AdvancesinComputedTomography,2019,8,1-9
http://www.scirp.org/journal/act
ISSNOnline:2169-2483
ISSNPrint:2169-2475
AReweightedTotalVariationAlgorithm
withtheAlternatingDirectionMethod
forComputedTomography
XiezhangLi
,JiehuaZhu
DepartmentofMathematicalSciences,GeorgiaSouthernUniversity,Statesboro,GA,USA
Howtocitethispaper:Li,X.Z.and
Zhu,J.H.(2019)AReweightedTotalVaria-
tion Algorithmwith theAlternating Direction
Method for Computed Tomography.Advances
inComputedTomography,8,1-9.
https://dx.doi.org/10.4236/act.2019.81001
Received:November23,2018
Accepted:January20,2019
Published:January23,2019
Copyright
c
2019byauthor(s)andScientific
Research PublishingInc.
This work is licensed under the Creative Com-
monsAttributionInternationalLicense(CC
BY4.0).
http://creativecommons.org/licenses/by/4.0/
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sociisnatoquepenatibusetmagnisdispar-
turientmontes,nasceturridiculusmus.Pel-
lentesquehabitantmorbitristiquesenectuset
netusetmalesuadafamesacturpisegestas.
Phasellusidmagna.Duismalesuadainter-
dumarcu.Integer metus.Morbipulvinarpel-
lentesquemi.Suspendissesedesteumagna
molestieegestas.Quisquemilorem,pulvinar
eget,egestasquis,luctusat,ante.Proinauc-
torvehiculapurus.Fusceacnislaliquamante
hendreritpellentesque.Classaptenttacitiso-
ciosquadlitoratorquentperconubianostra,
perinceptoshymenaeos.Morbiwisi.Etiam
arcumauris,facilisissed,eleifendnon,non-
ummyut,pede.Crasutlacustempormetus
mollisplacerat.Vivamuseutortorvelmetus
interdum malesuada.
Sedeleifend,erossitametfaucibuselemen-
tum,urnasapienconsectetuermauris,quis
egestasleojustononrisus.Morbinonfelisac
liberovulputatefringilla.Maurisliberoeros,
lacinianon,sodalesquis,dapibusporttitor,
pede.Classaptenttacitisociosquadlitora
torquentperconubianostra,perinceptoshy-
menaeos.Morbi dapibus mauris condimentum
nulla.Cum sociisnatoque penatibus etmagnis
disparturientmontes,nasceturridiculusmus.
Etiamsitameterat.Nullavarius.Etiamt-
inciduntduivitaeturpis.Donecleo.Morbi
vulputateconvallisest.Integeraliquet.Pel-
lentesque aliquetsodalesurna.
Nullameleifendjustoinnisl.Inhachabitasse
plateadictumst.Morbinonummy.Aliquam
utfelis.Invelitleo,dictumvitae,posuereid,
vulputatenec,ante.Maecenasvitaepedenec
dui dignissim suscipit.Morbi magna.Vestibu-
lumidpurusegetvelitlaoreetlaoreet.Prae-
sentsedleovelnibhconvallisblandit.Ut
rutrum.Donecnibh.Donecinterdum.Fusce
sed pede sit amet elitrhoncus ultrices.Nullam
atenimvitaepedevehiculaiaculis.
Classaptenttacitisociosquadlitoratorquen-
tperconubianostra,perinceptoshymenaeos.
Aeneannonummyturpisidodio.Integereu-
ismodimperdietturpis.Utnecleonecdi-
amimperdietlacinia.Etiamegetlacuseget
miultriciesposuere.Inplacerattristiquetor-
tor.Sedportavestibulummetus.Nullaia-
culissollicitudinpede.Fusceluctustellusin
dolor.Curabiturauctorvelitasem.Mor-
bisapien.Classaptenttacitisociosquadl-
itoratorquentperconubianostra,perincep-
toshymenaeos.Donecadipiscingurnavehic-
ulanunc.Sedornareleoinleo.Inrhoncus
leoutdui.Aeneandolorquam,volutpatnec,
fringillaid,consectetuervel,pede.
Nullamalesuadarisusuturna.Aenean
pretiumvelitsitametmetus.Duisiaculis.In
hachabitasseplateadictumst.Nullammo-
lestieturpisegetnisl.Duisamassaidpede
dapibusultricies.Sedeuleo.Inatmaurissit
amettortorbibendumvarius.Phasellusjusto
risus,posuerein,sagittisac,variusvel,tor-
tor.Quisqueidenim.Phasellusconsequat,
liberopretium nonummy fringilla, tortorlacus
vestibulumnunc,utrhoncusligulanequeid
justo.Nullamaccumsaneuismodnunc.Proin
vitae ipsumac metusdictumtempus.Namut
wisi.Quisque tortor felis, interdum ac, sodales
a,sempera,sem.Curabiturinvelitsitamet
duitristiquesodales.Vivamusmaurispede,
laciniaeget,pellentesquequis,scelerisqueeu,
est.Aliquamrisus.Quisquebibendumpede
eudolor.
Donec tempusneque vitaeest.Aenean egestas
odio sed risus ullamcorper ullamcorper.Sed in
nullaatortortinciduntegestas.Namsapien
tortor,elementumsitamet,aliquamin,port-
titorfaucibus,enim.Nullamconguesuscipit
nibh.Quisqueconvallis.Praesentarcunibh,
vehicula eget,accumsaneu,tincidunta,nibh.
Suspendissevulputate,tortorquisadipiscing
viverra, lacus nibh dignissim tellus, eu suscipit
risusantefringilladiam.Quisquealiberov-
elpedeimperdietaliquet.Pellentesquenunc
nibh,eleifenda,consequatconsequat,hen-
dreritnec,diam.Sedurna.Maecenaslaoreet
eleifendneque.Vivamuspurusodio,eleifend
non,iaculisa,ultricessitamet,urna.Mau-
risfaucibusodiovitaerisus.Innisl.Praesent
purus.Integeriaculis,semeuegestaslacinia,
lacuspedescelerisqueaugue,inullamcorper
dolorerosaclacus.Nuncinlibero.
Fuscesuscipitcursussem.Vivamusrisusmi,
egestasac,imperdietvarius,faucibusquis,
leo.Aeneantincidunt.Donecsuscipit.Cras
idjustoquisnibhscelerisquedignissim.Ali-
quamsagittiselementumdolor.Aeneancon-
sectetuerjustoinpede.Curabiturullamcorper
ligula nec orci.Aliquam purus turpis,aliquam
id, ornarevitae, porttitornon, wisi.Maecenas
luctusportalorem.Donecvitaeligulaeuante
pretiumvarius.Prointortormetus,convallis
et,hendreritnon,scelerisquein,urna.Cras
quisliberoeuligulabibendumtempor.Viva-
mustellusquam,malesuadaeu,tempussed,
temporsed,velit.Doneclaciniaauctorlibero.
Praesentsednequeidpedemollisrutrum.
Vestibulumiaculisrisus.Pellentesquelacus.
Ut quisnunc sedodio malesuadaegestas.Duis
amagnasitametligulatristiquepretium.Ut
pharetra.Vestibulumimperdietmagnanec
wisi.Maurisconvallis.Sedaccumsansollic-
itudinmassa.Sedidenim.Nuncpedeenim,
laciniaut,pulvinarquis,suscipitsemper,elit.
Crasaccumsaneratvitaeenim.Crassollici-
tudin.Vestibulumrutrumblanditmassa.
Sedgravidalectusutpurus.Morbilaoreet
magna.Pellentesque eu wisi.Proin turpis.In-
tegersollicitudinauguenecdui.Fuscelectus.
Vivamusfaucibusnullaneclacus.Integerdi-
am.Pellentesquesodales,enimfeugiatcursus
volutpat,semmaurisdignissimmauris,quis
consequatsemestfermentumligula.Nullam
justo lectus, condimentum sit amet, posuere a,
fringillamollis, felis.Morbinulla nibh,pellen-
tesqueat,nonummyeu,sollicitudinnec,ip-
sum.Crasneque.Nuncaugue.Nullamvitae
quamidquampulvinar blandit.Nunc sitamet
orci.Aliquameratelit,pharetranec,aliquet
a,gravidain,mi.Quisqueurnaenim,viverra
quis,suscipitquis,tinciduntut,sapien.Cras
placeratconsequatsem.Curabituracdiam.
Curabiturdiamtortor,molliset,viverraac,
tempusvel,metus.
Curabituraclorem.Vivamusnonjustoin
duimattisposuere.Etiamaccumsanligula
idpede.Maecenastinciduntdiamnecvelit.
Praesentconvallissapienacest.Aliquamul-
lamcorpereuismodnulla.Integermollisenim
veltortor.Nullasodalesplaceratnunc.Sed
tempusrutrumwisi.Duisaccumsangravi-
dapurus.Nuncnunc.Etiamfacilisisduieu
sem.Vestibulumsemper.Praesenteueros.
Vestibulumtellusnisl,dapibusid,vestibulum
sit amet,placerat ac,mauris.Maecenas etelit
uteratplaceratdictum.Namfeugiat,turpis
etsodales volutpat,wisiquamrhoncusneque,
vitaealiquamipsumsapienvelenim.Maece-
nassuscipitcursusmi.
Quisque consectetuer.In suscipit mauris a do-
lor pellentesque consectetuer.Mauris convallis
nequenonerat.Inlacinia.Pellentesqueleo
eros,sagittisquis,fermentumquis,tincidun-
tut,sapien.Maecenassem.Curabitureros
odio,interdumeu,feugiateu,portaac,nis-
l.Curabiturnunc.Etiamfermentum convallis
velit.Pellentesquelaoreetlacus.Quisquesed
elit.Namquistellus.Aliquamtellusarcu,
adipiscingnon,tincidunteleifend,adipiscing
quis,augue.Vivamuselementumplaceraten-
im.Suspendisseuttortor.Integerfaucibus
adipiscingfelis.Aeneanconsectetuermattis
lectus.Morbimalesuadafaucibusdolor.Nam
lacus.Etiamarcu libero,malesuada vitae,ali-
quamvitae,blandittristique,nisl.
Maecenasaccumsandapibussapien.Duis
pretiumiaculisarcu.Curabiturutlacus.Ali-
quamvulputate.Suspendisseutpurus sedsem
tempor rhoncus.Utquam dui, fringillaat, dic-
tum eget,ultricies quis,quam.Etiamsem est,
pharetra non, vulputate in, pretium at, ipsum.
Nunc semper sagittis orci.Sed scelerisque sus-
cipitdiam.Utvolutpat,doloratullamcorp-
ertristique,erospurusmollisquam,sitamet
ornareantenuncetenim.
Phasellusfringilla,metusidfeugiatcon-
sectetuer,lacuswisiultricestellus,quis lobor-
tisnibhloremquistortor.Donecegestas
ornarenulla.Maurismitellus,portafau-
cibus,dictumvel,nonummyin,est.Ali-
quameratvolutpat.Intellusmagna,port-
titorlacinia,molestievitae,pellentesqueeu,
justo.Classaptenttacitisociosquadlitora
torquentperconubianostra,perinceptoshy-
menaeos.Sedorcinibh,scelerisquesitamet,
suscipitsed,placeratvel,diam.Vestibulum
nonummy vulputate orci.Donec et velit ac ar-
cuinterdumsemper.Morbipedeorci,cursus
ac,elementumnon,vehiculaut,lacus.Cras
volutpat.Namvelwisiquisliberovenenatis
placerat.Aeneansedodio.Quisqueposuere
purusacorci.Vivamusodio.Vivamusvar-
ius,nullasitametsemperviverra,odiomau-
risconsequatlacus,atvestibulumnequearcu
eutortor.Doneciaculistincidunttellus.Ali-
quameratvolutpat.Curabiturmagnalorem,
dignissimvolutpat,viverraet,adipiscingnec,
dolor.Praesentlacusmauris,dapibusvitae,
sollicitudinsitamet,nonummyeget,ligula.
Crasegestasipsumanisl.Vivamusvariusdo-
lorutdolor.Fuscevelenim.Pellentesque
accumsanligulaeteros.Crasidlacusnon
tortorfacilisisfacilisis.Etiamnislelit,cur-
sussed,fringillain,conguenec,urna.Cum
sociisnatoquepenatibusetmagnisdispar-
turientmontes,nasceturridiculusmus.In-
tegeratturpis.Cumsociisnatoquepenati-
busetmagnisdisparturientmontes, nascetur
ridiculusmus.Duisfringilla,ligulasedporta
fringilla,ligulawisicommodofelis,utadipisc-
ingfelisduiinenim.Suspendissemalesuada
ultricesante.Pellentesquescelerisqueaugue
sitameturna.Nullavolutpataliquettortor.
Crasaliquam,tellusataliquetpellentesque,
justosapiencommodoleo,idrhoncussapien
quamaterat.Nullacommodo,wisiegetsol-
licitudinpretium,orciorcialiquamorci,ut
cursusturpisjustoetlacus.Nullaveltor-
tor.Quisqueerat elit, viverra sitamet, sagittis
eget,portasitamet,lacus.
In hac habitasse platea dictumst.Proin at est.
Curabitur tempusvulputate elit.Pellentesque
sem.Praesent eusapien.Duiselit magna,ali-
quet at,tempus sed, vehicula non, enim.Mor-
biviverraarcunecpurus.Vivamusfringilla,
enimet commodomalesuada,tortormetusel-
ementumligula,necaliquetestsapienutlec-
tus.Aliquammi.Utnecelit.Fusceeuismod
luctustellus.Curabiturscelerisque.Nullam
purus.Namultriciesaccumsan magna.Morbi
pulvinarloremsitametipsum.Donecutjus-
tovitaenibhmolliscongue.Fuscequisdiam.
Praesent tempuserosutquam.
Donecinnisl.Fuscevitaeest.Vivamusante
ante,mattislaoreet,posuereeget,conguevel,
nunc.Fuscesem.Namvelorcieuerosviver-
raluctus.Pellentesquesitametaugue.Nunc
sitametipsumetlacusvariusnonummy.In-
teger rutrum sem egetwisi.Aenean eu sapien.
Quisqueornaredignissimmi.Duisaurnavel
risus pharetraimperdiet.Suspendisse potenti.
Morbijusto.Aeneannecdolor.Inhac
habitasseplateadictumst.Proinnonummy
porttitorvelit.Sedsitametleonecmetus
rhoncusvarius.Crasante.Vestibulumcom-
modosemtinciduntmassa.Namjusto.Ae-
neanluctus,felisetcondimentumlacinia,lec-
tusenimpulvinarpurus,nonportavelitnisl
sed eros.Suspendisse consequat.Mauris a dui
et tortormattispretium.Sednulla metus,vo-
lutpatid, aliquameget, ullamcorperut, ipsum.
Morbieununc.Praesentpretium.Duisali-
quampulvinarligula.Utblandit egestasjusto.
Quisqueposueremetusviverrapede.
Vivamussodaleselementumneque.Viva-
musdignissimaccumsanneque.Sedatenim.
Vestibulumnonummyinterdumpurus.Mau-
ris ornarevelitid nibh pretiumultricies.Fusce
temporpellentesqueodio.Vivamusauguepu-
rus,laoreetin,scelerisquevel,commodoid,
wisi.Duisenim.Nullainterdum,nunceusem-
pereleifend,enimdolorpretiumelit,utcom-
modoligulanislaest.Vivamusante.Nulla
leomassa,posuerenec,volutpatvitae,rhon-
cuseu,magna.
Quisquefacilisisauctorsapien.Pellentesque
gravidahendreritlectus.Maurisrutrumso-
dalessapien.Fuscehendreritsemvellorem.
Integerpellentesquemassavelaugue.Integer
elit tortor, feugiat quis, sagittis et, ornare non,
lacus.Vestibulumposuerepellentesqueeros.
Quisquevenenatisipsumdictumnulla.Ali-
quam quis quam nonmetus eleifend interdum.
Nam eget sapien ac mauris malesuadaadipisc-
ing.Etiameleifendnequesedquam.Nulla
facilisi.Proinaligula.Sedidduieunibh
egestastincidunt.Suspendissearcu.
Maecenasdui.Aliquamvolutpatauctorlorem.
Crasplaceratestvitaelectus.Curabitur
massalectus,rutrumeuismod,dignissimut,
dapibus a, odio.Ut eros erat, vulputate ut, in-
terdumnon,portaeu,erat.Crasfermentum,
felis inportacongue, velit leofacilisis odio, vi-
taeconsectetuerloremquamvitaeorci.Sed
ultrices,pedeeuplaceratauctor,anteligula
rutrum tellus,vel posuere nibh lacusnec nibh.
Maecenaslaoreetdoloratenim.Donecmo-
lestie dolornec metus.Vestibulum libero.Sed
quiserat.Sedtristique.Duispedeleo,fer-
mentumquis,consectetuereget,vulputatesit
amet,erat.
Donecvitaevelit.Suspendisseportafermen-
tummauris.Utvelnuncnonmaurisphare-
travarius.Duisconsequatliberoquisurna.
Maecenasatante.Vivamusvarius,wisised
egestastristique,odiowisiluctusnulla,lobor-
tisdictumdolorligulainlacus.Vivamusali-
quam, urna sed interdum porttitor,metus orci
interdum odio,sitamet euismodlectus feliset
leo.Praesentacwisi.Namsuscipitvestibu-
lum sem.Praesent eu ipsumvitae pede cursus
venenatis.Duis sed odio.Vestibulum eleifend.
Nullautmassa.Proinrutrummattissapien.
Curabiturdictumgravidaante.
Phasellusplaceratvulputatequam.Maece-
nasattellus.Pellentesquenequediam,dig-
nissim ac, venenatisvitae, consequat ut, lacus.
Nam nibh.Vestibulum fringilla arcu mollis ar-
cu.Sedetturpis.Donecsemtellus,volut-
pat et,variuseu, commodo sed,lectus.Lorem
ipsumdolorsitamet,consectetueradipiscing
elit.Quisqueenimarcu,suscipitnec,tempus
at,imperdietvel,metus.Morbivolutpatpu-
rusaterat.Donecdignissim,semidsemper
tempus, nibh massa eleifendturpis,sed pellen-
tesquewisipurussedlibero.Nullamlobortis
tortorvelrisus.Pellentesqueconsequatnulla
eutellus.Donecvelit.Aliquamfermentum,
wisiacrhoncusiaculis,tellusnuncmalesuada
orci,quisvolutpatduimagnaidmi.Nuncvel
ante.Duisvitaelacus.Crasnecipsum.
Morbinunc.Aliquamconsectetuervariusnul-
la.Phaselluseros.Crasdapibusporttitorris-
us.Maecenasultricesmiseddiam.Praesent
gravida velitat elit vehicula porttitor.Phasel-
lusnislmi,sagittisac,pulvinarid,gravidasit
amet,erat.Vestibulumest.Loremipsumdo-
lorsitamet,consectetuer adipiscingelit.Cur-
abitur idsem elementum leo rutrumhendrerit.
Utatmi.Donectinciduntfaucibusmassa.
Sed turpis quam, sollicitudin a, hendrerit eget,
pretiumut,nisl.Duishendreritligula.Nunc
pulvinarcongueurna.
Nuncvelit.Nullamelitsapien,eleifendeu,
commodo nec, semper sit amet, elit.Nulla lec-
tus risus,condimentum ut,laoreeteget,viver-
ra nec, odio.Proin lobortis.Curabitur dictum
arcuvelwisi.Crasidnullavenenatistortor
congueultrices.Pellentesqueegetpede.Sed
eleifendsagittiselit.Namsedtellussitamet
lectusullamcorper tristique.Maurisenim sem,
tristiqueeu,accumsanat,scelerisquevulpu-
tate, neque.Quisquelacus.Donec etipsum sit
amet elitnonummy aliquet.Sed viverra nisl at
sem.Namdiam.Maurisutdolor.Curabitur
ornaretortorcursusvelit.
Morbitinciduntposuerearcu.Crasvenenatis
estvitaedolor.Vivamusscelerisquesemper
mi.Donecipsumarcu,consequatscelerisque,
viverraid,dictumat,metus.Loremipsum
dolor sit amet, consectetuer adipiscing elit.Ut
pede sem, tempus ut, porttitor bibendum, mo-
lestieeu,elit.Suspendissepotenti.Sedidlec-
tus sitamet purus faucibusvehicula.Praesen-
tsedsemnonduipharetrainterdum.Nam
viverra ultricesmagna.
Aeneanlaoreetaliquamorci.Nuncinterdum
elementumurna.Quisqueerat.Nullamtem-
porneque.Maecenasvelitnibh,scelerisquea,
consequatut,viverrain,enim.Duismagna.
Donecodioneque,tristiqueet,tincidunteu,
rhoncusac, nunc.Maurismalesuadamalesua-
da elit.Etiam lacus mauris, pretium vel, blan-
ditin,ultricies id,libero.Phasellusbibendum
eratutdiam.Incongueimperdietlectus.
Aeneanscelerisque.Fuscepretiumporttitor
lorem.Inhachabitasseplateadictumst.Nulla
sitametnislatsapienegestaspretium.Nunc
nontellus.Vivamusaliquet.Namadipiscing
euismoddolor.Aliquameratvolutpat.Nul-
lautipsum.Quisquetinciduntauctorau-
gue.Nuncimperdietipsumegetelit.Ali-
quamquamleo,consectetuernon,ornaresit
amet,tristiquequis,felis.Vestibulumanteip-
sumprimisinfaucibusorciluctusetultrices
posuerecubiliaCurae;Pellentesqueinterdum
quamsitametmi.Pellentesquemaurisdui,
dictuma,adipiscingac,fermentumsitamet,
lorem.
Utquiswisi.Praesentquismassa.Vivamus
egestasrisusegetlacus.Nunctincidunt,risus
quisbibendumfacilisis,lorempurusrutrum
neque,necportatortorurnaquisorci.Ae-
neanaliquet,liberosempervolutpatluctus,
pedeeratlaciniaaugue,quisrutrumsemip-
sumsitametpede.Vestibulumaliquet,nibh
sediaculissagittis,odiodolorblanditaugue,
egetmollisurnatellusidtellus.Aeneanali-
quetaliquamnunc.Nullaultriciesjustoeget
orci.Phasellustristiquefermentumleo.Sed
massametus,sagittisut,semperut,pharetra
vel,erat.Aliquamquamturpis,egestasvel,
elementumin,egestassitamet,lorem.Duis
convallis,wisisitametmollismolestie,libero
mauris porta dui,vitaealiquam arcuturpis ac
sem.Aliquamaliquetdapibusmetus.
Vivamuscommodoeroseleifenddui.Vestibu-
luminleoeuerattristiquemattis.Crasat
elit.Craspellentesque.Nullamidlacussit
ametliberoaliquethendrerit.Proinplacerat,
minonelementumlaoreet,eroselittincidun-
tmagna,arhoncussemarcuidodio.Nul-
laegetleoaleoegestasfacilisis.Curabitur
quisvelit.Phasellus aliquam,tortornecornare
rhoncus,purusurnaposuerevelit,etcommo-
dorisustellusquistellus.Vivamusleoturpis,
tempussitamet,tristiquevitae,laoreetquis,
odio.Proinscelerisquebibendumipsum.Eti-
amnisl.Praesentveldolor.Pellentesquevel
magna.Curabitururna.Vivamuscongueur-
na in velit.Etiam ullamcorper elementum dui.
Praesentnon urna.Sed placeratquam non mi.
Pellentesquediammagna,ultricieseget,ultri-
cesplacerat,adipiscingrutrum,sem.
Morbisem.Nullafacilisi.Vestibulumante
ipsumprimisinfaucibusorciluctusetultri-
cesposuerecubiliaCurae;Nullafacilisi.Mor-
bisagittisultriceslibero.Praesenteuligula
sedsapienauctorsagittis.Classaptenttaciti
sociosquadlitoratorquentperconubianos-
tra,perinceptoshymenaeos.Donecvelnunc.
Nuncfermentum,lacusidaliquamporta,dui
tortoreuismoderos,velmolestieipsumpurus
eulacus.Vivamuspedearcu, euismodac, tem-
pusid,pretiumet,lacus.Curabitursodales
dapibusurna.Nunceusapien.Doneceget
nuncapededictumpretium.Proinmauris.
Vivamusluctusliberovelnibh.
Fuscetristiquerisusidwisi.Integermolestie
massaidsem.Vestibulumveldolor.Pel-
lentesquevelurnavelrisusultricieselemen-
tum.Quisquesapien urna,blandit nec,iaculis
ac,viverrain,odio.Inhachabitasseplatea
dictumst.Morbinequelacus,convallisvitae,
commodoac,fermentumeu,velit.Sedinor-
ci.Infringillaturpisnonarcu.Donecinante.
Phasellustemporfeugiatvelit.Aeneanvar-
iusmassanonturpis.Vestibulumanteipsum
primisinfaucibusorciluctusetultricespo-
suerecubiliaCurae;
Aliquamtortor.Morbiipsummassa,imperdi-
etnon,consectetuervel,feugiatvel,lorem.
Quisqueeget loremnec elitmalesuadavestibu-
lum.Quisquesollicitudinipsumvel sem.Nul-
laenim.Proinnonummy felisvitaefelis.Nul-
lampellentesque.Duisrutrumfeugiatfe-
lis.Maurisvelpedesedliberotinciduntmol-
lis.Phasellussedurnarhoncusdiameuismod
bibendum.Phasellussednisl.Integercondi-
mentumjusto id orciiaculis varius.Quisque et
lacus.Phaselluselementum,justo atdignissim
auctor,wisi odiolobortisarcu,sed sollicitudin
felisfeliseuneque.Praesentatlacus.
Vivamussitametpede.Duisinterdum,nunc
egetrutrumdignissim,nisldiamluctusleo,
ettinciduntvelitnislidtellus.Inloremtel-
lus,aliquetvitae,portain,aliquetsed,lectus.
Phasellussodales.Utvariusscelerisqueerat.
Invelnibheuerosimperdietrutrum.Donec
ac odio nec neque vulputatesuscipit.Nam nec
magna.Pellentesquehabitantmorbitristique
senectus et netus et malesuada fames acturpis
egestas.Nullamporta,odioetsagittisiaculis,
wisi neque fringilla sapien, vel commodo lorem
loremidelit.Utsemlectus,scelerisqueeget,
placeratet,tinciduntscelerisque,ligula.Pel-
lentesque nonorci.
Etiamvelipsum.Morbifacilisisvestibulum
nisl.Praesentcursuslaoreetfelis.Integer
adipiscingpretiumorci.Nullafacilisi.Quisque
posuerebibendumpurus.Nullaquammauris,
cursuseget,convallisac,molestienon,enim.
Aliquamcongue.Quisquesagittisnonummy
sapien.Proinmolestiesemvitaeurna.Mae-
cenas lorem.Vivamus viverra consequat enim.
Nuncsedpede.Praesentvitaelectus.Prae-
sentnequejusto,vehiculaeget,interdumid,
facilisiset,nibh.Phasellusatpurusetlibero
laciniadictum.Fuscealiquet.Nullaeuante
placeratleosemperdictum.Maurismetus.
Curabiturlobortis.Curabitursollicitudin hen-
dreritnunc.Donecultriceslacusidipsum.
Donecanibhutelitvestibulumtristique.In-
tegeratpede.Crasvolutpatvariusmagna.
Phaselluseuwisi.Praesentrisusjusto,lobor-
tis eget, scelerisqueac, aliquet in, dolor.Proin
idleo.Nunciaculis,mivitaeaccumsancom-
modo, nequesemlacinianulla,quisvestibulum
justosemineros.Quisquesedmassa.Morbi
lectus ipsum, vulputate a, mollis ut,accumsan
placerat,tellus.Nullaminwisi.Vivamuseu
ligulaanuncaccumsancongue.Suspendisse
aclibero.Aliquameratvolutpat.Donecau-
gue.Nuncvenenatisfringillanibh.Fusceac-
cumsan pulvinar justo.Nullam semper, dui ut
dignissim auctor, orcilibero fringillamassa, b-
landitpulvinarpedetortoridmagna.Nunc
adipiscing justo sed velit tinciduntfermentum.
Integerplacerat.Pellentesquehabitantmorbi
tristique senectuset netuset malesuada fames
acturpisegestas.Sedinmassa.Classaptent
tacitisociosquadlitoratorquentperconubi-
anostra,perinceptoshymenaeos.Phasellus
tempusaliquamrisus.Aliquamrutrumpurus
atmetus.Donecposuereodioaterat.Nam
nonnibh.Phasellusligula.Quisquevene-
natislectusinaugue.Sedvestibulumdapibus
neque.
Mauris tempus eros at nulla.Sed quisdui dig-
nissimmaurispretiumtincidunt.Maurisac
purus.Phasellusaclibero.Etiamdapibusia-
culisnunc.Inlectus wisi,elementumeu, sollic-
itudinnec,imperdietquis,dui.Nullaviverra
nequeaclibero.Maurisurnaleo,adipiscing
eu,ultricesnon,blanditeu,dui.Maecenas
duineque,suscipitsitamet,rutruma, laoreet
in,eros.Uteunibh.Fuscenecerattempus
urnafringillatempus.Curabituridenim.Sed
ante.Crassodalesenimsitametwisi.Nunc
fermentum consequatquam.
Utauctor,augueportadignissimvestibulum,
arcudiamlobortisvelit,velscelerisqueris-
usauguesagittisrisus.Maecenaseujusto.
Pellentesquehabitant morbitristique senectus
etnetusetmalesuadafamesacturpiseges-
tas.Mauriscongueligulaegettortor.Nul-
lam laoreeturnasedenim.Donecegeteros ut
erosvolutpatconvallis.Praesentturpis.Inte-
germaurisdiam,elementumquis,egestasac,
rutrumvel,orci.Nullafacilisi.Quisqueadip-
iscing,nullavitaeelementumporta,semurna
volutpatleo,sedportaenimrisussedmassa.
Integeracenimquisdiamsodalesluctus.Ut
eget eros a ligula commodo ultricies.Donec eu
urnaviverradolorhendreritfeugiat.Aliquam
acorcivelerosconguepharetra.Quisque
rhoncus,justoeuvolutpatfaucibus,augueleo
posuerelacus,arhoncuspuruspedevelest.
Proinultricesenim.
Aenean tincidunt laoreet dui.Vestibulum ante
ipsumprimisinfaucibusorciluctusetultri-
cesposuerecubiliaCurae;Integeripsumlec-
tus,fermentumac,malesuadain,eleifendut,
lorem.Vivamusipsumturpis,elementumvel,
hendrerit ut,semperat, metus.Vivamus sapi-
entortor,eleifendid,dapibusin,egestaset,
pede.Pellentesquefaucibus.Praesentlorem
neque, dignissim in,facilisis nec, hendrerit vel,
odio.Namatdiamacnequealiquetviver-
ra.Morbidapibusligulasagittismagna.In
lobortis.Donecaliquetultricieslibero.Nunc
dictumvulputate purus.Morbivarius.Lorem
ipsumdolorsitamet,consectetueradipiscing
elit.Intempor.Phaselluscommodoporttitor
magna.Curabiturvehiculaodioveldolor.
Praesentfacilisis,augueaadipiscingvene-
natis,liberorisusmolestieodio,pulvinarcon-
sectetuerfeliseratacmauris.Namvestibu-
lumrhoncusquam.Sedveliturna,pharetra
eu,eleifendeu,viverraat,wisi.Maecenasul-
tricesnibhatturpis.Aeneanquam.Nulla
ipsum.Aliquamposuereluctuserat.Cur-
abiturmagnafelis,laciniaet,tristiqueid,ul-
tricesut,mauris.Suspendissefeugiat.Cras
eleifendwisivitaetortor.Phasellusleopurus,
mattissitamet,auctorin,rutrumin,magna.
Inhachabitasseplateadictumst.Phasellus
imperdietmetusinsem.Vestibulumacenim
nonsemultriciessagittis.Sedveldiam.
Integervelenimsedturpisadipiscingbiben-
dum.Vestibulumpededolor,laoreetnec,po-
suerein,nonummyin,sem.Donecimperdiet
sapienplaceraterat.Donecviverra.Aliquam
eros.Nunc consequat massaid leo.Sed ullam-
corper,loreminsodalesdapibus,risusmetus
sagittislorem,nonporttitorpurusodioneco-
dio.Sedtinciduntposuereelit.Quisqueeu
enim.Donecliberorisus,feugiatac,dapibus
eget, posuere a,felis.Quisque vellectusutme-
tustincidunteleifend.Duis utpede.Duis velit
erat,venenatis vitae,vulputatea,pharetrasit
amet,est.Etiamfringillafaucibusaugue.
Aeneanvelitsem,viverraeu,tempusid,
rutrumid,mi.Nullamnecnibh.Proinullam-
corper,dolorincursustristique,erosaugue
tempornibh,atgravidadiamwisiatpurus.
Donec mattis ullamcorper tellus.Phasellus vel
nulla.Praesentinterdum,erosinsodalessol-
licitudin,nunc nulla pulvinarjusto,a euismod
erossemnecnibh.Nullamsagittisdapibus
lectus.Nullamegetipsumeutortorlobortis
sodales.Etiampurusleo,pretiumnec,feu-
giatnon,ullamcorpervel,nibh.Sedvelelit
etquamaccumsanfacilisis.Nuncleo.Sus-
pendissefaucibuslacus.
Pellentesqueinterdumsapiensednulla.Proin
tincidunt.Aliquamvolutpatestvelmassa.
Seddolorlacus,imperdietnon,ornarenon,
commodoeu,neque.Integerpretiumsem-
perjusto.Proinrisus.Nullamidquam.
Namneque.Duisvitaewisiullamcorperdi-
amcongueultricies.Quisqueligula.Mauris
vehicula.
Curabiturnuncmagna,posuereeget,vene-
natiseu,vehiculaac,velit.Aeneanornare,
massaaaccumsanpulvinar,quamlorem
laoreetpurus, eusodalesmagna risusmolestie
lorem.Nunceratvelit,hendreritquis,male-
suadaut,aliquamvitae,wisi.Sedposuere.
Suspendisseipsumarcu,scelerisquenec,ali-
quameu,molestietincidunt,justo.Phasellus
iaculis.Sedposuereloremnonipsum.Pel-
lentesquedapibus.Suspendissequamlibero,
laoreeta,tincidunteget,consequatat,est.
Nullamutlectusnonenimconsequatfacili-
sis.Maurisleo.Quisquepedeligula,auctor
vel,pellentesquevel,posuereid,turpis.Cras
ipsumsem,cursuset,facilisisut,tempuseu-
ismod,quam.Suspendissetristiquedoloreu
orci.Maurismattis.Aeneansemper.Viva-
mustortormagna,facilisisid,variusmattis,
hendreritin,justo.Integerpurus.
Vivamusadipiscing.Curabitur imperdiet tem-
pusturpis.Vivamussapiendolor,congueve-
nenatis,euismodeget,portarhoncus,magna.
Proincondimentumpretiumenim.Fusce
fringilla,liberoetvenenatisfacilisis,erosen-
imcursusarcu,vitaefacilisisodioauguevitae
orci.Aliquamvariusnibhutodio.Sedcondi-
mentumcondimentumnunc.Pellentesque
egetmassa.Pellentesquequismauris.Donec
utligulaacpedepulvinarlobortis.Pellen-
tesqueeuismod.Classaptenttacitisociosqu
adlitoratorquentperconubianostra,perin-
ceptoshymenaeos.Praesentelit.Utlaoreet
ornareest.Phasellusgravidavulputatenulla.
Donecsitametarcuutsemtempormalesua-
da.Praesenthendreritaugueinurna.Proin
enimante,ornarevel,consequatut,blandit
in,justo.Donecfeliselit,dignissimsed,sagit-
tisut,ullamcorpera,nulla.Aeneanpharetra
vulputateodio.
Quisqueenim.Proinvelitneque,tristiqueeu,
eleifendeget,vestibulumnec,lacus.Vivamus
odio.Duisodiourna,vehiculain,elementum
aliquam,aliquetlaoreet,tellus.Sedvelit.Sed
velmiacelitaliquetinterdum.Etiamsapi-
enneque,convalliset,aliquetvel,auctornon,
arcu.Aliquamsuscipitaliquamlectus.Proin
tinciduntmagnasedwisi.Integerblanditla-
cusutlorem.Sedluctusjustosedenim.
Morbimalesuadahendreritdui.Nuncmauris
leo, dapibus sitamet, vestibulum et, commodo
id,est.Pellentesquepurus.Pellentesquetris-
tique,nuncacpulvinaradipiscing,justoeros
consequatlectus, sitamet posuerelectusneque
velaugue.Cras consectetuerlibero aceros.Ut
egetmassa.Fuscesitametenimeleifendsem
dictum auctor.In egetrisus luctus wisiconva-
llis pulvinar.Vivamus sapien risus,tempor in,
viverra in, aliquetpellentesque,eros.Aliquam
euismodliberoasem.
Nunc velit augue,scelerisquedignissim,lobor-
tiset,aliquamin,risus.Ineueros.Vestibu-
lumanteipsumprimisinfaucibusorciluctus
etultricesposuerecubiliaCurae;Curabitur
vulputateelitviverraaugue.Maurisfringilla,
tortorsitametmalesuadamollis,sapienmi
dapibusodio,acimperdietligulaenimeget
nisl.Quisquevitaepedeapedealiquetsus-
cipit.Phasellustelluspede,viverravestibu-
lum,gravidaid,laoreetin,justo.Cumsoci-
isnatoquepenatibusetmagnisdisparturient
montes,nasceturridiculusmus.Integercom-
modoluctuslectus.Maurisjusto.Duisvar-
iuseros.Sedquam.Craslacuseros,rutrum
eget,variusquis,convallisiaculis,velit.Mau-
ris imperdiet, metus at tristique venenatis, pu-
rusnequepellentesquemauris,aultriceselit
lacusnectortor.Classaptenttacitisociosqu
adlitoratorquentperconubianostra,perin-
ceptoshymenaeos.Praesentmalesuada.Nam
lacuslectus,auctorsitamet,malesuadavel,
elementumeget,metus.Duisnequepede,fa-
cilisiseget,egestaselementum,nonummyid,
neque.
Proinnonsem.Donecnecerat.Proinlibero.
Aliquamviverraarcu.Donecvitaepurus.
Donecfelismi,semperid,scelerisqueporta,
sollicitudinsed,turpis.Nullainurna.Integer
variuswisinonelit.Etiamnecsem.Mau-
risconsequat,risusnecconguecondimentum,
ligulaligulasuscipiturna,vitaeportaodioer-
at quissapien.Proinluctusleo iderat.Etiam
massametus,accumsanpellentesque,sagittis
sitamet,venenatisnec,mauris.Praesentur-
na eros, ornarenec, vulputate eget, cursussed,
justo.Phasellus neclorem.Nullam ligula ligu-
la,mollissitamet,faucibusvel,eleifendac,
dui.Aliquameratvolutpat.
Fusce vehicula, tortoret gravida porttitor, me-
tusnibhconguelorem,uttempuspurusmau-
risapede.Integertinciduntorcisitamet
turpis.Aeneanametus.Aliquamvestibulum
lobortisfelis.Donecgravida.Sedsedurna.
Maurisetorci.Integerultricesfeugiatligu-
la.Seddignissimnibhamassa.Donecorci
dui,temporsed,tinciduntnonummy,viverra
sit amet, turpis.Quisque lobortis.Proin vene-
natistortornecwisi.Vestibulumplacerat.In
hachabitasseplateadictumst.Aliquampor-
tamiquisrisus.Donecsagittisluctusdiam.
Nam ipsumelit, imperdiet vitae,faucibusnec,
fringillaeget,leo.Etiamquisdolorinsapien
porttitorimperdiet.
Craspretium.Nullamalesuadaipsumut
libero.Suspendissegravidahendrerittellus.
Maecenas quislacus.Morbi fringilla.Vestibu-
lumodioturpis,temporvitae,scelerisquea,
dictumnon,massa.Praesenteratfelis,por-
tasitamet,condimentumsitamet,placer-
atet,turpis.Praesentplaceratlacusaen-
im.Vestibulumnoneros.Utcongue.Donec
tristiquevariustortor.Pellentesquehabitant
morbi tristique senectus et netus et malesuada
famesacturpisegestas.Namdictumdictum
urna.
Phasellusvestibulumorcivelmauris.Fusce
quamleo,adipiscingac,pulvinareget,mo-
lestiesitamet,erat.Seddiam.Suspendisse
erosleo,tempuseget,dapibussitamet,tem-
puseu,arcu.Vestibulumwisimetus,dapibus
vel,luctussitamet,condimentumquis,leo.
Suspendissemolestie.Duisinante.Utso-
dales sem sit amet mauris.Suspendisse ornare
pretiumorci.Fuscetristiqueenimegetmi.
Vestibulumeroselit,gravidaac,pharetrased,
lobortisin,massa.Proinatdolor.Duisac-
cumsanaccumsanpede.Nullamblanditelit
inmagnalaciniahendrerit.Utnonummyluc-
tuseros.Fusceegettortor.
Utsitametmagna.Crasaligulaeuurna
dignissimviverra.Nullamtemporleoporta
ipsum.Praesentpurus.Nullamconsequat.
Maurisdictumsagittisdui.Vestibulumsol-
licitudinconsectetuerwisi.Insitametdiam.
Nullammalesuadapharetrarisus.Proinlacus
arcu, eleifendsed, vehicula at,conguesitamet,
sem.Sedsagittispedeanisl.Sedtincidunto-
dioapede.Seddui.Nameuenim.Aliquam
sagittislacusegetlibero.Pellentesquediam
sem,sagittismolestie,tristiqueet,fermentum
ornare,nibh.Nullaet tellusnonfelis imperdi-
etmattis.Aliquameratvolutpat.
Vestibulumsodalesipsumidaugue.Integer
ipsumpede,convallissitamet,tristiquevi-
tae,temporut,nunc.Namnonligulanon
lorem convallishendrerit.Maecenas hendrerit.
Sedmagnaodio,aliquamimperdiet,portaac,
aliqueteget,mi.Cumsociisnatoquepenati-
busetmagnisdisparturientmontes, nascetur
ridiculusmus.Vestibulumnislsem,dignissim
vel,euismodquis,egestasut,orci.Nuncvi-
taerisusvelmetuseuismodlaoreet.Crassit
ametnequeaturpislobortisauctor.Sedali-
quamsemacelit.Crasvelitlectus,facilisis
id,dictumsed,portarutrum,nisl.Namhen-
dreritipsumsedaugue.Nullamscelerisque
hendreritwisi.Vivamusegestasarcusedpu-
rus.Utornarelectussederos.Suspendisse
potenti.Maurissollicitudinpedevelvelit.In
hachabitasseplateadictumst.
Suspendisseeratmauris,nonummyeget,
pretiumeget,consequatvel,justo.Pellen-
tesqueconsectetuereratsedlacus.Nullam
egestasnullaacdui.Doneccursusrhoncus
ipsum.Nuncetsemeumagnaegestasmale-
suada.Vivamusdictummassaatdolor.Mor-
biestnulla,faucibusac,posuerein,interdum
ut,sapien.Proinconsectetuerpretiumurna.
Donecsitamet nibhnecpurusdignissimmat-
tis.Phasellusvehiculaelitatlacus.Nullafa-
cilisi.Crasutarcu.Sedconsectetuer.Integer
tristiqueelitquisfelisconsectetuereleifend.
Crasetlectus.
Utcongue malesuadajusto.Curabiturcongue,
felisathendreritfaucibus,maurislacusport-
titor pede, nec aliquam turpis diam feugiatar-
cu.Nullamrhoncusipsumatrisus.Vestibu-
lumadolorseddolorfermentumvulputate.
Sednecipsumdapibusurnabibendumlobor-
tis.Vestibulum elit.Namligula arcu,volutpat
eget,laciniaeu,lobortisac,urna.Nammol-
lisultricesnulla.Crasvulputate.Suspendisse
atrisusat metuspulvinarmalesuada.Nullam
lacus.Aliquamtempusmagna.Aliquamut
purus.Prointellus.
Vestibulumanteipsumprimis infaucibusorci
luctus et ultrices posuere cubilia Curae; Donec
scelerisquemetus.Maecenasnonmiutmetus
portahendrerit.Nuncsemper.Crasquiswisi
utloremposueretristique.Nuncvestibulum
scelerisquenulla.Suspendissepharetrasollic-
itudinante.Praesentatauguesitametante
interdum porta.Nunc bibendumaugue luctus
diam.Etiam nec sem.Sed eros turpis,facilisis
nec,vehiculavitae,aliquamsed,nulla.Cur-
abiturjustoleo,vestibulumeget,tristiqueut,
tempusat,nisl.
Nullavenenatisloremidarcu.Morbicur-
susurnaaipsum.Donecporttitor.Integer
eleifend,estnonmattismalesuada,minulla
convallismi,etauctorlectussapienutpu-
rus.Aliquamnullaaugue,pharetrasitamet,
faucibussemper,molestievel,nibh.Pellen-
tesquevestibulummagnaetmi.Sedfringilla
dolorveltellus.Nuncliberonunc,venenatis
eget,convallishendrerit,iaculiselementum,
mi.Nullamaliquam,felisetaccumsanvehic-
ula,magnajustovehiculadiam,eucondimen-
tumnislfelisetnunc.Quisquevolutpatmauris
avelit.Pellentesquemassa.Integeratlorem.
Nammetuserat,laciniaid,convallisut,pul-
vinarnon,wisi.Crasiaculismaurisutneque.
Crassodales,semvitaeimperdietconsequat,
pede purussollicitudin urna,ac aliquammetus
orciinleo.Utmolestieultricesmauris.Viva-
musvitaesem.Aliquameratvolutpat.Prae-
sentcommodo,nislacdapibusaliquet,tortor
orci sodales lorem, non ornare nulla lorem quis
nisl.
Sedatsemvitaepurusultricesvestibulum.
Vestibulumtinciduntlacusetligula.Pel-
lentesquevitaeelit.Vestibulumanteipsum
primisinfaucibusorciluctusetultricespo-
suerecubiliaCurae;Duisornare,erateget
laoreetvulputate,lacusipsumsuscipitturpis,
etbibendumnislorcinonlectus.Vestibu-
lumnecrisusnecliberofermentumfringilla.
Morbinonvelitinmagnagravidahendrerit.
Pellentesquequislectus.Vestibulumeleifend
lobortisleo.Vestibulumnonaugue.Vivamus
dictumtempordui.Maecenasatligulaidfe-
liscongueporttitor.Nullaleomagna,egestas
quis,vulputatesitamet,viverraid,velit.
Utlectuslectus,ultriciessitamet,semper
eget,laoreetnon,ante.Proinatmassaquis
nuncrhoncusmattis.Aliquamlorem.Cur-
abiturpharetraduiatneque.Aliquameutel-
lus.Aeneantempus,felisvitaevulputateia-
culis,estdolorfaucibusurna,inviverrawisi
nequenonrisus.Fusceveldolornecsapien
pretiumnonummy.Integerfaucibusmassaac
nulla ornarevenenatis.Nullaquissapien.Sed
tortor.Phasellusegetmi.Crasnunc.Crasa
enim.
Quisque nisl.In dignissim dapibus massa.Ae-
neansemmagna,scelerisquenec,ullamcorper
quis,porttitorut,lectus.Fuscedignissimfa-
cilisistortor.Vivamusgravidafelissitamet
nunc.Nampulvinarodiovelenim.Pellen-
tesquesitametest.Vivamuspulvinarleonon
sapien.Aliquameratvolutpat.Ut elementum
auctormetus.Maurisvestibulumnequevitae
eros.Pellentesquealiquamquam.Donecve-
nenatistristiquepurus.Innisl.Nullavelit
libero,fermentumat,portaa,feugiatvitae,
urna.Etiamaliquetornareipsum.Proinnon
dolor.Aeneannuncligula,venenatissuscipit,
porttitorsit amet,mattis suscipit, magna.Vi-
vamusegestasviverraest.Morbiatrisussed
sapiensodalespretium.
Morbicongueconguemetus.Aeneansedpu-
rus.Nampedemagna,tristiquenec,porta
id,sollicitudinquis,sapien.Vestibulumblan-
dit.Suspendisse ut augue ac nibh ullamcorper
posuere.Integereuismod,nequeateleifend
fringilla,augueelitornaredolor,veltincidunt
purusest idlacus.Vivamusloremdui, commo-
doquis,scelerisqueeu,tinciduntnon,magna.
Cras sodales.Quisque vestibulum pulvinar di-
am.Phasellustincidunt, leovitaetristiquefa-
cilisis,ipsum wisiinterdum sem,dapibussem-
pernulla velit vel lectus.Crasdapibusmauris
etaugue.Quisquecursusnullainlibero.Sus-
pendisseetloremsitametmaurismalesuada
mollis.Nullamidjusto.Maecenasvenenatis.
Doneclacusarcu,egestasac,fermentumcon-
sectetuer,tempuseu,metus.Proinsodales,
seminpretiumfermentum,arcusapiencom-
modomauris,venenatisconsequataugueur-
nainwisi.Quisquesapiennunc,variuseget,
condimentumquis,laciniain,est.Fuscefacil-
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inelit.Proinmolestievariusnibh.Vivamus
tristiquepurussedaugue.Proinegestassem-
pertortor.Vestibulumanteipsumprimisin
faucibusorciluctusetultricesposuerecubil-
iaCurae;Classaptenttacitisociosquadl-
itoratorquentperconubianostra,perincep-
toshymenaeos.Vestibulumorcienim,sagittis
ornare,eleifendut,mattisat,ligula.Nulla
molestieconvallisarcu.Uterostellus,condi-
mentum at,sodalesin,ultricesvel,nulla.
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sedmassacursuscursus.Pellentesquepulv-
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quamarcurhoncusdui,idtemporlacusjusto
etjusto.Proinsitametorcieudiameleifend
blandit.Nunceratmassa,luctusac,fermen-
tumlacinia,tinciduntultrices,sapien.Prae-
sentsed orci vitae dolorsollicitudin adipiscing.
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vestibulumdolor.Pellentesquealiquam,diam
eget eleifendposuere, augueerosporttitorlec-
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sit ametmauris.Cras commodo, urna ut port-
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pedenibhpretiumtellus, atdignissimleoeros
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Fusceposuere,tellusegetviverramattis,er-
attellusportami,atfacilisissemnibhnon
urna.Phasellusquisturpisquismaurissus-
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Vestibulumanteipsumprimis infaucibusorci
luctusetultricesposuerecubiliaCurae;
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gue.Morbimauriswisi,aliquetvitae,dignis-
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eudiamrhoncushendrerit.Vivamusutelit.
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rhoncusnequeeudui.Etiamvitaemagna.
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Vestibulumgravida.Morbimattisliberosed
est.
Abstract
Avarietyofalternatingdirectionmethodshavebeenproposedfor
solvingaclassofoptimizationproblems.Theapplicationsincom-
putedtomography (CT)performwell inimagereconstruction.The
reweighted schemes were appliedinl
1
-norm andtotalvariation min-
imizationforsignalandimagerecoverytoimprovetheconvergence
ofalgorithms.Inthispaper,wepresentareweightedtotalvaria-
tionalgorithmusingthealternatingdirectionmethod(ADM)for
imagereconstructioninCT.ThenumericalexperimentsforADM
demonstratethataddingreweightedstrategyreducesthecompu-
tationtimeeffectivelyandimprovesthequalityofreconstructed
imagesaswell.
Keywords
ComputedTomography,NonmonotoneAlternatingDirection
Algorithm,ReweightedAlgorithm
1.Introduction
Inmanyapplications, theproblemof recoveringasparse unknown vec-
tor xfroma setofmeasurementsbis presented.The l
1
minimization,
min
x
kxk
1
subjecttoAx= b,(1)
whereArepresentsameasurementmatrix,hasbeenwidelyusedin
solving the problem [1–5].This method has a unique sparsest solution
xundercertainconditions[6].However,thel
1
-normregularization
providesalesssparsityrepresentationthanthel
0
-normregulariza-
tionandmaylosesomedetailedfeaturesandcontrast.Onemajor
improvedl
1
minimizationalgorithmforfindingthesparsestsolution
efficientlyisthereweightedl
1
minimization[7].Somemodification-
softhereweightedl
1
-minimizationalgorithmshavebeenstudiedin
recentyears.Amoregeneralformofthereweightedl
1
-minimization,
DOI:10.4236/act.2019.81001Jan.23,20191AdvancesinComputedTomography
X.Z.Li,J.H.Zhu
inwhichtheweightsareraisedtoacertainpowerbetween0and1,
wasconsideredin[8,9].Numericalexperimentsintheareasofsparse
signal recovery,the errorcorrectionin sampletransitionand thetotal
minimizationforsparseimagegradientshaveshownthattheperfor-
mance ofrecoveryusing reweightedl
1
minimization isbetter thanthe
regularl
1
minimization[7].
Incomputedtomography(CT),algebraicapproachesforimagere-
constructioninvolvessolvingaconsistentsystemoflinearequations
Φf+e= u,(2)
where Φ isan m×n
2
projection matrix, fR
n
2
represents a 2Dn×n
image tobe reconstructed,eanadditive noisewith||e||
2
εfor some
knownε>0,anduR
m
thenoisyprojectiondata.Forlimited-
view reconstruction,the systemis underdetermined withmn
2
and
hasinfinitelymanysolutions.OptimizationbasedIterativemethods
areusuallyusedtofindanoptimalsolutionrepresentingtheoriginal
image.
TheimagefinSystem(2)canbereconstructedbysolvingthefol-
lowing total variation (TV) minimization with an l
2
-norm constraints,
min
f
||f||
TV
subjectto||Φfu||
2
ε,(3)
where||∗||standsfor||∗||
2
forsimplicity.Tomographyimagescan
beapproximatelymodeledtobeessentiallypiecewiseconstantsothe
gradientsaresparse.As||f||
TV
isthel
1
-normofthegradientoff,
theTVminimizationisalsoknownasthel
1
-minimizationmethod.
Ageneralizedl
1
greedyalgorithminthecompressedsensingframe-
work[10]wasintroducedtoincorporatethethresholdfeatureofthe
l
1
greedyalgorithm[11]andtheinverselyproportionalweightsused
inthereweightedl
1
-minimizationalgorithmforCT.Erroranalysisof
reweightedl
1
greedyalgorithmfornoisyreconstructionwasstudied
in[12].Thereweightedl
1
-normwasincorporatedintonon-localTV
minimizationtostreghtenthe structuraldetails andthe tissuecontrast
andthustoenhancetheCTreconstructingperformance[13].
Thealternatingdirectionmethod(ADM)isavariantoftheclas-
sicaugmentedLagrangianmethodforstructuredoptimization.For
aconvexoptimizationproblemwherevariablesappearingseparate-
lyarecoupledintheconstraint,theminimizationwithrespectto
allvariablesaretime-consuming.TheADMminimizestheaugment-
edLagrangianfunctionwithrespecttovariablesseparatelyviaiter-
ationtoreducecomputationtime,widelyusedinl
1
-normminimiza-
tion [14,15].An ADM with fastconvergence under certain strong con-
ditionswasproposedin[16].Aunifiedalternatingdirectionmethodby
majorizationminimizationwasproposedanditsconvergencewasan-
alyzed[17].Thenonmonotonealternatingdirectionmethod(NADA)
wasproposedforsolvingaclassofequality-constrainednonsmooth
optimizationproblemsandappliedtothetotalvariationminimiza-
tioneffectively[18,19].TheadvantageoftheNADAliesinthatthe
objectivefunctionisnotrequiredtobedifferentiablewhilereducing
thecomputationcomplexityandimprovingtheefficiency.
Inthispaper,wepresentareweightedTValgorithmusingADM
inCT.Therestofthispaperisorganizedasfollows.Notationsand
preliminaryareintroducedinSection2.Areweightschemecorpo-
rationintothetotalvariationalgorithmusingNADAisdeveloped
inSection3.NumericalexperimentsinSection4showtheadoptof
DOI:10.4236/act.2019.810012AdvancesinComputedTomography
X.Z.Li,J.H.Zhu
weightsimprovestheperformanceoftheTValgorithmusingNADA
incomputationaltimeanderrormeasurements.Finallyadiscussion
inSection5concludesthepaper.
2.Preliminary
In thissectionweintroducenotations andreview approaches forsolv-
ingaTVminimizationproblem(3)usingthereweightschemeand
ADMintheliterature,respectively.
Inthecaseofsparsesignalrecovery,itisextremelyimportantto
identifythelocationsofnonzeroentriesofasparsevectorx.Aftera
few iterations, thelarger entries in magnitudeare naturallyconsidered
to be nonzero entries.The process should concentrate onother entries
smallerinmagnitude.Theideaofthereweightedl
1
minimizationis
tomaketheweightssmallfortheentriesofxlargerinmagnitude
andtheweightslargeforthosesmallerinmagnitude.Inotherwords,
the weights are roughly inversely proportional to the magnitude of the
previous iterate solution.So reweighted l
1
minimization speeds up the
convergenceofrecovery.Theminimizationproblem(1) ismodifiedas
min
x
kWxk
1
subjecttoAx= b,
whereatthek-thiterationW=diag{w
(k)
1
,...,w
(k)
n
2
}isthediagonal
weightmatrixwitheachw
(k)
i
=
1
ε+||x
(k1)
||
andε>0.
InthecaseofimagereconstructioninCT,thetotalvariationofan
imagefisdefinedas
||f||
TV
= k∇fk
1
=
X
n
2
i=1
||D
i
f||,
where fis thegradient off,and D
i
is aforward differenceoperator
atapixeliinbothhorizontalandverticaldirections.Thegradi-
entfofanessentiallypiecewiseconstantimagefissparsesoa
reweightedschemecould beadoptedto speeduptheconvergence and
improvetheefficiencyofimagereconstruction.Thus,thetermtobe
minimizedin(3)isrevisedas||WOf||
1
,whereatthek-thiteration
W= diag{w
(k)
1
,...,w
(k)
n
2
}withallw
(k)
i
=
1
ε+||D
i
f
k1
||
.
Minimization(3)withconstraintscanbeexpressedasthemini-
mizationofanaugmentedLagrangianfunctionas,withLagrangian
multipliersλ
i
,auxiliaryvectorsv
i
,andregularizationparametersβ,
µ>0,
L(f,v,λ) =
β
2
||Φfu||
2
2
+
X
n
2
i=1
{kv
i
k+λ
T
i
(D
i
fv
i
)+µ||D
i
fv
i
||
2
},
(4)
whereλ
i
,v
i
R
2
.Givenf
k
,v
k
,andλ
k
inthek-thiteration,ADM
minimizesL(f,v,λ)withrespecttofandvseparately,thenupdates
λ.Inotherwords,ADMiteratesas
f
k+1
= argmin
f
L(f,v
k
k
);
v
k+1
= argmin
v
L(f
k+1
,v,λ
k
);
λ
k+1
= λ
k
+β(Df
k+1
v
k+1
).
DOI:10.4236/act.2019.810013AdvancesinComputedTomography
X.Z.Li,J.H.Zhu
3.ReweightedTVAlgorithmUsingNADA
InthissectionwedevelopanewreweightedTValgorithmusingNADA
forsolvingthefollowingminimization
min
f
||WOf||
1
subjectto||Φfu||
2
ε,(5)
wherethel
1
-norm||WOf||
1
isaweightedtotalvariationoff,andW
isaweightmatrixwithallpositivediagonals.
Dealing with the constraint as usual ADM, we first rewrite the min-
imizationabovewithapenaltyparameterµ>0as
min
f
||Wf||
1
+
µ
2
||Φfu||
2
,
or
min
f
X
n
2
i=1
||v
i
||+
µ
2
||Φfu||
2
subjecttow
i
D
i
f= v
i
,1 in
2
.
(6)
ApplyingLagrangianvectorsλ
i
R
2
,for1 in
2
,wedefine
L
w
(f,v,λ)
=
µ
2
||Φfu||
2
+
X
n
2
i=1
(||v
i
||
2
λ
T
i
(w
i
D
i
fv
i
)+
β
2
||w
i
D
i
fv
i
||
2
).
(7)
Thus,Minimization(6)isconvertedtominimizingtheobjective
functionin(7),
min
f
L
w
(f,v,λ).(8)
Forgivenf
k
,v
k
,andλ
k
,theproposedalgorithmiteratesas
f
k+1
= argmin
f
L
w
(f,v
k
k
),(9)
v
k+1
= argmin
v
L
w
(f
k+1
,v,λ
k
),(10)
λ
k+1
= argmin
λ
L
w
(f
k+1
,v
k+1
).(11)
Inthisnewalgorithm,amodifiedreweightedschemeisincorporat-
edafterafewnon-weightediterationsinordertospeedupthecon-
vergenceofthealgorithmandimprovetheefficiency.Thestandard
reweightedschemeadoptsamatrixW= diag{w
1
,...,w
n
2
},wherew
i
is
essentially inversely proportional to thenorm ||D
i
f
k
||.In thisproject,
thevaluesof w
0
i
sneed toberescaledso thattheweightsoffourterms
in the objective function L
w
(f,v,λ) remain closeto the case whereW
istheidentitymatrix.Thestandardreweightedschemeismodified
tocalculatingavectorwwithw
i
=
1
ε+||D
i
f
k
||
(1in
2
)and
rescaling wbyafactor r=
2
mean(w)
.Thereweighted matrixWin this
paperisobtainedthisway.
Minimizations(9) and(10) aresolvedbyadopting thenonmonotone
line search scheme in theframeworkof NADA though thereis anextra
reweighted matrixWinthe objectivefunction L
w
(f,v,λ).Thedetail-
sareasfollows.Afterchoosingadirectiond
k
=
∂f
L
w
(f
k
,v
k
k
),
searchastepsizes
k
uniformlyboundedabovesuchthatL
w
(f
k
+
s
k
d
k
,v
k
k
)C
k
s
k
δ||d
k
||
2
2
,forascalarC
k
and0<δ <1.
Thenupdatef
k+1
= f
k
+s
k
d
k
andcomputeascalarC
k+1
suchthat
DOI:10.4236/act.2019.810014AdvancesinComputedTomography
X.Z.Li,J.H.Zhu
L
w
(f
k+1
,v
k
k
)C
k+1
C
k
.Combinedwithsolutionof(10),the
sequence{L
w
(f
k
,v
k
k
)}isboundedabovebyamonotonicallynon-
increasingsequence {C
k
}though {L
w
(f
k
,v
k
k
)}itself isnot decreas-
ing.TheadvantagesoftheNADAremainsothatL
w
(f,v,λ)isnot
requiredtobedifferentiableandthattheusageofthenonmonotone
linesearchschemereducesthecomputationcomplexity andimproves
theefficiency.
Tofindv
k+1
= argmin
v
L
w
(f
k+1
,v,λ
k
)in(10),wesolve
∂L
w
∂v
i
=
v
i
||v
i
||
+λ
i
βw
i
D
i
f+βv
i
= 0.
Sov
i
(1+
1
β||v
i
||
)=w
i
D
i
f
λ
i
β
.Itfollowsthat||v
i
||=||w
i
D
i
f
λ
i
β
||−
1
β
.Thus,wehave
v
i
= max{||w
i
D
i
f
λ
i
β
||−
1
β
,0}
w
i
D
i
f
λ
i
β
||w
i
D
i
f
λ
i
β
||
,1 in
2
.(12)
Similarly,thesolutionof(11)isgivenby
λ
k+1
i
= λ
k
i
+β(w
i
D
i
f
k+1
v
k+1
),1 in
2
.(13)
Nowwepresent anew algorithmtosolvemin
f
L
w
(f,v,λ) usingNA-
DA.
Algorithm(ReweightedTotalVariationAlgorithmUsing
NADA)
1.inputΦ,u,ε
2.initializeβ,µ,f
0
,v
0
0
,ε,tol,k,maxit
3.performafew iterationsofregularTValgorithm withoutweight
usingNADA
4.whilek<maxit
4.1setareweightedmatrixW
4.2updatef
k+1
=argmin
f
L
w
(f,v
k
k
)byNADA
4.3updatev
k+1
i
by(12)
4.4updateλ
k+1
by(13)
4.5iferror<tolthenoutputf
k+1
,stop
4.6increasek= k+1
end
4.NumericalExperiments
Inthissection,thereweightedTValgorithmusingNADAforCT
reconstructionis implemented inMATLAB.ThereweightedTVmin-
imization(5)(or(8))iscomparedwiththeregularTVminimization
(3)(orminimizationof(4))fortheirperformances.Thenumerical
experimentsareconductedwiththe 2DShepp-Loganphantom [20]of
size128×128onanIntelCorei73.40GHzPC.TheMATLABcode
isdevelopedbasedonthesoftwarepackageTVAL3[19].
In each test, arandom matrix Φ R
m×n
2
(m0.3n
2
) isgenerated
and u= Φf+eisset,where thenoise e= 0.05*mean(Φf)*randn(m).
The parametersare takenas β= 2
4
and µ= 2
6
.The values of εin the
weightmatrixis0.01.Ifthe relativeerror||ff
recon
||
F
/||f||
F
ofthe
DOI:10.4236/act.2019.810015AdvancesinComputedTomography
X.Z.Li,J.H.Zhu
reconstructedimagef
recon
inFrobeniusnormislessthantol=0.05
thentheiterationisterminated.ForthereweightedTValgorithm,
15iterationsofthestandardtotalvariationminimizationaresetas
theinitialsolutionforthereweightedTV-minimization.Theaverage
results from 100 testsof the proposed algorithm andnon-weighted TV
regularizationareusedinthefollowingtableandfigure forevaluations.
Let f
ave
denote the average of the pixel values ofa 2D image f.Ex-
perimental results are also evaluated using the root-mean-square error
(RMSE),thenormalizedrootmeansquaredeviation(NRMSD),and
thenormalizedmeanabsolutedeviation(NMAD)whicharedefined
asfollows:
RMSE=
||ff
recon
||
F
mn
,
NRMSD=
||ff
recon
||
F
||f
ave
f||
F
,
NMAD=
P
m,n
|f(m,n)f
recon
(m,n)|
P
m,n
|f(m,n)|
.
Thesemeasurementsreflectdifferentaspectsofthequalityofthe
recoveredimages.RMSEevaluatesthereconstructionqualityona
pixel-by-pixelbasis.NRMSDemphasizeslargeerrorsinafewpixels
oftherecoveredimage.NMADfocusesonsmall errorsintherecovered
image.
TheoriginalandreconstructedimagesareshowninFigure1.The
experimentalresultsaresummarizedinTable1.TheCPUtimeis
measuredbyMATLBbuilt-infunctions.ThevaluesofRMSE,N-
RMSD, and NMAD forthe reweighted TV minimization are improved
fromthecorrespondingvaluesfortheTVminimization.TheCPU
timeissaved34.9%.Thenumericalexperimentsdemonstratethat
thereweightedTVminimizationissuperiortotheregularTVmini-
mization.
Figure1.Shepp-Loganphantomandreconstructedimages.
Table1.Experimentaldatawithshepp-loganphantom.
AlgorithmCPUTimeRMSENRMSDNMAD
TVUsingNADA14.4sec0.00850.00880.0353
ReweightedTVusingNADA9.4sec0.00660.00690.0294
Therearethreeimportantparametersinthisproposedalgorith-
m:β,µ,andrescalingfactorrforw.SimilartoaregularNADA
withoutreweight,thevaluesofβandµshouldbebetween2
4
and
2
13
.Therescalingfactorrisadoptedbetween
1
mean(w)
and
4
mean(w)
intheexperiments.Itisachallengingquestionhowtooptimizethe
reconstructedimagesbyreweighedTVusingNADA.Moreextensive
numericalexperimentswillbeperformedinfutureinvestigation.
DOI:10.4236/act.2019.810016AdvancesinComputedTomography
X.Z.Li,J.H.Zhu
5.Conclusion
ThereweightedTVminimizationhasshownitscapabilityofspeed-
inguptheconvergenceofsparseimagereconstructioninCTsince
itwasproposedmorethantenyearsago.InrecentyearstheADM
waswidelyappliedtoaclassofequality-constrainednonsmoothop-
timizationproblemsandparticularlyimagereconstructioninCT.As
farastheauthorsareaware,theADMhasn’tbeenusedtosolvethe
reweightedTVminimizationintheliterature.Thispaperdevelopsa
reweighted totalvariation algorithm usingthe ADMmethod forimage
reconstructioninCT,asthemajornoveltyofourwork.Theusageof
the nonmonotone line search scheme reduces complexity and improves
efficiency.Numericalsimulationindicatethattheproposedalgorith-
mcaneffectivelyincorporatethereweightedschemeandimprovethe
efficiencyforimagereconstructioninCT.
ConflictsofInterest
Theauthorsdeclarenoconflictsofinterestregardingthepublication
ofthispaper.
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