52
HAL Id: inria-00073161 https://hal.inria.fr/inria-00073161 Submitted on 24 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. Image Classification Using a Variational Approach Christophe Samson, Laure Blanc-Féraud, Gilles Aubert, Josiane Zerubia To cite this version: Christophe Samson, Laure Blanc-Féraud, Gilles Aubert, Josiane Zerubia. Image Classification Using a Variational Approach. RR-3523, INRIA. 1998. <inria-00073161>

Image Classification Using a Variational Approach

Embed Size (px)

Citation preview

Page 1: Image Classification Using a Variational Approach

HAL Id: inria-00073161https://hal.inria.fr/inria-00073161

Submitted on 24 May 2006

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.

Image Classification Using a Variational ApproachChristophe Samson, Laure Blanc-Féraud, Gilles Aubert, Josiane Zerubia

To cite this version:Christophe Samson, Laure Blanc-Féraud, Gilles Aubert, Josiane Zerubia. Image Classification Usinga Variational Approach. RR-3523, INRIA. 1998. <inria-00073161>

Page 2: Image Classification Using a Variational Approach

ISS

N 0

249-

6399

ap por t de r ech er ch e

THÈME 3

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Image Classification Using a Variational Approach

Christophe Samson — Laure Blanc-Féraud — Gilles Aubert — Josiane Zerubia

N° 3523

Octobre 1998

Page 3: Image Classification Using a Variational Approach
Page 4: Image Classification Using a Variational Approach

Unité de recherche INRIA Sophia Antipolis2004, route des Lucioles, B.P. 93, 06902 Sophia Antipolis Cedex (France)

Téléphone : 04 92 38 77 77 - International : +33 4 92 38 77 77 — Fax : 04 92 38 77 65 - International : +33 4 92 38 77 65

!"#$&%('')#+*

,.-0/132465870-9;:<>=?258@BADCE<GF0/9IH.JK<>@0LNMPORQS/T<GF0UBADVW1XJ3JX9S2ZYF[\9]/4_^`AbaG5G2T13<G@09&c9S/F0[013<

degf6hjik)lnmporqi6st`u6qPvxw`o?egw`hWhjiTyzhIt`uevogi|vhItS~i_|w`oog_i_|gu_w`ootSvxtSogu_i_

swSPi6q$sPvXtSot

$tS0w`sPqisi_uegi6suegioBkkl.u6qw`sijlnrtS~i_

&_]¡ i6si6voD|\¢£i&si_i6o>q¤t;¥tSsPvXtSqPvxw`otS¦Rhjw>i6¦Ri6¥w`qi_§qw?vhItS~i&u6¦Xt`v©¨utSqPvxw`ou_w`ª¦xi_¢«vqPe¬tSoi_8~iTyzsi_i6sP¥Gvo~jsi6~ª¦XtSsPv­NtSqPvxw`o¬swGu_i_®degi8vxu6si6qiotSqPªsiZwS¯u6¦Xt`°yv©¨utSqPvxw`o²±Kvz®³i®(qw´tSqPqPsPvªqi¡t§¦XtSi6¦µqw¶it`ue·v©¸i6¦3¹Iet`&¦xi_ºqw»qPegi¼i6¥i6¦xw`hji6orq?wS¯hItSo>½"sw`tSv¦vxPqPvxu¾vhItS~iu6¦Xt`v©¨utSqPvxw`o¿hjwGi6¦x|EªqjstSsi6¦½¶qwÀ¥tSsPvXtSqPvxw`otS¦w`ogi_®´mpoqPegi¬¦Xt`Pq;i_ut`i|«qPegi¬¥`tSsPvXtSqPvxw`otS¦ZtSswt`ue·et`ÁswN¥i6oºvqÁiTÂIu6vxi6ogu6½¿voºqPegi?¨i6¦xwS¯i_8~iTyzsi_i6sP¥>vo~Ãsi_Pqw`stSqPvxw`oD®¾mpo¡qPevxZtS0i6s¢Ri&t`»tu6¦Xt`Pv©¨utSqPvxw`o»utStSv¦vqp½¡¢«evxueu_w`o>qPsPvªqi_¤qwÃsw¥Gvxi&vhItS~i_¤u_w`hW0w`ªog"wS¯egw`hjw`~i6ogi_w`ªgWsi6~vxw`og¢«vqPe¶si6~ª¦XtSsPv­i_0w`ªoggtSsPvxi_|Gt.si6~vxw`oI0i6vo~iT¨gogi_It`Äti6qEwS¯v©¸i6¦x0i6¦xw`o~vo~qwµqPegitShjiu6¦Xt`®degiw`ªog8ogi_¤wS¯$w`ªs)hjwGi6¦vxt`i_¶w`o¶qPegi&¢£w`sPÅ8Wi6¥i6¦xw`i_"w`o¶qPegi&et`iIqPstSogPvqPvxw`ogqPegi_w`sP½¼vo¡hji_uetSovxu_®;degiWsw`0wi_§tS¦~w`sPvqPehvxµ¯Æt`q|bit`P½Ãqw¾vhW¦xi6hji6orq|EtSog§iTÂWyu6vxi6o>q®Ç¡iu_w`hWtSsiw`ªssi_Pª¦q.w`o?0w`qPe?P½GorqPegi6qPvxu¤tSog?tSqi6¦¦vqiZvhItS~i_$¢«vqPe¬qPegi)w`ogi_w`qtSvogi_r½t¤PqwGuet`PqPvxuhjwGi6¦DªgPvo~Itj\w`qPqsi6~ª¦XtSsPv­NtSqPvxw`oD®ÈÉÊ0ËÌ)Í ]ÎnÏ tSsPvXtSqPvxw`otS¦hjw>i6¦z|0u6¦Xt`v©¨utSqPvxw`oD|+¦XtSi6¦¦vo~g|0et`i)qPstSogvqPvxw`ogµqPegi_w`sP½|i_8~iTyzsi_i6sP¥>vo~&si6~ª¦XtSsPv­NtSqPvxw`oD|ghWvovhWv­NtSqPvxw`oD|tSqi6¦¦vqivhItS~i_®

Ð+ÑGÒÓÔÕÖÒ×zÔTسÕzÙbÚ]Û ÜÄÛTÝ]ØÞÙ°ß`ÝNÔTà]àSáâBã.äÁåå6æ_ç\èéäêëâà]ØíìTÙîÕzïÆس×zá+Ý]Ù0é\ØíðîÙîñKêNÔòSóSØíÒ\Ü\à_×zØÞòÔTôÞØÞï°ë_õTåçPõöBéØíðîÙèÙPÝ]Ùî÷.æ_ëø8ä\Ü\éè0ù

Page 5: Image Classification Using a Variational Approach

P¤\«Æ '# %('')#+* !"#00E

tSogWu_i;stS0w`sPqWogw`ªgWs_i6o>qw`ogjªo´hjwGf6¦xi;¥`tSsPvXtSqPvxw`oogi6¦.i_PqPvog æXtu6¦Xt`Pv©¨utSqPvxw`o§ vhItS~i_t¥i_uWsw>u_i_Pªgi¤s6~ª¦XtSsPvxtSqPvxw`o§s_i6sP¥`tSo>qZ¦xi_u_w`orqw`ªs®\togw`qPvxw`o)iEu6¦Xt`Pv©¨utSqPvxw`o6qtSo>qbtSsDotSqPªsiÄ8vxu6sf6qi±Kvz®³i®BtSqPqPsPvªgi6sDªoZ¦XtS0i6¦ uet>ªgiEv©¸i6¦iĦ vhItS~iN¹T|`v¦>iT¸8vxPqiÄiÄogw`hsi6ª8¸hjwGf6¦xi_i£u6¦Xt`v©¨utSqPvxw`o)tSsBtSsw>uegiÄsw`tSv¦vxPqi|hItSvx)¦xi_ZhjwGf6¦xi_)¥`tSsPvXtSqPvxw`oogi6¦xWtS0w`sgtSorq¤u_iIPª`Pi6qw`o>qstSsi_®Ri_i6sPovxf6si_tSoog_i_N|¦ÞtSswGuegi?¥`tSsPvXtSqPvxw`oogi6¦¦xiÃt¼hjw`orqPs¬w`orqÁiTÂIut`u6vq?gtSogW¦xi?ut`8si?i;¦Xt¼si_PqtSªstSqPvxw`o vhItS~i_&t_¥i_u¾sPvxi¾i6o u_w`hWqi?i_j8vxu_w`orqPvo>ªvq_® tSogWu_i;qPst_¥tSv¦z|£ogw`ªgIt]°w`ªqw`ogªo»sw>u_i_Pªgi&u6¦Xt`Pv©¨utSqPvxw`o¶i6sPhji6qPqtSo>q³w`qi6ovsªogi&w`¦ªqPvxw`o¡¯Æw`sPhj_iIijs6~vxw`ogegw`hjw`~f6ogi_Rw`orq£¦xi_¯ sw`orqPvxf6si_Rw`orq£s6~ª¦vxf6si_.±Kªogis6~vxw`o&6qtSo>q£T¨govxi$tSsE¦³i6ogi6h¦xii_£v©¸i6¦xRtStSsPqi6otSo>q ¦Xth6hji$u6¦Xt`iN¹T®\t«°ªgPqPv©¨utSqPvxw`oÁqPeg_w`sPvrªgiµiogw`qPsi$hjwGf6¦xisi60wi.PªsR¦xi_£qPst¥`tSª8¸;i0i_u6qPªg_«gtSogR¦xiµut`8siµi_Rsw`¦xf6hji_iqPstSogvqPvxw`og«i$et`i_i6o¼hj_utSov>ªgi® !ÞtS¦~w`sPvqPehji"rªgi)ogw`ªgµsw`0ww`ogi_Pqµsi6¦XtSqPv¥i6hji6orq)stSvxii6q.¯zt`u6v¦xi hji6qPqPsi$i6o&wGi6ª¥>si®$#.w`ªgÄu_w`hWtSsw`og£¦xi_Äs_ª¦qtSqÄw`qi6orªgPªs£i_EvhItS~i_RP½>o>qPeg6qPvrªgi_i6q«tSqi6¦¦vqtSvsi_$t_¥i_uu_i6ª8¸&swG8ªvqtSsªo;hjw>f6¦xiPqwGuet`qPvrªgit_¥i_uµs6~ª¦XtSsPvxtSqPvxw`oiw`qPq®% Í Ë '& § ( wGf6¦xi¤¥tSsPvXtSqPvxw`oogi6¦z|\u6¦Xt`v©¨utSqPvxw`oD|bqPeg_w`sPvxijijqPstSogPvqPvxw`ogiWet`i_|s6~ª¦XtSsPvxtSqPvxw`oÃt_¥i_uZs_i6sP¥`tSqPvxw`o¬i_u_w`orqw`ªsN|hWvovhWvxtSqPvxw`oD|gvhItS~i_$tSqi6¦¦vqtSvsi_®

Page 6: Image Classification Using a Variational Approach

"!#$%'&()(%$*+,.- k

¿$E

/ 021 Í Î\+3 Í 1 4

5 6 Í & É _8 É É 1 78® #.w`qtSqPvxw`o®Z®®Z®®®Z®®Z®®Z®Z®®®®Z®®Z®Z®®Z®®®Z®®Z®®Z®®Z®®®Z®® 8®Þ sw`¦xi6h PqtSqi6hji6orq(®®Z®®Z®Z®®®®Z®®Z®Z®®Z®®®Z®®Z®®Z®®Z®®®Z®® 8

9 : 1 Î É ;( &K Ë2< >= 1 Ë? 3 & &3K]Î+= É>Í Ê´Í'@BA =b É S 1 +33+3 Í 1 Ck8® i6¦XtSqi_¢£w`sPÅ8Á®®®Z®®Z®®Z®Z®®®®Z®®Z®Z®®Z®®®Z®®Z®®Z®®Z®®®Z®® Dk8®Þ $otS¦xw`~½Á¢«vqPe?vhItS~iZu6¦Xt`Pv©¨utSqPvxw`o ®Z®®Z®Z®®Z®®®Z®®Z®®Z®®Z®®®Z®®

4 < &KE3GFB8+3 Í 1 @ 1 +3 Í 1 & /)4g® sw`0wi_;¯ ªogu6qPvxw`otS¦ ®Z®®Z®Z®®®®Z®®Z®Z®®Z®®®Z®®Z®®Z®®Z®®®Z®® 6g®Þ $w`ªq«qPegiZu6¦Xt`Pv©¨utSqPvxw`o?qi6sPhIH®®Z®®Z®Z®®Z®®®Z®®Z®®Z®®Z®®®Z®® g®Þk JGw`ªog8ogi_$wS¯qPegisw`0wi_¾hjwGi6¦¼®Z®®Z®Z®®Z®®®Z®®Z®®Z®®Z®®®Z®® g® ( vovhWv­NtSqPvxw`oÃtSog?tS¦~w`sPvqPeh ®®®®Z®®Z®Z®®Z®®®Z®®Z®®Z®®Z®®®Z®®

7 KML AEÉ 3 É 1 &R É &3W 1 Î Í A 33 Í 1 5'58® i_u6sPvqPvxw`o¬wS¯BqPegiPqwGuet`PqPvxuZhjw>i6¦;®®Z®Z®®Z®®®Z®®Z®®Z®®Z®®®Z®® 8®Þ NB¸8i6sPvhji6o>qtS¦bsi_Pª¦q ®Z®®Z®Z®®®®Z®®Z®Z®®Z®®®Z®®Z®®Z®®Z®®®Z®® k8®Þ8® J>½GorqPegi6qPvxuvhItS~i_ ®Z®Z®®®®Z®®Z®Z®®Z®®®Z®®Z®®Z®®Z®®®Z®® S8®Þ8®Þ J8tSqi6¦¦vqivhItS~i_¼®®Z®Z®®®®Z®®Z®Z®®Z®®®Z®®Z®®Z®®Z®®®Z®® k,O

P <Í 1 & \+3 Í 1 4%9

ä\äàQ#RTSæR

Page 7: Image Classification Using a Variational Approach

-)$.GT ('- T $ B % T$* B&.$T .$T

8# b$

Ï tSsPvXtSqPvxw`otS¦DtSswt`ue?tSog&BtSsPqPvXtS¦ v +i6si6orqPvXtS¦ N!>ªtSqPvxw`oÀ± N£¹Ähjw>i6¦x£et_¥iµPegwN¢«oqw?i&iTÂIu6vxi6orq)¯ w`s¤t¾¢«vxi&¥`tSsPvxi6qp½§wS¯«vhItS~iIswGu_i_Pvo~¬sw`¦xi6hj¤Pªgue´t`Zsi_Pqw`stSqPvxw`otSog¬i_8~i)i6qi_u6qPvxw`oÞ8|%88|\|b+OG|kSg|k|k,OG|k8|)8 Ö|w`s.PetSi)i6~hji6orqtSqPvxw`o¼¢«vqPe¼t`uTyqPv¥i$u_w`orqw`ªs88|88|>k Ö® #.i6¥i6sPqPegi6¦xi_|>qPegiogw`qPvxw`o;wS¯bu6¦Xt`Pv©¨utSqPvxw`oD|G¢«evxueÁu_w`ogPvxq£wS¯t`Pv~ovo~tµ¦XtS0i6¦qw)it`ue;PvqiwS¯btSoIvhItS~i|Get`ÄstSsi6¦½Wi_i6o&vorqPswG8ªgu_i_Ivo;tµ¥`tSsPvXtSqPvxw`otS¦¯Æw`sPhª¦XtSqPvxw`o·± u_w`orqPvo>ªgw`ªg¤hjw>i6¦xT¹µhItSvo¦½À0i_utSªgijqPegiIogw`qPvxw`o»wS¯u6¦Xt`et`)t8vxu6si6qiotSqPªsi® ( tSor½¤u6¦Xt`Pv©¨utSqPvxw`o&hjw>i6¦xÄutSojiR¯Æw`ªogWvoWqPegi¨i6¦xIwS¯0PqwGuet`qPvxutSswt`uegi_± 8vxu6si6qijhjwGi6¦x¹T|¢«vqPe»qPegiIªgijwS¯ ( tSsPÅw¥¡$tSogw`h"!vxi6¦x± ( #!E¹.qPegi_w`sP½§t`¯ w`svo8yqtSogu_i;vo$Dg|R8|«8|||k8|Ekk Ö® J>qPsPªgu6qPªstS¦$tSswt`uegi_Át`WP¦vqPqPvo~g|Ehji6sP~vo~tSog?si6~vxw`o?~swN¢«vo~&hjwGi6¦x% )D Ö|tSog¾¯Æi6¢ hjwGi6¦x.t`.tWu_w`hvotSqPvxw`oÀwS¯EPqtSqPvxPqPvxutS¦BtSogi6qi6sPhWvovxqPvxu«qi_ueovxu_ O8|`r get_¥itS¦xwµi_i6oWi6¥i6¦xw`0i_¤¯ w`svhItS~iu6¦Xt`Pv©¨utSqPvxw`oD®'&Rªq|qw;w`ªsµÅGogw¢«¦xi_8~i|+¥i6sP½¾¯Æi6¢(¢£w`sPÅ8µet_¥i0i_i6o¼u_w`og8ªgu6qi_¼vo¼qPegi)¨i6¦x¼wS¯Ru6¦Xt`v©¨utSqPvxw`o>½&qPegiZªgiwS¯¥tSsPvXtSqPvxw`otS¦bhjwGi6¦x®( wPqwS¯BqPegivhItS~i_$¢£itSsiZitS¦vo~&¢«vqPeogi_i_¾qwW0iµsi_Pqw`si_?i_utSªgiwS¯qPegisi_i6ogu_iwS¯Eogw`vxi®(¦xw`qµwS¯Ä¥`tSsPvXtSqPvxw`otS¦hjwGi6¦x$et¥ii_i6oÃi6¥i6¦xw`i_¬¯Æw`ssi_Pqw`stSqPvxw`o¼¢«vqPe¼i_8~ii6qi_u6qPvxw`oD®.mpo?hItSor½ÃtS¦vxutSqPvxw`og|Dt`$tSqi6¦¦vqivhItS~vo~g|qPegi)~wtS¦\vx$qw&w`qtSvoÀtIsi6~vxw`oi6~hji6orqtSqPvxw`o¡voÀt`8vqPvxw`oÀqwÁsi_Pqw`stSqPvxw`oD®degisPvogu6v¦xi¤vx.t`i_¼w`o¼tSsPqPvqPvxw`ovo~¾qPegivhItS~i¤vo¡8v +i6si6orqtSsit`I±Kvz®³i®vo¡8v 0i6si6o>qu6¦Xt`i_¹T|bit`ue»tSsitÁ0i6vo~uetSst`u6qi6sPv­i_§r½t?¯ÆitSqPªsi®»degi&¯ÆitSqPªsi¾u6sPvqi6sPvxw`o ¢£itSsiÁvo>qi6si_Pqi_´vo"vx¤qPegi¾PtSqPvXtS¦«8vxPqPsPvªqPvxw`o¿wS¯vo>qi6ogPvqÖ½®.¯u_w`ªsi|w`qPegi6s$8vxu6sPvhWvotSorq.¯ itSqPªsi_$qPetSo?vo>qi6ogPvqÖ½¬utSo?0iªgi_¾>½u_w`o8yvxi6sPvo~jPªvqtS¦xiµtSstShji6qi6s®ÇºvqPevo¾qPevxR¯KstShji6¢Rw`sPÅ|gtsi6~vxw`o¾¢«v¦¦00iµiT¨gogi_t`qPegii6qwS¯v©¸i6¦x0i6¦xw`o~vo~jqwWqPegiZtShjiu6¦Xt`|tSog?t¤u6¦Xt`«vxuetSst`u6qi6sPv­i_?>½&tSstShji6qi6s¦voÅi_§qwqPegiÁPtSqPvXtS¦£8vxqPsPvªqPvxw`o¶wS¯«vorqi6ogvqÖ½¡¯Æw`sZvogPqtSogu_i¼±KhjitSo¶tSog¶PqtSoggtSs»iTy¥GvXtSqPvxw`oÀv©¯Ä¢Ri¤hItSÅijt)(ZtSªgPvXtSo§t`PªhWqPvxw`o¹T® i6ogu_i|0¢£i¤vorqPswG8ªgu_iIt&ogi6¢(¯ ªogu6qPvxw`otS¦¯Æw`s£u6¦Xt`Pv©¨utSqPvxw`o;vogu6¦ªg8vo~Wsi_Pqw`stSqPvxw`o;¢«vqPeÁi_8~i.si6~ª¦XtSsPv­NtSqPvxw`oD®Ädegi¨gotS¦si_Pª¦q«tSsiqPegi)vhItS~iwS¯Äu6¦Xt`i_|qPegisi_Pqw`si_ÃvhItS~itSog¬qPegi)hItSÃwS¯Äsi6~ª¦XtSsPv­i_¼8vxu_w`o>qPvorªvqPvxi_®degi&w`¦ªqPvxw`o¶¢«v¦¦£0iWhIt`iÁwS¯egw`hjw`~i6ogi_w`ªg)si6~vxw`ogN|w`s)u6¦Xt`i_N|\i6tSstSqi_»r½¡etSsPsi6~ª¦XtSsPv­i_?w`ªoggtSsPvxi_N®mpo"hji_uetSovxu_|£hItSor½´tSªqPegw`sWet¥i¾PqPªg8vxi_"qPegi¾PqtSv¦vqÖ½"wS¯.½GPqi6hjWu_w`o>qtSvovo~§i6¥>yi6stS¦\vogPqtS¦xiu_w`hW0w`ogi6orq.qPesw`ª~eÃqPegi Ï tSoÃi6sǧttS¦x°y tSeo8y v¦¦vXtSsÃqPegi_w`sP½wS¯Bet`iqPstSogvqPvxw`og%|%OG|8|8|Dk8|D6|D8|D|k88|k|g|r8|g|r Ö®£degi_iu_w`hW0w`ogi6orq$hIt_½0iµ¦v>ªvxet_¥Gvo~&8v +i6si6orq$¦xi6¥i6¦xwS¯Ei6ogPvqp½;8vxqPsPvªqPvxw`oD®RdegiPqtS¦xiZu_w`o8¨g~ªstSqPvxw`oÃvxswN¥i6ojqwµ0i«u_w`hWw`ªogjwS¯egw`hjw`~i6ogi_w`ªgEsi6~vxw`og£i6tSsi6qi_Wr½¤PetSsPWvo>qi6s°¯Æt`u_i_Äet_¥Gvo~hWvovhItS¦\¦xi6o~qPeD®Ç¡isw`0wiqwItS¦½ÁqPegi_isi_ª¦qqwjt¤¥tSsPvXtSqPvxw`otS¦bhjwGi6¦+¯Æw`s«vhItS~iu6¦Xt`Pv©¨utSqPvxw`oD®!vsPq|rqPegisw`¦xi6h PqtSqi6hji6orqRvxÄu_w`ogvxi6si_+®Ç¡iqPegi6o&si_utS¦¦qPegi Ï tSoÁi6sÄǧttS¦x°y tSeo8y

* éä * Ü

Page 8: Image Classification Using a Variational Approach

"!#$%'&()(%$*+,.-

v¦¦vXtSs¼qPegi_w`sP½¼wS¯£et`iqPstSogPvqPvxw`ogvo>qPsw>8ªgu_i_¡voÀhji_uetSovxu_®Ç¡ijtS¦xw;iT¸gtShWvogi¤qPegisi6¦XtSqPvxw`ogevÁ0i6qÖ¢Ri_i6o;qPevx£hjwGi6¦+tSog&vhItS~i.u6¦Xt`Pv©¨utSqPvxw`oD|qPsi_Pvo~qPegi.PvhWv¦XtSsPvqPvxi_«wS¯qPevxqPegi_w`sP½Á¢«vqPeÃtsi6~vxw`o?i6~hji6orqtSqPvxw`oÃvogu6¦ªg8vo~&t¤si_qw`stSqPvxw`osw>u_i_®mpoÃqPegi)ogiT¸8qi_u6qPvxw`oD|0¢£isw`0wiWtjogi6¢ ¯ ªogu6qPvxw`otS¦Ei6¥w`qi_¼qwIvhItS~iWu6¦Xt`Pv©¨utSqPvxw`oÀvot`8vqPvxw`o?qwjtSo?tSovxw`qPsw`vxuZPhjwGw`qPevo~IutStSv¦vqÖ½® &R½IvorqPswG8ªgu6vo~Átogi6¢·qi6sPhsi6¦XtSqi_qw¤qPegiµsi6~vxw`oi6~hji6orqtSqPvxw`oD|g¢RiµvhWwiZt¦xi6¥i6¦Du_w`ogPqPstSvorq|git`ue¦xi6¥i6¦+0i6vo~Wsi6¦XtSqi_¾qwtP0i_u6v©¨u$u6¦Xt`N®\Ç¡iµtS¦xw8vxu6ªgtSw`ªqRqPegi$i6et¥>vxw`ªs«wS¯DqPevx£qi6sPh®Edegi.tS¦~w`sPvqPeh ¢RiªgiqwÁhWvovhWv­i¤qPevx$¯Kªogu6qPvxw`otS¦vx.t`i_Ãw`o¼qPegietS¦©¯3y >ªt`8stSqPvxu¤hji6qPegw>Ãsi_i6o>qi_¼vo+OG|8|8)8Ö®mpo¶qPegiI¦Xt`Pq¤i_u6qPvxw`oD|¢RiÁqi_Pq)qPegiÁsw`0wi_»hjw>i6¦«w`o"P½>o>qPegi6qPvxutSog»tSqi6¦¦vqi&vhItS~i_N®degi_isi_Pª¦qµtSsiZiT¸tShWvogi_¼tSogu_w`hWtSsi_¬¢«vqPe?qPegi)w`ogi_w`qtSvogi_¬¢«vqPeÃtWPqwGuet`qPvxutSswt`ue?t`i_?w`o?t¤\w`qPq$hjw>i6¦z®

#E gEE degiZogw`qtSqPvxw`oªgi_¾egi6sit]¯ qi6s$tSsiµqPegiµ¯Æw`¦¦xw¢«vo~(i6ogi6stS¦Dogw`qtSqPvxw`o vxtSo?w`0i6o¾w`ªogi_¬Pªgi6qwS¯ |"! tSog$#¼tSsiZi6¦xi6hji6o>qwS¯%'&)( +* $,|-/. ( 002143 *2565656* 007178 ,9vx«qPegi~st`8vxi6o>q$w`0i6stSqw`s|:;< (#=, .?> @BA & 04C0217D vxqPegi8v¥i6sP~i6ogu_iw`0i6stSqw`s$tS¦vxi_qwE#B® degiZdbw`qtS¦ Ï tSsPvXtSqPvxw`o"± d Ï ¹RwS¯F#ÃvxiT¨gogi_?t`)

GIH (#=, .KJML N O # N.?P)!QR4JSL # 5T:;< (VU,XWYU[Z]\ &^ ( +* ,`_ba :]N U(dc, NMe?fg5mz¯#hZ]\i&2( ,ÄqPegi6o GIH (#=, .?j L N - # N ®k H ( ,Evx«qPegiPt`u_iwS¯\¯ ªogu6qPvxw`og¢«vqPe &Rw`ªogi_qw`qtS¦ Ï tSsPvXtSqPvxw`o¶± & Ï ¹ ÞS

k H ( , .lR #¼ªgueqPetSq GIH (#=,nmporq g5devxPt`u_i)u_w`orqtSvog$qPegi¯ ªogu6qPvxw`og¢«evxue¼utSo0iZ8vxu_w`o>qPvorªgw`ªgµqPesw`ª~e¬u6ªsP¥i_N|tSogqPegi6siT¯Æw`siÄqPevxPt`u_i£vxbPªvqtS¦xiįÆw`s\qPegi£PqPªg8½ZwS¯¯Ksi_iR8vxu_w`o>qPvorªvqÖ½)¥`tSsPvXtSqPvxw`otS¦sw`¦xi6hj$t`«qPegiZi6~hji6orqtSqPvxw`o¬w`ogi®

ä\äàQ#RTSæR

Page 9: Image Classification Using a Variational Approach

8 -)$.GT ('- T $ B % T$* B&.$T .$T

degiµi6sPvhji6qi6s.wS¯\qPegiZi6q vxiT¨gogi_?t` L ( , . G+H (F,E¢«egi6si §vxqPegiuetSst`u6qi6sPvxPqPvxu¯Kªogu6qPvxw`o¬wS¯®

(r,vxqPegi a f 8vhji6ogvxw`otS¦ tSªgw`s ¶hjit`PªsiIwS¯|btSogÀv©¯£qPegi¤0w`ªoggtSsP½ wS¯ vxhjw>w`qPe¾¢Riet_¥i

( , . L (r, . G+H ( , 5£¦Xt`Pv©¨utSqPvxw`oogw`qtSqPvxw`o .lR4Pg A & qPegii6qwS¯Bvqi_| % si6si_i6orqqPegiZ¦XtSi6¦btSqPqPsPvªqi_?qwWqPegiPvqi P | qPegiZi6qwS¯t`8hWvxv¦xi¦XtSi6¦xN|! _" :#.%$ tSog % Z | \ @ vx«qPegi ; 9'&¤u6¦Xt`uetSst`u6qi6sPv­i_?>½¾t¦XtS0i6¦Dvo ®

$i6hItSsPÅG) Ç¡iµ¢«v¦¦bw`hji6qPvhji_ªgi #ÃvogPqit`wS¯F#(dc,E¢«egi6oogwWu_w`o8¯ ªgPvxw`ovx«0wPv¦xi® Ç¡i¤ªgi¤qPegi¤qi6sPhWvogw`¦xw`~½)(pPvqi+*¤w`s,(Öv©¸i6¦-*|bi6ogw`qPvo~ÃtÁw`vo>qZwS¯RqPegi¤vhItS~i|\i6¥i6ovo¬qPegi¤u_w`orqPvo>ªgw`ªgut`i)voÀw`si6sµqwIhItSÅiWtÁu_w`oogi_u6qPvxw`oÀi6qp¢£i_i6oÀqPegi)¥`tSsPvXtSqPvxw`otS¦u6¦Xt`Pv©¨utSqPvxw`oñKtSsPqPvqPvxw`ovo~vojsi6~vxw`og/. @ ¹tSogj8vxu6si6qiPqwGuet`PqPvxu¦XtSi6¦¦vo~;±KtSs°yqPvqPvxw`ovo~Ãvo´u6¦Xt`i_ \ @ ¹T®À¦¦RqPegiÁi6¦xi6hji6orqjwS¯$t¬si6~vxw`o. @ i6¦xw`o~¼qw¬qPegi;tShjiu6¦Xt`®

Ç¡iµ¢«v¦¦bi6vqPegi6st½0(pu6¦Xt`Pv©¯K½;t¤Pvqi+*w`s1(îtSqPqPsPvªqi)t¦XtS0i6¦Dqw¤qPevxPvqi+*® 24365798;: <4 =8>:?8 ( wPqRwS¯0qPegivhItS~i_Ä¢Ri.tSsi$itS¦vo~)¢«vqPe;tSsiu_w`sPsPªqi_&r½j8v +i6si6orqw`ªsu_i_RwS¯0ogw`vxi®degi$u_w`¦¦xi_u6qi_&gtSqt)u_w`orqtSvo&¯ztS¦xivo8¯Æw`sPhItSqPvxw`ogRqPetSqÄhªgq£0isi6hjwN¥i_&w`sRwS¯Kqi6ogi_+® bi6qt`PªhjiqPetSq.qPegi)ogw`vxi@BAM DC Ã|±da .FE w`sHG¹T|et`tI¢«evqi~rtSªgvXtSoÃ8vxPqPsPvªqPvxw`owS¯Bvorqi6ogvqÖ½®ÄmÖ¯I@ÁvxtSo?t`8vqPv¥iogw`vxi|qPegiµ¦vogitSsi6~st`gtSqPvxw`oÃhjwGi6¦+vxwS¯qPegi¯ w`sPh

J (dc, . #(dc,o@ (dc, *LK c]Z +* ±°N¹

¢«egi6si J i6ogw`qi_qPegi$u_w`¦¦xi_u6qi_;gtSqt8®Edegi$~wtS¦+wS¯bqPegisi_qw`stSqPvxw`o¾sw>u_i_£vxRqwsi6qPsPvxi6¥i# ¯ sw`h J qPesw`ª~e´qPegi;hWvovhWv­NtSqPvxw`oºwS¯$qPegiÁ¯ ªogu6qPvxw`otS¦HMF(#=, . j LON #(dc, J (dc,QPR : c¯Æw`svogPqtSogu_i®Àdevx¤sw`¦xi6h vx)~i6ogi6stS¦¦½»v¦¦©yzwi_ ±Kvo"qPegiÁi6ogi;wS¯ t`gtShItSs$Þ8 K¹T|

* éä * Ü

Page 10: Image Classification Using a Variational Approach

"!#$%'&()(%$*+,.- O

tSog;qPegiZsi6~ª¦XtSsPv­NtSqPvxw`o?qPegi_w`sP½Á¦xit`«qwIt¢£i6¦¦©yz0wi_;¯Æw`sPhª¦XtSqPvxw`oD®£degiµt`Pvxuvxit¤wS¯Etsi6~ª¦XtSsPv­NtSqPvxw`o¼hji6qPegwG¬u_w`ogvxPqwS¯EvhW0wPvo~;w`hjiu_w`ogPqPstSvo>qµw`o?qPegiw`¦ªqPvxw`oh#B|tSogw¬qPegi¾t`u_i¾wS¯µt`8hWvxPv¦xi¾w`¦ªqPvxw`ogWvx¤si_PqPsPvxu6qi_+®!gw`sjvogPqtSogu_i|ÄqwÀw`qtSvo¿PhjwGw`qPew`¦ªqPvxw`og|Ät¬devÅw`ogw¥¶si6~ª¦XtSsPv­NtSqPvxw`o´qi6sPh j L N - #(dc, N R : c utSo"i&vorqPswG8ªgu_i_"vo¶qPegi¯ ªogu6qPvxw`otS¦ M rÖ®NÄ8~i_µtSsi)sw`tS¦½?qPegi)hjwPq$vhWw`sPqtSo>q¦xw¢ ¦xi6¥i6¦\¯ÆitSqPªsi_$vo¬vhItS~iswGu_i_Pvo~g|0tSogtSsiªgPªtS¦¦½WiT¨gogi_Át`tPetSsP&qPstSogPvqPvxw`o;wS¯0vo>qi6ogPvqÖ½I¦xi6¥i6¦z®degi6siT¯ w`si|>qPegisi_Pqw`stSqPvxw`ohji6qPegwGh)ªgPqZ0ijtS¦xiWqw¾si6hjw¥ijogw`vxiW¢«ev¦xiWsi_i6sP¥Gvo~qPevo§qPsPªgu6qPªsi_®jdvÅGegw`ogw¥si6~ª¦XtSsPv­NtSqPvxw`o§vog8ªgu_i_tSo¼vxw`qPsw`vxuWPhjwGw`qPevo~g|btSog¼i_8~i_tSsiogw`qsi_i6sP¥i_+®)mpoÃqPegi¨i6¦x;wS¯b¥tSsPvXtSqPvxw`otS¦DtSswt`uegi_|ghItSor½ÁtSªqPegw`set¥i.¢£w`sPÅi_¾w`otSovxw`qPsw`vxuµPhjwGw`qPevo~t`vo 88|8|8+OG|kS|k8|`k,OG|k|])88¯Æw`sBiT¸gtShW¦xi®BdegiRhItSvoWvxitvxBqwµiT¨gogit¯ ªogu6qPvxw`otS¦¢«egwihWvovhWv­NtSqPvxw`o¡swN¥>vxi_ZtIsi_Pqw`si_¼w`¦ªqPvxw`o¼¢«vqPeÃsi_i6sP¥i_Ài_8~i_®µovxw`qPsw`vxuhjw>w`qPevo~;hªgq.0i¤tS¦xi¤qwIsi6hjwN¥i¤ogw`vxi¢«vqPegw`ªqZPhjw>w`qPevo~;qPegi¤i_8~i_®u_w`hWhjw`otSovxw`qPsw`vxu«Phjw>w`qPevo~hjwGi6¦8vxt`i_Ww`o¤qPegi£hWvovhWv­NtSqPvxw`o&wS¯qPegi£¯ w`¦¦xwN¢«vo~¯ ªogu6qPvxw`otS¦ 88|+r|++OG|k,O

M(#=, . J L N #(dc,I J (dc, P R : c o R J L ( N - #(dc, N , : c 5 ±Æ¹

degi ¯Kªogu6qPvxw`oD| A C £|Bet`WtSq¤¦xit`Pq¤qPegi&qÖ¢£wïÆw`¦¦xw¢«vo~Àsw`0i6sPqPvxi_± i_i !v~g®N¹ Bv©¯`Z vx«u6¦xwiµqw¤­i6sw| ( ,Evx >ªt`8stSqPvxuZw`sogitSsP¦½rªt`8stSqPvxuI± .-)µi+i_u6q¹T|tSog´¯ w`sW¦XtSsP~i¾¥tS¦ªgi_jwS¯T| ( ,vxj¦vogitSsIw`sIPª8yz¦vogitSs¬± ) (%$ .2.$Ài+i_u6q¹T®"Ç¡itS¦xw¾et_¥i&qPegiWt`PvxuÁt`PªhWqPvxw`og)w`o K Z"¬| ( , (b, . ( , . ( , u_w`orqPvo>ªgw`ªgP¦½´8v 0i6si6o>qPvXtS¦xitSog K 8| ( ,8®Àdegi ¯Kªogu6qPvxw`ogjtSsiÁªgPªtS¦¦½u6¦Xt`Pv©¨i_¶vo¶qp¢£w¡utSqi6~w`sPvxi_N|EqPegi¾u_w`or¥iT¸´w`ogi_jtSog¶qPegi;ogw`ogu_w`or¥iT¸ w`ogi_®ÀǺegi6o vxu_w`o>¥iT¸0|qPegi?qPegi_w`sPvxutS¦$PqPªg8½¿tS0w`ªqjqPegi?hWvovhWv­NtSqPvxw`oºwS¯ M ¦xit`Iqw¡si_Pª¦q&Pªgue t`qPegiÀiT¸8vxPqi6ogu_i¡wS¯¤hWvovhWv­i6s® $o·qPegi¡w`qPegi6setSog+|.¯Æw`s?t¶ogw`ogu_w`or¥iT¸·¯ ªogu6qPvxw`oD|µqPegiqPegi_w`sPvxutS¦\PqPªg8½Ivxh)ªguehjw`si8v©ÂIu6ª¦q)±Kogw`o¾ªov>ªgi6ogi_wS¯qPegiµhWvovhªh|v©¯vqiT¸Gvxq¹T®#.i6¥i6sPqPegi6¦xi_|ogw`ogu_w`or¥iT¸¯Kªogu6qPvxw`ogEtSsiwS¯ qi6oWªgi_0i_utSªgiRqPegi6½ªgªtS¦¦½sw¥Gvxi0i6qPqi6ssi_ª¦q®d\tS¦xisi_i6orq$8v 0i6si6o>q ¯Kªogu6qPvxw`og$u_w`hWhjw`o¦½¾ªgi_+®

degi ( ªh¤¯ w`sGy J>etSe¬¯Kªogu6qPvxw`otS¦ ÞkvxÅGogw¢«oÀt`$qPegihjwPqµP½GorqPegi6qPvxuu6sPvqi6sPvxw`oï w`s$qPegii6~hji6orqtSqPvxw`oswGu_i_®bÇ¡iEutSoµ¨goghItSo>½.ª¦vxutSqPvxw`ogBtS0w`ªqDqPegihItSqPegi6hItSqPvxutS¦>w`ªogGyogi_«wS¯\qPevxhjwGi6¦z|8¢«evxue¾0i6¦xw`o~«qwqPegiµ¢«vxiµ¥`tSsPvxi6qp½ÁwS¯\¯Ksi_iµ8vxu_w`o>qPvorªvqÖ½¾sw`¦xi6hjN|t`¢£i6¦¦Bt`hItSor½ÃtS¦vxutSqPvxw`og%Þk|gg|%88|kS|k|k,OG|g)8 Ö®mpo¬t`8vqPvxw`oÃqwWsi_Pqw`stSqPvxw`oD|qPegiogw`qPvxw`o¤wS¯i6~hji6o>qtSqPvxw`oD|vz®³i®\~i6qPqPvo~µtSovhItS~iÄtSsPqPvqPvxw`oD|vxbvo>qPsw>8ªgu_i_+®degiRiT¸Gsi_Pvxw`owS¯BqPegi ( ªh¤¯ w`sGy J>etSe?¯Kªogu6qPvxw`otS¦Dvx)M(# * k , . J L N #(dc, J (dc,QP R : cEo R JML! N - #(dc, N R : c o ( k , * # Z k H ( ,b±Æk¹

ä\äàQ#RTSæR

Page 11: Image Classification Using a Variational Approach

-)$.GT ('- T $ B % T$* B&.$T .$T

0

ϕ

edge-preservingsmoothing

|grad(f)|

edge-preserving

!v~ªsij "!#$%&'(%*)

¢«vqPe k i6ogw`qPvo~)qPegii6qEwS¯+0w`ªoggtSsPvxi_$±KªoÅGogw¢«o&¯Ksi_i«8vxu_w`orqPvorªvqPvxi_T¹T®Ädegi¨gsPqÄqi6sPhvo§±Æk¹tSvhj$tSqÅi_i6vo~jqPegiw`¦ªqPvxw`o?u6¦xwiµqw¤qPegigtSqt8®EdegiZi_u_w`og;qi6sPh|¢Ri6v~erqi_?r½qPegisi6~ª¦XtSsPv­NtSqPvxw`o&tSstShji6qi6s |`sw¥Gvxi_£tµPhjwGw`qPevo~swGu_i_Ew`oWegw`hjw`~i6ogi_w`ªg£tSsit`N®degi.¦Xt`Pq«qi6sPh ¦xit`qw0w`ªoggtSsPvxi_«et_¥Gvo~jhWvovhItS¦+¦xi6o~qPeD®JGw|8qPegiµiT¸8i_u6qi_¾w`¦ªqPvxw`ovxBu_w`hW0w`ªogWwS¯egw`hjw`~i6ogi_w`ªgRtSsit`i6tSstSqi_¤r½PetSsP¤si6~ª¦XtSsPv­i_ji_8~i_7Btvxi_u_i6¢«vxiu_w`ogqtSorqw`¦ªqPvxw`oD®degi)hWvovhWv­NtSqPvxw`o¡wS¯ M¬vx.tI8v©ÂIu6ª¦qµqt`Pži_utSªgiqPegii6q k vxªoÅGogw¢«oD®ÄmÖoÁqPegi$¨i6¦xÁwS¯b¯Ksi_iµ8vxu_w`o>qPvorªvqÖ½;sw`¦xi6hj|8qPegi_w`si6hj«wS¯,++yÖu_w`or¥i6sP~i6ogu_iutSo0iZªgi_qw;tSsw_¸8vhItSqi0M?¢«vqPeÀtIi >ªgi6ogu_iwS¯Äi6¦¦vqPvxuZ¯ ªogu6qPvxw`otS¦x|+t`¯Æw`svogqtSogu_ivo Þkg|8g|gkS|gk8|)8 Ö®

mpot`8vqPvxw`o?qw¤qPegiµsi_Pqw`stSqPvxw`oÃtSogi6~hji6orqtSqPvxw`o¬sw`¦xi6hj|¢Ri¢RtSo>qqwWiT¨gogit¥tSsPv©ytSqPvxw`otS¦£hjw>i6¦RtS¦xw¾i6¥w`qi_¡qwu6¦Xt`Pv©¨utSqPvxw`oD® i6si6voD|\tu6¦Xt`Pv©¨utSqPvxw`o§swGu_i_hjitSogt¡i6~hji6orqtSqPvxw`oº¢«vqPeºt¡¦XtSi6¦¦vo~g|vz®³i®ºi6¥i6sP½ i6¦xi6hji6orqÁwS¯t¼tSsPqPvqPvxw`o0i6¦xw`o~§qw¡qPegitShji)u6¦Xt`N®$devx.hjwGi6¦\h)ªgPqµ¦xit`?qw&qPesi_i¤Pvhª¦qtSogi_w`ªgµswGu_i_i_7«i6ogw`vxvo~g|+i_8~isi_i6sP¥`tSqPvxw`o?tSog¾u6¦Xt`v©¨utSqPvxw`o»± $ .' ,8¹T®EmÖo¾w`qPegi6s¢Rw`s|8¯Æw`s«it`ue?Pvqi P¢RiµhªgPqitS¦xiµqwWi6qi6sPhWvogiv©¯ P vxtogw`vxP½&v©¸8i6¦z|8v©¯ P vxtSoi_8~ii6¦xi6hji6o>q$tSog;¢«evxueu6¦Xt` P 0i6¦xw`o~qw-»±Kvz®³i®¢«evxue¬vx«qPegiµ¦XtS0i6¦% Z wS¯ P -]¹T®!w`sqPegitSÅiZwS¯PvhW¦vxu6vqÖ½|¢£ihItSÅiZqPegiµ¯ w`¦¦xwN¢«vo~&t`PªhWqPvxw`og)

½G0w`qPegi_PvxZ± ¹

. NÄt`ueÃPvqiµvxuetSst`u6qi6sPv­i_¬r½;vq«vo>qi6ogPvqÖ½¾¦xi6¥i6¦Ä±K~si6½;¦xi6¥i6¦3¹T®. NÄt`ueÀu6¦Xt`+\ @0/ @BA & et`t (ZtSªgPvXtSo¼8vxPqPsPvªqPvxw`oÀwS¯Ävorqi6ogPvqp½21 (43 @ * 5 @ ,¢«egi6si"3 @ tSog 5 @ tSsi.si_P0i_u6qPv¥i6¦½IqPegi.hjitSotSogÁqPegi.qtSoggtSs¾i6¥>vXtSqPvxw`o¾wS¯

* éä * Ü

Page 12: Image Classification Using a Variational Approach

"!#$%'&()(%$*+,.-

qPegiu6¦Xt` \ @ ®BÇ¡i¢«v¦¦+ªgiµ¥i_u6qw`s$ogw`qtSqPvxw`o3 .lR 3 @ g @6A & tSog 5].lR5 @ g @BA & 5

. degi¤o>ªh0i6s $ tSogÃqPegi¤tSstShji6qi6s¥i_u6qw`s 3´tSog 5 wS¯£qPegiWu6¦Xt`i_µtSsiÅ>ogwN¢«o»±Kvz®³i® +(#.$ 2 )u6¦Xt`Pv©¨utSqPvxw`o¹T®

devx$hjwGi6¦\si6¦vxi_µw`oÃqPegihWvovhWv­NtSqPvxw`o¡wS¯£tIogw`ogu_w`or¥iT¸¬¯Kªogu6qPvxw`otS¦«± w`s T .$ ¹¢«egwi~i6ogi6stS¦D¯ w`sPh)ª¦XtSqPvxw`o?vx7

MF(#=, . J L N #(dc, J (dc, P R : c o R J L ( N - #(dc, N , : c o @ R J L H (#(dc,XW&3 * 5 , : c 5 ±K>¹degi¨gsPq$qÖ¢£wÁqi6sPhj.sw¥Gvxi¤tjsi_Pqw`stSqPvxw`oâ«vqPe¼i_8~iTyzsi_i6sP¥`tSqPvxw`oD®µdegi)¦Xt`Pqqi6sPh Hvo ±K>¹vx¤t¾0w`qi6orqPvXtS¦vog8ªgu6vo~¡tu6¦Xt`Pv©¨utSqPvxw`o¶u_w`ogqPstSvorq®ÃmÖq)qtSÅi_)vo>qwÃt`u_u_w`ªo>q¤qPegivo>qi6ogPvqÖ½;tSogjqPegi$tSstShji6qi6s«wS¯DqPegi$u6¦Xt`i_E¯Æw`sÄqPegi.u6¦Xt`Pv©¨utSqPvxw`o;wS¯bit`ue4(Öv©¸i6¦-*Yc\® Het`$~w`qt`hItSo>½hWvovhItjqPetSoÃqPegi)orªh0i6swS¯Äu6¦Xt`i_µtSog?vhW0wi_µtW¦xi6¥i6¦Bu_w`ogPqPstSvorqw`oÀqPegiIw`¦ªqPvxw`oD®jdegiWsitS¦EtSstShji6qi6s tSog @¬0i6sPhWvqZqwt`S°ªgPqZqPegi¤¢Ri6v~erq¤wS¯it`ueqi6sPh®mpo qPegi?ogiT¸GqÁi_u6qPvxw`oD|¢£i¬¢«v¦¦$i_i?qPetSqIqPevx&hjw>i6¦$vxIu6¦xwi?qw¡qPegi?et`iqPstSogPvqPvxw`ogsw`¦xi6h PqPªg8vxi_¾vohji_uetSovxu_®

!¶$ )# ¿* # 0*# '*) #$ $

degi Ï tSoi6s&ǧttS¦x°y tSeo8y v¦¦vXtSsqPegi_w`sP½wS¯µet`i?qPstSogPvqPvxw`og&et`ÁPegw¢«oºqw¡0iiTÂIu6vxi6o>q¤¯ w`s¤qPegi;PqPªg8½§wS¯$hji_uetSovxutS¦P½8Pqi6hjhIt`iÁª´wS¯µt?hItSqi6sPvXtS¦u_w`hWw`ªog´wS¯vogqtS¦xiZet`i_®£devxhItSqi6sPvXtS¦hIt_½;0iZi6vqPegi6s$w`¦vx?w`s¦vrªvx+®!gw`s$qPegiZtSÅiZwS¯Eu6¦XtSsPvqÖ½|¢RiµiT¸G0wi.qPegi.qPegi_w`sP½&u_w`ogu_i6sPovo~WqÖ¢RwWet`i_|tSog¾tSo¾iT¸Gqi6ogvxw`oÁqwqPegi ( et`i_«ut`iutSo¾0i$¯ w`ªogÁ¯Æw`s«vogPqtSogu_iµvo O Ö®Ede>ªg|8qPegiµqÖ¢RwWvogPqtS¦xiet`i_«hIt_½Ási6si_i6orq$i6vqPegi6stÁuegi6hWvxutS¦£tS¦¦xw½Àt` J aB ! ; a ! Ö|+w`sµqPegiW8vxPqPsPvªqPvxw`o¡¦xi6¥i6¦xwS¯«tgªvx Þk8Ö®degi¤u_w`hWhjw`o¼sw`¦xi6hvoÃqPegi_i¤8v 0i6si6o>qtS¦vxutSqPvxw`ogZvx$qwÁuetSst`u6qi6sPv­ijqPegiqtSv¦vqÖ½wS¯Pªgue¶t;P½Gqi6h®devxqPegi_w`sP½¬¦xit`Zqw¾qPegiWi_u6sPvqPvxw`o§wS¯qPegiWPqtS¦xiju_w`o8¨g~ªstSqPvxw`ogtSog&qwZqPegiµtSotS¦½8PvxwS¯DqPegivo>qi6s°¯Æt`u_i.0i6qÖ¢£i_i6o;qPegi$qÖ¢Rw)et`i_Ä¢«ev¦xi.qPegi.P½Gqi6hsit`uegi_vqPqtSv¦vqÖ½®!vsPq|BqPegi Ï tSo´i6s)ǧttS¦xPy tSeo8y v¦¦vXtSs´qPegi_w`sP½§vx)sPvxig½»i_u6sPvi_+®Àdegi6oD|B¢RiÁ¢«v¦¦qPsi_.qPetSq.v©¯R¢£i¤u_w`ogPvxi6sqPegivhItS~i¤vorqi6ogvqÖ½Àt`µt&hItSqi6sPvXtS¦z|+qPevxµhji_uetSovxutS¦EqPegi_w`sP½si6hItSvogªgiT¯ ª¦0¯ w`s«vhItS~iZi6~hji6orqtSqPvxw`oD®

ä\äàQ#RTSæR

Page 13: Image Classification Using a Variational Approach

D -)$.GT ('- T $ B % T$* B&.$T .$T

W

u0

ba

!\v~ªsi) ( , ('% ( )

8;7 =8[63 <Rw`ogvxi6sµtW8½GotShWvxutS¦BP½Gqi6h hIt`iwS¯Ätjvo~¦xigªvx¢«egwi (.vg¯Ksi_i)i6ogi6sP~½|0i6sªovq«¥w`¦ªhji|gvxt¯Kªogu6qPvxw`o H wS¯\qPegii6ogvqÖ½¾8vxPqPsPvªqPvxw`o ! ® Het`«qp¢£wWhWvovhIt8|tSogvx$ÅGogw¢«oÀt`.t ,. .%( .%b± i_i%!v~g®$¹T®devx gªvx¬vx.Pªwi_qw&iªogi6svxw`qPegi6sPhItS¦\u_w`og8vqPvxw`ogµtSogu_w`orqtSvogi_?vo¬t¤0w`ªogi_?w`hItSvo ®EdegiZPqtS¦xiZu_w`o8¨g~ªst]yqPvxw`og«wS¯\qPevx«P½Gqi6h tSsiµw`qtSvogi_;r½Áw`¦¥Gvo~)qPegi$¯Æw`¦¦xw¢«vo~j¥tSsPvXtSqPvxw`otS¦0sw`¦xi6h $±Æt`¯Æw`siT¸gtShW¦xivo O8|g8|gD|gk8|8g|8rK¹

"!$#%& ( ! , *%' ( ! , .?j L)(+* N - ! (dc, N R o & H ( ! (dc, ,-, : c *¢«vqPe ! Pª8Pi_u6qqw¤qPegiZu_w`ogPqPstSvo>q7 j L ! (dc, : c . h* ±Æ¹

¢«egi6si vxÄtSojw`0i6ojw`ªogi_IPªgi6qEwS¯ ¢«vqPe+vguevqP­.u_w`orqPvo>ªgw`ªg£0w`ªoggtSsPvxi_|GtSog vxZqPegi¤qw`qtS¦EhIt`ZwS¯qPegi gªvx+® N - ! (dc, N si6si_i6orqZqPegij~st`8vxi6orqhjwG8ª¦ªgZwS¯ ! (dc,®H Ap C ºvx)t?u_w`orqPvo>ªgw`ªg)ogw`o»ogi6~rtSqPv¥ij¯ ªogu6qPvxw`o§¢«vqPe¶iT¸t`u6qP¦½¼qp¢£w¬hWvovhIt]_tSog/.;± _ m0.6¹µPªgue§qPetSq H/(_, . H (.7, . 8® !ªsPqPegi6sPhjw`si| H vx rªt`8stSqPvxuÁtSsw`ªog_¾tSog1.]|tSog?vx~swN¢«vo~tSq$¦xit`Pq$¦vogitSsP¦½¬tSq.vo8¨govqÖ½|+i_iZ¯ w`s$vogPqtSogu_i ÞDb¯Æw`s$iT¸8¦vxu6vqu_w`og8vqPvxw`og.w`o H ± devxqPªg8½¾tS¦xw¤egw`¦x¯Æw`sMH A 32C £|t`vqvxiT¸80wi_;¯Æw`s«vo8yqtSogu_ivo OG|D8|8 K¹T®Edegi Ï tSoi6sǧttS¦x°y tSeo8y v¦¦vXtSs?qPegi_w`sP½;et`«¦xi_¾qw¤vo>qPsw>8ªgu_iqPegi&0i6sPqPªsPtSqPvxw`o´qi6sPh * N - ! N R |B¢«vqPe * PhItS¦¦z|B¢«evxue u_w`o>qPsPvªqi_¤qw¼w`¦¥iIqPegiI¯ztSv¦ªsiu_w`ogu_i6sPovo~WqPegi.ªov>ªgi6ogi_«wS¯bqPegiµw`¦ªqPvxw`o ! wS¯qPegi$sw`¦xi6h 4 "! # j L H ( ! (dc, , : c¬¢«vqPe! Pªgue?qPetSq j L ! (dc, : c . ®

* éä * Ü

Page 14: Image Classification Using a Variational Approach

"!#$%'&()(%$*+,.-

Rw`ogu_i6sPovo~»qPegi¾PqPªg8½´wS¯$qPegi¾sw`¦xi6h tSog´vqjw`¦ªqPvxw`ogIt` * C R|tSªqPegw`sIwS¯Þ8|D|k88|bªgi_qPegi +DyÖu_w`or¥i6sP~i6ogu_iqPegi_w`sP½¾i6¥i6¦xw`i_?r½ i (.vxw`sP~v ÞkÖ®ÄmÖ¯ tSogH ¥i6sPv©¯ ½ÁqPegiµsi6¥Gvxw`ªgP¦½;si_u6sPv0i_u_w`og8vqPvxw`og|qPegiZhItSvosi_Pª¦qtSsi %& ++yÖu_w`or¥i6sP~i_qw % ^ ¢«vqPe

% ^ ( ! , . L (. & , v©¯ ! (dc,`Z R _W . g t8®³i®orq w`qPegi6sP¢«vxi|

¢«vqPeB. & . R c Z ! (dc, . _ g tSog iT¨gogi_Ãt`7 . E "! J & & H (VU ( P , , N U ( P , N :SP W UÁvxi_u_i6¢«vxi \ & |SU( f , . _|SU ( f , . . 5

mz¯ ! £vx.tji >ªgi6ogu_iwS¯hWvovhWv­i6sµwS¯%&Pªgue?qPetSq ! Ru_w`or¥i6sP~i_.qw ! ^ vo]% & ( ,|qPegi6o ! ^ vxtWw`¦ªqPvxw`o?wS¯qPegiµsw`¦xi6h

!# L (. & , ¢«vqPe ! Z k H ( , *H ( ! (dc, , . _ 5 56*j L ! (dc, : c .

o>½Ii >ªgi6ogu_i ( < X,ªgueÁqPetSq %' ( < X, e J a P _ba mpq K * |>t`8hWvqt)Pªgi >ªgi6ogu_iu_w`or¥i6sP~vo~Ivo]%'&7( ,.± u_w`hWt`u6qPogi_vo %'&)( ,î¹T®

#.w`qi«qPetSqEqPegisit`u6qPvxw`o8yÖ8v ªgvxw`o&sw`¦xi6h si6¦XtSqi_jqw \vxEt`i_jw`oIqPegi¯ w`¦¦xwN¢«vo~ N± (µvo­_ªsP~`y btSoggtSªÃi >ªtSqPvxw`o¹

. * f * H ( , * ¢«egi6si £vx«qPegii6¥w`¦¥>vo~jqPvhjiZtSstShji6qi6s 5 ±8¹

( tSor½WtSªqPegw`s£et_¥iPqPªg8vxi_jqPevxÄ N|rt`Evo Þ8|6|`r Ö|>tSogjet_¥iPegwN¢«oIqPetSqEqPegiw`¦ª8yqPvxw`o (dc * ,BwS¯£±8¹Ät` * C vxÄvxi_u_i6¢«vxi.u_w`ogPqtSo>q£¢«vqPeÁqÖ¢Rwt`8hWvxv¦xi¥tS¦ªgi_`_WtSog .]®( w`si_w¥i6s|`qPegiRi6¥w`¦ªqPvxw`o¤wS¯gqPegiÄvo>qi6s°¯Æt`u_i£i6qp¢£i_i6o¤qPegi£qÖ¢£w$si6~vxw`og R c Z (dc * , . _ gtSog R c"Z (dc * , . . g vxZt $ E, $pyzqÖ½G0i |+8|+kD8|g|r Ö|0vo8y8ªgu6vo~Át¤u_i6sPqtSvo?si6~ª¦XtSsPvqÖ½wS¯BqPegiµvorqi6s°¯zt`u_i®

degiÃw`ª¦xiTyz¢£i6¦¦µ0w`qi6orqPvXtS¦ H utSoºi?vorqi6sPsi6qi_ t`;tÀ¦xi6¥i6¦µu_w`ogPqPstSvorq;qi6sPh|¯Æw`suTyvo~ ! qwWqtSÅi)w`ogiZwS¯qPegiZqp¢£wj¥tS¦ªgi_ _Áw`s'.]®Rdevxu_w`ogPqPstSvo>qet`.tSo?vogu6sit`Pvo~&¢£i6v~erqt` * C 8®degi$0i6sPqPªsPtSqPvxw`o;qi6sPh * N - ! N R et`«tZsi6~ª¦XtSsPv­NtSqPvxw`o¾i+i_u6qRw`oÁqPegi$w`¦ªqPvxw`o !

ä\äàQ#RTSæR

Page 15: Image Classification Using a Variational Approach

-)$.GT ('- T $ B % T$* B&.$T .$T

>½Ãt¥w`vx8vo~¾qPegi¯Æw`sPhItSqPvxw`o¡wS¯£Pvo~ª¦XtSsPvqPvxi_ZtSogÃ>½?si_PqPsPvxu6qPvo~¾qPegi¤Pt`u_i¤wS¯£w`¦ªqPvxw`ogN®devx¤0i6sPqPªsPtSqPvxw`o´qi6sPh ¥`tSovxegi_jt` * C 8®¡devxWPqPªg8½¡vx¤si_i6orqi_"egi6si6vo"vo"qPegiw`ª¦xiTyz¢Ri6¦¦Eut`i|i6¥i6oÃv©¯Äi6¥i6stS¦tSªqPegw`s.et¥i)¢Rw`sPÅi_¼w`o¬qPegi)hª¦qPv¦xiTyz¢Ri6¦¦Äut`i¤tSogh)ª¦qPv¦xigªvx Þg|'OG|+k|8g|8r Ö®

n 7] l : 68 7 < < Ç¡i¤utSoÃqPstSogP0wiqPegi_i)si6¥Gvxw`ªg.¢£w`sPÅ8.qw&vhItS~iswGu_i_Pvo~&¯ w`stIsi6~vxw`o¼i6~hji6o8yqtSqPvxw`o tSog»si_Pqw`stSqPvxw`o´swGu_i_® Di6q R i;tSo´w`i6o¶0w`ªogi_¶w`hItSvoD|£tSog#A C t&¯Kªogu6qPvxw`oÀqPetSqZsi6si_i6orq¯Æw`svogPqtSogu_iWqPegi¤vo>qi6ogPvqÖ½¡wS¯it`ue§vqi® Rw`o8yvxi6sµtj¯ itSqPªsi¤u6sPvqi6sPvxw`oÀwS¯£u6¦Xt`v©¨utSqPvxw`oD|+w`o¦½?t`i_¬ªw`oÃqPegi¤8vxPqPsPvªqPvxw`o¡wS¯Evorqi6o8yvqÖ½|tSog¡¦xi6q)qPegijvhItS~ij0iIu_w`hW0w`ªog§wS¯qp¢£w¬si6~vxw`og. & . R c Z #(dc, . _ g tSog. R . R c Z #(dc, . . g ®Di6q J A C i6ogw`qiIqPegiIw`gi6sP¥i_»gtSqt8|\u_w`sPsPªqi_§r½tSot`8vqPv¥iµ¢«evqi (tSªgvXtSoÁogw`vxi,@|8Pªgue¾qPetSqqPegi.¦vogitSsi6~st`gtSqPvxw`o¾hjwGi6¦+vxRwS¯\qPegi¯Æw`sPh J (dc, . #(dc,o @(dc, *K c]Z ® Di6q£0iµqPegi.¯ w`¦¦xwN¢«vo~&P½8Pqi6h

C %' (#=, *%& (#=, .?j L ( * N - #(dc, N R o & H (#(dc, , , : c *¢«vqPe]#ÃPª8Pi_u6qqw¤qPegiZu_w`ogPqPstSvo>q7 j L (#(dc, J (dc, , R : c e 5 R

± O¹

¢«egi6si 5 vxqPegiPqtSoggtSs¾i6¥GvXtSqPvxw`owS¯\qPegiµ¢«evqi (ZtSªgPvXtSo¾ogw`vxi @|gtSog H vxt`si6¥Gv©yw`ªg¦½Ii_u6sPv0i_ñ i_iµPªgi_u6qPvxw`o;k8®N¹T®Edegi.u_w`ogPqPstSvo>q«vo¼± O¹ÄvhW0wi_RqPegi$w`¦ªqPvxw`o #?qwsi6hItSvo¬u6¦xwiqw¤qPegiZw`gi6sP¥i_?gtSqt8®«u_u_w`s8vo~µqw$qPegiRsi6¥>vxw`ªgsi_Pª¦q|t` * C qPegiRw`¦ªqPvxw`oE#jwS¯+u_w`o>¥i6sP~i_qw.t$PhjwGw`qPei6~hji6orqi_ÁvhItS~i.¢«egwiµPvqi_£w`o¦½Ii6¦xw`o~¤qw . & w`s. R ±Æt8®³i®í¹T|GqPegi_iqp¢£w¤si6~vxw`ogR0i6vo~i6tSsi6qi_¼r½¼PetSsPÀsi6~ª¦XtSsPv­i_§i_8~i_I±KqPegiWhItS¦¦xi_Pq * |0qPegiWPetSsP0i_PqqPegi¤qPstSogPvqPvxw`ogT¹T|tSogqPegii6qwS¯0w`ªoggtSsPvxi_et¥>vo~;thWvovhItS¦b0i6sPvhji6qi6s)± i_i!v~g®Äk¹T®

Ǻegi6o * vx¨¸8i_+|¦xi6qiT¸tShWvogi&qPegij¯ ªogu6qPvxw`otS¦%&X(#=,.qw?~v¥iÁtSo»vo>qi6sPsi6qtSqPvxw`o wS¯«qPegivogªgi6ogu_iwS¯Bit`ue?qi6sPhw`o¾qPegiZw`¦ªqPvxw`o?wS¯ C %& (#=,®

%' (#=, .?J L ( * N - # N R

! "$#&%(')#*+%(,!-.*+&/

o f * H (#=,

"(102"43,!-5&*+#%(-6*

, : c ±Æ¹

degiZ¨gsPqqi6sPh vo»±Æ¹si6¥i6o>q.qPegiw`¦ªqPvxw`o¬¯Ksw`hqPegi)tStSsPvqPvxw`o¼wS¯Eev~e?¦xi6¥i6¦\8vxu_w`o8yqPvo>ªvqPvxi_®jmpoÀqPegi¤¨i6¦x¡wS¯vhItS~iWsi_Pqw`stSqPvxw`oD|bqPevxZqi6sPh vog8ªgu_i_tPqtSoggtSs¡dvÅGegw`ogw¥si6~ª¦XtSsPv­NtSqPvxw`o·± uT¯P®&i_u6qPvxw`o¶8®Þ?tSog rK¹T®ÁdevxZsi6~ª¦XtSsPv­NtSqPvxw`o¶swG8ªgu_i_tSo¡vxw`qPsw`vxu

* éä * Ü

Page 16: Image Classification Using a Variational Approach

"!#$%'&()(%$*+,.- k

Ω

R2

R1

R1

R1

antiphase boundaries

!v~ªsik % (% % )hjw>w`qPevo~g®#.i6¥i6sPqPegi6¦xi_|bqPegiWi_u_w`ogÀqi6sPh¯Æw`su_i_qPegiWw`¦ªqPvxw`o§qw¾qtSÅiWw`o¦½¼qPegi¤qÖ¢£w¥tS¦ªgi_ÄuetSst`u6qi6sPv­_vo~¤qPegisi6~vxw`og . & tSog4. R |rqPegi_iqp¢£w)¥`tS¦ªgi_N|b_¤tSog .]|GtSsiqPegi¦XtS0i6¦xwS¯O. & tSog . R ® i6ogu_i|Äi6¥i6o»v©¯qPegi&si6~ª¦XtSsPv­NtSqPvxw`o vxPqPsw`o~g|qPegi&¦xi6¥i6¦«u_w`ogPqPstSvorq Hegw`ª¦x¾i6sPhWvqtS¦¦bqPegiZPvqi_«qwWPqt½Áª8ÂIu6vxi6orqP¦½u6¦xwiµqwWqPegiZ¦xi6¥i6¦x_&tSog .N®degiu_w`or¥i6sP~i6ogu_i.w`o * vog8ªgu_i_RtSoIi6¥w`¦ªqPvxw`o&wS¯+qPegi«¢Ri6v~erq£wS¯+it`ue&qi6sPh®degi«¥tS¦ªgiwS¯ *vxEvovqPvXtS¦¦½ev~e¼±'(Öev~e *«vojqPegii6ogiTogw`o&ogi6~¦v~itS¦xi¹T®Edegi6siT¯Æw`si|qPegi«si6~ª¦XtSsPv­NtSqPvxw`oswGu_i_vx¨gsPqP¦½¼si6w`ogi6stSo>q®Á. * C |bqPegiWvogªgi6ogu_iIwS¯qPegiW¦xi6¥i6¦Ru_w`ogPqPstSvorq)vxvogu6sit`vo~À¢«ev¦xi¾qPegi¾PhjwGw`qPevo~Àqi6sPh vxW¥`tSovxevo~g®¶$w`ª~e¦½»P0itSÅGvo~g|EqPegi¾u_w`or¥i6s°y~i6ogu_iw`o * 0i6sPhWvq$vo¬qPegiZ¨gsPq.qi6gqwIsi6hjw¥iqPegiogw`vxitSog?qPegi6oD|¢«egi6oÃqPegi)vhItS~ivxPª8ÂIu6vxi6orqP¦½;si_Pqw`si_+|¢RiutSoPqtSsPqqPegiu6¦Xt`Pv©¨utSqPvxw`o?¢«vqPe¬i6ogw`ª~e?u_w`o8¨i6ogu_i®( w`si_w¥i6s|b¯ sw`h qPegiWsit`u6qPvxw`o8yÖ8v 0ªgPvxw`o»sw`¦xi6hu_w`sPsi_P0w`og8vo~?qw§±8¹T|b¢£ijÅGogw¢qPetSqqPegi$vorqi6s°¯zt`u_iµ0i6qÖ¢£i_i6o . & tSog . R et`«t $ $ (>6®devx£sii_u6qqPegiµ~i_w`hji6qPsPvxutS¦si6~ª¦XtSsPv­NtSqPvxw`o¬w`0i6stSqi_w`oqPegii_8~i_®degiu_w`ogPqPstSvo>qvo"± O¹hjit`Pªsi_µqPegi8vxu6si6tSogu6½?0i6qÖ¢Ri_i6oÀqPegi)w`¦ªqPvxw`o¡tSogÃqPegigtSqt8®JGw|0qPegi)¨gotS¦BvhItS~i¤¢«v¦¦Bi)vxi_u_i6¢«vxiWu_w`ogPqtSo>q¢«vqPe¼si6~ª¦XtSsPv­i_Àw`ªoggtSsP½"±Ket_¥Gvo~?thWvovhItS¦bi6sPvhji6qi6s¹T|tSog¾¢«v¦¦+qtSÅivorqwÁt`u_u_w`ªorq$qPegiZt`u_u_w`sgtSogu_iZ¢«vqPeqPegiZgtSqtji6q®

ä\äàQ#RTSæR

Page 17: Image Classification Using a Variational Approach

6 -)$.GT ('- T $ B % T$* B&.$T .$T

))b)

n 243'< 8 S n 7&«t`i_?w`oqPegiZsi6¥Gvxw`ªgi_u6qPvxw`ogN|¢£ivorqPswG8ªgu_iZqPegiZ0i6sPqPªsPtSqPvxw`o?tSstShji6qi6s * vo¶±K>¹>½ÁiT¨govo~M (#=, .?JSL N #(dc, J (dc, P R : c

#*+# *)&/

o * R J L ( N - #(dc, N , : c &5*,!+#*%(,!- *+&/

o @ R* JSL H (#(dc,XW&3 * 5 , : c "(102" 3,!-5*+#%(-2* */

5 ±Æ¹degiu_w`sPsi_Pw`og8vo~Isw`¦xi6h¢£i)tSsiµvorqi6si_Pqi_?vovx«qw¨gog?t¤w`¦ªqPvxw`o #¼PªgueqPetSq

# . ^ ( C M (#=, , 5 ±°D¹

degij¨gsPqqp¢£wÃqi6sPhjwS¯)±Æ¹T|\v©¯ * vx¨¸8i_+|tSsiIqPegi&w`ogi_wS¯«qPegiIsi_Pqw`stSqPvxw`o¶¢«vqPe"i_8~iTysi_i6sP¥`tSqPvxw`oï ªogu6qPvxw`otS¦«±Æ¹T®µdegi¦Xt`Pq.qi6sPh vxµtI¦xi6¥i6¦u_w`ogPqPstSvorqµvogPvsi_Ã>½?qPegi Ï tSoi6sµÇ§ttS¦x°y tSeo8y v¦¦vXtSsÀqPegi_w`sP½ÃwS¯£et`iqPstSogPvqPvxw`og|+ªgueÀqPetSq H tSqPqPst`u6qqPegi¥tS¦©yªgi_wS¯ #(dc,µqw¢tSs¤qPegiI¦XtS0i6¦xwS¯qPegi&u6¦Xt`i_®?mpo»w`ªsWtSswt`ueD|Evogu_iÁt?u6¦Xt`E\ @ vxuetSst`u6qi6sPv­i_;r½Wvq£hjitSo 3 @ tSogÁPqtSoggtSsÁi6¥>vXtSqPvxw`o 5 @ |>¢£i$et_¥i.uegwi6o;qwi_v~o \ @>½&qPegiZ¦XtSi6¦ 3 @ ® i6ogu_i|qPegii6qwS¯Et`8hWvxPv¦xi¦XtS0i6¦x vx«qPegiµ¥i_u6qw`s 3®mpot`u_u_w`sgtSogu_i¢«vqPe;qPegiµer½Gw`qPegi_vxI( ,ÄPqtSqi_Ávoi_u6qPvxw`o8®Þ8|8qPegiµ0w`qi6orqPvXtS¦ H A C hªgPqRqtSÅivo>qwt`u_u_w`ªo>qqPegi (tSªgvXtSoI8vxPqPsPvªqPvxw`o;sw`i6sPqp½jwS¯DqPegiu6¦Xt`i_® Hnet`vq $ hWvovhItw`ojqPegi¥tS¦ªgi_ 3 @ PªgueIqPetSq H (43 @ , . * K]; ®Ç¡iet_¥iu_w`ogqPsPªgu6qi_&tµ0wSyqi6o>qPvXtS¦H PªgueÁqPetSq H Z$\i&7(d ,ÄtSog H vxÄvxi_u_i6¢«vxi$tSstS0w`¦vxu® !sw`h !v~g®Bg|tSsw`ªog3 @ |%H vxwS¯qPegiµ¯Æw`sPhIH (dc, . @ (dc, . D D (dc 3 @ , R |8¢«egi6si @ vxt¤0wPvqPv¥i¢Ri6v~erqPvo~u_w`ogqtSorq®degi6oD|¢£i)u_w`ogPqPsPªgu6q$qPegi rªt`8stSqPvxuvxi_u_i6¢«vxi @ (dc, . @ @ (dcD Q @ , R qPetSqPw`vog @ tSog @ =& si_P0i_u6qPv¥i6¦½ÀtSqqPegijw`vo>q @ tSog @ ±! @ vxt (#$* #" 2G¹T®degijtSstShji6qi6s$ @ |% @ | @ | @ ± @'& ¹tSog Q @ tSsiju_w`hWªqi_À¯ sw`hqPegijiT¸8si_Pvxw`o¡wS¯ @ tSog @ =& ®( @ Pw`vog @ tSogB @ =& ±\ &2(d$,Æyîªogu6qPvxw`o¹Pªgue¬qPetSq*) @ * @+ vxt`PhItS¦¦t`0wPv¦xiµqwj~i6q$t¤0w`qi6orqPvXtS¦ H ¢«egwitSstSw`¦vxu_ @ tSsi,(Ösi_w`hWvotSo>q *vo?u_w`hWtSsPvxw`o¢«vqPe§qPegijtSstS0w`¦vxu°ªogu6qPvxw`og® !w`s)it`ue¶i6qwS¯tSstShji6qi6s[(43 * 5 ,|b¢Rij~i6q)t¾8v +i6si6orq0w`qi6orqPvXtS¦ H ® H vx >ªt`8stSqPvxu)tSsw`ªog¬vqhWvovhItÁtSogvx~swN¢«vo~I¯zt`Pqi6s$qPetSo?¦vogitSsP¦½tSq«vo8¨govqp½§± iT¸G0i_u6qi_¾sw`0i6sPqPvxi_¯Æw`s H iT¨gogi_?voi_u6qPvxw`o¬k8®N¹T®Ç¡iÁt`PªhjiIqPetSq vxt¾si6~ª¦XtSsPv­NtSqPvxw`o¶¯Kªogu6qPvxw`o ±Æt`qPegi&w`ogi_)wS¯dtSD®?¹µet¥>vo~¬qPegiuetSst`u6qi6sPvxPqPvxu_.iT¸G0wi_;voi_u6qPvxw`o¬8®ÞItSogPegw¢«o¬w`o !\v~g®®bi6q * 0i¨¸8i_+®E hWvovhWv­i6swS¯ M |Gv©¯bvq«iT¸8vxPq|>vx«t)w`¦ªqPvxw`o # tSqPvx°¯K½Gvo~qPegi.iT¸80i_u6qi_sw`0i6sPqPvxi_7si_Pqw`si_+|+i_8~iTyzsi_i6sP¥i_ ±Kv©¯ vxµ8v 0i6si6o>q$¯ sw`h t (pdvÅGegw`ogw¥"*)si6~ª¦XtSsPv­Nt]yqPvxw`o§¯Kªogu6qPvxw`oD|Bi_i&d\tSD®?¹tSog§¦xi6¥i6¦©yÖu_w`ogqPstSvogi_+®?. * C 8|B8ªsPvo~¬qPegij¨gsPq)qi6g

* éä * Ü

Page 18: Image Classification Using a Variational Approach

"!#$%'&()(%$*+,.-

µ µi

W

P(x) P(x)

Q(x)

i

k i

i+1

i+1

α p βi i i

i

!v~ªsi % ' % ' & ' (( % ' )

wS¯\qPegiu_w`or¥i6sP~i6ogu_iqPegiµ¢Ri6v~erqwS¯\qPegi.¦xi6¥i6¦Du_w`ogPqPstSvo>qqi6sPh vx rªvqi.vogPv~ov©¨utSo>q w`o¦½qPegijsi_Pqw`stSqPvxw`o§swGu_i_wGu_u6ªs®&$ZqPsi_i_¼vo§qPegijsi6¥>vxw`ªg)i_u6qPvxw`oD|t` * i_u6sit`i_N|¢RiWsw`~si_Pv¥i6¦½Ã~i6q)t;wS¯ qi6ogi_§tSovxw`qPsw`vxuI8v ªgPvxw`oÀ¢«ev¦xijstSvxPvo~qPegiju6¦Xt`v©¨utSqPvxw`oswGu_i_®#.w`qiqPetSqqPegiWw`¦ªqPvxw`oÀ¢£ijtSsi¤¦xwGw`ÅGvo~;¯ w`sµvxZt;u_w`orqPvo>ªgw`ªgZ¥tSsPvXtS¦xi±Æt`qPegivorqi6ogPvqp½¦xi6¥i6¦0¯Æw`s«iT¸tShW¦xiN¹tSog;ogw`qt)8vxu6si6qiw`ogiZt`«vot¤Pqw>uet`PqPvxuµ¦XtS0i6¦¦vo~ D|0|0|Ö®degi$u6¦Xt`Pv©¨utSqPvxw`o;vx£vhW¦vxu6vqP¦½IiT¨gogi_IqPesw`ª~eÁqPegi$¦xi6¥i6¦u_w`ogPqPstSvo>qqi6sPhvoÀ±Æ¹B¢«evxuetSqPqPst`u6qqPegiZt`8hWvxPv¦xi¥tS¦ªgi_wS¯BqPegiw`¦ªqPvxw`oqwN¢RtSs$qPegi $ ¦XtSi6¦xwS¯BqPegiZu6¦Xt`i_®

ÁÉ § vo¶qPegij¯Æw`¦¦xw¢«vo~g|¢RiÁ¢«v¦¦Rw`hWvqqPegi&u_w`o>¥i6sP~i6ogu_i;tSstShji6qi6s * vogiT¸Gvo~ #B|qPe>ªg # ¢«v¦¦Diµsi6¦Xt`u_i_r½ #B|i6ogw`qPvo~jqPegiZw`¦ªqPvxw`o¬wS¯qPegihWvovhWv­NtSqPvxw`oÃwS¯/M ®n 5' 8 7 <=< 1 ? 8!3":

bi6qPªhWhItSsPv­ijqPegi¤sw`0i6sPqPvxi_wS¯qPegiW¦xi6¥i6¦Äu_w`ogqPstSvorq0w`qi6orqPvXtS¦ H ¢£ijiT¸8i_u6q|\t`uTyu_w`s8vo~)qwZqPegit`PªhWqPvxw`og ( ,\¢RihIt`i$w`oIqPegiu6¦Xt`i_|>tSogI¢«vqPeIsi_P0i_u6qEqwZqPegi Ï tSoi6s«Ç§ttS¦xPy tSeo8y v¦¦vXtSs¬qPegi_w`sP½® Ã É Í%@ 3 1 3 ¡+ H qtSÅi_$vorqw;t`u_u_w`ªorq.qPegiZo>ªh0i6sµtSog?qPegiZtSstShji6qi6swS¯qPegiu6¦Xt`i_®

8®EÍ &38+3 Í 1 Í'@ += É 3 1 3 §+ ¯Æw`s $ u6¦Xt`i_| H h)ªgPq¤et_¥i $ hWvovhIt¬w`oi6¥i6sP½¬Pvqi P ªgue¼qPetSqr#( P ,vxu6¦xwiqw 3 @ ®)mÖoÀw`qPegi6s¢Rw`s|+qPegi¦XtSi6¦wS¯«t&u6¦Xt`\ @ vxqPegiu_w`sPsi_P0w`og8vo~jhjitSo 3 @ ®«u_u_w`s8vo~jqwWi_u6qPvxw`oÃk8®|gtsi6~vxw`oB. @ ¢«v¦¦+0i

ä\äàQ#RTSæR

Page 19: Image Classification Using a Variational Approach

8 -)$.GT ('- T $ B % T$* B&.$T .$T

iT¨gogi_&PªgueÁt`. @ . R chZ #(dc, . 3 @ g tSogÁit`ue;i6¦xi6hji6orq«wS¯6. @ 0i6¦xw`o~£qw \ @ ®k8® : & É Í%@ += É 3 1 3 ¡+ H vxbu_w`ogPqPsPªgu6qi_Pªgue)qPetSq H (43 @ , . * K ;.lf565 $ ®g® 33N3 3 1 8+3 Í 1 Í'@ += É '&K É H Pegw`ª¦x¤i«tS¦xiqwµ8vxu6sPvhWvotSqi«qp¢£wu6¦Xt`i_et_¥Gvo~¬si_P0i_u6qPv¥iÁ8vxPqPsPvªqPvxw`og 1(43 @ * 5 @ ,ZtSog 1 (43 * 5 ,.v©¯ 3 @. 3Ái6¥i6o»v©¯ 3 @vxIu6¦xwi?qw 3ÃtSog¿i6¥i6oºvo¿qPegi?ut`i¢«egi6si 5 @ tSog 5 ¬tSsi¬PªgueºqPetSqIqPegi?qÖ¢£w8vxPqPsPvªqPvxw`oget¥i)t¢«vxiZwN¥i6sP¦XtSvo~ÁtSsit±Kvz®³i®hWv©¸i_8vxPqPsPvªqPvxw`ogT¹T®

w`vorqÁjqw¬ktSsiWqtSÅi6o§vo>qw¬t`u_u_w`ªo>q)qPesw`ª~e»qPegiju_w`ogPqPsPªgu6qPvxw`o"wS¯ H ®&degi6sijiT¸Gvxqi6¥i6stS¦b¢Rt½G.wS¯Äu_w`ogPqPsPªgu6qPvo~¾tW0w`qi6orqPvXtS¦ H tSqPvx°¯ ½>vo~&w`vo>q)qwIk8|ªq.tWvxi_u_i6¢«vxi>ªt`8stSqPvxu \ & (d$,b0w`qi6orqPvXtS¦i_i6hjEqwµiqPegiit`vxi_PqEw`ogi|ri_P0i_u6vXtS¦¦½¤Pvogu_iqPegii6sPv¥tSqPv¥iH vxEvxi_u_i6¢«vxi¦vogitSs®BÇegi6oj¢Riet¥i $ u6¦Xt`i_N|qPegiiT¸8si_Pvxw`oÁwS¯ Hnvx.± i_i !v~g®>¹

H/(dc, . 3 3 (dcD 3 & , R v©¯ c e & 8 8 (dcD 3 , R v©¯ c & D D (dc 3 @ , R v©¯ @ & e c e @ @ @ (dc Q @ , R v©¯ @ e c e @

±°N¹

w`vorq¾»vxÁqPegi?hjwPq;i6¦vxutSqi® Ç¡i?¢tSorq¾qw¶iT¨gogiÃt§0w`qi6orqPvXtS¦ H et_¥Gvo~ $ ¢£i6¦¦x±K0w`vorq&N¹µPª8ÂIu6vxi6orqP¦½ÀhItSsPÅi_§qww`qtSvo $ si_P0w`ogi_®IdevxZhjitSogZqPetSq)¢£iW¢tSorqqwt¥w`vxItµu_w`o8¨g~ªstSqPvxw`oIvoW¢«evxue&t.¢£i6¦¦vxevxi6ojr½qPegiw`ogi«wS¯Dt.ogi6v~er0w`ªsPvo~)u6¦Xt`N®\d\wv¦¦ªgqPstSqi£Pªguetegi6ogw`hji6ot.tSogqwiT¸8¦XtSvoegwN¢»qw$w¥i6su_w`hjiRqPevx\8v©ÂIu6ª¦qp½|`¦xi6qBt`PªhjiqPetSq¾¢RiÀitS¦Z¢«vqPe iT¸gt`u6qP¦½ k¶u6¦Xt`i_¢«vqPe·si_Pi_u6qPv¥i¼tSstShji6qi6s¶±KhjitSo )qtSoggtSsi6¥GvXtSqPvxw`o¹ Z±°8® D+8® D¹T|±°k8® D 88® D¹$tSog¶±°88® Dg® D¹T® bi6q.iT¸gtShWvogiqPegi)vogªgi6ogu_iwS¯EqPegi tSstShji6qi6s$ @ ¢Ri6v~qPevo~¬it`ue rªt`8stSqPvxu @ wS¯ H ± i_i±°N¹P¹T®Wdegi¤¢«vxi6ovo~?wS¯qPegi¦xwGutS¦BtSstSw`¦vxu @ vxµi6qi6sPhWvogi_À>½?qPegi¤stSqPvxw D D ®degi)¥tS¦ªgiWwS¯ 5 @ vx.¨¸i_+|DtSogqPegiZ¥`tS¦ªgiZwS¯ @ vxi6¦xi_u6qi_¾>½;qPegiZw`i6stSqw`s® JGw|t¤¥tSsPvXtSqPvxw`o?wS¯ @ vog8ªgu_i_.t¤¥tSsPvXtSqPvxw`owS¯qPegi&¢«vx8qPe´wS¯qPegiI¢Ri6¦¦R¯ w`sPhji_»r½¡qPegi&tSstS0w`¦vxuD @ Pªgue»qPetSqqPegiÁhjwPqqPegi&¥tS¦ªgiwS¯ @ |RqPegiotSsPsw¢Ri_PqÁqPegi¢£i6¦¦z® !v~g®·¼si_i6o>qIqPegiÃk]yz¢£i6¦¦.w`qi6o>qPvXtS¦ H ¯Æw`sj¢«evxue & . R . ./f5 8®$degiu6¦Xt`i_¢«egwi¦XtS0i6¦x¤±KhjitSo¬¥tS¦ªgi_¹tSsiÁ8® DÁtSog»k8® D&tSsiogitSsP¦½Ãogw`qZ8vxPqPvo~ªvxPetS¦xi® &R½Ãi6qPqPvo~ & . R . f 5 tSog . f5 ;¢Ri¤hItSÅi¤qPegiqp¢£w?¢£i6¦¦xi6hji6sP~i|tSog§¢£ijutSo¶i_iWw`o !v~g® 8¾qPetSqZqPegijqp¢£wogi6v~e>w`ªsPvo~¼u6¦Xt`i_)tSsiogwN¢ 8vxPqPvo~ªvxPetS¦xi®

* éä * Ü

Page 20: Image Classification Using a Variational Approach

"!#$%'&()(%$*+,.- +O

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

y

8 10 12 14 16 18 20x

W with w1=w2=w3=1

!v~ªsij % ' $ ( ' ' %) &) , '(% & , £tSsi & . R . .lf5 8®

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

y

8 10 12 14 16 18 20x

W with w1=w2=10 and w3=1

!v~ªsi8 % ' $ ( ' ' %) &) , '(% & , £tSsi & . R .lf 5 ItSog .lf5 8®

ä\äàQ#RTSæR

Page 21: Image Classification Using a Variational Approach

-)$.GT ('- T $ B % T$* B&.$T .$T

( N #.m J C m ( #( N E NJ#JGm #(

±Æt¹ u_w`ogPqPstSvo>qvo¾qPegisw`¦xi6h ±Æ¹ C i6otS¦qp½Ávo>qPsw>8ªgu_i_?voqPegi.¯ ªogu6qPvxw`otS¦R±Æ¹

±K¹ 0w`qi6orqPvXtS¦D¢«vqPe䢣i6¦¦x C 0w`qi6orqPvXtS¦b¢«vqPe $ ¢Ri6¦¦x)± u6¦Xt`i_T¹

± uN¹ 0i6sPqPªsPtSqPvxw`o?qi6sPh * N - ! N R C si6~ª¦XtSsPv­NtSqPvxw`o?qi6sPh * ( N - # N ,dtS¦xij !sw`hqPegiµet`iqPstSogPvqPvxw`og«qPegi_w`sP½;qw¤qPegisw`0wi_¾u6¦Xt`v©¨utSqPvxw`o¯ ªogu6qPvxw`otS¦n+ n n 8;<=<$r 8 3' < 8F:p4 8;7dtS¦xi§¾PªhWhItSsPv­i_IqPegi¾qPstSogP0wPvqPvxw`ogj¢Ri¾hIt`i;¯ sw`h hji_uetSovxu_¼±Æ¹T|E¯Æw`sW¢«evxue¢RiWet_¥ijqPegi_w`sPvxutS¦Äsi_Pª¦qtSqZw`ªs8vxP0wtS¦z|+qw¾vhItS~iju6¦Xt`Pv©¨utSqPvxw`o§qPesw`ª~e¡qPegijhWvov©yhWvov­NtSqPvxw`o"wS¯)±Æ¹T®¾Ç¡iIsw`wiIqw¬iT¸gtShWvogiIiT¸8qPstS0w`¦XtSqPvxw`og;±Æt¹qw¶± uN¹vo»dtS¦xi¬Ir½8vxqPvo~ªvxPevo~jqPegiZw`ogi_«¢«evxueÃtSsihItSqPegi6hItSqPvxutS¦¦½¤°ªgPqPv©¨i_;¯ sw`h qPegiegi6ªsPvxPqPvxuw`ogi_®

y¯Ksw`hu_w`ogqPstSvorqqw¤0i6otS¦qÖ½Áy

mpo OG|R8|k8|£Ö|qPegi¬hItSqPegi6hItSqPvxutS¦Zsi_Pª¦q¾tSw`ªqÁqPegiÃqPegi_w`sP½wS¯et`i?qPstSogPvqPvxw`ogsi6¦½»w`o¶qPegi +DyÖu_w`or¥i6sP~i6ogu_i¾qPegi_w`sP½®§$ogi;wS¯$qPegi&hItSvo¿t`8¥tSorqtS~i_WwS¯ªgPvo~¼qPevx¤ÅGvogwS¯¥tSsPvXtSqPvxw`otS¦u_w`or¥i6sP~i6ogu_iÁstSqPegi6s)qPetSo"w`o¦½¡u_w`or¥i6sP~i6ogu_i¾wS¯hWvovhWv­i6sN|BvxqPetSq)qPegisi_ª¦qsi6hItSvo?ªoguetSo~i_?¢«vqPe¼tWu_w`orqPvo>ªgw`ªgµi6sPqPªsPtSqPvxw`oÃw`o?qPegiµ¯ ªogu6qPvxw`otS¦ Þk Ö®£mpotSsPqPvxu6ª¦XtSs7v©¯ ( ! , + C ^ ( ! , t` * C qPegi6o ( ! , o j L N ! (dc, J (dc,QP R : c + =C ^ ( ! , o j L N ! (dc, J (dc,QP R : c t` * C

¢«vqPe J A C tj¯ ªogu6qPvxw`oÃvo % R (d$,|+i6ogw`qPvo~¾qPegi)gtSqt8®mÖoÀt`8vqPvxw`o¼qw&qPegiuetSs°yt`u6qi6sPvxqPvxu_wS¯hWvovhWv­i6ssi_i6orqi_vo¬i_u6qPvxw`oÃk8®|hWvovhWv­i6s+# £wS¯±Æ¹¢«v¦¦DqtSÅiZvo>qwt`u_u_w`ªo>qqPegi)w`gi6sP¥i_?gtSqt J ® i6ogu_iZqPegi6siZ¢«v¦¦D0it¤qPst`iTyÖw ¡0i6qÖ¢Ri_i6o¬qPegisi6~ª¦XtSsPvqp½wS¯BqPegiw`¦ªqPvxw`o¬tSogvq«¦xi6¥i6¦bwS¯Et`u_u_w`sgtSogu_iZ¢«vqPeqPegiZgtSqt8®

yE¯ sw`h¢£i6¦¦xqw $ ¢Ri6¦¦x«y

( wPq¤wS¯µtSªqPegw`s¤et¥i¾PqPªg8vxi_¶qPegiÁw`ª¦xiTyz¢Ri6¦¦ut`i |B8|6|k8g|br|\ Ö® JGw`hji;wS¯

* éä * Ü

Page 22: Image Classification Using a Variational Approach

"!#$%'&()(%$*+,.-

qPegi6het_¥iZqPsitSqi_?qPegik]yz¢£i6¦¦But`it`vo g|r Ö|tSogqPegiZ~i6ogi6stS¦\ut`iu_w`sPsi_Pw`og8vo~qw $ ¢Ri6¦¦xvxsi_i6orqi_?r½ &«tS¦xwWvo O Ö® &«tS¦xwWet`$iT¸Gqi6ogi_?qPegiZsi6¥Gvxw`ªgsi_Pª¦q$qwqPegi¤ut`i¤wS¯ ! A C 2;tSogï w`st&w`qi6o>qPvXtS¦H A 20C et_¥Gvo~ $ ­i6swSyz¥`tS¦ªgi_hWvovhIt8®

y¯Ksw`h rªt`8stSqPvxu0i6sPqPªsPtSqPvxw`o?qw¤si6~ª¦XtSsPv­NtSqPvxw`oêgPvo~It ¯ ªogu6qPvxw`o¾y

bi6qogw`qi A DC »ªgueqPetSq)± i_ij±Æ¹P¹ (dc *X!* * -E! , . %' ( ! , 5

$¢Ri6o¬tSog J>qi6sPo>i6sP~ Þk Det_¥iswN¥i6o?qPetSq«¯Æw`stSor½;u_w`or¥iT¸ ¯ ªogu6qPvxw`o?wS¯ * - ! ± w`qPegi6s¢RitSÅi6s«u_w`og8vqPvxw`og«w`o tSsi$si rªvsi_ÁvoÁqPegi6vstSsPqPvxu6¦xiN¹T| ! |>qPegi$w`¦ªqPvxw`o;wS¯I t` * C 8|qPv¦¦et`µqPegiWtShjiWuetSst`u6qi6sPvxPqPvxu_)qPetSoÀqPegiWw`ogi_si6¥Gvxw`ªgP¦½¼iT¸G0wi_¬¯ w`stSoÀvxw`qPsw`vxu0i6sPqPªsPtSqPvxw`o * R N - ! N R ±KqPegiÀsi_utS¦xi_ 0i6sPqPªsPtSqPvxw`o wS¯¾±Æ¹P¹T® wN¢£i6¥i6s| h)ªgPqet¥i±ÆtSq)¦xit`Pq¹ZttSstSw`¦vxu&0i6et_¥Gvxw`ªs)¢«vqPe»si_Pi_u6q)qw * - ! tSq)vo8¨govqÖ½®¬derªg|v©¯ vxqPegidvÅGegw`ogw¥»si6~ª¦XtSsPv­NtSqPvxw`o"¯Kªogu6qPvxw`o·± i_iÁd\tS¦xi¹ZqPegi6o * R ( N - # N , . * R N - # N R tSog»qPegiqPegi_w`sP½?egw`¦x®mÖ¯ vx.tSogw`qPegi6s.si6~ª¦XtSsPv­NtSqPvxw`o¼¯ ªogu6qPvxw`oD|qPegiqPegi_w`sPvxutS¦w`ªog8ogi_.wS¯$±Æ¹vx$tSow`i6osw`¦xi6h®

n nQ: M K 7)63 :bi6qi_u6sPv0iqPegiPqPstSqi6~½;¢£iet¥it`w`qi_qw¤hWvovhWv­iÁ±Æ¹R¢«vqPesi_P0i_u6q«qw #B®mÖ¯#"vx)thWvovhWv­i6sWwS¯Z±Æ¹T|qPegi6o´t;ogi_u_i_tSsP½¡u_w`og8vqPvxw`o"vxqPetSq)qPegi&i6sPv¥`tSqPv¥iÁwS¯«qPegi¯ ªogu6qPvxw`otS¦ M .tS¦vxi_»qw]#´vx)i >ªtS¦Äqw?­i6sw®$Pvo~¬qPegi Nª¦xi6s°y \tS~stSo~i¾i rªtSqPvxw`o¶qwu_w`hWªqiZqPegiZi6sPv¥`tSqPv¥iwS¯/M |¢£iw`qtSvoqPegiµ¯Æw`¦¦xw¢«vo~IP½8Pqi6h

N #(dc, J (dc,QP`o R H (#(dc, , * R : ;< ( C 1 R C 1 - #(dc, , . K c Z +*04C 10 . w`o +* ±°¹

¢«egi6si a vx)qPegiÁw`ªqÖ¢RtSs"ogw`sPhItS¦qw¬qPegiI0w`ªoggtSsP½§w`hItSvo |EtSog : ;< i6ogw`qi_qPegi8v¥i6sP~i6ogu_i¶w`0i6stSqw`s® Di6q?si6hItSsPÅ·qPegi§ogw`o¦vogitSsPvqÖ½ wS¯¤qPegi¶ N vo ±°¹tS¦qPegw`ª~eH (#(dc, ,\vxÄvxi_u_i6¢«vxi¦vogitSs£¢«vqPeIsi_P0i_u6qÄqwi#(dc,R± i_i)±°N¹P¹T®devxEogw`o¦vogitSsPvqp½Ivx£8ªgiqw.qPegiR¯zt`u6qBqPetSq¢£i«w.ogw`qet_¥iqPegi btS¦Xt`u6vXtSo&w`i6stSqw`s # .K: ;< ( - #=,|¢«evxueWvog8ªgu_i_tSoÁvxw`qPsw`vxuPhjwGw`qPevo~± dvÅ>egw`ogwN¥Isi6~ª¦XtSsPv­NtSqPvxw`o¹T|ªqtSotSovxw`qPsw`vxuµPhjwGw`qPevo~Ww`8yi6stSqw`sRPªgue&qPetSqÄqPegi~st`8vxi6orqRvoIqPegi8v¥i6sP~i6ogu_i$vxE¢Ri6v~erqi_Ár½¤qPegi$u_w>iTÂIu6vxi6o>q C R C ®d\w§w`qtSvo tSoi_8~iTyzsi_i6sP¥Gvo~»swGu_i_|«t`u_u_w`s8vo~¶qw +O Ö|¢RivhW0wi¾qPegi¾¯Æw`¦¦xw¢«vo~u_w`ogqPstSvorqw`o

ä\äàQ#RTSæR

Page 23: Image Classification Using a Variational Approach

D -)$.GT ('- T $ B % T$* B&.$T .$T

±Æt¹ 9 ^ 9 R 9 . & WtSog mpq |±K¹ 9 9 R 9 . 8|± uN¹ 9 R 9 P½>hji6qPsPvxutS¦tSog¾PqPsPvxu6qP¦½¾i_u6sit`Pvo~Iw`o ) SW o q ) ®Rw`og8vqPvxw`o´±Æt¹Rvxsi6¦XtSqi_?qwItSo¬vxw`qPsw`vxuhjw>w`qPevo~Áw`oegw`hjw`~i6ogi_w`ªgtSsit`|g¢«ev¦xiÁ±K¹¦xit`qw¬si_i6sP¥iÁi_8~i_¯ sw`h iT¸8u_i_v¥i&PhjwGw`qPevo~g® Rw`og8vqPvxw`o·± uN¹0i6sPhWvqqw¼t_¥w`vxvogqtSv¦vqPvxi_®bi6q * i$¨¸i_+®degiZtSswt`ue¾¢£iZt`w`q«qwhWvovhWv­i M E¢«vqPe;si_Pi_u6qqw #¬vxRt`i_;w`oqPegi - # $ $ $,¬hji6qPegw> +O|0 Ö®devxhji6qPegwG¼i6sPhWvqµqw¾vhW¦v©¯K½qPegiÁhWvovhWv­NtSqPvxw`oºwS¯ M |Är½"t_¥w`vx8vo~¡qPegi¾8v©ÂIu6ª¦qPvxi_Wsi6¦XtSqi_"qwÃqPegi;ogw`o´¦vogitSsPvqp½¶wS¯qPegiI N voº±°¹T®ÁdegiIt`PvxuWvxitvxZqw¾vo>qPsw>8ªgu_i;tSo»tSª8¸8v¦vXtSsP½À¥`tSsPvXtS¦xiI¢«evxue§¦xit`qwj¨gog?hWvovhWv­i6sµwS¯$±Æ¹qPesw`ª~e¬qPegisi_w`¦ªqPvxw`oÀwS¯B¦vogitSsµ N ³® Di6qt`PªhjiZqPetSq vxtI¯ ªogu6qPvxw`oÀ¥i6sPv©¯K½Gvo~u_w`og8vqPvxw`ogI±Æt¹qwÀ± uN¹T®mpoÃqPegi¤si_u6vqi_¼siT¯Æi6si6ogu_i_|0vqet`µ0i_i6oswN¥i6o?qPetSq utSo0iµ¢«sPvqPqi6o¬t`7

( , . !) . R o (.7, + ±°k¹

¢«egi6si§qPegiÀ¯ ªogu6qPvxw`o vx?i6sPv¥i_ ¯Ksw`h (.7, . U( (VU , &2(.7, , .4(VU , &2(.7,Á¢«vqPeU( , . ( ,;± i_i +O8|£ ¯Æw`sjhjw`si?i6qtSv¦x¹T|tSog vxIPqPsPvxu6qP¦½ u_w`o>¥iT¸0®.&vxÁtSotSª8¸8v¦vXtSsP½j¥tSsPvXtS¦xiµiT¨gogi_;vo¼±°k¹RtSogÁ¢«egwi$¥`tS¦ªgi.vxR~v¥i6o;r½jqPegi$¯Æw`sPhª¦Xt . . 9 R 9 ®«u_u_w`s8vo~&qw¤qPegietS¦©¯ rªt`8stSqPvxusi6~ª¦XtSsPv­NtSqPvxw`oD|¢£iZ~i6q7

M (#=, . M ^ (# * .7, ±°6>¹

¢«vqPe

M ^ (# * .2, .?J L N #(dc, J (dc, P R : c o * R JML ( .4(dc, N - #(dc, N R o (.)(dc, ,-, : co @ R* H/(#(dc, , : c 5 ±°¹

M ^ (# * .7,|utS¦¦xi_WqPegi .' T .$ |vxu_w`or¥iT¸W¢«vqPeWsi_P0i_u6qqw.qPegitSª8¸Gv¦vXtSsP½¥tSsPvXtS¦xi.±K¢«vqPe$#¾¨¸i_¹RtSog;vxi_u_i6¢«vxi rªt`8stSqPvxuµ¢«vqPe¾si_i_u6qqw #»±K¢«vqPe .«¨¸8i_¹T®«u_u_w`s8vo~qw»qPegi¼u_w`og8vqPvxw`ogw`o |qPegiÃ¥`tS¦ªgi_;wS¯ .¬tSsiÃutS¦xi_¯Ksw`h »qw |$tSogºvo·w`ªs¾ut`iqPegi ¯Kªogu6qPvxw`o¡¢Rijªgi¤vx)Pªgue§qPetSq . f ± i_iId\tS¦xi&¹T®&degiW¥tSsPvXtS¦xi .)hItSsPÅGZqPegi¦xwGutSqPvxw`o§wS¯8vxu_w`orqPvo>ªvqPvxi_¤Pªgue»qPetSq3.4(dc, ;v©¯'c§0i6¦xw`o~Zqw¬tSo§i_8~i&tSog .4(dc, fw`o¶egw`hjw`~i6ogi_w`ªgItSsit`® !sw`h ±°6>¹T|qPegi&hWvovhWv­NtSqPvxw`o wS¯ Mº¢«v¦¦R0ijqPstSogwi_¶vo>qwqPegiIhWvovhWv­NtSqPvxw`o´wS¯HM ^ ¢«vqPe»si_P0i_u6qqw(# * .7,®M ^ vxZvxi_u_i6¢«vxirªt`8stSqPvxu&vo #¿tSog

* éä * Ü

Page 24: Image Classification Using a Variational Approach

"!#$%'&()(%$*+,.- 8

( , 9 R 9 u_w`or¥iT¸8vqÖ½Su_w`og8vqPvxw`og¤±Æt¹qw± uN¹tSqPvx°¨i_dvÅGegw`ogw¥ r R ½i_]ogw

dbw`qtS¦ Ï tSsPvXtSqPvxw`o rk N N &R 9 ½i_]ogw

(i6hItSo ( u £¦ªsi Þ8 9 & 9 & & 9 ogw]½i_

i60i6sPq DitSer½ Þ,O ( f o R , && 9 ogw]½i_(.si_i6o Þ ( ( , , ! " 9 R 9 #$

½i_]½i_ ½Gi6sMJ>ªs°¯zt`u_i_ +O f o R f &

R&% & 9 ½i_]½i_d\tS¦xi JGw`hjii_8~iTyzsi_i6sP¥>vo~ ¯Kªogu6qPvxw`og®

u_w`o>¥iT¸jvo .]®BdegitSª8¸Gv¦vXtSsP½¤¥tSsPvXtS¦xi'.£vxsi6¦XtSqi_IqwqPegisi_i6ogu_iwS¯D8vxu_w`orqPvo>ªvqPvxi_|tSogH (#(dc, ,Ävx«vxi_u_i6¢«vxiZ¦vogitSs)±K¦xw>utS¦¦½Á¦vogitSs¹¢«vqPe?si_Pi_u6q«qwE#(dc,®!Di6qogw`qi

H ' !() ' (#(dc, , . E ) ' !() ' #(dc,ho k ' *(+) ' + *¢«vqPe ' !() ' tSog k ' !() ' Z î !sw`h ±°N¹$¢Ri¤ogw`qPvxu_i¤qPetSqqPegiiT¸8si_Pvxw`oÀwS¯BH ' !() ' ±Kvz®³i®qPegiµ¥tS¦ªgi_wS¯ ' !() ' tSog k ' !() ' ¹Ri6i6og$w`o¾qPegi¥`tS¦ªgiwS¯F#(dc,®&R½¡qPegi&ªgi;wS¯ Nª¦xi6s°y btS~stSo~i¬i rªtSqPvxw`oD|EqPegi&hWvovhWv­NtSqPvxw`o wS¯ M ^ ¢«vqPe"si_P0i_u6q)qw #¦xit`)qw¬t»±K¦xw>utS¦¦½g¹µ¦vogitSs N®degijhWvovhWv­NtSqPvxw`o´wS¯HM .¢«ev¦xi * C vxw`0i6stSqi_qPesw`ª~e?qPegitS¦~w`sPvqPeh

, C vovqPvXtS¦v­i #, C vovqPvXtS¦v­i *

, C si6itSq- M ^ (# * .7,Ä¢«vqPe #¨¸i_.4(dc, . C 1 R C 1 - C M ^ (# * .2,£¢«vqPe .¨¸8i_+|vz®³i®w`¦¥iqPegiZ NY#(dc, * R :;</. .4(dc, - #(dc,02o ' !() ' #(dc, .%J (dc,I k ' !() ', C ªorqPv¦bu_w`o>¥i6sP~i6ogu_i)w`o$#

, C i_u6sit`i *degi¤ N si_Pª¦qPvo~Á¯ sw`h C M ^ (# * .2,«¢«vqPe ..¨¸i_Ãvxµw`¦¥i_êgPvo~;tÁu_w`o`°ª~rtSqi~st]y8vxi6o>q$tS¦~w`sPvqPeh®

É &R8_3Kb+3 Í 1 degi.ev~egi_Pq«qPegi.o>ªh0i6swS¯\¦xwGw`g«w`oÁqPegiµi_u6sit`Pvo~jwS¯ * |GqPegiµu6¦xwi_PqqPegi.¥`tS¦ªgi_«wS¯\qPegi

ä\äàQ#RTSæR

Page 25: Image Classification Using a Variational Approach

-)$.GT ('- T $ B % T$* B&.$T .$T

w`¦ªqPvxw`o´qw¬qPegiÁhWvovhIt 3 @ wS¯ H ® !gw`s¤st`u6qPvxutS¦sit`w`og± u_w`hWªqtSqPvxw`otS¦qPvhjiN¹¤tSog¯ sw`h iT¸80i6sPvhji6orqtS¦rªtS¦vqtSqPv¥i.si_Pª¦qN|r¢Ri$Pqw`ÁqPegiu_w`o>¥i6sP~i6ogu_iµiT¯Æw`si * sit`uegi_R­i6sw®degi6siT¯Æw`si|qPegi¥tS¦ªgi_wS¯BqPegiZw`¦ªqPvxw`oÃtSsiogw`q$tS¦¢Rt½G$PqPsPvxu6qP¦½¾i rªtS¦+qw¬±Kªq¥i6sP½¾u6¦xwiqwr¹¤qPegi;¦XtSi6¦x 3 @ ®´mpo w`si6sIqw¡tS¦¢t_½8I~i6qIt¼0i6s°¯Æi_u6qW¦XtSi6¦¦v­i_¿vhItS~i¾¢£i¾hItSÅiqPegi¯Æw`¦¦xw¢«vo~;w`0i6stSqPvxw`oÃw`o¬qPegiw`¦ªqPvxw`o¼wS¯qPegisi6¥Gvxw`ªg.tS¦~w`sPvqPeh R¯Æw`s$it`ueÃvqi P wS¯qPegiw`¦ªqPvxw`oD|v©¯ Q @ e #( P , e Q @ =& ± i_i!v~g®>¹£qPegi¦XtS0i6¦bwS¯ P vxi6q«qw 3 @ ® 'I#0E$&# ° · \$')#

Ç¡isi_i6o>qZiT¸G0i6sPvhji6orqtS¦si_ª¦q.w`o¼0w`qPeÀP½>o>qPegi6qPvxuWtSogÃtSqi6¦¦vqi)vhItS~i_N®Ç¡iWtS¦xwu_w`hWtSsiqPegiZsi_Pª¦q$sw¥Gvxi_¾¯ sw`hqPegi¥`tSsPvXtSqPvxw`otS¦hjwGi6¦D¢«vqPe?qPegi)w`ogi_$w`qtSvogi_?r½ªgvo~?t;Pqw>uet`PqPvxu¤hjwGi6¦B¯ w`su6¦Xt`Pv©¨utSqPvxw`oD®¤Ç¡i¨gsqZi_u6sPvi¤qPevxPqw>uet`PqPvxu¤hjwGi6¦z®degi6oD|B¢Ri&u_w`hWtSsiÁqPegi&si_Pª¦q¢£iÁw`qtSvo"r½¡qPegi&hWvovhWv­NtSqPvxw`o´wS¯±Æ¹µªgPvo~ÃqPegiÁiTyqi6sPhWvovxqPvxutS¦~w`sPvqPeh si_i6orqi_ÃvoÃi_u6qPvxw`o?g® &qwjqPegwi)w`qtSvogi_?¢«vqPeÃqPegiPqwGuet`qPvxuhjwGi6¦z®

8;< 3 !n ? 8 <4 F <4 : 4 8;7degiPqw>uet`PqPvxuhjwGi6¦ªgi_¤egi6sit]¯Kqi6sÄvxBt`i_jw`oWqPegi ( tSsPÅw¥W$tSogw`h !vxi6¦x± ( #!E¹qPegi_w`sP½|EtSog»hji6orqPvxw`ogi_¶¯Æw`svogqtSogu_iIvo$D8|£8|£Ö®?devx)hjwGi6¦£swN¥>vxi_WtSo¶vhItS~iu6¦Xt`Pv©¨utSqPvxw`o´r½´t`Pv~ovo~§tæXtS0i6¦«qw¼it`ue¿Pvqi®¶degi;hItSvo¿8v +i6si6ogu_i_W0i6qÖ¢£i_i6o¿qPegiqw>uet`PqPvxuµhjw>i6¦btSog;qPegiµ¥tSsPvXtSqPvxw`otS¦Dw`ogiZtSsiB¨gsPqt¤8vxu6si6qi.¯ stShji6¢£w`sPÅÁvogPqit`wS¯tu_w`o>qPvorªgw`ªgw`ogi|8i_u_w`ogÁqPegi$ªoÅ>ogwN¢«o¾vxRtSo;vhItS~iµwS¯D¦XtS0i6¦xtSogÁogw`qRqPegi.vorqi6ogPvqp½|gtSog¨gotS¦¦½;qPegiZtS¦~w`sPvqPeh ªgi_;qwWhWvovhWv­iZqPegii6ogi6sP~½;vxqw>uet`PqPvxu®bi6q tSog 0i¤iT¨gogi_§t`µ0iT¯ w`si® Di6q \ 0i¤qPegiWi6qZwS¯u6¦vrªgi_ZªgueÀqPetSq¢£iIiT¨gogit&¨gsqµw`si6s ( !¿hjwGi6¦±Kvz®³i®¢£iW¢«v¦¦qtSÅiWvo>qwt`u_u_w`ªorq¤t;PvqiWtSogÀvq.¯ w`ªsogitSsi_qogi6v~e>w`ªsT¹T®¬ ~¦xw`tS¦8vxu6si6qi&¦XtSi6¦¦vo~ %(u_w`ogPvxPqwS¯$t`Pv~ovo~Àt?¦XtSi6¦ % Z qwit`ue¬Pvqi P Z ® &R½ÁqPegiªgiZwS¯ &«t_½i_$qPegi_w`si6h|gqPegiZi_PqPvhItSqi_?¦XtSi6¦¦vo~%¿vxw`qtSvogi_qPesw`ª~e?qPegiµ¯ w`¦¦xwN¢«vo~IsPª¦xi

% . ( % , #25. E( %Y, E( %',. ( E( %Y, E( %', , * ±°8¹

¢«vqPe i6ogw`qPvo~¤qPegiw`gi6sP¥i_;gtSqt8|8tSogI>½Iw`hWvqPvo~ ( ,B¢«evxue¾w>i_£ogw`qi6i6og&w`o%$® E( %',vx)Pªwi_¼qw¾iWªgue§qPetSqE( %Y, . E( , % ,¤±K¢«evqijvor¥tSsPvXtSorqogw`vxiN¹T|¢«egi6si % vxqPegi¦XtS0i6¦\t`Pv~ogi_;qwWqPegivqi P |gtSog;¢«vqPe

* éä * Ü

Page 26: Image Classification Using a Variational Approach

"!#$%'&()(%$*+,.- k

E( , % , . f E 5 ( ( 3 , R

E5 R , 5 ±°+O¹

degiZsPvxw`shjw>i6¦+vx$tW\w`qPqsi6~ª¦XtSsPv­NtSqPvxw`o¬PªgueqPetSq7E( %', . ( G ( % )* % ,-,¢«vqPe (_ * .2, . f v©¯F_ . .o f v©¯F_ . . ±°¹

¢«egi6si G vx;qPegi¼qi6hW0i6stSqPªsiÀtSstShji6qi6s® mpo·t`u_u_w`sgtSogu_i¡¢«vqPe±°8¹T|$qPegiÀi_qPvhItSqi_¦XtS0i6¦¦vo~ %¿vxw`qtSvogi_¾r½¾w`¦¥Gvo~

% . % ( %', * tSog %vxt¤~¦xw`tS¦\i6ogi6sP~½;ªgueqPetSq7% ( %', . fG (

A & E 5 o ( , 3 , R

E5 R o ( % )* % ,-, 5 ±°¹

.Pvo~t ( $± ( t]¸8vhªh t;wPqi6sPvxw`sPv3¹u6sPvqi6sPvxw`oD|bqPegijw`qPvhWv­NtSqPvxw`o¡vxhIt`ijr½¼Pvh¤yª¦XtSqi_¶tSoogitS¦vo~?¢«vqPe»t ( i6qPsw`0w`¦vx)tS¦~w`sPvqPeh®;mpo§t`8vqPvxw`o§qw¾qPegiIorªh0i6stSog¡qPegitSstShji6qi6swS¯gqPegiEu6¦Xt`i_|NqPegiÄsi rªvsi_voªqBtSsiDqPegiEvovqPvXtS¦rqi6hW0i6stSqPªsi|`qPegiÄi_u6sit`Pyvo~j¯Æt`u6qw`s$~w¥i6sPovo~ÁqPegiZqi6hW0i6stSqPªsi)i_u_i6o>q|tSog?qPegiZ¥tS¦ªgiZwS¯ ®£degiqi6hWi6stSqPªsihIt½i_u6sit`i«P¦v~erqP¦½®Bdegi¥tS¦ªgiRwS¯ ¾i60i6ogw`oWqPegi£¦xi6¥i6¦gwS¯egw`hjw`~i6ogi6vqÖ½W¢Ri«iT¸G0i_u6q®degiµhWvovhWv­NtSqPvxw`oÃwS¯±°¹qPstSv~erqP¦½;¦xit`«qwWt¦XtSi6¦¦v­i_vhItS~i|¢«egi6sit`«vo¾qPegi¥tSsPv©ytSqPvxw`otS¦Z¯KstShji6¢Rw`sPź¢Ri¼sw`~si_Pv¥i6¦½±Æt` * C ¹&~i6qÃt§¦XtS0i6¦¦vo~g® mpo w`si6s¾qw"~i6qu6¦Xt`Pv©¨utSqPvxw`o¡si_Pª¦q)8vsi_u6qP¦½¼u_w`hWtSstS¦xi&qw;qPegijw`ogi_sw¥Gvxi_À¯ sw`h qPegi¤¥`tSsPvXtSqPvxw`otS¦hjwGi6¦z|B¢£iÁvhW0wi&qPetSq)qPegiÁ¦XtS0i6¦xi% qtSÅi&qPegi6vs¥tS¦ªgi_)vo R 3 @ g @BA & ?|3 @ 0i6vo~ÃqPegihjitSo¬wS¯qPegiu6¦Xt`i \ @ ® 8!3 Q: 8 = 7O3#8;< 7 <Ç¡iet_¥iqi_qi_WqPegi«sw`0wi_¤¥`tSsPvXtSqPvxw`otS¦hjw>i6¦w`ojw`qPej½>o>qPegi6qPvxu$tSogWsitS¦tSqi6¦¦vqi~si6½Ã¦xi6¥i6¦EvhItS~i_®&degiWsi_Pª¦q)w`o¡P½>o>qPegi6qPvxujvhItS~i_tSsiWhjitSovo~`¯Kª¦¦Äqw¾¥`tS¦vxgtSqijqPegihjwGi6¦\tSog¾qwÁtSotS¦½>­iZqPegiZ8v 0i6si6ogu_i_$¢«vqPe?qPegiPqw>uet`PqPvxuhjwGi6¦\i_u6sPv0i_¾vo?i_u6qPvxw`o8®®Ä¦¦+si_Pª¦q¢£i6siZw`qtSvogi_¬w`o¬t;88 (à u_w`hWªqi6s®ÁÉ § mpo¾qPegiµ¯ w`¦¦xwN¢«vo~ Ï µ¢«v¦¦Di6ogw`qiqPegiµ¥`tSsPvXtSqPvxw`otS¦u6¦Xt`Pv©¨utSqPvxw`o¬tSog J

ä\äàQ#RTSæR

Page 27: Image Classification Using a Variational Approach

S -)$.GT ('- T $ B % T$* B&.$T .$T

qPegiqw>uet`PqPvxuZw`ogi®

mpo¡w`si6sZqw;sPªo¡qPegijsi6¥>vxw`ªg¥`tSsPvXtSqPvxw`otS¦£tS¦~w`sPvqPeh|\¢£ijogi_i_ÀvoªqtSstShji6qi6sqPetSq¢RiµiT¸G0i6sPvhji6orqtS¦¦½¾t`Sîªgq®BÇ¡i¨gsq«t`Pªhji.qPetSqRtSstShji6qi6s 3| 5 tSogÁqPegi.orªh0i6s $wS¯Eu6¦Xt`i_$tSsiZÅGogw¢«o"±Ksi6¦vhWvotSsP½¬i_PqPvhItSqPvxw`oÃw`stSstShji6qi6s.~v¥i6o?>½?tSo¬iT¸8i6sPq¹T®mpot`8vqPvxw`oÁqwZqPegi¥`tS¦ªgi_RwS¯ | @Wvo¼±Æ¹B¢£i$vorqPswG8ªgu_iµtst`u6qPvxutS¦tSstShji6qi6s ¢«evxuetSvhjtSqsi_utS¦vo~IqPegi~st`8vxi6orq.hjw>8ª¦ªg7 C ®£devxtSstShji6qi6s.utSo¬ii_i6o¼t`$t¤qPesi_egw`¦x¨¸8vo~WqPegi¥tS¦ªgiµ¯ sw`h ¢«evxueÃt¤8vxu_w`orqPvorªvqp½vxt`vhWv¦XtSqi_¾qwjtSo?i_8~i®Ç¡iqPegi6o¬PqtSsPq¯ sw`h * .lf tSogÁi6¦xi_u6q£vqÄP0i_i_IwS¯bi_u6sit`Pvo~g®BÇ¡ivovqPvXtS¦v­iI#;qwZqPegi$gtSqti6q J ® !gsw`hiT¸80i6sPvhji6orq|r¢£iet¥iogw`qPvxu_i_IqPetSqE¢£iogi_i_Wqwvogu6sit`i«qPegi~¦xw`tS¦si6~ª¦XtSsPv­NtSqPvxw`oÁqi6sPh±Ksi_qw`stSqPvxw`o¶tSog¡¦xi6¥i6¦Ru_w`ogPqPstSvorq¹Zvz®³i®&qPegiW¦Xt`PqZqp¢£w¬qi6sPhj)wS¯Z±Æ¹µ¢«ev¦xi * C ;vo»w`s°yi6sqwW~i6q0i6qPqi6sorªhji6sPvxutS¦PqtSv¦vqPvxi_µtSog¾qwIt_¥w`vx?qPegiZsitS0itSsi6ogu_iwS¯ogw`vxit`qPegisi_qw`stSqPvxw`o¬qi6sPh ¥`tSovxPegi_N®£Çºegi6o?vq$vxogw`q$hji6orqPvxw`ogi_+|qPegii6¦xi_u6qi_ ¯ ªogu6qPvxw`o?vxqPegiw`ogiZsw`wi_;>½ (µi6hItSo¬tSog ( u £¦ªsiI± i_idtS¦xi)WtSog Þ K¹T®

7 5 / Ê 1 += É +3K 3 § É degiR¨gsqogw`vxP½½>o>qPegi6qPvxuvhItS~iu_w`ogPvxi6si_WvxwS¯+Pv­i .v©¸i6¦x± 6uegi_uÅ vhItS~iN¹tSog¤u_w`o>qtSvog¯Æw`ªsBu6¦Xt`i_B¢«egwitSstShji6qi6sÄtSsi~v¥i6oWvoWd\tS¦xik8®Bdegi¢«evqi#(tSªgPvXtSoogw`vxi«vorqPswG8ªgu_i_jvx£PªguejqPetSq£qPegiMJ #. vxD& ±K¥tSsPvXtSogu_istSqPvxwr¹T®degi`yz¢£i6¦¦0w`qi6orqPvXtS¦Ç vo·±Æ¹µvx)v¦¦ªgPqPstSqi_"w`o!\v~g® OG® !v~g®¬?si_i6o>q)qPegiÁ8v +i6si6orq¤w`ªqPªqvhItS~i_¢Riw`qtSvo¯Ksw`h qPegiZhWvovhWv­NtSqPvxw`oÃwS¯±°¹ EqPegisi_qw`si_¾vhItS~i&± J #.µ,OG®ÞW&¹T|qPegiZhItSwS¯8vxu_w`orqPvorªvqPvxi_¬±KvhItS~i&wS¯$tSª8¸8v¦vXtSsP½¡¥tSsPvXtS¦xi .4¤iT¸Gqi6sPotS¦R0w`ªoggtSsPvxi_¹tSog§qPegi Ï ¢«evxueÀvxµu_w`hWtSsi_ÀqwÁqPegi J .®0m֯Ģ£i¤u_w`hWªqi¤qPegi¤0i6su_i6orqtS~iWwS¯ÄhWvxu6¦Xt`Pv©¨i_¼v©¸i6¦x|qPegi Ï ¦xit`Zqw D8®Þk tSog¼qPegiJ qw D8®Á®WdevxZP¦v~e>q8v +i6si6ogu_iWhItSvo¦½Ãu_w`hji_¯ sw`h qPegiWPhItS¦¦Bvxi_u_iWwS¯£¢«evqi¤ >ªtSsiWw¥i6sP¦XtS0i_¼r½ÃqPegiqPsPvXtSo~¦xiItSogÃqPegiWu6vsu6¦xi± i_iqPegiWu6¦Xt`Pv©¨utSqPvxw`o¡si_Pª¦qZw`o !v~g®W¹T®WdegiWu_w`hWªqtSqPvxw`otS¦ÄqPvhjijvxW¾i_u_w`ogµ¯Æw`sµqPegiÏ tSog O`;i_u_w`og.¯Æw`sµqPegi J .® !\v~g®¤ÁPegw¢ZqPegi J #.tSogÃorªh)i6s)wS¯£hWvxu6¦Xt`v©¨i_v©¸i6¦x«i6¥w`¦ªqPvxw`o¢«ev¦xi * i_u6sit`i_®BÇ¡iutSoi6qi_u6q«qPegiPqi6¾w`o * ¯ sw`h ¢«evxueqPegi J #$tSog&qPegio>ªh0i6s«wS¯\i6sPsw`sRtSsi.PqtS¦xi¤±KqPevx£utSo;i6qi6sPhWvogi.qPegi$qw`vo~)¥tS¦ªgi$wS¯ * ¹T®E$o!v~g®ÀD?¢£i&Pegw¢ qPegiIvogªgi6ogu_iÁwS¯qPegijÅGvog§wS¯ ¯Kªogu6qPvxw`o"w`o Ï ²si_Pª¦qÁ± i_iId\tS¦xi¹T® !gw`s$t¤dvÅGegw`ogw¥;si6~ª¦XtSsPv­NtSqPvxw`oD|i_8~i_.tSsiw¥i6sPhjwGw`qPegi_+®£ÇvqPe?tWu_w`o>¥iT¸ |gqPegi6siqPv¦¦btSsihItSor½;hWvxu6¦Xt`v©¨i_¾v©¸8i6¦xw`oqPegi0w`ªortSsPvxi_® &Ri_q«si_Pª¦q$tSsiµswN¥>vxi_?¢«vqPeqPegiµªgiwS¯\ogw`ogu_w`or¥iT¸ ¯ ªogu6qPvxw`og|i6¥i6ov©¯qPegiµqPegi_w`sPvxutS¦bw`ªog8ogi_wS¯BqPegiµhjwGi6¦+vxtSow`0i6o¾sw`¦xi6h®

* éä * Ü

Page 28: Image Classification Using a Variational Approach

"!#$%'&()(%$*+,.-

u6¦Xt` hjitSo 3 @ PqtSoggtSsi6¥GvXtSqPvxw`o 5 @j±K¦Xt`uÅ; rªtSsi_¹ 8® D 8®Þ¾±KqPsPvXtSo~¦xiN¹ 8® D 8®Þk¾± 8vxPÅg¹ 6r8® D 8®Þ?±K¢«evqiZ rªtSsi_¹ +O`8® D 8®Þ

dtS¦xi&k ZBtSstShji6qi6s¤wS¯RqPegij?u6¦Xt`i_µ¯Æw`s (puegi_uÅ"*¤vhItS~i®ÁdegijPqtSoggtSs§i6¥>vXtSqPvxw`o»vxqPegitShjiµ¯ w`sit`ue?u6¦Xt`|tSog;vx«qPegiZw`ogiZwS¯BqPegiµ¢«evqi%(tSªgvXtSoogw`vxi®

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

80 100 120 140 160 180x

W potential - "check" image

!v~ªsi OS£w`qi6orqPvXtS¦ H ¯Æw`s 6uegi_uÅ vhItS~i® J>vogu_iZqPegi)PqtSoggtSsÃi6¥>vXtSqPvxw`o¬vxqPegi)tShji¯Æw`sRtS¦¦0qPegi.u6¦Xt`i_|>qPegi$¢£i6¦¦xet_¥i.qPegi.tShji$¢«vx8qPeD®Edegi0itSÅ8£i_i6h qw)i$ogw`ogPhjwGw`qPeD|ªqqPevx«vx8ªgiµqwWqPegi¦xw`q$8vxu6si6qPv­NtSqPvxw`oD®

ä\äàQ#RTSæR

Page 29: Image Classification Using a Variational Approach

8 -)$.GT ('- T $ B % T$* B&.$T .$T

d\w`sPv~votS¦ ogw`vxP½À± J #.DW&¹

NJGd.$d«m° #si_Pqw`si_§± J #.µ,OG®Þ`&¹ hItS¬wS¯B8vxu_w`o>qPvorªvqPvxi_

D J#JGm !Bm Rd«mP #Ï J

!v~ªsi¡ ´$i_Pª¦q¾w`o (puegi_uÅ *ÀvhItS~i® Ï tSsPvXtSqPvxw`otS¦tSstShji6qi6s7 . f | . f |@ . 56f | OÀvqi6stSqPvxw`og;w`o * ® J>qw>uet`PqPvxuÃtSstShji6qi6s7 . Eb56f |DDD§vqi6stSqPvxw`og¾w`svhª¦XtSqi_¬tSoogitS¦vo~g® Rw`hWªqtSqPvxw`otS¦bqPvhji¥tSsPvXtSqPvxw`otS¦ `®|Pqw>uet`PqPvxu O``®

* éä * Ü

Page 30: Image Classification Using a Variational Approach

"!#$%'&()(%$*+,.- ,O

20

25

30

35

40

45

50

55

60

2 4 6 8 10 12 14 16 18 20

num

ber o

f erro

rs

iteration number (epsilon)

Error evolution: check image

’PlotErrors’

20

22

24

26

28

30

2 4 6 8 10 12 14 16 18 20

SNR

val

ue in

dB

(var

ianc

es ra

tio)

iteration number (epsilon)

SNR evolution: check image

’PlotSnr’

!v~ªsi. #$ªh0i6swS¯DhWvxu6¦Xt`Pv©¨i_Iv©¸8i6¦xµ±Kqw`¹ÄtSog J #$º¥tS¦ªgi)±K0w`qPqw`h&¹¢«vqPeÁsi_P0i_u6qqw¤qPegiZorªh0i6s$wS¯Bvqi6stSqPvxw`ogw`o * ¯Æw`s 6uegi_uÅ vhItS~i®

ä\äàQ#RTSæR

Page 31: Image Classification Using a Variational Approach

-)$.GT ('- T $ B % T$* B&.$T .$T

( , . R u_w`o>¥iT¸»± dvÅ>egw`ogwN¥¹

( , . ( ( , , Eu_w`o>¥iT¸¡± (µsi_i6o¹

( , . 9 & 9 Bogw`ogu_w`or¥iT¸»± (µi6hItSo ( u £¦ªsiN¹!v~ªsi¼D £¦Xt`Pv©¨utSqPvxw`o wS¯ (puegi_uÅ"*ÁvhItS~iÁ¢«vqPe 8v +i6si6orq ¯Kªogu6qPvxw`o ± i_i;d\tSD®»¹T®#.w`ogu_w`or¥iT¸¾¯ ªogu6qPvxw`og$sw¥GvxiZ0i6qPqi6ssi_Pª¦q$i6¥i6o?v©¯qPegi6si)tSsiogwIqPegi_w`sPvxutS¦bswGwS¯Eu_w`o8yu_i6sPovo~¼qPegiÁw`¦ªqPvxw`o"vo»qPegi&ogw`ogu_w`or¥iT¸"ut`i® Rw`or¥iT¸»¯Kªogu6qPvxw`og¦xit`»qw¬wN¥i6sPhjw>w`qPesi_ª¦q7B¢£i~i6q$gtShItS~i_?i_8~i_®

* éä * Ü

Page 32: Image Classification Using a Variational Approach

"!#$%'&()(%$*+,.-

degiµi_u_w`ogÁvhItS~iµvxwS¯v­iµ8 8)v©¸8i6¦xZ± T~8s.vhItS~iN¹T|gtSogÁqPegi.ogw`vxP½&¥i6sPvxw`o;vxªgue;qPetSq£qPegi J #. vxi rªtS¦qw;D&µ®8devxP½>o>qPegi6qPvxu.vhItS~i.u_w`hji_R¯ sw`h qPegi.gtSqtSt`iwS¯$qPegi (µ m*JGm J|tSog¶¢Rt`Wu6sitSqi_"qwÃqi_PqWi6~hji6orqtSqPvxw`oºtSog´u6¦Xt`Pv©¨utSqPvxw`o´hjwGi6¦x®devxjvhItS~i;vogu6¦ªgi_jqÖ¢RwÀhItSvo¿8v©ÂIu6ª¦qPvxi_®´degiÁ¨gsqjw`ogi;vxWqPegi;vsPsi6~ª¦XtSsPvqÖ½ wS¯.qPegietSiZw`oÃ0w`qPqw`hsPv~erq.etSogPvxi®degii_u_w`og¬w`ogiZvx.8ªgiZqwjqPegi)~si6½¾¦xi6¥i6¦b~st`gtSqPvxw`ow`o¼qw`¼sPv~erqetSogvxi®ZÇ¡isi_i6orqsi_ª¦q.¯ w`sµqÖ¢Rw¾8v 0i6si6o>qi6qµwS¯Äo>ªh0i6s)tSogÃt]ystShji6qi6s$wS¯Bu6¦Xt`i_®

!vsPq|¢£i¼et¥iÀi6¦xi_u6qi_ºqPesi_iÀu6¦Xt`i_Á¢«egwi¼tSstShji6qi6s?tSsi¼PªhWhItSsPv­i_ vo·d\tS¦xig®degiWu_w`sPsi_P0w`og8vo~?k]yz¢£i6¦¦E0w`qi6orqPvXtS¦ÄvxPegw¢«o§w`o !v~g®Á DqPegi¤¢£i6¦¦x)tSsi¤¢«vxijtSogu6¦xitSsP¦½;i6tSstSqi_+®£$o !\v~g®«8|8¢RiZutSoi_iµqPetSq«qPegi Ï hItSotS~i_P¦v~erqP¦½;0i6qPqi6s«qPetSoqPegi J wGi_qPegi¤~si6½Ã¦xi6¥i6¦E~st`gtSqPvxw`o± uT¯P®¤qw`ÀsPv~erqZetSogvxiN¹T|\tSogÀ¦xit`qw;¦xi_µvs°ysi6~ª¦XtSs«0w`ªoggtSsP½ÁqPetSoÁqPegiw`ogiµwS¯ J .® (.¦xw`tS¦¦½| Ï ¿si_Pª¦qtSsiµu6¦xwi.qw¤qPegi J ¿w`ogi_N|ªqqPegiu_w`hWªqtSqPvxw`otS¦\qPvhjivxr¤i_u_w`og«¯Æw`s«qPegi Ï ·tSog?Wi_u_w`og¯ w`s«qPegi J .®

degi6oD|0¢Ri¤et_¥i¤uegwi6o¡i6v~erqZu6¦Xt`i_W± i_i¤dtS¦xi¤¹T® !sw`hqPegiW]yz¢£i6¦¦0w`qi6orqPvXtS¦EPegwN¢«ow`o !v~g®Wk8|0¢£i¤si6hItSsPŬqPetSqu6¦Xt`i_.orªh)i6s 88| OÁtSogÀ;tSsiogitSsP¦½?ªog8vxPqPvo~ªvxPetS¦xi®degiu6¦Xt`Pv©¨utSqPvxw`o?si_ª¦qtSsisi_i6o>qi_?w`o !v~g®6g®EÇ¡iZutSoogw`qPvxu_iZqPetSq¯ w`s«qPevx$i6¦v©yutSqiu_w`o8¨g~ªstSqPvxw`oD|>qPegi¥`tSsPvXtSqPvxw`otS¦hjw>i6¦gvxÄhªgueIhjw`siiTÂIu6vxi6o>qµ±ÆtSog¤¯zt`Pqi6s¹qPetSoIqPegiqw>uet`PqPvxuhjwGi6¦z®degi J ´vxRhIt`i.wS¯t¦xw`qwS¯D¯ztS¦xisi6~vxw`og|G¢«evxueÁvx«tu_w`ogi rªgi6ogu_iµwS¯qPegi.sw¸GvhWvqp½;wS¯qPegiu6¦Xt`i_®BmÖ¯\¢Riµsi_8ªgu_i |8¢«evxue¾vxqPegiµ¢Ri6v~erqwS¯B\w`qPq«si6~ª¦XtSsPv­Nt]yqPvxw`oD|G¢Ri~i6qR¦xi_£¯ÆtS¦xisi6~vxw`og|GªqR¢£i.~i6qtZ¦xw`qRwS¯\vxw`¦XtSqi_IhWvxu6¦Xt`Pv©¨i_Áv©¸8i6¦xµ± 8ªgi.qwqPegiogw`vxiN¹T®«degi Ï i6sPsw`s.tSsihItSvo¦½PvqPªtSqi_?w`o¬w`ªoggtSsPvxi_µwS¯qPegiZvsPsi6~ª¦XtSsµPetS0iw`o;0w`qPqw`h ¦xiT¯Kq|GªqqPevxvxqPegi.i_qRqPst`iTyÖw Bv©¯\¢Riµ¢RtSo>q«qw¤si6hjw¥iµqPevxhWvxu6¦Xt`v©¨i_v©¸i6¦x|b¢£ijPetS¦¦£gtShItS~ijqPegiji_8~i_®&degiju_w`hWªqtSqPvxw`otS¦RqPvhjiWvxID8ji_u_w`og¯Æw`sqPegiÏ tSog 8,OGi_u_w`ogB¯Æw`sEqPegiMJ .®G$o !v~g®R¢£iPegwN¢ qPegii6¥w`¦ªqPvxw`oÁwS¯0i6sPsw`sRtSogJ #$¢«ev¦xi * i_u6sit`i_N® #$w`qiÁqPegiÁ¥tSsPvXtSqPvxw`o"wS¯.qPegi J #$ i6¥w`¦ªqPvxw`o´0i6qÖ¢Ri_i6o´vqi6stSqPvxw`ogjkDtSog?)D¤0iT¯ w`siZqtSv¦vqÖ½®«devxu_w`sPsi_w`og$qwjqPegistSo~iZ¯ w`s¢«evxueÃqPegiZsi_Pqw`si_?vhItS~i0i_u_w`hji_tWvxi_u_i6¢«vxi¤u_w`ogPqtSo>qµvhItS~iÁ±KqPegii6qµwS¯£t`8hWvxPv¦xi)¥`tS¦ªgi_µwS¯ #(dc,u_w`or¥i6sP~i_qwjqPegiZi6qwS¯B¦XtSi6¦xT¹T® J>vogu_iqPegi J #$ vxu_w`hWªqi_?¯Ksw`h qPegiu_w`hWtSsPvxw`o?¢«vqPeÃw`sPv~votS¦vhItS~i¬±K~si6½¬¦xi6¥i6¦x¹T|DqPegiJ #$¥`tS¦ªgijtSqZPqtSv¦vqp½¼vxµogw`q)tS¦¢Rt½GqPegi¤hIt]¸Gvh)ªh®NEP0iTyu6vXtS¦¦½Iegi6si6voD|Gvogu_i¢Ri.w)ogw`qsi6qPsPvxi6¥itS¦¦qPegi$¦xi6¥i6¦xwS¯bqPegi~st`gtSqPvxw`o¾wS¯b~si6½I¦xi6¥i6¦x7BvoqPegiPqtSv¦vqÖ½w`hItSvo¬i6~voovo~&t]¯ qi6svqi6stSqPvxw`o¼kD8| #Ãvx«vxi_u_i6¢«vxi)u_w`ogPqtSorq$¢«vqPe?i6v~erqt`8hWvxPv¦xi¥tS¦ªgi_|>¢«egi6sit`RqPegi$w`sPv~votS¦0ªoogw`vxP½jvhItS~i$et`«kZ~si6½W¦xi6¥i6¦x± i_i !v~g®8tSog !\v~g®«+OI¹T®

ä\äàQ#RTSæR

Page 33: Image Classification Using a Variational Approach

kD -)$.GT ('- T $ B % T$* B&.$T .$T

u6¦Xt` hjitSo 3 @ PqtSoggtSsi6¥GvXtSqPvxw`o 5 @ DD8® D 8®Þ8 8® D 8®Þ8k 8D8® D 8®Þ8

dtS¦xiZ£tSstShji6qi6sµwS¯qPegi)kWu6¦Xt`i_¯Æw`s1(Ö~8s *vhItS~i®Rdegi)PqtSoggtSs?i6¥GvXtSqPvxw`o?vxqPegitShji.¯ w`sit`ue¬u6¦Xt`|gtSogvx«qPegiw`ogiZwS¯BqPegi¢«evqi%(tSªgvXtSo¾ogw`vxi®

0

10

20

30

40

50

60

70

100 120 140 160 180x

W potential - gdr image - 3 classes

!v~ªsiB\w`qi6o>qPvXtS¦ H ¯ w`s T~8svhItS~i$¢«vqPeÁk)u6¦Xt`i_® J>vogu_i$qPegi$PqtSoggtSsÁi6¥GvXtSqPvxw`ovxqPegitShjiµ¯Æw`stS¦¦DqPegiu6¦Xt`i_|8qPegiZ¢£i6¦¦xet¥iqPegiZtShjiµ¢«vx8qPeD®

* éä * Ü

Page 34: Image Classification Using a Variational Approach

"!#$%'&()(%$*+,.- k8

ogw`vxP½;gtSqt± J #$ D¤&¹ 0i6s°¯Æi_u6qu6¦Xt`Pv©¨utSqPvxw`o

Ï J

!v~ªsiZ i_ª¦qB¯ w`sRt.ogw`vxP½WP½>o>qPegi6qPvxu(Ö~8s *«vhItS~iu_w`orqtSvovo~Wku6¦Xt`i_® Ï tSsPvXtSqPvxw`otS¦tSstShji6qi6s) . | . f 8|!@ . 5 |D8vqi6stSqPvxw`ogw`o * ® J>qw>uet`PqPvxuµtSstShji6qi6s7 . 5 8|\DDD¤vqi6stSqPvxw`og«¯Æw`sPvh)ª¦XtSqi_¬tSoogitS¦vo~g® Rw`hWªqtSqPvxw`otS¦\qPvhjiÄ¥`tSsPvXtSqPvxw`otS¦ $r`®|qw>uet`PqPvxu µ`®

ä\äàQ#RTSæR

Page 35: Image Classification Using a Variational Approach

k -)$.GT ('- T $ B % T$* B&.$T .$T

u6¦Xt` hjitSo 3 @ PqtSoggtSsi6¥GvXtSqPvxw`o 5 @ DD8® D 8®Þ8 8® D 8®Þ8k k8® D 8®Þ8 6r8® D 8®Þ8 6r8® D 8®Þ88 88® D 8®Þ8O 8D8® D 8®Þ8 8k8® D 8®Þ8

dtS¦xi) £tSstShji6qi6sµwS¯qPegi)Wu6¦Xt`i_¯Æw`s1(Ö~8s *vhItS~i®Rdegi)PqtSoggtSs?i6¥GvXtSqPvxw`o?vxqPegitShji.¯ w`sit`ue¬u6¦Xt`|gtSogvx«qPegiw`ogiZwS¯BqPegi¢«evqi%(tSªgvXtSo¾ogw`vxi®

0

10

20

30

40

50

y

80 100 120 140 160 180x

W potential - gdr image - 8 classes

!v~ªsiÃk W\w`qi6o>qPvXtS¦MH ¯Æw`s T~8sIvhItS~iI¢«vqPe ?u6¦Xt`i_® Di6q)ogw`qi&qPetSqqPegi&u6¦Xt`i_¢«egwiµhjitSo¥`tS¦ªgi_.tSsiW88® D8|b8D8® DjtSogÀ8k8® D8|tSsiµogitSsP¦½;ªog8vxPqPvo~erªvxPetS¦xi®

* éä * Ü

Page 36: Image Classification Using a Variational Approach

"!#$%'&()(%$*+,.- kk

ogw`vxP½;gtSqt± J #$ D¤&¹ 0i6s°¯Æi_u6qu6¦Xt`Pv©¨utSqPvxw`o

Ï J

!v~ªsiZ6i_ª¦qB¯ w`sRt.ogw`vxP½WP½>o>qPegi6qPvxu(Ö~8s *«vhItS~iu_w`orqtSvovo~Wu6¦Xt`i_® Ï tSsPvXtSqPvxw`otS¦tSstShji6qi6s) . 8| . | @ . 5 |8kµvqi6stSqPvxw`ogw`o * ® J>qw>uet`PqPvxutSstShji6qi6s7 .f5 G8|>DDD.vqi6stSqPvxw`ogB¯ w`sBPvh)ª¦XtSqi_ItSoogitS¦vo~g® Rw`hWªqtSqPvxw`otS¦gqPvhji\¥`tSsPvXtSqPvxw`otS¦ D8_®|qw>uet`PqPvxu 8,OG_®

ä\äàQ#RTSæR

Page 37: Image Classification Using a Variational Approach

kS -)$.GT ('- T $ B % T$* B&.$T .$T

600

800

1000

1200

1400

1600

5 10 15 20 25 30 35 40 45 50

num

ber o

f erro

rs

iteration number (epsilon)

Error evolution: gdr image (8 classes)

’PlotErrors’

20

22

24

26

28

30

5 10 15 20 25 30 35 40 45 50

SNR

val

ue in

dB

(var

ianc

es ra

tio)

iteration number (epsilon)

SNR evolution: gdr image (8 classes)

’PlotSnr’

!v~ªsiµ #$ªh0i6sBwS¯hWvxu6¦Xt`Pv©¨i_v©¸i6¦x«±Kqw`¹\tSog J #."¥`tS¦ªgiµ±Kw`qPqw`h&¹b¢«vqPe¤si_P0i_u6qqw¤qPegiZorªh0i6s$wS¯Bvqi6stSqPvxw`ogw`o * ¯Æw`s T~8sµvhItS~i¢«vqPe¬¤u6¦Xt`i_®

* éä * Ü

Page 38: Image Classification Using a Variational Approach

"!#$%'&()(%$*+,.- k

f 5* vqi6stSqPvxw`oÃw`o *

f *Wvqi6stSqPvxw`o?w`o *

E * vqi6stSqPvxw`o?w`o *

G *Wvqi6stSqPvxw`o?w`o *!v~ªsi;8 JGw`¦ªqPvxw`o # ± w`o¬qPegi¦xiT¯ q.etSogvxiN¹$tSogÀt`w>u6vXtSqi_ÃhItSÀwS¯£8vxu_w`o>qPvorªvqPvxi_± w`o&qPegisPv~e>q£etSogPvxiN¹B¯ w`s£qPegi T~8svhItS~i¢«vqPe¾u6¦Xt`i_Rt` * i_u6sit`i_®degiev~egi6sqPegi)orªh)i6swS¯Evqi6stSqPvxw`o¼w`o * |qPegi)¦xw¢Ri_Pq.qPegi¥tS¦ªgi)wS¯ * ® !sw`h vqi6stSqPvxw`o"qwÁkD qPegisi_qw`stSqPvxw`o»qi6sPh vxw`hWvotSorq|qPegi6o´t]¯Kqi6svqi6stSqPvxw`o kD¾vq¥`tSovxegi_¢«egi6sit`qPegi&¦xi6¥i6¦u_w`ogqPstSvorqqi6sPhvogu6sit`i_± i_iI±Æ¹P¹T®ä\äàQ#RTSæR

Page 39: Image Classification Using a Variational Approach

k8 -)$.GT ('- T $ B % T$* B&.$T .$T

G G * vqi6stSqPvxw`o?w`o *

G * vqi6stSqPvxw`o?w`o *

G *Wvqi6stSqPvxw`o?w`o *

* vqi6stSqPvxw`o¬w`o * ±3¨gotS¦Dsi_Pª¦q¹!v~ªsi+OS Rw`orqPvorªtSqPvxw`o"wS¯ !\v~g®?88® !sw`h vqi6stSqPvxw`o»kD8|bqPegiW¥tS¦ªgiWwS¯RqPegiW¦xi6¥i6¦£u_w`o8yqPstSvorqqi6sPh vx?Pª8ÂIu6vxi6orqP¦½ ev~e qw"¦xi6q¬tS0itSs¾qPegi¡u6¦Xt`v©¨utSqPvxw`o sw>u_i_|.vz®³i® qPegisi_qPsPvxu6qPvxw`o wS¯.t`8hWvxPv¦xi;¥tS¦ªgi_¯ w`s #(dc,qwÃqPegi;i6qWwS¯¦XtS0i6¦x±KvqvxWit`½¡qw¼i_iÁqPevxegi6ogw`hji6ot;w`oÃqPegiqw`¼sPv~erqµetSogPvxi~st`gtSqPvxw`o§tSsit¹T®degi¨gotS¦F# A ^ si_Pª¦q.vx.qPegiw`ogiZsi_i6orqi_§±Kvo¬u_w`¦xw`s¹w`o0w`qPqw`h ¦xiT¯KqwS¯ !v~g®«6g®

* éä * Ü

Page 40: Image Classification Using a Variational Approach

"!#$%'&()(%$*+,.- k,O

7 5 5 8 É & &33 É 3 ¡ É Ç¡iet¥iqi_Pqi_jqPegi«sw`wi_Wu6¦Xt`Pv©¨utSqPvxw`oIhjwGi6¦w`oJG£.d¿vhItS~i_EswN¥>vxi_W>½Wu_w`ªs°yqi_½ÁwS¯BqPegi!si6ogue J>t`u_iZ$~i6ogu6½À± # NJ¹T®

degi¨gsPq$8 8Wv©¸8i6¦x«vhItS~isi6si_i6orq.t ªque¼tS~sPvxu6ª¦qPªstS¦tSsit± i_i !\v~g®¹T®$o iT¸8i6sPqjet`&iT¨gogi_ qi6o¿u6¦Xt`i_|Rit`uew`ogiet_¥Gvo~¶tSotS~sPvxu6ª¦qPªstS¦.vorqi6sPsi6qtSqPvxw`o±K¢«vo>qi6s¢«egitSq|+0i6sPhItSogi6orq~st`®®®í¹T®)degitSstShji6qi6sZwS¯ÄqPegiWu6¦Xt`i_tSsisi_i6o>qi_ÀvodtS¦xi 88®degi Ï tSog;qPegi J tSsiPegw¢«ow`o !v~g®«8®!#.w)ogw`vxi$vxvo>qPsw>8ªgu_i_+|tSogÁqPegisi6~ª¦XtSsPv­NtSqPvxw`o¬i6sPhWvqqw¤egw`hjw`~i6ov­iZ0i6sPqPªsPtSqi_¬tSsit`®

degi?i_u_w`og´vhItS~ivxIwS¯µPv­i¬8 8¼v©¸i6¦x¬± i_i !v~g®ºD¹WtSog´¢£i?et_¥i¬iT¨gogi_¯Æw`ªsu6¦Xt`i_¢«egwijtSstShji6qi6s;± i_iId\tS¦xi O¹µet¥ij0i_i6o»i_PqPvhItSqi_»vo DÖ®;degi Ï tSog J tSsiI~v¥i6o´w`o !v~g®À8® !v~g®À¬Pegw¢¤qPegiÁi6¥w`¦ªqPvxw`o wS¯qPegiÁsi_Pqw`si_¶vhItS~i#¶tSog¼qPegi¤hItS¡wS¯R8vxu_w`orqPvorªvqPvxi_ .¢«ev¦xi * i_u6sit`i_®Ç¡i¤utSoÀogw`qPvxu_iWqPetSq¢Ri¤hItSÅii6ogw`ª~e´vqi6stSqPvxw`ogIw`o * qw¬~i6qjt¬vxi_u_i6¢«vxi¾u_w`ogPqtSo>qWvhItS~i # ± u6¦Xt`Pv©¨utSqPvxw`o¹T® !gsw`hqPegi¤i6¥w`¦ªqPvxw`o§wS¯ .¢RiWsi6hItSsPÅ?qPegi¤sw`~si_Pv¥i¤si6~ª¦XtSsPv­NtSqPvxw`o»w`o¼qPegi¤0w`ªoggtSsPvxi_7.qPegiev~egi_qqPegii_u6sit`i¢Z® s® q® * |8qPegiPetSsP0i_PqqPegii_8~i_®

ä\äàQ#RTSæR

Page 41: Image Classification Using a Variational Approach

k -)$.GT ('- T $ B % T$* B&.$T .$T

u6¦Xt` hjitSo 3 @ PqtSoggtSsi6¥GvXtSqPvxw`o 5 @ Sg® 8 8® 8 O`k8®Þ 8® Dk 8®Þ 8®Þ8 k8®Þ 8® 8 DD8®Þ +OG®Þ,O8 8®Þ 8®ÞO 8®Þ 88® 6)88® 8 k8®Þ8 8®Þ 8®ÞD 8®Þk 8®Þ

dtS¦xi 8 µtSstShji6qi6s)wS¯RqPegi¾DÁu6¦Xt`i_µ¯ w`sµqPegi JG£.dvhItS~iWwS¯t ªque»tS~sPvxu6ª¦qPªstS¦tSsit8®.degihjitSogtSog¬PqtSoggtSs¼i6¥GvXtSqPvxw`og.et_¥ii_i6oÃsw¥Gvxi_Ãr½qPegi !si6ogue J>t`u_i$~i6ogu6½À± #MNJ¹T®

!v~ªsi¤ Ä$sPv~votS¦JG£.d vhItS~iµwS¯t ªque?tS~sPvxu6ª¦qPªstS¦btSsit;±3¸8k)uetSoogi6¦z|g8]8v©¸i6¦x¹T®

* éä * Ü

Page 42: Image Classification Using a Variational Approach

"!#$%'&()(%$*+,.- k

Ï

J

!v~ªsi )i_Pª¦qZ¯ w`sqPegi JG£.dtSqi6¦¦vqiWvhItS~iIwS¯t ªque"tS~sPvxu6ª¦qPªstS¦tSsitu_w`o8yqtSvovo~ÃDju6¦Xt`i_® Ï tSsPvXtSqPvxw`otS¦\tSstShji6qi6s7 . | . | . 56f |0kWvqi6stSqPvxw`og.w`o * ®J>qwGuet`qPvxuWtSstShji6qi6s7 . 5 |D)DD;vqi6stSqPvxw`ogZ¯ w`sZvhª¦XtSqi_§tSoogitS¦vo~g®"Rw`hWªqt]yqPvxw`otS¦bqPvhji¥`tSsPvXtSqPvxw`otS¦ µk`N®|PqwGuet`PqPvxu 8D`N®ä\äàQ#RTSæR

Page 43: Image Classification Using a Variational Approach

)D -)$.GT ('- T $ B % T$* B&.$T .$T

u6¦Xt` hjitSo 3 @ PqtSoggtSsi6¥GvXtSqPvxw`o 5 @ kD8®Þk 8®Þ8 k,OG® 8®6k 88®Þk ®Þkk 8®Þ ®Þ,O

d\tS¦xi OSEBtSstShji6qi6s.wS¯qPegiju6¦Xt`i_¯Æw`s«qPegiZi_u_w`og JG£.d vhItS~iZi_PqPvhItSqi_vo DÖ®

!v~ªsiD JGi_u_w`og?w`sPv~votS¦ JG£.d·vhItS~i&±Æ8]8Wv©¸i6¦x¹T®

* éä * Ü

Page 44: Image Classification Using a Variational Approach

"!#$%'&()(%$*+,.- g

Ï

J

!v~ªsi8£i_Pª¦q«¯ w`sqPegii_u_w`og JG£.d tSqi6¦¦vqiZvhItS~iZu_w`o>qtSvovo~&&u6¦Xt`i_® Ï tSsPvXt]yqPvxw`otS¦rtSstShji6qi6s7 . | . f 8| . 5 |Dvqi6stSqPvxw`ogw`o * ® J>qw>uet`PqPvxutSstShji6qi6s7 . 5 G8|g)DDvqi6stSqPvxw`og¯ w`svhª¦XtSqi_¬tSoogitS¦vo~g®ä\äàQ#RTSæR

Page 45: Image Classification Using a Variational Approach

r -)$.GT ('- T $ B % T$* B&.$T .$T

vqi6stSqPvxw`oÃ

vqi6stSqPvxw`o»D

vqi6stSqPvxw`o»i_Pqw`si_;vhItS~i ( tS?wS¯B8vxu_w`orqPvo>ªvqPvxi_

!v~ªsiZ i_u_w`og JG£.d u_w`orqtSvovo~&Wu6¦Xt`i_® N¥w`¦ªqPvxw`oÃwS¯\qPegiµsi_Pqw`si_vhItS~iZtSogqPegi£hItSWwS¯8vxu_w`o>qPvorªvqPvxi_E¢«ev¦xii_u6sit`Pvo~ * ¢«egi6o¤vogu6sit`i_qPegiRorªh)i6sEwS¯vqi6stSqPvxw`ogN®

* éä * Ü

Page 46: Image Classification Using a Variational Approach

"!#$%'&()(%$*+,.- rk

¿)

Ç¡iÁet¥i;si_i6orqi_¿t¬¥tSsPvXtSqPvxw`otS¦«hjw>i6¦¢«evxue sw¥Gvxi_|EqPesw`ª~e´qPegi;hWvovhWv­NtSqPvxw`owS¯£tjogw`ogu_w`or¥iT¸¬¯Kªogu6qPvxw`otS¦z|0tju6¦Xt`v©¨utSqPvxw`oÀtSogÃtSoÃi_8~iTyzsi_i6sP¥>vo~Ási_Pqw`stSqPvxw`oD®.devxhjwGi6¦gvxt`i_jw`oWqPegi«si6~ª¦XtSsPv­NtSqPvxw`oÁqPegi_w`sP½WtSogjhji_uetSovxutS¦et`iqPstSogPvqPvxw`og£qPegi_w`sP½®mpqqPegi_w`sPvxutS¦Bw`ªog8ogi_$egw`¦x$vo?qPegiut`iwS¯Äu_w`or¥iT¸si6~ª¦XtSsPv­NtSqPvxw`oD|0ªq.iT¸G0i6sPvhji6orqtS¦si_ª¦qPegwN¢qPetSqogw`ogu_w`o>¥iT¸¡si6~ª¦XtSsPv­NtSqPvxw`o"¦xit`Zqw0i_Pq¥GvxPªtS¦£tSog rªtS¦vqtSqPv¥iIsiTyª¦q®BdegiµtS¦~w`sPvqPeh ªgi_&qwhWvovhWv­i.qPegi.sw`wi_I¯ ªogu6qPvxw`otS¦+vxit`P½jqwvhW¦xi6hji6orqtSog§¯zt`Pqi6sqPetSo´tPqwGuet`qPvxu&w`ogiI¯ w`ssi_Pª¦q¤tSq)¦xit`q >ªtS¦vqtSqPv¥i6¦½»u_w`hWtSstS¦xi®¬Ç¡iet¥i¤iT¸Gevvqi_¼tIi6¦vxutSqi¤u_w`o8¨g~ªstSqPvxw`o¿± u6¦Xt`i_$et¥>vo~ÁogitSsP¦½¬tShjitSstShji6qi6s¹¯Æw`s¢«evxue)qPegi£¥tSsPvXtSqPvxw`otS¦8tSswt`ueWvxDhjw`siRiTÂIu6vxi6orqBqPetSo)qPegiRPqwGuet`PqPvxu£hjwGi6¦rªgPvo~µ\w`qPqsi6~ª¦XtSsPv­NtSqPvxw`oD®ÁdegiWogiT¸8qPqi6g¢RiIi6o>¥>vxtS~ijvo¡qPevxZsw`hWvxPvo~?¢£w`sPŧtSsiµ¨gsPqqw;vo8yu_w`sP0w`stSqiZtSotSªqw`hItSqPvxui_PqPvhItSqPvxw`owS¯u6¦Xt`£tSstShji6qi6s)±KªogPªi6sP¥Gvxi_;u6¦Xt`Pv©¨utSqPvxw`o¹T|i_u_w`og´¢£i¬u_w`ª¦x´vorqPswG8ªgu_i¬tÀi6¦ªsPsPvo~"w`sWsi_u_w`ogPqPsPªgu6qPvxw`oºsw>u_i_® degi6oD|¢£i¢«v¦¦iT¸8qi6ogqPevxhjw>i6¦\qwjqPegih)ª¦qPvxP0i_u6qPstS¦ut`iÁ±K¢«vqPe¬8vsi_u6qµtS¦vxutSqPvxw`og$qwIhª¦qPvtSogtSqi6¦¦vqiZgtSqtjtSog¾u_w`¦xw`svhItS~vo~>¹T®

n)7 8 8;: 8 = < qPegi¤tSªqPegw`sZqPetSoÅ?qPegi !si6ogue"J>t`u_iW~i6ogu6½"± #MN J¬egw`hjiTytS~i)tSq - ( 44 T .$¹Ä¯ w`ssw¥Gvx8vo~IqPegi JG£.d tSqi6¦¦vqiZvhItS~i_.tSog¾qPegi%(µm JGm J¤±KÇ¡w`sPÅ>vo~ (µsw`ªw`o Ï vxvxw`oD|NmphItS~i|J>v~otS¦>tSogmÖo8¯Æw`sPhItSqPvxw`oZswGu_i_Pvo~ - ( 44 .#T .$¹EwS¯ #. JI¯Æw`s«qPegiT~8sµqi_Pq«vhItS~i&®

ä\äàQ#RTSæR

Page 47: Image Classification Using a Variational Approach

-)$.GT ('- T $ B % T$* B&.$T .$T

_#E

J®b$¦¦xi6o"tSogg®$tSeoD® (_ hWvxu6swu_w`vxuIqPegi_w`sP½À¯Æw`stSo>qPvet`iÁw`ªoggtSsP½§hjw`qPvxw`otSogvq\tS¦vxutSqPvxw`o¤qwtSo>qPvet`i£w`hItSvou_wtSsi6ovo~=*® & 2 2 $|,OSDD8|D,O`8®

Þ E®g$¦¥`tSsi6­|g® y E® +vxw`og|tSogg® y ( ® ( w`si6¦z® (_mphItS~ii6¦xi_u6qPv¥ihjw>w`qPevo~&tSog?i_8~ii6qi_u6qPvxw`oÃ>½?ogw`o¦vogitSsZ8v 0ªgPvxw`o*® & .$. &M% .G6|±Æk¹ ÞSr888|+8®

Þk E®E$hswPvxw§tSog Ï ®Ed\w`sPqw`si6¦¦vz®($sw_¸8vhItSqPvxw`owS¯.¯Kªogu6qPvxw`otS¦xÁi6i6og8vo~»w`oîªhWg.r½Ãi6¦¦vqPvxu¯Kªogu6qPvxw`otS¦xZ¥>vXt ++yÖu_w`o>¥i6sP~i6ogu_i+*® $ &(,(% - X|rk ÞDk88|bD8®

E®$hswPvxwtSog Ï ®dbw`sPqw`si6¦¦vz®>(]$ojqPegi$tSsw¸GvhItSqPvxw`o;wS¯0¯Kªogu6qPvxw`otS¦xRi60i6og8vo~w`oZ°ªhWgÄ>½rªt`8stSqPvxu|8i6¦¦vqPvxu¯Kªogu6qPvxw`otS¦x *® %. x|r)8]y & Dk8|8®

Þ J®Do~i6ogi6o>q¤tSog ( ® N®(.ªsPqPvoD®0( ( ª¦qPvet`iIqPegi6sPhjw`hji_uetSovxu_¤¢«vqPe§vo>qi6s°¯Æt`u6vXtS¦PqPsPªgu6qPªsi8®N¥w`¦ªqPvxw`owS¯BtSo&vxw`qPegi6sPhItS¦+vo>qi6s°¯Æt`u_i *® &M$*.- B% .- &M% X|D ÞkkkGk8|b8®

8 (¤®ª0i6sPq$tSogÄ® Ï i_i®I( ¥tSsPvXtSqPvxw`otS¦bhji6qPegwG¾vovhItS~isi_u_w¥i6sP½"*® & T$ &M .G6|8kS±Æ¹ Sr,O`8|,OG®

O J®&«tS¦xw® ( ( vovhItS¦vorqi6s°¯zt`u_iu6sPvqi6sPvxw`oW¯Æw`sBet`iqPstSogPvqPvxw`ogEvoWhWv©¸8qPªsi_EwS¯ tSeo8y v¦¦vXtSs gªvx*® &M> #. .>$. %$ 6| OS 8,OGD8|DD8®

Þ (¤® &«tSsP¦xi_| E® &Rsw`ogtSs+|+tSog¬® N®JGw`ª~rtSovx8vx®O( !\sw`o>q$sw`tS~rtSqPvxw`o¼¯Æw`ssit`u6qPvxw`o8y8v ªgPvxw`o i >ªtSqPvxw`ogÁwS¯ZvxqtS¦xi¬qp½>0i+*® &M> #T .'$ %$ 6|$ O`rr88|8®

Þ (¤® &i6¦¦xi6qPqPvovz| ( ®Bt`w`¦vovz|tSog.® Ï i6s8vz®;( #ªhji6sPvxutS¦hWvovhWv­NtSqPvxw`o¾wS¯0~i_w`hji6qPsPvxutS¦qÖ½Giµsw`¦xi6hjsi6¦XtSqi_qwWutS¦xu6ª¦ªgwS¯B¥tSsPvXtSqPvxw`og *® |,OSÞ8G,O`8|\8®

D ( ®&i6sPqPegw>+| ®ZtSqw|)J®EªD|tSogg® i6sPªvXt8® ( &«t_½i_vXtSovhItS~iu6¦Xt`v©¨utSqPvxw`oªgvo~( tSsPÅwN¥ÁstSogw`h ¨i6¦x *®) "! M( |+6±K>¹ ÞGk8|88®

)®&R¦XtSÅiZtSog® vxi6sPhItSoD®,! G $ #.$#2°® ( ®³m®³d®Esi_|D,OG® .®³)® &Rw`ªhItSoÀtSog ( ® J>etSvsw®1( h)ª¦qPvxutS¦xistSogw`h¨i6¦x¬hjwGi6¦b¯Æw`s&«t_½i_vXtSovhItS~iÄi6~hji6o>qtSqPvxw`o *® "!>$ #> ) $ |k 8+OOG| ( tSsueÁSg®

* éä * Ü

Page 48: Image Classification Using a Variational Approach

"!#$%'&()(%$*+,.- r

k E® &Rsw`ogtSsÁtSog !®i6vqPvxueD®>(N$o&qPesi_iTyzet`i$w`ªoggtSsP½Whjw`qPvxw`o;tSogjqPegivo~ª¦XtSs¦vhWvq;wS¯¤t¡¥i_u6qw`s°yz¥tS¦ªgi_ (.vo­_ªsP~`y btSoggtSª i >ªtSqPvxw`o *® &M$*.- , .- &M% x|0SÞkGk,O`8|bk8®

6 (¤® tS~votS¦D® (EJ>qiT¯ÆtSoItSog i6¦xiTy J>et¢ qÖ½Git`£t`P½>hWqw`qPvxu¦vhWvq£wS¯qPegiet`iTyƨi6¦xi rªtSqPvxw`og *® - E|k±°N¹ Þ,OG88|8®

g®_Ç(®tSeo¤tSog g®+N® v¦¦vXtSs+®=( !si_iEi6ogi6sP~½ZwS¯t«ogw`o>ªov©¯ w`sPh(P½8Pqi6h®mT®mporqi6s°¯zt`u6vXtS¦¯Ksi_ii6ogi6sP~½"*® - . ('- ..6|g±°N¹ ÞG8,OG|\8®

8 Ï ® t`i6¦¦xi_N|S)® µvhWhji6¦z|tSog (¤®EJ8tSvsw®=( (µi_wGi_PvxuÄt`u6qPv¥i£u_w`orqw`ªs *® ' .$.%% M( .$ ! G.|±°N¹ 88 O`8|b,OG®

+O \® £etSsP0w`oovxi6s|E® &£¦XtSoguTy !B6stSªg+|(W®$ªi6sPq|+tSog ( ® &«tSsP¦XtSªg+® ( i6qi6sPhWvovxqPvxui_8~iTyzsi_i6sP¥>vo~¼si6~ª¦XtSsPv­NtSqPvxw`o vo"u_w`hWªqi_"vhItS~vo~=*® !#$ #B ) $*+ . |%8±Æ¹ ÞGk8| !gi6sPªtSsP½¡,OG®

® i_u_w`h0i_|£® ( w`sPsPvxN|£tSog g® i6sPªvXt8® (EJGw`hji;vhWsw¥i6hji6o>qIqw &«t_½i_vXtSovhItS~ii6~hji6orqtSqPvxw`oD®+ÄtSsPqw`ogi[hjwGi6¦¦vo~g®±Kvo¬¯ si6ogue¹ *® !>$ TT' %©|6±K>¹ Þk,O`kGk8|b,OG®

® i_u_w`h0i_|£® ( w`sPsPvxN|£tSog g® i6sPªvXt8® (EJGw`hji;vhWsw¥i6hji6o>qIqw &«t_½i_vXtSovhItS~i£i6~hji6o>qtSqPvxw`oD®`ÄtSsPq\qp¢£w Du6¦Xt`v©¨utSqPvxw`oD®8±Kvo¯Ksi6ogue¹ *® !#$ . .' %©|6±K>¹ ÞkkGk8|b,OG®

ÞD m® !gw`ogi_utWtSogE®8d\tSsPqtSs® (degi.~st`8vxi6orq«qPegi_w`sP½;wS¯bet`i.qPstSogPvqPvxw`og¯Æw`s«P½8Pqi6hj¢«vqPeWqÖ¢£ww`qi6o>qPvXtS¦g¢£i6¦¦x*® $ B- +T2 $->|_±°N¹ ÞD8|+8®

Þ8 J® (µi6hItSo´tSog ® y N® ( u Dªsi® ( &t½i_PvXtSo¶vhItS~iÁtSotS¦½8Pvx ¤tSo´tS¦vxutSqPvxw`o"qwPvo~¦xiegw`qw`o¬i6hWvxPvxw`oqw`hjw`~stSer½"*® $ GT M( 2X|ǧt`Pevo~qw`o .|$hji6s®%J>qtSqPvxPq®$w>u®68|+8®

Þ J® (µi6hItSo¡tSog (¤®0i6½Gogw`¦x®O(Rw`ogPqPstSvogi_Ãsi_Pqw`stSqPvxw`o§tSogÃqPegi)si_u_wN¥i6sP½ÃwS¯R8vx°yu_w`orqPvorªvqPvxi_*®' !>$*#% T$ &M% .G ).-) ' . )T% & |gk Þk8,OGkk8|b8®

Þk N® i(.vxw`sP~vz® (Rw`or¥i6sP~i6ogu_iµsw`¦xi6hjR¯Æw`sE¯ ªogu6qPvxw`otS¦x«w`sRw`0i6stSqw`s *® $* M- ' .$.%% T B T' 2- > $ &M% ,$ |D,O`8®

ä\äàQ#RTSæR

Page 49: Image Classification Using a Variational Approach

)8 -)$.GT ('- T $ B % T$* B&.$T .$T

ÞS N® (µvªgPqPvz® > $ T) %> ) E$. î® &RvsPÅ>eSªgi6s|Sg®

Þ \® y g® (.si_i6oD®( &«t½i_PvXtSo¿si_u_w`ogPqPsPªgu6qPvxw`o¯ sw`h i6hWvxPvxw`oqw`hjw`~stSer½¿gtSqt¼ªgvo~hjw>8v©¨i_N ( tS¦~w`sPvqPeh *® !%# > |ÞS Gk8|D8®

Þ8 g® t`gtShItSs+® 2 $ .- (%$* . T $ (#$. M.$.' #> tS¦xi $ov¥i6sPvqp½;Esi_|+k8®

Þ,O d® i60i6sPq&tSog® DitSe>½®?(_ ~i6ogi6stS¦v­i_ N ( tS¦~w`sPvqPeh ¯Æw`s&k &«t_½i_PvXtSosiTyu_w`ogPqPsPªgu6qPvxw`o´¯ sw`h w`vxw`ogtSqtêgPvo~ (.vg¤sPvxw`s *® !'#M + | S GD8|b8®

Þ ( ® Zt`|Ä)®ÇºvqPÅ>voD|ÄtSog ®Bdbi6sP­w`0w`ª¦xw® (EJ>otSÅi_ Át`u6qPv¥iu_w`o>qw`ªsWhjw>i6¦x*®' .$.%% M( .$ ! |+Þk8Gkk8|D,OG®

Þ ® ZtSqw® . T, $ T> $ T # . G. ( $ $ %, $&(,(% .' ) ! $.%T- -`®e qPegi_Pvx| $ov¥i6sPvqZi #$vxu_iTy JGw`evXtj$orqPv0w`¦vx|DSg®

ÞkD J®µvxuegi6ot`tSh½|)® µªhItSs|\®g$¦¥i6s|®>dtSoogi6ortSªh|tSogÁ® Äi6­_­_v s® (Rw`o8y¯ w`sPhItS¦u6ªsP¥`tSqPªsi w¢ b¯ sw`het`iqPstSogPvqPvxw`og£qw)t`u6qPv¥i$¥>vxPvxw`o *® &M$ .- % .- &M% x|0kSÞ,O`GkD8|88®

Þk8 J® btSÅGehItSotSotSog ® i6sPvoD® (EJ>vhª¦qtSogi_w`ªg&tSstShji6qi6sIi_PqPvhItSqPvxw`otSog¿i6~`yhji6orqtSqPvxw`o¼wS¯ (µvgstSogw`h¨i6¦xªgPvo~ÁPvhª¦XtSqi_ÃtSoogitS¦vo~=*®M !>$*# .$. &M .G ,T-) M' T .% 6|\ O`G8k8|$ª~ªgPqZ8®

Þk ® ( tS¦¦Xt`8vz| ®³)®JGi6qPevXtSoD|tSog &µ® .® Ï i6h)ªsPvz® (N¥w`¦ªqPvxw`otSsP½Ã¯ sw`orq.¯Æw`s.qw`0w`¦xw`~½vogi6i6ogi6o>qPetS0iWhjwGi6¦vo~ÃtSog§si_u_w¥i6sP½"*®Ámpo $ - $ !D|0tS~i_kk8|%J>qw>uÅ>egw`¦h| J>¢£i_i6oD|bSg®

Þkk &.® ( tSo`°ªotSqPetSog¿)® £egi6¦¦XtSt8®( $ogPª0i6sP¥>vxi_"qiT¸8qPªsii6~hji6o>qtSqPvxw`oªgvo~( tSsPÅwN¥&stSogw`h ¨i6¦xhjw>i6¦x *® !>$*# .$. &M% .G ,.- ' . ).%6|Dk O`rr8| ( t_½À8®

ÞkS® ( tSsueD®;( Ï vxPªtS¦gsi_u_w`ogqPsPªgu6qPvxw`o&ªgPvo~¥`tSsPvXtSqPvxw`otS¦hji6qPegw> *® ) ! M( |+D ÞkDGk8|b8®

Þk ® ( tSsue tSog ( ® w`­_vxw®(_ ¥tSsPvXtSqPvxw`otS¦$hji6qPegwG´¯Æw`sjqPegisi_u_wN¥i6sP½¿wS¯PhjwGw`qPew`ªoggtSsPvxi_ *® +) "! " M( |+ OD OG8|b,OG®

* éä * Ü

Page 50: Image Classification Using a Variational Approach

"!#$%'&()(%$*+,.- O

Þk8 E® ( wG8vxut8® (_degi$~st`8vxi6orqqPegi_w`sP½IwS¯\et`iqPstSogvqPvxw`ogtSog&qPegi$hWvovhItS¦vo>qi6s°¯Æt`u_iu6sPvqi6sPvxw`o *®&M$*T- % .- &M% x| k6r8|\,OG®

Þk,O g® y ( ® ( w`si6¦tSog J® JGw`¦vhWvovz® ,!$.% 2- ) .' , °®&£vsPÅGeSªgi6s|+8®

Þk ® ( ªh¤¯Æw`s¬tSog ® J>etSeD® ( &w`ªoggtSsP½i6qi_u6qPvxw`o¬r½¾hWvovhWv­_vo~&¯Kªogu6qPvxw`otS¦x *®mpo $*+ % M( .$ ! G. .$. B >|>J8tSo !stSogu6vxu_w|8®

Þk .®+$¢Ri6o»tSog¼\®J>qi6sPo>i6sP~g® ( #$w`ogu_w`o>¥iT¸Ã¥tSsPvXtSqPvxw`otS¦sw`¦xi6hjZ¢«vqPe¡tSovxw`qPsw`vxui6sPqPªsPtSqPvxw`og *® > $ &M% !- $ T- &(,( #6|8± O¹ OD OG8|b8®

)D d®Dt¥>¦vx8vxWtSog ¤® yÖd® Dvxw¢Z®4(mÖo>qi6~stSqPvo~¬si6~vxw`o»~sw¢«vo~ÀtSog§i_8~iIi6qi_u6qPvxw`o *® $*+ - M( .$ ! .$. B >|\8®

g !R®$i6vqPvxue tSog ® ( ® JGw`ogi6s® (desi_iTyzet`i?w`ªoggtSsP½"hjw`qPvxw`og&ªogi6sIu_w`ogPqtSorq¥i6¦xw>u6vqPvxi_®DtSsPq)w`ogi )degiW¥`tSovxevo~?Pªs°¯zt`u_iWqi6ogPvxw`o§¦vhWvq *® $ - T T $-G|D8± µ¹ Þk,OG88|88®

r g®>ªvogPqi6voD|8\®#J>qi6sPor0i6sP~g|gtSog g® &µ®i6¦¦xi6s® ( !t`Pq£sit`u6qPvxw`oD|P¦xw¢ 8v 0ªgPvxw`oD|tSogu6ªsP¥iZPegw`sPqi6ovo~=*® & " M&()(% -#.TT|r 8kk8|\8®

rk E®bªg8voD| J®B.Pegi6s|tSog N® !tSqi6hWvz®D( #.w`o¦vogitSsqw`qtS¦£¥`tSsPvXtSqPvxw`o¶t`i_§si6hjw¥tS¦tS¦~w`sPvqPeh *® - .|>8D ÞG88|b8®

\® J>qi6sPor0i6sP~g®B( Ï i_u6qw`s°yz¥`tS¦ªgi_¶¦xwGutS¦ÄhWvovhWv­i6sWwS¯ogw`ogu_w`o>¥iT¸»¥tSsPvXtSqPvxw`otS¦sw`8y¦xi6hj *® - ..6|g8 O`GD,OG|b8®

r \® J>qi6sPo>i6sP~(tSog²Ç(®³\® i6vhji6s® ( Dw>utS¦IhWvovhWvxi6s§wS¯t qPesi_iTyzet`i¿tSsPqPv©yqPvxw`o»sw`¦xi6h ¢«vqPe"qPsPv¦xi)îªogu6qPvxw`og*® $*+ - T T $->|S± µ¹ DD,O`k8|\Sg®

)8 J®dbi60w`ª¦z|Ä® &R¦XtSoguTy !B6stSªg+|(W®ª0i6sPq|>tSog ( ® &tSsP¦XtSªg+®>( Ï tSsPvXtSqPvxw`otS¦tSswt`ue¯ w`si_8~iTyzsi_i6sP¥Gvo~Wsi6~ª¦XtSsPv­NtSqPvxw`oªgPvo~Wu_w`ª¦xi_¾ N ³ *® !'# ) $*+ . | O8±Æk¹ Þk,OGk,OG|\8®

O \®'.®`dbi_w| (¤®,J8tSvsw|rtSog &µ®`)®SǧtSogi6¦¦z® (£sitSqPvo~Zu_w`oogi_u6qi_Wsi6si_i6o>qtSqPvxw`ogÄwS¯u_w`sPqPvxutS¦b~st_½;hItSqPqi6s$¯ w`s¯ ªogu6qPvxw`otS¦ ( $mE¥GvxPªtS¦v­NtSqPvxw`o *®B !%# + |+8±8¹ ÞG8k8|\,OG®

ä\äàQ#RTSæR

Page 51: Image Classification Using a Variational Approach

r -)$.GT ('- T $ B % T$* B&.$T .$T

r )® #®+dvÅ>egw`ogwN¥§tSog Ï ® ¤®D$si6ovoD® # ( 2 (%$ . . °®&Çvogqw`otSog;Çv¦xi6½|D,OOG®

r J®.® e>ª;tSog&)®Eªv¦¦xi®>(mporqi6~stSqPvo~si6~vxw`oI~sw¢«vo~jtSog&i_8~ii6qi_u6qPvxw`o *® !#$ # , .$ &M% ,.-) ' . )T% 6|\±Æ¹ ÞS GDD8|b88®

* éä * Ü

Page 52: Image Classification Using a Variational Approach

Unité de recherche INRIA Sophia Antipolis2004, route des Lucioles - B.P. 93 - 06902 Sophia Antipolis Cedex (France)

Unité de recherche INRIA Lorraine : Technopôle de Nancy-Brabois - Campus scientifique615, rue du Jardin Botanique - B.P. 101 - 54602 Villers lès Nancy Cedex (France)

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France)Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38330 Montbonnot St Martin (France)

Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - B.P. 105 - 78153 Le Chesnay Cedex (France)

ÉditeurINRIA - Domaine de Voluceau - Rocquencourt, B.P. 105 - 78153 Le Chesnay Cedex (France)

ISSN 0249-6399