HAL Id: inria-00070421
https://hal.inria.fr/inria-00070421
Submitted on 19 May 2006
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Comparison of video dynamic contents without feature matching by using rank-tests
Alain Lehmann, Patrick Bouthemy, Jian-Feng Yao
To cite this version:
Alain Lehmann, Patrick Bouthemy, Jian-Feng Yao. Comparison of video dynamic contents with- out feature matching by using rank-tests. [Research Report] RR-5586, INRIA. 2005, pp.15. �inria- 00070421�
ISSN 0249-6399 ISRN INRIA/RR--5586--FR+ENG
a p p o r t
d e r e c h e r c h e
Thème COG
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
Comparison of video dynamic contents without feature matching by using rank-tests
Alain Lehmann — Patrick Bouthemy — Jian-Feng Yao
N° 5586
Mai 2005
Unité de recherche INRIA Rennes
IRISA, Campus universitaire de Beaulieu, 35042 Rennes Cedex (France)
Téléphone : +33 2 99 84 71 00 — Télécopie : +33 2 99 84 71 71
*!+%,-./&0, +'1324!5!
6798;:=<?>@BACD8E<<GFIHJ8;KMLM:ON)P?Q.RTSKMA+@BCVUTFXWE:98;<YZ[@B<\^]38ER
_G`badcefg*hjilk4monqprascesnutdvxwxy{z|p}z|~n
I}vedpG qk4_
u{;vqpu4e}est$`bes}t$`{eyE[xxxVixz0Ji 3{wesn
'sd) b¡xx¢ _G`{z£n¤res;vqpub}esnredyBp$nybv)¥xes¦0y{ed§tdz¨esyBp4z©nrnrz¨cz£¦£xrz¨pmªcesnq«{re¬Eep®edesy¥oz©4edv
nredwcedyBp}ns¯°^etdvxy{nrz©4edI¦£v4tdx¦;nr{)p}z¨vx±Op}edc;v}x¦{4eMnrtdrz£4prv}ns¯_G`{edm}e¤tvy{nqz©4esreMp}v¬Ee¤}esx¦¨z£²s±
p}z¨vy{n v~xyª«byb³oybv)y5´B¬b«4putd¦£nrnq±9nrEestz¨µ;tubz£nqpr}z¨¬{«4prz£vxy5¯[_G`be*nrz¨cz£¦£xrz¨pm3vx~5p®v¥oz£4esv3nredwc.esyBp}n z£n
ts¦©t«b¦©)p}es.¬om3es¥)¦£«{)p}z¨ybwxybb}vx{rz©)p}eGnqp}prz©np}z£tuz£n}nq«{es.~¶}vxc·}xyb³p}esnqps¯+9p 4voesnIybvpX}es¸B«bz£}e
xyom¹c)p}t$`{z¨ybwºvx~p}`be»¦¨v4ts¦G~¶espr«b}esn¬Eep®edesyDpr`be¼p®v½tvxy;nqz©4ed}es¥oz£4esv½nqeswxcedyBp}ns´Gy{Dtdxy
bes¦¾z¨pr`»4z¨¿EesresyBpuyB«{c¬Eedv~+tvc.{«4preM¦£v4tdx¦T~¶es)p}«b}esnGz£yp}`beJp vnredwc.esyBp}ns¯+À{«brpr`besrcvx}ex´
v«bcesnq«b}ez£nªnred¦¨~Á±9ybvx}c¦£z¨²ses¹`{z£t$`?¦£¦£v)un~¶vªnqz£cb¦£et«be»z£yBpredw}prz£vxy5´®y{Âed¥edyDvxy4±9¦£z¨ybe
bbpres»td¦£nrnq±Ã4edEedy{bedyBp¤tvc¬bz£y{)p}z¨vy¼v~+pr`be.4z¨¿EesresypJ4esn}t}z¨bprvx$ns¯Gk4prz©n~tÄp}vx}m'}esnr«b¦|p$n`{¥e
¬Eedesy¼vx¬4p$z£ybesvxyreM¦¾¥oz©4edvnres¸B«bedy;tesn®~¶vGp v.cvp}z¨vyes¥xesyp}estdvxwxy{z|p}z¨vy'b}vx¬b¦£edcns¯
Å»ÆBÇTÈ}ÉÊ )Ëx¢ bz£n}nqz£cz¨¦©}z¨pmªcesxnr«b}ex´o$yb³p}esnqps´b¥oz©4edvnres¸B«besy{tetvc{}z£nrvxy
¼ +u3!( 0 Ä!&
¢ f eG$bEvxrp+b MnqesyBpre®«bybeybv«b¥xes¦¨¦£e ceMnq«b}e4enqz£cz¨¦©}z¨p GedyBpr}eG4esn[n s¸B«besy{teMn¥oz© dv
nrz£c.{¦¨e s¥)¦£«bedM¯[gytdvxy{nrz£bad}eGbesnXbesn}t}z¨4p}ed«b$nInr{)p}z¨vx±Op}edcEvx}ed¦©nd¯5æ©nXnrvxyBpXtdvxy{nrz£ d sn[tvcc.e
besn M¦£z£n})p}z¨vy{n5«bybe®4z£nqpr}z£¬b«4prz£vxyz¨y{tdvxybyo«beXcxz£n0~¶vxy;tÄprz£vxy34eX¦©ut¦©xn}nreX4e tdvxyBpresyo«{n0tdvxy{nrz£ d dex¯
nqz£cz¨¦©}z¨p X4e 4ed«n s¸B«besy{teMn¾¥oz£ svuesnqpts¦©t«b¦ deI;5¦ s¥)¦£«{)p}z¨vy«{ybe nqp}prz©np}z£¸B«beIz©n}nq«beI«by
p}esnqpI4eG}xybw{¯5æ4yeMnp[{n+y stdesn}nrxz¨}e ¸B«beG¦£esn+4esn}t}z£4pres«b}n¦£v4td«4esn[4es«3n s¸B«besy{teMnnrvxz£edyBp+cz£n
esy.tdvx}reMnqEvxy;by{tdeXybzb¸B«beG¦£esn+ybvc¬b}esn+4eubesn}t}z¨4p}ed«b$n+tdx¦£td«b¦ Mn+by{n+¦£esn[bed«3n s¸B«besy{teMn[nqvz¨esyBp
swx«¾¯eb¦£«{nd´4y{vpr}e*cesnr«bre*eMnpybvrc¦£z©n se«bprvxcprz©¸«{edcedyBps´btde¸«{z5;esrcedp«4p}z¨¦£z©nqesu4esn
p}est$`by{z£¸B«beMnJnrz£c.{¦¨eMn*;v«b*¦z¨yBp dwx$)p}z¨vy 4e4z¨¿ sresyBp}n4eMnrtdrz£4pres«b$nxz¨y{nrzI¸B«be4esntdvxc¬{z¨y{xz£nrvxy;n
be*tdesnubed}ybz¨es}n ~¶vy{tÄp}z¨vy¼beJ¦©t¦©xn}nreedp¤xbx4p sesnGedy¦£z¨wybex¯esn snr«b¦¨p}p}nn})prz©nq~z©nrxyp$n®vxyBp p
v¬4presyo«{n+nq«b+4eMnn s¸B«besy{teMn¥Bz©4esv des¦¨¦£esn;v«b[4ed«brv¬b¦£adcesn4eGreMtvyby{z©n}nrxy{teX s¥ dy{edcedyBp}n
¦£z sn«cvx«b¥edcedyBpEed"!«5¯
#Ê
M
È
¡ $
¢
cesnr«b}e*be*nrz£c.z£¦©}z|p x´opreMnp¤be}xybw{´4tvc{$z©nqvy'4en s¸B«besy{teMnG¥Bz© sv
= >
3!0 +&X!
k4z¨y{tdep}`be.xcvx«byBp¤vx~Xc«b¦¨prz£ces4z©'{)p}'z£n$bz©4¦£mwrv)z£ybw{´E«4p}vxc)p}z£t3nqm4nqprescnJ}eybeses4eM¼prv
{rv4teMnrn¤pr`{z£n*`o«bwexcvx«byBp*v~®{)p}{¯3_G`besred~¶vx}ex´Tp}`be4ed¥ed¦£vxbcedyBp*v~®c.edpr`bv4bn*`{z£t$`^}e.x¬b¦¨e
p}v3}estdvxwybz¨²se¤nredcyBprz©tdx¦¨¦£mnqz£cz¨¦© pr`{z¨ybwBn®z£nGt}«{tdz£x¦{p}v3`{xy{4¦£e¤p}`besnre*b)p$b¯Iko«{t$`c.edpr`bv4bn®c3«{np
¬Eexbb¦£z£ts¬b¦£e¤z£yªp}`betdvxyBpre op v~5¥oz©4edv.b)p$¬{nqeMn[p}vwx}vx«{ªnqz£cz¨¦© ¥Bz©4esv3nredwxcesyp$n[p}vxwxedpr`bess´prv
n})p}z£nq~¶m¸«{ed}z¨eMnd´prv3e opr$xtp `bz£wx`b¦£z£wx`Bp}n@?¶ex¯w{¯£´oz¨ynqEvxrp ¥oz©4edvBnA´vIprv.nq«{c.cxrz£²deu¥oz©4edvBnd´B¬b«4p¦©nrv
z£ypr`betvyBpre opGv~0¥oz©4edvnr«br¥edz£¦¨¦©y;teup}v4ep}estÄpunr;eMtz©¦Epmo;eMnGv~0es¥xesyp$n®`bz£t$`b}vx¬{x¬b¦£m.}es¸B«bz£}e
`o«bcxyªz¨yBp}ed}¥xedyBp}z¨vy5¯CB¤edy{tdex´op}`beJt$`;¦£¦¨esybwxz£ybwb}vx¬{¦¨esc·p}v.nrvx¦£¥xez£nXprv.cnqescxyBprz©td¦Etvy{tes4p}n
xy{¦¨v)G±9¦¨es¥xes¦E¥oz©4edv.~¶eM)pr«{reMn4«be*prv.p}`be es¦¨¦¨±9³By{v)ynredcyBprz©tJwxED£x´%F4´{HG9¯
_vEedr~¶vx}c·*}estvwxybz¨prz£vxy3p}nq³T´vxybeu`;xn+p}v4eµ;ybepr`{euresb}esnredyBp}prz£vxy.v~Epr`beu¥oz©4edv3nqeswxcedyBps´
`{z£t$`.z£n+vx~Ttdr«;tz©¦oz¨cEvxrp}xy{te¯_ vtdvxcb¦£edcedyBp}xrmxbb}vt$`besntdxy3¬Ee4z|¿Ted}edyBp}z£pres5¯+_G`{ed}e
xre®p}`bevxybeMn+`bz©t$`3e Bp}}tÄp[wx¦£vx¬;¦o~¶espr«b}esn~¶}vxcpr`be`{vx¦£e®¥oz©4edvnredwc.esyBps¯+_G`bedmxrenrz¨cb¦£eGprv
z£cb¦£edcedyBps´x¬b«4p[cmJ`;¥xeX{rv¬b¦¨escnz¨pr`.tvc.{¦¨e nrtdedybeMnd¯uy.e bcb¦£eX~¶vcvxprz£vxy}estdvxwybz|p}z¨vy
«;nqz£ybw»wx¦£vx¬{x¦+xbb}vt$` z©n!D¨MHGX`besre¼nqz£cb¦£e.y{vxy4±9{$cedpr}z£t.4z©np$y{tde.ceMxnr«bre.¬{nqeM vy
`{z£nqprvwx$cn¤vx~®nr{)p}z£x¦[y{predcEvx$¦[z£yBpredy;nqz¨pm»wx$x4z£edyBp$n`;xnJ¬;esedy^«{nres¾¯.ÃyIDHG9´0nr;vqp¥oz£bedvn
`;¥xe¬;esedy½t$`{x}tÄpresrz£²deMz¨pr`½b}vx¬;¬bz£¦¨z©nqprz©t3cvxprz£vxyºcvobed¦©n*prvts4p}«bre.p}`be'4vc.z£y{xypz£cwe
cvxprz£vxy8?¶z=¯ex¯pr`be®tdces}Gcvp}z¨vy%A¾xy{*pr`{eX}esnrz£4«;¦xn}tesybeIcvp}z¨vy5¯_G`beMnqe cvobed¦©n0}eI¦£esxryBp¾~¶}vxc
w¦¨v¬{¦v4tdtd«br}edy;te3nqp}prz©np}z£tsn*tvxcb«bpresVv)¥xes¤p}`be.`bvx¦£e3¥oz©4edv¼nredwxcesypM¯ oz£c«{c ¦¨z£³xes¦¨z£`bvov4
tdrz¨presrz£vxy esre*«{nresz¨pr`bz£ynq«bEed}¥oz£nrest¦©xn}nrz|µ;ts)p}z¨vyn}t$`bedcex¯
gy½p}`bevpr`{ednqz©4ex´+p}`bed}exre'pr`be¦£vots¦®bb}vt$`besn`bz£t$`Âe opr$xtp~¶espr«b}esn~¶}vxc nq;)prz£v±
p}edcEvx$¦£¦¨m¦¨v4tdx¦¨z£²deM¼reswxz£vxy{nprvªv)¥ed$tvxceJpr`{eb}vx¬b¦£edcnuvx~+pr`be3wx¦£vx¬{x¦0b{rvBxt$`5¯G4z¨§ªt«b¦¨pm
vx~p}`besnre»¦£vots¦uxbb}vxt$`{esn.z£ns´®`bv) es¥xess´Xpr`{ppr`benredwc.esyBp}nªxre¼ybvº¦£vxybwed}edb}esnredyBpreM¬om
nrz£ybwx¦£e'~¶eM)p}«bre¼¥estÄp}vxM´I¬b«4pªºnrepv~¤~¶eM)p}«bre¼¥estÄp}vx$nd¯Â¤n^tvxy;nqeM¸«{edy{tdex´Ip}`be»tvc.;}z£nrvxy¹vx~
nredwcedyBp}n¾z©n5y{vu¦¨vybwxes0nqpr$z£wx`Bpq~¶vrG$xy{*z©n5ybvrc¦£¦£mxt$`{z¨es¥xesJ¬omJcp}t$`bz£ybwp}`beX¦£v4tdx¦~¶espr«b}esn
¬Eep®edesy¼pr`be3b}votdesn}nqeM¼¥oz£4esv'y;pr`be.b)p$¬{nqe¯ÃyJDG=´TªnrepJv~I¦£vots¦nq{ted±Op}z¨ce4eMnrtdrz£4p}vx$n
~¶vreMtvwxybz£²dz£ybwcvprz£vxy{pqpresry;nu}e*breMnqesyBpresy;es¥)¦£«{)p}es¾¯X_G`{ec)p}t$`{z¨ybwvx~p}`be~¶espr«b}esn
z©nI4vybeuz£ywx}edeM4mcyby{edM´x`besreMxn[4z¨¿EesresyBp 4z©np$y{tdec.eMxnr«b}esn+`{¥e¬;esedypreMnp}es¾¯+ z£4esvn[vx~
`o«bcxycvp}z¨vytÄp}z¨¥oz¨prz£esn`{¥e¬Eedesy¼«{nqeM~¶vxGpr`{e*presnqp}nxy{wxvov4ª}estdvxwxy{z|p}z¨vy$)p}esn®`{¥xe*¬Eedesy
}edEvxrpreM¾¯
Ãy¹v«bcep}`bv4¾´+ e'`{¥xeªtdvxy{nrz£bed}esVp v¼pmoEesnv~cvprz£vxy¹besn}t}z¨4p}vx$n*~¶vx3t$`{$xtpred}z£²dz£ybw
bmBy;cz£t[tdvxyBpresyBp0v~4¥oz£bedvunres¸B«bedy;te[z¨pr`¦£vots¦)~¶eM)p}«breMnd¯_G`be+µ{$npvxybeItdtv«byBp}nT~¶v¾pr`{eXnq;)prz£v±
p}edcEvx$¦{es¥xv¦¨«4p}z¨vyªv~¾p}`beJz¨yBpresreMnp ;vz¨yBp}n®`bz©t$`ªz©n nrnr«bces.p}v3¬Ee¤}ed¦©)p}esp}vp}`be¤p}}estprvx}mvx~
p}`becv)¥oz¨ybw*v¬4estp}ns¯_G`beunrestdvxy{3p}rz£esnp}vtdbpr«b}eGpr`bez£yBpresy{nqz¨pmvx~;p}`becvp}z¨vy5´xy{. eutÄpr«;¦£¦¨m
bbpres½pr`be¼nrtdedybecvprz£vxyc.v44es¦ vx~DGGprv v«b¦£v4tdx¦®nrep}«b5¯º°Ve`{¥xe4esnrz£wxybeM½yvrz£wxz£y{x¦
bz£n}nqz£cz¨¦©}z¨pm cesnq«b}eª`bz©t$`¹4voesnybvp3ybedeM¹y½e o{¦¨z©tz¨p3{xz¨}z£ybw»v~p}`be'¦£v4td¦I~¶es)p}«b}e'¥xeMtÄp}vx$n
vx~Xp}`bep v»nredwc.esyBp}ny;Vtsyº4esx¦[z|p}`º»bz|¿Ted}edyBpyo«bc¬Eedvx~Gtdvxcb«4p}esV~¶espr«b}e¥xestprv}nJz£y
p}`beªp®v nredwc.esyBp}np}v ¬Eetdvxc{}es5¯9p3z©n¬;xnresºvxyÂnqz£cb¦£enqp})p}z£nqprz©tdx¦Xp}esnqp.xy{ºz£n3esnqmVprv
tdvxcb«4p}exnGz¨puz¨yo¥xv¦¨¥esn®$yb³oz£ybw3v;es}prz£vxy{n®vyb¦¨m¯I_G`bexn}nq«{c.bprz£vxyz£ns´4pr`;)pu3~¶es)p}«b}e¥x¦¨«{e*nrep
z©nuwedybes}preM¬om¼y»«by{³By{v)ynp}vot$`;xnqprz©tb}votdesn}nd¯_G`be¬{xnrz£tz£4eMz©nup}`bedy¼p}vªpreMnp`bep}`bedpr`be
p®v¼~¶espr«b}eª¥)¦£«be'nrep$nxrewedybes}presV~¶}vxcpr`benrxceªb}votdesn}nvxybvpM¯Àb«brpr`besrcvre´0®e'tdxy
tdvxc3¬bz¨y{ep}`be4z¨¿EesresyBpbesn}t}z¨4p}vx$n®z¨y»t¦©xn}n±Ã4es;esy{4edyBpux{4p}z¨¥eJGmx¯
KLKNMOPPQR
n[pr`{e4eMnrtdrz£4prz£vxyvx~Tcvxprz£vxyed¥edyBp}nIz£n *ybvxy4±=pr}z£¥Bz©¦{brv¬b¦£edc¼´nrz£ybwx¦£eu4esn}t}z£4prvIz£nXz¨y;4edeM
y{vpnr«4§ªtz£edyBpy{º¼nqedpv~4esn}t}z£4prv}nJ`{n*prv¼¬Eeª«{nres¾¯ ®vovnqprz£ybw DHGX`{xn¬;eMtvxce¼Evxb«b¦©
cedpr`bv4J~¶vx«4p}vxc)p}z£t+~¶eM)p}«breInred¦£estprz£vxy*vxtvxc3¬bz£y{)p}z¨vy5¯ Buv)®ed¥edM´p}`beIwxesybed$¦£z£²s)p}z¨vy¤p}vuc«{¦|p}z|±
td¦£nrn¤t¦©xn}nqz¨µ;tdprz£vxy¼z©nuybvxp¤vx¬o¥oz¨v«{ns´bed¥edy»z¨~[nrvxcez£yo¥xeMnp}z¨wB)prz£vxy;nG`{¥xe¬Eedesy«by{4esqp$³edy5¯ D<
´G=¯
bz£n}x4¥)xyp$wevx~p}`besnre»¬EvBvBnp}z¨y{w¹¦£wxvx}z¨pr`bcn.z£nª`bv) es¥xes3p}`bedz£'tdvxcb«4p$)prz£vxy;¦£¦¨mÂe 4;esy{nrz¨¥e
¦£esxry{z¨ybwnp$wxe¯*°Ve.`{¥xe34eµ{y{es nrz¨cb¦£ecep}`bv4 `{z£t$`z©n*¬b¦£ep}v¦£esxrypr`be¸B«{¦£z|pm¼vx~Ipr`be
z£y{4z£¥oz£b«{¦{besn}t}z¨4p}vx$nprv4z©nrtdrz£cz¨y;)preu*wz¨¥edyt¦©xn}n~¶rvc p}`berescz£ybz¨y{wvxybeMnd¯_G`bz£nX4esn}t}z¨bprvx
¸B«{x¦¨z¨pm'z©n¤pr`besy»e 4b¦£vxz¨presp}v'4eM4«{te.b}vxEed¤®edz£wx`Bp}z¨ybwp}v'tvc¬bz£ybep}`be4z¨¿Ted}edyBpJ4eMnrtdrz£4p}vx$n
z£ypr`bebesnrz¨wybesnp$)p}z£nqprz©tdx¦Tpresnqps¯
_G`be}edcxz¨y{bedv~+p}`bz£nures;vqpz©nvx}wxybz¨²sesxn®~¶v¦¨¦£v)und¯ _G`bet¦©xn}nqz¨µ;ts)prz£vxy~¶$ced®vx}³ªtvc3±
{rz©nqz£ybw3pr`benqp}prz©np}z£ts¦54z©n}nqz£cz¨¦©}z|pm'c.eMxnr«b}eJxy{ªp}`bet«{eJz£yBpreswx$)prz£vxypreMt$`bybz©¸B«be*z£nGz£yBpr}vob«{teM
z£y½koestprz£vxy½4¯Ãyºk4estÄp}z¨vy < ´5pr`{e.z£yBpresreMnpEvxz£ypnred¦£estprz£vxy^np$wxey;p}`beªtvy{nqz©4esreM¦£vots¦[cv±
p}z¨vy»~¶espr«b}esn}eb}esnredyBpreM¾¯Àz£y{¦£¦£mx´Ee 4;esrz£cedyBp}x¦}esnr«b¦¨p}n¤v~Icvp}z¨vyed¥edyBpt¦©xn}nrz|µ;ts)p}z¨vy}e
}edEvxrpreM'z£y»koestprz£vxy ¯ f vxy;t¦£«{4z£ybwrescxr³4nG}e*wxz£¥xesy'z£y»k4estÄp}z¨vyo¯
0¼ !ª +G" 1
!#"%$'&(*)+!,-."/)01)+$'23)4)456$'2798.:;)+7*<)+=>()?5@$A
_G`{e»p}nq³Âv~cvprz£vxy?es¥xesypª}estdvxwybz|p}z¨vy?tdy ¬EeVnredesy np}`beb}vx¬b¦£edc v~t¦©xn}nrz|~¶moz£ybw¹½wxz£¥xesy
¥oz©4edv½nredwc.esyBptdtdvx$4z¨y{w p}v¹nqvc.e¼b}esbeµ{ybeM t¦©xn}nqeMn
c ∈ C¯ _0v½xt$`{z¨es¥xepr`bz©np$xnr³¹ exre
{res¥Bz£vx«;nq¦£mwz¨¥edyªnred¥xes}x¦be bxc.{¦¨eMn
s~¶vx esxt$`ªtd¦£nrns´~¶vxX`{z£t$`ª®eu³oybv)Dpr`{et¦©xn}nIcesc¬Eed$nq`bz£5´
z=¯ex¯
C(s)∈ C¯_G`bz©nnqedpv~be bcb¦£esnz¨¦£¦¬EeI~¶«{qp}`bed+4edy{vpreMxn0¥oz£4esvbp}x¬{xnre
S¯_G`be {rv¬b¦¨esc vx~es¥xedyBpreMtvwxybz¨prz£vxyDtdyD¬Ee~¶vrc3«b¦©)preMÂn3pr`benresx}t$`½~¶vpr`be¼cz£ybz¨c3«bc vx~Jº4z©nrnrz£c.z£¦©}z|pm
~¶«{y{tÄp}z¨vy
ΦB
C(r) = arg minˆ
c∈CΦ(r|c) ? A
`{ed}eªz©np}`beªp}esnqpres¹¥Bz©4esvnqeswxcedyBps¯»°^e'y{v) }e~¶vrc3«b¦©)prep}`betd¦£nrnrz¨µ;tdprz£vxyºb}vx¬b¦£edc xn
}ep}rz£ed¥)x¦bb}vx¬b¦£edc¼´xz=¯ex´x®ep}rmprvµ{y;.pr`bec.vBnpXnrz£c.z£¦©Xnredwc.esyBp
s∗z£yvx«bX{)p}x¬{xnre
S y{¬;xnre p}`bet¦©xn}nrz|µ;ts)p}z¨vy'vxypr`{et¦©xn}nGv~
s∗¯
C(r) =ˆ C(s∗), `besre s∗= arg min
s∈SΦ(r, s|C(s)) ?O A
tpr«{x¦¨¦£mx´4®etvy{nqz©4es p}`be*pr`{reseJ¬Eesnqp¤nredwxcesyp$n®«{nqz£ybwc)vrz¨pm¥xvxprz£ybwp}vz¨y{tdreMxnre¤p}`be}vx¬b«{nqpq±
y{esn}nd¯Ãytdxnre®pr`{pX¦£¦4pr`{reseunqeswxcedyBp}n+¬Eed¦£vxy{wp}v4z¨¿EesresypXtd¦£nrnresns´)p}`beut¦©xn}n+v~Epr`be¬Eesnqp nredwc.esyBp
z©n+t$`bvnredy0¯0Ãy.vx$4edprvµ{y{pr`beGcvnqp[nqz£cz¨¦©[nredwxcesyp e®`{¥eXp}vJ4edµ{ybeJ4z£n}nrz¨cz£¦£xrz¨pm*c.eMxnr«b}e
~¶vup®vªwxz£¥xesynqeswxcedyBp$n
r y; s `bz©t$`cm¬;e.t¦©xn}n¤4es;esy{4edyBpM¯J_G`bz©n¤td¦£nrn4edEedy;4edy{tdm»tdy
¬Ee«{nqprz¨µ{eM¬om'p}`be~tÄp¤p}`{)pybvp*¦£¦0wxz£¥xesy»4esn}t}z¨bprvx$n`{¥e*prvª¬Ee3t$`;$xtÄp}ed}z£nqprz©t~¶vxJ¦£¦0wxz£¥xesy
td¦£nrnresns¯LB¤v) es¥xess´®e`{¥xe®prv*esy{nq«{re®pr`;)pIp}`beu4z¨¿EesresypXbz£n}nqz£cz¨¦©}z¨pmc.eMxnr«b}esn[}etvc.;$¬b¦£e
z£ypresrcnGv~pr`besz¨¤cxwxybz¨pr«;4enq«{t$`pr`;)p¤nqeswxcedyBp`bz©t$`¼z£nGcvx}enqz£cz¨¦©Gp}`{yvxy{eJvx~+xybvp}`bed
td¦£nrnGx¦£nrvwxep$nnqc¦£¦£ed¥)¦£«bex¯
CEDCK+CGF
®e~¶vre¼®etsy 4eµ;ybe»^bz£n}nqz£cz¨¦©}z¨pm½c.eMxnr«b}ex´X®e¼`{¥xeprv½nr;eMtz¨~¶m¹`bv) eresb}esnredyBp
wz¨¥edyº¥oz£bedv nredwc.esyBps¯»¤n3np$)preMºz¨y¹pr`beªz£yBpr}v44«{tÄp}z¨vy5´+ e'«{nqe'¦¨v4tdx¦I~¶eM)pr«{reMn*prv t$`;$xtÄp}ed}z¨²se
p}`benqeswxcedyBptvyp}edyBps´5pr`{p*z©n¼nqedpv~I~¶eM)pr«{re¥)¦£«besns¯_G`betvxy;nqz©4ed}es¦¨v4tdx¦Inq;)prz£v±=presc;v}x¦
besn}t}z¨4p}vx$nz£¦¨¦®¬;ez¨yBp}rv44«{tdes½z£yÂkoeMtÄp}z¨vy
<
¦£vxy{wz|p}`½pr`{e'p}est$`bybz©¸B«bep}vVnred¦£estÄp3pr`benq;)prz£v±
p}edcEvx$¦£¦¨mª¦£v4tdx¦¨z£²deM'}edwz¨vy{n®v~z£yBpresreMnp`besreJpr`{esnre¦¨v4tdx¦54esn}t}z£4prv}nG}e*tvxcb«bpres5¯
n[vxybenrz¨y{wx¦£eGbesn}t}z¨4p}vx+z©n+ybvpInr«4§ªtz£edyBp[prvtdx4pr«{re pr`beutdvxcb¦£e y{vprz£vxy.v~TJc.vxprz£vxy.ed¥edyBps´
®ez¨y;4edeM^tdvxy{nrz©4ed¼nrepv~
d= 1. . . L 4z¨¿Eesresyp¦£v4tdx¦I4esn}t}z¨bprvx$ns¯B¤edy{tdex´ e¦©nrv`;¥xe3prv nrEestz¨~¶mª3®m.prvtvc¬bz£ybe¤pr`be4z©nrnrz£c.z£¦©}z|pm¥)¦£«besn v~0pr`be4z¨¿EesresyBpG4eMnrtdrz£4p}vx$nd¯°Ve*tvy{nrz£4es®
®edz£wx`Bp}esnq«bc¼´Ey{pr`{e4z©nrnrz¨cz£¦£xrz¨pmªcesnq«{re*¬Eep®ededy¼p®v¥oz£4esvªnqeswxcedyBp$n
r xy{ sz©nGµ{y{x¦¨¦£m wz¨¥edy¬Bm
Φ(r, s|(C(s)) = XL d=1
ωd(C(s))Td(r, s) ?< A
`{ed}e
Td
z©n.nrz£c.z£¦©}z|pm.preMnpu`{z£t$`z£nGybv) 4eµ{y{esz¨ypr`{e*nr«b¬{nres¸B«besyBpGnr«b¬{nrestprz£vxy»4¯xy{
ω z£n
td¦£nrnq±9bedEedy{4esyBpG®edz£wx`Bp}z¨ybw.`bz©t$`¼z¨¦£¦¾¬;ee 4b¦£xz¨y{esz¨y¼nr«b¬{nrestÄp}z¨vy4¯< ¯
77@5 !,5 -; )4 $
Ãy½cep}`bv4V«;nqz£ybw¼¦¨v4ts¦I4eMnrtdrz£4prv}ns´0¥oz£bedv»nqeswxcedyBpz©nresb}esnredyBpreM¬om
$"
vx~ m ~¶eM)p}«bre
¥estprvx$nJv~G4z¨cesy{nqz£vxy ?Áp}`beªyo«bc3¬;esv~®tdvxy{nrz©4ed}es^4esn}t}z£4prv}n0AÄ´¾`besrecz£n*p}`beyo«bc¬Eedvx~
nred¦£estpreMz¨yBp}ed}esnqp Evxz£yp$nXz£yp}`be¥oz©4edv.nqeswxcedyBpM¯°?z¨pr`vpr`{ed® v}bns´x®exreJtvy4~¶rvyBpresz¨pr`'pr`be
{rv¬b¦¨esc·vx~tvc{}z¨y{w*p v.nqedp}n v~¾~¶espr«b}e¤¥)x¦¨«beMn
Θr,Θs
v~ybvp®ybestdesn}nrxrz£¦¨m3es¸B«{x¦Enrz¨²se
mr, ms
¯
ÃytvxyBp}}npIp}v3tdvxc{}z£ybw~¶espr«b}eu¥estÄp}vx$nd´pr`{etvc{}z£nrvxyvx~0nqedp}n v~¾~¶eM)pr«{re¥xestprv}nXz£n ybvp
v¬o¥Bz£vx«;nd¯+ ;vBnrnrz¨¬{¦¨e*Gmz£n®p}v.eMnp$¬b¦£z©nq`tvx}}esnr;vy{4edy;teJ¬;edp esedypr`beJ~¶espr«b}esn®vx~0pr`{e*p vnqedp}n
¬om¼{xz¨}z¨y{wpr`be.vxy{esn`bz£t$`V}eybeM}esnqp¤z¨y ~¶espr«b}e3nr{xtde ?Onqese3Àz¨w;¯ ?6A Aį B¤v) es¥xedM´;pr`besre.}e
f(r,1)f(r,2)f(r,3)f(r,4) f(s,1)f(s,2) f(s,3)
D ! "
$##
%M!"& M'
#
f(r,1) f(r,2) f(r,3) f(r,4) f(s,1)f(s,2) f(s,3)
)(*+
"%,"M!"&-
"&./
!M!%
Àz£wx«b}e
B
棦¨«;np}}prz£vxyv~Evn}nrz¨¬b¦£e*cbbz£ybwBnG¬;edp esedy~¶espr«b}enqedp}ny;'p}`bedz£b}vx¬b¦£edcn
nred¥ed$¦b{rv¬b¦¨escnX«{nrz£ybwnr«{t$`'cxbbz£ybw{¯+9~0®eu}esnqpr}z£tpXpr`{ec{bz¨y{wnq«;t$`pr`{p®eMxt$`~¶eM)pr«{rez£n
«;nqeM3vyb¦£m3vy{teGp}`bedyp}`bed}e¤}eunrvxceG~¶espr«b}esn[`{z£t$``{¥eGybvpX¬Eedesyªtvy{nqz©4esreM?Oxn+p}`bep®vnqedp}n
bvybvxp `{¥epr`be*n}cenrz¨²se AXy{`{edy{tde®e¤¦£vovnre¤nqvce¤v~¾p}`beJe opr$xtpresªz¨y4~¶vrcprz£vxy5¯ Buv)®ed¥edM´
z¨~0®ex¦¨¦£v)?c«b¦¨prz£b¦£eJnrnrz£wxybcedyBp$nIp}`be{rv¬b¦¨esc v~34edwedybes}presªcxbbz£ybwcm}z©nqe`bed}ex¦¨¦
~¶eM)p}«breMn*v~Gvxybeªnrepxrenrnrv4tz©)preMz¨pr`½vybenqz£ybwx¦£e.~¶espr«b}ev~®pr`bevxpr`besnrep ?nredeªÀz£w{¯»6?¶¬%A2Aį
_G`{eJtdvxy{nres¸B«besy{teJz£nGxwz£yªp}`{)p®eJbv3ybvxputvxy;nqz©4edG¦£¦E~¶espr«b}esns¯[ ~¶«{qp}`bedEvn}nqz£¬bz£¦¨z¨pmcm¬;e
p}v»tvxy;np}}xz¨y pr`bec{bz¨y{w¼nrvxces`bv)nr«{t$` p}`{)px¦¨¦+~¶eM)p}«breMn}e3eM¸B«{¦£¦¨mVtvxy;nqz©4ed}es5´¾¬b«4pz¨pz£n
KLKNMOPPQR
f(r,1), . . . , f(r,mr)
f(s,1), . . . , f(s,ms)
Àz£wx«b}e3
B
_G`be3¥oz£4esv'nres¸B«bedy;tesn}etvy{nqz©4esreMprv'¬Eep}`be}esnr«b¦¨p¤v~Ixy«byb³oybv)y»b}vx¬;¬bz£¦¨z©nqprz©t
{rv4teMnrns¯f vy{nqeM¸B«bedyBpr¦£mx´M®eIxn}nq«bce[p}`{)p0pr`be e opr$xtpres~¶eM)pr«{reMn0}e[}es¦£z£²s)p}z¨vy{nTv~{xy*«byb³oybv)y5´
¬{«4p¤t¦©xn}n±Ã4es;esy{4edyBpu4z©np}rz£¬b«4p}z¨vy5¯
y{vpvx¬o¥oz¨v«{n `bv)?prv4v.p}`{)pM¯IÀb«brpr`besrcvre´B®eJ v«b¦£`{¥euprvts¦©t«b¦©)p}e¤p}`be4z£nqp}xy{te*¬Eep®edesy
es¥xesrm^;vBnrnrz£¬b¦¨e{xz¨.v~u~¶eM)p}«breMnp}vµ{y;¹p}`bey{es}esnqp~¶eM)p}«brez¨ypr`bevp}`bednrepM¯º_G`{z£n.z©n.¸B«bz¨pre
e 4;esy{nqz£¥xexy{'`{edy{tde~¶«{qp}`bedu4$¬{t$³v~nr«{t$`¼y»{brvBxt$`5¯
Ãy{nqpreMxvx~5esnqp}¬{¦¨z©nq`{z¨ybwtdvx}reMnqEvxy;4edy{tde¬;edp esedyp}`be¤es¦¨esc.esyBp}nIvx~Tpr`be¤p vnrep$nd´B etvxy;nqz©4ed
p}`beG¥)¦£«besnxn}es¦£z£²s)p}z¨vy{nv~;«byb³oybv)y.4z£nqpr}z£¬b«4prz£vxy
D?Onqese®Àz£w{¯[ A¯CB¤edy{tdex´prv4eMtz©4e®`bep}`bed
p®vnredwxcesyp$n[M´nIxrevx~Tpr`be¤n}ceutd¦£nrn[z¨p z©nInr«4§ªtz£edyBpIp}v4estdz£bex´B`bep}`bed[p}`be¤¥)x¦¨«beMnI}eu4$y
~¶}vxc pr`{e3n}ce«byb³oybv)ybz£nqpr}z¨¬{«4prz£vxy»v¤ybvxps´EzO¯ex¯Gp}esnqprz£ybwªp}`be`omo;vxpr`beMnqz©n
H0:Dr≡ Ds¯ 5ep
«;n®µ{}nqp¤tvy{nrz£4es®pr`betsxnreJvx~nrz¨ybw¦¨e*¦£v4td¦5besn}t}z¨4p}vx?
L= 1A¯
_G`be*p®v±Ãn}cb¦£esn °?z¨¦©tv 4vxy¼uxyb³B±Ãko«{c(_0eMnpuz©n. es¦¨¦¨±9³By{v)y D Gy{np$)p}z£nqprz©tdx¦¾c.edpr`bv4prv
p}esnqpGpr`bz©n`omo;vxpr`beMnqz©n
H0 ~¶vxnrts¦©G¥)¦£«besns¯[_G`bz©nGybvxy4±9{x}xc.edpr}z£tupreMnpu`{n pr`be4¥)yBp}xwxeprv¬;e
bz£nqpr}z¨¬{«4prz£vxy4±=~¶}edeuy;¥xvxz©bnp}`beuµbp v~5b}estdz£nrecvobed¦=¯[k4;eMtz¨µ;tdx¦¨¦£mx´xpr`be¤presnqp nqp}prz©np}z£tz©nIwz¨¥edy5´
z£y».ybvx}c¦£z¨²ses~¶vrc¼´b¬om
Wd=
12(mr+ms) mrms
12Xmr
j=1
R(mr+ms)j
mr+ms+ 1 −1 2
? A
`{ed}e
R(mr+ms)j
z©n+pr`{eu}xyb³vx~;p}`be
j±=pr`¥)¦£«bev~Tp}`beµ{$np[~¶eM)p}«brenqedpIz¨yp}`be¤tvc¬bz£ybeM~¶eM)p}«bre nrepM´z=¯ex¯pr`be;vBnqz¨prz£vxyz£yp}`beuvx$4esreMnqeM¸«{edy{tdev~5¬Evpr`'¥x¦¨«{e¤nqedp}ns¯[_G`be¤4z©np}rz£¬b«4p}z¨vyªv~Tp}`bz£nXpreMnp
nqp}prz©np}z£tsnuz£¦¨¦+tvyo¥xed}wxe*~¶v
mr, ms→ ∞prv'pr`be N(0,1)±Ã4z©np}rz£¬b«4p}z¨vy»z|~[p}`be.`BmoEvp}`besnrz£n
H0 z£n
~¶«{¦|µ{¦£¦£es¾¯[9~+ybvpM´4pr`{e¥x¦¨«{eJz£¦£¦5¬;e*~xG~¶rvc#²desrv;¯
B¤edy{tdex´X®e»4eµ;ybe
Td = Wd2 xnxyz£y{4z©td)p}vxv~J4z£n}nrz¨cz£¦£xrz¨pmx¯nVtdvxy{nres¸B«besy{tev~pr`be tdvxyo¥xesrwedy{tde*reMnq«b¦¨p?z|~
H0 z©n~¶«b¦¨µ{¦£¦¨eMAÄ´
Td
z£nz¨y;4edEedy{bedyBpvx~[pr`be3cxwxybz¨pr«;4evx~[pr`be.tvxc{xreM
~¶eM)p}«bre¥)¦£«beMnd¯'_G`{z£n{rv;esqpmz©nz£c;vqp$yBp`bedy½®etvxy;nqz©4ed»td¦£nrnq±Ã4edEedy{bedyBp esz¨w`p}z¨y{w
nz¨px«4prvc)prz©tdx¦¨¦£m^edy{nr«b}esnpr`beybvx}c¦£z¨²M)p}z¨vy^vx~upr`{e4z©n}nqz£cz¨¦©}z|pm^¥)¦£«beª~¶v.x¦¨¦Gtvy{nrz£4esreM
besn}t}z¨4p}vx$nd¯
.)+!#$ -;"A) 7@ 0 5.!#$A7>5 0 & 2)4=@$A 5.=
npr`be°?z£¦£tdv ovy uxyb³B±Ãko«{c _0eMnp*z£n*vyb¦£m4eµ{y{es»~¶vnrts¦©*b)p$b´T etdxybybvxp*xbb¦£m¼pr`be.preMnp
bz¨}estpr¦£m^vy½pr`{e4eMnrtdrz£4prvp}«bb¦£ex¯ºÃy{nqpres¾´+ e'pr}esp3est$`¹¦£vots¦®4esn}t}z¨bprvx.nqes{$)p}ed¦£m^nxy
z£y{4z£¥oz£b«{¦54eMnrtdrz£4p}vxuy;'®e*z¨¦£¦¾pr`{edy¼z¨yBp}rv44«{tde.tdvxc3¬bz¨y{es¼4z©nrnrz¨cz£¦£xrz¨pmcesnq«b}ex¯
CEDCK+CGF
F
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0 5 10 15 20
P[T|incorrect]
P[T|correct]
0 0.1 0.2 0.3 0.4 0.5 0.6
0 5 10 15 20
P[T|incorrect]
P[T|correct]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 5 10 15 20
P[T|incorrect]
P[T|correct]
Àz£wx«b}e <
B
bxc.{¦¨eMn~¶vxpr`betdvxy{4z¨prz£vxy;¦ob}vx¬{x¬bz¨¦£z¨prz£esn
pd(t|c)?¶wresedy¦£z£ybe Ay;
pd(t|c)¯ ?reM¦£z¨y{e A v¬4p}xz¨y{es~¶}vxc#p}`be3{)p}nqedp}ns¯®¤¦¨¦v~+p}`bedc nq`{v)j{rvybvx«by;tesEesx³p¤²desrvªxy{»ª4eMt}esxnrz£ybw
y;)pr«{reJ~¶vxz£y{tdreMxnrz¨ybw.¥)¦£«besns¯
_G`be.nqz£cb¦£esnqpGm'p}vtvc¬bz£ybep}`be3bz£n}nqz£cz¨¦©}z¨pmvx~X¦£¦0z£y{4z£¥Bz©4«{x¦besn}t}z¨4p}vx$n®vx«b¦©»¬Eeprv
«;npnr«bc p}`bedc¼´0z=¯ex¯£´«bybz¨~¶vx}c ®edz£wx`Bprz£ybw;¯B¤v) es¥xedM´¾ybvxpx¦¨¦Ibesn}t}z¨4p}vx$nJ`{¥xe3prv¼¬Eeªt$`{$xt±
p}ed}z£nqprz©t~¶v¦£¦Etd¦£nrnresn®y{'`bedy;tex´4«bybz¨~¶vx}c esz¨w`p}z¨y{wcm}¬bz|p}}xrz£¦£m.4eswx$x4eupr`be¸B«{¦£z¨pm.vx~
p}`bebz£n}nqz£cz¨¦©}z¨pm.ceMxnr«bre¯Ãy;np}esx5´®eup}rm.prv3¦¨eM}ypr`be*4z©nrtdrz£cz¨y;)prz£¥xe¤;v)®ed v~5est$`ªz£y{4z£¥Bz©o±
«;¦I4eMnrtdrz£4prv
d ~¶rvc pr`bep}}xz¨ybz£ybw¼b)p$xy{^4es4«;tebrv;esJ®edz£wx`Bp}z¨ybw~¶«by{tprz£vxy
ω ~¶vvx«b
bz£n}nqz£cz¨¦©}z¨pmc.eMxnr«b}ex¯
¤yz¨y{bz£ts)prv¾~¶vxTp}`beXxbb}vxb}z£presybesn}nEv~44esn}t}z¨bprvx
dz£nTp}`beXnr«{tdtdesn}nTb}vx¬{x¬bz¨¦£z¨pm
P[tc< tf|d]
p}`{)pGp}`beJpresnqpu}esnr«b¦|p
tc
~¶vGnredwc.esyBp®{xz¨moz£ed¦©4z£ybw.tdvx}reMtÄp®td¦£nrnrz¨µ;tdprz£vxy'z©nnrcx¦¨¦£ed®p}`{ypr`be
¥)x¦¨«be
tf
~¶vx+¤nredwxcesyp{z£mBz£ed¦©4z£ybwu~¦©nqe®vxy{ex¯0°^e µ{}nqpz£yBpr}v44«{teXpr`beGtvy{4z¨prz£vxy{x¦{rv¬{¬bz£¦£z|pm
bedy{nrz|pmvx~0p}`beJpresnqpunp$)p}z£nqprz©tdn
Td
wxz£¥xesyªpr`{e*td¦£nrnresn v~0pr`beJpreMnp}esnqeswxcedyBpG{z£M¯+vx}eb}estdz£nred¦£mx´
®etvy{nqz©4es p}`be*p®v.tsxnresnG`besreJedz¨pr`besu¬Evpr`¼nredwxcesyp$n}evx~p}`benrxce*td¦£nrn
cvx®p}`{)puvybe*z£n vx~td¦£nrn
c y{p}`bevp}`bedvybe*z£n " v~+t¦©xn}n
c´4zO¯ex¯
pd(t|c) =P[Td(r, s) =t|C(r) =C(s) =c]
pd(t|¯c) =P[Td(r, s) =t|C(r) =c, C(s)6=c]
?O A
B¤edy{tdex´4p}`benq«;tdteMnrnG{rv¬{¬bz£¦£z|pmªvx~4eMnrtdrz£4prv
dwxz£¥xesyt¦©xn}n
c tdy¬Eetd¦©t«{¦£presxn
ko«{tst
(d|c) =P(tc< tf|tc∼pd(·|c), tf ∼pd(·|¯c))
= Z ∞
0
pd(tc|c) Z ∞
tc
pd(tf|c)dt¯ fdtc
?A
_G`be.y{x¦¨m4nrz£n¤v~[p}`be3escbz¨}z©td¦`bz©np}vxwx$cnvx~[pr`be3p®v'tvy{4z¨prz£vxy{x¦4z©np}rz£¬b«4p}z¨vy{n?¶vx¬bp}z£ybeM
~¶}vxc pr`be¼p}}xz¨ybz£ybw½nrep0A.`;xnnr`bv)yp}`{)pp}`bed}e»z©nª^brvybvx«{y{teMEesx³)pª²sed}v ?OnqeseÀz£w{¯ < Aį
Ãy;nqbz£}es¹¬om DG9´X®e'esnqprz£c)p}ep}`besnre¼4z£nqpr}z£¬b«4prz£vxy;n3«;nqz£ybwºczop}«brev~* z¨$xtb«b¦©nrexy{xy
e 4;vybedyBprz©¦5bz£nqpr}z¨¬{«4prz£vxy
b
pd(t|c) =λd(c)δ0(t) + (1−λd(c)) 1
βd(c)eβd−t(c) (t>0) ?1FA
KLKNMOPPQR
`{z£t$` z¨yBpr}v44«{tdesnnqvc.e^z£c.{¦¨z©tz¨pnqcvovp}`bz¨y{w`bz©t$` b}ed¥edyBp}nv)¥xesqµ{pqprz£ybwprvÂpr`beºb)p$b¯#_G`be
eMnp}z¨cpre~¶v
λz©n¤4edµ{ybeMxnGp}`be~¶$xtprz£vxy¼v~In}cb¦£esn
t <1¯®_G`beesnqprz£cpre~¶v
β z©n¤nrz¨cb¦£mpr`be n}cb¦£ecesxy»v~I¦£¦0¥)¦£«beMn
t≥1¯_G`be~¶vrc3«b¦©.~¶vx
ˆ
pd(t|c)¯ z£ny{x¦¨vwx«be¯®_G`{ees¸B«{prz£vxy~¶vx¤pr`be nr«{tstesn}n®b}vx¬{x¬bz¨¦£z¨pm'tdxyybv) ¬;ets¦©t«b¦©)p}esxn
B
k4«{tdt
(d|c) =λd(c) + (1−λd(c))(1−λd(¯c)) βd(¯c) βd(c) +βd(¯c)
?A
_v'³eded¼p}`beµ;y{¦4z£n}nrz¨cz£¦£xrz¨pmceMxnr«breMnutvc{$¬b¦£extdrvBnrn¤4z|¿Ted}edyBptd¦£nrnresns´{®e`;¥xe*prv
y{vx}cx¦¨z£²de*p}`besnrenr«{tdtdesn}nb}vx¬{x¬bz£¦¨z¨prz£esns´{nrvpr`{pupr`{enq«{c v)¥edux¦¨¦4esn}t}z¨bprvx$nGz©nuvxy{ex¯ Buv)®ed¥edM´
n pr`be4z¨¿Ted}edy{tdesn®v~pr`beMnqe*b}vx¬;¬bz£¦¨z¨prz£esnGxreJ}pr`besnqc¦£¦=´4 e{b¦¨me oEvxy{edyBprz©¦5nqpr}ep$t$`p}z¨y{w
{rz£vx®prvp}`beybvrc¦£z£²s)p}z¨vy5´BzO¯ex¯£´
ωd(c) = 1
Zeα(d|c) z¨pr` Z =X
d
eα(d|c) ?A
`{ed}e ®e escbz¨}z©td¦£¦£m~¶vx«by;
α= 10prv*nq«4§ªtdz¨esyp}¦¨m3np}redp}t$`p}`be$)pr`{ed+nqc¦£¦b4z¨¿Eesresy{teMnv~{p}`besnre
{rv¬{¬bz£¦£z|p}z¨eMnd¯
! '*!& +
<5>$)4"A)4 $ 7@5>$ )? ) $A7>5
_G`{ez¨yBp}ed}esnqp¤nqes¦¨eMtÄprz£vxy»cedpr`bv4¼`bz©t$`» e`;¥xe«{nres¼z£y»vx«b¤e 4;esrz£cedyBp}n¤z£nu¬{xnres¼vy¼'nqz£cb¦¨e
tdrz¨presrz£vxy'prvªnr;vxpvx«4pp}`bereswxz£vxy{n®z¨pr`».`bz£wx`¼4edy{nrz¨pm'v~+nrtdedybecvxprz£vxy¼xtÄp}z¨¥oz¨pmx´4z=¯e¯¨´
A(p, t) = 1
|F|]{(q, τ)∈ F(p, t) | |F D(q, τ)|> γ} ?MA
`{ed}e
F z©nuªnr{)p}z¨vx±Op}edc;v}x¦¾ybedz£wx`o¬Evx«b}`bvov4'z£y{4v) ?ex¯w{¯
15×15×3AGy{p}`be
F D z©npr`{e
p}edcEvx$¦+~¶$ce4z|¿Ted}edy;tex¯'°Veªtsyº¦©nqv»xtstvc.v4bpre3pr`betdxnre.v~¼c.v)¥oz£ybw¼tdces}¬om»µ{$np
tdvxcEedy{n})p}z¨ybw^p}`be4vxcz£y{yBpz£cxwxe¼cvp}z¨vy ?ex¯w{¯£´ }ed{reMnqesyp}es¬omy?)§ybe¼cvp}z¨vy c.v44es¦A
`{z£t$`tdxy¬;eu«;nq«{x¦¨¦£m3nrnr«bcesprv¬;eu}ed¦©)p}esprvp}`beutdxced$cvxprz£vxy5¯+_G`bedy0´z¨p tdxy¬;eutsy{tded¦£¦¨eM
¬omtdvxy{nrz©4ed}z¨y{w p}`be
DF D ¥x¦¨«{esn ?z©nqb¦©xtdesDÀb$ce ¤z¨¿Ted}edy{tde A.z£y{np}esvx~¤pr`{e
F D ¥)¦£«besns¯
f vy{tesry{z¨ybw3pr`{e*pr`b}esnr`bvx¦©¾´4®e*«{nqeM
γ= 80`bz©t$`}esnr«b¦¨p}nGz£y»3`{z¨w`b¦¨mnqes¦¨eMtÄp}z¨¥eb}v4teMnrns¯
_G`betÄp}«{¦5nred¦£estprz£vxy'vx~p}`be*;vz¨yBp$nz£nt$`bz£ed¥xeMªz£y»3wreses4m~nq`{z¨vy5´4`bed}e* enr«{tdtdesn}nqz£¥xes¦¨m
nred¦£estppr`{e'Evxz£yp.z|p}`¹c 4z£cx¦ tdrz¨presrz£vxyº¥)¦£«bex¯V_0v ¥vxz©Vpr`{p.¦£¦XEvxz£yBp}n3}e'nqes¦¨eMtÄp}es^~¶}vxc
x¬;v«4pp}`be'n}ceªEvnrz|p}z¨vy{nd´+y;º`bedy{tdex´prvesy{nr«breªpr`{ppr`besm^xreªnr«4§ªtz£edyBp}¦¨mºnq{reMx^v)¥xespr`be
`{vx¦£e¤¥oz£bedvnqeswxcedyBps´o®eJnr«{tstesn}nrz¨¥ed¦£m3cxnr³.pr`benr«b}rv«by{4z£ybwvx~0ed¥ed}mnred¦£estpreMªz£yp}ed}esnqp®Evxz£yBps¯
tstv}4z£ybw¦¨m´xp}`bet}z|p}ed}z¨vy
A(p, t)z©nGe 4b¦£z£tdz|p}¦¨mcv44z¨µ{es)~ÁpresGesxt$`nqes¦¨eMtÄp}z¨vy'z£y'p}`benqesy{nqepr`;)p
x¦¨¦
F D ¥x¦¨«{esnz¨yºªybesz¨w`o¬;v«br`{vBv4
F0 v~Ip}`benqes¦¨eMtÄp}esz£yBpresreMnpJ;vz¨yBp}e3nrepJprv²ded}v{¯ÃyVvx«b
e 4;esrz£cedyBp}ns´4pr`{z£ncnq³
F0 `;xnG¬Eededyt$`{vnredy¼nq«{t$`p}`{)pp}`be¬b¦£vot$³4n
B`{z£t$`»}e*z¨yBp}rv44«{tdesz£y nr«b¬{nrestprz£vxy
< ¯<
4vybvpv)¥xesr¦©0¯
_G`benqes¦¨eMtÄprz£vxy½{rv4teMnrnz£nnqprvbEes¹nnqvovyºn*pr`{et}z¨pred}z£vxyº¥)¦£«be~¦£¦©n¬;es¦¨v) ¼p}`b}esnr`bvx¦©
}ed¦©)p}z¨¥e*prv'pr`be.z¨y{z|p}z£x¦tdrz¨presrz£vxy»c 4z£c«bc¼´EzO¯ex¯£´
βmax(p,t)A(p, t)´¾`bz£¦£e®e«{nres
β = 0.01 z£y
p}`bee 4Eed}z¨cesyp$nd¯
CEDCK+CGF
0 35
0 384
t-dimension
x-dimension Nodding
0 70
0 384
t-dimension
x-dimension Drinking
Àz£wx«b}e
B
樦£«{nqpr$)p}z¨vy v~p}`be^;vz¨yBp»t¦£vx«{bn ?¶{rvxestÄp}z¨vy vxyBp}v¹pr`{e opq±9b¦©ybeHA~¶v'p}`be^td¦£nrnresn
ybv4b4z£ybw ?¦¨ed~Áp0A3y{ 4}z£yb³oz¨ybw ?¶}z¨w`pAįÃyDtdnqeªvx~ brz£yb³oz¨y{wÄ´0p}`benr«b¬4eMtÄpwx$¬{n*p}`be'w¦£nrn
vx~JG)p}edM´4rz£yb³4nªxy{Db«4p}n'z|p¬{xt$³Â`bz£t$` }esnr«b¦¨p}nz£y pr`beVnq`bv)y ¦£e~Ápr±=}z¨w`Bpq±9¦¨ed~Ápª{)prpred}y5¯j¤n
ybv4b4z£ybwJz©n«{np¤.}vp$)prz£vxyv~pr`be`{esx5´{nq«;t$`.{pqpresry¼tdxybybvxpG¬Eevx¬;nqesr¥es¾¯
"A!<) $ 7 "!, <5 7 "/*!#$A 7@5
_vt$`{x}tÄpresrz£²de.cvp}z¨vy¹tvxyBp}edyBpz¨y½¥oz©4edv;´z¨p3nredescny{)p}«b}x¦¨¦£mp}vnqvced`bv) 4esn}t}z£¬;ep}`bepr$)±
eMtÄp}vx}m¹v~pr`be»cv)¥oz¨y{wVv¬4estp}nz£yÂp}`be»z£cxwxe»nres¸B«bedy;tex¯ B¤v) es¥xedM´[p}`be»4ep}estÄp}z¨vy v~v¬4estp}n
xy{esnqprz£cprz£vxy¼v~pr`{edz£Gpr$)estprvrz£esn®z©nu.ybvxypr}z£¥Bz©¦¾{rv¬b¦¨esc¼¯Buedy;tex´{ e«{nrec«{t$`¼nrz¨cb¦£ed
besn}t}z¨4p}vxJ`bz©t$` p}rz£esnJp}v¼ts4p}«b}ep}`benq{prz£v±=presc.Evx$¦+es¥xvx¦£«4p}z¨vy v~ pr`bez£yBpred}esnqp;vz¨yBp$nd¯.°Ve
tdvxy{nrz©4edp}`be
<
(Evxz£yBp3t¦£vx«; ?nredeÀz£w{¯ Awxesybed$)p}es^¬om p}`benred¦£estpreMºz¨yBpresreMnp3;vz¨yBp}n3z£y½pr`be
¥vx¦£«bceJ~¶vx}c.eM¬Bmªpr`bez£cwxenres¸B«besy{te¯Xkoz£y{te®e}eJy{vpuz£yp}ed}esnqpreMz£ypr`bex¬{nrvx¦£«4pre*Evnrz|p}z¨vy
vx~Gpr`bec.v)¥oz£ybw»vx¬bestÄp$n ?Oxn®e'4v ybvpxn}nq«bcep}`be'¥oz©4edv nredwcedyBp}nprv¬Eex¦¨z£wxybeMAÄ´¬b«4p.vxyb¦£m
pp}`bedz£nr{)p}z¨vx±Op}edc;v}x¦es¥xv¦¨«4p}z¨vy5´¾®e3z£¦¨¦[}e~¶esJpr`becesnq«{resc.esyBp}nres¦£prz£¥¼prvpr`betdedyBpresJvx~
w}¥oz¨pmv~pr`beMnqe*Evxz£yp$nd¯
_vÂ4eMnrtdrz£¬Eepr`{eVEvxz£yBpt¦£vx«;¾´¤®e tvy{nrz£4es'~¶v'eMxt$` z£yBpred}esnqpEvxz£yBp
pi = (xi, yi, ti) pr`be
~¶v¦¨¦£v)z£ybw3ceMxnr«brescedyBp
ν(pi) = (xi−x)¯g(yi−y)¯h(ti−t)¯l ?qx A
vx~Ivx$4ed
o=g+h+l´T`besre?
¯
x,y,¯ ¯tA¤z©n¤p}`betedyBp}ed¤vx~Xwx$¥oz|pmv~ x¦¨¦z£yp}ed}esnqpEvxz£yBp}ns¯'B¤edy{tdex´
®etsytdx¦£td«b¦©)pre~¶es)p}«b}e¥xeMtÄprv~¶vx¤esxt$`»z£yBpresreMnp¤Evxz£yBp
pi
`bed}ep}`be4z¨¿EesresyBpJtvc;vybedyBp}n
v4eMnrtdrz£4prv}ntdvx}reMnqEvxy;p}v 4z¨¿Eesresyp.tvxc3¬bz£y{)p}z¨vy{nv~
g, h xy{ l¯Ãy½vx«be 4;esrz£cedyBp}ns´ e tdvxy{nrz©4edtvc¬bz£y{prz£vxy{nGz¨pr`
o= 1. . .6¯
¥edy»p}`bvx«{wx`¼pr`{esnrecesnq«b}edcesyp$n®ed}ez¨y{nrbz£reM¼~¶}vxc pr`be.td¦©t«{¦£prz£vxy¼v~Icvxcesyp$nd´{pr`besre
xreInrvxceXnr«b¬4p}¦¨e®4z¨¿Eesresy{teMnd¯9~b eI®vx«{¦£tdvxc{}e[¥oz£bedv¤nredwc.esyBp}n5«;nqz£ybwucvxcesyp$nd´z=¯e¯5p}`be nr«bc
vx~5pr`{e*tdvxcb«4p}es'¥x¦¨«{esns´B®eJ v«b¦£'r«{y'z£yp}vp}`be*b}vx¬b¦£edc pr`;)pGpr`beJ¥)}z£prz£vxyv~5p}`beJbz|¿Ted}edy{tdesn
vx~[c.vcedyBp}nv~I4z¨¿EesresyBpvx$4es}nuxreybvpJes¸B«{¦=´{`{z£t$`cx³xesnupr`be.4eµ;ybz|p}z¨vyv~Ixy b{rvbrz©)p}e
bz£nqp}xy{tec.eMxnr«b}eub}vx¬b¦£edc)p}z£tx¯[_G`bz£n `bv)®ed¥edXz©n®t}«{tz©¦;~¶vx pr`be*t¦©xn}nXbedEedy{4esyBp®®edz£wx`Bprz£ybw;¯+
KLKNMOPPQR
M
nrestdvxy{V4z¨¿Eesresy{te.z£np}`{)p*pr`{eµ{$nqp*v}bedJ¬{}m4tedyBp}rz©tcvxcedyBp$nJxre¦£Gmon¤²desrv¬om4eµ{y{z|p}z¨vy
xy{4vybvxpImoz¨es¦£xyBm3z¨yb~¶vx}cprz£vxy5¯ÃyªtvyBpr$xnqps´)p}`beµ{$npIv}bedIcesnq«{resc.esyBp+®e¤tdvxy{nrz£bed[`{¥e
nr`bv)y«bprv¬Ee4z£n}t}z£c.z£y{prz£¥xe!?¶¬om'¦£vovx³oz¨y{wpGpr`be®edz£wx`Bprz£ybwBnA¯
7,$A 7@5 5>$ )45 <$'& 5 7 "/*!#$ 7@5
_G`{ecvp}z¨vy z£yp}edy{nrz¨pm ?¥xes¦¨v4tz¨pm,Az©n*ybvxpr`bes¤z£c.Evxrp}xyBpJnrvx«b$te3v~Iz£y4~¶vx}c)prz£vxyprvt$`;$xtÄp}ed}z¨²se
cvxprz£vxyes¥xedyBp$nd¯9~X®etvxy;nqz©4ed~¶vxJe bxc.{¦¨eªG¦£³oz¨ybwy{V'}«byby{z¨ybw;es}nrvxy0´;p}`be.pr$)eMtÄprvrm
besn}t}z¨4p}vxXtdvx«b¦©b}vx¬{x¬b¦£mybvxpX¬;e¥xesrm.4z|¿Ted}edyBpM´`besreMxnI¥xed¦£v4tz¨pmB±=}ed¦©)p}es.4esn}t}z¨bprvx nq`bv«b¦©
¬Eex¯ °^e^`{¥xe bbpres pr`beºn}tedy{e cvp}z¨vy t$`{x}tÄpresrz£²sprz£vxy z£yp}rv44«;tes ¬Bm DG*prvDvx«{¦£v4td¦
xbb}vt$`5¯_G`beJ`bz©np}vxw}xc·vx~5pr`be*tvy{nrz£4esreM¦£v)G±=¦£ed¥ed¦{cvxprz£vxy~¶eM)pr«{reMnXz©n®ybv¦£vxybwedGtvc.{«4preM
v)¥edXpr`be*`bv¦¨eJ¥oz£4esv.nredwcedyBps´4¬b«bpGz£y.¬b¦£v4t$³
Bi vx~nrz¨²se 32×32×5nq«b}}vx«by{bz¨ybwpr`beJz£yp}ed}esnqp
Evxz£yBp
pi´b`bed}eJpr`betdvxy{nrz©4ed}es'c.vxprz£vxy~¶espr«b}ez£n®p}`be¥xes}xwxeMybvx}c¦{v) cxwxybz¨pr«;4e
¯ v(p, t) =
P
q∈W(p)k∇I(q, t)k2· |vn(q, t)| max
|W|η2,P
q∈W(p)k∇I(q, t)k2 ?qM A
`{ed}e
W z£n® 3×3ybesz¨w`B¬Evx«{r`bvov4z£y{4v)´
η2z©n®ybvz£nre¤}ed¦©)p}es.pr`{reMnq`bv¦£'y{
vn= −∂I/∂t||∇I|| ¯
¤wxz¨y5´B®eJtdvx«b¦©ªxtstvc.v4bpre¤tsced$cvp}z¨vy5´oz|~yomx´o¬omtvxy;nqz©4ed}z£ybwpr`{e}esnrz£b«{¦Eybvx}c¦{v)
cxwxybz¨pr«{bex¯
n*brv;vBnqeMz£yIDG9´¾p}`bz£nbz£nqpr}z¨¬{«4prz£vxyVz£n*cv44ed¦£esVz¨pr`ºczopr«b}e4z£nqpr}z£¬b«4prz£vxyVv~Gz¨$xt
{«b¦£nre3p¤²desrvE?tvr}esnrEvxy{4z£ybwp}vªpr`{e3nrmBc3¬;v¦¨z©tnqp}pre ybvªcvxprz£vxy y;'tdvxyBprz£yo«bvx«{nu{rp}ed4±
}esnredyBp}z¨ybwªpr`be3reM¦0cvxprz£vxy¥)¦£«beMnd¯uÃyVtvxyBp}}npup}v DHG9´;p}`betvxyBp}z¨yo«bv«{n{xqpJz£ncvobed¦£es»z¨pr`
¦£vxwx±=y{vx}cx¦44z£nqpr}z£¬b«4prz£vxy0´¬Eests«{nreGpr`besre`{ed}evx¬{nred}¥))prz£vxy;n+`bed}eGpr`be¤²ded}v±ÃtesyBpred}esh*«;nrnrz£xy
}esnqpr}z©tÄpreMprv
(0,∞)GxnIy{v3¦£vxybwed®nq«bz¨p}x¬b¦£e?OnqeseÀz¨w;¯I A¯9ptdxyª¬Eee o{¦£xz¨ybeMª¬om.pr`be~xtp p}`{)p p}`be¬b¦£v4t$³onGxre*b¦©xtdesvxy¼reswxz£vxy{n®z¨pr`¼$)pr`{edG`bz£wx`¼cvprz£vxy¼xtprz£¥oz|pm¯°^ewxedp
P[v|Bi] =λδ0(v) + 1−λ v√
2πσe−
1 2
„log v m σ
«2
(v>0)
?q
< A
_G`{eJc oz£c«{c(¦£z¨³ed¦£z¨`bvov4esnqprz£cprz£vxyv~pr`be;$cep}ed$nXz©n
ˆλ = 1 M
X
v∈B
(v≤ε)
?q A
ˆ
m = exp(ˆµ) z|p}` µˆ= 1 M¯
X
v∈B,v>ε
logv ? A
ˆ σ2 =
1 M¯
X
v∈B,v>ε
log2v
−µˆ2 ?qsA
`{ed}e
M z©npr`be.prvxp}x¦yo«bc3¬;es*v~ n}cb¦£esnJz£ypr`{e.¬{¦¨v4t$³
Bi y{ M¯ z£np}`beyB«{c¬EedJv~®n}cb¦£esn
z¨pr`
v > ε= 0.1¯
CEDCK+CGF
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
0 2 4 6 8 10 12 14
P(vres)
vres
the three typical types of distributions of vres
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0 0.5 1 1.5 2 2.5 3 3.5 4
Type I Type II Type III
Àz£wx«b}e
B
k4vxce®pmB{z£ts¦4e bxc.{¦¨eMn+v~;v4tst«b}resy{teG4z©np}rz£¬b«4p}z¨vy{n
P[¯v|B]v¬4p}xz¨ybeM~¶}vxc pr`beGp}esnqpres
¥oz©4edvbp}x¬{xnresns¯
*!?3! 00 ®*!
_G`{e'Eedr~¶vx}cxy{tev~uv«b3cedpr`bv4¹`;xn3¬;esedyes¥)¦£«{)p}es¹vy½p®vV4z¨¿EesresyBp3¥oz£bedvV{)p}x¬{xnresn~¶vx
p®v½ed¥edyBp'}estdvxwxy{z|p}z¨vyD{rv¬b¦¨escnd¯ nªp}`beVtsced$ºz£nªµ4es z£y ¬Evpr` tdxnresns´G e^4v¹ybvxp'`{¥e
p}v tvc;esy{nrpre~¶v3tsced$¼cvprz£vxy0¯¼À{vxpr`beed¥)¦£«{)p}z¨vy½ e'«{nqeM½»¦£es¥xed±=vybe±9vx«bpJ¥)¦£z©b)p}z¨vy
nqpr$)p}edwm
B
xt$`¥oz©4edv¼nqeswxcedyBpJ`{xn¬;esedy^t¦©xn}nqz¨µ{es¬{xnres vxy»p}`be3}edcxz¨ybz£ybwvxybeMnd¯_G`benqedpJvx~
¦£v4tdx¦04esn}t}z£4prv}nGz£y{t¦£«{4eMnG~¶vx¤esxt$`¼z£yBpred}esnqpuEvxz£yp?¶vx¬{¦¨v4t$³,A®pr`{epr$)eMtÄprvrmªbesn}t}z¨4p}vx$nG«b¼prv
v}bed
6y;pr`{ep}`b}ede¤cvxprz£vxy'z£yBpresy{nqz¨pmª4eMnrtdrz£4prv}n
λ, m, σ ?vx®{rp®vx~5pr`{edc#xtstv}4z£ybwprv3pr`be e 4;esrz£cedyBp}nts}rz£es'vx«4pAį
; )4 $A-."A):;)+7 !#$! 0 ! <)
_G`{e µ;}nqpItvxy;nqz©4ed}es3b)p$¬{nqetdvxy{nrz£nqp}nvx~;`o«bcy.wxeMnp}«breMn ?nredeÀz¨w;¯Aį[_G`bed}e}e®nrz34z¨¿Eesresyp
td¦£nrnresns´¾z=¯e¯¨´ nr`{³oz£ybw'vybe n*`beMx Ä´ ybv4b4z£ybwÄ´ t¦©bbz£ybw`{y;bnÄ´ y{nr®ed}z¨ybw'pr`beb`{vxybe y;
4}z¨y{³Bz£ybw®pred įJ¤¦¨¦pr`{esnre3wesnqpr«b}esn`{¥xe3¬;esedy^td}}z¨eM¼vx«4pnqes¥xes}x¦5prz£cesnJ¬omnqes¥xesy 4z¨¿Eesresyp
nr«b¬4eMtÄp$nd¯._G`be.prvp$¦[nrz¨²se.vx~Ipr`{eªb)p$¬{nqe3z£nbx¥oz©4edv¼nredwxcesyp$nd¯.Àb«brpr`{ed}c.vre´¾z|p`{nprv¼¬;e
y{vpreMªp}`{)pu¥oz©4edvBnG}eJx¦¨¦¾}estdvx$4esª~¶rvc pr`ben}ceJ¥oz¨es;vz¨yBp¤y;ªp}`benq«b¬bestÄp¤xre*tesyBpred}esz£y
p}`beªn}t}ededy0¯ ¤n¦£¦[pr`bewesnqpr«b}ev~ pr`be'n}cet¦©xn}n*v~Gyºz£y{4z£¥Bz©4«{x¦Xnr«b¬4estpreMnqesc¬b¦£esVest$`
KLKNMOPPQR