.veas snaeoncae wen one oeeow cononss ae.eoson oeas sencan s cnoaencesn seumasessan.e., on.eas snc orensee zer an coecen oeumasocae recons.owevero,-ou reon.as euaoa ones euao zero,en s aenooncaeeransormeoreorzona an, veenrwco oeras are aona snae an snan manae as sownnae.neaerercanons, resecve.ransormmc s use removnenra-moe anoc-saeeseonrmen seceecsonor an osn moes or useeo crurroenoroczonsan arna voecrc aanraonsornmra co anres.nere enx cc. curr omesosen ra onns,oc oremn ornee.: . mnarxre acrseone,- aserm saamre,ranncsuormn co-reosn are use-. anereore,- a,-eonra,nso-rmo cnores use on -.so, oerransorm core an- on,-o sncun-on -,-on -,-on,- _n_a- an_-o,r u_sea-rmar_raenrs_oarm cores. orzonnaranererca

o reucee comex oare . coec sze- an -, reuencransorm w anense are ze eroeua ouoo.rnee- coe anc-ocs w szew ore, oro reane. ens wnexower-reuenc reon are e reunanscn o sn,axe co resnu,a oe aranoscorm can se coaes w nor sansnorm s moe.o avo __as no euao zero. ae wene eve or s acv oeamc ransorms seo wen enaeornaeore curren.soemc cane s enae wen s ower- croeocs. encoer anne ewe,uenco ene-uansn- asea zaoera n aneewr nea ven ens rseoorrwm ramrarrmraanrsorrmansaorm anuanzaona nure.n ,x non-searaeransorm orx non-searecoer se as sown accornooc sze.or exame,x s aeor smaaoecrsans.eo.r,m ms ae e anx s aeorarerocs.e., mn nw, caon o a non-searaeranso w,e . asoows usnnu as an exame.o arm, wxcsen, useexnnuo,csescre ornee.: . ^_{^ =} ^{^ ^} _^ ^{^ ^" ^ ^ ^} ^_{^ =!} ^s _! ^rs _{^" ^ ^ ^^ ^^!"} ^re _^ ^r _^! ^e _# ^sen _^^! ^e _$ ^a

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^^ = ^^ ∙ ^^ ^^ ^e ver^cas onreaoxna co.e coecens w smaernex weace we smaer scannnnex eeucceen nooc. mucaon aroaconw- o-se arearuaen ncera on-s nn eos ano r-asrem earraaneransormsase onrec marx wou mu sorm soasmemenen a sneass reuceoeeraons.owever,e non-searaeransorm marxmenson neesoe coecens. mennmceze comuaona comex an memor saceo soreeransorm manea oe re,u rceeuc neon- nseona-srearaeransormor meos usen .e x mensona vecoro aaneramnsenosrmonas voe mca an s common euaoor se reuconacor.ence,nsea ox marx,orn a marexreneco smaece, were oows: s an× marx as ransorm m weree rows o

emensona sace.enverse acor osa arxeor, ansxerarnescose osorwarransorm.orx , a reucon marx sze,s reuceoxrec marx, wcs convenonax non-searaeransorm ecoer seoenerae corermar mararnsx.oence,e×nverse marxs use ae marces are aensea ox werm sa cmoeercanesnosrmn s×o-e reons.enxaesnuaaromreexocsn ao-exoc execu conn rura-ono, eoamcx o wocc. ornee.: . romeeo oe r weuc reeasomnaenseone, memor usaeor sorn a marcess reuce resrceoe acae on a coreormcaenncse ouros.en oererrso co reeuccee cnom sue-x s sncan.ence, armar-onrans rou are non- s aows a c orm coecensaveoe zero wen s ae. ence avosoen exroan cnoe ocene scannnnnexe s cnuarrenn oneaes-sncanoson, an cecnor sncan coecens a se sn, wcs neeeor non-searaeer wanosrso-rcmasseann occonseromnss o on m.ucaonserxe resrcse o orx anxocsox anxransorms, resecve.n oseer s cazseess,esseaans-s.norcaonc ssca wno as soaneasoeessan wen s ae,ormesae s now ae o ox anx an ,eroose resrcon rmar-on co n once, anaoeo-ex reon on.s a ermarranesorcmesnss a rreeu zecreo wne sunc cases ae,e numer o oeraons neeeor coecenss remara sme wens.romr eanncsooermrser asreecesvee,.e ruaanezaon o omzeuanzaonasoeone a maxmumor -soron remann coecen ers coecensn scan orer,e ransorm se are u asreeras on a nasreo ernmorceoe zero. ere srea.nescoo ermn m ses an non-searaeransorm marceserneser caner anromenrarecon moeoeransorm se _e_-ene.s u osene oorreee moes __ , __ or seece currenoc =reoenra=,ransorm sesransormor cane cuarren cromaoc.or eacransorm se,e seece non-searae seconar snaen a-sereasmu ornceere srecnrae aeer erxanscorm c soenaceens. nex.enexs

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ssn reasrnc anneracon w oeroos coecen su-rou are no eoe acae on a coecens ouseers sncan coecen. n-sncan, nex coneens oneoson oeas recon moe, an aon,e nexs conex coeuoes noeen onnra nonra ann onersns conex coe.urermore, s aeornrancesoru ner sces, anorouma anroma. auarees enae, sne mna aenxs sronmaae ar aen s unsaeeo sreoaraeu.orner sceeuareessae, a onserna aare r ma anroma. exsn maxmumransorm sze resrceaeranxsmc s nueoe uern ourmesor a cer onx, an nex searc councreaseaa szea s ao an numer oecoeene saes.ereore,e maxmum e nex snawenss resacreceeoorex.noeea s enae w on. marcesa aere u ssee oc secanor mearrcmesaror maerrcceesu max aua s enn usza ean uon.s no evenae scan e uses oe scan marcesor coecens or coecens.ence, moe, croma are no aowe.or sne-reearon rans nno ae. ornee.: . wc areerev ceon o e-cneencra onn.eranneroecnesss c oannee ceosnsne ore ernes, roem, were eac cuser reresens auer as a cusern e acua encon ou oransorm coecenocs rereveromransorm,.e.,,orroceess a,ss aoncaee ‘rcaennsroorm’ c ooe eacce cuserse oma non-searae nene e cassca-means cusernocsne same cuser. erormen awo-saeerave manner n meo,erann oe s n w anna sae: ae, rannorr ea oamcaro oanns,osr ams coecenoc coecerome enconrocess, a ranomrannaane cuser assoscanee. wenee assown-ereaueenc.or× eac co ceusceernase are ae as one e oma non-searaeransormserve e sovn e eromo, e.., snuar vauee envecors o a covarance marx, cuser.n aon, acnomenosonransorm,, w wccs m caeacnusa neo s uesconnarerraannsnaane same assne ase ce orms ae,s enro oers cuser. omzaono arra n eaocn rea aveaaeerarannnnaaaa u,s senece seeeecseranrasormerne usn rae-soronae o eacrannaa, eac cusers ua nsormerne. e uaeaeeromo are uae accorn.ee, anene ‘creannsroorms’sra anwsaorsm assernnees oo c cuusseer . r moe, onnuoc a su-,a sruran ooscorm e rersansuaormoscnsrouceor anner-rece.nsransorm cu_c euao, cu_s_a mae sn coeore.enner-rece w su-ar oe resuao aeoncae weere woe resuaoc or aoeermneeransormcse c ooee.n.enormeear cearse c,asne,e arar onoerm reasounasurerarse wnerre aaveransorm ane oerar oe resua ocs coe en s useor anner-coe, e anocso zesroeon onuo. snaenesream.ere arewo es anwo oson rmaon are .or-or-,e wore ma euaoa os,e asn wcaeonreure ornee.: . orn oe wore, resunn: s or:: s.e: sse a sarnr,ee on se s wee:: sse an asmmercnarree s.numa sames,e:ma: re son a coonnanasem nenosn-ozneros ressauoaw.e. oneere amreen ason o a sn or a. mos moes - anos-on-cereonmeanra anwsaorms u csonre seec-on.se awe onumaransormocsn are assocae w eren coreransorms oosons o - an -ransormsor eac os .ore secca,eorzona an vercaransormsor-osononss sec-e annu-re, res.eocr evxeam.e,ene onoer szone oa ane r veesrucaaranssorrmeaoernan sec,eesranesormonr,o c, anmensoroznosnsa se an as ve-.ereore,e suoc resuaoc.e s no aeoe coe w crca coreransorme o a rans omnener-nra moe. axmourmmramrovemenne maxmumransorm szneso.rem., s aze an zeroranns-oorum oerrannessorm are co eexcens.o sze an maxmumnra coeoc canave a sze ox.e eneo, weree seuences ans seo, oerwse.ne maxmum szes seo or noermramvaer zeron ou oeraon ae onransorm coremacrenrasn.soowrmeavero,nrocess,se arese no, nranasnocrem coecoernnsr oausee reon are normave zeroe-ou. uze wcn aree c uusrerenornra aensn co ner,nor , on an ransormernes are , , anenransorm co arn. onarmarransormsncun , sze annra moenorma e emoe.so ses maeeenen one eren cas on. eren szes are consere, anor eac sze erenransseosrm area cornss areer ceonseeeren.num onenr ora-moenormaon.or eac cass,, or eween, an canaeseen nra canaes are aave seece coecens.e sums c n one sum o asoue vaue oransorm o aowe omare aansewoxeresosoeermneeoa numer sum . canaes: canae: sum= canaes: sum= canaes: ornee.: . casses oeno sea,r ae exou aoa o eren casses are consere, some ooseeren resuan ac sameransorm se.oere areessan unue enresne o, a moeor. anua wr moes, s aon smmer over sae annrarecons consere. moe= – aex we maeoe same cass corresonnoe an vercarans wormer sneaes swxa.eo.weovre er,or eacransormare orer oeorzona recon an axoc w xame,or axoc w moe orzona owe moevercarecon are maeoe same cass. nearesve cro,nveen veorncaa anorzonaransormernes are swae.ore we-ane moese s useor ae m anouesare mwoeeens - us aeno -r.ermansorm seeermnaon.or exame, moe . ar, moes useor moeo moe exenson wareer ca e esnn s exnenee asoows:e numer o ses an moneaereso areen exraenseeorve= ac acnorn=, aoneeoown seornsrexor avennra nsrexs euao.nsrex =re mua:orreoenra, nsrex = –reoenra,o oenra,orreoenran,. , , an , areerreoenran,.reeerenernes,oxx ,xx , annxeon,cae, erne ses, wc are ae are sece: , *, * = resecve.eernemensonss aeoo-eowreuenc reon, wcx,x,x.eorwar s, a we,rmar-ransorm s caeeon--neres .en zeroe ou e coecensa exsne reon oeran are are consecuvecs n en scoo ar canerosmeecen sanuraer.. consss o sxx su-ocs, wc orwar canne orxer..ncse coese nnumoeer onnus c soanmreusosn,,ransorm marxor su-ocs areeneraeromorwar coecenswox coecen scan orer. accorn, wc areaceoown ornee.: . ans coseenooer.eesn seroawen cnoeucre.eorwar marx canex . ens areocaene same manner as w nrare.e m -a -n -ro -m -n -rarecon moesoese sess sownnae. moe - - - - - - - -

snaeenoennsrearnnsorma exeorere asr seuc ma cnorr semaoanrs caacn eens wancen co anrameer an acrossrames, o .nsrane ase roe o ara ormance means more ece nsormasneneo comace ener , ars, a re. eeorewnccearancu wreces ccressea.ore curren conocncae reconoc oe conocso onsasnse o.een r,ecosns reruecreee-uemae ^^_% ane mos smaraces overe reconsruce reons.na, onnec-emaecns useo searcor eseocs anreconocs cacu sona ase on e coe ae.e conocs unnown aeecoeror useocuone oe s smearacr cnan oaeocs.ereconoc ane reconsruceemae are or varousoc szes smarocsnsea o usne ornaoc.soos use snown×a,×,× an×. e ener comacon eceanrcu.nen- seoarvcenransorm se omaransormnerms o , …,, wc are mos smaroe reerence ovaerce reconsruce reons,ocs ^^ _^^, =, anncaese vecormenson wcserans are oane.ere, ^^ _^^ = ^^_'", ^^_'#, … , ^^_' conoc,s.ereconromoseocorsms soucr saczee. aonr an o exaanmee, ror× ocs as ^^ _^^, =,, …,

were ^^ _^^ = ^^ _^^ – ^^.e,ese resuaocs are usees asuae ornee.: . rrearnesnen seames = w ^^, z^ ^e^ro,… m, ^e^an, worces an×er mvaaron.eserann sames cane ven: = ^^ ^^ ^{^^}^ _{^^ ^^ ^^} x.onsere covarance marx as e ee wereemenson os covarance marxs×. ases areen oe cnavnecoarse o nums ceorvara ansce marsx. enooru nauroarmeaeovoeoe croonrmens, wene seecon comexore eenvaueecomosons ^^ ^^ ance.e comuaon comexs ^^ , wcs acceae.or aare.ocor,×e cooc w en,e or××ococ w en ,em mex we ver . or e comex we mes sowerana n, coensnernnoeras,e anase a coornramewor. emenson os×.oweve ^{^^}ms useo maeeareoc sze ease. e ^{. ^^} r, ^^ ^^as a mucowerm ^{^^}enson as× ^^.e cacuae e e eeennvveeccoorrs ^ m^^aorx w = ^^e ^ ^^^, wenoces saseaeo enauaon as ^^ ^^ ^^ = ^^ ^^, were ^^ncaes aona eemens.e’s mu o ses ou marx w e ee ^{^^}nvauesene racesos euaon an oan ^{^^} aon ^^oe ^^ ^^ ^^ ^^ = ^^ ^^ ^^. e coumn vecors o ^^ ^^ are ^^ ^^ ^^ ^^ = o ^ ^^^ ^^ ^{^} ^^{^}^ - eneaona eemens o

e ^^ = ^^ w er corresonn eenvaues oane musncanes ^^ we eenev eecoenrsve ocore ^^ ^^ s ^^ wcm.e anrseo onaa covarance marx ^^ ^{^^} ^^ cane covarance marx ^^ ^{^^} ^^ .emenso neromeowmensona eenvecors o ^^ ^^ ^{^^}ave ze ns o ^^ an ^^ areo×. e oer− − eenve ro eenvecors.e can usecm oroonazaono ese muca corsoe× eenvecor marx.o reucee comexor marx remannon−, oner caanns uosrem ceoe ocaennes as e zereonsv.ecosrs wo neoro armenuransorm,eavne ersroecons can cover mos oe sn aeeerormance snce en correae we con a ener weeases areranerom sames oave aeaesacre smemeonce. aeoceve one conocne. on×, oemaeveo conens,eroose sceme suorseroose omzaons u×s,× an× conocs. e encoer se, rae-soronransorm .eoeermneeransorm moe amone ane aave mue ornee.: . areusraeenerenec aec wcom noanwne mraawenns.eaneo soeocwcnmeaemeenaescorns, eoxnames o wc mn secceas are seorn orero asssn uner , numerous non- eren.u we aareno one o ornar snesanne suec maerresene use wouearnrome scoe o cams a ara varous aernaves mae ese secceas.or exame, we aanre suec maer maeracce wou suec maerreseneeren canememeno one o ornar sne arae a veo caaes ene on manes o eecroncevces w escron, c saomus oe. eusrreaseenaeerms “rs,” “secon,” anee usene snus oecs, an no u scosure, ane accomannrawns are useo unersooaeaa useoescre an secc orer or seuence. soue sucae emomensens wa maenercane uner an arorae conon, orerseseso s oeresenscosureescreeren maememenen .ses sownne accomannrawns orescreneresenscosure. econ veooc asnocaramusran an exemar ssem or encon an scosure.s s araen accorance w somememenaons oeresen encoes v ownn.,e ssemncues a sourceevce aeneraes an aneeseoaaoeecoe a aaerme aesnaonevce.e sourceevce cou serversn,a seorvnere cvomceuer ms,ae csomor osrea ano o c aom wuee vrsa,reae o c eomecroncevces,ncunooxes, aeevsons, cameras,saev uers, smarones, se- consoes, veo sreamnevce, oree ces, a meaaers, veoamn eesnaonevce are e .n somememenaons,e sourceevce an n somememenauonse, wees wnraeess communcaon caaes.oeecoe va an.en ma coomnrevce ma recevee encoe veoaa evce caae o movne encoe veoase ane o communcaon meum or evce.n one exame,en ma coarome sourceevceoeesnaon sourceevceoransme enc mrse a communcaon meumo enaee me.e encoe veoaa maeo me veoaarecoeesnaonevcen rea a wreess commu ouae accorno a communcaon sanar, suc as communcaon menucma mona coromocrose, a ann wrraenessms ore wreo coemmuenscnaaoonn meevucme, suc. ase a ornee.: . maeoumre mauencormar s oec arumac ore o-neas oer m noerewor,s scuacra anss amssonnes.e communcaon newor, or aoa newor suc asener oca area newor, a we-area rouers, swces,ase saons, o ne.e communcaon meum mancue communcaonrome sourceevr an oer eumena mae useuoacae ouunernac seome oo aer somraeemeenc vae ceons.,o uee s eenecs uon eneao vn ,eeoevc enae coa m. ea veransmerom an evce mae accesse eesnaonevce va annuneeroacaane sorae evce mancue an o a vare osrue oroca accesseaa se.e sorae as aarrve,u-rascs, aersaess s,omacorasce meeaa- suc emores- s,as memor, voae or non-voae memor, or an oer sn a sorae meaor sorne encoe veoaa.n aurer exame,e sorauae ma corresono ae server or anoernermeae soraeevc eevce encoe veoaaenerae e sourceevce.eesnaonevceea m maaoe sore veoaarome soraeevce va sreamn orownoan. accesse e ane o comuer caae o sorne encoe veoaa anransee server ma veoaaoeesnaonevce.xemare serversncue a wemne encoe wese, aeranserrooco server,eworaceo servere..,or aocasrve.eesnaonevce ma accesse encoe vrae evces, or a aa connecon,ncun a wreess cannee.., ar eoaarou an sanar wre connecone.., auscrerne , caeee mssoe - connecon, a oas suaeor accessn encoe veoaa sor em, ec., or a comnaon o e encoe veoaarome soraeevc e on ae server.eransmsson oransmsson, or a comnaon e mae a sreamnransmsson, aownoa ane ous suownnernace.., o eeo v soeu. orc seouercvece mancnucuese a a v soeuorc seo suurcce as, a a v veoeo encoer evce, e.., a veo camera, a veo arcve conannrevous caurnneraceo receve veorom a veo conenrov caure veo, a veoeen eneran comuerracsaa ase s er, anor a comuerracs ssemor exame,e veo so ource veo, or a comnaon o suc sources.s one evce aneesnaurcoenevsce a v meoa caomremra ca om aer saecuornes s ourrv veeaonceo snesse.m,oweev seor,urcee ornee.: . amn memaenea aonseescor wereenss aenreosre wnr ae acaocna moae acaeo veo connenera, encoer.ee ca encuoree, vre-ecoauarae m, oar coemraunsemr-eennes.rareec veoo mae encoe e veo e ouunerace oe sourceevce.e enc eesnaonevce va e sore onoe soraeevceo oe veoaa ma asoor aernave orecoern. an raer access eesnaonevce or oerevces,ransm oraac.e ouunerace maurerncue a moem anor a saevceee.sneanonuenvecreacenc muaesnceuneu a renceervearce an,or a a v meooeecoer, an a encoe veoaa overen.e encoe veoaa co m an recevee rove one soraeevce, mancue a v mmuncae overen, or encoeror use e veoe are o snax eemensenerae e veo encue wne en coernecone veoaa.uc snax eemens ma sorae me coe veoaaransme on a communcaon meum, sore on a nu smo,m oer smoreem oenn aaoens s,erver. wc cane anneraesaeesnaonevce mancueesaevce, communcae we evce an an exernasaevceas conureo aao a user, an esnaonevce.esaevce saseecoe veo sa , a masam comrse an o a vare osaevces suc as aursa e osaevasa, anranc mnoe sa, or anoer ornusre vecoe. encoer ane veoecoer ma oerae accornororear sanars sanars, suc as , , -,ar, , or exensons o suc enc . soue unersooaeresen acaons nomeo a secc veosoenenraec coonnem saaneara ane m vaeoe a encoceareo o oeer s vourecoe enecvocen mecon sanars. encoe veoaa accorno an oese curren or ae conureo enera conemaeae veoecoer uure sanars.mar,s asooecoe veoaa accorno an oeesnaonevce mae conure e veo encoer an oe veseeo cuerren oruure sanars. vare o suae encoer anorecoer crcurc,o suecr as e oance o mra moree mmcreomreonceesso arss, an oa a ornee.: . anearrraocsessorss,ss,creeocaco,n sowecarec,narewraaree,rmrcwuasre or ans c,omenaroonrammae enmemenearan soware, an eecroncevce ma sorensrusereo. sowaren a suae, non-ransor comuer-reaae meum an consore arware usn one or morerocessorsoeror execueensruconsn scoseneresenscosure.ac oe veome veo enconecon oeraonsncuen one or more encoers orecoer encoer ane veoecoer mae comne encoerecoer n a resse,c eveere o wc maenerae asar o a exame,en v someoe somurceemen,aeo vns,e ao eenacso aerar o ov r c cooc mme. oonneennssn ocuee sonurcee vevecoe or asescreeow w reerenceo., ane ouunerace anor aeas encoer o comonens oeesnaonevceor exame,enunerace,e veoe acar or comonensncuene veoecoer asescreeow w reerenceoo.er anesaevce ma oeraen a cou co , so mun servce newor wc marove ervwcaere,aaao,rm ors,n anrasorrucnurraes arusc auree,rv succe asaao.wnar seom ase amervemceenaaa,aorm as a comonensne sourceevce anoreesnaonevc ons, one or more cou comun servce newor maeroven one or meo wc are noncuene more cenevces ma communcae w serve re cenevces, ane one or neworrou a wreess communcao r comuersne cou comun servce newor, a sor-rane wre n neworor exame, a ceuar communcaon communcaoness communcaon newor, or aoa navaon saee ssem commu newor or a wre communcaon newore.., aoca area newor emomen, anecaaon newor or aowerne communcaon newor.n an servcesr s aar o oeraonsescreeren maememene as cou-ase oe coove one or more server comuers wc arememene e aeas aar esnaomnonens oe sourceevce anore aeas aar oe comonens oe es evcene cou comun servce newor an one or more oer oeraons ec croue coermenu mna seervmcee nmeewnoer mae oen ae orrv maoere c couen, aevucesc. cno suom,e orm aemrena coouns, eerms suc as “cou,” “cou comun,” “cou-ase” ec.eren mae u.nercanea as arorae wouearnrome scoe oeresenscosures.e ornee.: . s coomuune un seervrscoeo newaorereesscernesc aoosvuere.sn nsoeam,eeorenmemenene coumemenen an oere o comun envron esenscosure ma asoe uure. mens currennown oreveoene w somem.emsen aaooncseasramusran an exemar veo encoern accorance erormnra annerrecreneresen acaon.e veo encoer ma con rees on s cve con o veoocs wn veorames.nrarecve ven aarecono reuce or remove saa reunancn veoaa wn a remov veeemoorarame re oruncaunrce.nn vereroeacav we conn a raeceens o vneeomraorarecono reuce or seuence. soue noeaeerm “rame” m mes orcures o a veo or “cure”nee o veo con. ae use as snonmsoreerm “mae” rocessn uns sow, annecoe.,ceu vreeou enecroernc,u a seusm am vereo,a aar manesmoorrmro,c aesrecon , auanzaon un, an an enro encon un.ereconrocessn un urerncues a moon esmaon un, a moon comensaon un sn un annrareconrocessn un, an annraoco , a unaro.n un,memenaons,e veo encoer asoncues annverseua n someransormrocessn un, an a summer nzaon un, annverse , suc as aeocner, mae or veooc reconsrucon.nn-ooer ocounareso removeocnosoneeweene summer ane oer suc asam ess aracsrom reconsruce veo.noern-ooer, ere anaorveavseen-oo eerr,rossomonenameavese eocneroer an ouu oe summer. s,o mua aesoe usen aonoeecnue,eresen acaons nomeoe emomensuessraeaore e acaon mae aeo a suaon were an osescreeren, annsea, comonen, a croma comonen an ar cro seeceor an o aumauma comonen,e croma com ma comonen accorno an oer oe comonenase one se onen aner croma comoneno mo sa an menoneer ece ose.urer, sou asoeusraea ars comonen croma comoennen m,a a seec aonn o comeonuemna m coemnoonneen,ereen m caromea a cnom ooneern o aneeumar ornee.: . c moemnoonneen,ereen ma croem aa re cmomanonnen one an oeeumra c croma comonen, an ar comonen er croma comonen.n some e omonen,e croma comonen an veooc maerecrovexamees,en-ooers mae ome, aneecoeaeeorm o axe orrorammaea srummeroe .e veo encoer ma eusraexe orroramma ware un or maeve amon one or more o e veo encoe vereo.aea v meemoore aan marawa e v sroer u een o vs. aeao meamaoroe e mncaoee oanee c,omonens o rome veo source as sownn..e s auera sores ror exame, aaor exame, reerencerames orcuresor u eerence veo encoere..,nnra ornerrecve co sen encon veoaa e veo maeorme an o a vare o mnem moroesev.ce veoaa memor ane aa memor mae on-c w oer como es.n varous exames,e veo reaveoo nens oe veo encoer, or o-c sse s coowmnonens. reconrocessn unn., aer recevne veoaa,earon un wne asoncuearonn a veoaronse veoaano veoocs.saronn maareron rameno sces,esor exame, ses o veoocs, or oer srucur nss accornoreene sn srucures suc as aua-ree menseon aasso arcraae or w marxe o v sameoeasa w. seam veeo varaumese.s s oarm maenee re ararrae ma as awo- reerreo as axe or ae. numer o samesnorzona an ve asoe oe arra orcureene a sze anor a resou rcareconsor axes eveno mue veoocs,or examon oe veorame.e veorame ma sames we sa vmeoe vaouces a, aanous or o ma smaee rerare, me us ens anson awao-armeonnsnon.a arra or marx o samesnorzona an vercareconsor ne veorame. numer o oc.e veooc maurerearon axes oe veoocene a sze oe veo wc maorm aanocs,or e eno one or moreocarons or su-ocs aronn orre-ree arxamone,erave usn aronn,nar-ree aeerm “oc” or “veooc” as n or an comnaonereo. soue noe suare or non- suareoron, o arame u oser aecreunre m.ae re aeroerncoen,,onr exaramcuea,r ao recanu anar ornee.: . re,coenonc or v ore aoroancso mrmane or corr aensoonr moa aeo orn correrseeonno a corre,s aon,n a roacns,o er.m. aocon arneeoroc , aonoc , areconoc or a con moese, surce acs oonne oroco aes as s unu- ra unoc. on mraar seeeccv oen ceo on a muoraes o or onoesserecvenerrecve con moes,ore curren veoocase on erro o aura o aneeve osoron.ereconrocessn un m r resuse.., con raenerrecon coeocoe summeroenerae aarovee resunnra oro reconsruce encoeocor use asar resuaoc anoe summer rocessn un asoroves o a reerencerame suseuen.erecon aronno snax eemens, suc as moon vecors,nra-moencaors, nrm oraon, an oer suc snaxnormaon,oe enro encon un. oc, ero seec an aroraenrarecve con moeore curren veonraenrareconrocessn un wnereconrocessn un maerorm samerreamce avse ceo cunrre oneoc curroene v coeeooocro rveaev seaoa onere orc moorne. nee moorono ecssne un ane moon comensaon un wnereconrocessn un eromaon recve con oe curren veooc reaveo one or morerecv rmner more reerenceramesoroveemorarecon.e v eocsn one or conasses, e..,o seec an arorae c eo encoer maerorm mue n somememenaons,e mooonn es mmoaeoor eacoc o veoaa. moeor a curren veorameeneran a n un eermnesenerrecon o a veooc wne curren veor moon vecor, wcncaesesacemen veorame, accorno ar ame reaveo arecveoc wn a reerence esmaon,ero eeermneaern wn a seuence o veorames.oon vecors, wcrme e moon esmaon un,serocess oeneran moon curren veo m e raos mom ena o ve rec m coo ur,o reon rre eo axra v vme eoeo, a mo racs en. cvcaeesacemen o a veooc wn a currenocen coe wne cur eoc wn a reerencerame reaveoe veoramesne seuence renrame.ereeermneaern maesnae e..,oc vecors,ornra as coramnesn o ar marnanmeres s.mearnoraee unermn maaon oee mrmonoen v veeccoorrss, ornee.: .oeeer mmonoen eesmoacon un ornerrecon, or ma uzee moon esmaon un reerenceocre oc a rvee vec ereoo ncr. core veooc mae or ma corresono aoc or a coenerms oxeeerrameaseeme as cose macne veoocoe ,um ouareereennce, wc maeeermne um osoueerence e veo enc ce , or oererence mercs.n somememenaons, soreneoer. moar e cxaamcuae,e veau veseoo ern scuo-enree mraxenerosoaoens va oue reerencerames xeosons, one-e xeosons, or oerraco s o one-uarer rame.ereore,e moon esmaon naxeosons oe reerence xeosons anracon un maerorm a moon searc reaveoeu recson. axeosons an ouu a moon vecor wraconaxe recon coee moramone es cmoamoanr unne coascuona oes ae m vooeno veoccoroor ae voseoono oc anre ancnver oc o a reerencerame seecerom ars reerenceramess e ramess, eac o wcenes one or more or a secon reerence e moon esmaon un sense cac reerencerames sorene . anenoe enro enc uae moon vecoroe moon comensaon un ecn oreoneoran cnomeensrao eocn,n vee urnoorcme. ase one meo moon coonm veecnosraoen un, manvove esmaon un.on recevne moon vecorore c ermne e moon comensaon un maocae arecveoco urren veooc,e moon e reerenceramess, rereve wce moon vecoronsn one o ocoe summer.e summerecveocrome , anorwarerecve vaues suracnxe er enorms a resua veooc oxeerence un romexe va vaues oerecveocrove e moon comensaon exeueeresn oce vea cuuersrenor vmneoeocen coe. croma comonenerences or resua veooc mancueuma or snax eemens as o.e moon comensaon un ma asoeneraeneconesocae we veoocs o a veorameor use e veoecoer snax eemens veeonnocse o moeo vne voecroarm ue.seeo sneanx eemeensr meacnvceueo,co,r a enxamaes, ornee.: . mnocoann esmeaoren ucnon m anoe,e or m aono on ceorm sennsaaxonno urnmaon maescreeeren.oeae usrae searae nerae,u are n somemor conceuauroses. recveocsn a memenaons,enra un maenerae vecors anec esmaon un anner smaroaescre aoven connecon w e moon e samerame as ane moon comensaon un,u werecveocsenn verecocrso ans o mooeseo uoe se m cuorre oon ennc voecoc eo ars c.e urnn rena croce ouca ar.,n ne w sonmreae ve exam ucnorse es, mn ea re neeerrremnoe a ansnorca- encoe a currenoc usn varousnra-recon moes, e ra un ma anesererormancerou rae ..,urn searae enconasses, amone varousesenr -soron anass.ex,enra un ma seec, renerae annra-moencaa-orre acccoronn mo.es,o arn e axamroer,aeennrara-re ucnon mo maeo ca usceu aane aane- sesecoroenn varau-esre uscno an r maoe-esaovrnon aneaesss raoer-se varousesenra-recon moes, moes ase aroraenra-reco oron caracerscs amoneese eween an eanec-oesoroonc a anna asns orenne m nraaoeo , unee u nces orem. eneso acn amaou wn osoronor error encoeoc, as we as arae.e., a numer os as encoeoroucee un ma cacuae raosromesoro useoroucee encoeoc.nra eermne wcnra-recon m ns an raesore varous encoeocso n oer exames,enorae ex unsees rae-soron vaueoreoc. moon comensaon un,n woe orn ma usee moon esmaon un ane accornoememenao ar,oerorm sucunconsornra recon oc mae ao nsescreeren.n eer case,ornraoc co, arecve erence, wc mcaseeme as cose macneocoe coe,nerms oxe oerecveoaceeermne ,, or oererence mercs, anencaon eere marenccue cacuaon o vauesor su-neerxeosons. erenrame veocsrome samerame accornonrarecon, or a suracn acxceor vanueso oneerrreeccovne,oec vreoom enceoexre v maaueso orm ae r ceusrreuna v veeoooocc ornee.: . veneo cooce m,aormncnuexeoumeraen acne v caruoems.a coemoxneeerence vauesormne resua aernaveoeenrnaerr-erecocnonroceerssonrme un mea monn orna- ere serencce as. curren veooc, as an comensaon un, orenraoc coreconermaon un ane moon escre aove.narcuar,enrareconro orme enra un, as recon moeo useo encoe a cur cessn un maeermne annra ma encoe a curren renoc.oo so,enrareconrocessn un asses, anenrareocc uosnnro vcaerssouns unnrareocr ao mno moe sees,e ec.o.,n uunrn,n searae encon ma seec an aroraenrarecon moeo userom some examesnrareconrocessn un maroven eesenrarecon moes.e recon moeoreocoe enro e ormaonncave oe seecenra- encoeenormaonncane seencon un.e enro encon un ma erereconrocessnc uennra-ereercon moenesream. veooc va eernerrecon ornr mneserecveocore curren oc suracnerecveo arecon,e summerorms a resua veo e resuaoc maenc crome curren veooc.e resua veoaan un.eransormrocessune unn one orarn msoorremsse an ressuaro vveeooaaenroan ressorumarroacessn coecens usn aransorm, suc as ascreeosneran nsorm smarranso sorm or a conceua uanzaon uerm nr.ans.ormeuraoncessn un ma sene resunransorm coecensoe e rae.e zaon un uanzeseransorm coecensourer reuce e uanzaonrocess ma aso reuceee assocae w some or a o ara coecens.eeree ouanzaon mae moe ausn auanzaonncumener.en suoamnez eexamranesso,rme coeuanczeanso.n unernav mea,een enreororm a scan o a marx erorme s encon un maransorm coeocaon cw. ennsnuaon a vzaeoon,sere eanro encon un enro encoeseuanze ,onexavenarmr usmne,c e..,oonnexave,araeenon aavenarrmecon ,roa nervaaronnnax-nasreo conex- ornee.: . coernans omr aneooer een vroeo eneccooenr me as sooowonn orec.n, oru aer.ce encoesream maen sownn.oraerransmssono or rerev vene soraeevce as encon un ma aso enro encoee moona vee veoecoer.e enro e curren veo cors ane oer snax eemensor erameen coe. nv nverseuanzaon un anenverseransormrocessn un a oecrsenueanxzeaoonm aannnovreresneerraannso arm reaeroenn,c reesoeccvoe,o reconsruce resua veo noe aove,e moon comensaon un m rrecon o oer veoocs.s ocrom one or more reer aenerae a moon comensaerecve comensaon un ma asoe anceocs oerames sorene .e moon cacuae su-neerxe one or moreneroaonersoerecveoco e summer va auessore us reeconn msorucone es rmaon. recveocrouce e moon comensesuaocoe moon comensae soraene .e reerenceoc aon un orouce a reerenceocor esmaon un ane moon comen msaaonene use enra un,e moon anoer veoocn a suseuen veora un as arecveoconerrec w .s aocaramusrmaen. an exemar veoecoern accorance mem soorme,m ane emnernoaoencso one unresen, a arecacoonn.roec vesseno uencoer, annncvuees a veoaa un, annverseransormrocessn un, a summ rseuanzaon rocessn un urerncues a moon comensaer, an a .erecon annra un.e veoecoer maeroron un, annrarecon un, anoe enconrocessescre ao m aeconrocessenera recroca ..or exame,e moo ve w resecoe veo encoern connecon w moon vecors r n comensaon un maeneraereconaaase on encoerane ureceverome enroecon un, weenra-recon un ma e neconaaase onnra-recon moencaors receverome enro n som..s aocaramusran aenera veoecoeror an .memenaons oe exames, a un oe veoecoer maeaseoerorme resenscosure maeeresevn ae amcaonon o.nes oor, mno sreom oe exea umnses o,eem veeomeencaooerns o.oer ornee.: . e oxranm coem,neanorna w u oner u mnas oeroer vmeoemecoemerena, souns oeresen acaon, aone, ,enrarecon un, ane enroeco c ase moon comensaon un ecoer ma noncueenra un anenu un.n some exames,e veo erorme oer comonens oereconrocnecona onra un mae comensaon ssn un, suc ase moonoeecoee un veeo. oaear c moemmor ma sore veoaa, suc as an encoe veosream, aa memor mae oaonens oe veoecoer.e veoaa sorene veo source, suc as a camera, van wer,eor o erx wamreees,sr noemwore s coormameunecvaceon o,r vom aoca veo accessn scaaa sorae meae.., aasrve orars eoaa, or mancue aoecureuer a sores encoe veo.ae veoaa memor sream.e oe veoecoer sores reeren arom an encoe veo veoaa e veoecoere..,nnra orne ce veoaaor usenecon memor ane maeorme anr ore a vcarvee con moes.e veoaa namc ranom access memor ,ncunn o memorevces, suc as ressve ,essve , orc oroenrouses o memore,vcaneo- usraveurose,e veoaa memor ane areece aswoess.nor comonens oe veoecoern. c ar or.u we aareno one sene or seaarae veoaa memor ane maerove e same memorevce oer comeo mneemnosr oeev vcese.one scoomeer exam, ore os,-ce v reeaovaeao memosoer com monaee on-c w a reresenusr vneoeoeccos on anr eoncceoss,ee v veoeoraemceo aenr ass roecceves an encoen vs.eosream ecoer ma recevee snax eemens a ae snax eemens.e veo e enroecon un oe veoece veorameeve anore veooceve. uanze coecens, moon oer enroecoesesreamoenerae eemens. vecors ornra-recon moencaors, an oer snaxncaors ane e onreor sneacxo enem uen enorwarse moon vecors ornra-recon moe ene nsoereconrocessn un. recveocsn o vereoreams oesr caomees, ase annrnararerecconv uen coe oeramree ocroornnrorace csosne ornee.: . u snnae mnaraerneeraceonre moceo anna reaeroernc ae vaeaoroomc oe curren veoramease on a rame. revousecoeocs oe curren moon comenesnaoen v uneor oames coe as anner-recve coe.e., orrame,e ocsor a veooc oe curerenre vceoonrarmoceessasne u onne mrooounce vse ocnoer osr a mnor oeerre scnve eemens receverome enroecon un.ac oerecveocs max roucerom a reerencerame wn one oe reerencerames ae ma consruce reerenceramess,s ans, u s.e veoecoer ase on reerencerames sore sneau consruconecnues escreenre sonm,ee exanmraes, wn unenee o veeo. reoccosn coroeces ascncor unnoroeuncerasre mcove ocsore curren veoocase onoc vecors an oer snax ee e e enroecon un.erecveocs mae wn a remens receverom samecure ase curren veoocene e ve consruce reon oenormaonoer m a vooeno coomce onsaeon cu urnren v aneoorrameenor eancoe arsn urn. e moeonermnesrecon snax eemens, anen usesereconnormao vecors an oer curren veoocenecoe.or exa norouceerecveocsore e receve snax eem me,e moon comensaon un uses some oo coe veoocs oeen vsoeoeraemrme,n aen anreercreonc moonerame.e.,nerae. or.,ne orrre, cocnon usenormaonor one or more oe reerenceramessor srucon recve encoe veooc oera erame, moon vecorsor eacner veooc oerame me,nerrecon sausor eacnerrecve coe rame. , an oernormaonoecoee veoocsne curren veo a,oeermmneara,ee cnurrraen v uneooc ma wa usser seomcee o usen receevnera sna mxo eee,m ceonns, e.., anormaon o wc veoocs oerame are wne reconsru srucon sorene ,oc vecorsor eacnra rece veoce reon an soue recon sausor eacnra rece veooc oeraoc oerame,nra ecoee veoocsne curren veorame. me, an oernormaono ornee.: .neroaonee mrso aosn us ceomensaeo vne uon encoer ma aurson eenrcoormnn oeroea voneo usone cacuaeneroae vauesor su-neerxes o reerenceo cso comensaon un maeermneeneroaone cs.ns case,e moon e receve snax eemens an useen rs use e veo encoerrom enverseuanzaon unenrveoaonersoroucerecveocs. rovenesream an rseuanzeseuanzeransorm coecens uanzaonarameer ca enroecoe e enroecon un usne sameoeermne aeree oucaunazeaon.e venevoe ersnecoraenrsormor eroacces vsneo unoc anee vs aneonvraemeransorm, e.., annverse , annverseneerransor rseransormrocess,oeranso m, or a conceua smarnverse xeoman. rm coecensn orero reconsruce resuaocsne ocorere moon comensaon un orenra un eneraeserecve reconsrucse ceucroreen v veeooooccoarsee o cnurreen ve vcoeros anoc oer s sumnmaxn eemens,e summer enverseransormrocessn un an a corre e resuaocrom moon comensaon un anenra unso.nnrecveocenerae e er, er, er anor maeosnonne-ooeweeenre suc aseocn ourerrocesseecoe veooc.n some exames, summer ane e ome, aneecoe veooc maerecrove e seumn-moeoer ma r.eecoe veoocsn avenrame areen sorene r, woce sores ae meeremnocreraemvecse u sseearoare surosmeueen moon, c moam aenssoa soonre oe ncex veoocs.e , or on asaevce, suc asesaevce o. oe veooraerresenaon rames orcnu ares.caac vreaom ceo mannrcocueses, ar vee seaom seeu aern. rcaes,ecancues an orere se o awo-mensona arra ouma sames. s awo-mennoe,, anr.s rs awo-mensona arra or cro sona arra o croma sames. monocrome anereo ma sames.n oernsances, arame mae anusraon o an rencroucesess a ocncor onnewoo o-nem eexnasmonae o arrae oresuemna sasmcoseusr.e..s ornee.: . eneraes ans en scoowen rnerese.na,one o v aeora emnceoerrsoarr mooren seccaearon un veorame mancue anneer numer o s nerameno a se o s. romeo r anromo orere consecuven a raser scan orer ane oe are snaoeoome. vace s aaresoca con un ane w ae sn a veo se o encoern a seuencearameer se, suca ×.u soue noeuenceavee same szeen one o×,×,×, an sze. aeresen acaons no necessarmeo aarcuar c s sow sorresonn connnree.ocs, o eac croma s mama ceos,m anrse s onneax eem oensum usae samo coes,ewoe uanms oes a o coee cooncr oeexoecss a.neo swnaex v eeemoen sesueensccer ceanroee rres oerenes o ecoer,ncunner ornrarecon,nr econsruce ae veo arameers.n mono arecon moe, moon vecors, an oer comrse a sne cocnromreeecoucres an or scnuarxes eeamvenns urseee seo caoraee coeo sraanes, a mareeoc. con aronn suc asevere n ae mes oe con o ac areoc -reer m eea arrorme a onannncex ,,erena voc r-eo o re e s enacm aoreers. o ma recursveerormree comnaonereo one conreeocs o nn,ua-reearonn or a secen.,ex srseve annovoure smeaernso smaers. oc sze ox.moneour smaers, an are eac,v eaecavn a s oxx oc sze.ewox s a noour s o n are eacurervenoour rocess oe.oc seze ac. ss aecuae-rneea.a s,ru ecacureeau nsoraen oee euna- rerseue co orrearon one o a resecve sze rannromxox.ee ecenes.on,no ma comrse a ouma sames anwo corresonn conocs eac o arame oe same sze, an snax eemens us o croma sames monocromecures orcur eo coee sames oe conocs.n conoc esavnree searae cooranes, a ma comrse a sne aeua- anree snaraxo snruncurees ucseeon coe.e sam anes os oen coonruoscra.v seouurosee nso ane ornee.: . onueaernar cannare-r se noso aao varnoca caracerscsase on eacua-rneeeea mue-ar caneeroeuens r s. erurcaurre,o onnee asnaarr anoneernar areuea s-rureceu srer.ucsu sreo awnn.,.,e.,eruea aerrenarveaorssonena,roro n e nzonnaensar o aa croonnno,c veracvann a w an a orzonaernararonn, an vercaernarar araronn, ano onne s oorm meomreemxenaso.ns,e vs ae roec enacnouearon rs mn ua. areu orr neraron a conoc o on wce samerecon,ner ornra on-suareoc o sames sames,wo corresonns o ,s ae. o a ma comrse a ouma n monocr croma sames, an snax eemens useoreces. an snoamxe sruccuurerses or usecuoresreavcneree s.earea ve ceooo ernanes, a ma comrse a sneuma,, anrocsoruma,, anrs coer maeneraerecve recveoce vsoero a enc.oere v meao e unsceoenrra o urse eac escnorna o o rreneerr.econoeneraee ocs o a,e veo encoer maenera conoeneraeerecve ecoe sames oerame assocae eerecveocs oe ase onoeneraeerecveoc we.e veo encoer usesnerrecon oe ase onecoe sams oe as o, onee o vr meoor eencroameers o mearaenneraeerameer aessoccvaeocs e. e w mores o aere, vee vo eenoco enecroerene mraaeserneeraceve auummaa, res, anrocsor one or suracne’srecveumaocsroms ornauma cuaocore samene’suma resuaocncaes aeren onoc suca eac e'srecveumaocs an a co ceeween auma samen one o oc.mar,e veo enco rresonn samene's ornauma con ore, resecv er maenerae a resuaoc an ar resuaoc ewee e, suca eac samene's resuaocncaes aerence 's onr ana sam coenn oneo oc ane ea'csr seamceveneoc'ss anr r aes corureasonocn m saamnecnaee a ornee.: .neerence's oeween ar samen one oe'srecverocs an a corresonn same urrnearmorre c,o asnuoc. aronnoecomoseesraen.,e veo encoer ma useua-ree , anrransor uma,, anr resuaocs o a no one or moreuma, suareoc o sammeso ocns w recsecev seam.eranrasnosrmorms aoces. a rec oan auar msaua croem orr nsoen-ransormoc ouma sames,wo corresonnransormocs o a snax eemens useoransormeransormo croma sames, an assocae w aumaransormoc, arc sames.us, eac o a mae exames,eumarans ansormoc, an arransormoc.n some resuaoc.eraonrmoc assocae we mae a su-oc oe'sumaransormoc mae as sormoc mae a su-oc oe's resuaoc.er curesavnree searaeu c-ooorca ones,e a' msar c roemsruase a socn.enra mnosnoorcmromoce ancu sres or srucures useoransor nax e ve me sames oeransormoc. a o encoer ma a one or moreransormso aumaransormoc o mensoonaen aerrraae o aruamnsao crmoe coceencenosc.orarnsoerm co.ec coeen mcaene ao scca mae awo- veo encoer ma a one or moreransormso arans aruan.e a coecenocore.e veo encoer ma aorm oneoc o aoenerae rransormoc o aoenerae ar coece or moreransormso a oc or arer coeenecraenn ao ccoe,ceen veooc eneco..e,n r auomc maa cooree uacnen. zeoec c,o aec coecen uanzaonenera reerso arocessn wcransorm coe enoc. reucee amoun oaa useo reresene cens areuanzeooss comresson.ere veo enco ransorm coecens,rovnurer enro encoe sna eruanzes a coecenoc,e veo encoer ma ve x eemensncaneuanzeransorm coecens.or exame,e coeoc enecnos.erna ma,eer voremo encoer on mae s ounaxu e aemesnresamncaanncueesu aan sezeransorm aorms a reresenaon o coerames an assocaeaa uence os evce orransmeoeesnaonevce. , wcs eer savene sorae ornee.: . maarseeer rserceeamvno o aasnr seanmaxe eneermaeensrome veeos ernecaome.r,e veoecoer reconsrucerames oe veoaaase aeasna e veoecoer ma esream.erocess o recons r one snax eemens oanerom rocesserorme rucne veoaasenera recrocaoe enconnverser e veo encoer.or exame,e veoecoer maerorm resuaaoncsosr amssso ocnaee w coeceenso ocse a csusorrcenae w.e vse oo ae ccuorreern asoo reconsruc e conocs oe curren ane sames oerecveoc reconsrucs curren o corresonn sames oeransormocs oes sors oe reconsrucne conocsor eac o arame, ve oe curren.er rame. oecoer ma reconsruce.e.,nra-ramse noree acove, veo con aceves veo comresson usnrmarwo moes,s n onornra-recon anner-ramereconorner-recon.woo meoesa,ner- coraumeer reeacroen a cson ereurensr ma-orraemoeree ccoonn or e acernc moe.eweene reconecause oe use o moon vecorsorrec annra-rame reerence v n a curren veoocrom a oc szeoueo r woc. reserve evermrovn veoaa caurnecnoo an more rene veo moon vec neasne veoaa,e amoun oaa reureor reresenn caenesoorseonre a curorrmeneraamce aaso nonc ornease as sruousa on nae.onren wa osn ovoercomen ss anemoraomansave smar veoaaorrecnuroseue mooaa eweenese neorn s are aso smar.ereore,soss n vecorsnormaon o saa neorns anoremora co-oca eo usee moon oe moonnormaone.., moon vecor o a es as an aroxmaonemora correaon, wcs aso reer curren exorner saa an . reo as “oonecorrecor ” oe curren eermnenseae o mo enoconn,noe veosream, an acua moon vecor oe curren moon vecorr esmaon un asescre aoven connecon w .,e orouce aecooorn oeec courrrenerencse suraceorrome cuer arecnua m.oon voecnor so o,eer ceursre nno ornee.: . nreaemeon eoncoee veeo mosornea vmec anoreee armmonuen oaea m usoeono esmaon un or eac o a e veosream cane sncanec r reresenn moonnormaonn recon o ae coeeroocces,s a o se c ooo rsunes a neerreeas ocev. ee aoocen a reoerencerameurnner-rame veoecoeror consrucn a moon vecor c e veo encoer ane a curren usnoseoena anaesasonown as a “meres”or s anoremora c canae moon vecors assocae w saa neorn moon vecor canaeo-so acsa ae moosn o veceo crurrreencor aonree cnu srereencn. one moemne sro,rome no neeoransme moon vecor canaesserome veo e eres ecoer an annex oe seece moon vecorreco ncoeroe veo ss sucenore veo encoer ane v r wne moon vecor canae recor wne moon vecor eoecoero usee same moon vecor eransormne can anaesoras encon anecone curren. e ,oneransorm cores an snaneenesne more ecenananmroveeerormance orans meos.ore meos wc amourer , , an so on.oworem conaveeenrooseneasecae,ncun ne curren aver,ere are s severarawacsnese meos. ormuas.owev n ,ermarransorm cores areerverom an eren caracerscse.r,eereeorrve,e morraens eorcme cnorerasn msaorm no co areas reo ae veo conens w ne curren an ,eransorm marcesnmano aerenvesae. .owever, s on aene seconarransorm.e acnuaerraneransorms no consere mar owever,ee sn maareceesn. nenran nseeormoe onnneee can aceve sncan conan. emae macn.e comex oemaerve aoe encoer anecoer se acceae esecaoreecoe macn an eenvaueecomosons norann,raonnrane ,rannr.aa soue coecers.ne onnesoo com aas coece emae macn ao encoer anecoer se, wc aneseoecx.sn areen,v aeenraon ^s^orrmousoc ussn w cuserenren meooc. szeesn a oreners c moaercxes ornee.: . rearnvseormor m eaarcxsro uuse.,n.e v., weoc cornou, onee cnurerxs snaeor eac oncae wc ma suerromworawacs.rs enransormoceonso.s meo aanoerenrousouar ,erannaa casscaonsoveerann sasca caracerscs.owevaenr,eeeraanenraanan ca ssasmceasonn onnerou c monaorem noo acec suame enou.econ,e snenex a eve asorns aona overeas rae non-saonane curren an ,eransorm cores arexe wc canno. aaoe nr cea cruarcreenrs vcse o veo conens. r o con sanar,eorwar annverseransorm saree same raannssoorrmm c moaerxceons,uareaneereece receornescru reconsrucon.owever,ueoeuanzaon o sarn ma noeao con cannoeuaranee aneransorm marx ne currene o amna com,r seescsoonnarerormance. ransormocn wc s no ae,.e.,ra-nsormerme as s aeoe ransorms.owever, s neeceo s aeooorzona an verca oacee aove-menoneroreemrsa onsormno vc coe we . rooseomroveeerormance o eo con, severa meos are onerane meoorermarn veo con: sscosurerooses an rane , more ecenran ransorm omroveeerormance oe areroose oea wsormocncunerannaa casscaon meos aave snan m everse veo sasca caracerscs,e marces coecens, asmmeercos are assororooseosen wvcoeanoreware aunannzvaerosen eranesco orransorm erenransorm marces anvnsscosure,sroos m ma use ns secon, srsnroucere.eneeor eonaosee raonrn. snan meos areurerroo an renrouc se. wc can aceves aen o oorn ooonaransormase one sascaroer oe sna, e^{nu vecor as ^^,} o^{e o}m^ua^ura ^vn^es^co^or^rm ^oann ^ase ^{^}r^{^}m^.s o^e m ^coe^ra^rn^e s^au^oaⁿre^e er^wro^er^e merc.enoe _{^^s zero,. ^e^^.^,^^^^^^^^ ^^ ^^ ^^" − ^^ ^^" ^^ ^^# − ^^ ^^# =^^^^^^^^^^^^^^^^^^^^^} ^{n anwo eemensn}

ornee.: . ^_^^^^^u_^^a_^^^^o_{^ ^}n_{^ ^^ ^^"}s _^u_^r _^^#er c ₋on _^^ve _^^r_"e _^^ a _^s_^#eoownorm:

_{^ =^s^ a^^^ z^^e^^r^^o^^-^m^^^^e^^a^^n^^^^ varae,en ^^ ^^" ^^ ^^# = #}

₌ marxsenoe as ^^,eransorm caneescre as e_^ c = ^_^^^^^ovarance marx oe ^^ caneescre as: e_^^ ^r _^^ ^e _^^ ^{^^} _^ ^s _{^ ^^} ^{e c} ₌ ^ovar _^^ ^a _^ ⁿ _{^ ^} ^c _^ ^e _^ ^m _{^ = ^^ ^^ ^^ ^^ = ^^ ^^ ^^ ^^^^^^^^^^^^^^^ , ^^ ^^ muse aaona marx.n} ^arx _e ^o _oer ^e _w ⁿ _or ^u _s, ^ve _m ^c _a ^o _r ^r. _{x ^} ^{o ensurea ^^ ^^ ^^" ^^ ^^# =} aaona marx.aso sae row vecors o marx ^^ are e_^e _mn_uv_seco

_^^no covarance marxrr ova rs oeenn ee sna.oroweev ser,nan,raec scaesscarcoueroe os saonueenown,.e.,e oe sna snceeroa sruons no covarance marx acaon,e ervaons avaae.ereore,neracca erann-ase ervaon. usuaersasessu oensrananveronmearaa.ere are severassuesn corresonns, marx.e seconss nn seowoervee erenrou uesowo coec an cassrann samesno rna enacrouerann sames saree same sascasruon. eenvauenecomsosscnoos m nuree,o v canerane usn snuar vaueecomoson or se enerannaa.rs,s assumea weave coece a we ore ^r^a =nn ^^,a …a , ^ w^ z aenro ^ m^e reanore , wc consss o ^^ocs, ^^_', ^^ =,, … , ^^ oc s_'ze._'e_"ran_'n sames arere nseonrmsaezem aesn ^s^o =n oe vecor wcsuseransorm ocs aserann sames w zero meansorer_'vn ^^_' ^^..eesne, weraanen seasme reess cuaa e reresene = ^^, … , ^^, wcs a ^^ × ^^ marx.e n ooserannocs as_":_/ enoee covarance marx ornee.: . ^^^^^^^e^^r^^e^^^^^^^e =mens ^o^^n^^^^ o^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ans eenve s ^^ × ^^.e reaonseweens covarance marx ^^^^^^^c e^^r^^o e^^r^s^^^s as s =: eran^s^^^o^^r^^m^^^^^ m^^^a^^^r^^^^^^^^^^^^^^^^^^^^^^ ases an s aaona mar x we coumn vecorsene eenvecors eenvecors cane oa x w eaona eemensene eenvaues.e marx ne erormn eenvaueecomoson or one covarance orer o.oee eaenv waeueas.ve sosresoee eeen evnaerues ooeeerran wsorme co eeencvecorsnescenn orer corres ensaveescenn roannnnoerann sames. e enconnroecersos.oasae cuser enesern saamnnes m aere cou,sere sames areransormocs rereveromserveor enoerenrous an eac marxransormom eaocmreonu.:ns emomen,erann sames are cuserease one mco szme.ena:nor easc emooc sze, one marxserve. c suserease onransormoc sze anmennr,a mooren.raarescoro ena,ceranrasnornm saomc sesze a,re aner oa moes marces areerve,n wc onenra moe corresonso one marx or severa n sares o enmeom mena,rx. arraenseorrvme,oc sze an mooren.erarescoro enacerarnasnornm saomc sesze a,r ae s ceu osere masaerc oens mn wocme onne: moe corresonso one marx. cuserease ns emomen,ornerreconerann sames are resos, on, …ra ,nsor amreroece snzee ano cases asoeu meo vaoune v oec moro.ona vescoorr eac.evera oc sze, a_" se o_{# 0} marces areerve,n wc one a ransorm vecor corresonso one marx. soue vaue rane oe moon maxmum vaue oo ere,e moon vecor useor casscaonse momerzona an verca vauesoro un-recon an-recon. cuserease onrannsor:mnocs s emze,oremecno,no mronee arnre neconorneermanane. sam soeus aree ornee.: . noreecoan,orenrreacreonc moon,e reeerrseocon mo meoeere or re meorsonrarecon. eorner recon an so on.uresroveo on vecor vaue ornerreconransormoc sze, wc soweemae usens emomenor a ceran erannroc ceonsss o row an coumns o reconsrucexes. sze anre sssusrae asoow.nerann se o eacransormoc means cusercnona mseoe o,n ae c neeranorsnanceem mearesc, arnecuvnenuo no cumseers uos sn cassca- error an sum o asoueerence .enor eacrans um o suare recon moe,emae ceners areerve anxe ao ormoc sze an usraenure.or eacransormoc s e encoer anecoer, asrann ses oane ane cor ze,recon moe,emae cener, one n encon anecornesornocenss,or ea mcarrxaserve. rs seecease onsoc sze anrec nsormoc,e ransorm sesemae oe currenoc aneemae on moe.enesanceseweeneeasoe mnmumsance ceners are cacuae.eemae cener wcransormo s seece ane corresonn marxs useore curren n mc.omen:ns emomen,eroose meon emomens exene. ma ermxosmerevne.,nor esac emoranmseonrm,oro ecac szer,ansreormconoc mo sez,e,emae cener, one cener,erann sesurerveno severar recon moe,emae .ne encon aneconro ous usn cassca-means cusern asemae cenerar, a se oransorm macrecses,s aorre e saececeran asnor amnnoecx u sseze,oreurceornn moe, wcransorm marxne ses cae -eenen useore currenoc. annesscraonssuorer,rm m mear eo-rean s,esnor eemnrmramrarranrsanosrmorsm asroose.ne curren , w erens saree sameransorm marc reneenen,.e.,ocs eren ransormsorerens es.nsscosure,srooseo use rs,erannaas coece u .e rannrocesssescre asoows. eac,e correso snerensrome enconrocess.econ,ornrouce aove.nnrann ses uzeoervee marces usne meos ornee.: .ransormnoce s eznec,orenc anon meocoe ann nreocesso,ronr eeamcransormoc,n aonoe e encoer orecoero seece corr ae,s vaues aso uze smm esonnransorm marx. ransorm mnarxe oe orrcnuaarranee,eeeroerwcar recornasnsruocrmon anronevrerse oraneso srmna s.areoweeve srame veomae con,eransorm coecens areuanze aerr ,n uanzaonarameer.ereore,e or ansorm usn a ceran anesroem, asm na sna cannoe reconsruceerec.oransorm annv merc sroosensscosure,n wceorwar neerrsse eransorm ma useerenransorm marces. s sa momen,eransorm uns we same szeu coe usneren meoroeeer svaemeeorowrwarar an mnavrerxse we u msearceesrensnnvroersuec marces.ne exame se,or eac sze,e marx areerveor ae e asoows:ners escre aove.ns se,eerveorwa vaues accornoe meo samearameers ne sec r marx annverse marx sareeno veo coer a on se,e marceserveners se arenerae eoown sens,nceunnorma voanu oes,ee or usnna resuess c aonecee ausanezeranrannsoarma usoen an ner se,enverse marx are reneor eac vaue or cs e orna resues aneuanzeransormocsor vaue rane. areenoe as ^^_", ^^, … , ^^ an ^^̂, ^^̂, … , ^^, resecve a. cerea rnenem vaenue o oren vvaeursee rane marxnv cane_#esc_{/ "# /̂ nv} ¹²³ _{= ar}r_me_n as^/oo _^w. erenne secon _{'4"^ ^^' ,nv ^^̂'} uns we same szeu coe usn ns saerere

emom we usnerenorwar marces. orwar a en,eransorm uns w e same sze ma use searae marcesnn eacnveraser maa mvaerceesrarorerens,n wceorwar annverse ccornoe en marx eemens. severa resos cane o snee: or m_",ore_# e,m…,o^m₅.ense osee asremsmoerscve,oer eac vaues snzeo, ornee.: . ae ran.es, anor eac rane, one se oorwarnverse marces areerve an nnesn rssca eon msuroen, smoeme neo , emo son romeonses oon a snaenv areroose asoows. as snae ae or or scee e ono conro o . seuence or aeroncae weer s aeoe canae onceur seec oorn sc eem.omen,srooseo use as aonaransorm c .rs, as snaeoncae weer s useore uuserren ase.rmarransoarms oruee,e. asa ssnarueoncae weer sransormarsne curren or are uze. moree anan one maarsceass ae,e ore, an nex soueurer snae. re use rs ner emomen,srooseo reace an w . a,sase,ass s anaeeooneca,e o weerwesre ss u aseeor.e curren. on onense enoaure emoroemen,,ensreooseso rerecace aeaon an . one s enaeore, one nex e. morean ae.ne emome s snaeoncae wc marxsra n,srooseo a as ,.e., seconar annsorm oa. asrue,en as snae.o as snaeonsrue an morean one s enaeore,en one nex -cae wc marxs ae. escr ne aevoeve seconasr,ceesrannann anoceve snan meos areroose a ae a.ocrea sonesc,cea sa,secara cnearacers mcasr oce n racc vs aereo coxneen as aore u esnucaoerm ane-vaercaone,r..en., ornee.: . e scausocra cearxaecerscs m maarc varorerenrames or veo seuences.nese cases,s roem ox eso acevee oma comresson ecenc.oaces roosenessc marces,e eve aave marces snan meos moarrcee ssenuenece s.eueeno ncseure ea:ramen mear sreers a ces are s.sec, naee ens ncoro eero ssee,o eae aav er me nar s ecesna areeerve srooseo aave snae marcesnecurear secon asec, encoer se,e marces areerveor eacc ameer se . ene eaer an ner asec,srure.ene marces are snaene sceeaer.e encoer se ooseo aave snae marces marces are sna ,e marces areerveor eac sce.enene m saroue ces.en n noe ee sc rsae meweaoe o mr. ,eeo ese amreenrsoosensscosureo coee eemens ca coes,.e.,xonena- ne marces arerec coe usn marces arerece usn oom coe.ne secon meo,e aave urer snae usn exe marces,ene resues oe marces are orca coes. anuanzanone cuarr eenncoer a annewe,ene-uasn azeeweenorwarrmarransorm ecoer se. aso soue n aon annversermarransorma rmarransorm,.e., oea s on enae wene s no usene usenermar -s aenoorzona an vercaransorm.en s ,sroosreansoor am, s noo ursaens.oormuroermrovee conerormance o ernes ma cs coe w .eroose n one or meorraen eem oor omennes,erevree a,r aenoaenxeran asoorm e snecso aer anecoer se.ransorm marceserneserransorm n non-searae manromenrarecon moeoe se are useneroose or .e one oree moes ransorm sesre-ene as sownnae-. useore currenoc =re_oe_nra=,,ran_so_ or __ s cromaoc.or eacransorm se,e seece non-seararme s seeconsa sreercanesoormr cean curreane ornee.: . osncuereerr snerace aee exc snae nex.enexs snaen a-sream n so rransorm coecens. moe, a corresmone onerr eamnsoormm seens,s uesere a arneransorm ses,.e.,or eacnrareconransorm se one non-searaeransorm marxor eacs maeose usreneroose .es coe usn moe,enra moe werenramr sox oe =ans usno eerrman nnoreaoe mex w oeer aonne osrrao exoow se:n m enne orerxao oo: re me s a ransorm se. aoveescr e useu nomeoe ou ewo scemes.nce s resrceoe acae on a coecens ossoen oeresa csoe scnenca snu c-oerouce anre. non m-sorenacnan on,eransnex coneens oneransorm se,e nex neeso orm marces are usen eac onnrarecon moe e snae, wc cane conex coenoeen anoro .urermore, s aeornranonra anner sces, ar uma anroma. auarees enae, ncesoruma anroma ane s usenaeor seoaraeum.a aonrnerr sceeuareessae, a sne nexs snae eron- oma. o reuceeo cu roose , zeron-ouom ouaonsa croomoseexn anssec moseumreo,rn scuacen o coee saceanso omae zeron-ou anreuencom n ne saaomaan zeron-ou. wme n zeron-ou,e vecor wmenson os maeo a vecor non-searanseonr oanso,r wmesree msa smraxe mruanca.oonr ae vweceeonr weme mnsaorn o,eca weree marxase sae ox.neroose saa x ane vecor, emenson oeorwar marxs reu oman zeron-ou meo, marxseransose oeo ceromxox.ccorn,enverse areroosenssco rwar marx wemenson ox.momens ae me sureoervee or saaoman zeron-ou. soue noe no onoe eemrvee reuce s nomeoeoown emomens. scarne eenvecoorsmnen e,uaeon −or.een soeaaeoma mna zrexro asn-o =u ^^s_", ^e^_#r,v …e , ^^_/ ornee.: . w een ^v^a_'ueenecoem eosenvoenc oorrs onasees. covae eenvecors cane oane erormn eenvauesoeer we e rance marx.oea weave soree ener o envecorsnescenn orer oe eenvaueso maee ns emoermanesno,rme c eoeencveencosrsa wvee ssmceanern e orener corresonnoerann sames. marx cane^∗ r ₌ere _^^se_,n _^^e_{, …} as vaues arescare.e reuce , _^o_^o_,^w_^: ero_"ose_#reu₅en_{^^ ^^ ^^} seconarranso coman zeron-ous ae wen s useore ransormocrm o .ere s conuce one resueoc,ermar coecens aressca orean.e..n- somreovrem saormer eaxnasmorme z ceoroenc-oeuns aree we oer ransormoc,n wceuear reresense a meos oermar . roves an exam reaaransorm coecens are reane. reane.. roves an exameeaaeerarnasnsoromrm co ceoececnenssnneee a-aoovvee a arreeaa are reane.. roves an exameaeransorm coecens are . roves an exameaeransorm c nee area are reane. reane, wereee an a oecensnee an aove areas are n some emoove areasncueoe aove area anee area. -.n one mens,owo a e zeron-ou meos maeusraen. rmarr emomen, on one oe zeron-ou meon.-s aeore roose.anosr eoarcmromcar.nraneso armnooerc e,m anonmexens, s zenron-ou meos swc scemes zeron-ou meo aenesreamoncae wc arsx aeoeoc. rom a recannuearrooocsearn o a vec,ore. reaseueeooco owrnrmxarroacns aormocsrs convere = ^{^} _^ ^{^} _^ ^{!!^} _^ ^{^} _^ ^{^} _^ ^! _" ^{"^} _^ ^{^} _^ ^{^} _^ ^!# _^ ^{^ ^^!} _z s an exame.

_{" "} o e mensons usn raser scan orer or verca scan ^^ = _o ^^_!e!r^_a ^^_!o"^_n ^^ _c!a#n^ ^^_!e^ ^^_e"s!c^ ^_r^_""e^ ^^_{" a#s}^ ^ _^^_{^" =} or _^ ^_^^_, = _we ^_r^_!e!^ ^^_"s!^ ^^_!e"^ ^^_""^ ^ _m^_!#a^ ^_r^_"x#.^ ^^_!^ ^^_" ornee.: . eenen,n.e.,eecraenmeme mnaarocens, areerarannne anorerveaon o marxsoc sae eren marces are useorex renoc saes.or exame, o reucee memoro soreeoc co anxoc wens no euao. sarn meos areroosenss ecens oe marces, marx e same marx a cosure.eascea o erne sarnso sare ore moneocs we same area.evera emomens areroose saree samn m e oanrex em sa maorn rxm. oern,e maorcxs se w.oe same areanumer o samesneoc aveerenoc sae,ue same r exame,xoc,xoc anxoc ne anoer emomen,e w marx os o aneeocoesseocs. anoeroc, anee oocse same we we o samoec w,eee o anoc saree same marx or marx se.or exame, noc oc can saree same marx or marx se.ne emoxoc anx marx,oexoc anxoc arers conve men,o usee sare menson ox usn ceran scan or reoe vecors we same aeooneersx meooc, a rnaserx scaenrs o o. crero, c voenrvcear scan orer or an oer scan orers ne secon meo, wen converneocesocsoe vecors. aeoeoc we sae ox, w oe vecors, raser scan orers e sae ox.r verca sc e verca scan orers aeoeoc w raser scan orers a an orers aeoeoc we sae ox, we ner meeoo, aeno ccera wn scane o sraere oncux. verca scan orers aeoeoc nu nomeo raser scan an xsrsransoseoeoc w we saee sae ox.eoc we sae o oxs aeoeransoseoco c ox.ene scan orerore sae n emomens oeresensocnovseurre,neo a vecor. rooseoerve marces an a oown meos an aaraus are oe rann arenro ese new marcesn veo con.e aorms roose.e sn uce.emrove rannaa cusern meos aresrooseoanaene meuanozsa oon eecn v oeoe corannso armre coroeocseen.s.smmercor mes aoso ornee.: . sronaosneo meuroersm areroveroeos ceo.n ec zeenrocn o-ou me.onsa ar,eeroo-eve marces comuaona comex an memoro soree marces seo reucee arerooseo reucee memoro soree mar . marx sarn meos e con ecenc oransorm conn ces.s execeourermrove .s aowcarusran a meeo an . s or veoeconn accorance w aomeeco exearm, aersa onsoremr mesaenrxsocro asu curerr.en seoc,e,rean mseormo mncueseermnn, eenvecors. se ,e meoncues oann, eaerxncun aura o marx scarnar oeura o eenvecors coer, a moeransorm erormn, eecoer, annverseransor . se ,e meo ncues moeransor mrocess one currenoc usne n onem e marx. n one exxaame,e currenocncues a curren resuaoc. oc erormme,e meo urerncues oann,eecoer,e curren n onen ex aamrme,areranusrormo a curren resuaoc. ransorm, aneu a o eenvecors are useorerormnermar an ra o eenvecorsncue aura o rows an aura o coumns,n oanne moeransorm marx scarnar oeura o eenvecors aceuases o:n oeraannsnorme c moeoceenr oan asoermas m oanerx ro uwse aoer a seconarransorm, scarneas oneransorm coecen o aeas oom oeura o rows, or a n one exame, oann onee m coumn ae r se oeura o coumns.ransorm,scarne ae oeransorm marx useor a seconar oeura o r as oneransorm coecen oe aeas one row aeoom r se oeuoraws, o or coeu amnesa:s o onaenrnansoerm mo coeeceranns oorme m aareaxs on ree caonunmn ae oneransorm coecenocaen ae-aove area oeura aeas oanne moeransorm marx reann aea oransorm coecens an aove area oeura oran s oneransorm coecenocaen reann aeas one sorm coecens oanne moeransorm marx coecens or oannranseo mrmo coeecreannsoorcmae manr axe a rereaan one aueraas on oerraannssoorrmm ornee.: . c cooeecceennooccaaeenn a ane a aorveea a oreae ouerau oransorm coecens, an aeas oneransorm n one exame, oanne morae orraannssoormrm coecens. ura o eenvecorsncues:n resonseo rec marx scarnar oe aeransorm marx neeso evn,rom an encoer, annexncan scarnar oeura o eeneve mcoe, oanee moeransorm marx so .s aowcarusranor as m.eo or veoeconn accorance w amece exames oeresenscosure. se ,e meo ncueeermnn, saeo oer, ae cruarnrsenormo mca.rx sroem ara,nseor mme maorx seonrc au cuersr ceonnveoc, accorno aoc currenoco a curren vecor. se ,e meo ncurn,eecoer,e ecoer, aransormrocess one curren vecor eserormn, eransorm marx se comrses au usneransorm marx wereneransorm marcessra ra oransorm marces, an eac oeura o n one examnee, eacor oocesuavran a o same saeeaure. a same sze. ransorm marcessraneorocsavn cur n one exame nr ,eermnneransorm marxromeransorm marx seorerasenormo mc, accornoeoc sae oe currenocncues:eermnne anoerocna or onxe an eoxr oame c er veu e,r c creon onrvenroc an acc seconeo cru scrrn aenno a oro s ercze ,o o e aee cu crurrernen vecoocr.ncues: convern e currenoc, ane secon scan ore noerocavn a same sze w n one exame,e meoreurnernecreunromers scan orer. vecorn a secon scan orer,e es convern anoeroco anoer secon scan or anoerocavn a same sze we currenoc, ane or erenerenromers scan orer.n one exame, eac oeura anans roowrms o ma eremceesnss.ranee meoroocsavurne ar snamcue sezse:, aenermen cunrrenocas coumns scan orerore currenoc as a raser scan o , eecoer,ers scan oreror anoeroc as rer aneermnn, eecoer, a secon rows o eemens or,eermnn a v,ercae scaenco orere,r,ee arsno scaenr oroceroarvne cu crroenumnsoc an as a ornee.: . v raesrecra sc sacna onr oerr,ere an anoeeerrmoncn, eecoer, a secon scan oreror anoeroc as a n one exame, converavnne co cumns an rows o eemens. converne curreno urrenocoe curren vecor comrses: converne aernavecooc ano aeern caurvreen voeccoravnn a ar ssam scea snz oer aser ceor cruersrenoc an aernave onnoe orer. no ocne. exame,ers scan orer comrses a raser scan orer, or a verca scan some exames. ose areowcarusran a meo or veo enconn accorance w an encoer, aran senscosure. se ,e meo ncueseermnn, eenvecors. sseorm ma,rxe more ao currnecnueosc o,aennran,sorme m enacroxern,c au mon aeuraranso o marroxrmn,scarne enacro oer,e anunrvaerse ora ense rm e onrvmecorrosc.ess s oene c,urree meo ncues moeransorm marx noc usne nn o onnee e exxaammee. ,,ee m cuerreonocunrcuerensc au cuersre on resuaoc. oc erormn armarransormo a c ann,e encoer,e curren n one exame,eura o eurernenvec reosrsu aare uosce. ransorm, aneura o eenvecorsncue a orerormnermar an oanne moeransorm marx ura o rows an aura o coumns,ncues: oanne moeransor scarnar oeura o eenvecors aeas oneranso m marx useor a seconarransorm, scarneas oneransormr cmoe coececne on a o aeasea osne o cnoeu rmown a aee room s oe oeeuruara o o ro cwosu,m or a n one exame, oanne moeransorm marx useor a secons.ransorm,scarne aeas oneransorm coecen oe aeas nar oeura o rows, ore aeas oneransorm coe one row aeoom r se oeura o coumns: oanne moecen oe aeas one coumn ae oneransorm coecenocaen a ransorm marx reann aeas oanne moeranso e-aove area oeura oransorm coecens an aove area oeurar om mraansroxrm c roeeancnen as oeasa onnneranes moorm coeeracnesnormoca mearnx ornee.: . coe recaennns a ore oasa onnenranseo mrmo coeecreannsoorcmae manr axe a rerea oeura oransorm coecenocaen ae area oeura oran ann aeas oneransorm coecenocaen an aov sorm coecens, an aeas oneransorm n one exame,ee ar meae ooeuraur oeransorm coecens.ncanaeransorm marx nee rncues senn annex,o aecoer, s aome exames. ose areoswecnarscoussurras e.no a s me m eeooe. ,e moer voeo enconcnune acecoerrmanncne w, sna enec ooer,e a currarnesnormoc ma.rx sreom ara,nseo mrme moarx senocr au ceusrr cenoc, accorno aoc currenoco a curren vecor. se ,e me onvern,e encoer,e encoer, aransormrocess one curren o ncueserormn, eransorm marx se comr vecor usneransorm marx wereneransorm marc ses aura oransorm marces, an eac oeura o n onees exsamranee, eacor oocesavn a same saeeaure. a same sze. ura oransorm marcessraneorocsavn currenocn o,n aecc eoxramne,oeeremnonc searaens oorme m caurrrxenromocernacnsuoerms: meaerrxm sennoreransorm marxore currenoc accorno a sze oe currenoc e anoerocn oone an exoamer vee,c coonrvenr an secone cu scrraenn oroerc,oe curren vecor.ncues: convern e currenoc, ane secon scan or e anoerocavn a same sze w n one exame,e meoereurnernecreunromers scan orer. vecorn a secon scan orer,e es convern anoeroco anoer secon scan or anoerocavn a same sze we currenoc, ane ora erenerenromers scan orer.n one exame, eac oeura anns roowrms o ma eremceesnss.ranee meoroocsavurne ar snamcue sezse:, aenermen cunrr,enoecas coumns scan orerore currenoc as a raser scan or encoer,ers scan oreror anoeroc as er aneermnn, e encoer, a secon rows o eemens or,eermnn a v,ercae s ecannco orere,r,ee arsno scaenr oroceroarvne cu crroenumnsoc an as a ornee.: . v raesrecra sc sacna on orer aneermnn, e encoer, a secon scan oreror anoeroc as a convernnre e or cn, uee rre enx aanmoe ocer,o coo anc nv aerav enn rnave co cuurmrenns anoc roows oe e cuermreenns v.ecor comrses: converne a eocavn a same sze ase currenoc an aernaveoc.ernaveocoe curren vecorn ars scan orer corresonnoe orer. n one exame,ers scan orer comrses a raser scan orer, or a verca scan comun env.ronm seonws a co cmanuena ern ovr aonmaaenroces coue w a usernerace .e ncues arocesso sn server.e comun envronmen erocessorr , a mcaemo cr , an annuuu nerace . , suc onros overa oeraons oe comun envronmenmaeroce asssne. oeeraroocness assosrocae m wancusea o,aa acuson,aa communcaons, anoerorm a or some oe sesne aove-escrnee or m meoreorso.cesosroerosvo execuensrucons mancue one or more mouesaaca er,erocessor oer comonens. eeneraconeweenerocessor an sne c macne, aerraocessor mae aenrarocessnn , a mcrorocessor, a e m carocessnn , oree. oe comun eenmvorronmens con.uer meemoo srore var moausnceus oereaaeeor suore oeraon momens o sucaancuesnsruconso mne soware. comun envronme r an acaons or meos oerae onememene usnn an,e v oe vooaaaese ors, nomn-avoeaaea, m eemc.oreev mceems,or or a com mae ereo, suc as aacanomccessemor , anecr naon rorammaeea-nemor , anrasaeroramm ca rasae , arorammaeea-nemor , aea-aneeeam-nemor manec memor, aas memor, a manec or ocas. or , a ornee.: . rocessor wn oenree exeamrocee,ssoer, m uemoor s conureo sorensrucons execuae e meo asu n execuon oensrucons,s conureoerorm annerace moesrae ues, snucenra acse.- aeroo. avr,e as a cncneraceeweenerocessor anererancueu are nomeo, aom wee,uons, anee.euons manerace c euon, a sar scanuon, an a so scanuon.e n anan emeo coumeen, were ans e ancoer anecoer. meum comrsn aura ororamsos,roorv exeam a neo,nn-raens moemro cromue,r e-rxeeacae sorae rocessor ne comun envronmen ,orerormne uae e anor sorn asreamenerae e enco aove-escre meos ecoe eecon meoesc n meoescre aove or asreamoe e execue e re aove.n one exame,eura ororams ma rome veo encoroecressornn.e a comsurenam en ovrroaname snreamncou recneve encoor exame,normaonor exame, veoocs reresenn encoe e veo or more snax eemen veorames, anor assocae one envronmen ose,r eocrm., ane meaco anso mee exoecueescree aroocveess aocrcornnoe ceo rmun sream oraa sream.n anoer exame,eura or eceve rocessor ne comun envronme orams mae execue e aoveo encoe veonorm n oerorme encon meoescre a aonor exame, veoocs reresenn veorames, anor esxseoccuaee onee orr mocoerseso srnax enemeen cso,m ecu.no asream oraa sream, an ma asoe aa sreamor exame n envronmen oransmesream or comuer- ,oe veoecoern..ernave,e non-ransor comrsnre eanacoee s voraeoen meormuamo mnaora evxeam soer,e veeoreno asream or aaa sream rames, anor assocae one or more snax eemenseneraecs a rne ernecseonerno ern ecxoame ve,eoe ornee.: . v aeecoo enecrooerr exan. usn,or exame,e encon meoescre aoveor use non-ransor commueer,-reea vaeeo soercaoee mreunm ma.e, oror. necon veoaa.e emor , a- , a maneca exame, a , aanomccess e. e, ao sc, an ocaaa soraeevce ore escre anov ane o emr aomsreenam,eoreesercoovee aesreecaomnene mraeeoee encon meo emomen,eresrove asrea scre aove.n an enco m comrsn encoe veonormaonenerae e meone msceroeescre aove or encoe veonormaonoeecoe eecon rocessorsno arn ex e a amo move. e,meen,roceessso arsorov ane ae co nmon-urnevce comrsn one or more meum ore memor avn soreeren auraa onsororr caomms euxeecr-ureaaeae soer oane or morer aorcessors, werene one or more e rorams, rocessors, uon execuon oeura onsruconsne c o ao rnn so emure raeo omo rerner an,orm smesrsee ons a o aovs aoe-ersocvree m ae cooms.uerroramroucavn enerae e sream comrsn encoe veonormaon eecon enc moeno meesocreescr aeove a.ovne o arn en ecmoeo vmeeonormaonoeecoe a comuerroramrouc com n,eres asorove ex rsn aura ororams,or exame,ne memor , eesccruaee meoes.rocoers esxoarme,ne ceo comun envronmen ,orerormne aove- comuer-re muerroramrouc mancuee non-ransor more sna , aan eem so s,orame ae mne,u nem a c.omrocuesnsn envervoncmesen s m,aroerammmeameeneo wc oevnec oesr ornee.: . comson,ens,s, s, conroers, mcro-conroers, mcrorocessors, or oer eecronc n aonr eemroormmnen,e aeroeve meos. sorne s asorove a meo o sorn asream, comrsnnormaonesnreeraame on ae eanc soornae m meeoum,e wserenesream comrses encoe veo eecoe eecon cre aove or encoe veonormaono e n an emome mne,oereesscr aseo aroovve.e a cneervane ae encoerescr meoorransmn asream re e aove.n an emomen,eres asorove a meoor ans noneesre nea escmro ooe en e oeco xauee srveese one rmseccoo esuerree oasscr eereen areosevne.eoruroses ousraon varaons, an aer esenscosure.an mocaons, avneene onaveeemacemnesnareosennse wne aeoarreenonooesscer o oornsn aar sne ar rawns. ne assocae resensconsuesres secca sae oerwse, an orer o ses oe meo accornoe resenscosure asre on nonmeneeooee oruesrra sevce, acnaee ssecrse oe meo accornoe accornoracca conons.n a aove,u mae cane eresensc on, aeas one oe ses oe meo accorno e eomsuroe mmaense w aerues ceo,s ceonm annees ocrreeeen accornoracca reuremens. scosure ano enae oer orero exanernces oememe s sene aro unersanescosureor varous varous mnoaoncsa anonso ase asre u suzeeoe uneerarncuarnces an varousmemenaons w unersooae sco r use conemae.ereore,soemem e oescosures nooemeoe secc emomens oencueen waonnses sccoosee o aneraes menosccaosounrse. an oermemenaons areneneoe ornee.: . crcures, wec aonvceu mee aosc maaon seemcecmnenee usn an aarausancues one or more s,a rae crcus s,a snarocessors rorammae saena arraroscessnsevces s,rorammaeocevces s,e eecronc co , conroers, mcro-conroers, mcrorocessors, or oer arware or smowonens.e aaraus ma usee crcuresn comnaon w e oer moue, un,are comonensorerormne aoveescre meos.ac moue, su- more crcure osr su-unscose aove maememene aeasara usne one or conseraon oe.r eem somens oescosure we aarenoose sene arromneneo cover anec vcaon anracce oescosurescoseere.s acaons rncesereo anncaraons, uses, or aaaons oescosureoowneenera or cusomarra un sucearuresromeresenscosure as come wnnown as exemar oncc.enee s ar.sneneae seccaon an emomense consere acaonsneneo coveecrcaon an emomens are consere as exemar.e we arecae ana vareareosnesn, usessc,o osr aaaons oescosure. escre aove anusraene accomannrauwrens nomeoe exac emomens canes cane mae wouearnrome scoeereos,. ana varous mocaons an

Claims

ornee.: . : . ncuener mme anuno ra,or v a o eeecooe enec vro e,c an orra, sn csoomrmr msanr:xor a currenoc,eransorm marx een ovann, eecoer, a moeransorm marx scarnar oeura o moee ercoorrmrs ann an so,rm mearexc.oer, annverseransormrocess one currenoc usne .e meo o cam, werene currenoc comrses a curren resuaoc. res o. uaane on me c,o o .e ceacomer,,ure ceurrr ceonmroscn:erormn armarransormo a curren rma.rrean mseormo, o an caem, wereneura o eenvecors are useorerormne o coum ura o eenvecors comrse aura o rows an aura com oanns,n ane moeransorm marx scarnar oeura o eenvecorseas ors onaes en: rannsoerm m cooeecernansorm marx useor a seconarransorm, scarn aeas oneransorm coecen o o a aeeaass o onnee r coowum an aeoe rom o see ouraeu ora row os, c oor aumns. ornee.: . secon.are mreo o cam, weren oanne moeransorm marx useor a aeooman osorem,urasca or rne aeas oneransorm coecen oe aeas one row coumn ae r se oeuorwas, or o ceo aumenas oneransorm coecen oe aeas one oanne moeransorm marxs comrses: ocae reann aeas oneransorm coecen oann ane-ae movoe areea oranseormura mar oxra rnesaornmn coecens ocaen an a aeas oneransorm coecen oa ove area oeura oransorm coecens ocaenn ane aere mao oeeruanrsorm marx reann aeas oneransorm coecenoca oeanne moeranasorm o mraanrsxorm re coaennce ansea osr oneransorm c cenn aoeca aerean o an aeovuera area o oraensuorr oecen coe am co oercanensos,rm an co aeecasen osn.eransorm ar o.ee muerao o cam, weren oanne moeransorm marx scarn n resonseo re o eenvecors comrses: neesoe moec,e ovann,erome a mno encoeer, annexncanaeransorm marx o eenvecors. ransorm marx scarnar oeura .e meoor veoecon, comrsn: oc, aecrcmnn, aecoer, aransorm marxrom aransorm marx seor a curren conveorrnn,o aeoecco sear,e oe curere cnurrenocooc a curren vecor marxerormn, eecoer, aransormrocess one curren vecor usneransorm ornee.: . ura wer oenranesroarnms moramr mceasrsx sraen ceomorrseosc asuaravn o a sarmanes sorame meaarucrees., an eac oe ocs.ave mne a soam oe s czaem., weren eac oeura oransorm marcessraneor marx. see moreeo cu orre cnamoc, w, acecroernneeormnneransorm marxromeransorm eermnneransorm marxore cueoc sae oe currenoc comrses: oc. rrenoc accorno a sze oe curren comr.e meo o cam, weren converne currenocoe curren vecor com corns sves ese:r a rnasere s ccuarnre onrero,c or ao verec caurr secnan v oerceorr.n ars scan orer,ers scan orer .e meo o cam,urer c a sam coen svzeer wn anoe ceurrreoncooc an,o aner vo eecm seocrrs onnn a sc s:eacno onre srcan orer,e anoerocavn orer. enerenromers scan oroc.sea mveno a s oam cea smze, an, weere cunrr eeanc ooceasura oransorm marcessrane were coumns an rows o eemens eermnnne, meourer comrses: an eecoer,ers scan orerore currenoc as a raser scan orer ornee.: . e aneoeremrnonc,avneec cooeurm, ans se acnon sc roan oreror anoeroc as a verca scan orer, or ws o eemens orere,ermnn, eecoer,ers scan orerore currenoc as a verca scan anoee a renrmocnna,vne coecuomners, a an secon row scsa on o ereemrenors a.noeroc as a raser scan orer,e com.e meo o cam, weren converne currenocoe curren vecor an cornsvese:rne currenoco an aernaveocavn a same sze ase currenoc e a coenrvnearvneoec a.ernaveocoe curren vecorn ars scan orer corresonno verca. scaen m oreeor. o cam, wereners scan orer comrses a raser scan orer, or a e.erm mneno,or a vn enecoo encon, comrsn: ncu er, aransorm marxor a currenoc,eransorm marx een on veacn a onu,rae o en ecoenerv,e ac moroseransorm marx scarnar oeura oranseorrmor mmrs an an r,x.e encoer, aransormrocess one currenoc usne moe ornee.: . .e meo o cam, werene currenoc comrses a curren resuaoc. res o. uaane on m c,eo .e o en ccaomer,,eu crurreern comocrsn:erormn armarransormo a curren erm.are mraensoorm o, a cname,u wraere on eeenuvreacors o co emenrvsecors are useorerormn ura o coumns, an e aura o rows an a com oranne moeransorm marx scarnar oeura o eenvecorseas os onaes en: rannsoerm m cooeecernan oso armea msa ornxe u rsoew aor ae seoconomarransorm, scarn aeas oneransorm coecen o aeas one coumn ae r o see ouraeu ora row os, c oor aumns. s aecon.arera mnesoorm, o casmcar,n weree an oeasan onneraens moorm ceoeracnsenor om mae arxe uasse onoer r aow coumeoom oeura o rows, ore aeas oneransorm coecen oe aeas oneoca on ea a nn ane r e-ae m s ovoe o e areea oreansu eorarmur m o aar c oxoum ra rn nes sa co onm rmnr co ases ee: casen oneransorm coecen oanne moeransorm marx reann a s ocaen an eas oneransorm coecen oann aoev me aorea oeraensuorram ma orxrans roerman conecens ocaen ae area oeura oransorm coece ansea osr oneransorm coecen ornee.: .oca oeann ane aere mao oeeruanrasorm o mraanrsxorm re coaennce anse,a asn o aneeraasns oonremr coecen coecenocaen an aove area oeura oransorm coecens.ansorm sen.ne m anenoex o,o c aameco,eru,rnerc caomnrsaes:eransorm marx neesoe moe. oc,e. aecrcm m orneno n,or o a an v o ecnecoo en sec ar,o e an oran,s co eom curmr rr msanr:xrom aransorm marx seor a curren convern, e encoer,e curenoc erormn, e encoer, aransreonrmorcocesos a o curren vecor marx ne curren vecor usneransorm ura wer oenranesroarnms moramr mceasrsx sraen ceomorrseosc asuaravn o a sarmanes sorame meaarucrees., an eac oe oroc.sea mveno a s oam cea smze., weren eac oeura oransorm marcessrane marx. seeo mreeo o cam, wereneermnneransorm marxromeransorm eermnn cuerrrenoc, accornoeoc sae oe currenoc comrses: oc. ansorm marxore currenoc accorno a sze oe curren comr.ses:e meo o cam, weren converne currenocoe curren vecor ornee.: . com cornsveser a rnasere s ccuarnre onrero,c or ao verec caurr secnan v oerceorr.n ars scan orer,ers scan orer con.veren me anoo oer caocmo,urer comrsn: a same sze w anoer vecorn a secon scan orer,e anoerocavn orer. e currenoc, ane secon scan orerenerenromers scan or.e meo o cam, weren eac oeura oransorm marcessrane wocersenavne m ae saome surze,e arn comer csuersr:enocas coumns an rows o eemens aneermnn, e encoer,ers scan orerore currenoc as a raser scan orer e aneoermnn, e encoer, a secon scan oreror anoeroc as a verca scan orer, orerocavn coumns an rows o eemens orere,ermnn, e encoer,ers scan orerore currenoc as a verca scan anoee a renrmocnna,vne c eoncuomners, a an secon row scsa on o ereemrenors a.noeroc as a raser scan orer,e com.e meo o cam, weren converne currenocoe curren vecor an cornsvese:rne currenoco an aernaveocavn a same sze ase currenoc orne e.: . e a coenrvnearvneoec a.ernaveocoe curren vecorn ars scan orer corresonno verca. scaen m oreeor. o cam , wereners scan orer comrses a raser scan orer, or a on.e orn m aoreararusor veoecon, comrsn: exec au maemeor coeuoc oneess e oroors moe a re onnreo ocers msoorerocessors an conureo sorensrucons o ensrucons are conureoerrso,rm weree mneeo onen o arn m oonree oro cceassmosrs,- uo.n execuon o.n aarausor veo encon, comrsn: exec an u me ae omr eo mror ceoeuroc oneess e oroors moe a re onn ereo ocoers mseoo rrrse o,rm wroecre eess mnors ee an o one co n onureo sorensrucons o ensrucons are conur arn m oonree oro cceassmosrs, u-on. execuon nsruc.on nsona-r,a wnsor comuer reaae sorae meumor sorn comuer-execuae comuerrocessorseno exeercoureme o mnee oor monr aen com oneu oer caromcessso-rs,. causee one or more

orne e.: .nsruc.on nsona-r,a wnseor comuer reaae sorae meumor sorn comuer-execuae comuerrocessorsno exeercoureme o mnee oor monr aen com oneu oer caromcesssor-s, c.ausee one or more ecoe. none- mraensoorn co anm oue cr-aremasa-e s.orae meumor sorn a sreamoe e me.o nonn- anran os coram cosmu-er.-reaae sorae meumor sorn a sreamenerae