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Морфологический анализ цветных (спектрозональных) изображений

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Ii?oieiae/aneee aiaeec oeaaoiuo (niaeo?iciiaeueiuo) ecia?aaeaiee.

Iuoueaa TH.I.

Iineianeee ainoaea?noaaiiue oieaa?neoao, Iineaa, ?inney

1. Aaaaeaiea

Oi?ioi ecaanoii, /oi ecia?aaeaiey iaeiie e oie aea noeaiu, iieo/aiiua
i?e ?acee/iuo oneiaeyo inaauaiey e(eee) eciaiaiiuo iioe/aneeo naienoaao
iauaeoia iiaoo ioee/aoueny ?aaeeeaeueii. Yoi ianoiyoaeuenoai ii?iaeaeaao
cia/eoaeueiua o?oaeiinoe a i?eeeaaeiuo caaea/ao aiaeeca e eioa?i?aoaoeee
ecia?aaeaiee ?aaeueiuo noeai, a eioi?uo ?aoaiea aeieaeii ia caaenaoue io
oneiaee ?aaeno?aoeee ecia?aaeaiee. ?a/ue eaeao, iai?eia?, i caaea/ao
auaeaeaiey iaecaanoiiai iauaeoa ia oiia ecaanoiie ianoiinoe, ecaanoiiai
iauaeoa ia i?iecaieueiii oiia i?e iaeiio?iee?oaiuo oneiaeyo inaauaiey, i
caaea/a niaiauaiey ecia?aaeaiiee iaeiie e oie aea noeaiu, iieo/aiiuo a
?acee/iuo niaeo?aeueiuo aeeaiaciiao e o.ae.

Iaoiaeu ii?oieiae/aneiai aiaeeca, ?ac?aaioaiiua aieaa aeanyoe eao oiio
iacaae, [1-5], aeey ?aoaiey ia?a/eneaiiuo caaea/, auee a iniiaiii
i?eaioe?iaaiu aeey i?eiaiaiey e /a?ii-aaeui ecia?aaeaieyi e ieacaeenue
aeinoaoi/ii yooaeoeaiuie, [5-11].

Iaaeaeo oai, ii iaiueoae ia?a aeaa ianoiyoaeuenoaa oeacuaatho ia
oeaeaniia?aciinoue ?ac?aaioee ii?oieiae/aneeo iaoiaeia aiaeeca oeaaoiuo
ecia?aaeaiee. Ai-ia?auo, a caaea/a iaia?oaeaiey e auaeaeaiey iauaeoa
iineaaeiee, eae i?aaeei, i?aaeaea anaai oeaaoii ioee/aaony io oiia.
Ai-aoi?uo, iienaiea oi?iu ecia?aaeaiey a oa?ieiao oeaaoa iicaieeo
i?aeoe/anee ono?aieoue yooaeo oaiae e aeeyiea iaii?aaeaeaiiinoe a
i?ino?ainoaaiiii ?ani?aaeaeaiee eioaineaiinoe niaeo?aeueii iaeii?iaeiiai
inaauaiey.

2. Oeaao e y?einoue niaeoiciiaeueiiai ecia?aaeaiey.

anee ioaa/athuea aio auoiaeiua neaiaeu anao aeaoaeoi?ia iaeeiaeiau:

.

.

.

. Aeey iaa

. (1)

Ionthaea neaaeoao

Eaiia 1. ss?einoue fe e oeaao (e ethaie aaeaeeoeaiie niane e(*)
eceo/aiee e1((),…,em((), m=1,2,… ii?aaeaeythony y?einoyie e oeaaoaie
neaaaaiuo.

a ethaie aaeaeeoeaiie niane eceo/aiee ia eciaieo ie oeaaoa, ie y?einoe
iineaaeiae.

, j=1,…,n.

Aeey anyeiai eceo/aiey e(*) iiaeii caienaoue ?aceiaeaiea

, (1*)

,

, j=1,…,n, e ia caaenyo iaiin?aaenoaaiii io niaeo?aeueiiai ninoaaa
eceo/aiey e(*).

.

. A oaeie oi?ia ?aaainoai (1*) i?aaenoaaeyao “aaeain eceo/aiee”.

, i,j=1,…,n.

, i,j=1,…,n.

.

(2**)

ii?aaeaeei ?aaainoaii

, (2)

, – (-ecia?eiua ooieoeee ia iiea c?aiey X, oaeea, /oi

.

. Eeann oeaaoiuo ecia?aaeaiee iaicia/ei LE,n.

iacuaaaony oeaaoiui ecia?aaeaieai, a oneiaea

(2*)

oneiaeai oece/iinoe ecia?aaeaiee f(().

, oaeiaeaoai?ythuea oneiaeth: y?einoue j(x)=1. *a?ii-aaeui aa?eaioii
oeaaoiiai ecia?aaeaiey f(*) aoaeai oaeaea iacuaaoue oeaaoiia
ecia?aaeaiea b((), eiathuaa a eaaeaeie oi/ea O oo aea y?einoue, /oi e
f(*), b(x)=f(x), x(X, e aaeue oeaao, ((x)=b(x)/b(x)=(, x(X.

3. Oi?ia oeaaoiiai ecia?aaeaiey.

o aaeoi?a f(x) iiaeao eciaieoueny aeeeia, ii iai?aaeaiea inoaiaony
iaeciaiiui.

.

iiaeao ieacaoueny iaeeiaeiaui.

Eae i?aaeei, neaaeoao o/eouaaoue iaiinoiyinoai iioe/aneeo oa?aeoa?enoee
noeaiu e o.ae. Ai anao neo/ayo oi?ia ecia?aaeaiey aeieaeia auoue
eiaa?eaioia ioiineoaeueii i?aia?aciaaiey ec auaeaeaiiiai eeanna e, aieaa
oiai, aeieaeia ii?aaeaeyoue ecia?aaeaiea n oi/iinoueth aei
i?iecaieueiiai i?aia?aciaaiey ec yoiai eeanna.

.

eciii?oiu e iaeeiaeiai aeaoaeueii oa?aeoa?ecotho noeaio, oioy e a
?aciuo oeaaoao.

.

, iineieueeo, anee iaeioi?ua aeaoaee oi?iu iauaeoa ia io?aaeaiu a
ecia?aaeaiee f(*), oi iie, oai aieaa, ia aoaeoo io?aaeaiu a g(*).

.

?anniio?ei oaia?ue aieaa iiae?iaii iiiyoea oi?iu aeey iaeioi?uo
oa?aeoa?iuo eeannia ecia?aaeaiee e eo i?aia?aciaaiee.

4. Oi?ia eoni/ii-iinoiyiiiai (iicae/iiai) oeaaoiiai ecia?aaeaiey.

, j=1,…,n, i=1,…,N, iai?a?uaiu. Iineieueeo niaeanii eaiia 2

, (3)

, – iai?a?uaiua ooieoeee.

. Aeey oaeiai ecia?aaeaiey i?eiai neaaeothuaa i?aaenoaaeaiea:

, (4)

aai /a?ii-aaeue aa?eaio

(4*)

, e oeaao ecia?aaeaiey (4)

(4**)

, i=1,…,N.

eiion:

. (4***)

v(a), i/aaeaeii, niaea?aeeony a n(N ia?iii eeiaeiii iiaei?ino?ainoaa

, (4****)

eioi?ia iaciaai oi?iie a(() a oe?ieii niunea.

, ia aeieaeai aucuaaoue iaaei?acoiaiey.

Ecia?aaeaiey ec eiiona(4***) eiatho oi?io, eioi?ay ia neiaeiaa, /ai
oi?ia a(() (4), iineieueeo iaeioi?ua ec ieo iiaoo eiaoue iaeii e oi aea
cia/aiea y?einoe eee(e) oeaaoa ia ?acee/iuo iiiaeanoaao Ai,
i=1,…………..,N. Oaeaea iiiaeanoaa ieacuaathony, ii nouanoao,
iauaaeeiaiiuie a iaeii, /oi e i?eaiaeeo e oi?iuaieth oi?iu ecia?aaeaiey,
iineieueeo iii io?aaeaao iaiueoa aeaoaeae oi?iu ecia?aaeaiiiai iauaeoa,
/ai ecia?aaeaiea (4). Yoi caia/aiea eanaaony e L(a(()), anee ?a/ue eaeao
i oi?ia a oe?ieii niunea.

.

Ecia?aaeaiea (3) eiaao ia eaaeaeii iiaeiiiaeanoaa Ai :

, anee e oieueei anee auiieiyaony ?aaainoai (4);

;

, i=1,……,N.

Aeieacaoaeuenoai . Ia iiiaeanoaa Ai y?einoue e oeaao ecia?aaeaiey (3)
?aaiu niioaaonoaaiii

, i=1,.…..,N.

, o.a. auiieiyaony (4).

. Iineaaeiaa ooaa?aeaeaiea i/aaeaeii (

Oeaao ecia?aaeaiey ii?aaeaeyaony eae yeaeo?iaeeiaie/aneeie naienoaaie
iiaa?oiinoe ecia?aaeaiiiai iauaeoa, oae e niaeo?aeueiui ninoaaii
iaeo/athuaai yeaeo?iiaaieoiiai eceo/aiey a oii aeeaiaciia, eioi?ue
eniieuecoaony aeey ?aaeno?aoeee ecia?aaeaiey. ?a/ue eaeao i
niaeo?aeueiii ninoaaa eceo/aiey, iieeaeathuaai iiaa?oiinoue iauaeoa e
niaea?aeauaai eae ?annayiiia oae e nianoaaiiia eceo/aiey iauaeoa.
Iineieueeo niaeo?aeueiue ninoaa iaaeathuaai eceo/aiey, eae i?aaeei,
i?ino?ainoaaiii iaeii?iaeai, iiaeii n/eoaoue, /oi oeaao ecia?aaeaiey
ianao eioi?iaoeeth i naienoaao iiaa?oiinoe iauaeoa, i aa oi?ia, a
y?einoue a cia/eoaeueiie noaiaie caaeneo e io oneiaee “inaauaiey”.
Iiyoiio ia i?aeoeea a caaea/ao ii?oieiae/aneiai aiaeeca oeaaoiuo
ecia?aaeaiee noeai aaaeiia cia/aiea eiaao iiiyoea oi?iu ecia?aaeaiey,
eiathuaai iinoiyiiue oeaao e i?iecaieueiia ?ani?aaeaeaiea y?einoe a
i?aaeaeao caaeaiiuo iiaeiiiaeanoa Ai , i=1,…,N, iiey c?aiey X.

Eoae, ionoue a niaeanee n eaiiie 3

, (5)

, ooieoeey gi(*) caaeaao ?ani?aaeaeaiea y?einoe

(6)

a i?aaeaeao Ai i?e iinoiyiiii oeaaoa

, i=1,…,N, (7)

i=1,.…..,N.

a oeaao ia Ai ?aaai

(7*)

Oi?io ecia?aaeaiey (5) ii?aaeaeei eae eeann anao ecia?aaeaiee

(8)

,

.

, i=1,…,N.

ia Ai ii?aaeaeyaony ?aaiui oeaaoo f(*) ia Ai, i=1,…,N).

. Iiiaeanoai, niaea?aeauaa ana oaeea ecia?aaeaiey

(9)

, niaea?aeauai ethaoth oi?io

, (10)

.

Aieaa oaeiaiia iienaiea oi?iu ecia?aaeaiey iiaeao auoue iieo/aii ia
iniiaa iaoiaeia aii?ieneiaoeee oeaaoiuo ecia?aaeaiee, a eioi?uo oi?ia
ii?aaeaeyaony eae iia?aoi? iaeeo/oaai i?eaeeaeaiey. A neaaeothuai
ia?aa?aoa aeaii i?aaenoaaeaiea oi?iu ecia?aaeaiey a aeaea iia?aoi?a
iaeeo/oaai i?eaeeaeaiey.

5. Caaea/e aii?ieneiaoeee oeaaoiuo ecia?aaeaiee. Oi?ia eae iia?aoi?
iaeeo/oaai i?eaeeaeaiey.

. Eia/a iiaeii n/eoaoue, /oi

(10*)

.

iiey c?aiey X.

ec oneiaey

. Oiaaea ?aoaiea caaea/e (11) eiaao aeae

, i=1,…,N, j=1,…,n, (12)

e eneiiia ecia?aaeaiea (4) caaeaaony ?aaainoaii

. (13)

ecia?aaeaiee (4), y?einoe e oeaaoa eioi?uo ia eciaiythony a i?aaeaeao
eaaeaeiai Ai , i=1,…,N.

.

.

. Aoi?ia ooaa?aeaeaiea neaaeoao ec ?aaainoaa

, auoaeathuaai ec (13). Iineaaeiaa ooaa?aeaeaiea neaaeoao ec ?aaainoa

,i=1,…,N auoaeathueo ec (12) e ?aaainoaa (1), a eioi?ii eiaeaen k
neaaeoao caiaieoue ia x(X. (

,[2].

(4***)

ecia?aaeaiey (4), a eiaiii

iiaeii ioiaeaeanoaeoue n oi?iie ecia?aaeaiey (4).

Aiaeiae/ii aeey /a?ii-aaeiai ecia?aaeaiey a(()

iiaeii ioiaeaeanoaeoue n oi?iie ecia?aaeaiey (4*), eae yoi naeaeaii a
?aaioao [2,3].

I?eia/aiey.

iiaeao auoue ?aaeeciaai iaoiaeii aeeiaie/aneiai i?ia?aiie?iaaiey, a
aeey iiiaeo caaea/ ii?oieiae/aneiai aiaeeca ecia?aaeaiee aeinoaoi/iui
ieacuaaaony eniieueciaaiea eeoue i?iaeoi?a I .

, iineaaeiaa, a naith i/a?aaeue ii?aaeaeyaony ecia?aaeaieai

,

, ii?aaeaeaiiue ?aaainoaii (13).

.

) aeieaeiui ia?acii i?aaieciaaiiie iineaaeiaaoaeueiinoe iicae/iuo
ecia?aaeaiee

(*)

A (*) iiaeii, iai?eia?, eniieueciaaoue oae iacuaaaioth en/a?iuaathuoth
iineaaeiaaoaeueiinoue ?acaeaiee [], oaeiaeaoai?ythuoth neaaeothuei
oneiaeyi

;

;

, niaiaaeaao n C.

[ ]. (

– ecia?eiuo) ?acaeaiee a eaiia (*).

Oiaaea

,

.

-ecia?eii, N=1,2,… (

Aii?in i oii, eaeei ia?acii iiaeao auoue iino?iaia en/a?iuaathuay
iineaaeiaaoaeueiinoue ?acaeaiee, ianoaeaeaaony a neaaeothuai ioieoa.

aii?ieneiaoeey f(*). Oae eae

, (14*)

=1,2,…,q. Oa oi/ee, eioi?ua niaeanii yoiio i?eioeeio iiaoo auoue
ioianaiu e ianeieueeei iiiaeanoaai, aeieaeiu auoue ioianaiu e iaeiiio ec
ieo ii i?iecaieo. O/eouaay yoi, oneiaeiny n/eoaoue, /oi caienue

, (14)

.

, a eioi?ii

(15)

, i=1,…,q, iiaeii auei n/eoaoue yeaeaaeaioiuie.

– caaeaiiua aaeoi?u Rn. ?aoaiea caaea/e

, eae anyeee iia?aoi? iaeeo/oaai i?eaeeaeaiey oaeiaeaoai?yao oneiaeth
F2=F, o.a. yaeyaony i?aeoi?ii.

, oi, eae iieacaii a [3], eneiiia ?acaeaiea X ninoieo ec iiiaeanoa

, e eiaao iaei iauaai n ?acaeaieai (14).

, i=1,…,q. Oiaaea

. (16)

), a /anoiinoe, ioiineoaeueii ia?aciaaiey oaiae ia f(*).

– ionoua, eee ioeaaie ia?u, niioaaonoaothuea ecia?aaeaiey eiatho aieaa
i?inooth oi?io.

.

oae, /oiau

.

Neaaenoaea 1.

.

e iia?aoi? I(3) e o.ae.

iiey c?aiey X.

, o?aaoaony ii?aaeaeeoue oeaao e ?ani?aaeaeaiea y?einoae iaeeo/oaai
i?eaeeaeaiey ia eaaeaeii Ai,i=1,…,N.

Aeey i?aeoeee, eae oaea auei ioia/aii, aieueoie eioa?an i?aaenoaaeyao
eeann ecia?aaeaiee (5), oeaao eioi?uo ia eciaiyaony a i?aaeaeao
iaeioi?uo iiaeiiiaeanoa iiey c?aiey, e caaea/e aii?ieneiaoeee
i?iecaieueiuo ecia?aaeaiee ecia?aaeaieyie oaeiai eeanna.

Caieoai ecia?aaeaiea (5) a aeaea

(17)

.

(18)

, a oi a?aiy, eae oeaao aeieaeai nio?aiyoue iinoiyiiia cia/aiea ia
eaaeaeii ec caaeaiiuo iiaeiiiaeanoa A1,…,AN iiey c?aiey X, (ni. Eaiio
3).

Oae eae

aeinoeaaaony i?e

, (20)

e ?aaai

(21)

Caaea/a (18) oai naiui naaaeaia e caaea/a

. (22)

. (23)

>0,

,

(17) yaeyaony ecia?aaeaiea

(24)

iia?aoi?a Oi (23), ioaa/athuee iaeaieueoaio nianoaaiiiio cia/aieth
ri,

; (25)

Iaaycea iaeeo/oaai i?eaeeaeaiey

(19*).

Aeieacaoaeuenoai. ?aaaionoai (24) e au?aaeaiea aeey Ii neaaeoao ec
(17),(20) e ?aoaiey caaea/e ia nianoaaiiua cia/aiey aeey iia?aoi?a Oi
(23). Iineieueeo Oi naiinii?yaeaiiue iaio?eoeaoaeueii ii?aaeaeaiiue
iia?aoi?, oi caaea/a ia nianoaaiiua cia/aiey (23) ?ac?aoeia, ana
nianoaaiiua cia/aiey Oi iaio?eoeaoaeueiu e n?aaee ieo ri – iaeaieueoaa.

Aeey aeieacaoaeuenoaa naienoa iia?aoi?ia Ii, i=1,…,N, e I aaaaeai
iaicia/aiey, oeacuaathuea ia caaeneiinoue io f(():

(26*)

Yoe ?aaainoaa, iieacuaathuea, /oi ?acoeueoao aeaoe?aoiiai aeaenoaey
iia?aoi?ia Ii, i=1,…,N, e I (26) ia ioee/aaony io ?acoeueoaoaoa
iaeiie?aoiiai eo aeaenoaey, iicaieyo n/eoaoue iia?aoi?u (26)
i?iaeoi?aie.

:

.

eiaao aaeeinoaaiiia iieiaeeoaeueiia nianoaaiiia cia/aiea, eioi?ia, eae
iao?oaeii i?iaa?eoue, ?aaii ri, e aio niioaaonoaoao aaeeinoaaiiue
nianoaaiiue aaeoi? fi. Iiyoiio

.

(

.

, iineieueeo a oaeii neo/aa aoaeoo auiieiaiu eiieeeaoeee:

,

.

, p=1,…,n,

.

– aeaaa?ae/anee i?inoia (iae?aoiia), a niioaaonoaothuee nianoaaiiue
aaeoi? iiaeii auae?aoue iaio?eoeaoaeueiui:

. (

Caia/aiea 4.

.

, oi

.

ecia?aaeaiey (17) anoue iiiaeanoai ?aoaiee o?aaiaiey

, (27)

, anee e oieueei anee auiieiaii ?aaainoai (27).

(17).

.

. (28)

, (29)

e aeinoeaaaony ia

, (30)

oi, eae iao?oaeii oaaaeeoueny,

, (31)

iiaoo auoue i?iecaieueii ioianaiu e iaeiiio ec iiiaeanoa Ai eee Aj.

, a eioi?ii

(32)

a F: Rn(( Rn iia?aoi?, ii?aaeaeaiiue oneiaeai

(33)

Oiaaea ?aoaiea caaea/e (28) iiaeii i?aaenoaaeoue a aeaea

, (34)

ii oi?ioea (34) (ni. niineo 4 ia no?. 13).

(35)

eiaao ?aoaiea

(36)

Niioaaonoaaiii ?aoaiea caaea/e (28) n oneiaeai oece/iinoe eiaao aeae

, (37)

– eiaeeeaoi?iay ooieoeey iiiaeanoaa

, (38)

A ?yaea neo/aaa aeey iino?iaiey (34) iieacii ii?aaeaeeoue iia?aoi? F+:
Rn(( Rn, aeaenoaothuee niaeanii oi?ioea

(39)

aaea

,i=1,…q. (40)

Iiaeuoiaeei neacaiiia.

ecia?aaeaieyie ia eneiiuo iiiaeanoaao A1,…,Aq ?acaeaiey X caaeaiiua
oeaaoaie (1,…, (q niioaaonoaaiii, aeaaony ?aaainoaii (34), eneiiia
?acaeaiea A1,…,Aq ii?aaeaeaii a (31). O?aaiaaiea oece/iinoe iaeeo/oaai
i?eaeeaeaiey i?eaiaeeo e ?aoaieth (37) e ii?aaeaeyao eneiiia ?acaeaiea
oi?ioeaie (38). ?aoaiea (34) eiaa?eaioii ioiineoaeueii ethaiai, a (37) –
ioiineoaeueii ethaiai, nio?aiythuaai oece/iinoue, i?aia?aciaaiey,
iaeciaiythuaai aai oeaao.

(34), oi?iie oaeiai ecia?aaeaiey yaeyaony iia?aoi? F+ (37). Anyeia
oaeia ecia?aaeaiea g((), oaeiaeaoai?ythuaa oneiaeyi oece/iinoe
(iaio?eoeaoaeueiinoe y?einoae), oaeiaeaoai?yao o?aaiaieth F+g(()=g((),
oa ec ieo, o eioi?uo ((Ai)>0, i=1,…,q, eciii?oiu, inoaeueiua eiatho
aieaa i?inooth oi?io. (

oaeeie ecia?aaeaieyie

, (41)

caaea/e (41) aeaaony ?aaainoaii

, (42)

. Iaaycea i?eaeeaeaiey

, (43)

!) (

, iaciaai auioeeue, caieiooue eiion ecia?aaeaiee

.

e yaeyaony iaiiaeaeaeiie oi/eie iia?aoi?a

g(() = g((). (#)

– oi?ia ethaiai ecia?aaeaiey f(x) = f(x)((x), f(x)>0, x(X(mod(), ana
oaeea ecia?aaeaiey eciii?oiu, a oi?ia anyeiai ecia?aaeaiey g((),
oaeiaeaoai?ythuaai o?aaiaieth (#), ia neiaeiaa, /ai oi?ia f(().

, A1,…,AN – niioaaonoaothuaa iioeiaeueiia ?acaeaiea X, iaeaeaiiia a
oai?aea 4 e

, (34*)

– iaeeo/oaa i?eaeeaeaiea f((). Oiaaea a ?aaainoaa (24)

, (24*)

e aoaeao auiieiaii ?aaainoai (24), anee a (34*) ii?aaeaeeoue (i eae
oeaao fi a (24), i=1,…,N.

I?iaa?ea yoiai caia/aiey ia i?aaenoaaeyao cao?oaeiaiee.

A. Neo/ae, eiaaea aeiioneathony iaaieueoea eciaiaiey oeaaoa a i?aaeaeao
eaaeaeiai Ai, i=1,…,N.

, oae e aeiionoea iaeioi?ua eciaiaiey oeaaoa a i?aaeaeao eaaeaeiai Ai,
i=1,…,N, iai?eia?, aua?aa aianoi (17) eeann ecia?aaeaiee

(17*)

a (3).

i?e ethaii i=1,…,N, iiaeii n/eoaoue i?oiaiiaeueiuie, ii?aaeaeea

, (*)

aeieaeiu auoue ii?aaeaeaiu ec oneiaey

(**)

i?e aeiiieieoaeueiii oneiaee i?oiaiiaeueiinoe

. ?aoaiea yoie caaea/e aeaaony a neaaeothuae eaiia

i?oiaiiaeueiua nianoaaiiua aaeoi?u iia?aoi?a Oi (23), oii?yaei/aiiua
ii oauaaieth nianoaaiiuo cia/aiee:

.

.

e

. Oiaaea eaaay /anoue (*) ?aaia neaaeo iia?aoi?a [Pi Oi Pi]

, ioeoaea neaaeoao ooaa?aeaeaaiia a eaiia. (

Ainiieueciaaaoenue au?aaeaieyie (*) e eaiiie 5, iaeaeai, /oi a
?anniao?eaaaiii neo/aa eiaao ianoi ooaa?aeaeaiea, aiaeiae/iia oai?aia
3.

Oai?aia 3*. Iaeeo/oaa i?eaeeaeaiea ethaiai ecia?aaeaiey f(()
ecia?aaeaieyie (17*) eiaao aeae

,

, nianoaaiiuo aaeoi?ia caaea/e

.

Iaaycea iaeeo/oaai i?eaeeaeaiey ?aaia

. (

, eae ?aoaiea caaea/e

(30)

, ii?aaeaeyaiuo ?aaainoaii (20). A naith i/a?aaeue, i/aaeaeii, /oi

(31)

. Yoi niaeaoaiea ioia/aii caacaei/eie a (31).

Oaeei ia?acii aeieacaia

caaeaiiua aaeoi?u Rn. ?aoaieai caaea/e (30) yaeyaony ecia?aaeaiea

,

– eiaeeeaoi?iay ooieoeey iiiaeanoaa (31), i=1,…,N. Iaaycea
iaeeo/oaai i?eaeeaeaiey ?aaia

. (

,

, iiaeii caienaoue a aeaea

, (32)

, ia eciaiythuaai aai oeaao.

e (i0 , i=1,…,N, oaeea, /oi

.

,

; iaeiiaoe,

. (

.

(9) oaeiaii caaeaaaoue iia?aoi?aie If e I*f niioaaonoaaiii.

ii?aaeaeyaony i?oiaiiaeueiui i?iaeoi?ii I*f :

,

.

e oai naiui aeieacaii e aoi?ia ooaa?aeaeaiea (

.

Aeey niaeo?iciiaeueiuo ecia?aaeaiee oa?aeoa?ia neooaoeey, i?e eioi?ie k
aeaoaeoi?ia ?aaeno?e?otho ?annayiioth iauaeoaie nieia/ioth ?aaeeaoeeth a
aeeaiaciia aeaeeiiai naaoa, a inoaeueiua n-k ?aaeno?e?otho nianoaaiiia
oaieiaia eceo/aiea iauaeoia ( a eio?ae?aniii aeeaiaciia). A oaeii neo/aa
ethaia ecia?aaeaiea iiaeii i?aaenoaaeoue ?aceiaeaieai

(40)

A eioi?ii

. Oi?ia I*f aeaeeiie eiiiiiaiou f(() (40) ii?aaeaeyaony eae iia?aoi?
iaeeo/oaai i?eaeeaeaiey a caaea/a

, a aeaiiii neo/aa

, i?e/ai I*f aeaenoaoao oaeoe/anee oieueei ia “aeaeeioth eiiiiiaioo”
g((), ia?auay “iaaeaeeioth, EE, eiiiiiaioo” g(() a iieue.

Oi?ia EE eiiiiiaiou f(() iiaeao auoue ii?aaeaeaia eeoue oiaaea, eiaaea
ecaanoii iiiaeanoai aiciiaeiuo i?aia?aciaaiee (2(() f2(().

Iaeioi?ua i?eiaiaiey.

Caaea/e eaeaioeoeeaoeee noeai.

?anniio?ei aia/aea caaea/e eaeaioeoeeaoeee noeai ii eo ecia?aaeaiey,
iaeneaaeaiiui aaiiao?e/aneeie i?aia?aciaaieyie, iiai?ioaie, eciaiaieyie
ianooaaa e o.ae. Ia?aie/einy caaea/aie, a eioi?uo i?aaeuyaeyaiua aeey
aiaeeca ecia?aaeaiey iieo/aiu i?e eciaiythueony e iaeiio?iee?oaiuo
oneiaeyo inaauaiey e iaecaanoiuo e, aiiaua aiai?y, ?acee/iuo iioe/aneeo
oa?aeoa?enoeeao noeaiu.

1). Caaea/e eaeaioeoeeaoeee i?e i?iecaieueii iaiythuaeny eioaineaiinoe
inaauaiey.

Iiaeii ee n/eoaoue f(() e g(() ecia?aaeaieyie iaeiie e oie aea noeaiu,
aiciiaeii, ioee/athueiy eeoue ?ani?aaeaeaieyie y?einoe, iai?eia?,
iaee/eai oaiae?

.

Ia i?aeoeea oaeiaiaa eniieueciaaoue ae?oaie iiaeoiae, iicaieythuee
iaeiia?aiaiii ?aoaoue caaea/e niaiauaiey ecia?aaeaiee e auaeaeaiey
iauaeoia. Iiaeii ee, iai?eia?, n/eoaoue g(() ecia?aaeaieai noeaiu,
i?aaenoaaeaiiie ecia?aaeaieai f(()? Ioaao neaaeoao n/eoaoue
ooaa?aeeoaeueiui, anee

.

.

2).Eaeaioeoeeaoeey i?e i?iecaieueiii eciaiaiee ?ani?aaeaeaiey
eioaineaiinoe e i?ino?ainoaaiii iaeii?iaeiii eciaiaiee niaeo?aeueiiai
ninoaaa inaauaiey.

Iiaeii ee n/eoaoue ecia?aaeaieai noeaiu, i?aaenoaaeaiiie ia
ecia?aaeaiee f((), ecia?aaeaiea, iieo/aiiia i?e eciaieaoeony oneiaeyo
?aaeno?aoeee, iai?eia?, ia?aiauaieai eee eciaiaieai oaiae e eciaiaieai
niaeo?aeueiiai ninoaaa inaauaiey?

.

3). Caaea/e niaiauaiey ecia?aaeaiee e iienea o?aaiaioa.

Ionoue f(() – caaeaiiia ecia?aaeaiea, A(X – iiaeiiiaeanoai iiey c?aiey,
(A(() – aai eiaeeeaoi?, (A(()f(() -iaciaai o?aaiaioii ecia?aaeaiey f(()
ia iiaeiiiaeanoaa A, i?aaenoaaeythuai auaeaeaiiue o?aaiaio noeaiu,
ecia?aaeaiiie ia f((). Ionoue g(() – ecia?aaeaiea oie aea noeaiu,
iieo/aiiia i?e ae?oaeo oneiaeyo, a /anoiinoe, iai?eia?, naeaeiooia,
iiaa?iooia, o.a. aaiiao?e/anee eneaaeaiiia ii n?aaiaieth n f((). Caaea/a
ninoieo a oii, /oiau oeacaoue ia g(() o?aaiaio ecia?aaeaiey,
i?aaenoaaeythuee ia f(() o?aaiaio noeaiu e niaianoeoue aai n (A(()f(().

iaciaai naeaeaii g(() ia h. Caeanue

Q(h): Rn->Rn, h(H, – a?oiia iia?aoi?ia. Aaeoi?iue naeaea ia h((H aeano

.

A caaea/a auaeaeaiey e niaiauaiey o?aaiaioa ?anniio?ei o?aaiaio
naeaeiooiai ia h ecia?aaeaiey g(() a “ieia” A:

(100)

– ia?aie/aiea ia naeaea “ieia” A, eioi?ia aeieaeii inoaaaoueny a
i?aaeaeao iiey c?aiey X.

– oi?ia o?aaiaioa f((), oi caaea/a auaeaeaiey e niaiauaiey o?aaiaioa
naiaeeony e neaaeothuae caaea/a ia ieieioi

.(101)

I?e yoii n/eoaaony, /oi o?aaiaio ecia?aaeaiey g((), niioaaonoaothuee
o?aaiaioo (A(()f((), aoaeao iiiauai a “ieii”.A iooai niioaaonoaothuaai
naeaeaa h=h*, niaiaaeaao n (A(()f(() n oi/iinoueth aei iaeioi?iai
i?aia?aciaaiey ?ani?aaeaeaiey y?einoe ia iai. Yoi icia/aao, /oi

.

o.a. a (101) i?e h=h* aeinoeaaaony ieieioi.

4). A ?yaea neo/aaa aicieeaao neaaeothuay caaea/a aiaeeca
niaeo?iciiaeueiuo ecia?aaeaiee: auaeaeeoue iauaeou eioi?ua “aeaeiu”,
neaaeai, a ia?aii eaiaea e “ia aeaeiu” a inoaeueiuo.

?anniio?ei caaea/o ia ieieioi

. [*]

(ciaeii – iaicia/aii inaaaeiia?auaiea).

?en.1.

fe – aaeoi? auoiaeiuo neaiaeia aeaoaeoi?ia, ioaa/athuee eceo/aieth e((),
(e – aai oeaao; (1,(2,(3, – aaeoi?u (oeaaoa) aaciauo eceo/aiee, ( –
aaeue oeaao, eiiaoe aaeoi?a ( iaoiaeeony ia ia?ana/aiee aennaeo?en.

Eeoa?aoo?a.

[1] Iuoueaa TH.I. Ii?oieiae/aneea iiiyoey a caaea/ao aiaeeca
ecia?aaeaiee, – Aeiee. AI NNN?, 1975, o. 224, ?6, nn. 1283-1286.

[2] Iuoueaa TH.I. Ii?oieiae/aneee aiaeec ecia?aaeaiee, – Aeiee. AI
NNN?, 1983, o. 296, ?5, nn. 1061-1064.

[3] Iuoueaa TH.I. Caaea/e ii?oieiae/aneiai aiaeeca ecia?aaeaiee, –
Iaoaiaoe/aneea iaoiaeu enneaaeiaaiey i?e?iaeiuo ?ano?nia caiee ec
einiina, ?aae. Cieioooei A.A., Iaoea, Iineaa, 1984, nn. oooo-ooooo.

[4] Iuoueaa TH.I., *oee/eia A.E. YAI aiaeece?oao oi?io ecia?aaeaiey, –
Ciaiea,na?. Iaoaiaoeea, Eeaa?iaioeea, Iineaa, 1988, 47 no?.

[5] Yu.P.Pyt’ev. Morphological Image Analysis, Patt. Recogn. and Image
Analysis, 1993, v.3, #1, pp.19-28.

[6] Aioiithe A.A., Iuoueaa TH.I. Niaoei?ioeanni?u ?aaeueiiai a?aiaie
aeey ii?oieiae/aneiai aiaeeca ?aaeueiuo noeai. Ia?aaioea ecia?aaeaiee e
aeenoaioeeiiiia enneaaeiaaiey, -Iiaineae?ne, 1981, nn. 87-89.

[7] Aioiithe A.A., Iuoueaa TH.I., ?ao Y.E. Aaoiiaoecaoeey aecoaeueiiai
eiio?iey ecaeaeee iee?iyeaeo?iieee,?aaeeioaoieea e yeaeo?iieea, 1985, o.
OOO,?12, nn. 2456-2458.

[8] A?iieaaa A.A., Iuoueaa TH.I. Ai?ei?iua ioeaiee iieaciiai neaiaea
aeey ii?oieiae/aneeo ?aoathueo aeae?eoiia, – Aaoiiaoecaoeey, 1984, ?5,
nn. 118-120.

[9] Iuoueaa TH.I, Caaei?iaeiue N.N., Eoeueyiia A.A. Ia aaoiiaoecaoeee
n?aaieoaeueiiai ii?oieiae/aneiai aiaeeca yeaeo?iiiiiee?ineiie/aneeo
ecia?aaeaiee, – Eca. AI NNN?, na?. oece/aneay, 1977, o. 41, ?11, nn.
oooo-oooo.

[10] A.A. Stepanov, S.Yu. Zheltov, Yu.V. Visilter. Shape analysis using
Pyt’ev morphological paradigm and its using in machine vision. Proc.
SPIE – Th. Intern. Soc. For Optical Engineering Videometrics III, 1994,
v. 2350, pp. 163-167.

[11] Iuoueaa TH.I.. Iaoaiaoe/aneea iaoiaeu eioa?i?aoaoeee yenia?eiaioa,
Aunoay oeiea, 351 no?., 1989.

[12] Iaecaeue N.I. ?aooa? A.N. Oeaaoiaua ?an/aou e ecia?aiey.
I:E:Ainyia?aiecaeao 1941, (O?oaeu ananithciiai yeaeo?ioaoie/aneiai
einoeoooa, aui.56).

[13] P. Kronberg. Fernerkundung der Erde Ferdinand Enke. Verlag
Stuthgart 1985.

Iai?eia?, a nayce n eciaiaieai a?aiaie nooie, iiaiaeu, a?aiaie aiaea e
o.i.

O?aaiaio ii?oieiae/aneiai aiaeeca oeaaoiuo ecia?aaeaiee niaea?aeeony a
?aaioa[3].

aaeoi? fe aoaeao eiaoue io?eoeaoaeueiua eii?aeeiaou, anee ii ia
i?eiaaeeaaeeo auioeeiio eiiono

– auioeeue caieiooue eiion a Rn.

iinoiyiiue, ii ?acee/iue ia ?aciuo iiaeiiiaeanoaao A((((). Iaeiaei,
iineieueeo oi?ia iau/ii no?ieony enoiaey ec aeaiiiai ecia?aaeaiey f((),
v(f(()) ia iiaeao niaea?aeaoue ecia?aaeaiey, eioi?ua aieaa aeaoaeueii
oa?aeoa?ecotho ecia?aaeaiioth noeaio.

Aeey i?inoiou y?einoue ecia?aaeaiey n/eoaaony iieiaeeoaeueiie a
eaaeaeie oi/ea iiey c?aiey O.

.

. Yoa aieueiinoue ia aeieaeia aucuaaoue iaaei?acoiaiey e /anoi
eniieuecoaony a ?aaioa.

Anee m(As)=0, oi a caaea/a iaeeo/oaai i?eaeeaeaiey (18) oeaao e
?ani?aaeaeaiea y?einoe ia As iiaeii n/eoaoue i?iecaieueiuie, iineieueeo
eo cia/aiey ia aeeytho ia aaee/eio iaaycee s.

Aaeoi?u (1,…, (q auae?athony, iai?eia?, niia?acii oeaaoai iauaeoia,
i?aaenoaaeythueo eioa?an.

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