WO2024152003A1 - Methods and apparatus for transform training and coding - Google Patents

Methods and apparatus for transform training and coding Download PDF

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Publication number
WO2024152003A1
WO2024152003A1 PCT/US2024/011479 US2024011479W WO2024152003A1 WO 2024152003 A1 WO2024152003 A1 WO 2024152003A1 US 2024011479 W US2024011479 W US 2024011479W WO 2024152003 A1 WO2024152003 A1 WO 2024152003A1
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Prior art keywords
conex
secon
ransormerne
moe
veo
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French (fr)
Inventor
Ning Yan
Xiaoyu XIU
Wei Chen
Hong-Jheng Jhu
Che-Wei Kuo
Changyue MA
Xianglin Wang
Bing Yu
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Beijing Dajia Internet Information Technology Co Ltd
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Beijing Dajia Internet Information Technology Co Ltd
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Priority to CN202480007689.2A priority Critical patent/CN120530625A/en
Publication of WO2024152003A1 publication Critical patent/WO2024152003A1/en
Priority to US19/065,843 priority patent/US20250211789A1/en
Anticipated expiration legal-status Critical
Ceased legal-status Critical Current

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Classifications

    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04N—PICTORIAL COMMUNICATION, e.g. TELEVISION
    • H04N19/00—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals
    • H04N19/10—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using adaptive coding
    • H04N19/102—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using adaptive coding characterised by the element, parameter or selection affected or controlled by the adaptive coding
    • H04N19/12—Selection from among a plurality of transforms or standards, e.g. selection between discrete cosine transform [DCT] and sub-band transform or selection between H.263 and H.264
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04N—PICTORIAL COMMUNICATION, e.g. TELEVISION
    • H04N19/00—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals
    • H04N19/10—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using adaptive coding
    • H04N19/134—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using adaptive coding characterised by the element, parameter or criterion affecting or controlling the adaptive coding
    • H04N19/157—Assigned coding mode, i.e. the coding mode being predefined or preselected to be further used for selection of another element or parameter
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04N—PICTORIAL COMMUNICATION, e.g. TELEVISION
    • H04N19/00—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals
    • H04N19/10—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using adaptive coding
    • H04N19/169—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using adaptive coding characterised by the coding unit, i.e. the structural portion or semantic portion of the video signal being the object or the subject of the adaptive coding
    • H04N19/17—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using adaptive coding characterised by the coding unit, i.e. the structural portion or semantic portion of the video signal being the object or the subject of the adaptive coding the unit being an image region, e.g. an object
    • H04N19/176—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using adaptive coding characterised by the coding unit, i.e. the structural portion or semantic portion of the video signal being the object or the subject of the adaptive coding the unit being an image region, e.g. an object the region being a block, e.g. a macroblock
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04N—PICTORIAL COMMUNICATION, e.g. TELEVISION
    • H04N19/00—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals
    • H04N19/50—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using predictive coding
    • H04N19/593—Methods or arrangements for coding, decoding, compressing or decompressing digital video signals using predictive coding involving spatial prediction techniques

Definitions

  • ornee . - caoneresen acaonsase uon an camsroro..rovsona on,”eo o.nanuar,,, ene, “e eenroes o an wcarsaunscorororareansorm reerreannn an uroses.
  • veoeconncunc o soamnen em,o amenecso oer, aeruersaen oscosaurreurove a meoor ransormernesor a curren nen-overansormransorme ocn annra moe an seecn,eecoer, a ernes.rneore currenocnenra moeromeura o ransorm veo enconn a, sencocnu ans:ec o, saonmne e,mo anm eenncos oer, aeruersaen oscoasruruenerno-voev ae meranosoromr ornee.: .
  • sr eannasoermorernee csurorren a cuorrcen anoc seenc ann,nra meo eeeermnn, e encoer,a e currenocnenra moerome ncoer, a ransormerneor n ar asec, some emomenusr oae oresransormernes.
  • comuer reaa sxe s asoraece s momeeu emmoorm seonrsn o coemrueseern-exscosurerove a non-ransor execurmee one or more comuerro ecuaensruconsa, wen ero cessors, causee one or more comuerrocessorso n me a soeve anccor asneco, soem seec eon asec.
  • ransor comuer momens oeresenscosurerove a non- accorno -reaae sorae may be used to generate a code for oeresenscosurerove a non- accorno -reaae sorae.
  • ac sco,ra soem mee emumoomens oeresenscosurerove a non- accornoe s r sorn asreamoeecoe e meo soecon asec.
  • eaeescron aere un exearmsooes ona ano aree noor reesorncveen oeraeersecsrenonsc aonsuree.oown ornee.: . seccaon,e aucscoramea enxnrawns, wc arencororaen an consue aar os escron, serveo exaanmeesr cnocnssesen o wesceosurerseenscosure an,oeer w e veoocsn.
  • eecs a.ua-reuesraaeas s corumcuorseonus ora an,e en. resuu osraeesvson rocess oe asecen., an . usraesveos aron o a conoc searonnes eresensc..s oson,e anransormen accorance w some exames o scosure..osurse.e or n accorance w some exames oeresen eresensc.s anusraon oe or n accorance w some exames o .osusre.
  • aove area are rean eaeransorm coecensnee- area are reane.e n accso an rn acc ancuo esr wraancoe sn w o ome a en so x em axmaem exa eemaes oeraensreosremn coescocseunres.ne aove . s anusraon o an exames oaeeresenscosure. are reanen accora ransorm coecensnee area .
  • aove areas are reane, wereeone o a ann e axaomve aereaasnceuraensoorm ceoe acensnee an arean accorance w some exames ove area anee .s anusraon o oocerarenssenose asncosure.
  • orancse a wa sorammemusermaenna aon csom ouenre esennvronsmcoensure c.oue w a user wc areeuesrerancee wn noew acceom maanennearawon ss.eccmemenaons, emomens o numerous non-mn seccea neoowneaeescron, maerr s are seorn orero asssn unersanne suec aernaveesse mnaeeer uesen..uor ex wamee, aphone ao oanreen oo or onnar sne ara varous e suec maerreseneeren cane e o ornar sne ara a veo caaes.
  • n aono - wcaseen em orm eeco oen aueransorm muen seece srcanesmoermss ursoemoer resua co.neo newnern arnoucnra coeocs.
  • uses - an -.ae sowseassunc eransorm marces are n oreroeee oroona oeranosnso orm me seece .
  • coecens aore re zeurcoee oeu comorexe o-ar ane sze--oc ans w s-z,ewre oruenecr,a onrsoorm w reane an.e euao.ne coecens wnexower-wingc reon are e re sn ,e resua o aoc cane coe wransorm s moe.o avo _un_anac o snax con,eransorm s as no snae wene eve or s acvase noo eruae co zero.oeamc ransorms seo wen enaeorn urren.soemc cane s enae wen s ower coeocs.
  • aow nevoenr-,seear naeransorm soasmemenen a sneass reuceo mnmze comuaona comone-xsear aane mraenmsoorrm s maacerxom soenson neesoe coecens.ence, reuce non-sea reeransorm manea oe reuce n raeransormor meos usen .e x mensona veocn-osreoar aaneramnsenosrmonas voe mcaor ann aerse cnom smon euaoor se reucon
  • armar-o ecen su-rou are non- s aows a cononn oneransormn coexe scneansnav oenoee zaesro-s wnencanoss aon, aen. ornee.: .
  • rae-soron remann coecens are enorceoe zero. orers coecensn scan orer,e ransorm se aerree u asreeraon nasorm srea.nescoo ermn ma sesn anrom non-esenarraareercanosno mrmo meaorceesraenrneser canere-ene.
  • ncenex ssn reasrnc anneracon w oeroos coecen su-rou are no eoe acae on a coecens ouseers sncan coecen.
  • 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.
  • axmummramnsrovemenne maxmumransorm szeo.rem., s aze an zeroranns-oorum oerrannessorm are co eexecens.o sze an maxmumnra coeoc canave a sze ox.e m neo, weree seuences ans seo, oerwse.ner axmum szes seo or normramvaer zeron marransormaonrocess,eres no e ou oeraon ae onransorm coecens.owever, s ae, nranasnocrem coecoernnsr oau csee reon are normave zeroe-ou.
  • uze wcn aree c uusrerenornra aensno ner c,nor , on an ransormernes are , , anenransorm o aren. onarmarransormsncun , sze annra moenormao emoe.so ses maeeenen one eren cass n.
  • 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.
  • 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 .
  • su-ocs consss o sxx su-ocs, wc orwar canne orxer.
  • .ncse coese nnumoeer onnus c soanmreusosn,,ransorm marxor su-ocs areeneraeromorwar coecenswox coecen scan orer.
  • 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.
  • e anuarnra moes moeso are conveno sownn., amon - an moeso are we-anen na anuarnra moes, an moes -o anar moemoen. a ra moes.n aonoe anuarnra moes,e aso aen nrecurren moemoen. o are ocsn suanrcee s a.
  • anourear snera moes mae suare an recanuar veoocs,esesanar an m cca,oro e anuarnra moes are aso su oes, anuarnra moes amon sae o a veooc sas oreor eacoc sae.en a recanuaroc veooc m es a ceran conon, annex o a we-anenra moe oe convenona anauaer anara mveoe reeceermveneromee v veeoo eneccooeerr us acnco ar mnaon annex o a as sownnaeeow.as,or non-suareocs,e we reaons e veo encoer usnenexes oe co -anen
  • a veo source a veo encoer evce, e.., a veo camera
  • a v e a source suc as a veo caurnneraceo receives veorom ae vo aerco cvoen ceonnarnovnerr,e avnouosr a c caomureue vrerao, a vcs seoseen eneran comuerracsaa ase source veo, or a comna emor exame,e veo sources a veo camer on o suc sources.s one evce aneesnaone a o a secur surveance ssem,e sourcememenaon vce maorm cameraones or veoones.owever,e an mae asescreneresen acaon mae aca
  • anoraac.e ouunerace maurerncue a moem anor a eesnaonevcencuesenunerace, a veoecoer, an a enscoae vevceeoa.a oveernuennerac.ee m enaconecu vee ao recaeav ceorm anmuonrc aa meo oevm an receiveee rove one soraeevce, mancue a vare o snax eem eren, or encoeror use e veoecoernecone veoaensenerae e veo encue wne encoe veoaaransme on a a.uc snax eemens ma sorae meum, or sore on ae se communcaon meum, sore on a wc can
  • en scuocer as an,e ve,oecoe-r ma oerae accornororear san ,ar, , or exensons o suc encoars. soue unersooaeresen acaons nomeo a secc veosnenecon sanar an mae acaeo oer veo enconecon sanars.
  • vare o eac maememene as an o a naro scuessaorse encose,r anocraeocnoere ccrccunre, sruace as ornceu ors more msc,roerocersosorrs, a aerras s,screeoc, soware,arware,rmware or a ammae enmemenearan soware, an eecroncevc n comnaonsereo.
  • one or more cou Chroservce new anoreesnaonevce wc are noncuene more cenevces ma comomrun mcaaee wrov serevern co omne oure mrsorne ceen couevc ceosm, anune s oenrvec oer ornee.: .
  • n neewwoorr ar souor-ra an was weersese csosm commumnucnaon neworor exame, a ceuar communcaon communcaon newor o caon newor, or aoa navaon saee ssem communca r a wre communcaon newore.., aoca area newor emomen, aeon newor or aowerne communcaon newor.n an servc as aar o oeraonsescreeren maememene as cou-ase oee csomrovoneens o onee o sro murocree seervvceer com anueorrs wec a aeraesm aeamre onee come aoneenas aar esnaonevcene cou mor servce newor an one or mor s
  • sooumenesuesscrraeeeareonr anensea, coem aonecna, aon m caromea a comeoneon a a snu aaorn c wroemrea a cnom oosneens ac secoercenor an o aumauma comonen,e croma comonen aner croma c o an oer oe comonenase one seece ose.urer, so omoneno mo sa an menoneeren mae an oeuma c u asoeusraea ars comonen croma comonen, a seco omonen,e croma comonen aner comonen,e crn comonen menoneeren mae an oer oeuma meno oma
  • v.eoeaa memosr au mearae o soarense re,eorern ecxeam ve, aaor exame, reerencerames orcuresor usen enco eo encoere..,nnra ornerrecve con moes.en veoaa e veo maeorme an o a vare o memorevces.
  • n.a,r aer receives veoaa,earon un wne asoncuearonn a veoonse veoaano veoocs.saronn maarero rameno sces,esor exame, ses o veoocs, or oer srucure asnsocanes ws aceco vrenooaar.eeen ve seoranme srsuc ourre msa suce a rse aaruea- asre aewo- ornee.: . remeerrnesonoa as a arraxe or o mra arex.
  • reeren ar eoc or veooc mae or corresono aonce,ror exame,o an reconn or aransormn anor mae or correseeonno a, a, a ocs,o er.m. aoonreeo corresonn ran c , aonoc , areconoc or a con moesec , surce ac a s oon nno e orroco aes as s unu- ra unoc.
  • curren veora or exame mancaeesacemen o a veooc wn a currenocemne o croecur we reanveeo cu arrernecravmee.oce wreene ar rmenereencearearmne m reaaveoe veoramesne seuence asrames orrames.enra un esnae e..,oc vecors,ornra conn a manne maeermne vecors, e moon esmaon un orne r smaroeeermnaon o moon vecorsoeermneeoc vecor.
  • erenrame accorn ome samerame accornonrarecon or a suracnxe onerrecon,e veo encoer maorm a resua veooc eno cooce m, vaues oerecveocromexe vaues oe curren veooc ve ormnxeerence vaues.exeerence vauesormne resua eannrcaureecoonuma an croma comonenerences.
  • aernaveoe rocessn un manra-rec a curren veooc as an comensao ner-reconerorme e moon esmaon un ane moon escre anov uen.
  • orenraoc coreconerorme enra un as recon moeon usearoc euncaor,e ae cunrrreanreocc.onooro scoe,ssnen urnare mcaoneroecrmessnne annra ma encoe a currenoc usn varousnrarecon moes, e..
  • anenrareconrocessn un or a ,urn substitutee encon ma seec an aroraenrareco moe seecon un,n some examesnrareconrocessn n moeo useromeesenrarecon moes.e recon moeoreo un marovenormaonncave oe seecenra- encoeenormaonnccoe enro encon un.e enro encon un ma ererecoannrocees ssenec uenra-recon moenesream.
  • vaero vecoeoereconern accorance memor
  • an enroecon un areconrocessn u cues a veoaa un, annverseransormrocessn u n, annverseuanzaon rocessn un urerncues a moonn, a summer, an a .erecon annra un.e veoecoe comensaon un, annrarecon un, anoe enconroces r maerorm aeconrocessenera recroca ..or ex sescre aove w resecoe veo encoern connecon w moon vecorsa rmeceev,eero mmooen e cnormoenseacoonn un un m,a weenereanerar-erecocnona uanase m on
  • oaear c moemmoor ma sore veoaa suc as an encoe veosream, aa memor mae oannens oe veoecoer.
  • veoaa sorene veo source su e,or exame,rome soraeevce,rom aoca veo accessnc assca a caamae srao,ra vae m werea oer.
  • a reresens veoocs o an encoe veorame an oe veosream ecoer ma receiveee snax eem assocae snax eemens.e veo e enroecon un oeens ae veorameeve anore veooceve.
  • oeramree ocroornnrora coe un maeneraereconaaor a veooc oe curren veora cessn snaenrarecon moe an reerenceaaromrevo mease on a rame.
  • aenscr sneaxro eeermeesns useo coee uns o a coeoc oxes anowe veo seuence cane r oerenes o ecoer,ncunner ornrarecon,nrare econsruce ae veo arameers.n monocrome con moe, moon vecors, an oer comrse a cures orcuresavnree dissolvee cooranes, a mareeoc.
  • sne conreeoc an snax eemens useo coee sames oe con o co acnevere aeeocer mearorme anxoc o sames.
  • oc sze ox.moneour smae a s o rs an are eacvenoour s oxxococ sze s.ze.ewox s an are eacurervenoour ornoecess oe.
  • s a recanuarsuare or non-suareoc o sames sanm wecs,weo s caomrreesroencnon,ns oer o crronmraa, sasm aese, a.n sn oax a ee ma comrse a ouma n monocromecures orcuresavnr mens useoreces.
  • aronnoecomose sraen.,e veo encoer ma useua-ree , anrran euma,, anr resuaocs o a no one or moreuma, suareoc os saomrmeso ocns w recsecev seam.eranrasnosrmorms aoces.
  • a oenerae auma coeceno ansormso aumaransormoc o mensona arra oransorm co core coecenoc mae awo- veo encoer ma a one oer mcoernesr.ansroarnmssorom a coeracnesnor mmaoec a s oca aaruaonen.eraee ornee.: . arra cnoseormcenoco oc aoreoen.eraee a vero co eencoceerno mcao are one.
  • oc ereneran a coecenoce.., auma coecenoc, a coecen uan or ar coecenoc,e veo encoer mauanzee coecenoc.
  • oc szeorreserven eveeramsrnovne v veeooaaa cae aumrnounec onoaoa re anure moroer re renreese vnneo ornee.: .
  • ormunas.eo cwurerveenr,ee arnveran,soermr cmoraers mraans norm cores areerverom an eren caracerscs.ereore, mor o aao ae veo conens w ne curren an e ecenransorm cores remanoenvesae. .owever, s on ae,neransorm marcesn are acuaraneransorms no consere.
  • non-saonar caracerscs o veo s arexe wc canno aaoeransorm mnarex cuorreunar vaneeoe coener s co ean cne ranrs.,eorwar annverseransorm saree sameransorm coecens,eerec reconsrucecoonns crauncnoon.eouwareavnere,eu aenoeeranusanozaon o sarn ma noeaoe om rm marx n a comressonerormance.
  • wc can aceves anr e onou m oc a oorn o raonsoonramraannsnormermass oe m oenane s suaares ercraroer oe sna, e nu , ve .
  • ornee .ransormnoc an s ezme aonmnerna, mooren.raaresocr eoanc,reanrsann sames are cuserease on areerve,n wc onenra moe cor ormoc sze, a se o marces one ma resonso one marx or severanra moes sare nr ax.
  • ra emomen,ornerreconerann sames are cuserease on , sor , m ... , oc are sze re a e n ne e o a ca s s o s ue v e a m u o e o o n v m ec o o o r. n v a ec s or or eac .
  • sesurerven oc sze,recon moe,emae cener,erann encon aneconrooc seesvse,raor eraocusr uasnsnor cmassocca s-zme,earnes ccusoner mno as .ne ar, a se oransorm marces are seece a e,emae cener marxne ses useo n annex useourerncae wcransorm -eenenrrem caurrenoc.
  • moes moes,.e.,nor esr acscnoe ras-urn ree,en coenrrea mnoes oe or mnarces areraneorerennrareconrane.ere,enrare rarecon moe se, one marces ses snrouce aove, we-acnoen mnoraer seec maon c monosess o are m uorezeano orn reecnararecon moes.
  • nsnscaonsuren someeo emoon ners emom mens on snan areroose asoows. as snae ae oen,srooseo a eve ono conro o . seuence orc r or sforementionederoncae weer s aeoe canae oneur se .ec oor rn sc s ee ,m.omen,ass sronoaseeoo unseca as aonaransorm curren. asrue,e as snaee woeer s useore use asermarransorm oe.
  • ene marces are snaenncoeer se,eaeer mnarec seesc aorne aesrvece, ornee.: . encsoeror sosee,oe aa mvearc sesn aareeerveo mra earccescnuecurearameer se .
  • ene eaer an ner asec,srore.ene marces are snaene sworter.e encoer se, oseo aave snae marces marces are sna e marces areerveor eac sce.enene m saroue ces.en n noe ee sc rsae meweaoe o mr.
  • ao ose or mae useu nomeoe ousveeesecrres cwoeo scceenme ssu.-nrcoeu are nons-s resnrccaen,oe ancaexe c on a coecens oson oeas sncan coecen.
  • morean oner oneens oneransorm se,e nex nee ansorm marces are usen eac onnrarecon mo soe snae, wc cane conex coenoeen anoro e.urermore, s aeornranonra anner sces, are snae seuamraae an.orronmear.
  • wmenson o, n-ou,e vecor wmenson os maeo a vecor non-sea weres smaeran.or a vecor wmenson o,eca wereeraer manasroxrmassee s maaer ox mxu.ncaeonroeosweee snaeaoma mna zrerxo anne vecor, emenson oeorwar marxs reuceromxox.c -ou meo, marxseransose oeorwar marx wem corn,enverse areroosensscosureoervee or enson ox.momens ae meooervee reuce saaoman zeron-ou.
  • seconarranesorromo osere.uenceromean zerosn co-nouuces a onee w resen s useoreransormocs oane.n somermarrans ueoc,ermar coecens arescare..-ro orm coecens aree we oerransormoc,n wceuear rerveseen sosmee e axreaamae zerarnosnor-mou co meecoens os arermar . roves an exameaeransorm coecensnee e reane. reane..
  • roves an exameaeransor -aove area are reane.
  • roves an exa m coecensne aove area are .
  • rove meaeransorm coecensnee area are reane.
  • reane wereese an a enxam aoeve aareasencraunseorom coee acoevnes arenee an aove areas are n some emomens,owo a e zeron-ou mea anee area. -.n one emomen, on one oe zeron-o os maeusraen.
  • rmarransormoc.ne anoer emomeu meon.-s aeore roose.or eacrmarr n, zeron-ou meos swc scemes zeron-ou meo ansormoc, annexs snaenesreamoncae wc arsx aaeroeoc. rom a recann uearrooocsen o a,e resueoc orrmarransormocsrs convere ⁇ ⁇ !! ⁇ ⁇ ⁇ ⁇ v ⁇ ecor.aeeoownxoc as an exame.
  • soc sae eren marces are useorexoc anxoc w es.or exame, o reucee memoro soree coecens oeen ms no euao. sarn meos areroosensscosure.eas arces, marx e same marx amoneocs we sacea o erne sarnso sare ore marx sarn. me area.evera emomens areroose ornee.: .
  • aeoeoc we sae ox, we ver cors, raser scan orers e sae ox.r verca scan ore ca scan orers aeoeoc w raser scan orers aeoeocrs aeoeoc we sae ox, we verca scann oreerrs a meeo, aone ceroa w cn s wcane o sra eere o sancuxn.u nomeo raser scan an xsrsransoseoeoc we sae oe ox.xe.neoc we sae o oxs
  • ooewremver,emeenraooonse marx sarn meos nome ma sea a meos are aso aowe.or exame, we uoce neesoereransoses wsareranse. ernesernverevous secneosns,eoerc eaocnoc sze annra moe, on one ransorm one sneraneso arnm aernee.
  • ernes seecevera emomens areroveo snae snax eemensor w mue ceran sze,nw oonn oe. emoermneens, aree cearseve o.wos sownernessrove,asoreocs w rsns conex coe w conex. s ennaae,eeow,ns emomen,e conexonc seconnsen coe w narazee w vacue erneo 0 nse uxze.
  • sx noa ennaaone w ceran sze,wo er, , soreocs ersns conex c nes areerve.s sownnaeeow,ns emomen, ass moe,.e., euaoe w conex. s enae,e seconnsen coe w narze varouea ono 0 nceaxe wc ernes uze. sx noa ennaaone ocs w ceran sze,ree ,ernes areerve.s s ,asore emomen,ersns conex coe w conex.
  • ennanaeeow,ns ns areen coe w conex an, resecve 0 oncae e,e secon anr narze vaue 1 o 2 0 nex wc ernes uze.
  • s nxoa ennaaoen ocs w ceran sze,ree ,ernes areerve.s sownnae,easore emomen,ersns conex coe w conex.
  • s enae,e seocow,ns en coe w conex 1 .erns coe w 0 ass moe,.e., euaroan.ns ornee.: . narze vaue o 00 n 1ex sxanaon e nrnoe enas uese ocs w ceran sze,our e,rnes are ,asore emomen,ersns conex coe erve.s sownnae eow,ns ns areen coe w conex w conex 0 .
  • s aso snan meuroersm areroveroeos ceo.n ec zeenrocn o-ou me.onsa ar,eeroo-seeveo re mucaercees ornee.: . c aoremruoaosoenao co remuceexe a mnem moermoor soor seoreee ma marces. marx sarn meos e con ecenc oransorm con rces.s execeourermrove some exames. ose areoswecnarscoussurrae.nn a s me eeo an ,e moer v.
  • oeoeconcnunes ac ocoarnance w ecoer, aura oarunen-overansorm ransormern n, a annra moe. se ,e meoncues seecn, esor a currenocn ore currenocnenra moerom eecoer, a ransormerne n some exames,e meoeuurraer onc ransormernes.
  • eermnnaerss se ncues one ooown ses:n resonseo eermnnaersnnss a,eermnnae s no enaen resonseo aseeransorm ne seconns, seecners ransormerne ns, se erne orn resonseoeermnnaersns ane secon nc snomee secon ransormerne ase ransormerne.
  • ransorm exames eura o ransormernes comrse ars vaue comerrsnees, a a sresconn, a secornansonr,m aner ane,r ann are one orransormerneenarze conex, a secon conex, an a more conexs comrse ars e seconns co r conexersns conex coe usners conex er conex, orn ceoxe cone a usnass meo seceo.nn s coomneex exa amnese,ern cnoes conn aex coases w means euaroa oncae w moe n some exame c ernes uze, eermnnaerss, se ncues one ooown ses:n resonseo eermnnaersnnss,eermnn
  • aecoer a mes,e meo urerncues: snan, e encoer ano one ooown acas:na resnoncsaeeso weeermernnease enaes e annaeer,o sremnn,, ee e ennccooeerr, e a asrue, an snan, e encoer anoeecoer, a ernene , resonseoeermnnae s no enae, sen,e encoer, x orn n some exames, se ncues:n resonseoeer e a asase.
  • suc ase oeraons a era oeraons oe Chevronmenmaeroce ssocae wsa,aa acuson,aa communcaons, anoerorm ass onr s.omee orocees ssoersn mea aonvce-ueesc ornee or m morerocessorso execuensrucons mancue one or more moues eos.oreover,erocessor oer comonens.e aacaeeneraconeweenerocessor an sne c ma rocessor mae aenrarocessnn , a mcrorocessor, a ec mneme,o arracas cornoceussrenon , oree.
  • roramm a aeroeara-mmnaeemeoar-nemo,r anrasaerorammaeea-nemor manec memor , aea-nemor , a n one, ex aamase m,eme morem, aor manecs o cro oncuas.
  • rocessor wereerocessor reo sorensrucons execuae e meo asus , uon execuon oensrucons,s conureoerorm annerace moerae ues, snucenra acse.- aeroo.
  • avr e as a cncne wraecee,euweoenns, aenroceessore.
  • anererancueu are nomeo, aomeuon, a sar scanuon, an a s euons manerace can meum comrs enme ao co o scanuon.e n an u umraeen, w oere an ros e r anc amsooe s,rr oo a rvn exeec am ao neoe ,nr.
  • sornesream on aa soras ae msoeruomv wee are mneoe os sreoarmn co amrssreesam en,c coomrsnnormaonenerae e encon meoescre aove or e e veo eecoe eecon meoescre aove. ncoe veonormaono ornee.: .
  • aarausancues one or more s,a snarocessnevces srae crcus s,a snarocessors rorammaeae arras s, conroer,rorammaeocevces s,e eecronc s, mcro-conroers, mcrorocessors, or oer a comonens.
  • aaraus ma usee crcuresn comnaon w e oer morwuaer,e u onr s,o orw suare-u cnomosnceonsseo aroevreo mrmaneem aeomveeescre meos.ac moue, su- more crcures ne aeasara usne one or conseraon oe.r eem soecmcenaso oescosure we aarenoose

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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: obtaining, by a decoder, a plurality of Karhunen-Loève Transform (KLT) transform kernels for a current block in an intra mode; and selecting, by the decoder, a KLT transform kernel for the current block in the intra mode from the plurality of KLT transform kernels.

Description

ornee.: .
Figure imgf000003_0001
Figure imgf000003_0002
Figure imgf000003_0003
Figure imgf000003_0004
Figure imgf000003_0005
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Figure imgf000003_0007
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Figure imgf000003_0009
-
Figure imgf000003_0010
Figure imgf000003_0011
Figure imgf000003_0012
Figure imgf000003_0013
Figure imgf000003_0014
Figure imgf000003_0015
Figure imgf000003_0016
Figure imgf000003_0017
Figure imgf000003_0018
Figure imgf000003_0019
caoneresen acaonsase uon an camsroro..rovsona on,”eo o.nanuar,,, ene, “e eenroes o an wcarsaunscorororareansorm reerreannn an uroses. ceor a ao oresoa c vomeouser ssuore a vare o eecroncevces, suc asaeevsons, meaaers, veoamn,a coens cooems,u smerasr,oanes c,am vereaos,eea recornevces,a sreamnevces, ec.e eecroncevcesransm an receveco onre orenecrwnsee cvoces, veo a veoaa across a communcaon newor mmuncae evce.ueo ame , anor soreea veoaa on a sorae resources oe soraeeavncwe, ve coa caocn o mae ceom usmeunoca comon newor anme memor coo onne or more veo con sanarseores communcae orres ssoree. voero exaama acec,o vrneo - sacnenacrsnceuoeoenrsaeeo.o,nvanc,onxoraonesoe , curexerr e eoon .,ovn meose..,neoru-recon, cnora-nr,e orcoen, ore.eeoe coanaeen aervaana uzesreconnerenne veoaar. mnemo czonne armasaoo cnosmor e o reunanc rae, we avon o vesse voeuoaaa.no aorma uses aower nm aorsme ansse oeresenscosureroveorransormrann an con. veoeconncunc:, o soamnen em,o amenecso oer, aeruersaen oscosaurreurove a meoor ransormernesor a curren nen-overansormransorme ocn annra moe an seecn,eecoer, a ernes.rneore currenocnenra moeromeura o ransorm veo enconn a, sencocnu ans:ec o, saonmne e,mo anm eenncos oer, aeruersaen oscoasruruenerno-voev ae meranosoromr ornee.: . sr eannasoermorernee csurorren a cuorrcen anoc seenc ann,nra meo eeeermnn, e encoer,a e currenocnenra moerome ncoer, a ransormerneor n ar asec, some emomenusr oae oresransormernes. veoecon.e aarausncu enscosurerove an aarausor or mo es one or morerocessors an a memor coueoe one wererenroec oenseso orrs m anore conroceusrseorso sorensrucons execuae e one or morerocessors, e meo a , uon execuon oensrucons are conureoerorm or veo enncc co aor noun .r ao es aeeca,r ra ssoum as see n ec cm. uomens oeresenscosurerove an aaraus one or mo es one or morerocessors an a memor coueoe rocessors,r weerroecnessoers on aen or c moonreurroecessoor sorensrucons execuae e one or moreoerorme meo s, uon execuon oensrucons are conure comuer rena aae s as a orec acco e,r s monmee eomeo semcoenns a osece.resenscosurerove a non-ransor execue umor sorn comuer-execuaensruconsa, wen erorme one or more comuerrocessors, causee one or more comuerrocessorso n m aeo accornoers asec. comuer reaa sxe s asoraece, s momeeu emmoorm seonrsn o coemrueseern-exscosurerove a non-ransor execurmee one or more comuerro ecuaensruconsa, wen ero cessors, causee one or more comuerrocessorso n me a soeve anccor asneco, soem seec eon asec. ransor comuer momens oeresenscosurerove a non- accorno -reaae sorae meumor sorn asreamoeecoe e meoransor conm aenue ers ase r-reaa acs. ee sco,ra soem mee emumoomens oeresenscosurerove a non- accornoe s r sorn asreamoeecoe e meo soecon asec. eaeescron aere un exearmsooes ona ano aree noor reesorncveen oeraeersecsrenonsc aonsuree.oown ornee.: . seccaon,e aucscoramea enxnrawns, wc arencororaen an consue aar os escron, serveo exaanmeesr cnocnssesen o wesceosurerseenscosure an,oeer w e veoocsn. accsor aanocce wa sraommemusraemnen an exema.r ssemor encon anecon emen.s aocaramusranaons oeresenscosure. somem an exemar veo encoern accorance w omememen.aonss a oocereasernamscosure. s usran an exemar veoecoern accorance w w somem.aon emenss a a ononsue osrraese eonnre os se aco nesnu ser crea. osuroec.aram o a veoecoern accorancen accorance w. ssom anemusreameonna oo aow-reuenc non-searaeransorm rocess . rou arenso ocearreasmenssucsosure. aroneno mue veoocs oeren szes anra snaesown a accroarmaencse r wecurs svoeme rmereesmenenaaonon os o araemresenscosure.arcuar,. usraeseneran an encoe o s,. eecs a.ua-reuesraaeas s corumcuorseonus ora an,e en. resuu osraeesvson rocess oe asecen., an . usraesveos aron o a conoc searonnes eresensc..s oson,e anransormen accorance w some exames o scosure..osurse.e or n accorance w some exames oeresen eresensc.s anusraon oe or n accorance w some exames o .osusre. some exames aowcar o exorne non-oca correaonsn accorance w . oeresenscosure. accorance w somse a enxamusersa oone or aensenuarscnorsaurer.econ moesn an n ornee.: . accorance w. somes e axnamuessr oaone oreseneemae useorransormoc cusernn neornem.aes, ann accuosrraancoen w or saomnensc exaoasu creu.sernase onrecon moe an . s anusraon o an examames oeresenscosure. aove area are rean eaeransorm coecensnee- area are reane.e n accso an rn acc ancuo esr wraancoe sn w o ome a en so x em axmaem exa eemaes oeraensreosremn coescocseunres.ne aove . s anusraon o an exames oaeeresenscosure. are reanen accora ransorm coecensnee area . nsc aen wus sroame exames oeresenscosure. aove areas are reane, wereeone o a ann e axaomve aereaasnceuraensoorm ceoe acensnee an arean accorance w some exames ove area anee .s anusraon o oocerarenssenose asncosure. sarnn accorance w some ex nra moe manor marx .s aowcaruasmraens o a meeroesenscosure. exames oerese or veoeconn accorance w some meoor ve.oesn co anoswccosu uasrre. raeusrnan. a me anono arcc voeo encon corresonnoe resenscosu rance w some exames oenerace,n acc.re. orancse a wa sorammemusermaenna aon csom ouenre esennvronsmcoensure c.oue w a user wc areeuesrerancee wn noew acceom maanennearawon ss.eccmemenaons, emomens o numerous non-mn seccea neoowneaeescron, maerr s are seorn orero asssn unersanne suec aernaveesse mnaeeer uesen..uor ex wamee, a warene ao oanreen oo or onnar sne ara varous e suec maerreseneeren cane e o ornar sne ara a veo caaes. memene on manes o eecroncevces w ornee.: . escron, c saomus oeeusrreaseenascoseureerms “rs,” “secon,” anee usene snus oecs, an , ane accomannrawns are useo u no useoescre an secc orer or seuence. soue suncersoaoea emeoamaen usse onesre wsaen mascoesurnercane uner an arorae conon, orersesesose sownne accomannrawenessc orreescrereenn maeeresmenemscenoseurn areoc-seransorm e. n ,are s w -reuenceron r oc-szeransorms, uo×n sze, are enae, wcsranmsaorrm u coseeuceonrs aree zrer roeeso ouuonor veeor, e.., an seuences. reuenc w ane euao, soa onansormocs w szew ore, oro exame, eower-reuenc coecens are reane.or s euaoro an, o×nraenseorm coocum, wns ora anssoermo ccoe wcen asn are aes.emoce, wen euao, oneo rows oransorm coecens ar, wens useor aareoc,e enr aree.enransorm s moesransorm ss removenranesoormc ss us meo we.oeu zeron aso ou su anor vsa cuoens.un aon,ransorm szen, suca encoeraseex o c rae maxransorm szeeenn one nee o s oose uo-en or-en ueransormeeconeccmoremoreenaraonns. n aono - wcaseen em orm eeco oen , aueransorm muen seece srcanesmoermss ursoemoer resua co.neo newnern arnoucnra coeocs. uses - an -.ae sowseassunc eransorm marces are n oreroeee oroona oeranosnso orm me seece . ouanze more accuraeaneransorm marcesn .arox,eeereansnoerrmme maarec veasu aeres aeer caonesocrmenes c aoreeoceanves w-n.e- rane, aerorzona an aer vercaransorm,
ornee.: .
Figure imgf000008_0001
nra an , eveor weerns ae en en a a s oncae wen one oeeow c or no.ere, s ae onoruma.e snans se umasessaononss ae.eoson oeas sncan coecenore nsee zeron-.e., on.eas sncan coecen oeumasocae recons.owever,ou reon.a as euao zero,en s aeno oncaeeransor s euao one,enwo oeras are aona snae an snan manmaee asor sowenornzonaae an. vnerecaerercanosnos,rm res seeeccvoen.orranso arnm msc moe ors usee curr reenmovoncsennrara-mooce an anoocn-sraa aneeneern exencces. currenoc on s useoroorzona an vercaransorm cores s on,en marxrecson,-rmarransorm co .en comesoransorm n aree a res are use.ereore, aeransorm cores use on -.so, oseer sraame,ncun-on - an -,-on,-on an- on,-on - annso-rm, coresncun-on -,-on -,-on,- __a _or u_sea-rmar_raenrs_oarm cores. nraner
Figure imgf000008_0002
ornee.: .
Figure imgf000009_0004
coecens aore re zeurcoee oeu comorexe o-ar ane sze--oc ans w s-z,ewre oruenecr,a onrsoorm w reane an.e euao.ne coecens wnexower-reuenc reon are e re sn ,e resua o aoc cane coe wransorm s moe.o avo _un_anac o snax con,eransorm s as no snae wene eve or s acvase noo eruae co zero.oeamc ransorms seo wen enaeorn urren.soemc cane s enae wen s ower coeocs. encoer ann-re ewe,uenco ene-uansn- asea zaoera n aneewr nea vennsoorrwmarrmarransorm anuanzaona nure.n ,x non-se ersermarransormaecoer se as sown accornoo araeransorm orx non-searaeransorms ae e ancx sze.or esx aamee,oxra s aeor smaocs.e., mnw, caon o a non-searaerarnserormo,c wsc.e., mnw,e . asoows usnnu as an exame.o a xsen, useexnnu,sescre ^ ^^^!! ^ ^^^!" ^^!# ^^! oc ^^ "" ^^ " ） ^^ ^^ sr ^^ s ^
Figure imgf000009_0001
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Figure imgf000009_0003
^^ = !" !#^!$ "! "" "# "$ #! #" ## #$ $! ^^$" ^^$# ^^$$（- ornee.: .ransorm coeec noenn-searaeransorms cacuae as ^^ = ^^ ∙ ^^ , were ^^ncaese suseuen re-oran vzeeco ars, anxosc a usxnera sncsanonrmn marx.ex coecen vecor ^^s oraona co.e coece oreroraocorzona, vercanex ns w smaernex weace we smaer scannnnex eeucceen nono-c. ow-serearaeransorm m uenc non-searaeransormsase onrec marx wuouc mauon aereoraaconos. aow nevoenr-,seear naeransorm soasmemenen a sneass reuceo mnmze comuaona comone-xsear aane mraenmsoorrm s maacerxom soenson neesoe coecens.ence, reuce non-sea reeransorm manea oe reuce n raeransormor meos usen .e x mensona veocn-osreoar aaneramnsenosrmonas voe mcaor ann aerse cnom smon euaoor se reuconacor.ence,nsea ox ma ace, were oows: rx, marxecomes an× marx as ran weree
Figure imgf000010_0001
） s osro or ace.enverse acm msa arxeor, anxrec marx, wcs convenonax nxon- , a reucon marx sze,s reuceoxrec ma searaeransorm ecoer seoe rx.ence,e×nverse marxs use ae marces are aneeranes ceoare ormxar wransoerm sa cmoeercanesnosrn×o-e reons.enxaesnuaaromreexoc m se conuraon, eac o wc ee oe reucemensn ao-exoc excun r-oomxoc. rom o w reasonaseone,r moremmaonrce usaroe.onr o srorerno a reuce co marcess reuce resrceoe acae on a coecens ouseers co mex s sncan.ence, armar-o ecen su-rou are non- s aows a cononn oneransormn coexe scneansnav oenoee zaesro-s wnencanoss aon, aen. ornee.: . ceneccen avoors sen excarna c cooeecceenns sc aan snenccnoe curren esn, wcs neeeor e wors-caseann o nseromnss on. non-searaeransormsorx anxocsox o a mnuxcaronserxe resrcse ose cases,eas-sncan scanosonasoeessan ansorms, resecve.n oer szesessan.orocs w a sae ox anx an wen,eroosse a reserc,oormesae s now ae on once, anaoeo- n rmar-on coecens a ex reon on.s a ermarransormss rreeu zecreo wen s ae,e numer o oeraons neeeor coecenss remara smne suc cases.rom encoerersecve,euanzaon o omzeuanzaonasoeone a w maexnmum ransorms areese. rae-soron remann coecens are enorceoe zero. orers coecensn scan orer,e ransorm se aerree u asreeraon nasorm srea.nescoo ermn ma sesn anrom non-esenarraareercanosno mrmo meaorceesraenrneser canere-ene. one oree mo sorm se __ s useore currenoces __ , __ or seeceore curren cromaoc.or eacranso=reoenra=,ransorm sesransorm canaesurer sece e exrm se,e seece non-searae seconar snaen a-sream onceernra aerransocrm co senaceens. nex.enexs
ornee.: . ncenex
Figure imgf000012_0001
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 oes, as wncaeorne. or. ornee.: . oe wore, resunn: s or:: s.e: sse anar sreen, o sn we smeae:: sse an asmmercnarree s.numa sames,e:: re son a coonnanasem nenosn-ozneros ressauoaw.e. oneere amreen as mono o a sn or a. s moes - anos-on-cereonmeanra anwsaorms u csonre seec-on.se awoeo onumaransormocsn are assocae w eren coreransorms.o sons o - an -ransormsor eac oson re secca,eorzona an vercaransor s secenure.or exame,eorzona an verca srmsor-osons- an -, resecve.en one se oe resuaransormeaoernan sec,eesranesormonr,o c, anmensoroznosnsa se an as verc-.ereore,e suoc resuaoc.e s no aeoe coe w coma coreransorme o a ransor nener-nra moe. axmummramnsrovemenne maxmumransorm szeo.rem., s aze an zeroranns-oorum oerrannessorm are co eexecens.o sze an maxmumnra coeoc canave a sze ox.e m neo, weree seuences ans seo, oerwse.ner axmum szes seo or normramvaer zeron marransormaonrocess,eres no e ou oeraon ae onransorm coecens.owever, s ae, nranasnocrem coecoernnsr oau csee reon are normave zeroe-ou. uze wcn aree c uusrerenornra aensno ner c,nor , on an ransormernes are , , anenransorm o aren. onarmarransormsncun , sze annra moenormao emoe.so ses maeeenen one eren cass n. eren szes are consere, anor eac sze erenranseosrm area cornss areer ceonseeeren.num onenr ora-mnoenormaon.or eac cass,, or eween, an canaeseennra canaes are aave seece coecens.e sums co one sum o asoue vaue oransorm o aowe mare 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 - - - - - - - -
Figure imgf000016_0002
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 ^^ ^^, =,, …,
Figure imgf000016_0001
were ^^ ^^ = ^^ ^^ – ^^.e,ese resuaocs are usees asuae ornee.: . rrearnesnen seames = w ^^, z^ ^e^ro,… m, ^e^an,ore ervaon.eserann sames cane ven: ^^ ^^ ^^ wcs an× marx.onsere covarance marx as eenvecor w = s oer ^e^ ^^ ^^ s ce^ ovamraenncseo- mna orx.so cro nvaaurraancema maervxeos co×nen.s, wenasese a sreeecenon oe oer caen eaenev nauumeerecom asososn enso ^u^ ^^or.oero×oeorormance.e comuaon comex ^^ , wcs acceae.or aareoc,e c c w en ,e comex s w en ,eme comex we ommeexs so wwere ver .or× ocnoeraen anaor×oc,en n coens coernramewor. eme ns, aas aor ^^ms useo maeeareoc sze ease. e eenvnescoonrs o ^^^os.× = ^.^ ^^ ^o^w, wevecr, ^^ ^^as a mucowermenson as× ^^.e cacuae ^^ ^^ ^^ ^ sase euaon as aues were^ ^^ =n ^^ ^c^aes-, eenv e eenvecor marx we ^^enoeseaona marx we ^^e ^^n ^^eaona eemens.e’s mu o ses ouaon ^^oe ^^ ^ ^r^a ^c^ =es ^^ ^o^ ^^s e-ua.on an oan eenvauesee ^^ c^ou ^m^ ^n^ v =ec ^o^r ^s^ o ^^ ^^ ^^- aree eenvecors o ^^ ^^ ^^ w er corresonn oane musnnceanesao ^^ wena e eeenm ev eeecnos enr osve o ma corerces ^^. s ^^ wcmee an ^ rs^ eo = ona ^^ ^ a c^ no.varance marx ^^ ^^ ^^ cane covarance marx ^^ ^^ ^^ .eme eromeowmensona eenvecors o ^^ ^^ ^^ave zero eennvseocnosrs o. ^^e a cnan ^ u^se arecmo or×o.onazae oer− − eenvecorsoe× eenvecor marx.o reuc ono ese mucaon, one can usee oane een ee comexor marx remann−ransorm coecens as zerosv.ecosrs wo neoro armenuaeraenseorrmor,meaanvcne ersroecons can cover mos oe sna ener we e snce en correae we conoc. eases areranerom sames ornee.: . oave aeaesacreoesmmaeevmeenoe con aenes,oecroevoese on sceem coe snuoorcsnereo. m oznao×n,s u×se,o×ee armnne×er caonsonrm mocose. ose o amonee enc aoner se ae,a raev-e msuoronransorm e xeen.en,rare usecs aon w e-anenraoes e curr se o reerence sames neorn a curren .e., aove ner eenerec oronesroeseen cunrr neanura voeoreescec saameosr o veeo c cuorren.owever,o caure , a numer o anuarnra moess ex nenn resouons, e.., usraes aaram onra moes aseneenner aonmen.son .. e anuarnra moes, moeso are conveno sownn., amon - an moeso are we-anen na anuarnra moes, an moes -o anar moemoen. a ra moes.n aonoe anuarnra moes,e aso aen nrecurren moemoen. o are ocsn suanrcee s a. auea,narernarreearon srucures aen ,eses veo uneua recanuar veoocs aso exsorenrareconn .ueo seece wrom anee anu oar onnera mveone vseoorocere,n varooucs s seas oes. anourear snera moes mae suare an recanuar veoocs,esesanar an m cca,oro e anuarnra moes are aso su oes, anuarnra moes amon sae o a veooc sas oreor eacoc sae.en a recanuaroc veooc m es a ceran conon, annex o a we-anenra moe oe convenona anauaer anara mveoe reeceermveneromee v veeoo eneccooeerr us acnco ar mnaon annex o a as sownnaeeow.as,or non-suareocs,e we reaons e veo encoer usnenexes oe co -anenra moes are snae maeonexes oe we-a nvenona anuarnra moes, wc are ensurna aoa n nenra moes e veoecoer aerenarse,us anuarnra moes amumonere.e., an ouarnrnar mao moeses.e.s, unecaannaer m, aone,eenra m mooee an meos uncane.s a resu, aoo ecenc o snannra moe con rovn a conssenesn acrosserenoc szes. ss aceve we ornee.: . moe ae sows a man reaonseweennexes o convenona anuarnra ,s w anerennexe rse o we-anenra moesorenrarecon oerenoc saesn oc saeserecsens a wnex oes a v oeoo ccon, avn reresens ae oe veooc. rao anuarnra moesenona mnoeexses o we-anenra , , , - - ,
Figure imgf000019_0001
econ veo.osc asnocaraaeramn accuosrraannce a wn ex semar ssem or encon an scosure.s sownn.,e ssemncues aom soeurcmeeevmenaons oeresen encoes veoaaoeecoe a aaerme aesnaonevcece.e saourecneraes an aneesnaonevce ma comrse an o a we vare o eecroncevcese,necvucen ornee.: . coouo sxeervs,ers, seravere ceovmsuonesr,s, caemseroas orao comuers,ae comuers, smarones, se- consoes, veo sreamnevce, ,saevces, a meaaers, veoamn eesnaonevce ar oree.n somememenaons,e sourceevce an n somememeen eauonse, wees wnreess communcaon caaes.oeecoe va an.en maa coomnresvce ma recevee encoe veoaa evce caae o movne encoe veoaare ane o communcaon meum or evce.n one exame,en ma comrsoeme sourceevceoeesnaon sourceevceoransme encoe veo a communcaon meumo enaee me.e encoe veoaa mae mouae aaccarecoeesnaonevcen rea a wreess communcaonroo orno a communcaon sanar, suc as communcaon meum ma comcros, anransmeoeesnaonevce.e aoreuenc sec e an wreess or wre communcaon meum, suc as a meum m rum or one or morescaransmssonnes.e communcaon newor, oar aoromaa nre owo ara scuce- asasee nneewronre., suce c aosm amuoncaca aroena m neewuomr, m a we-area rouers, swces,ase saons, or an oer eumena mae use ancue communcaonrome sou uoacae n so rceevceoeesnaonevce. ouunerme oermemenaons,e encoe veoaa maeransmerom an evce maaceo a soraeevce.useuen,e encoe veoaane sorae evce manec accesse eesnaonevce va annunerace.e sorae as aarrue an o a vare osrue oroca accesseaa sorae mea suc emoresv-e,us-r,aassc ms,emoaersaess s,omacscea-n a sorae meaor so r, voae or non-voae memor, or an oer suae ma correson rne encoe veoaa.n aurer exame,e soraeevce encoe veoao ae server or anoernermeae soraeevcea maoe sore veo aenerae e sourceevce.eesnaonevce ma accesse e aneaarome soraeevce va sreamn orownoan.ee server ma v weoaao o ceomesunera coanaeev oce sor.nxeme eanrcoee s verveeorsanac aune ara wnsem sernveree. en.,coore aoecases, arvee.raensesernaroooncoevce s merave ar,cceesswoer encoacee veooraaaerou aenvc seasn, oarr a ornee.: . warae co cnonnencecono,nnec.u.,n aa wrueesscsr cearnnenee.., a, careee msse - connecon, a oas suaeor accessn encoe veo oem, ec., or a comnaon o e encoe veoaarome s aa sore on ae server.eransmsson oransmsson, oraeevce mae a sreamnransmsson, aownoa ane ous o sr a uow co nnm ernnao ace.n ., o eeo v soeu. orc seouercvece mancnucues a veo source, a veo encoer evce, e.., a veo camera, a v e a source suc as a veo caurnneraceo receve veorom ae vo aerco cvoen ceonnarnovnerr,e avnouosr a c caomureue vrerao, a vcs seoseen eneran comuerracsaa ase source veo, or a comna emor exame,e veo sources a veo camer on o suc sources.s one evce aneesnaone a o a secur surveance ssem,e sourcememenaon vce maorm cameraones or veoones.owever,e an mae asescreneresen acaon mae acaeo veo connenera, e caeuore w,reree-scsa anureor, w orre co acaons. encoer.e encoe veoa muer-enerae veo mae encoe e veo e ouunerace oe soura maeransmerecoeesnaonevce va e sore onoe so ceevce.e encoe veoaa ma asoor aernave orec raeevceoraer access eesnaonevce or oerevces,ransmoern. anoraac.e ouunerace maurerncue a moem anor a eesnaonevcencuesenunerace, a veoecoer, an a enscoae vevceeoa.a oveernuennerac.ee m enaconecu vee ao recaeav ceorm anmuonrc aa meo oevm an recevee rove one soraeevce, mancue a vare o snax eem eren, or encoeror use e veoecoernecone veoaensenerae e veo encue wne encoe veoaaransme on a a.uc snax eemens ma sorae meum, or sore on ae se communcaon meum, sore on a wc canne s aonmenmeraemeenasoans,rv eveer. cees anna aonn exevercnea msaancevuceeeass caonevurce, comamoun ac uaseer, w an meae csomnarosne aenv eo a c oe a v.aree osaseavceevcess sucas as aeecuoe vrseao ornee.: . se oasa, aeasmasa, anranc mnoe sa, or anoer ornusr sean vv arecoe. s, en scuocer as an,e ve,oecoe-r ma oerae accornororear san ,ar, , or exensons o suc encoars. soue unersooaeresen acaons nomeo a secc veosnenecon sanar an mae acaeo oer veo enconecon sanars. enera conemaeae veo encoer oe sourceevce mae conureo ecnoerae v coenoemaaa acecoraneo v aneo oecoeesre cur orene oresunuareo snanevarces. mmaare, cons asooecoe veoaa accorno an oese curren oruure sanars ure e veo encoer ane veoecoer . vare o eac maememene as an o a naro scuessaorse encose,r anocraeocnoere ccrccunre, sruace as ornceu ors more msc,roerocersosorrs, a aerras s,screeoc, soware,arware,rmware or a ammae enmemenearan soware, an eecroncevc n comnaonsereo. sowaren a suae, non-ransor com e ma sorensruconsore arware usn one or uer-reaae meum an execueensruconsn scose morerocessorsoerorme veo enconecon oeraonsncuneresenscosure.ac oe veo encoer ane veoecoer mae comenen en ocnoee orr mecooree erncoers orne aco rers, eer o wc maenerae asar o a n somememenao esecveevce. exame,e veo s ns, aeas aar o comonens oe sourceevce or ases ource,e veo encoer or comonensncuene veo encoer o comocnrenes oeoewe wsna reoenrenecveceo.or, e axnamee, oueunnuernaecreace an,oer v aeeoase acoaerr an or ceomsoanenesvncceue man oee vraeeone ac cooeur com asuesncr seervceeo nwew wor r weecren mceaoro., soware,aorms, anornrasrucure, suc asoware as ae ve ervceaa, ornrasrucure as aervceaa. rvceaa,aorm as a comonensne sourceevce n somememenaons, one or more cou comun servce new anoreesnaonevce wc are noncuene more cenevces ma comomrun mcaaee wrov serevern co omne oure mrsorne ceen couevc ceosm, anune s oenrvec oer ornee.: . n neewwoorr, ar souor-ra an wer weersese csosm commumnucnaon neworor exame, a ceuar communcaon communcaon newor o caon newor, or aoa navaon saee ssem communca r a wre communcaon newore.., aoca area newor emomen, aeon newor or aowerne communcaon newor.n an servc as aar o oeraonsescreeren maememene as cou-ase oee csomrovoneens o onee o sro murocree seervvceer com anueorrs wec a aeraesm aeamre onee come aoneenas aar esnaonevcene cou comun servce newor an one or mor s oe escreeren maememene e one or more cenevc e oer oeraons e cou comun servce newor mae arva es.n somememenaons, eerms suc as “cou,” “cou comune cou, auc cou, or ar cou.nercanea as arorae woue ,” “cou-ase” ec.eren mae use soue unersooaeresensarnrome scoe oeresenscosure. comun servce neworescrcosures nomeoenmemenene coumemenen an oere o ce aove.nsea,eresenscosure ma asoe uure. omun envronmens currennown oreveoene w somem.emsen aocaramusran an exemar veo encoern accorance erormnra annaonsescreneresen acaon.e veo encoer ma con rees onerrecve con o veoocs wn veorames.nrarecve ven veoram sea oracruerec.nonero rreeuccev oer c roemnove re saesa on reemunoarnacrne vceoonaoa re wun a removeemora reunancn veoaa wn aacen veorames or ce or seuence. soue noeaeerm “rame” mae use as s cures o a veo or “cure” nonmsoreerm “mae” sn soewnen o v.e,o ceon. rocessn un, aec veo encoerncues a veoaa memor, arecon , auanzaon uoecureuer , a summer, aransormrocessn un urerncues a mno, an an enro encon un.ereconrocessn un annrarecoon esmaon un, a moon comensaon un, aaron un,memenaons,ne vroceeoss enn un, an annraoco un.n someransormrocessn un, acnoe ar sum amseorncuores v aneonveorcse reucaonnszrauconon u.nnn,- aononverseer ornee.: . o,c sucou asn aareesooc renmoveer,o mcaneesso arsaocnesroemw reeecnonse summer ane oer suc asameavese er,rossormucoe veo.noern-ooer, er anoraven-oo er , ma ansoenea umseen aavoenoseeeecnocune,eerresoen aer anc oauonus o noem sumemeore. em sooumenesuesscrraeeeareonr, anensea, coem aonecna, aon m caromea a comeoneon a a snu aaorn c wroemrea a cnom oosneens ac secoercenor an o aumauma comonen,e croma comonen aner croma c o an oer oe comonenase one seece ose.urer, so omoneno mo sa an menoneeren mae an oeuma c u asoeusraea ars comonen croma comonen, a seco omonen,e croma comonen aner comonen,e crn comonen menoneeren mae an oer oeuma meno oma comonen aner croma comonen, an ar comonen er cneromeare conm maonene a. rnem soamnen exa omne oes,eeumn-ao coomoenresn m,aee croma comonen an veooc maerecrove e summeroe . ome veeo, a enneecoeaeeorm o axe orrorammaearware un or maeve a coer ma eusraexe mon one or more o e orrorammaearware uns. e veo veoaa memor ma sore veoaaoe encoe e comonens o rome v enceoo seorurc.ee a vs seoownaanne. v.eoeaa memosr au mearae o soarense re,eorern ecxeam ve, aaor exame, reerencerames orcuresor usen enco eo encoere..,nnra ornerrecve con moes.en veoaa e veo maeorme an o a vare o memorevces. vn veoaa memor ane aa memor mae on-c w oer comonens arous exames,e veo reaveoo oe veo encoer, or o-c reconrsse oc s c esoo swm nno unnens. n.a,r aer recevne veoaa,earon un wne asoncuearonn a veoonse veoaano veoocs.saronn maarero rameno sces,esor exame, ses o veoocs, or oer srucure asnsocanes ws aceco vrenooaar.eeen ve seoranme srsuc ourre msa suce a rse aaruea- asre aewo- ornee.: . remeerrnesonoa as a arraxe or o mra arex. o s naummeesr o w sam samese vaues. samene arra ma asoe oe arra orcureene a s norzona an vercareconsor axes eveno mue v ze anor a resouon oe veorame.e veorame ma e veooce aoanocss or m,aor eexa rmeare,e usn aronn. sames w same vaues, aou as awo-mensona arra or marx o samesnorzona an o smaermensonane veorame. numer o oc.e veoo vercareconsor axes oe veoocene a sze oe veo wc ma c maurerearoneno one or moreocarons or su-ocs aronorm aanocs,or exame,erave usn aronn,nar-ree aeerm “noc or” orr “ve-eroeeoc” aasr useonnere onr a mna coem anoaroonn,nereaor.cu saro au recaen noue suare or non- suareoron, o arame or acure. reeren ar ,eoc or veooc mae or corresono aonce,ror exame,o an reconn or aransormn anor mae or correseeonno a, a, a ocs,o er.m. aoonreeo corresonn ran c , aonoc , areconoc or a con moesec , surce ac a s oon nno e orroco aes as s unu- ra unoc. on mraar seeeccv one o aura oosserecvenerrecve con moes,ore curre e con moes or one o aura o aneeve osoron n veoocase on error resuse.., con raenerrecon co .ereconrocessn un marovee resunnra oro recons eocoe summeroenerae a resuaoc anoe summer rocessnru ucne en acsooerovoces sorn uasxe a esemaern os, a su recer aesn mceoroanm vee suseuen.erecon aronnormaon, an oer suc snax cors,nra-moencaors, n orero seec an aroraennorramaon,oe enro encon un. oc,enrareconrocessn u recve con moeore curren veonrarecve con oe currenn wnereconrocessn un maerorm samerame ase cur veooc reaveo one or more neorocsne un ane moorenocoe coeorove saarecon.e moon esmaon recve con on ceom curerennsa vone uonoc w reanveeor oenec oron morroecesresnc uvneocesronrm onen oerr ornee.: . m cooren reearsesnecse, e.ra.m,eos soeecrov ane aemroorrarecon.e veo encoer maerorm mue n somememenaons,ea meo coonn e moeor eacoc o veoaa. moeor a curren veoramee smaon un eermnesenerrecon o a veooc wne c neran a moon vecor, wcncaesesacemen veoram urren veorame reaveo arecveoc wn a reerence esmaon,e,e arccoormreno aer meoeeornm ensemaaoenrn un wn, as seeuenrcoece oss v oeoenerarames.oon vecors, wc esmae mo n moon moon vecor,onor veoocs. curren veora or exame, mancaeesacemen o a veooc wn a currenocemne o croecur we reanveeo cu arrernecravmee.oce wreene ar rmenereencearearmne m reaaveoe veoramesne seuence asrames orrames.enra un esnae e..,oc vecors,ornra conn a manne maeermne vecors, e moon esmaon un orne r smaroeeermnaon o moon vecorsoeermneeoc vecor. rrecon, or ma uzee moon esmaon un r ceoeerenceocre oc a rveeereoncceorarmee vaeoseoemce m aas coes oer m maac cornresoen voeo aoocco or ae ,nuemrm os ouarxeeeerreennccee, wc, o mr oaereeeerremncnee mercsu.mn o somesomueemenaereonce e veo encoer ma cacuae vauesor su-neerxeosons ns, sorene .or exame,e veo encoer mane o reerencerames xeosons, one-e xeosons, or o roae vaues o one-uarer rame.ereore,e moon e erraconaxeosons oe reerence xeosons anr smaon un maerorm a moon searc reaveoeu recson. aconaxeosons 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 o e ramess, eac o wcenes one or more re r a secon reerence e moon esmaon un sense cacuae m erencerames sorene . anenoe enro encon un. oon vecoroe moon comensaon un ornee.: . ecn oreoneoran cnomeensraeocn,veeroorcmease one meo moon coonm veecnosraoener un, manvove esmaon un.on recevne moon vecorore curren mne e moon comensaon un maocae arecveoco wce moon v veooc,e moon e reerenceramess, rereveerecveocro ecoronsn one o ocoe summer.e summer enorms am reseu , anorwarerecve vaues suracnxe vaues oerecveocroa veooc oxeerence un romexe vaues oe curren veoocen cve e moon comensaon roma comeonenxeereernecnecse o vrauoes.ormen moe oe. c on re csomuaen vsaeoon uonc ma manc ausoeuemna or snax eemens assocae we veoocs o a veorameor use e eraenecone veoocs oe veorame.e snax eemens manc veoecoer snax eemensenne moon vecor useoen erecue,or exame,ncanerecon moe, or an oer snaxnormaonescr veoc, anas moon esmaon un ane moon comensaon un maeeerenn.oeae usrae searaeor erae,u are n somem ceomnceeuauroses. recveocsn a mannnaons,enra un maenerae vecors anec esmaon un ane mer smaroaescre aoven connecon w e moon e samerame ase curoon comensaon un,u werecveocsenn vecors as oose renocen coe an we vecorsen reerreo asoc reco o moon vecors.narcuar,enra un maeermne annra- e n moeo useo encoe a currenoc.n some exames,enra un ma anncoees a cuerrreneroormcan ucsen vroaurous rane-ra-sroerocno ann moes, e..,urn searae enconasses, amone varousesenra-recon m ass.ex,enra un ma seec, enerae annra-moe oes, an aroraenra-recon moeo use an rae-soron vaues usnn ac raaoer- acscoorronn an.aosrs exoarmee v,areousnreasen unra-re mcaon ca mcouae an seecenra-recon moeavnees rae-soron caracer es, moes ase aror scs amoneese ae-saenra-recon moeo use. eween an encoeoroonc a anna asns orenenraa, uneenceormeneso acn amaou wnas o encoseoroonrooruc eerrore ornee.: . enco uneo mca, as ca wceua ase a raoraserom.e., ae numsoerr oons asn u rsaeoroucee encoeoc.nra eermne wcnra-recon moe exse esore varous encoeocso n oer exames,enra un maes u rsaee-esoron vaueoreoc. moon comensaon un,n woe ornar,oe moon esmaon un ane accornoememenaonsescr rorm sucunconsornra recon oc mae aocaseem eeren.n eer case,ornraoc co, arecve erence, wc maeeerme as cose macneocoe coe,nerms oxe oerecveoc mancune ,, or oererence mercs, anencaon eererecvee coaccusaron o vauesor su-neerxeosons. erenrame accorn ome samerame accornonrarecon, or a suracnxe onerrecon,e veo encoer maorm a resua veooc eno cooce m, vaues oerecveocromexe vaues oe curren veooc ve ormnxeerence vaues.exeerence vauesormne resua eannrcaureecoonuma an croma comonenerences. aernaveoe rocessn un manra-rec a curren veooc, as an comensao ner-reconerorme e moon esmaon un ane moon escre anov uen., orenraoc coreconerorme enra un, as recon moeon usearoc euncaor,e ae cunrrreanreocc.onooro scoe,ssnen urnare mcaoneroecrmessnne annra ma encoe a currenoc usn varousnrarecon moes, e.. un asses, anenrareconrocessn un or a ,urn searae encon ma seec an aroraenrareco moe seecon un,n some examesnrareconrocessn n moeo useromeesenrarecon moes.e recon moeoreo un marovenormaonncave oe seecenra- encoeenormaonnccoe enro encon un.e enro encon un ma ererecoannrocees ssenec uenra-recon moenesream. veooc va eernerrecon on eermneserecveocore curren oc suracnerecve rnrarecon,e summerorms a resua veo e resuaoc maencu ocrome curren veooc.e resua veoaan un.eransormrocessne unn one orarn msoorremsse an ressuaro vveeooaaenroan ressorumarroacnesssornm ornee.: . c someacrernanss uosn aransorm, suc as ascreeosneransorm or a conceua uanzaon uerm nr.ans.ormeuraoncezsasnon u unn muaan sezneseer raensusornm croaensocrem coecensoe e rae.euanzaonrocess ma aso re nsourer reuce e coecens.eeree ouan uceee assocae w some or a o arameer.n some exam zaon mae moe ausn auanzaonncuneu es,euanzaon un maenerorm a scan o a marx erorme sanzeransorm coecens.ernave,e enro encon un maransorm coeocaon cw. ennsnuaon a vzaeoon,sere eanmro encon un enro encoeseuanze ,onexavenarr usmne,c e..,oonexavearaeenon aavenarrmecon ,roannervaa,nax-ase conex- con or anoer enro encon meooo orecnue. ronnnro eransmeoe veoecoer as sownn e encoesream maen sownn.oraerransms ., or arcvene soraeevce as eneco curnren un v ma aso enro esnocnoeo oer m reoreovna vecorse an veoe oeceor sernax. eeeme ennsroornverseuaneeo zanvr oea nrms aeee nuan nn czoaeon. un anenverseransormrocessn un a ocnexeomanovreresneerraannso arm reaeroenn,c reesoeccvoer,reoc reconsruce resua veo noe aove,e moon comensaon un m on o oer veoocs.s ocrom one or more aenerae a moon comensaerecve coamcueanesa suo reerenceocs oerames sorene .e moon c n- un ma aso a one or moreneroaonersoerecveoco en seuerxe vauesor usen moon esmaon. r mmer ase reconsruce resuaocoe moon comensae soeracevneeocrou.cee reereen mceooonc co mmaensaenone u unseoeronruace a reerenceocor esmaon un ane moon comensaon un as are un,e moon anoer veoocn a suseuen veorame. cveoconerrec ornee.: . w somem.emensa aonosc oaerraemsenu asracnao ann. exeem vaero vecoeoereconern accorance memor, an enroecon un, areconrocessn u cues a veoaa un, annverseransormrocessn u n, annverseuanzaon rocessn un urerncues a moonn, a summer, an a .erecon annra un.e veoecoe comensaon un, annrarecon un, anoe enconroces r maerorm aeconrocessenera recroca ..or ex sescre aove w resecoe veo encoern connecon w moon vecorsa rmeceev,eero mmooen e cnormoenseacoonn un un m,a weenereanerar-erecocnona uanase m on eneraereconaaase onnra-recon moencaors receve a econ un..s aocaramusran aenera veoecoerrome enromemenano snosm oe exeamreseesn, a a unca oon.e vso,eon soecmoee erxam meas,eemaseeo meor an nerorm.e resenscosure maeve amon one or more oe uns oe aons oe exame,enra un maerormememenaons oer veoecoer.or orn comnaon w oer uns oe veoecoer, esen acaon, aone, ,enrarecon un, ane enroec suc ase moon comensaon un ecoer ma noncueenra un anoen un.n some exames,e veo erorme oer comonens oereconroucnecona onra un mae comensaon ssn un, suc ase moonoeecoee un veeo. oaear c moemmoor ma sore veoaa, suc as an encoe veosream, aa memor mae oannens oe veoecoer.e veoaa sorene veo source, su e,or exame,rome soraeevce,rom aoca veo accessnc assca a caamae srao,ra vae m werea oer. w., areeasss nerwveo orr caormmusnc.aoen v o veoaa, or mancue aoecureuer a sores encoe veoaaroeoaa memor sream.e oe veoecoer sores reerence ve m an encoe veo veoaa e veoecoere..,nnra ornerrec oaaor usenecon memor ane maeorme an o a varvee co on me mmoes.e veoaa namc ranom access memor ,ncunncron orevces, suc as ressve ,essve , or oerouses o memore,vcaes.neoo-r ornee.: . coumsroanevnes ouroese v,eeo vecoeoeraa mnemo.r or an.ue we a areareencoe on aes swoesnnc areaaraee v meemooraae mvecmeso.rn som ane exaemes,e m vaeoe e or s aroav meemore s mamae mee omorevce oer comonens oe veoecoer, or o-c reaveoose co n-c w urneeconrocess,e veoecoer receves an emnconens. a reresens veoocs o an encoe veorame an oe veosream ecoer ma recevee snax eem assocae snax eemens.e veo e enroecon un oeens ae veorameeve anore veooceve. uanze coecens, veoecoer enroecoesesreamoenerae eemens.e enroe mcooonn u vnecors oernnorrwa-arresceo mno moon veecnorsca ororns,ra a-nre oer snaxncaors an oer snax eemensoereconrocessn u con moe recveocensne o vereoreams oesr caomees, ase annrnararerecconv uen cone. oeramree ocroornnrora coe un maeneraereconaaor a veooc oe curren veora cessn snaenrarecon moe an reerenceaaromrevo mease on a rame. usecoeocs oe curren moon comenesnaoen v uneor oameesr ceoeco as anner-recve coe.e., orrame,e ocsor a veooc oe nrocessn un rouces one or morerecve eeme curren veoramease one moon vecors an oer snax roucnes receverome enroecon un.ac oerecveocs mae ma consrruocm ae re reereernencecerarmamee wssn, onse o ane resere,n ucsenrameeauss c.oe veoecoer ase on ree nsruconecnues nre sonmcerames sorene . escreeree exames, wene veoocs coe accornoenra moe ocsore cunr,rene vnreaooc unase o oneocre veccoorsn anroc oessern sn unax eemreonsuc reecsevrecve e enroecon un.erecveocs mae wn a reconsruce rerom samecure ase curre eon oe e m n veoocene e veo encoer. normaonor a vooeno coomce onsaeon cu urnren v aneoorrameenraarsn une moeoner vmecnoesrs arne ocoenr ornee.: . s cunrreanx e vemeeons,o acnenen useecsoee.reorc exoanmneo,rmea moonorouceerecveocsore e receve snax eemensoeerm on comensaon un uses some oo coe veoocs oe v ne arecon moee..,nra ornerrecon usenormaonor one or more oeoe rreameree,n acnenraemreresscoorneramraeme, meoe.o.n, ve ocr, consrucon recve encoe veooc oerame,nerrecon sau orsor eacner veooc oerame, an oernormaonoec sor eacnerrecve coe rame. oee veoocsne curren veo a,oeermmneara,ee cnurrraen v une ma use some oe receve snax eemens, e.., anormaon o wc veoo ooc wasrece usnenra moe, consrucon sorene ,oc vcs oerame are wne reconsruce reon an soue recon sausor ecorsor eacnra rece veooc oerame,nra ecoee veoo eacnra rece veooc oerame, an oernormaononeroaonee mrsoc aosnsn us ceome ceunrsraen eo vn v e ue onorame. encoer ma aurson eenrcoormnneroaon usne cacuaeneroae vauesor su-neerxes o reer oe veoocso comensaon un maeermneenero enceocs.ns case,e moon e receve snax eemens an use aoners use e veo encoerrom enverseuanzaon unenenrvoaonersoroucerecveocs. rovenesream an erseuanzeseuanzeransorm coecens uanzaonarameer cac enroecoe e enroecon un usne sameoeermne aeree uae e veo encoeror eac veoocne veoramer ouanzaon.enverseransormrocessn un aes annverseraannssoorrmm, er.oc.,es asn,onvereseranso,rm an conevercseneerransorm, or a conceua smarnverse xeoman. ensn orero reconsruce resuaocsne ocore ceurrrene m voeon comensaon un orenra un eneraeserecve recons oocase one vecors an oer snax eemens,e summer erucsecoe veoocore curren veooc summne resuaocrom monovne crsoemraennssaoromn urnocessn an unenr aan a c uonrreso.nnnn-oreocveeroc sucen aesraeeocne ornee.: . er,ourere,rrocesseere acnoeor veo maoc.eno ssomoene exameweese,neen s-uomomerer an me e ome, aneecoe veooc mae a .eeco recrove e summeroe reerence e veoocsn avenrame areen sorene , wc sores a memorraemvecse u sseearoare surosmeueen moon, c moam aenssoa soonre oe nceoxe v veoeoocors.ae , or on asaevce, s erresenaon n acauc v aesoesaevce o.. rames conrocess, a veo seuencecancues an orere se o awo- omrencsuorneas. ararca oramume ma saamnceus.erees a samwo-e amrreanss,oneano arera,, anr.s rs awo-mensona arra or croma sames. o croma sames. monocrome anereorencues o n oernsances, arame mae anusraon o an r n onewo-mensona arra ouma sames..s s sownn.oc,ess accorno one exame oeresenscosure. eneraes an encoe e veo encoeror more seccaearon un veorame reresenaon o aramersaronnerameno a se o s. romeo r ma anncruome aonnoeoero nmum.earc o ss o araerrees coonseccau cvoenn u an ras aenr scaen w orer a anee osne a veo ar see suennaceeaveee v saemoe en sczoeerenn on ae s oeuenc×ear,ame×er s,e, s×uc, ana sz×e..u soue noeaeresen acaons no necessarmeo aarcuar corr s sownn., eac ma comrse one ouma sames,wo sameseson one c coonnrreeeeooccss. oe c sronmaax e saemmeness, aenscr sneaxro eeermeesns useo coee uns o a coeoc oxes anowe veo seuence cane r oerenes o ecoer,ncunner ornrarecon,nrare econsruce ae veo arameers.n monocrome con moe, moon vecors, an oer comrse a cures orcuresavnree searae cooranes, a mareeoc. sne conreeoc an snax eemens useo coee sames oe con o co acnevere aeeocer mearorme anxoc o sames. aronn suc asnar-reearonannce,,erena vr-eore eencaoreronn m,au raec-urrseeveareornornm orree a ornee.: . cosmenaceonnereo. on,ee coxnreeocss orseve annovoure smeaerns,o e samcaearvns. oc sze ox.moneour smae a s o rs, an are eacvenoour s oxxococ sze s.ze.ewox s an are eacurervenoour ornoecess oe. e acss aecuae-rneea.a s,ru ecacureeau nsoraen oee euna- rerseue co orreseonarnono ma o co am rersseec ave s ozeu rmana sanmreosm anxwo coorrxes.onene coneoccse o cnrom.a, eac o arame oe same sze, an snax eemens useo coe sames monocromecures orcuresavn e sames oe conocs.n conoc an sn ree searae cooranes, a ma comrse a sne ae ax srucures useo coee sames oe conoc. soue noe one ua-reearonnecen . an s onorusraveuroses an uaernar cane s noso aao varnoca caracerscsase on eacua-rnn eeeear a m-uree-ar caneeroeuens r s. erurcaurre,o onnee sarone aua-ree srucure ann.,ere areveo anar anernarree srucure.s sown e ,.e.,uaernararssearonnes o a conocavn a w an a orzonae onn,orzonanararonn, vercanararonn, nrn saormeamreomnenn,a ano vercaernararonn. ano one or mo ns,e veo encoer maureraron a conoc o o rexs. s a recanuarsuare or non-suareoc o sames sanm wecs,weo s caomrreesroencnon,ns oer o crronmraa, sasm aese, a.n sn oax a ee ma comrse a ouma n monocromecures orcuresavnr mens useoreces. an snax sruc ee searae cooranes, a ma comrse a sneuma,, an ures useorece.e veo encoer maeneraerecve e vreooc esncooreurma, ma, an users o eac oe. recveocsor a.e v nrarecon ornerreconoeneraee ocs o a,e eo encoer usesnrareconoeneraeerecve ecoe sames o veeraom een acsosoecrae m waeneerae.ee vreeco evnecooecrs o usesenerreasec oonn ornee.: . ooeeneraeaseer oencecvoeeoc sams oe as o, onee o vr meoor eencroer maeneraeerecveocs e. ames oeranerame assocae w mores o aere, vee vo eenoco enecroeneraesrecveuma,, anrocsor one or suracne’srecveumaer maenerae auma resuaocore samene’suma resu ocsroms ornauma conoc suca eac e'srecve aocncaes aerenceeween auma samen one o oc.ma umaocs an a corresonn samene's ornauma con ore, rerse,cvee v,e souc encoae erac s maameenneraee a's res resuauaococ ann acare ress auaeroc eween a samen one oe'srecve ocs an a correson ence 's orna conoc an eac samene'sr resuaocn m saamene erenceeween ar samen one oe'srecverocs an ncae ane's or a corresonn same urnearmorre c,o asnuoc. aronnoecomose sraen.,e veo encoer ma useua-ree , anrran euma,, anr resuaocs o a no one or moreuma, suareoc os saomrmeso ocns w recsecev seam.eranrasnosrmorms aoces. a rec oan auar msuare or non-ransormoc ouma sames,wo corresonnran a comrse a snax eemens useoransor sormocs o croma sames, an assocae w aumar meransormoc sames.us, eac o a mae ex ansormoc, aransormoc, an arransormoc.n some resamuaes,oce.umearansraonrsmormoco acsso mcaaee w a su-eoc o maee a' ssu- reoscu oaeoc.'sumaransormoc mae a su-oc oe'sr resuaoc.n monocro er curesavnree searae cooranes, a ma com mecures or srucures useoransorm rse a sneransormoc an snax e veo encoere sa mmae as o oenera onrs moormreroc. a oenerae auma coeceno ansormso aumaransormoc o mensona arra oransorm co core. coecenoc mae awo- veo encoer ma a one oer mcoernesr.ansroarnmssorom a coeracnesnor mmaoec a s oca aaruaonen.eraee ornee.: . arra cnoseormcenoco oc aoreoen.eraee a vero co eencoceerno mcao are one. or moreransormso a oc ereneran a coecenoce.., auma coecenoc, a coecen uan or ar coecenoc,e veo encoer mauanzee coecenoc. reucezaeon ameonuenra o reearsao us aerocoes rsenre wsencraensroanrmso crome ccoens areuanzeooss comresson.ere veo encoeruanzes a ecens,rovnurer enro encoe snax eemensnca coecenoc,e veo encoer ma veo encoer mae neuanzeransorm coecens.or exame,e coecens.n rorm one snax eemensncaneuanzeransorm aorms a rereasen,aeo vn oeo co enecorearmes m ana a osusocua aesarae,a wmcasn ecueres sa av seeuence os evce orransmeoe ne sorae er recevn asesrnaonevce. maarseesreamo o eamenerae e veo encoer,e veoecoer reconsruce an snax eemensromesream.e veoecoer ma e rames oe veoaaase aeasnar one snax eemens oanerom rocesssreearmo.rmeerocesse o v reecoo ennscroucern.eo vr eexoamaae,seene vraeo reecrocaoe enconnverseransorms one coecenocs assoca coer maerorm resuaocs assocae wes oe cue w s o a curren o reconsruc e conocs oe curren a rren.e veoecoer aso reconsrucs curren o corresonn sames ne sames oerecveocsors oe reconsrucne conocs oeransormocs oes oe curren.er rame. or eac o arame, veoecoer ma reconsruce.e.,nra-ramse noree acoovne, vorenor cao-renc aconev aens vneeor- corammeresrseonc usonnorrmarwo moes,s noea coue reare as eernra-ramerecon o ner-recon.wo moes,ner-ramerecon conrues mor r ar moe.eweene reconecause oe use o moon v eoe con ecencannra-rame reerence v ecorsorrecn a curren veoocrom a ueo woc. oc szeorreserven eveeramsrnovne v veeooaaaa, cae aumrnounec onoaoa re anure moroer re renreese vnneo ornee.: . m coaoenn veecsoorseonre a curorrmeneraamce aasoncreases susana.ne wa o overcomns anemoraomansave smar no on arou o neornsnoe saa eweenese neorn veoaaorrecnuroseue moon vecorsnormaon s are aso smar.ereore,sosseo usee moon oe o saa neorns anoremora co-ocaes as an aroxmaonemor mao coorrneanoornm, waocnes. a.,so m roeeornre vecoo ars “ oo a counrrenecor exorner saa an . recor ” oe curren eermnenseae o mo enocon ensm,naoone un veosream, an acua moon vecor oe curren moon vecorrecor oe asescre aoven connecon w .,e oro curren s suracerome acua moon vecor oe curren neeo encuocee ae mooonone vcecororeeerremncnee e morooen c eusrremnaon. unonor s eoa,ceres no ramenoe veosream ane amoun oaa useor reresen o a e veosre n moonnormaonn eame cane sncanecrease. recon o a coeroocces,s a o se c ooo rsunes a neereocvee aoocen a reoerencee vraemoe eurnner-rame veoecoeror consrucn a moon vecor cana ncoer ane a curren usnoseoena can esasonown as a “meres”or s anoremora co ae moon vecors assocae w saa neorn moon ve -ocaes oe curren anen seecn one memerrome no neeocorarn csamnaee moso asn a ve mcooro cna vnecoarerescsoerorrome cuer vreneo.on so,eres ecoer an annex oe seece moon vecorr encoeroe veo ss sucenore veo encoer a ecor wne moon vecor canae recor wne moon ne veoecoero usee same moon vecor eransormn vecor canaesor encon anecone curren. e ,oneransorme core an aseenesne more ecenananmroveeerorman s an snan meos.ore meos wc amourer , , ance so o onr.ansoworemve cr,oenre aareve see sneverroaosreawnacesnaseseec maee,oncs.un ornee.: . ormunas.eo cwurerveenr,ee arnveran,soermr cmoraers mraans norm cores areerverom an eren caracerscs.ereore, mor o aao ae veo conens w ne curren an e ecenransorm cores remanoenvesae. .owever, s on ae,neransorm marcesn are acuaraneransorms no consere. e seconarransorm.e nermar owever,ee sn maareceesnnenran nseeormoe onnneeerve a cano aceeve en sconerc aann ceoconer san. emae macn.e comex oemae macn an eenvaueecomoso e acceae esecaoreec ns norann,raonnranaaes co,ecreaonenr. emaaa seo muacen co aecoe ers.ne onnesoo comex.n , aeransormocs w ncoer anecoer se, wc aneseocs areenveno ^^ro erenoc szes arers coece erveor eacrou.n ve us usn cusern meo.en one marxsransorm marxs u o con, onenexs snaeor eac oncae wc ma s se,.e., wcroue currenransormoceonso.s meo aanuoerreormenwroouraswoacuasr.anerse,aeerarannnn saama ceasssn ocnaeonrouso coveerann sasca caracerscs.owever,erannaa c normoe same enou.econ,e snenex a eve asscaonn mae no accurae ne curren an ,eran assoormrn corse aona overeas. non-saonar caracerscs o veo s arexe wc canno aaoeransorm mnarex cuorreunar vaneeoe coener s co ean cne ranrs.,eorwar annverseransorm saree sameransorm coecens,eerec reconsrucecoonns crauncnoon.eouwareavnere,eu aenoeeranusanozaon o sarn ma noeaoe om rm marx n a comressonerormance. ransor e curren an , seconarransormerme as s aeoeransormms.ocn wc s no ae,.e.,-s aeooorzona an verca oowaceveer,e aos neeceoreransormoc coe we . rooseomroveeerveo-rmmeanncoen oeroenm vs oeo conn v:eo cosns,c seovseurrae mreooosess a arne ornee.: . oranneerane, more e meceonroarnsoermrmarransorm omroveeerormance oe areroose oe ocncunerannaa casscaon meos aa a weverse veo sasca caracerscs,e marces coecveen ss,n aasnmm meercos are assororooseosen wvcoeanoeeuanzaon eec oransorm erenransorm rwar annverseransorm ma use ns s marces anvnsscosure,srooseo enae or . snan meoecon, srsnroucere.eneroose rann an res areurerroose. wc can aceves anr e onou m oc a oorn o raonsoonramraannsnormermass oe m oenane s suaares ercraroer oe sna, enu , ve . ^e c ^^.^,^ o ^r as ^^,e ouu vecor o a or merc.enoe ^^s zero s ^^.e correaoneween anwo eemensn u^^a^^^^o ^^n ^^ ^^s" − ^^ ^^" ^^ ^^# − ^^ ^^# =^^^^^^^^^^^^^^^^^^^^^（-） ^^^^^^^^^^^^^ ^^ ^^ ^^ ^u^r ^^er c −on ^^ve ^^re ^^ a ^s^e =o^^^o^^w^^^^n^^^^^^orm:
Figure imgf000039_0001
" # " # ^^ ^^^^^^^^^^（-）en ^^ ^^" ^^ ^^# = oow:nsscos"ure,#e = marxsenoe as ^^,）eransorm caneescre as ^^^^^^e^^^ c^^o^^v^ ^a^r =an ^c^e ^^^ m^^^a^^^r^^^x^^^^ o^^^^^^^^e^^^ ^^^^^^ c^^a^^n^^^^^^e^^^^^^^（-） ^^^^^^^^^^^^^ ^^ ^^ = ^^ ^^ ^^ ^^ = ^escre as: ere ^^se covarance marx o ^ ^ e ^ ^^ n ^ u ^ v = ec ^ o ^ ^ r ^ . ^^ o ^^ e ^^ n ^^ s ^^ u ^^ r ^^ e ^^^（ a- ^ ） , ^^ ^^ mausrx.e aaona marx.ne oe ^ ^^ ^^" ^^ ^^# = aaona m r wors, marx ^^ musransorm marx ^^no raso sae row vecors o marx ^^ are eenvec covarance marx ova ors o marx ^^. oerenn ee sna.oroweev ser,nan,raec scaesscarcoueroe os saonue ceovnaorwannc,e. me.a,rxe ornee.: . o acea sonn,ae snceeervaroonas usuasrauseon onsra nnon avaae.ereore,neracca erann-ase ervao romaa.ere are severassuesn corresonn m n.ersssuesavenerann seowoervee erenrous,n eacarxr.oue seecroannnssu seamseosw soar ceoeec sa amne s caassscarasnrnu saomn.esno eenvauenecomsran osscnoos m nuree,o vene carnaerane usn snuar vaueecomoson or s wee ore ^r^a =nn ^^,a …a , ^ w^ z aenro ^ m^e reanreosernn nseaa. em,e wrs nsoc, n o conss a esss vsu e ome ^^oac wse, ^^'a,v ^^e = co,ec, …e , ^ a^ oc s'ze.'" ' sames are norma cor wcsuseransorm ocs as
Figure imgf000040_0001
w ze ze as ^^' = ^^' ^^.en, weaeese resua e reresene = ro meansorervn .eserann sames can ooseran " … , ^^/, wcs a ^^ × ^^ marx.eenoee covarance marx ^^^^^n^^^^^^^^^^o^^c^s = as: ^^^^ ans eenveecreemenson^^^ o^^^^^^^^^s^^^ ^^^^^^^ ×^^^^ ^^^^^.^^^^^^^^e^^^ r^（ea-on）seweens covarance marx ^^^^^^^^^^^o^^r^s^^^s as =:^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ases anesre asaoenaran msaorrmx m warx w^^^^^ eae^^^ o c（ noau-m） ene vecorsene eenvecors eenvecors cane oane eror mensene eenvaues.e marx.oea mn eenvaueecomoson or one covarance orer oe eenv waeueas.ve sosresoee eeen evnaerues ooeeerran wsorme co eeenvecorsnescenn orer corresonnoerann sames. censaveescenn e enconnran roen cersos.oasae cuser enesern saamnne meo,e sames areransormocs rereveromserveor eac s are cuserenoerenrous an eac marx aor eanc an emr ocoou szm. e,en on,eera mnanr sxamseersv aere. cuserease oneransormoc sze. ornee.: .ransormnoc an s ezme aonmnerna, mooren.raaresocr eoanc,reanrsann sames are cuserease on areerve,n wc onenra moe cor ormoc sze, a se o marces one ma resonso one marx or severanra moes sare nr ax. ransormocn s ezme anomen,o mrone.erarescoro enacerarnasnonrm saomce ssze a,re a s ceu osere mase on areerve,n wc one mo arces n n an e corresonso one marx. ra emomen,ornerreconerann sames are cuserease on ,sor , m … , oc are sze re a e n ne e o a ca s s o s ue v e a m u o e o o n v m ec o o o r. n v a ec s or or eac . ran e s ve o r r a m resos a" se o# 0 marces areerve,n wc one asoue vaue r oc sze, corresonso one marx.ere,e moon ve ane oe moon vecor vaue oorzona an verca cor useor casscaonse maxmum n an emomen,orner vareuecsoro un-recon an-recon.ransormoc sze,reco onerann sames are cuserease on recon,erecon mon moe an neornemae. soue noeaornra recon moe re eere reersonrarecon. eornerrecon,e on erso moe or moon vecor vaue ornerreconrecon an so sz.e, w.c csonrsosvs oeo r sowow anee cmae usens emomenor a ceranransormoc er oumns o reconsrucexes. sze anrecoann mnoe,rocees nsesuosrrnaeem asaoeosw ar.enveerannno se c ous eearcs usranns coarmoc means cusernase on a ceransance merc,ncunu n ssca- error an sum o asoueerence .enor eacomeo sum o suare recon moe,emae ceners areerve an ransormoc sze an usraen..or eacransormoc xe aoe encoer anecoer, asrann ses oane ane cor sze,recon moe,emae cener, one n encon anecornesornocenss,or ea mcarrxanserve. rs seecease onsoc sze anrecosormoc,e ransorm sesemae oe currenoc aneemae cen moe.enesanceseweeneeasoe mnmumsances s ners are cacuae.eemae cener wcransormoc. eece ane corresonn marxs useore curren ornee.: . ,or eacnr aanns emormomocen s,zee,rroeocseon m meooe,enm emaoemens exene.n emomen ns emomen,or eacransorm cener, one marxserve. sesurerven oc sze,recon moe,emae cener,erann encon aneconrooc seesvse,raor eraocusr uasnsnor cmassocca s-zme,earnes ccusoner mno as .ne ar, a se oransorm marces are seece a e,emae cener marxne ses useo n annex useourerncae wcransorm -eenenrrem caurrenoc. nsscosure,er-ransorm ane ranso eenenrmarransormsroose.ne curren , w erens sarerme s mameeorasn,soermr mmaarrcersa.nsnormss arsecosurne,eesnerno,o.e.,ocs eren ransormsorerens.e rannrocessses seo use rs,erannaas coece usnerensrome en cre asoows. eac,e corresonnrann ses uz conrocess.econ,ornrouce aov eoervee marces usne meosrans nee. encon a nocrom neconrocess,or eacransormoc,n aonoe e eer oorce scze,recon moe an neornemae,s vaues aso uze ooero seece corresonnransorm marx. moes,.e.,nor esr acscnoe ras-urn ree,en coenrrea mnoes oe or mnarces areraneorerennrareconrane.ere,enrare rarecon moe se, one marces ses snrouce aove, we-acnoen mnoraer seec maon c monosess o are m uorezeano orn reecnararecon moes. ane .ereore,nsscosure, or we-anenra mnuarocn emomen, on non-we anuarnra m oess consere.n oneraneor eacoc szen oes are uzeor ,.e., marces are rannrom wc eac marx corresonso onenrarecon moe anenra mo.enra moes we aneor oneoc,e corresonn non-we no onee esm uozemo seece corresonn marx. .e., eac marxsraenne,oor ween arna m noone-w rane annuroamrn -ra~ mo.es aerree uore,ze aorocs, ornee.: . coorwear w an wnve-anenra moe,e corresonn marxsene an uzeor smemrseer. ransorm mnarxe oorcnuaarranee,eeeroerwcar recornasnsruocrmon anronevrerse oraneso srmna saree same veomae con,eransorm coecens areuan .owever,n uanzaonarameer.ereo ze aerransorm usn a ceran anem asnroe re,e orna sna cannoe reconsruceerec.oransor m, asmmerc sroosensscosure,n wceorwar s sarenn e saeve merrsse o ermran waoso rmrmen m,a mae us rre xan wsoerr emen us uenrasn wsorm m erenen sa var emc rsee e ss.zeu ma corce usneren meooerveeorwar annverse marcessn es.ne exame se,or eac sze,e marx areerve rouce asoows:ners escre aove.ns se,eerve or ae vaues accornoe meo samearameers ne orwar marx annverse marx sareeno veo co secon se,e marceserveners se arenerae eoown sere ans,nceunnorma voanu oes,ee or usnna resuess c aonecee ausanezeranrannsoaa usen an ner se,enverse marx are reneor eac vaue rmocs e orna resues aneuanzeransormocs or vaue rane. areenoe as ^^, ^^, … , ^^ an ^^̂, ^^̂, … , ^^, resecvoer a. cerea rnenem vaenue o oren vvaeursee rane marxnv can"e#esc/ "# /̂ nv123 = arrmen as/oo ^w. erenns seare secoen sam emeno'4 vem" rseen^, ^^' ,env mararn ^ xs^̂'orm uns we same szeu coe usn ner emomen,eransor wm ue usnerenorwar marces. orwar annverse marce ns w e same sze ma use searae marcesn eacar sarorerens,n wceorwar annverse ccorn moaeav oene ore mreonre m emarox ememens. severa resos cane se: ens oe asmmerc,or eac sze, ranes, anor eac ",#,…,^5.ese resosvee vauesno ae. rane, one se oorwarnverse marces areerve an ornee.: . nsnscaonsuren, someeo emoon ners emom mens on snan areroose asoows. as snae ae oen,srooseo a eve ono conro o . seuence orc r or sceeaeroncae weer s aeoe canae oneur se .ec oor rn sc s ee ,m.omen,ass sronoaseeoo unseca as aonaransorm curren. asrue,e as snaee woeer s useore use asermarransorm oe. asruencae weer sransormarsne curren or are u an asase,e ore, an nex ze. morean one marces are use rs,nears em s soou naemene o,ur nse carr eo s wosn eeae eo. r reaces useo arne cur w . asase, s aeoe, ren. neour em oerwse s ae. on one s enaeoromen,srooseo reace an an . one s enaeoree, o,ene srec aeoe. morean ae.ne emomen,ne nexs snaeoncae wc marxsransorm o . assrooseo a as ,.e., seconar an asrue an morean orunee,en as snae.o as snaeoncae wc marx s enaeore,en one nex e- aevoe s ae. n ve seconasr,ceesrannann escre.ore secca,erane an marocces aerveex sen aano me encooser ar aenroose an racca acaons,e sasca caracers ecoer.n e s cs o veo conens are usuame-varan,.e., causocra cearxaecerscs m maarc varorerenrames or veo seuences.nese cases,s roem oxe marcese,soe aceveevee o aamave comres msaonrc eesc sennca.no maceeos roosensscosure:ners asec,srooseo a s marcesne seuencearameer se . e e ave snae ore seuence.ene marces are snaenncoeer se,eaeer mnarec seesc aorne aesrvece, ornee.: . encsoeror sosee,oe aa mvearc sesn aareeerveo mra earccescnuecurearameer se . ene eaer an ner asec,srore.ene marces are snaene sceeaer.e encoer se, oseo 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 amreenroosensscosureo coee eemens ca coes,.e.,xonena sne marces arerec coe usn marces areaere uscne u -oom coe.ne secon meo,e aave urer sn snexe marces,ene resues oe marces are orca coes. anuanzanone cuarr eenncoer a annewe,ene-uas aeeweenorwarrmarransorm ecoer se. aso so nzaon annversermarransorma rmarransorm,.ue noea s on enae wene s no usene usenermarrea.,nsorm-,s aes nno uose.orozounrae arnm verorvceaera cnosonrm.eroernm s ,nsesr moaoseeraon ae or oneeroveran,s ance o er anormenocxes c aoeo w encoer an.eecroose n one or more emomens,ere areoa ransor oer se.ransorm marceserneserra m ses an non-searae manromenrareconsorm se are useneroose or .e one ore n moeoeransorm sesre-ene as sownnae. ore curerenoc moes=re_oe_nra=, __ or __ s use cromaoc.or eac ,ransorm ses seeceore currensurer sece er eaxnsocrm s se,naee seece nonn-seex.araeen seecxonsa srnaranesonrm canae onceernra moe, a corresone a oer a-sream n somnerra r en amso nsoorm rmm co seenes,ce s uen seres. a arne oransorm ses,.e.,or eacnrareconransorm ses ne non-searaeransorm marxor eacs maeoe usreaneroose .es coe usn moe,enra moe rx =nsnorrmanoeex wneraooowe:n mneraoo:e s ornee.: . werenram sooeusee nnorea moeer aonrxsenexore ransorm se. ao ose or mae useu nomeoe ousveeesecrres cwoeo scceenme ssu.-nrcoeu are nons-s resnrccaen,oe ancaexe c on a coecens oson oeas sncan coecen. morean oner oneens oneransorm se,e nex nee ansorm marces are usen eac onnrarecon mo soe snae, wc cane conex coenoeen anoro e.urermore, s aeornranonra anner sces, are snae seuamraae an.orronmear. sc aeuaerueaesre eenasesa,e, a snnecesorumnae axnsroma an useorouma anro snae eron-ouma. rooseo r,e zucee comuaona comex ane memor sace o coecens oe zeron-ou eron-ou o sroosensscosure,ncune saaoman n aner seaueancomoamna zner zoeron-ou. wmenson o, n-ou,e vecor wmenson os maeo a vecor non-sea weres smaeran.or a vecor wmenson o,eca wereeraer manasroxrmassee s maaer ox mxu.ncaeonroeosweee snaeaoma mna zrerxo anne vecor, emenson oeorwar marxs reuceromxox.c -ou meo, marxseransose oeorwar marx wem corn,enverse areroosensscosureoervee or enson ox.momens ae meooervee reuce saaoman zeron-ou. soue noe n one emomen,e s noorme seaoaeomoown emomens. scarne eenvecorsn euaon −.e an zeron-ouserve w ^^ene eenvecors noee marx as = ^^", ^^#, … , ^^/ eenva'ueec ases.e eenvecors cane oane erormn eenvauesooemeors won oer eenv oencorsen coveasrcaenncen ma orrxer. ooee eaen wveave soree ener oeransorm coecensaveescenn o aueso maee ns emomen,e eenve rer corresonnoerann sames. marx cane∗ r cors w smaer eenvaues arescare.e reuce =ere ^^s"e,n ^^e#, … as , ^o^5o,^w^^^: ^^ ^^ ornee.: . seconarranesorromo osere.uenceromean zerosn co-nouuces a onee w resen s useoreransormocs oane.n somermarrans ueoc,ermar coecens arescare..-ro orm coecens aree we oerransormoc,n wceuear rerveseen sosmee e axreaamae zerarnosnor-mou co meecoens os arermar . roves an exameaeransorm coecensnee e reane. reane.. roves an exameaeransor -aove area are reane.. roves an exa m coecensne aove area are . rove meaeransorm coecensnee area are reane. reane, wereese an a enxam aoeve aareasencraunseorom coee acoevnes arenee an aove areas are n some emomens,owo a e zeron-ou mea anee area. -.n one emomen, on one oe zeron-o os maeusraen. rmarransormoc.ne anoer emomeu meon.-s aeore roose.or eacrmarr n, zeron-ou meos swc scemes zeron-ou meo ansormoc, annexs snaenesreamoncae wc arsx aaeroeoc. rom a recann =uearrooocsen o a,e resueoc orrmarransormocsrs convere ^^!!^ ^^^ ^^^ ^ v^ecor.aeeoownxoc as an exame. rer: s co ^^n"v!e^ ^r^ ! "e" "^ ^^" ! o# # ^ a ^^ v" ! ecz o or 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^!#^ ^^"#^ ^^!^ ^^" eenen,n.e.,eecraenmeme mnaarocens, areerarannne anorerveareonn ooc s maaarrxx.soc sae eren marces are useorexoc anxoc w es.or exame, o reucee memoro soree coecens oeen ms no euao. sarn meos areroosensscosure.eas arces, marx e same marx amoneocs we sacea o erne sarnso sare ore marx sarn. me area.evera emomens areroose ornee.: . saree samne one em maorxm oern,e maorcxs se w.or eexa same areanumer o samesneoc aveerenoc sae,ue same marxme,xoc,xoc anxoc ne anoer emomen,e w os o aneeocoessee soacms. anoeroc, anee oocse same we w ooec w,eeneoc o anoco cacn sa srearee seam saeme ma mraxr oxr or m maraxrx se se..nore e exmamoem,enx,oc anx marx,oexoc anxoc arers convereoeo usee sare menson ox usn ceran scan orers vecors we same aeooneersx meooc, a rnaserx scano o. crero, c voenrvcear sceano ocresr or an oer scan orers ne secon meo, wen converneocsoe voee vecors. aeoeoc we sae ox, we ver cors, raser scan orers e sae ox.r verca scan ore ca scan orers aeoeoc w raser scan orers aeoeocrs aeoeoc we sae ox, we verca scann oreerrs a meeo, aone ceroa w cn s wcane o sra eere o sancuxn.u nomeo raser scan an xsrsransoseoeoc we sae oe ox.xe.neoc we sae o oxs aeoeransoseoco conver e scan orerore sae n emomens oeresenscosure,enoo ac vescorrs. raser scan or verca scan.n o convereo a vecor w sze ox,ee sne exame emomen o marx sarn,or aoc w w sze ox asusraarerans w ,eocsrsransoseo aoc ver en. anen convereo a vecor usn raser scan or ocear sacnasn.osee coorcres.osn snownn mar.x,seenxe woceeo wrera,nseosn ans cnra moenra moe o m,.e.,ereconreconsromoom-eo a oe wransosneoc,n aonoe cane o ove-r.owever, aer canes,.e.,enra moe eoc sae,ereconrecon asoransose s caneo m.ereore,e corresonn marx oe ecoer, weoncesoecn we w saen ora mxoe an mnnrsae mao oe m m.sr ao rmecaneuerasrecocve a ones ornee.: . ueseoseanreraen w m,arex. saee m oeransoseoc ane maenra moe msneeconn on “e emoormen,e- wnee-aane nraenn orraa e m m s,oo ”ees m e ar mea nsoe e cronvse nre arseooorws. asescre oows: a moe mserve as merwss oranar moe,en m = m escrenneee, e a mno =er e –m momen, we-anenra moes are consereor as asoows con “ ore-nenraoes,”e maenra moe mserve s m: es -= oranar moe,en m = m erwse m= -,en m =-m ses sou, me = noe – mane aove examememenaon,eocsransoseosn oarmereamnensa won,. ooewremver,emeenraooonse marx sarn meos nome ma sea a meos are aso aowe.or exame, we uoce neesoereransoses wsareranse. ernesernverevous secneosns,eoerc eaocnoc sze annra moe, on one ransorm one sneraneso arnm aernee. canowever,ueoevers oe resue sna caracerscs, roose ,nsscosure mnou caeure aernee s seneacons mreuoonss.ourermrovee erne seecon,or eacoc sze rove.ne mue a annra moe, morean one ransormernes erneaeerv.e an uazes.rueo,r senan,erneenexasusrresr snaeoncae weer ss useore currenoc. snaeoncae wc erne sze annrna m soomee esxa nmoems oe.eresenscosure,e numer o ernesor eacoc ere so,e numer o ernes oerenoc sze mae ernesn a.reorer exvaemoer,orcees w sezren oes arxee.rveorocs w sze ox we ornee.: . ernes seecevera emomens areroveo snae snax eemensor w mue ceran sze,nw oonn oe. emoermneens, aree cearseve o.wos sownernessrove,asoreocs w rsns conex coe w conex. s ennaae,eeeow,ns emomen,e conexonc seconnsen coe w narazee w vacue erneo0nse uxze. sx noa ennaaone w ceran
Figure imgf000050_0001
sze,wo er, , soreocs ersns conex c nes areerve.s sownnaeeow,ns emomen, ass moe,.e., euaoe w conex. s enae,e seconnsen coe w narze varouea ono0nceaxe wc ernes uze. sx noa ennaaone ocs w
Figure imgf000050_0002
ceran sze,ree ,ernes areerve.s s ,asore emomen,ersns conex coe w conex. ows ennanaeeow,ns ns areen coe w conex an, resecve0oncae e,e secon anr narze vaue 1o2 0nex wc ernes uze. s nxoa ennaaoen
Figure imgf000050_0003
ocs w ceran sze,ree ,ernes areerve.s sownnae,easore emomen,ersns conex coe w conex. s enae,e seocow,ns en coe w conex1.erns coe w 0ass moe,.e., euaroan.ns ornee.: . narze vaue o00n 1ex sxanaon e nrnoe enas uese ocs w
Figure imgf000051_0001
ceran sze,our e,rnes are ,asore emomen,ersns conex coe erve.s sownnae eow,ns ns areen coe w conex w conex0. s enae,e secon anr narze vaue1 ano0n2,e rxesecve. sx noan eanaone ocs w
Figure imgf000051_0002
ceran sze,our e,rnes areerve. ,asore emomen,ersns conex coe w cone s sownnae eow,ns en coe w conex.erns coe wx0.as s enae,e seconns narze va1ue o0nex s moe,.e., euaroa. sx noan eanaone
Figure imgf000051_0003
rooseoerve marces an a ese, aaraus are oe rann arenrouce. new marcesn veo con.e aorms roose.e snan meo emrove rannaa cusern meos aresrooseoanee s o n veo con areroose.smmerc meo rooseo uanzaon eec oeransorm coecens. or s aso snan meuroersm areroveroeos ceo.n ec zeenrocn o-ou me.onsa ar,eeroo-seeveo re mucaercees ornee.: . c aoremruoaosoenao co remuceexe a mnem moermoor soor seoreee ma marces. marx sarn meos e con ecenc oransorm con rces.s execeourermrove some exames. ose areoswecnarscoussurrae.nn a s me eeo an ,e moer v. oeoeconcnunes ac ocoarnance w ecoer, aura oarunen-overansorm ransormern n, a annra moe. se ,e meoncues seecn, esor a currenocn ore currenocnenra moerom eecoer, a ransormerne n some exames,e meoeuurraer onc ransormernes. rom an encoer, a aancaes we ues: recevn, eecoer an ecoer, one ooown acs:n reson er s enae anerormn, e eecoer,ae s enae,se anoeermnnae asrue,eermnn, ernenex orn reso recevn, eecoer anrome encoer, a ecoer,ae s non esneaoeeermnnae asase,eermnn, e eermnnae asr.n some exames, se ncues:n resonseo currenocne ue, seecn, eecoer,e ransormerneore n somen exraam moese,asee oneernenerneexnex. mzaon seermne usn aae-soron n some exeacmnesue. sze annra m , a numer oeura o ransormernesor eacoc corr oes nome.n some exames,eura o ransormernes sze,eseonrso s onzee osr moreerenrsromocse o se aconrs s szzee, a ann one or more seconocs o a secon ocsserenrom a secon num ars numer oe one or morers exames, er oe one or more seconocs.arcuar,n some numers.ers szes ×,ers numers,e secon szes ×, ane secon ns, anen o snoem oer e mxoarmeenss, coerresonerone onen orex ms anarze vauencun one or more erne an an s seocmone examersa,nseormuraerne oenraarno zsre eor cmoneerxnse.sncue ars ransorm ne one or more conexsncu vauencues arsn an a secon coe usners conex anee se aconrs connse cxon anex a c soeceon w coneex secoen crsonenx,s o cro cnoeex ornee.: .nn acaea wss moe.n some exames,en coen aass moe means euaroa o nc some eexranmese uze. eermnnaerss, se ncues one ooown ses:n resonseo eermnnaersnnss a,eermnnae s no enaen resonseo aseeransorm ne seconns, seecners ransormerne ns, se erne orn resonseoeermnnaersns ane secon nc snomee secon ransormerne ase ransormerne. ransorm exames,eura o ransormernes comrse ars vaue comerrsnees, a a sresconn, a secornansonr,m aner ane,r ann are one orransormerneenarze conex, a secon conex, an a more conexs comrse ars e seconns co r conexersns conex coe usners conex er conex, orn ceoxe cone a usnass meo seceo.nn s coomneex exa amnese,ern cnoes conn aex coases w means euaroa oncae w moe n some exame c ernes uze, eermnnaerss, se ncues one ooown ses:n resonseo eermnnaersnnss,eermnnae s no enaen resonseo ase ransorm ane seconns, seecners ransormerne anerns, seerencenn rees seocnosneoeerrmannsnormaernee arssens,ranes seocromnernnes or,ne resronseoreanersmornmnernae aseresns,e seconns, anerns, seecn n some exa ransormerne. ransormerne, a seconmes,raensourrmaern oe, arransorrmanseorrnmese cronme, arnse a aourrs orran msoorrem conerenxes coemrnaserz aer vsa cuoen coexm,r as ses arsn, a seconn, an arne one conex coe usners co econ conex, an ar conexersns anerns c nexe seconns conex coe usne secon conex exam onex coe wer conex, or coen aass moe.n some uzee.s,en coen aass moe means euaroa oncae wc ernes eermnnn saomee exrasmens,s se,eermnncnuesa onee oosow non en saees:nn r reessoonnsseeoo ornee.: .reanesrmornmnernae aseersnrsan,soerm seceornnenn rses,o annseoeerns, seecners seconns, anerns, seecne seconermnrnaersns,eransormernen resonseoeermnna ansormerne ase ns, seecner ransormerneersns,e seconns, aner eermnnaersns,e s ase ransormerne orn resonseo ransormerne ase raneconns, anerns, seecneour some exames. os aeorwecsenarsucsors saor urnm e a ae n mrnee. no corresor veo enconn accorance w econusraen.. se ,e meoonenncceu wese meoor veo aura oarunen-overansorm rans oann, an encoer, moe. se ,e meoncue ormernesor a currenocn annra currenoc. se ,e meseermnn, e encoer,a s enaeore ore curre oncues seecn,e encoer, a ransormerne nn somoce exnaenra moeromeura o ransormernes. aecoer, a mes,e meo urerncues: snan, e encoer ano one ooown acas:na resnoncsaeeso weeermernnease enaes e annaeer,o sremnn,, ee e ennccooeerr, e a asrue, an snan, e encoer anoeecoer, a ernene , resonseoeermnnae s no enae, sen,e encoer, x orn n some exames, se ncues:n resonseoeer e a asase. seecn, e encoer,e ransormerneoremnnae s enae, one currenocnenra moease mzaonne srnoemen eexex cam. neuse,.ee ernenexseermne usn aae-soron oss o veoua aan ecnueenera mnmzese amoun osoron n some examse amoun oaa reureo encoee veo,e rae. s cze annra moes noes,m a neum.ner s oomee exaumraes, oeuraranso ormernerasnsoror emacernoecs sozrer,eseonrso s onzee osr moreerenrsromocse o se aconrs sze an one or more seconocs o a secon ocsserenrom a secon numer oe s onzee, o arn mo are sresco nnumeorc os.aer oncuea orr m,onre somrse ornee.: . e nxuammeress,.ers szes ×,ers numers,e secon szes ×, ane secon ns, anen o snoem oer e mxoarmes,e ernenexs anarze vauencun one or more n some exameens,s corresono one or more conexs. erne an a secon ranseormuraerne oenraarnzseorm vaueernencsuncesu ae arsrsn anr aan sescorm ne one or more conexsncue ars conex an a secon c on coe usners conex ane seconns co onexersns conexn aass moe.n some e nex coe we secon conex, or coencae wc ernxames,en coen aass moe means euaroa o eermnnna somee exames s ne u os, e sz naee. e, senncueesrs onen o asonow resnon sseeos:n resonseo ransormerne ase ransormerne, seners seecners , orn resonseo seecne secon ran n as ane seconn as senersn as ane secon sormerne ase ransormerne,ransormenrn seo,m ae se ecxoanmes,raensourrman as ern o. e, an aranrsormrearnnes comrse ars vaue comrses arsn, a seconn, an sormerneenarze conex, a secon arne one or more conexs comrse ars e conex, an ar conexersns conex coe usners conex e secron connexs, c oorn ceoxe cone a usnass meo seceo.nn s coomneex exa amnese,ernns conex coe w means euar coen aass moe rmn soa oncae wc ernes uze. ee ome exames, se ncues one ooown ses:n resonseo rannsnormaerene asse no enaranes,o sremnernee, sersnn aesrsn rens aosnse ano seee scners n resonseo seecne secon ransormerne as econn as ersn as,e seconn as, anern as e ransormerne, sen ransormerne ase ransorme orn resonseo seecner anern a rne, senersn as,e seconn as,ransormenrn seo,ms ae se e. cxoanmes,raensourrmaern oe, arransorrmanseorrnmese cronme, arnse a aourrs ornee.: . orran msoorrem conerenxes coemrnaserz aer vsa cuoen coemrses arsn, a seconn, an arne one conex coe usner x, a secon conex, an ar conexersns anerns s conexe seconns conex coe usne secon conex exame conex coe wer conex, or coen aass moe.n some uze.s,en coen aass moe means euaroa oncae wc ernes eermn some exames, se ncues one ooown ses:n resonseo rnnae s no enae, senersn asn resonseo seecners aneansorrmner anse asn reesonseranos soermecnerne,e se secnonersranns aosrm,eer sneeconn as,ransormerne, senersn as,e seconn as, anern a aseo seecner ransormerne ase ransormerne sn resonse e seconn as, anern as orn resonseo se , senersn as, erne ase ransormerne, senersn as ecneour ransorm as. ,e seconn as, anern comun env.ronm seonws a co cmanuena ern ovr aonmaaenro coue w a usernerace .e ncues arocessor cessn server.e comun envronmen erocessor , a mcaemo cronros, o avn annuuu nerace . , suc ase oeraons a era oeraons oe comun envronmenmaeroce ssocae wsa,aa acuson,aa communcaons, anoerorm ass onr s.omee orocees ssoersn mea aonvce-ueesc ornee or m morerocessorso execuensrucons mancue one or more moues eos.oreover,erocessor oer comonens.e aacaeeneraconeweenerocessor an sne c ma rocessor mae aenrarocessnn , a mcrorocessor, a ec mneme,o arracas cornoceussrenon , oree. oe comun envronm sore varouses oaao suore oeraon momens o sucaaen .e memor mancuereeermne soware. comu ncuesnsruconsor an acaons or meos oerae onememen envronmen , veoaases,maeaa, ec.e memor mae ereo,n suec as us an aancean oo vmoacece osrs none-vmooare memor, aenvcese,c orrc aa comrnasaaone ornee.: . roramm,a aeroeara-mmnaeemeoar-nemo,r anrasaerorammaeea-nemor manec memor , aea-nemor , a n one, ex aamase m,eme morem, aor manecs o cro oncuas. rocessor wereerocessor reo sorensrucons execuae e meo asus , uon execuon oensrucons,s conureoerorm annerace moerae ues, snucenra acse.- aeroo. avr,e as a cncne wraecee,euweoenns, aenroceessore. anererancueu are nomeo, aomeuon, a sar scanuon, an a s euons manerace can meum comrs enme ao co o scanuon.e n an u umraeen, w oere an ros e r anc amsooe s,rr oo a rvn exeec am ao neoe ,nr. n-raens moemro cromue,r e-rxeeacae sorae rocessor ne comun envronmen ,orerormne aove-euae e anor sorn asreamenerae e encon meoescre ao scre meos ecoe eecon meoescre aove.n one exame, ve or asreamoe e execue erocessor ne comun envronmeeura ororams ma rome veo encoer n. asream oraa snreao receveor exame,normaonor exame, veoocs reresenn en mncun encoe veo or more snax eemens, e coe veorames, anor assocae one envronmen oec., an ma asoe execue erocessor ne comun sream oraa rormeecon meoescre aove accornoe receve roces sream.n anoer exame,eura ororams mae execue e aovesoor ne comun envronmen oerorme encon meoescre assocae en ocnoee or v meooren sonrmaxa eoenmeonrs e,x eamc.en, vo aeosorecams re oresenn veorames, anor execue erocessor ne comun envro raa sream, an ma asoe aa sreamor exame,oe veoecoernnmen oransmesream or comuer-reaae sorae meum maa ..ernave,e non-ransor comrsn encoe veonorma ve soreeren asream or aaa sream rames, anor assocae one or moreo snnoarx e exeammenes, veneeoraeocs a rne ernecseonerno ern ecxoame veo veo encoern. usn,or exame,e encon meoescre aovee,e aecoeror exame,e veoecoern. or. necon veooar uas.ee ornee.: . noenm-roarnsor co,m aue-r-reaa, ae m saornae meum mae,or exame, a , aanomccess e. ecae, ao sc, an ocaaa soraeevce ore escre an an emomen,eresrove asreamenerae e encon meo emomen,oveer oers arovsreeam aosreeamec coomersne enceocoen v meoenoorescre aove.n an encon meoescre aove or encoe veo maonenerae e meoesc normaonoeecoe eecon rocessorsnr o arne ex e a amo move. e,meen,roceessso arsorov ane ae co nmon-urannsoevce comrsn one or more meum ore memor avn soree r comuer-reaae sorae or morerocesso ren aura ororams execuae e one rorams, are conrs,u wreeroeneroerm onee o ar movoer-eesrcorcessors, uon execuon oeura onsruconsno arn so emraeo omrernan,smesrseons o as aorsovree m ae cooms.uerroramroucavn enerae e encon m eam comrsn encoe veonormaon eeco eoescre aove or encoe veonormaonoeecoe a comuerronram mreoouc coemscrrsen a aouvrea.n o anro ermamos,more enxa,meeres asorove execuae erocessor ne comun envron ,ne memor , escre meos.or exame,e c men ,orerormne aove- comuer-re omuerroramrouc mancuee non-ransor more sna , aan eem so s,orame ae mne,u nem a c.omrocuesnsn envervoncmen maememene w one or s, s, s, conroers, mcro es s,rorammaeocevces comonens, -conroers, mcrorocessors, or oer eecronc n aonr eemroormmnen,e aeroeve meos. sornesream on aa soras ae msoeruomv, wee are mneoe os sreoarmn co amrssreesam en,c coomrsnnormaonenerae e encon meoescre aove or e e veo eecoe eecon meoescre aove. ncoe veonormaono ornee.: . eneraen aen e enmcooermeesncr,eer aeosve a.sno anr eomvoem aen m,eoeresorransmn asream recevn asreamoeecoe eecoerescre asorove a meoor ans noneeneescrooen e oxauesrveese onrmscoesureoaseereen a senreosevne. sceosoruroses ousraon varaons, an aernavememenaons we a ure.an mocaons, avneene oeeacnsreseneneoarenoose o ornar sne ar rawns. reonescrons ane assocae resensconsuesres sse ocncan sae oerwse, an orer o ses oe meo accornoe resenscosure eneoeusrave, ane ses oe meo accornoe accorresn are nomeoe orer seccaescre aove,u mae cane eenosracca conons.n aon, aeas one oe ses oe meo accorno ec eomsuroe mmaense w aerues ceo,s ceonm annees ocrreeeen o arcceorron exoaracca reuremens. scosure ano enae oers sene aro unersanenernces oememenaons anoes uzee unernrnces scosureor varous varous mocaons as are sueoea an varousmemenaons w unersooae scoe oes rcuar use conemae.ereore,soememenaonsscose cosures nooemeoe secc emomens oencue wne sco ana mocaons an oermemenaons areneneoe crcures, wec aonvceu meee o aose c maarese on sen emsc cecmos neunre ee. usn an aarausancues one or more s,a snarocessnevces srae crcus s,a snarocessors rorammaeae arras s, conroer,rorammaeocevces s,e eecronc s, mcro-conroers, mcrorocessors, or oer a comonens.e aaraus ma usee crcuresn comnaon w e oer morwuaer,e u onr s,o orw suare-u cnomosnceonsseo aroevreo mrmaneem aeomveeescre meos.ac moue, su- more crcures ne aeasara usne one or conseraon oe.r eem soecmcenaso oescosure we aarenoose sene arromneneo cover an varaonns, an usesr,ac orc ae oaaeonssc oosureessccoosseureeroe.owsn aecaenoenras ornee.: . orr cnucsoemsarereroac ancenncue arn. sucsnearuresromeresenscosure as come wnnown as exemar on.e eneae seccaon an emomense consere acaons seccaon an emomens are consere as exemar.e escre ao wnen ve anee ao urse c rco aave eer an na var e acea cormeosnes an, use nnssc,o o rsr au a wrea ns na s,oons anm o aee vaos roec u eoxsaucre e.momens canes cane mae wouearnrome scoeereo. s mocaons an

Claims

ornee.: . o.a mnen,o aor v: ecoeoer,e acourna, co omrasrn: or a curreno unen-overansorm ransormernes s cn annra moe an romeeeucran, oeecraonsero,r amernerasn.sormerneore currenocnenra moe re.ceven me,o oe ceacomer, anurer comrsn: enae a rom an encoer, a aancaes weer s ernormnn re,sonseeoecoeeerrm, onnen ooaowen acs: ae s en asrue,eermnn, eecoer, or ae, an recevn, eecoer anrome encoer, a ernenex aen resson nsoe eonaeeerm.nnae asase,eermnn, eecoer, ore cu.rrene meocon o ceanmra m, woeereron seecn, eecoer,e ransormerne n resonseoeermnnaemeaurasru oe, sransormernes comrses:ransormerneore currenocnenra moeasee oencne, eerneeconere,x.e on .e meo o cam, wereneura o ransormernes corresono szee osr moreerenrsromocse o se aconrs s szzee, a ann a oners or n mumoree sre ocone onoec osr o mo are secrosn sozce,sers erenrom a secon numer oe one or more seconocs. s szes. ×e m, aenoe o se ccoanm nu,m weerresn.ers szes ×,ers numers,e secon ornee.: . one or m.oree mnse, ano oe c oanem or, m woreerenns ceorresonerneo onnee oxr mso are cnoanrzeexs v.aue comrsn .e meo o seconeura o carmans,o wrmereenrn:es comrse ars ransormerne an a ee onnra ean ors rzeorm mo vreaueern ceomrses arsn an a seconn ersns co cnonexex cos ceom usrnse aersrs conex an a secon conex e seconns coe we secon con coexn,ex or c aonen aass moe. ore cu.rrene meocon o ceanmra m, woeereron seecn, eecoer,e ransormerne ooown se meura o ransormernes comrses one enae ns r:esonseoeermnnaersns,eermnnae s no ers ranns reosrmonseernoee asermennraanersns ane seconns, seecn n resonseoeermnnasoermerne or e secon ransormerne ase ransorrsmnernse. ane seconns, seecn . ransee ormur maeo erne o o ca , an armanrs,o wrmereen rran: ness comrse ars ransormerne, a secon enarze vaue comrses arsormerne e one or more conexs comrse arns, a co sneceoxn, a sne,c aonn a crn ersns conex co onex, an ar conex e seconns conex ceo usners conex erns conex coee w usneer sec coonne cxo,n oerx co aen wn aass moe. ornee.: . ore cur.rene meo o cam, weren seecn, eecoer,e ransormerne ooown seocnenra moeromeura o ransormernes comrses one enae ns r:esonseoeermnnaersns,eermnnae s no ers ranns reosrmonseernoee asermennraanersns ane seconns, seecn ns, seecnn reseo sencsoenoeermnnasoermrsernnes,e seconns, aner n r ransormerne ase ransormerne or ns, seecneseonsreoeerramnsnonrmearnee asrsensr,anseo sremconernen.s, aner ranse. ormuera me erne oo o ca , arramns,o wrmer ranseeronn re: ms ceormner, asne a arosurransraonrsmormerneer,n ae secon ee onnea orrze mo vrea cuoen ceoxmsr cosems arsersn, a seconn, an arn ersns conex co ars conex, a secon conex, an ar conex e se e usners conex ecronnns cso cnoenxex co coee w usneer sec coonne cxo,n oerx co aenn aass moe. ore cur.rene moeco o cam, weren seecn, eecoer,e ransormerne ooown se nenra moeromeura o ransormernes comrses one enae ns r:esonseoeermnnaersns,eermnnae s no ns, seecnn reseonrsseoerearnmsnornmearnee arsns,e seconns, aner n resonseoeermnnase ransormerne ns, seecne secon ransormernee arssens,raen sseocromnernnes, aner ornee.: . ns, seecnn reseonsreoeerramnnnaersns,e seconns, aner ns, seecnn reseoonusreoeerrms anno snrme ormarne erne as e arsse ensr,ans raneo s srem ocrmonerne ernn o e.sr, aner . meoor veo encon, c ernes oora ann, an encoer, auraomr osna:runen-overansorm ransorm eerm cunrrnen,oce enn acnnra moe seecn oer,a s enaeore currenoc an romeura, oe enrcaonsero,r amerneras.nsormerneore currenocnenra moe enae s.naen m,eoe o en ccoamer an,uor aere ccoomerr, asn:aancaes weere s e arnorm ann re,sonsee eoncoeeerrm, onnen ooaowen acss: en n a rsesruoen,s aeno senearmnn,nae enecoer ans nooae ene, ae se econ , seer,, ae enecrnoeern,eex or a asase. n, e encoer,e ore cur.rene moecon oe cnarma mo, weeroremn seeecurna, e encoer,e ransormerne n resonseoeermnnae s enae o, seecrnan,sorme eenrnceose cromrses:ransormerneore currenocnenra moease one ernen,ex.e o one or m.oree mrseooc os c oa amrs, s wzeer aenn oneeura o ransormernes correson szeserenrome secon sze, an ars n ourm more seconocs o a secon sze,ers erenrom a secon numer oe one or more seceorn ooec ons.e or morersocss ornee.: . secon sz.ese m ×eo, a on caem seco,n w neurmeneresr.s szes ×,ers numers,e comrsn. onee m ore moore on csa, amn,e w oneere onr moerense cronreresnonexso a onen oarr mzeore va couneexs. e meo o cam seconeuraranso o ranso,r wmeerernne:s comrse ars ransormerne an a ee onnea orrzerm mo vrea cue oern n ce eoxmsr cosems arrsn an a seconn ersns se ars conex an a secon conex e seconn cosn coenxe cxo ceoe usn weer ssec coonne cxon aenx, or coen aass moe. ore currene moecon oe canmra mo, weerroemn seeecunra, oe encoraenrs,oremerneransormerne ooown se s comrses one ns n r: resonseoeermnnae s no enae, senersn as erne, seneesrsonsne aos se aencne seecrs ransormerne ase ransorm ne resrsonsne aos se aencne seeo co sen nconn as o n as.rr erne, sen ansormerne ase ransorm ranse. ormuera me oo o caramnsor,m weerrenen:s comrse ars ransormerne, a secon enarezrene v,a an ar ransormerne e one or more cuoen ceoxmsr cosems arsers arns, a co sneceoxnn, an arn ersns c , a secon conex, an ar conex e seconnosn coex coe usners conex erns conenxex co coee w usneer sec coonne cxo,n oerx co aenn aass moe. ornee.: . ore cur.rene moecon o cam, weren seecn, e encoer,e ransormerne ooown ses enra moeromeura o ransormernes comrses one n r:esonseoeermnn , senn resonseo se ae s no enae, senersn as erne ecners ransormerne ase ransorm erne, senne e resrs rsonsn ne a aos s se a ,ence sne se econeco sencnonn as as, anranesorrmerne ase ransorm n resonseo seecner ransormernn as or erne, senersn as,e seconn as, anerne a ass.e ransorm e.e meo o cam, weren: ransormuraerne o, arransormransernes comrse ars ransormerne, a secon enarz ormerne, an aour ransormerne e e vaue comrses arsn, a seconn, an arn e onrse orn mosr ceo cnonexexs comrse ars conex, a secon conex, an ar conex e seconns cone cxo ceoe usners conex erns conex coe w usneer sec coonne cxo,n oerx co aenn aass moe. .e meo o cam, weren seecn, e encoer,e ransormerne ooroeo cwurnren seocnenra moeromeura o ransormernes comrses one ns n r: reessoonnsseeoo seeeecrmnnnearseranss noorm enaernee, a ssenersn as erne, senersn as,e seconn e ransorm n resonseo seecne secon as, anrern as erne, sener ansormerne ase ransorm n ressonsne aos,e seconn as, anern as erne, senersn as se,ece sneconern as, anranseormrernne a ass oer ransorm orne e.: . erne, senne resrsonsne aos se,ece sneconeounr as, anranseormrernne as as.e ransorm on.e onr m aorarausor veoecon, comrsn: execua a meemore o ceonuroc e oreess mooors ree a ro on cnees osorr ms,o wreerreocnessoer osn aen or c monoreurreoceoss sororensrucons o ensrucons are conureoerorme meon an one o camss,- uo.n execuon on.e on aarausor veo encon, comrsn: a memro mror ceouroceessoorse a on execuae e one o corn moruererne or morerocessors an conureo sorensrucons o ensrucons are ocoesseorrso,rm weree mneeo onen o arn m oonree oro cceassmosrs, u-on. execuon nsrucon.s naon,- wranesn eoxrec cuoemuer o rneea oar meo sreor caome mueeumor sorn comuer-execuae comuerrocessorsoerorme meon an one o carmrsoce-sso.rs, causee one or more nsrucon.s naon,- wranesn eoxrec cuoemuer o rneea oar meo sreor caome mueerumrocoerss soorsr,n cau csoemuee or-neexe ocuae comuerrocessorsoerorme meon an one o cams -. r more ecoe. e no mne-raonsnor an c oom cuamers-re-aa.e sorae meumor sorn a sreamoe e me.o nonn- anran oso cram cosmu-er.-reaae sorae meumor sorn a sreamenerae
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