image contour

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Chapterl

CONTOUR EXTRACTION

1.1 Overview

In this chapter we briefly summarie previ!us meth!"s !# c!nt!ur e$tracti!n.Theree$ist tw! main %el"s !# research #!r the &'(!b)ect c!nt!ur e$tracti!n pr!blem*The!b)ect c!nt!ur #!ll!win+ ,OC- !r se/uential meth!"s an" the multiple stepc!nt!ure$tracti!n ,0CE !r parallel meth!"s. 2ater3 they "evel!p an !b)ect(!riente"c!nt!ure$tracti!n al+!rithm ,OCE. Als! there e$ists an al+!rithm which initiate" athir" class !#c!nt!ur e$tracti!n al+!rithms3 sin+le step parallel c!nt!ur e$tracti!n ,4CE.

-inally3 the c!mparis!n between the OCE ,5 $ 5 win"!ws3 4CE ,6 $ 6 win"!wsan" c!nt!ur e$tracti!n base" !n ,& $ & win"!ws is per#!rme".

T! present the results3 the c!mparis!ns between these c!nt!ur e$tracti!n meth!"sare"!ne usin+ the number !# arithmetic !perati!ns versus number !# e"+es.

At the be+innin+ !# this chapter3 we summarie previ!us e"+e "etecti!n3 an"!b)ect

rec!+niti!n.

1.& E"+e 'etecti!nThe m!st imp!rtant stu"y %el" !# c!mputer visi!n is the "etecti!n !# !b)ect ,&(

"imensi!nal ima+e as e"+e p!ints ,pi$els !# a 6("imensi!nal physical !b)ect.

Thec!rrectness an" c!mpleteness !# e"+es c!nstitutes is an essential t!!l t!e$tract !b)ectc!nt!urs an" !b)ect rec!+niti!n 768. -!r simplicities an" #acilitates !# ima+eanalysis3 thee"+e "etecti!n is very imp!rtant 798.

Once a lar+e chan+e in ima+e bri+htness has !ccurre"3 usin+ the e"+e "etecti!nwe cane$tract an" l!calie p!ints ,pi$els. The relati!nship between a pi$el an" itssurr!un"in+nei+hb!urs interprets the e"+e "etecti!n. The pi$el is represente" as an e"+ep!int i# the

re+i!ns ar!un" a pi$el are n!t similar3 !therwise3 the pi$el is n!t appr!priatet! be

rec!r"e" an e"+e p!int.

1.6 :rey(level metrics

'etecti!n3 rec!+niti!n an" e$tracti!n !# &'(!b)ects in +rey;level ima+es are themainimp!rtant pr!blems in c!mputer visi!n. 0!"el c!nstructi!ns an" trainin+s as wellasc!mputati!nal appr!ach #!r a better an" parallel implementati!n in bi!l!+icallye$plicit

neural netw!r< architectures are "iscusse" in many b!!


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applicati!ns !# shape analysis sprea" !ver the scienti%c an" techn!l!+ical areain which

the values !# scale #!rm a hierarchy #r!m the smallest t! the lar+est spatialscale 71=8.

The imp!rtant pr!blem in +rey;level ima+e an" c!nt!ur analysis is e"+e"etecti!n.E"+es characterie !b)ect b!un"aries. ! they are use#ul #!r se+mentati!n3re+istrati!n3

an" i"enti%cati!n !# !b)ects in a scene in i"eal ima+es.

The chan+e in intensity "e%nes the e"+e an" its cr!ss secti!n has the shape !# aramp.Usually3 "isc!ntinuity characteries an i"eal e"+e lieare l!!


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respectively. The +ra"ient e/uati!n is B 1G +* +* !r #!r simplerimplementati!n

usin+ B

+3 +3 79&8.

E"+e l!cati!n is +iven by a pi$el l!cati!n when the p!int value ,$3 y e$cee"ss!methresh!l". The e"+e map "escribes the l!cati!ns !# all e"+e p!ints. The number!#imp!rtant thin+s +ives interpretati!n h!w the thresh!l" value is selecte" by the"esi+n"ecisi!n such as ima+e bri+htness3 c!ntrast3 level !# n!ise3 an" e"+e "irecti!n.!metimes

it is very use#ul t! w!r< !ut the in#!rmati!n !# the e"+e "irecti!n +iven by

H arctan7; 8 ,1.56y

E"+es "rives t! b!un"aries in ima+es an" represent areas with str!n+ intensityc!ntrasts3 a varyin+ intensity #r!m !ne pi$el t! the !ther. It is very essentialt! re"uce theam!unt !# "ata an" eliminate the useless in#!rmati!n by e"+e "etecti!n !# anima+e3while


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"i##erences !r Eucli"ean "istances. Abs!lute value c!mputati!ns are easyimplementati!nan" #aster !perati!ns when c!mpare" t! s/uare an" s/uare r!!t !perati!ns. Jencet! +et a#aster c!mputin+ the +ra"ient ma+nitu"e3 sum the abs!lute values !# the+ra"ients in theX ,wi"th !r h!ri!ntal an" in the K ,hei+ht !r vertical "irecti!ns isper#!rme".

There are a pannel !# +ra"ient !perat!rs use" t! "etect an e"+e. Ly r!tatin+ the


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spatial +ra"ient measurement !n an ima+e. The e"+es are !#ten "e%ne" by thehi+hli+htre+i!ns !# hi+h spatial +ra"ient. In +eneral3 the input an" !utput !# the!perat!r is a +rey(scale ima+e. The estimate" abs!lute ma+nitu"e !# the spatial +ra"ient !# theinput ima+eat a p!int there#!re3 !##ers #!r each p!int3 the pi$el representati!n values inthe !perat!r

!utput.

The!retically3 the !perat!r is ma"e up !# a pair !# &V& c!nv!luti!n mas


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II=:::

Y,'

-i+. 1.6 !bel Cr!ss c!nv!luti!n mas


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"i##erent


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implementati!n !# e"+e "etecti!n is "!ne. The paralleliin+ e"+e "etecti!npr!cess is!btaine" by splittin+ the ima+e int! se+ments an" then "etectin+ the e"+esin"epen"entlyin each !# the se+ments 768. >e sh!ul" be care%il an" ma


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win"!ws 7518.

!me b!!


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tartin+p!int

-i+. 1. ? Ob)ect c!nt!ur #!ll!wer ,OC-

ll

OC- al+!rithms have #!ur main "rawbac


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the meth!"!# c!nnectin+ ,tracin+ these e$tracte" e"+e elements is se/uential.

2i


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the binary

value !# a pi$el p!int ,i3).

tep6* The e$tracte" c!nt!ur e"+es are s!rte" an" st!re" !r !ptimie" acc!r"in+t! the

applicati!n re/uirements. The e$tracti!n pr!ce"ure is sh!wn in -i+. 1.S.

Ima+e is #rame" byD er!sOCE ,!b)ect Q3 Ei+ht rules is!riente" c!nt!ur applie"e$tracti!nE$tracte" e"+es iss!rte"

-i+. 1.S Ob)ect(!riente" c!nt!ur e$tracti!n ,OCE meth!"

The e$tracte" c!nt!urs #r!m !b)ects usin+ this pr!ce"ure are near the ima+eb!un"ary.

!3 the !b)ects within !ne pi$el "istance #r!m the ima+e b!r"er are n!t cl!se"an" that isinterprete" why the ima+e sh!ul" be #rame" with er!s t! ensure all !# thec!nt!urs arecl!se". -i+. 1.a an" -i+. l.b sh!ws the e$tracte" e"+es with!ut #ramin+ ,!ne!# thee$tracte" c!nt!ur is n!t cl!se" an" a#ter #ramin+ the ima+e with at least tw!

un"er+r!un" pi$els ,all e$tracte" c!nt!urs are cl!se" respectively.

a b

-i+. 1. OCE pr!ce"ure ,a >ith!ut c!rrectin+ the #irst step3 an",b A#ter c!rrectin+ the %rst step

15

1.5.5 in+le step parallel c!nt!ur e$tracti!n 4CEG meth!",6$6 win"!ws

There are tw! al+!rithms] the %rst !ne 7518 which uses an S(c!nnectivity schemebetween pi$els3 an" S("irecti!nal -reeman chain c!"in+ 76=8 scheme is use" t!"istin+uish all ei+ht p!ssible line se+ments c!nnectin+ the nearest nei+hb!urs.This

al+!rithm uses the same principle !# e$tracti!n rules as the OCE al+!rithm. Thesec!n"al+!rithm 7518 uses a 5(c!nnectivity scheme between pi$els3 an" 5("irecti!nal-reemanchain c!"in+ 76=8 an" 76&8 schemes are use" t! "istin+uish all #!ur p!ssiblelinese+ments. L!th al+!rithms use a 6$6 pi$els win"!w structure t! e$tract the!b)ectc!nt!urs by usin+ the central pi$el t! %n" the p!ssible e"+e "irecti!n whichc!nnects thecentral pi$el with !ne !# the remainin+ pi$els surr!un"in+ it.

The #irst al+!rithm is +iven e$actly the same e$tracte" c!nt!urs as the OCE

al+!rithmsan" is #aster ,6.S times #aster while the sec!n" al+!rithm +ives similarc!nt!urs3 but n!ti"entical an" is #aster ,5.& times #aster 7518. The %rst al+!rithm e$planati!ns


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arementi!ne" as #!ll!ws

The e"+es can be e$tracte" by applyin+ the "e%niti!n which says that an !b)ectc!nt!ur e"+e is a strai+ht line c!nnectin+ tw! nei+hb!urin+ pi$els which haveb!th ac!mm!n nei+hb!urin+ !b)ect pi$els which have b!th a c!mm!n nei+hb!urin+ !b)ectpi$el an" a c!mm!n nei+hb!urin+ un"er+r!un" pi$el 75=8. Ly this "e%niti!n3 n!e"+escan be e$tracte" %!m the three #!ll!win+ cases*

1( The win"!w is insi"e an !b)ect re+i!n i# the all pi$els are !b)ect pi$els]

&( The win"!w is insi"e a bac


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then a,i3)V;a,i3)!r&W P e"+e = @i#b,i3) an" b,i13) an" b,i13)(1 an" 7n!t 7b,i3)(188

then a,i3),;a,i3) !r &1 P e"+e 1 @i#b,i3) an" b,i3)(1 an" 7b,i13) !r b,i13)(18 an" 7n!t 7b,i(13) !r b,i(13)(188then a,i3),;a,i3)!r&& P e"+e & @i#b,i3) an" b,i3)(1 an" b,i(13)(1 an" 7n!t 7b,i(13)88

then a,i3),;a,i3) !r &6 P e"+e 6 @i#b,i3) an" b,i(13) an" 7b,i3)(1 !r b,i(13)(18 an" 7n!t 7b,i3)1 !r b,i(13)188then a,i3),;a,i3)!r&5 P e"+e 5 @i#b,i3) an" b,i(13) an" b,i(13)1 an" 7n!t 7b,i3)188

then a,i3),;a,i3)!r&9 P e"+e 9 @i#b,i3) an" b,i3)1 an" 7b,i(13) !r b,i(13)18 an" 7n!t 7b,i13) !rb,i13)188then a,i3),;a,i3)!r&? P e"+e ? @i#b,i3) an" b,i3)1 an" b,i13)1 an" 7n!t 7b,i13)88

then a,i3),;a,i3)!r& P e"+e @@

1.5.9 C!nt!ur e$tracti!n base" !n &$& win"!wsThis al+!rithm is mainly use" #!r +rey(scale ima+es 7518. It uses a smallerwin"!w #!rc!nt!ur e$tracti!n3 i.e. &$& win"!w pi$els an" their structure an" pi$elnumberin+ are

sh!wn in -i+. 1.11.

4! 41

461Z*

-i+. 1.11 4i$el numberin+ #!r &$& win"!ws

The pr!cesse" pi$el is the "ar


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= The ei+ht rules !# e"+e e$tracti!n are applie" an" are c!"e" usin+ S("irecti!nal

chain(c!"e as sh!wn in 2istin+ 1.6.

2istin+ 1.6

Implementati!n !# the ei+ht rules #!r c!nt!ur e$tracti!n ,&$& win"!ws`]`&6*#. ...31.6.1] 6([]GiG33. . ..ii]*1[._^G

1P#,b,1[1 b,i.1[ 18 , b,i(131[H b,i(131G(1 -l ,b,i(131[H b,i31[(1

then a,1;13)V;a,i(13)Uh,1;1.) U &W P e"+e = @i#,b,i(1"[ b,i.1[(1fl,b,i(1.iHb,i(131(1

then a,1;13)V;a,i(13)Uh,i;1.) U & P e"+e 1 @i# , b,i.) b,i3)(1 18 ,b,i.1[H b,i(1.1G 18 ,b,i3)H b,i(1.1(1

then a,i3),;a,i Uh,i.) U & P e"+e & @i# , b,i b,i(1.1G(1 18 , b,i.[H b,i(1.1

then a,i3),;a,i3) Ub,i3) U &6 P e"+e 6 @i# , b,i.)(1 b,i(1.1G(1fl ,b,i.[(1 H b,i 7l ,b,i.i(1 H b,i(1.1

then a,i3)(l,;a,i3)(1 Ub,i3)(1U &5 P e"+e 5 @i# ,b,i.)(1 b,i(1.1 #8,b,i3)(1Hb,i.1[

then a,i3)(1,;a,i3)(1 Ub,i3)(1U &9 P e"+e 9 @i# , b,i(1.1(1 b,i(1.1 fl ,b,i;1.)(1H b,i.i(1 -l ,b,i;1.)(1H b,i.1[

then a,i(l3)(1 ,;a,i(13)(1 Ub,i(13)(1U & P e"+e ?@i#,b,i(1.i(1b,i.i #8,b,i;1.i(1Hb,i.)(1

then a,i(l3)(1 ,;a,i(13)(1 Ub,i(13)(1 U & P e"+e @

1

1.9 C!nt!ur "escript!rs:enerally3 !b)ect "escript!rs are re#erre" t! !b)ect representati!ns which +iveways !#

"escribin+ !b)ect pr!perties !r #eatures 7&68 an" 7&58. Ob)ect rec!+niti!n hasvery

1S

imp!rtant area in !r"er t! !btain %$e" !b)ect representati!ns t! a#%netrans#!rmati!ns3

such as translati!ns3 r!tati!ns an" linear scalin+ 7&98 an" 7&?8. 0anyrepresentati!ns suchas !b)ect c!nt!urs are !btaine" usin+ &' shape #eatures !# 6';!b)ects 7&6837&83 an"7&8. -eature e$tracti!n an" analysis are s! easy that it +ives an imp!rtanta"vanta+e !#such appr!ach.

It is necessary t! learn t!!ls ab!ut analysis an" statistics shapes #!rresearches3en+ineers3 scientists an" me"ical researchers. tatistics an" analysis !# shapesare use" aste$tb!!< !# special t!pics c!urse #!r a +ra"uate;level in statistics an"

si+nalDima+eanalysis 7568 an" 7998.

The "escripti!n !# the e$tracte" c!nt!urs n!rmally use the chain c!"e3 i.e. the


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$ an" yc!;!r"inates !# the startin+ p!int #!ll!we" by a strin+ !# chain c!"esrepresentin+ thec!nt!ur e"+es. The c!nt!urs are "escribe" in Cartesian representati!n3 the $ an"y c!(!r"inates3 !r p!lar representati!n3 usin+ a len+th !# the line I #r!m !ne p!intt! the ne$tin se/uence an" the an+le =1 between every tw! lines 75&8 an" 7558. There isan!ther type

!# p!lar representati!n which "escribes the c!nt!ur by usin+ r3 ,the len+th #r!mare#erence p!int insi"e the c!nt!ur t! the c!nt!ur vertices an" H3 ,the an+lebetween the

tw! lines r3 an" rm.

1.? C!mparis!n between the c!nt!ur e$tracti!n al+!rithmsThe c!mparis!n is ma"e between the #!ll!win+ three al+!rithms*= C!nt!ur e$tracti!n ,CE] it will be re#erre" t! as the %rst al+!rithm ,!r &$&win"!ws.

= 4CE meth!"] it will be re#erre" t! as the sec!n" al+!rithm ,!r 6$6 win"!ws.= OCE meth!"] it will be re#erre" t! as the thir" al+!rithm ,!r 5$5 win"!ws.

The c!mparis!n is ma"e with respect t! spee" an" number !# c!nt!ur e"+es. Thebinary test ima+es are illustrate" in -i+. 1.1&.

Table 1.13 Table 1.& an" Table 1.6 present the c!mparis!n between the threeal+!rithms with respect t! the number !# !perati!ns versus the number !# e"+es#!r

Circle3 Rectan+le an" E letter c!nt!urs respectively.

1

,a ,b

,C

-i+. 1.1& Linary ima+es ,a Circle3 ,b Rectan+le3 an" ,c E letter

Table 1.1 C!mparis!n between the al+!rithms #!r Circle ima+e

1& 61 91 6?66 1S66 6191& 5651& 91=S

11?5 161= 996? S?199 S?6=5=? ?696 11116 19S?1S 1S?915

a l+. ; al+!rithm3 NE ; number !# e"+es3 AE ; all e"+es an" N= is the number!# !perati!ns

Table 1.& C!mparis!n between the al+!rithms #!r the Rectan+le ima+e

?6 116 1?9 1S?=6 1669= &1?S= &9655= 6&9=6

1S?6 115&9 6SS99 519? 5?5?


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&&6?S 91&SS S15 969 13&69

al+. ; al+!rithm3 NE ; number !# e"+es3 AE ; all e"+es an" N= is the number!# !perati!ns

&=

Table 1.6 C!mparis!n between the al+!rithms #!r the E letter ima+e

&9 9 6 161 1?=655=& 59916 9?6 9?=5 S91?? 1=5?

S=& 1951 &S9&? 51 9??5& 9??5&1&6S& 19 &159 &19 61&5 5=9=

al+. ; al+!rithm3 NE ; number !# e"+es3 AE ; all e"+es an" N= is the number!# !perati!ns

e$tracte" c!nt!urs e$tracte" c!nt!urs e$tracte" c!nt!ursusin+ %rst al+!rithm usin+ sec!n" al+!rithm usin+ thir" al+!rithm

-i+. 1.16 E$tracte" c!nt!urs usin+ the three "i##erent al+!rithms

The %rst c!lunm !# -i+. 1.16 sh!ws the e$tracte" c!nt!urs by the %rst al+!rithm.Thesec!n" c!lumn !# -i+. 1.16 sh!ws the e$tracte" c!nt!urs by the sec!n" al+!rithman" thethir" c!lumn in -i+. 1.16 sh!ws the e$tracte" c!nt!urs by the thir" al+!rithm.

-i+. 1.15 sh!ws the c!mparis!n between these al+!rithms with respect t! thenumber

!# !perati!ns versus the number !# e"+es #!r the binary ima+es as sh!wn in -i+.1.1&.

&1

number !# !perati!ns ,NO

15===== (16===== (1&===== (

1 1===== (1====== (===== (S===== (===== (====== (9===== (5===== (6===== (&===== (1===== (A


=


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&[= 6= 5= 9= =[= 1= == =

number !# e"+es ,NE

,a

Rectan+le ,5$5

,6X6

,&$&

59==== (5&9=== (5===== (69=== (69==== (6&9=== (6===== (&9=== (

&9==== (&&9=== (&===== (19=== (19==== (1&9=== (1===== (

9=== (

11111111111111111= 19 6= 59 9= 9 = 1=91&=16919=191==19&1=&&9&5=&99=

number !# e"+es ,NE

,b

E letter ,5$5

,&$&

9==== (&9=== (


=d

=

Q+;.)

,6X61 1 1 1 1 1 1 1 1 1 159 s! 19 s! 1=9 1&= 169 19= 1?9 19= 19

number !# e"+es ,NE

,?

-i+. 1.15 Number !# !perati!ns versus number !# e"+es usin+ the all al+!rithms#!r the ,a Circle3 ,b Rectan+le3 an" ,c E letter


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&&

The results presente" in -i+. 1.15 sh!w that the al+!rithm which ha" use" thesmallestsie !# win"!ws is the #astest al+!rithm ,as in Circle an" Rectan+le. J!weverthe results

!btaine" usin+ theses al+!rithms #!r the E letter ima+e is n!t l!+ical.

The e$periments are repeate" #!r the binary ima+e !# 2ibya map shape ,see -i+.1.19.-i+. 1.1? sh!ws the pl!t between the numbers !# !perati!ns versus the number !#

e"+es #!r 2ibya map c!nt!ur.

-i+. 1.19 Linary ima+e #!r the 2ibya map

=9=====

=======_ 99=====9======59=====5======69============&9=====&======19=====1======9=====


1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1= 9= 1== 19= &== &9= 6== 69= 5== 59= 9== 99= === =9= == 9= === =9= ===

number !# e"+es ,NE

-i+. 1.1? Number !# !perati!ns versus number !# e"+esusin+ the all al+!rithms #!r the 2ibya map

&6

The c!mparis!n between the teste" binary ima+es #!r the number !# !perati!nsversus

the number !# e"+es is sh!wn in Table 1.5.

Table 1.5 C!mparis!n between the al+!rithms #!r the all test ima+es

91=S 6&9=6 1=5? 15&=9S? 5 ?5? 9 ??5& 66S&=1S?915 1&69 5=9 = 95S96

*1 l+. ; al+!rithm an" N= is the number !# !perati!ns


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The results presente" in Table 1.5 sh!w that the c!nt!ur e$tracti!n usin+ &$&win"!wsis #aster than the !ther al+!rithms an" the c!nt!ur e$tracti!n usin+ 6$6 win"!wsis #asterthan the thir" al+!rithm ,5$5 win"!ws #!r the all test ima+es e$cept that #!r Eletterima+e. !3 t! !btain l!+ic results an" +!!" c!mparis!n between "i##erent win"!ws#!r

c!nt!ur e$tracti!n3 the n!n;c!mple$ shapes are teste".

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