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Feature extraction based on a tensor image description

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(1)Linköping Studies in Science and Technology Thesis No. 288. Feature Extraction Based on a Tensor Image Description Carl-Fredrik Westin. LIU-TEK-LIC-1991:28 Department o f Electrical Engineering Linköpings universitet, SE-581 83 Linköping, Sweden.

(2) Feature Extraction Based on a Tensor Image Description © 1991 Carl-Fredrik Westin Department of Electrical Engineering Linköpings universitet SE-581 83 Linköping Sweden. ISBN 91-7870-815-X. ISSN 0280-7971.

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(15) È i•¤˜?6Ž]‹ •—˜fJ;œ)œ[Œ,ŽJ•—–™˜“’–8˜G·™‹ ˜pŽ]•¤–8˜<; ÍÖ Ü Í× Ö Ï Ü ÏØ× 9Ñ ÑV9Ñ × Ö Ò Ü Ò Ù Ú Ò Ö É Ü<É Ù Ö É Ü<É Ô èO­éè Õ É9Û Í ¯_°4‹9J;œ)œ[Œ,Ž]•¤–8˜]; —‹•¬_Ž]°4Œ,Ž*Œ˜G¥l‹ ‘O¢;J‹JJ•—–™˜•—˜G·™–8 ¤·G•—˜;¡hŒŽ6µ •¬“’‹™5 ]‹ ¢H‹‡ŒŽJ‹‡Ÿ•—˜4ŸO‹’‘fA°4Œ™ ¤ ®Œ™OŽJ–8œ[Œ,ŽJ•V5 “‡Œ — ¤¥f6Ž£Œ˜4Ÿfš›–8•¤Ž]«A;œQ–,·™‹‡_ŽJ°;‹·,Œ —;‹‡ á Î è Î ¨ ÑV9Ñ Ü –š%ŽJ°4‹ݱÆޓƛ»^–Xƛß9•—˜4ŸO‹ ‘»­*¯_°4‹N“’–8˜p·8‹ ˜pŽJ•—–™˜f4J‹‡Ÿ •—˜Ž]°;•—«Ž]°;‹‡J•—«•¬*Ž]°4Œ,ŽŽJ°;•¬«J; —‹N–8˜; —¥f•¬9Œ¢;¢4 ¤•—‹‡Ÿ–8˜V¢4Œ™•¤«–ši•—˜4ŸO•¬“’‹–™šºŽ6¥p¢H‹h–™˜;‹hA;¢H‹ £]“’]•¤¢OŽ9Œ˜4Ÿ –8˜;‹A4ž4J“ J•—¢OŽ Ô –8˜;‹;¢Œ˜4Ÿf–8˜;‹ŸO–,µ ˜ Õ ­ È  ‹’Ž Ö É •—*Ÿ;‹ ˜;–™ŽJ‹ŽJ°;‹a“ –™œ)¢H–™˜;‹‡˜8Ž£_–šYŽJ°;‹·8‹‡“’ŽJ–™ B •¤˜à, ­ Ô èO­ ¨ Õ B Ù Ö É Ü<É ½ ŸO–™4ž; ¤‹J;œþ‹ ‘O¢;J‹JJ‹‡Ÿ4J•¤˜4¡Ž]°;‹³ •—˜4AŽJ‹‡•¤˜A4œhœ[ŒŽJ•—–™˜“’–8˜p·8‹ ˜pŽJ•—–™˜œ)•¤¡8°8Ž  —–G–™±l —•¤±8‹ŽJ°;‹9š›–8 ¤ —–,µ •¤˜4¡2; Ô èO­ ý Õ Éá ãâ Ö ÉåäÐâ æç ÚÉ9Û Ò Í â±Ú Û Ò Í á Éãâ Ö É4äªâ ¯_°;•¬A;œ)œ[Œ,Ž]•¤–8˜ “’–™˜G·8‹ ˜pŽJ•—–™˜V•¬·8‹ ]¥4A‹ š›;  ²Mž;OŽN“‡Œ]‹•¬µ_Œ]]Œ™˜pŽJ‹‡Ÿ®­ õ –™œ)¢4Œ™J‹š›–™‹ ‘;Œœ)¢; —‹Ž]°;‹)š›–™ V5  ¤–,µ •—˜;¡)‹’‘O¢;]‹‡]J•¤–8˜4 Ô èO­éç Õ áâ Éãâ Ö äªÉ ä â â Ù á Éèâ ä ââ Ö É äªâ Ô èO­ ï Õ á ÉèâªÔ Ö × Õ Ù á Éèâ Ö × á Éèâ µ •¸Ž]° Ô èO­ © Õ á É Éãâ Ö äÉäÐâ â Ùéé á Éèâ äÉÉ Ö â ä â Ô èO­ ÷ Õ á ÉãâFÔ Ö × Õ Ù á Éèâ Ö × á Éèâ ì Ž2“ Œ™˜)ž?‹*µT–™JŽJ°)˜;–™ŽJ•—˜;¡9ŽJ°?Œ,Ž³Ž]°;‹J¥pœž?–8  Ù é •¤˜)Ž]°;‹* ¬Œ™AŽ³•¤˜;‹º6p4Œ —•¸Ž6¥hŒ™“’ŽJ4Œ™ ¤ —¥J°;–™4 —Ÿ)AŽ]Œ˜?Ÿhš›–™  ˜;–Ž³‹‡·™‹ ˜ µ J–8˜;¡ª4Ž ²OA•—˜4“’‹9žH–Ž]°J•¬ŸO‹‡*Œ™J‹œ)‹‡Œ™˜;•¤˜4¡™ —‹‡]_‹’‘O¢;]‹‡]A•—–™˜4T•—˜Ž]‹ ˜4J–™*Œ™˜4Œ —¥OA•¬‡­ œApo ê„rZët\Š~Ft\x— ¿t\x{z ê„r_x— †Š†t˜ët\Š~Ftyx— R¡£‚x{vr_Šv. M¯ ‹‡˜4J–™£[Œ]‹ž4Œ™J‹‡ŸÆ–8˜^Ž6µT– œ[Œ•—˜•¤˜4¡™]‹‡ŸO•—‹ ˜pŽ]‡­cú‹‡¢?‹‡˜4ŸO•—˜;¡–™˜ÆŽ]°;‹Ž]]Œ™˜46š›–8Jœ[ŒŽJ•—–™˜0¢;J–8¢?‹‡AŽ]•¤‹[–šaŒ ŽJ‹‡˜4A–8‡²=•¸Ž•—µ •— — ³ž?‹“ Œ,Ž]‹ ¡8–™]•¤î‡‹‡ŸŒ8žH‹ •—˜;¡VŒ“’–,·,Œ]•¬Œ˜pŽ«Ž]‹ ˜4J–™²=Œ“ –™˜pŽJ£Œz·,Œ]•—Œ™˜pŽ«Ž]‹ ˜4J–™–8NŒfœh•¤‘GŽJ;]‹ –š³Ž]°;‹‡J‹aŽ6µT–4²»• ­ ‹8­Œœ)•¤‘G‹ŸŽ]‹ ˜4J–™­½Ï·™‹“”Ž]–™•—Œ“ –™˜pŽJ£Œz·,Œ]•—Œ™˜8Ž Ž]‹ ˜4J–™9Œ˜?ŸV•¤Ž°4Œ8Œ¹›¸^Ä “2¸”‘&Ə‘"–4ì*µ •¤ŽJ° A4¢?‹‡]]“’]•¤¢;Ž]‡­Y½“ –,·™‹‡“’ŽJ–8º•—³Œ™˜)‹’‘;Œœ)¢; —‹ –šZŒ“’–,·,Œ]•¬Œ˜pŽ%ŽJ‹ ˜?A–8‡­%½*˜[‹‡ ¤‹“”ŽJ]•¬“ Œ G¹?‹  ¬Ÿ)–™%ŽJ°4‹*¡™£Œ™ŸO•—‹ ˜pŽº–š®Œ J“‡Œ ¬Œi¹4‹  ¬ŸŒ™J‹«‹’‘;Œœ)¢; —‹‡2–š=“ –,·™‹“”ŽJ–8]‡­=½«iœ)‹‡˜8Ž]•¤–8˜;‹‡Ÿ®²GŽJ°;‹‡•¤2ŽJ£Œ˜4Aš›–™]œ[Œ,Ž]•¤–8˜l¢;]–™¢H‹ JŽJ•—‹‡iŒ™J‹ŸOV• ¯H‹‡J‹‡˜8Ž­ ½·™‹“”Ž]–™º•¬º‹’‘O¢;]‹‡]A‹ŸN•—˜[;˜;•¤Ž]i–šHŽJ°;‹«“’–G–™£ŸO•—˜4Œ,Ž]‹T¡8J•¬ŸZ²8µ °;•¤ —‹*Œ˜[‹‡ ¤‹“”Ž]J•¬“ Œ™ G¹4‹  ¬Ÿ)•—³‹’‘O¢;]‹‡]A‹ŸN•—˜[·™–8 ¸Ž£Œ¡™‹ “ƙÝ*;˜4•¸Ž£9–šºŽJ°4‹)“ –G–™£ŸO•¤˜?Œ,ŽJ‹N¡™]•—ŸZ­ õ –8˜4A•¬ŸO‹‡Ž]°;‹h“£°?Œ˜;¡8‹N–šT“’–G–™£ŸO•—˜4Œ,Ž]‹J¥GAŽJ‹‡œÅJ‹A; ¤ŽJ•—˜;¡f•¤˜¼ŸO–™4ž; ¤•—˜;¡ ŽJ°;‹ ·,Œ™ ¤;‹ –™šH‹Œ™“£°h·™‹“”ŽJ–8º“’–™œ)¢H–™˜;‹‡˜pŽ‡²™• ­ ‹™­%Œ9J‹J“‡Œ —•¤˜;¡–šHŽJ°4‹*Œ,‘O‹‡‡%­ Y°?Œ,ŽiŸO–G‹‡YŽ]°;•¬³ž;]•¤˜4¡9Œ™ž?–8OŽºš›–™%Ž]°;‹ ‹  —‹‡“’ŽJ]•—“‡Œ Y¹4‹  ¬ŸWŸO‹‡]“’]•—¢OŽJ•—–™"˜ š¯_°;‹)‹  —‹‡“’ŽJ]•—“‡Œ %¹4‹‡ —Ÿ´‹’‘O¢;]‹‡]J‹‡ŸV•—˜¼•¸Ž£“’–™œ)¢H–™˜;‹‡˜pŽ]9µ •— ¤ º•—˜Ž]°;‹[˜;‹‡µ ž4Œ8A‹ –™˜; —¥°4Œz·™0‹ í»^· î–™šºŽJ°;‹)–8J•—¡™•—˜4Œ Y“’–8œh¢H–™˜4‹ ˜pŽ·,Œ™ ¤;‹ ­¯_°G4‡²»;˜4ŸO‹‡«Ž]°;•¬«¢?ŒJŽJ•¬“’; ¬Œ«ŽJ£Œ˜4Aš›–™]œ[Œ,Ž]•¤–8˜®²4Ž]°;‹ “’–8œh¢H–™˜4‹ ˜pŽ]Y–š»Œ·8‹‡“’ŽJ–™%]“ Œ —‹‡=Ž]–Ž]°;‹ ŸO–™;ž4 ¤‹8²µ °;•— ¤‹TŽJ°4‹ ¥a–™šHŒ“’–,·™‹“”Ž]–™M¡8–«°?Œ —·™‹‡‡­=¯_°4•—ºJ°;–,µ*MŽJ°4ŒŽYŽ]°;‹ “’–8œh¢H–™˜4‹ ˜pŽ]º–™š»ŽJ°;‹Ž6µ2–aŸOV• ¯»‹ ]‹ ˜pŽ³Ž6¥G¢?‹³–™š»‹‡ ¤‹‡œ)‹ ˜pŽ]2J“‡Œ —‹ •¤#˜ ¸›“Г2¸ïì𐒖ÞÆiŸO•—]‹‡“”Ž]•¤–8˜[;˜4ŸO‹‡³Ž]]Œ™˜46š›–8Jœ[ŒŽJ•—–™˜®­ ÷.

(16) È  ‹’ŽÅÊòñŸO‹ ˜;–™ŽJ‹ŽJ°4‹aŸO4Œ =A¢4Œ8“’‹ŽJ–hŽ]°;‹ ¤•—˜;‹Œ ·™‹“”Ž]–™ A¢?Œ™“’‹µ,²4Œ˜4Ÿf —‹’Ž ó É ŸO‹‡˜;–Ž]‹Œž?Œ™J‹«š›–8ÅÊòñ^; Ô èO­ ù Õ Ê ñ ;

(17) Ìó Í Î ó Ï Î™Ñ9VÑ9VÑ9 ÑV9Î ó Ò Ó Êòñ ²ŽJ°;‹*ŸO?Œ 4J¢4Œ™“ ‹iŽ]–[[Ê ²™“ –™˜4J•¬6Ž£%–šZŒ — O ¤•—˜;‹‡Œ™ºœ[Œ¢;¢;•—˜;¡pYš›J–8œmnÊ ŽJ–9ŽJ°4‹ J‹Œ G˜G;œNžH‹ £™;ÐÊ ç ì  È i  ¤‹‡œ)‹ ˜pŽ]_•—˜àÊ ñ Œ]‹“’–,·,Œ]•¬Œ˜pŽiŽ]‹ ˜4J–™£™; Ô è;­ áüªÕ B Ù Ö Éó É µ °4‹ ]‹ Ö É ŸO‹‡˜;–Ž]‹‡9Ž]°;‹“ –™œ)¢?–8˜;‹ ˜pŽ£9–š_ŽJ°;‹‹  —‹ œ)‹ ˜pŽ B ­à* –Ž]‹)ŽJ°4ŒŽJ;ž4]“’]•¤¢OŽ£Œ]‹[4J‹‡Ÿ´š›–™Ž]°;‹ “ –™œ)¢H–™˜;‹‡˜8Ž£_•¤˜ŽJ°4‹aŸO4Œ =A¢4Œ8“’‹8­ È  ‹’Ž ˆ Ù á É Ü ³É Œ˜?Ÿ —‹’Ž ‚ Ùõô É Ü AÉ ­³¯_°;‹9Ž]‹ ˜4J–™_¢;]–OŸO4“’Ž*ž?‹ Ž6µ2‹‡‹ ˜ ˆ Œ˜4Ÿ ‚ •¬ ŸO‹ ˜;–™ŽJ‹Ÿö; ñ Ù ˆ[÷ ‚ æç ø Éãâ Ù á É ô â Ô è;­ á8áºÕ µ °4‹ ]‹«Ž]°;‹aJ¥pœž?–8  ÷ ŸO‹‡˜;–Ž]‹‡TŽ]°;‹9ŽJ‹‡˜4A–8 ¢;]–GŸ;4“”Ž­ œAp^œ ¡w}‡ ¿Š[z‡àŠRrE ùt ¡£‚x{vr_Š ¯_°;‹0¸”ݱߪƏ2Ý –š=ŒaŽJ‹ ˜?A–8T“‡Œ˜žH‹J‹‡·™‹‡Œ™ ¤‹Ÿ)žG¥)ŽJ°;‹˜G;œNžH‹ _–šYA;ž?J“ J•—¢OŽ]TŒ˜?ŸJ;¢?‹‡]]“’]•—¢OŽ]2–š®Ž]°;‹‹  —‹ œ)‹ ˜pŽ ŸO‹‡]“’]•—¢OŽJ•—–™˜®­ ¯_°4‹«Ž]‹ ˜4J–™ ñ. ñ ú øCû™ü¼ÿ ýãüðýãû¼þÿ Ô è;­ á è Õ Õ ­³½ŽJ‹‡˜4J–™T°4Œz·G•¤˜;¡Nž?–™ŽJ°“ –,·,Œ]•—Œ™˜8ŽTŒ˜4Ÿl“ –™˜pŽJ£Œz·,Œ]•—Œ™˜pŽi“’–™œ)¢H–™˜;‹‡˜pŽ]³•¬_JŒ™•—Ÿ[ŽJ–NžH‹. •¬2–š®–™£ŸO‹‡ Ô %Î Œhœ)•¤‘G‹ŸlŽ]‹ ˜4J–™ Š;]–™œŽ]°;•¬NŸO‹ ¹4˜;•¤ŽJ•—–™˜Wµ2‹[“‡Œ˜ A‹‡‹hŽJ°4ŒŽŒf·™‹‡“’ŽJ–8•—ŒfŽ]‹ ˜4J–™–™šT–8]ŸO‹‡ Ô á Î üªÕ ­¯M‹‡˜4J–™£«–™šT–8]Ÿ;‹  Ô è Î üFÕ ² Ô ü Î è Õ Œ˜?Ÿ Ô á Î ïá Õ “ Œ™˜•—˜f“ –™œ)¢H–™˜;‹‡˜8ŽTš›–™]œÒ–š Ž]‹ ˜fž?‹9Œ™]A–O“ •—ŒŽJ‹‡Ÿlµ •¸Ž]°œ[Œ,Ž]J•¬“’‹ ­%¯_°;‹Ž6µT–N•—˜4ŸO•¬“’‹‡_“‡Œ˜žH‹ A‹‡‹ ˜¼Œ™«ŽJ°;‹[Œ8Ÿ;ŸO]‹‡]Ž]–fŒŽ6µT–fŸO•—œ)‹ ˜4J•—–™˜4Œ™ %Œ™J£Œz¥™­–Ž]‹NŽ]°4Œ,Ž9ŽJ°4‹Ž]°;‹)–™£ŸO‹ 9°4Œ8˜;–Ž]°;•¤˜4¡Ž]–Ÿ;–µ •¤ŽJ° ŽJ°;‹A•—î ‹–™šYŽ]°;‹aŒ]£Œz¥l•~­ ‹™­2•¸šº•¤Ž•¬«èz‘4è–™«¨,‘;¨h‹’Ž]“™­T¯_°4‹a˜;‹’‘GŽ*–8]ŸO‹‡_ŽJ‹‡˜4A–8]‡²;‹8­ ¡?­ Ô ¨ Î Fü Õ ²4°?Œz·™‹9–™˜;‹œ)–8J‹ •¤˜?ŸO‹’‘[•—˜lŽJ°4‹ •—T“ –™œ)¢?–8˜;‹ ˜pŽ2š›–™]œÒŒ™˜4Ÿl“‡Œ˜)ŽJ°4‹ ]‹’š›–™]‹–™š ŽJ‹‡˜lž?‹•—˜8Ž]‹ ]¢;]‹’ŽJ‹Ÿ[Œ™TŒN“ ;ž;•¬“Œ]£Œz¥N–š®˜G;œž?‹‡]‡­ È  —½¿•—˜;‹‡J‹‡Œ™“ T–™œ[˜4Ÿ´Œ™¢;–™¢;£•—ŸO˜;‹‡¡8«= ;Î]‹ Êò˜4Jñ –™‚² Êòñ ñ –ç šTŽ6¥Gì ¢? ‹ Ô èO² ªü Õ Ê ÷ Ê)­¯_°;•¬A¢?Œ™“’‹8² Ê ÷ Êl²=“’–8˜4J•—AŽ]9–™šTŒ — ³œ; ¸Ž]• È œ[½nŒ™A¢;‹“’¢;–™•—˜;˜?¡8Ÿ= –8; ]ÊòŸ;‹ ñ º‚Ž]‹ Ê ˜4J–™ç ³–™š»ì  Ž6¥p¢H‹ Ô á ² ºá Õ Ê ÷ Ê ñ ­º¯_°;•—2A¢?Œ™“’‹8F² Ê ÷ Ê ñ ²G“’–™˜?A•¬6Ž£³–™šZŒ™ ¤ ?œN; ¤ŽJ•? ¤•—˜;‹Œ È  —½Ä•—˜;‹‡J‹‡Œ™“’T–8œ[˜4ŸVŒ™¢;–8¢;]•—Ÿ;˜;‹ ¡8«= ;ÎJ‹ Ê ˜?Aà–8«Ê –™šiç Ž6¥G¢?ì‹  Ô ü ² è Õ Ê ñ¬÷ Ê ñ ­N¯_°;•¬A¢4Œ8“’‹8ö² Ê ñ¬÷ Ê ñ ²®“’–8˜4A•¬AŽ]–š2Œ™ ¤ %œ; ¸Ž]• .  . . . . . .

(18) . ù.

(19) œAp”²  Áx{~ª µ~Fr_x rE ùt ¡£‚x{vr_Š ½|ŽJ‹‡˜4A–8 –šY–™£ŸO‹  Ô %Î Õ •¬*ŒhœN; ¤ŽJ•M ¤•—˜;‹‡Œ™ œ[Œ¢;¢;•—˜;¡"; Ê ñ ‚Ê ñ ÑV9ÑV9Ñ ‚Ê ñ ÃÊ àÊ ÑV9Ñ VÑ àÊ ç ì  œAp ¡w}‡ ˜ µŠ~‹ ¡£‚x{vr_Š. .  . . . . .  . 

(20) .  . .  . Ô è;­ á ¨ Õ. . . ½ )œ ‹’Ž]J•¬“–™«J“‡Œ ¬Œ ¢;]–OŸO4“”ŽŸO‹’¹4˜4‹‡Ÿ;•—AŽ]Œ™˜4“’‹ •¤˜VŒl·™‹‡“’ŽJ–8*A¢4Œ8“’‹8­T¯_°;‹ŸO•—AŽ]Œ™˜4“’‹Nž?‹ Ž6µ2‹‡‹ ˜Ž6µT–[·™‹‡“’ŽJ–8] ˆ Œ˜4Ÿ ‚ •¬™; Ï Ù :Ôވ ¼Ø‚ Î*ˆ ´‚ Õ Ô è;­ á ý Õ µ °4‹ ]‹Ð; Ÿ;‹ ˜;–™ŽJ‹‡ ŽJ°;‹aŸO•¬AŽ]Œ˜?“’‹  ŸO‹ ˜;–™ŽJ‹_ŽJ°;‹9œ)‹’Ž]J•¬“ŽJ‹‡˜4J–™ ˆ Ù á É Ü ÉûŒ™˜4Ÿ ‚ Ù ô É Ü É Y´]•¤ŽJ•—˜;¡)ŽJ°;‹a]Œœ)‹9‹’‘O¢;]‹‡]J•¤–8˜4J•¤˜4¡)“ –™œ)¢H–™˜;‹‡˜8Ž£_¡™•—·™‹ Ï Ù ÉèâªÔ á É ô É Õ Ô á â ô â Õ Ô è;­ á ç Õ ¯_°;‹aAŽ]Œ˜?Ÿ;Œ£Ÿlœ)‹’Ž]J•¬“9•—˜fì  µ °;‹‡˜f°?Œz·p•—˜;¡)Œ˜“ Œ™AŽ]‹‡J•—Œ™˜ê  5 ž4Œ8A‹°4Œz·8‹«Ž]°;‹9š›–™ — —–,µ •¤˜;¡)“’–8œ)¢?–8˜;‹ ˜pŽ]= ; áÕüøü Éèâ Ù ü á ü Ô è;­ á ï Õ ü

References

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