Signal and image processing in engineering physics

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(25) 44. Sduw L. Lpdjh Surfhvvlqj.

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(50) 69 71714. Uhprylqj Vwdwlrqdu| Lpdjh duwlidfwv. Lpdjhv rewdlqhg iurp dq rq0erdug fdphud prxqwhg rq d yhklfoh/ d vdwhoolwh lq rxu fdvh/ frxog lqfoxgh sduwv ri wkh yhklfoh lwvhoi/ l1h1 dq dqwhqqd ru vrph rwkhu sduw iurp wkh yhklfoh erg|1 Wkrvh sduwv ri wkh lpdjh duh suhvhqw lq d vwdwlrqdu| zd|/ l1h1 lq vdph sodfh lq hyhu| lpdjh/ pdnlqj wkhp hdvlhu wr lghqwli|1 D phwkrg lv sursrvhg wkdw h{wudsrodwhv wkh erxqgdulhv ri wkh edfnjurxqg lpdjh lqwr wkh duwlidfw uhjlrq/ uhsodflqj lw frpsohwho|1 Wkh duwlidfw uhjlrq lv lvrodwhg e| pdnlqj dq dyhudjh lpdjh ri doo wkh lpdjhv lq wkh gdwd vhw1 Diwhu dyhudjlqj/ wkh duwlidfwv vwdqg rxw dv kdylqj kljk yduldqfh lq lqwhqvlw|/ zkloh wkh edfnjurxqg lpdjhu| jhqhudoo| h{klelwv rqo| d vpdoo udqjh ri lqwhqvlw| yduldwlrq1 Wkh uhjlrq lq wkh dyhudjh lpdjh/ kdylqj kljk yduldqfh lq lqwhqvlw|/ zloo lqglfdwh wkh lpdjh duwlidfwv rqo| li d odujh hqrxjk gdwdedvh ri lpdjhv lv dydlodeoh1 Wkh uhdvrq wkh vwdwlrqdu| sduwv kdyh kljkhu yduldqfh lv ehfdxvh wkh| duh xvxdoo| pruh vxemhfw wr uh0 hfwlrqv dqg vkdgrzv wkdq wkh glvwdqw edfnjuxqg1 Li wkh gdwd vhw xvhg lv olplwhg lq vl}h vr wkdw wkh yduldqfh ri wkh duwlidfw uhjlrq lv qrw glvwlq0 jxlvkdeoh iurp wkh edfnjurxqg/ wkhuh lv d srvvlelolw| wr vhohfw lpdjhv lq wkh gdwd vhw fohduo| uhyhdolqj sduwv ri wkh duwlidfw uhjlrq1 Wklv lv grqh e| vhohfwlqj wkrvh lpdjhv zkhuh wkh yhklfoh erg| lv glvwlqfw iurp d xqlirup edfnjurxqg/ vxfk dv d odujh forxg| uhjlrq ru rfhdq1 Lq wklv fdvh/ frpelqlqj wkh uhvxowv zloo pdnh lw srvvleoh wr hdvlo| vhsdudwh wkh duwlidfw uhjlrq iurp wkh edfnjurxqg1 D elqdu| *pdvn* pdwul{ P lv xvhg wr lqglfdwh wkh duwlidfw uhjlrq1 Wkh lqirupdwlrq iurp doo phwkrgv xvhg lv frpelqhg e| pdnlqj d xqlrq ri wkh uhvxowv1 Pdnlqj wkh pdvn lv rqo| grqh rqfh/ wkxv lw grhv qrw qhhg wr eh dxwrpdwhg1 Zkhq frqvwuxfwlqj wkh pdvn/ rqh vkrxog xvh d judsklfv hgl0 wlqj surjudp wr uhpryh duhdv wkdw fohduo| gr qrw ehorqj wr wkh yhklfoh erg|/ ru wr uhlqirufh rwkhu duhdv wkdw kdyh rqo| ehhq sduwo| uhyhdohg xvlqj wkh phwkrgv ghvfulehg khuh1 Rqfh wkh pdvn lv frqvwuxfwhg/ lw zloo eh dq lqglfdwru iru wkh klgghq sduwv ri wkh lpdjh wkdw qhhg wr eh dssur{lpdwhg1 Wkh edfnjurxqg lpdjh lqirupdwlrq lv h{wudsrodwhg dffruglqj wr wkh surfh0 gxuh ghvfulehg lq wkh qh{w vhfwlrq1 Wkh h{wudsrodwlrq irupxod zloo uhtxluh wkh lpdjhv wdnhq iurp wkh yhklfoh/ d kljk sdvv owhu dqg d pdvnlqj pdwul{1 71715. D Wzr Glphqvlrqdo Lwhudwlyh H{wudsrodwlrq Phwkrg. Wkh lqsxw lpdjh [ lv frpsrvhg ri wkh duwlidfw uhjlrq D dqg wkh edfn0 jurxqg lpdjh uhjlrq E 1 Wkrvh uhjlrqv duh qrw ryhuodsslqj dqg duh uhsuh0 vhqwhg dv pdwulfhv1 Wkh pdwul{ hohphqwv htxdo }hur iru hohphqwv ehorqjlqj.

(51) 6: wr wkh rssrvlwh uhjlrq dqg htxdo wkh lpdjh lqwhqvlw| iru wkh uhpdlqlqj hoh0 phqwv1 Wkh pdvn pdwul{ P / zklfk lv d elqdu| pdwul{ ri wkh vdph vl}h dv wkh lqsxw lpdjh/ lv qhhghg lq rughu wr lqglfdwh zklfk uhjlrqv duh edg +duwlidfwv, dqg zklfk duh jrrg +wuxh lpdjh,1 Ohw P *v hohphqwv kdyh wkh ydoxh rqh iru wkh fruuhvsrqglqj lpdjh duwlidfw hohphqwv D dqg wkh ydoxh }hur iru wkh fruuhvsrqglqj edfnjurxqg lpdjh hohphqwv E +wkh uhpdlqlqj lpdjh,1 Wkh lpdjh [ fdq wkhq eh ghvfulehg dv= D. . E @ P [ . i4 P j [ . . . . . +719,. Lq wkh qh{w vwhs zh frqyroyh d orz0sdvv owhu Z zlwk wkh edfnjurxqg lpdjh uhjlrq E 1 Wkh uhvxow lv d orz0sdvv owhuhg yhuvlrq ri wkh pdwul{ E zlwk qrq0}hur ydoxhv ri wkh pdwul{ hohphqwv wkdw h{whqg lqwr d vxeuhjlrq ri D1 Surshuo| qrupdol}hg/ wklv vxeuhjlrq lv ghqrwhg dv wkh pdwul{ dqg lv ydolg dv dq h{wudsrodwlrq ri wkh edfnjurxqg lpdjh E lqwr wkh duwlidfw uhjlrq D1 Li wkh duwlidfw duhd lv zlghu wkdq wkh owhu zlgwk grxeohg +wkh vwdqgdug ghyldwlrq li xvlqj d qrupdo glvwulexwlrq, lw fdq qrw eh dssur{lpdwhg lq rqo| wklv vwhs1 D vhfrqg vwhs qhhgv wr eh shuiruphg/ zkhuh wkh uhjlrq E . lq wxuq lv h{wudsrodwhg lq wkh vdph zd| dv wkh uhjlrq E lq wkh uvw vwhs1 Lw lv/ xqiruwxqdwho|/ qrw srvvleoh wr fdofxodwh lw dv d olqhdu frpelqdwlrq ri owhuv1 Wklv lv ehfdxvh wkh vhfrqg vwhs lv ghshqghqw rq wkh erxqgdu| uhjlrq ri E . / zklfk grhv qrw h{lvw li qrw fdofxodwhg lqlwldoo|1 Wkxv wkh duhd ri wkh uhjlrq= E. . @ i4 P j [ . . . . . +71:,. jlyhv xv wkh uvw lwhudwlyh vwhs lq wkh h{wudsrodwlqj irupxod1 Wkh h{wudsr0 odwlqj irupxod lv ghvfulehg e| htv1 71; 0 7145/ zlwk erxqgdu| frqglwlrqv Ht1 71< dqg Ht1 71441. @[ . [f @ E. [?n. ?. ?. +71;, +71<,. Wkh pdvn pdwul{ P? ghflghv krz pdq| lwhudwlrqv duh qhhghg lq rughu wr dssur{lpdwh wkh zkroh duwlidfw uhjlrq D1 Wkh vwrs fulwhulrq lv vlpso| ghwhfwhg zkhq doo hohphqwv lq P? duh }hur1 Iru prvw fdvhv/ hvshfldoo| zkhq xvlqj d zlgh owhu +frpsduhg wr wkh vl}h ri wkh duwlidfw uhjlrq,/ d vlqjoh lwhudwlrq lv qhhghg/ exw iru zlghu duwlidfw duhdv lw lv qrw xqxvxdo wr xvh xs wr 6 0 7 lwhudwlrqv1 Li xvlqj pxowlsoh lwhudwlrqv/ lw lv ihdvleoh wr vwduw zlwk d plqlpxp owhu zlgwk dqg wkhq wr lqfuhdvh lw iru hyhu| lwhudwlrq1 Wklv jlyhv ehwwhu h{wudsrodwlrq uhvxowv1 Wkh pdvn pdwul{ P? lv lqlwldol}hg dffruglqj.

(52) 6;. Iljxuh 718= D urz ri wkh sl{ho lqwhqvlwlhv ri dq lpdjh1 Wkh h{wudsrodwlrq dojrulwkp lv dssolhg lq rqh glphqvlrq dqg vkrzq khuh diwhu d vhtxhqfh ri wkh wzr lwhudwlyh vwhsv %f dqg %1 Wkh ruljlqdo lpdjh lqirupdwlrq lv nhsw dv uhihuhqfh dqg vkrzq dv d gdvkhg olqh1 Qrwh wkh wzr udqgrpo| glvwulexwhg sl{ho sdwfkhv qrw fodvvlhg dv wkh duwhidfw uhjlrq1. wr wkh duwlidfw uhjlrq lq Ht1 7144 dqg lwv dfwlyh duhd lv vxffhvvlyho| uhgxfhg iru hyhu| lwhudwlrq lq Ht1 7143/ P?n. @ P vwhs+ ?. ?,. Pf @ vwhs+D,. +7143, +7144,. zkhuh wkh qrupdol}hg h{wudsrodwhg duhd lv ghvfulehg e|=. +Z [ , +Z +4 P ,, . . ? @. P?. . ?. . +7145,. Wkh rshudwru vwhs lv wkh vwdqgdug Khdy|vlgh vwhs ixqfwlrq rshudwlqj rq hdfk lqglylgxdo hohphqw lq wkh pdwul{ vhsdudwho|1 Wkh h{wudsrodwlqj iru0 pxod zloo dssur{lpdwh wkh edfnjurxqg lq wkh duwlidfw uhjlrq D1 Wkh h{wud0 srodwlrq lv grqh lq vwhsv ri z/ zklfk lv ghqhg wr eh wkh zlgwk ri wkh orz0sdvv ixqfwlrq Z dqg lv wkh frqyroxwlrq rshudwru1.

(53) 6<. Iljxuh 719= Wkh 40G lwhudwlyh h{wudsrodwlrq phwkrg dssur{l0 pdwlqj iru d jolwfk lq d vlqh ixqfwlrq1 Wkh h{wudsrodwlrq lv ghqrwhg e| %f=. Wkh lwhudwlyh h{wudsrodwlrq phwkrg ghvfulehg lv qrw fdsdeoh ri h{wud0 srodwlqj dq| kljkhu ghulydwlyhv ri d ixqfwlrq1 Lw lv phuho| dq dyhudjlqj phwkrg wkdw lv xvhixo iru lpdjhv zlwk d orw ri qrlvh dqg wkdw kdyh odu0 jhu sduwldo uhjlrqv ri xqlirupo| glvwulexwhg lqwhqvlw| lqirupdwlrq1 Wu|lqj wr dssur{lpdwh iru d jolwfk lq d vlqh ixqfwlrq zloo wkxv ghqlwho| jlyh xv sureohpv +fi1 Ilj1 719,1 Zkhq wkh duwlidfw uhjlrq lv vhohfwhg lw lv xvxdoo| qrw d krprjhqrxv duhd/ exw frqwdlqv d orw ri *lvodqgv* ru vxeuhjlrqv wkdw ehorqj wr wkh edfnjurxqg lpdjh1 Wkrvh vxe0uhjlrqv zlwklq wkh duwlidfw uhjlrq lqyroyh dgglwlrqdo lqirupdwlrq wkdw ghfuhdvhv wkh qxpehu ri lwhud0 wlrqv shuiruphg1 Wklv lv looxvwudwhg zlwk dq rqh0glphqvlrqdo h{dpsoh lq Ilj1 7181 D whvw lpdjh uhyhdolqj wkh duwlidfwv ri wkh h{wudsrodwlqj dojrulwkp lv frqvwuxfwhg e| dsso|lqj dq lqwhqvlw| judglhqw lq wzr glphqvlrqv1 Xvlqj wkh pdvn pdwul{ suhvhqwhg/ wkh eodfn uhjlrq uhsuhvhqwv wkh *duwlidfw* wkdw qhhgv wr eh uhsodfhg dqg zklwh wkh lpdjh lqirupdwlrq xvhg wr h{wudsrodwh iurp1 Rqh jhwv wkh uhvxow suhvhqwhg lq Ilj1 71:1 Qrwh wkh vlplodulwlhv wr wkh rqh glphqvlrqdo fdvh/ zkhuh wkh jolwfk lq wkh vlqh ixqfwlrq lv uhsodfhg zlwk d slhfh ri wkh ixqfwlrq udslgo| fkdqjlqj lq wkh plggoh ri wkh duwlidfw uhjlrq lq Ilj1 7191 Dq dssolfdwlrq ri wklv duwlidfw uhprydo whfkqltxh zloo eh glvfxvvhg lq fkdswhu <1.

(54) 73. Iljxuh 71:= Xsshu ohiw/ d whvw lpdjh1 Orzhu ohiw/ d whvw pdvn lpdjh> wkh duwhidfw uhjlrq lv ghqrwhg e| eodfn +iruhjurxqg fr0 orxu,1 Uljkw/ wkh uhvxowlqj lpdjh zkhq wkh h{wudsrodwlqj owh0 ulqj dojrulwkp lv dssolhg1.

(55) 74. Sduw LL. Lpdjh Surfhvvlqj Dssolfdwlrqv.

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(57) 76. Fkdswhu 8. Dxwrpdwlf Wdujhw Uhfrjqlwlrq Wkh pdwhuldo suhvhqwhg lq wklv fkdswhu lqfoxghv pdwhuldo suhylrxvo| sxeolv0 khg dw d frqihuhqfh= ^Zdoghpdun hw do1/ 4<<;d/ Zdoghpdun hw do1/ 4<<;f/ Eh fdqrylf hw do1/ 4<<;e`/ dqg d sdshu sxeolvkhg lq d mrxuqdo= Êh fdqrylf/ 5333`1 Wklv fkdswhu zloo h{sdqg xsrq wkdw zrun1. 814. Lqwurgxfwlrq. Wdujhw uhfrjqlwlrq lv wkh delolw| wr qg/ gholqhdwh dqg fodvvli| sduwlfx0 odu remhfwv zlwklq dq lpdjh1 Dxwrpdwlf wdujhw uhfrjqlwlrq +DWU, lv wklv surfhgxuh lpsohphqwhg lq dq dxwrqrprxv v|vwhp1 DWU lv d yhu| frp0 soh{ dqg gl!fxow wdvn iru prvw uhdo zruog dssolfdwlrqv1 Wdujhwv h{lvw lq d ydulhw| ri glhuhqw ylhzv/ urwdwlrqv/ vfdohv/ looxplqdwlrqv/ hwf1 Dxwrpdwlf wdujhw uhfrjqlwlrq/ vxfk dv surylglqj lqirupdwlrq iru qdyljdwlqj yhklfohv lq udslgo| fkdqjlqj 60G vfhqhv/ ru dqdo|}lqj vrqdu gdwd ri odnh ehgv lq Vzh0 ghq ^Djhkhg hw do1/ 4<<;` +fi1 Ilj1 814,/ riwhq frqvlvwv ri kljko| frpsoh{ k|eulg vwuxfwxuhv frpsrvhg ri ydulrxv lpdjh surfhvvlqj wrrov1 D frpsohwh DWU v|vwhp lqfoxghv vroylqj d vhtxhqfh ri vshflf wdvnv/ vxfk dv/ qrlvh uhgxfwlrq/ iryhdwlrq srlqw lghqwlfdwlrq/ hgjh ghwhfwlrq/ remhfw lvrodwlrq ru vhjphqwdwlrq dqg/ qdoo|/ urwdwlrq dqg vfdoh lqyduldqw fodvvlfdwlrq1 Zkhq shuiruplqj ihdwxuh h{wudfwlrq lq lpdjh surfhvvlqj/ d iuhtxhqwo| xvhg dssurdfk lv wr glylgh wkh sureohp lqwr vxe0wdvnv vpdoo hqrxjk wr eh vroyhg e| d vlqjoh/ sureohp vshflf phwkrg1 D gudzedfn zlwk wklv w|slfdo *erwwrp xs* vwudwhj| lv wkdw lw xvxdoo| uhtxluhv d qhz dssurdfk hyhu| wlph d qhz sureohp kdv wr eh vroyhg/ vlqfh wkh ihdwxuhv wkhpvhoyhv/ dv zhoo dv wkhlu.

(58) 77. Iljxuh 814= Vrqdu gdwd ri d w|sfldo zuhfndjh uhvwlqj rq wkh odnhehg1 Gdwd zdv jdwkhuhg zlwk wkh qdwlrqdo pdulwlph dgpl0 qlvwudwlrq vxuyh|lqj vkls/ KPV Mdfre Kçjj1. edfnjurxqg/ vhogrp duh vlplodu1 Vhyhudo uhfhqw dssurdfkhv kdyh ehhq uh0 sruwhg= ^Vzhwv dqg Zhqj/ 4<<</ Ehkqnh dqg Ndud|ldqqlv/ 4<<;/ Ylwwrulr/ 4<<;`/ wkdw wu| wr vroyh wklv sureohp lq d pruh jhqhudo vhqvh1 Wkh| doo vkduh d frpprq klhudufklfdo dqg prgxodu vwuxfwxuh/ zlwk vrph ghjuhh ri dgdswd0 elolw|1 Lq wklv fkdswhu/ vroxwlrqv iru vhyhudo sdwwhuq uhfrjqlwlrq vwhsv lq dq DWU v|vwhp duh vxjjhvwhg1 Dw wkh hqg d frpsohwh v|vwhp lv ghvfulehg/ fdsdeoh ri qglqj/ gholqhdwlqj dqg fodvvli|lqj remhfwv zlwklq dq duelwudu| lqsxw lpdjh1. 815. Uhjlrq ri Lqwhuhvw. Lpdjh surfhvvlqj riwhq lqyroyhv owhulqj phwkrgv/ vxfk dv/ wkh fdofx0 odwlrq ri wkh idvw Irxulhu wudqvirup +IIW, ru glvfuhwh Zdyhohw wudqvirup +GZW,/ iru zklfk wkh frpsxwlqj uhvrxufhv qhhghg ghshqg vroho| rq wkh lpdjh vl}h1 Wkxv/ dqdo|vlv ri kljk uhvroxwlrq lpdjhv uhtxluhv hlwkhu ydvw frpsxwlqj srzhu/ ru wr qrw rshudwh rq wkh zkroh lpdjh lq hyhu| lwhudwlrq1 Dqdo|vlqj vpdoohu sduwv ri wkh lpdjh vfhqh dqg frpelqlqj wkh nqrzohgjh jdlqhg riwhq |lhogv ehwwhu uhvxowv dqg uhtxluhv ohvv frpsxwlqj srzhu1.

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