Title Page
 Table of Contents
 List of Tables
 List of Figures
 Object extraction by the gradient...
 Object extraction by the contour...
 Graph theory approach to picture...
 Feature extraction
 Experiments and conclusions
 Biographical sketch

Title: Object extraction and identification in picture processing
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00097624/00001
 Material Information
Title: Object extraction and identification in picture processing
Physical Description: xiii, 140 leaves. : ; 28 cm.
Language: English
Creator: Lin, Peter Pie-teh, 1945-
Publication Date: 1972
Copyright Date: 1972
Subject: Optical pattern recognition   ( lcsh )
Scanning systems   ( lcsh )
Photographic interpretation   ( lcsh )
Electronic digital computers   ( lcsh )
Electrical Engineering thesis Ph. D
Dissertations, Academic -- Electrical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
Thesis: Thesis -- University of Florida.
Bibliography: Bibliography: leaves 137-139.
Additional Physical Form: Also available on World Wide Web
General Note: Typescript.
General Note: Vita.
 Record Information
Bibliographic ID: UF00097624
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000577276
oclc - 13948416
notis - ADA4971


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Table of Contents
    Title Page
        Page i
        Page ii
        Page iii
    Table of Contents
        Page iv
        Page v
    List of Tables
        Page vi
    List of Figures
        Page vii
        Page viii
        Page ix
        Page x
        Page xi
        Page xii
        Page xiii
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
    Object extraction by the gradient method
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
    Object extraction by the contour analysis
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
    Graph theory approach to picture processing
        Page 71
        Page 72
        Page 73
        Page 74
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
        Page 84
        Page 85
        Page 86
        Page 87
        Page 88
        Page 89
    Feature extraction
        Page 90
        Page 91
        Page 92
        Page 93
        Page 94
        Page 95
        Page 96
        Page 97
        Page 98
        Page 99
        Page 100
        Page 101
        Page 102
        Page 103
        Page 104
        Page 105
    Experiments and conclusions
        Page 106
        Page 107
        Page 108
        Page 109
        Page 110
        Page 111
        Page 112
        Page 113
        Page 114
        Page 115
        Page 116
        Page 117
        Page 118
        Page 119
        Page 120
        Page 121
        Page 122
        Page 123
        Page 124
        Page 125
        Page 126
        Page 127
        Page 128
        Page 129
        Page 130
        Page 131
        Page 132
        Page 133
        Page 134
        Page 135
        Page 136
        Page 137
        Page 138
        Page 139
    Biographical sketch
        Page 140
        Page 141
Full Text

,C IECT E.:'.P CTI,: I r rrir i T ii- T, '-
I N ir' TU'.E ''. UE3 I.',I


Peter rc i-t:h L;n

. Dis ,': rLation P'resented t. the CradJuat.: Ciouncii olI
thei-C ni 'elx si Tr Floriida
in Partial uifi l ineiC-n of r. h: rl .c. rren-ers c r t:h
Degree nf Do['ctor of Philos crph



To m, F !i t p .

AC> I "'l LE E'iE:J .

This i.oril; I.a supported in part Lb: thr. Ilati..nal Sci-

.nce Foundati-.n ', raint ,iK-2"n tlhe Ilffic,:. of fIaval Research

Count ract ;jI.il. 1 --bS.-A -1)1 fIl, 1, thie i nter for Inf orrpatics

Re searchh and the i .aduat,: ..choo! of tlie iUriversit.. of

Florida. This financial as:sista-nce is iratetfull acni] -

e dged.

The -uthor 1-o1.lj lil.e to ha tal he members of his

supcr i 1-.',.rv conirni ttLe and dcpartPri.ntal facu]i L, representa-

ties; for the fine jiob the.- diJ it reading and there fore

imprrovin'. the presentation madC in this dis sertation. Dr.

Ra :m ond i.. H-cl .'tt from the riledical School should be

-icno..:l ded for the informant ion on medical aspects.

For his tir;-e ind effort in directing the rescaarcli, the

author uoiuld especial ,' lil:.e to tl-inl. Dr. Juliush T. Tou.

Without his S tl.li1at1in idens, his invaluable 5su Ip st ionS,

his honest crit icism and his enlightening guidance, this

dJisertatiorn would not ha.ve bcen possible. The author's

utmost thanl.sr i r.lit g o to him.

Hliss lean Ponian should be acknor'ledgced for proofread-

in; .


ACK:L;'.'L rcE: iEF T;:TS . ..



AF.STPACT . . . .

C h apte i

i I NT PROD IUCT I O . . . . . . .

1.1. .1 Suitci- of e's, arch in their
Arc 3 ':f Picture 'r -ces:- n' . .
1. .uiiimi3ar of thtse Se -arina in
Chat r . . . . . .

M1ETHOD . . . . . . . .

2.1. Loundal' Seecments Findinr . .
2.2. Corbininc EuInd'rT- Se'ier-'s
t Fur the EC I.PJ Ir, CL ni c,.iurs
of Obic . . . .

A .LYSI S . . . . . . . .

3. 1. Som cie Funji-danental Concept
in Finr ir Pict r . . . .
3.2. Contour Firdiric in .lulti-
level Pic ur . . . . .
3.3. In- luCsion Relati i- on .rIronc
Contours . . .
3.4. Object E.:traction b. C.'ipar]son .

PROC' SS . . . . ....... . .

.1. Some Graph Theiory F.ackgroiundls
and the Proporties cf i nlT . .




:.: i

( II

TABLE OF C,:i E1 NTE I c..n t i ruid)

F' 3a,:.

J.2. Th :e pr sent a i.ri .f a
[i 1 1 it -e d F'c ri, b a
,.'e ; h [i . . . . S
t FEATUJErE"T F'.CT:. ........... 9.
E rEATIUP.E EYTF.T IT IK2 ... .. . . . 11

5.1. irne Fundari:. ntai L..cal Feature 91
5.2. Cl. al Features . . . . . 102

I E. r'EF. i;Et;T 5 .u.D cOiNCLLJ -'iOC i: . . . 106

S.]. E.:p r ntii s ith ni' rr: ,-_,'Ch .ro -irs T[-
Pict .]r-- . . . . . . 106
.2. ExperiF-ents ,n t -rin Cl
Pictures . ... ..... . 119
6 .5. E:.: p r i ientcts :n E lo d Cell
Pi c: t u res- . . . .. . .. 126
,6.. C-Lnclus ions -rnd FuLrther research. 132

E i PBL I,','_ PH. .- .F' . . . . . .. . .. . 13"

FBI CF. ',.PH HICAL ETCH . . . . . . . 140


Table Fage

5.1 Tabli c f area, x:-increvmien and
v- inr cre en . . . . . 92

5.2 able of nmrment . . . . . . 93

3. Table cf rimoirrent of inertla and
product of iriert a . . . . .

6.1 Local featnLure of ni.ucei iii
Figure 6.12 . . . . . .. . 124


Fheur- Pa'ge

i.1 A scene of tt.o' cube . . . 1

1.2 A bloci diara'il of a gen.rai pictul re
proce ssing t ; .i . . . . ..

1.3 E:xampleE off (' l iheyig. onal grid and
1.' r.ct.angul r grid . . . 4

1.J .A bLinar' picture of nurimeral "t." . . 6

1.S A digiti:ed picture of a portion of
3 si n ce l picture . . . . 8.

1.6 Linlag amon faces . . . . .. 1i

1.7 Several inhibited linl: . . . . 11

1.8 An example of the decio':imos ition of
three-diriensional objects . .. . 13

1.9 Local ccmnpleteness of It, . 17

1.10 [he graph of a vPDL. . . . . .. . 20

1.11 An example of the tructurel descrip-
tion tof a pictIure . . . . . 22

1.12 Criiar ar for the : clas of pictures of
chromiosoris and three ex::arrpl-eF of
chrome, ome s . . . . . . . 2

2. 1 The -nc lhborin:: points of 'a picture
point p and the octal chain codes . 2S

2 The enhanced picture of Figure .4 . 33

2.2 Flowu diagram for searching for
tentative b-un.Jdary paths . . . . 34

2.4 ilistogram of I-valtuei of Figure 2.3 . 35

LIST OF FI .LIREC (c:.nr inru-. d

Figure F'P e

S.E Flow diagram for filling 1 o ar-s between
tentative bo:undar. paths . . . . 37

2.6 E::amfples cf dec:cn',:rati,:.n from i 1 n:1r n-
lunct i :n bIoiLiu dar'. i :,,i nt iri.l I i 1 olrinc -
Lion pointrL on th i-' iin rl ..' . . ?.

2.7 F lo dli'?.r'im for :rderin:- noriri ir; uliar
paths ars found i unc tio:n areas . . . 41

S.8 Flow di igr ai for findirn: all the
Siis lc t contour s nid l : r, r i .:,r
bound ar. in an isoi'ate'd pictr ure . .. . 4

2.9 F amir. i1lc o.f '.a ovi c r aippinz (b :-if-
foldin. nd I'c tc...h irn.: .. . . . . 15

2.10 F;3 arrples of loo: pingl nodcsc . . . . 45

2.11 Fl:i. dia raiL for coi: 'ilriing pith: in
forillin; thr,: boundiar: c:,nt:I.urS :f
ob iects . . . . . . . . 47

3.1 The bin.i'r piictures transformed from
Figure 1.5 . . . . . . . 2

F.-2 FlI:,', dia, rami for finding' conto:,urs in
a multi-lev 1el picture . . ... . 6

3. 5 The labeled 7ricture r.btained fr..imi
Figure 1.5 . . . . . 64

35. St'atc dia.3r-an for finding the inclu-
s in relate ion a .. ne count: our u .. .. .6t'

5.5 The Hasse graph reprresenting t1he
inclus irn relation .a rr:ong contours
in Fi ure l.S . . . . . .

3.6 The- Il'-sc graph obt-irni d through the
delction nf small 1 iito.rs ... . 69

5.7 The Hasse c-raph -obtained through trh:
deletion of similar contcur. . . . 0

4.1 An example of an undirected finite
graph . . . . . . . . . 7


LIST OF FiGURi ES (,:conlt nu,.dl

Figure Page

4.2' ;n e.,nample of a simp l. gr.4 h . . 3

4.3 An :exampei of a 3 ..eiphted riph . . 79

4. I ,\n tl..T of the i.elphted : an li sho'.n
in Figure 4.3 .. .. . . . . . 79

4.5 Floi. diacra: m for finding an '1ST of
S.1 i.hted graph . . . . . . 81

4.6 Flo': liac'rar for finding radial paths
ind writers in a tr'c. . . . ... 5

4.7 Thri:e Lethods for connecting pictur'-
po ints . .. . . . . . . 33

5.1 X-:, coordinate system and the eight
possible line s~ements ..ith the
corresponding octal chain codes . ... 91

5.2 Principal axis direction . . . . 101

6.1 Anr 8-lev.l picture of a hIu1man
chromos j.:-m .e . . . . . . . 107

6.2 Th-e nlanced picture of Figure 6.1 . 103

0.3 The boundair. picture obtained from
Fii-ur.- o. b;, the gradient method .. . 109

6.4 The smoothed difference function of
thc bounndary sho'.n in Fivure- 6.4 . .. ill

6.5 Th. labeled picture obtained fron
Fieure 6.1 . . . ... . . . 112

6.6 ihe boundary picture obtained from
Figure 6.1 by the contour analysis . 113

6.7 The smooth,-d difference function of
the contour shoh .n in Fig re 6 .6 . . 114

6.8 An :s15T obtained from Figure 6.1 . . 116

6.9 The four radial paths in Figure 6.3
obtain-d by deleting small branches . 117

LIST OF FIGURES (conti nui. cl

Figure- Page

6.10 Ar. S-level pictur-. of a pc'. tion .of
i s.in cell picture . . . . . 12

6.11 Th labil d pictuij obtained frocr
F i gure 6.i u . . . . .. . . 121i

S.12 The boundaar. picture obt ainri:-l from
rfi ure 6 .10 bv.' th,- conto:.ir ir, 1at ,'l i . 123

6.13 Flo': di. aciari of findlint the dis tribu-
ti :n of objects in Fi.ure r.10 .. . 125

6.14 An 'S.T used to .Ic;cribe the dist ibu-
tion of obj cts in Figure 6.1 . .. . 127

6.15 The clusters of :b1-cts. in Figure
6 11 . . . . . . . . 1

6.16 ,An F-level picture oi f a pcrt ion o
a blood celi picture . . . . 129

6.17 Flowi diagrarm for ccunt inr e o.ver-
lpping blood cells . ... . . . 131

6.18 ihe hitcocra of the intensities .f
a blood cell pict .ur. . . . . . 133

Abstract of [iis :ertatl ion Pre-.scntcd to: the Gra.duate Council
of the Uini rsitv % of Flt orid' in Partial Fulfillmrcnt of rthe
Requircments for the [iDeree of Docto:r of Phii osophyh

OBJECT EXTF', IA T O;'i ,,_DiE, i EjT iF CAT iONl
if PICTl. i.E PR[OCE:E.':.i;G

Peter Pei teh !in

:la' rch, 19"2-'

Clhi i rir ni : Dr T L l 1i.r' T t.-'o,
'i r 1 .ep'rti.,ent E1,-ctric.ial Engineering

Anr. co,,mput, ri:. i picture process ing system can ccncjr-

ally be div idedJ into four maior units: a picture .Jigiti cr,

an b.ject e:.:tiact.o a feature ext a ictor and 3 classifier.

This dis.-ertatin is concerned mTainlyv with ne appionches to

object extract icn ind f nature extract ion.

Thrcc informant ion handling methods have been developed

which mai. be used to m.echanize the extraction of objects

fre"t mi ulti e l c1 i cture These methods are those of the

gradient .anail.sis, the contour analiysis and the gr'ph theory

approach. in the gradicint an:aly'jis method, a locally optimal

threshold i usd to, fin t in tlic bounda3 r points. In this ncu

approach, high efficiency is achieved because the tentative

boundary paths are sin.uitanerous.- fuLind. Then, after filling

the p ips lono the boundaries or at the intersections and

removing the tail boundary segments are funr.d. P:ulcs are

set to. combine the bound.- scemnii rnt;: in order tor decompoise

the overlapping, self-f.:,idin,- and t.oclClin objects in an

are3. picture ( Th contour inali., rr,-tho-d i: .I:. 'clopeJd on

the assumpt .i tha-t the thres-hoJld usedJ for tran f rin i

ui lt -ic- el pic tiri to a b in r," picture i- .pp ::imi i tel '

c.n c'stint in a wJind ri' .) This rie thl-:d permit .er *;~- ce. f7 ll

object c :.- racti:,on for i mult i-le. el picture ,ith selec ti ve

delctio n if n:nbo.,urid r contours In the criph the:r-,r ajp-

pro.ech, a rmult -Ic -l pic:t re a.- transfered t". i.ei ahted

graph. An MIF 'rlinimal 'pi inning F.ore t of the iei hted Fraph

is then found. ,'- finding n r l -e 1pi ncipal rpa ths : -f a tri.: in

tlhe '1 th the sl: eton o f thc ohicct corr p:nd n t tr hi, tree

can then be fl-:,und.

The bound r T cont our ff4 an object ha been encoded b.

a sequence 'i:f octal chain codes. A local feature e:t r tc r

has been designed to find the area, cpntro:'id, shape, princi-

pal a.3.is direct ion and the el.ntation inde.d .f an :,bi.ect i th

the i:noiledge ocf t he sequence of :,c ct l chain c,:,des onf th;

bo und ,r'i contorur. A glo.bIal f .atu rc e.tra.ctor h i bc-n .1 -

signcd to find the inclusicon rel rionsi-hilp a moring '. cts and

the distribution .',f 'obiccts in a picture. Thc nc l i .n

rela: ionship is relprc rented hby la S i., gra'iph. 1hie di:tribu-

tion of ohli cts r.i,.-" be rcpresentc d h an 'IST I.'linimral :pan-

nin- Tree).

The nc.'l,' desiigned objic t cx:tracrtor ind the feature

extractor mI ctliOh s I'hav be.'n t st:d bh" .nal.sis of the

ir, f orrin't ion in the picture of chromos on s., :kin cells ind

blooJ cells. in eval at ion of chtromo- om picture the

maiior tasks;- are to identify _nd c'ta3 ori:e All chrc, mosorrmes.

In anal,;sis of the his tol:.ogi l skin cell ph-toriicrc,, raphs.

the problem is to find tthe tr c:ture :of cell: in epidermis

in ordcr to detecct the degree of the m li :nanc, of poi sible

tumors. in CL V lu.lti :n of tlihe bl:o_-c.d cell photomicrn,, raphs,

the coal is tr. obt iin the his toc.r'ir, of the blood cell photo-

intensitie in ori er to reveal critical diac no:st ic inforrni-

tion. In each of these three evaluation tc s t very> promis-

ing results .ere achieved by the use of combinations of the

nei. techniques. .\ more complete cormputc ri d picture proces-

Sin; s .te m is suggested s an extension of tlhie- newly

developed. technique ..

x iii



Ficture procei:: g a proce wh12ich t ran c :.r;..n

to de -cript ions. F.:ir c..: ple, i.h ,. n a picture prc r.- c 'or

"see- a writing "E ", it : h uld le able t tell that it i

th,- Cihiriese character for "s:upn." h'h n ,. pi.:tur: prF,'o c ssc.or

"see-.:" the scene as sbhoi n in Fi.; e 1.1, it sh:uid tL. ll

that th: re 3re tc: .:i.uces, and E ,in the sic ne w he-e cub-

A is in front of cube P.

Fi Liure .1. A scene -*f t -' cubh, .

The prrce s sor performs ti.o rain function : the firs t

is "to see" and the second i; "to pive the description."

from iwh.t i_.= "scin." "1' see" 1 the proce'aE usual ly called

ob.C ict e .t actionn fr.mFi the scen e. "T i cicc the descript i -n "

from i hat has been "seen" includes feature e...trc: r ion and

identif-cartio.n. In general, human be ing .i r. he bc t pi -

ture proc, es.s:or- up t:, the present. 'ine dr:ial'.i. of man's

ability.- ja a pictfire prc oce scr i that his visu.il s'/ tem is

oeaLs; ti red. tlechan actionn :,of the picture p rciccssin be-

ciam pss ib l aftc-r the in cnticn of the modern compu ter.

This mechaniriat in is i'r': desiirable a3 it frees manpower

from routine visual t'- i .

There are two- princip -al type.r of pictiurre encountered

in ever'-Ida life. i'ne is the picture of three-drmern sional

object s. This type of picture is the protection of the

three-di iens onnal object on a picture pI.-ne. The projection

i su ppo:s d to: exhibit the depth information. Sc ',r. il re-

se Archer: 1- have conducted research dealing g i th this

t' pi of three-dimensio-nal picture. The other type of pic-

ture is tw.o -dimcn sion,1a Twr-di-;enrsi-onal pictures are

i thi artifici .l pictures iuch as characters and

maps, or n'-tural imaigcs those dep:h inforimat ion is not

impor i t ant L .nd :i liro- t c:inncit be seen in the picture planes,

such as pictures of particle tracks in the bubble cham-

bcr fingerprints and cell images. From

here on, "picture processing" means the mechan iat icn of

picture processor unless otherwise specified.

A picture procc ssing syS '.:rt carn rInerail be divid.d

inco four parts: a picture d r i .i:er, an ob I ct *-.tr. r ct.-'r,

a fc tur. e:.tractor arid a. c la s i fi r The picture di ii : er

andl the object e:,:tractor perform the funcr.ion oi f "j eeing .

The featur-e .'.r rac r and the c la ; if ier perfrn the func-

tion o:f ".ivinc th. descr iptions. Future 1.. is a lloc.

dia gram of r encral picturE pro cer iir: s te m.

optical picture oict eture cli-
picture di:ltl:er e. t ,t r .. t ractor ti' r

Figure 1.2. A blocd: dia.:ram of a j -n-ra picture proYc. s-
si n s ,s t o.

1 1. Suriuy ,:f r'eseavch in the ares
of Picrilie 'roccss in:

A brief surve,- of the are.-a of picture process ine is

presented in this s-ect :in.

1.i.1. Picture Di,'it :er

A picture diciti :e-r transformer the data of an iima-.-

to a digitized form -hich is accessib le b,. i dicita) com-

puter. .An optical pic ure can be repre sic:tcd mathenat ic ally

as a rsal function f on a pict.uro plane i1, i.hich is a simple ,

connected subset of the real lane f: D-R, ..here P is the

set of the intensity, values of the picture points. Th-re

are ti.o principal W is to quu.inti :e 1 3 picture plane:

the he . ornail grid and the rect.ian i llar rid. Figure 1.3

cho'.s the t ,o) types of grid He:x:a,;on l crids have the

adi.'ant t e of having si:: ne i hhborirn: picture points Chich

.ir nearest to p, for eve pr. picture point r. The-, h.iv- the

dr'ihbai c of heing bha4ed :on an uncomirmonri, n.on-orthoganal co-

ordinate s t',';tm. The rect.ir:uil.ar :rldi contain only fCoIur

neighboring picture point- whichh are near t z to p, for

ever.,'e ic:ture pui t p, but i is very eas. to a.cce s ever'

Dicture pc int.

i a) b)

Figure 1 .3. E::nples of a) he::auonal erid and Ib) rec-
tanCul r grid.

Ihe rectangular grid forms an orthogonal coordinate

s-,sten. Hence the picture plane becomes I = I,. X I,,

where i and [ are subsets of the integer set. Froan here

on, all digitized pictures ui:e rectangular grids. The in-

tensitie. of a qiuantized picture are quantized into n

levels. llIsual ly n is s t. equal to 2k because this maximizes

storage efficiency .ithinir the bit -orilenrt:-d di Lital computers.

A -level picture is called a I -b it picture f i = 1.

the dit'iti- ed picture is called a binar., picture. An n-le el

diciti:ed picture is a mapping g: i-U. where i = 0 ,1 .

n-l1 i1 the sc t of anti: 2d intense it'- %a l e A diciti:ed

picture can also be r- rprcsented as a matri::. The location

of a picture point is stccified b'" the lo.cati:on of the el -

ment in the niatrix. The intensit,'- cf a pictu re point is

indiciat.e hv the v.altue of the correspond.irn. element in the

matri .. Fi igur 1.4 is a binai picture of a nui meral "6"

represented as this Ai.itrix. form.

A picture digitizer performs a transformiat ion from an

f mapping to a ni-hj mapping g. 3 comp i t picture Jdiilt1:iti T

system, F IDAC (.Pictorial IData Acquis ition '.oiuuter I .as

been implen,-mntJd at the C i1 (Center for informatics F:esearchl

in the University of Florida. The PIL'C, C uhich is a mnodifi-

catic n of the FlPAi '1 1 L(Film Inrput to ieital Automatic

Compputeri 'y'stemi, is one of th-- better picture jigiti-:ers

available today. Tt consists of a i:RT, ti.n lens-s.:, a photo-

multiplier, an a-J converter and a scan control intit. Th

digiti:ed pictures are stored on a maonietic tapF.' The

PIDAC can alternately, be inrtrfacud iith a digital computer

to store the digitized form from a picture The ma::imum

spatial resolution of tlih riIDAC is 1.24i':0 spots along the

long a.is and 800 lines p-r 35 riu film. The ma::imum digi-

tized level of the PILAC is = 64. V r:,' good -lc'vel

II 1 1 1 1
1 1 1 1

Si1 1 1 1
1 1 1 1 I

1 1 1 1 1

Figur': 1.1. A binary picture of numeral "6."

p i c t r can bl a 3 l ved bL'. the f' L.-C. Th. sc in ir. I p-:.- d

cf the PiPAC 1I .3 se pictuir. Figure 1.5 is portion of

an ,- level s i n cell picture obta ined fiom F'PI.['C.

Once a dci t i ed pictur, ir 1 bta jined, the pitItre j :r dat

arc Lthein a: c ible b, : Jic tal ccmiput'rs Th- nie.:t prO :-.

is the c.t:tr ctli ei of ob -i c s from th': picture.

1.1.2. Ob ject E:::t ract io.:n

There are mainly three methods iis.d j in extract in ob-

ject; frori the S.:ene The firs r method fipd; th; b,:,unjdarli z

of ':bh! iccts :nd thcn de-co posn*er u :bi ect fr,: m bouTnd r aries.

Thie secc.nd i'th.d finds th- thr hol.J t h t.t rat n sfe:r r r.ulti-

level picture to a bI- nar.. pictur.- and th,-n fii:nd the crn-

tours of the, b narv picture es The third metlid.J finJs the

clIust-rs in a picture and c,,nsid:r: :iach cluster ras an ob-


There are tiwc miarin app roaches used to find the b.:,o nd ja-

ries o' objects. One approach finds the Cre haiced pictiur.-

first and then find then in bo-und aries Th,: .-nhan.edJ pictiureui

can be found either from the spatial doniiin *'r fri.r, the

spatial frequency. -Jo~ni n. To find the enhanced ,ictures

directly froi i the picture plane (spatial do:,niini th- m,:-:t

frequent ly used Ime thods are th- gradient m.- thbe .d '1 ) and

the I. placian method.1 I Rv. th gradienci t methodj, e-ach,

picture point in an enhanced picture is set to have a value

equal to the gradient c.f intensity nt that picture point.

i 1 2
1 1 :

1 -1 : 1
1 34

4 -I 4 1
4 5 4 .1

-1 ;* ;'

:11 1

Figure- 1.5. A digit ied L picture of prt f rtin skin
cell pictLire.

Since the data are dipir i:ed. app c.::.iarat ion 0: f gr adi,. nt 13

SS1 e "

( l'i*1,j:.1- ; i ,j*1ll' ] i ,ilic is a v er,' good apr.rc. xi a-

tion e..;cept that is 'are root caIcul tiOi i- iivol ed.1

Appro '.inati:,rn of the Liplaciain furicti.or, is ric.d f:or

*di iti :e dta, s..c h 3a 1 = li-1, 'i. ii i +

S.l i j-I p i, 1i 1 i.,i T her function L then reilrc-s. e ts

the enh.Inced [ictiLre. Thie- 'nIlan d picture i.i I normal,

h-ive high values 'it the b:ndrl iri s. To find! the ,erhha.rc j

picture from the spitial f re .c u i-r. ,' dicrin,11' tie'. picture

f is trains, fcrred tc. a Fou.ri er s .pictrumr F fir : A I.ieh-

pass filter H is appl ied tco enh.in_.e v:ilues c f F 'it hiit- l

frequencies rel ti ve t,: those at Ic.I. freque.ci:es. I[he in-

verse FourieI trans: f: rmr.ti::, n cf Fli is the cc.rres- pondirI e.n-

hanced picture. After the enhanced picture is found. a

threshold is then sc t to find the b,.-u dar, points. Eo ui dar:.

polnits are con tccted b"- a nrulti-step prices s l ThI other

approach finds tlic boundaries b.- -s.,ue : ff a iiatci ed filter

ihich can extract the bcundir iies directly frism the pictrl.ie

e. The pirpo -se cf the ed ;e cper ator used bv Huech:l 1 i

to fit aan ideal ed'.e : lem ,ent to a, emir, rical 1. obtair ed

edge element. In scariiinrc the picture i iihen in ed': is

found bv the edge opel.rator, scarnninn; is interrupted and the

edge is traced until lost.

After the boundaries in a picture are found, objects

are to be extracted. C.uim:an did the work or, c.::trctrinr

thrc: -di i'-nsi,:.rin l :ob ccts The main idea u-sed tc:, extract

the thrcL-d-imdr ni l.on l ob L cts 3s based on tl-r a prior

kn,.i-wledge of the possibility' of tc-.c faces belonging to an

ob iec t. A vertex i? in general l a point :.f inte rsectio.rn f:f

t\;:, ,.r mr.u e bounJdaries .-of regio,:ns ,A pr.:. ran -.EE his been

built to e :x minei the co.nfi; uriat ic.n : f lines nrcetin,; at the

vcrte. to. obtain evidence relevant to '.hetlth r the regions

involve : d bi lonn; t.: s:iir: object. Tic t ,-pes of links, stron.i

links and iweak links, are used. Figure 1 shi:nu the

lin:kagc :.f facesi at several vertice A s l id line implies

a str.'rig lin k and a dotted l in implies a weak link. Fig-

uire 1.: sh.nt.s the links which art: inhibited.

A rg ion is defined as a surface bounded b; simn ly

closed curves. A nucleus is a set of regions. nq :o nuclei,

A and I-. are iin .ed if ithe reg ions a and b are linked where

a E A and b i B. Three rules are set t.. link: the nuclei.

First rule: If tjwo nuclei are linked b t r .. r more

strong links, they are merged into a larger

nuc le s.

Second rule: If nuclei A and B are joined by a strong and

a weak link, they are merged into a new


Third rule: If nucleus .A consists of a single region, has

one link with nucleus FS and no links ,ith any

other nucleus, a-nd B are merged.

"F:rk" Ar ro'."

Figure 1.6. Linkage amrn.:,n, faces.

"Lc: 9'



"r!*,1:ch inc T' s"


Fipurc 1.'. Several inhibited links .

\ el

The firf r rule is applied on the picture r, peatedlv until

it iS. no 1roncer possi ble to com'ibin, nuclei The second and

thirJ rules ar then applied success ivil'.l..

FiLgure 1., is an example of the dJ cormpo.sition of three-

,Jir ensLion .1 1 ob iict5 in step 1, verv' nucleus correspornds

to a re .-i n: for e:.amolr.1.:, nucld e i F. and C corr,-sp.rnd to

rgi. ns a, h and c, respe.cti''el,. There are to strong

link: c cornn,-ctin nuclei A .and E. rne lini. comes fr.-o the Y

intersection of r -e ion- a, I_ and c. The other linl. comrLe

from the srroT' intersectiorn of region ,3 a an b. All other

linI.s are Ji rived in the ie asa e v. S tep 2 is the strai.ht-

for':w rd aipll icat ion of the three r-ulr: to combine nuclei.

Thc regions correspond to nuclei in a group on an object,

for e:x:imple regions a, b and c form an object. It is ob-

io-lou that 'iu:rman's irreth dJ can be applied only to the pic-

tures of thire- -dimencsicnal *.obicts.

ThI second method uses thresholds. to transfer a multi-

level picture, and then finds the contours of the binary

pictures as o-bie-ct b.,ounndarie.s. Pretitt i11 used the local

minimia af the optical dJnsit\.Yfrequencv dis tribution of a

picture as the thri-shold; to find the backerocund levels,

c't.)-,pla.sm l r:- l5 and nucleus levels.

The third method is the clustering method. s3hn(16)

proposed a method to group points into objects by the

clusterinc method, i;hich is graph theory oriented. Th is

cluste.ring method is iimoitivated b) the perception of t:o-

Sn .r- - . . .

Sr 7- r- - -.

... J ^ ...

Figure 1.S. An e.x:ample of the decoi:.rpo ition of thrLL -
Jimiension-lal obiectS.

dimen si:rn. .l p: in t s ,t 3s S.par it "c;-es tits. The pr inci-

ple of e!r'upinri used is "pri:c:iiiTii 't 3 Jescribh bd y h 'rt-

heimner. 1") The proposed methr.od is applicable to binary

pictures. For a binary picture every picture point i:ith

gre I va. lue 1 is a vertex. Picture points havir.rn ; re.

value 1 are called :Lbject points. The connection. between

ob :. c p.cints is cal led aJ n edge. u. ii ht is ass iglne, to

eve ede. It is c.equal to the EiucliJdean Ji staice bet.ceen

the coi respon-,r diri object po ints An tIST lili .inim.31 ipainin;

Tree) T is JefineJ as a spanr.in i tree of C. t.lose i; ighl't is

mininiui r air-mn all spanni 3r ng tre.: of 6'c.S'ome ': d in the

Ml.T c n be deleted by us ir g 3 factor as the measure of thei

s in i ficant ed:e inconsris tency. The IS'T is then clustered

to a fo.re t. Ever., tree in the forest t clusters together all

the point' in o:,ne :vbjiect.

S1 3. F a ure r e E:-:t ra c t io

Feature C::t ract ion stroneil ,'. depends on the type of

pictures han i led. li,:'%w bi_2 the feature set should he de-

pends on the purpose of ihaindlling the picture.

There are two main' types ocf features. One is the local

feature which depends on individual abiects in the picture.

Arca and centroid were presented by Freeman. 18) Eden (4

has p ropo ed the fundamental st r T-:-.e as the features of

handiwritte.n F n lish characters. Topolo i cal features are

proposed by Tcu and Gonzalc: (S for clhir:actcriznin2 hand-

w.ritten characters. Topolofical features have been used

I 9 I1 'P jI
for auto:jma tic f incerprint intcrpr:etatio ,\ ske leton

has been proposed to describe indirc.- t 1.' the ,shape .:f o:b-

jects.. A skeleton can be thought of as a er.r-i l i:d .axi

of s r.'m ,tr' of ita ocb ect. At first the concept ..as applied

to the b ir a :r pie. t rur: n d .Ps If, '.n t anar i Fhii -

brick id ,nJ the-rz h .'dv: d c\ l.:-ped al :r ithmis to fi nd the,-

skl: c t;r.ns in binary pictures. Levi ge nerall:ed the

concept to the miulti le'el pictures b: defining a r'ne dis-

tance functions which t' .l thIe L rc'.: lc\'el intens11tie irnt,

considerati-.n. Ledle.' lul us-d th ratio .f the rnu inbh r of

concavities to the number f se me tnt of the hournd ir'' i s

the only. fcat.,r. in d.:tecrine the mintrtic cells.

The other e l o bal fe tur .

Global features are he one-s t-.hich reveal the in t rrelati:.:n-

ship ainon o'bic tcT in t ihi: picture. *'_.o)mCtim:3 in cb-ect can

be discribh.- i in tcrm s :f fund :;iental c'n-,nrionn ntnr I.lob.a

features cai al 's be uzcd ro Jdescribe the intl rrelat r i:nsh ip

amc-ng fundamental cc.mrrponents c f an obii ect. [nc]lusllon rel'i-

rion among regions can De found b/ A: . 24 :'.'eral lin-

guistic descriptions have been used to describe the l,-,bal

features. tar siimhan used :-yntax:-directed hierarch'.

labeling to describe the particle i tracks in thi bubble:

chamber. Shat. pr p .:sel d a PF'rIL '- (PF ctur.- Dc cr, ipt i,:n

language) which may be the most formal and useful linuistic

approach in picture processing u.p to d.at,. L ngi stic ap-

proach has the advantage :f Jdscrbiing the picture fo:rmal.v.

Th- PDL I'Pictull're Description Lan uag1tc is 3 picture

or graph algcbr- *:ver the set of primitive structural de-

scription:z r..Jr the e.: r tions *, .*; *, and '. Fig-

uri: 1.9 sn ,:u the lical ciorrpleteness of the oip.erations

[*, -, :.:. *]. Flements in the i'DL are c,,nc; id ered equal if

they ai e qui\'alent. The, equii talent relation i deJfined as-

1. 1: is .ea: l:. equ ivalent to if there ex:i ts.; an rso-

imorphi:ni betL.een rraph. o'f 1 and u, :uch that the corres-

punrLdinr e jdce hi\e identical n.iimts.

2. ''1 ; equii lent to S, if (a S1 is ,, eakly equivalent to

,, and (b) tail li ) = tail(ls, and head (r l) = head(Sj .

A number of useful algebraic properties are given below:

1. Each of th,: binary operators i; a .occia3tive.

. is the *:,nl, coimmut-ative operator, x and are "ueaklv"

C ommuiMl t t \ve.

3. The unar, operator

(a' 3cts a3_ comiplemi entat ion in a Boolean algebra.

f', SJi) = ('-t "V 1-Si "

'bi obc s a "de t organ's law" iw'ith respect to :- and -

(- l": 1x',' ) = ((-S, l- (~ )

I- 1 -S .) = ((-5 )::(- ))

(c In\ol tion:

((- S-s ) = ) .

n nc at at tI on Ile- c rip. i-1-.r,



a b




b b





Figure 1.). Ir Lcal coupleteneres of f s

4. The ope-rator.

f: .1 ( ' I) = .

(t,.) . I b s = ( 1" b , I er;e OL, is a b inarv

pe r 3 tor

5. The null point t pr iiiti e

Ia) = .

Id ) L = .

P,- Lu inr sg Jm r:.f the algebraic properties of PDL to

'ioec unar: :pc: rartc-r and label design.ators as far as p,:,s-

sible i thin an expression, a stand.ird form- f(S) PDL of an

expr esion : can be obtained. f(S) is defined bv:

if ( = 1 = \ = I -

= [-..' iS I primritive ( 1 then f(S) =

Ve 1f ,

if S = 1 '' bS l b( 'x, -, then fr ) = l; l) f S


if S = -(, then ffS = flg(S) f


if S = (-(S1 S )), h then fl'S) = f((-,f( )) ffl-s)).

if S = (-(S -S~, .l, then f(S) = (f(f-S ) )xf((--S ) i)


if = 1' '51 0 i 0 *,-,.,' the n

fl.- f(= ifl ( i 1 1 i f(i* 5.


if S = f-l'-U 1 i l then f (. = f i .

if 5 = 1 [ 1' {. = f 11}, then flc l = tlI f l. '-l, 1 11

S1 ;:e

if = i ('S1 t lin f iS = f i '5 11

A valid PFri. e::pressioi I vPDI i s the uonc Nihose ;tan-

dard form is such that if I..'p app,.: r: ini it urie or i-re

times for some primiti-e p and labcl i their p i' i p appears

once and onlv once Cutside the scope ,:f a ,

The graph Jei sc ihe bd : a \PPL 's i.- d finL- bt.' th,-- fol -

lowin g al orithm:

1. Transform S into standard form bL, appl.'i n, the func:ti:on f.

". Replace each expression of the form I p 'l b, a new. priimi-

tivi p' This remri.v. all operators.

3. Generate: the connectivit.- graph of the resultiin, cx-

press ion.

4. Connect the tail and head nodes of each edge p to the

corresponding nodes of p

5. Eliminate all edges of the fo.rmi p

The above algorithm furmnli.l defines the meaning of r1belcd

c...pr-essions and the operator. Figure 1.10 shows the Craph

of a vPDL.

step 2 ((r + b) (b + 3) c) a



step 4 c


step S i

Figure 1.10. The graph of a vPDL.


It has been proved that an. vPDL describes? unique

primicive conrnectivitv and rd ar connected s-t o:f pr r iti L e -

canr be effectively described b:." vPDL. It has ailo becen

Shoin that the or L i n i'tail i of picture c r.n be at ain',

con\'vln 1 nt p lace.

The scit of rule i or gra m ai r G; that cenerar te.: i. r-: cr ibe-:

the claC :if picture? F' '.'ill be a t..'pe : ccl rte.\ t- fr: I

phrase structure gramIFa.r with the follo.in. restr cti onrs.

Each production i of t he form:

5 pd1 f pd ,|pd1 1 ... p. I n 1 1.

where S is a non-termir-il .-mbol arid p.l1 is an.. PDL expres-

sio:n r ith the additicr that n rn-terminal sr imbo ls are allo.-

able replacements for primitiVc class names. ':entnrces of

L(GI will consist of PDL c pr -si.:-n : thls, the class of

terminal s,.',bihcls c f C .ill be a sub.et :f

(*, ,-,*,-, .1',.} \(primitive class names) labelbl d .si -i tor )

Each grammar G will haie one distincuishedr non-terninal

s:,rbol from which L(G i may be generated; the s*.'-bol on the

left part of the fir_'t production c.f C .ill be the distin-

guished symbol.

The hieraichic structural description H~ C) of a pic-

ture C,'IG ha':ing primiitive structural description TSIC) LI'G

is defined as the parse of T (CI according, to G: H- C) is

convenient tly rcprces ntcd as a parenthe ies-fre tree 4

simple- exa.mplc of PDrI. description of a house i;- ivcn in

House liv I *I i, )i T ri ni;1 i l

Triangle I did. d il h'i

L('i =[ ('h* I-m i I I' Idr. dr.rl I)'1


1a ,,) L I i ,

T I c.

FI- ':i :

dm\ h

.and priimit i,.-s

vm I

cl c-, c
C= (7 hd dm) h))
I' \- + i'h~ 1 ) .l ,d - ( 1. d dn, h))

// Tri in le

v *
p dm

.ple Vnd prse of a "house"
db) Ex.anples and parse of a "house"

Figure 1.11. An e-inmple of the structure descriptions
of a picture.


Figure 1.11. N:o te that all thr e pictures f-, house e in

Figure 1.i1 have the -iisme primitive tructurs1 de. sc:riptir ns.

,h ich carn be a.ic: epted bi the r amrii-i r .

PDL can d-.-cribe Ver. i el th, interrel. 1 ti cnsIi ip be-

t..en pri iti 1 1 s, but it does not have th- a.bilil,' to find

the prim tiv i Fiigurc 1.12 sh:, s three pi.:ctures of chr:.io -

some_ z- 1nd the- accomipan,'i, rrairia'-r to de-cribe the pi,:tijrre.-.

G: Chromosome }I r'

K: \v 1+-i liv+ 1 li -\lI :

El p- *p

pr i iv

M q :3

T (C 1 (' + Lp) v F v % F. .
T(C.~ = I(p.v*p) i* F+p v

T (C '.I = ip *'. p) (p+* 'v

Fi;,ure 1.12. Crammar for the cl-,s of picture. of chrromia-
sOiC and three examples of chr,.omsF on, Me .

it is seen from Figure 1.12 that the PDL .entences,. which

describe htiman chrciros oiLc. re \er. iniple ex.press ions

The mi-Tin problmrn in li.indli:g: the picture of chrnmo_-.on-r -= is

to find the primitive-'.

1.1. -. C i as i cf i t ion

*.nce g':o d 'set of features has been e\tracted i-an:
i i. i
cl 3 sification techniques 5 are available. If the set

.*f feature s .*f different categories are linearl: separable,

linear clas-if ic.ation can be used; othcrisi .- rncn-linear

classificatin -. should b[e us:ed. ul t i le el class ifica-

tin 'i i s-ome:times used. F:r lin uist ic d.e criptirn .:.f
fe. tu r ,-* ; ir 3T .
fe.- ture a .r 3iTiri 4 c ,n be- des i ned tn accept a

sentencL- :nlv if de cribs a p ic r.re *:,f s:,mie specific

ca te::r A :rammar can then ser'.te the purp ose- otf classi-

fyvin o :bjectts. Fo:r c xa'riplF :ny picture hiavin a. PDL ey-

pression '..hich is acceptable by the grammar shlon in Figure:

1. 1 2 is. class s ififed as a c hr.om s. me. Notc- that a31 three

pictures :hih:.in in Figure 1.12 will be acce ptc-d as chromc -

.imes. The main problem of d si ening a gr i iammAr is that

it has to be complete in the sense that it slh:uld be able

to accept all picture- in 3 c3te .ry. Here uc like to em-

pha.i:e that in crder to have a good result r on the classifi-

caticon, a god ; et of features i required. If the fea-

tures' set :s poor, no r.atter hC,' g:o.cd the cl3assification

technique is the result i.ill be of poor quality.

1.2. Summary of the RF;mairtinE Cha-pters

Chapters II and Iii present tu,, different methods of

object e..:tactio.n. Chapter II uses the gradient method to

find the enhanced picture Boundary points are found by

r.apti''el., thresho: ld in the ,r adients. Pounrd.arv pFaths i.i1i

be found in the process 'f findi ng the boundar.t points.

Gaps ill then be filled in and the bound ar :'eemerint are

then found. S cci. al la''s ate used to c.c:rbir c the bo.undarv

se.grientt to form thr boundai ie- of indiv idi:ll objects.

\C'rla pp'] c ri s lf- folding arid touchirin objects are dc-cum-

poscJ. The gr.aJ i.nt uisd are inte.etsi r t- her than real

numbers, such as tl-.se used b. R:oberts Hence less

storage is required. The tlhre,-slold is adap tiv rather than

fix.ed. It is then less sensitive to the rinie. Chapt, r

III presents the contour aral.sis ri.. tl::,od. This r., thod ".'s

mocti ated in p..perimrt ntine .'ith area picture dasta b.y so.-i -

ing jiffcrent lev els of a picture in a disr, ia unit. The

main idea is that in a Isriall iindoi. section inr the picture.

the threshold for transferring the picture to: a binar. pic-

ture is aIpproxiimatel constant. The thresholds ate adap-

tive rath er than fixed, such as those usej b:- Pre itt. The

result of this method is \cry successful, es c.ciall',- for

the area pictures. Chapte- 1i'V disc s.1 Se the graph t ch:or

approach to the picture proce-ss;ing. cS.el:-tons of ohiects

can be found bv this approach. Further t lc.re t ica 1 research

should bc done in this areas.

Claipter 1 presents th,- extract ion of r-se'eral ir.portantr

features for area pictures. Are. centroid, shape, princi-

pal a.is direction and elongation index of an object aqe

the local ft'e ture-s discuss d in this cha te-r. Inc lu:ion

re l t 3 ion :iaione ri:c b. :c rts and the distribute on o: f objects in j

picture .are the .lcbal features presented in this chapter.

Chapter VI dis;cuis es the e:.cperimencrs i.'t Li:medical

images b' t he abcvhe methcds and u ':c5 st f.ir:hir reseac Lh.

Chrc;.rins.one 5s. in cell and b ::lood cell pictures 're anal ':c .

In ex.periri.rIntin i.it chriomo.ome pictures, the mainn pr:bl

i- finr in: cihr:m:s: ies in .a rpictur,.. In e::pwrr mencinc with

s in cell 7:ctii. es, tihe r in prollemrr is to det.-ct the tumor, .

In e:,:reriir'ienr in ith biodi cell pictures, the main prcblem

is t.: find th- histc2:ram of the inrtens; itie All the ex-

rerirenrrs sho,-:.: very pr.om ising results. It is hoped that

further r- search can produce a more s-ophisticated imape ana-

1v: i ..i t e m.

CIiAiTEF 1ii


I'hen hu man hi ngs l1..: at an. scene the inrpact in f:,r-

mnticn i.e c r is th' hihapes of the ob h ect in th, .scene.

Th:- infurmiation r \'.- linii the s h.nes of th- b 'bl t are th.-

boundari c If the ob iects ir ovc rl a: re.l it i : : p s ib i:

to use the information :of b:oun.idariec adnd ,rc'.' in tns it ie t_

decompose objects. At i.e can ve ry eas il imriiCine, :'ne ..3

to find the bouir, darics is to i-c ti'iU fact that usually th'

boundri arises corsis t of thcsc poirit: h .i in ', e r,' thi h chain:

of intLcn=iti s front their r n i;hb rc.r Ui in2 t lh prop- rt:. t tui

find the bo-undarie : i.s called the gr'idient methd, bec a u e

the chance of intensities is mea. sured bv' the cr-tdi ent. ihe

gradient method is '.er:., p~od if there are high coCntrast: ir

the boiindarics, cn n if there Xli-t soUr i nu unnl iformr dis tribu-

tion of the intensities in the obiects.

Sone definitions ivill be intr-.duced before cettinc

into the croblben.

Dc-finition 2. 1.--A p I:int p. in the picture plane I is

in S-nc i hboTing point of the picture point p in I if and

onlv if -0:dip.p i.l:, here d is the Euclidcan distn nce func-

tinn. In ordei to miake. their l at r d lisc L ion easier, the

S -neighborirn: points : f pF ate laceled as shoi.-n in F i Lure 2.1.

The s;e of all S-neigiihhb rinc points o:f p is denoted as I'F.I.

The oc1t.1 chain cod.- thcl h Lniico.jde the line secent from p

t,: p1 1 t. recuse th i v. il l 'es ran.e from 0 to 7, thev

codeJ is an .:ctal c.:de.

It is obI:vio:.us that anv curve in the diciti:ed picture

is appr' ii-ted I'.. 3 cquence of line segrients whichh j.-in

their p-iint to their p-neichborinc points. Hence an'- curve

in th,- I Lt itt ed picture cain be represented b.. a sequence of

octal chain cod.e- and. their start point of tri e cu rv'e. Let C

be a cuive represented bL, the chain cci.dcs c ... c arnd the

start point c. The reverse of the curve C can then be repre-
rented b.- the sequence C = I( -41 C. 1 .and the

start point ,:*i uhich is the enj roint of the curve C. Be-

causct of the small stcrare requ iid to stole the chain coJe

.rnd tiec aus it is easy. to manipulate, this technique is used

throuChoult the di sertat ion t enco.de the curve in the digi-

tized picture.

P P6 P

6 5
/ I

P P-
1 P, 5

Figure 2.1. The S-neighborine points of a picture point p
and the octal chain codes.

2.1. B.:und ir. 'r,: ii,.'rnts Findin,

Phys ic 3l ., a hbioundar, point is a picture, point i whichh

ha.d a hi h incre i c-- of intrens itv from its n ei;c bor. r T -

rite of chanrie ,:of inticn it fr:m T [ t r he:re 1 an 8-

neighbo:rin, pirnt of p, c:'n bE: r-v3li3ted b' a dif fervent i t;.r,

whichh i de-fined as h p,p. i = L i i i] 'd. i ,, p h i re

9 is the picture- funicticir, nd d i; th; Euc id:.- n dis ranic-.

It is .. c i- ;' sct rn from ligur 2 .1 that di p, pi = 1 if 1 i

even and d( ,pi. = if i i s d. The h* function can ti-h n

be rede fined .1 h*i" ,p i = t'"([ lpl-gr l. I ril ], i.here E' i

= 0 if i s :dd and El. il 1 if i 13 ev n. Th, '' funct i:n

is defined t.', the followiinc m:iior inL t:ible.

I' g p

'^ -in-l' -(in- ... -1 0 1 ... n-2 n-1

E ( i) 0 -(In -1 1 '.*. -(n -2 I, ,. .. -1 ',' 1 . .In .7 I- n -1 1 -T
1 -(n-l) -(n-f ... -1 0 i ... n- i n-l

The situation n : f a drop of intensity frr.. pi t. p i~

not under :crrisidt Tr: t ion, i:c au thi s situation C .'ill be con-

sidered as an. increase f ir it ns- it .' f'rin p to p He-nce .c

can define an h*' func:tioin thich is. cquAil to' thl-: h* function: r

when the vialuC of h i- -qual to. c.r greater than .

The value of h-"' is sct equal co 0 I.lhen the val'ie- o, f h' i1

less than I. F:.r in n-l ve\cl picture th rc: arie r2n- p:-os'.ible

values of h** function = (0,1 ,.... 2nr becnu:e the rnrge

of the h'" fi.irct i .n is disciete there is on e-to-one map-

pini h"*' fr rrio F. = ranc ierh' rl onto 'I .here [b is- : subset

of the intecer nu!'-.ier and h**" pFreseres thce orrdering .of

the element;. h'-' -rue-s [he purpose of q .iar, i:in the

r nce of h '' A net. finctiioni h can then Fe defined as the

c. ,mposliti:r.n of h"'' and h' that is, h = h **.h This

fiinctiorn i- a rimeasure of the quanti:cd rate of increase of

inteisitics. F ie carn .rite the h function in terms of a 'm'

function, h(p.p.' = T[1 p 'l-g(p.. ,E'i' Fo r n = S, the

function :ran be repre ei tcd bv. the fo'll ouing mapping t able:

gipi -g(Pi
-c -5 -4 -5 -1 0 1 2 3 4 5 6

E (i P 0 0 0 0 0 0 0 1 3 5 u S 1 ) 11

1 0 0 0 0 0 0 0 0 2 4 7 9 12 1 3 14

The magnitilde of the gradient at a picture point p is

defined as the maix-iial increase ocf intensity from the neigh-

boring po-inrt to the- picture point. The f.:llowing definition

is then yielded.

Definition 2.2.--The gradient b (or sometimes will be

called h-talue) at a picture point p of the picture g is

defined as

b(p) = max (h(p,pi ,
ic[O ,1.... ,7

Iiiere tihe i function is defined earlier.


The enhanced picture of a diil- i ed pictur.- is

digitized picture Nith the intensity at ever; picture point

equal to the gradien It at thr- corresponding picture point in

th. g pictur--. The enhiriced picture of an n-level picture

is of 2n-l level. it .ill be seen 1 -iter that .:.ne e' tir:

value is required t. ijdentif.- the ho undary points. It is

then obvious that a L. bit picture will 'vield a -.*1 bit en-

han d picture. Th enhanced picture :of Figure 1. 1hich

ij a binar.' pictrurc of ri nui:ral "6" is s Lho-.n in F1i-.re 2.2.

Because ocf the unavcidabli n.'.ise Appe Arinrc in the pic-

ture. there is no ,U3. to finJr the real boundaries in I:re

step. I'We can breal: the procc s of fi nding boundr ari.e- int t:

several sters. The first step is to find all those points

which can quite pcssEibly, be boundary points. These points

are called tentative boundary: points. The fact th:t thl--

boundar., points of a bon.dar. path are connctedJ can be used

in the process :'f finding tentati ve boundaries.

The fc-llo,'-ing section details the scheme of finding

the tentative boundary: paths. Octal chain coJes Are used

to encode the paths.

2.1.1. Tentative F.ound .rv Plth '-earchin-,

Thi eti-rh'd of finding the tentative boundary paths is

based on the principle that : tentative b LouInd:ir' point is a

point which has a gradient greater tbAn the gradients of

the neichborin. nonhound r- points and is conrncted t- some

other t rntative bound.;r-, points. Figure 2.3 sho'.: the flow

chart used to --se tch the. tent itive bou ndary paths.

The input to thi? tentative bourndir' path finder is

tlie r.i di iti ced pic.ture. The outr'it i=:

I. Tih tentative boundary r, i.:ture whichh is a i.n.ir. picture

having g v alue -1 jt th.- tentative boundj'r.,' point ind I else-

whe re.

2. A list of tentative boindar' paths. F:.r each tentative.

b unrdar path it hjs

23. the start point : f th= tent't tive boundary p t:i h,

2b. the len,'th of the tentative boundar.t path,

2c. a -.is uence of octal chain coJdes ihic-I encode the


2d. an indicator ihich denotes whether the tentative

bcundar.- path is clo.sd or open, and

2e. the end point of the tentative boundary path if

the indicator denotes that it i; an open path.

The threshold 1 in the flow chart, which is Lised to

pick up the first point in a tentative boundiari.' path, is

usuall:. dc.cided bh the rfollowind g method.

First, find the his togram of tlh b values in the pic-

ture, i.hich is a plot of number of picture points rhosc

gradients are greaater than or equal to a b-value. Figure

2.4 is the histocram of the enhanced picture shown in Fig-

ure 1 .3. To deternint which is i from this histogram, find

Figure '.'. The enhanced picturc nf Fiure 1.1.


I--- rFi r, d t'..

S /. 1i r- t I. p.. hIlt

'Jot': p' i thE .7 r. Irt h ing
the l ..' 'i l o ''.i Jl "rr ,:,iL
I(D!L0~ I i a. the r ti it pE inII.
S.:e t a r.X I inJ i. = I

NOt, tnec p LInt .'- r I that i t
: .. arila m l t- \'il iie ar:r.: "* 'p 'l .1

crf *:nj p.:,irt 'r

'..J in tre .:...:tj l :.:t blp '' = -1 in.: I = i .
Ch rn co fd fr-n '.1 in ri the I.L 1- lc,.nt *..F [re
p" tc thc L rtrt .-,iiuenr,:e thle -,c a :lh 3iin
pc'in in ri e C: J3ie froi.1 r t*.* r

lS-c bi .'-! = 'lp -' : ;';,I I "l I" l n c}

no S. c p=:., and


b,:.r!n, Doint o. ,
te s' rt /
\"-s. poirint

Figure '.5. Flowt diagr m for scarching for tentative
bound Lr,' paths.

the greatest di.:,p in the his.rtcgr. m from --1 t,, in this

example 0 1.111 be s t nas 1.

600 r,

- 0o r


2 b

Figure -.4.

Histocrajm of b- values cf FiCure 2.3.

Searchirn the tcntit i v bouLndary paths of F LIur' 2.2

yields tw.o tentative boundary. paths:

path 1 :

start point = (2,9 ,

length = 52.

octal chain codes = 44-12212122123 114 3 3122222101000

7077"6o 7o665665636565,

closed contour.

a t h

start p-jint = 116,1 i ,

leri:th = in ,

octal chain c:des 11 3 0lui 5,

close d conr t:I r.

in this spec ial e:.am ple the tenr at ei t bo. und'.r. paths

are the final bournd' J r ri n to.rs. in inmot t t the practical

ca.es, because of the e. istence: ,:f rn,:.i-. o erl appin self-

foldiri,: and touichine, there w il 1 e ist g p- s betL- een tentative

'boundar, p-aths arnd t .ils :.f itern t ti ve boundary. paths. The

fo:lloi irn .ecticrns ji rs.is the str tez-'ies otf s -lvirin these

p r oh 1 m.- .

-.1.2. Proncedur,' of ii in.. 'as a irrd
Pe tte ri ii i .: i. ne ci:c ,- ent .

T. fill the g~-sap b.e ti,' en tentat, ie boundary paths orin

has t.: i ecS rmir. n the ex:.tr ime pints.. itf tentative boiindiary

paths. For an e:xtrir e point p of a tentative bo~tilnrd ry path

C. let be the nearest tent ti\ve bou.ind ar, point t ihich is

not on C or is on C and hias 'more than five points from p

alo r. C. If the Euclidean distance beticen p a nd p' is less

than 3, p is then co:nnected to p' thr, ugh the shortest path.

If p' is an a:ytr r'ne point :,f a te n t at i\ bhournda ry path C',

tentative boudaJi ries C and C' will then be combined. If p'

is not an extreme point, p' will then be an intersec tion

node. Figure 2.5 shoi,s the ilot. diagrai fur filling gaps.

1 f

Lr 3[ r ri -. ~

r.atfh; -;lr 3


theP wrI n P

r ~p I,. .-- rr~,-,
kit vap~cn to in .2th


i--- tr .111 point -i 3 id
newtcnIt b.'uri I-


tp 1 rpt P il.I p.
T) 1L1UP1' I- W E.

Figure 2.5.

Flai diasrgm fOr fiIling gap; beIjcmen
tentative boundary p.Drhs.

A bo-njar. egnr-l t is defined a; a b-ounrdar, path be-

ti.' n i.ccessive ir unction nor Jes I *:r a boundary pr th i.hich

doi1 not have function nodes on it. .A tail of a boundary '

ri the bhouindary, r th bet.e r, an r c tr -m..:x point .hi-ch is not

a junction node, and a junct inci- node. Hi nce:- the main pro-

cedure ijr de terri ni n buundar1,. s- rrntm is to. order the

junct i-ui node.'s .ialon the bouri.dair par. ths. Once boundary.

Soe uFn.cri ts arc d etcrmined, th,-- bouiid'r., segim,-nts j;oinr--d at a

iinctiion ncde can eas il', be noted. because only contours

of objects ar, of int rest, all tails iill bt- erased.

2.. Combininc Eou.nda:. Se.e:ments to Forn
the Loun.dar. ,'.:i5 On:urs .'b ie cts

Sormet ime s a po int p on the b2oundarvy iill degenerate

into sev eral ii nction nodes after appl'ying the process

stated in the pre'. i:ous sectior. to the digiti d pict,, rCs.

It happens mrrot often 'vhen p is a real junction pcint. Let

p be a boundary point i.hicl de-.enerates into V: junction

nodes n n1 ,n. p = {nl ... n is the complete set of

function nodes dJeenerated from the point p. The ideal

cases (i.e., no degeneration) are:

1. point p is not a junction point and N i empty, and

2. point p is a junction point and the cardinal number

of j 1-. s 1.

Figure 2.6 sho.ws sonme degcnceraticon cases. In the de-

generation cases, there must exist singular paths connecting

the dceencra:ted iun:ction nodes. Let N be the set of all

junction nodes in the p cttrc. Th.: dcgenerate l lat icn F.

is defined on such that nrin' whic r: i, n'c ., if t lI r-

exists a sequence o f int.-r' :cticr n n = no e ,n ,,..., n = n'

satisfying tl.; condition that there i is a sin;.ilar :--e ent

bh-.twein junction nodes ni and ni+1, = I . ,: Ti: d. -

genc liatc rc I a icn P is obvious l 1 n quiv'alenC'. i-'1 .tion.

The c-qu ivalI-ncc ri lat i:nr, F: ci n thus partition .1 inltI t 3 C0l-

lection of eq ,uvalence .: 1 sses. Eve-ry equivalence cl ass

is then a complete set of function nodes deCI erat'i-e r d fr.ri

somr- p int. n the pro:e s in i tiavc to find the sin uilar

segments first, then i.c car: decide ''which junction r..:d s ftorti

a complete sct. As one touild c::pcct ti rmost ricf onL-:,i

and c .siest way to dc tern-in, if a bounda'r7 se~rcent is singu-

a3r is b. the length of the se-wn t.


ideal (b) de-cnk-ate

Figure 2.6. Exanrnples of dic e.necraticn from ia ) a noniuniction
bour.d.ry point and (b i a junction point on the

A rhr,:shcld u is assigned such that if a boundary seg-

ment 1s of length le ss than it is l s fied as 3 singu-

lar sc Egim nt: other .i e it is- nonsin uliar. Let N be :~i cqui-

vAlence class induced b'- th, relation, i: :nd let 5, b th

;s t of 2_11 FirLul r c rse entic rs conn-:-ctin t, the Luinction

nod,:s in :. i;'. ) forms a n function area. Let E Lhe the set

of 31a nons nj ula r s Leri ts co'nne, tin o o the junction n dc. s

in ,i. The crdc rirn of the 1ci:iennts in F is very ic- eful in

cc-mbinin, baounJar .segments

It is imr'portant to point out that c.nly thi chain codes,

ihich encode th,. line s5Lc men t connect ing the junction nodes,

are us:d to detect the ordering of the ncns incularr boundary"

segment.; around the j inct ior. area_. The ordering of segments

aiouinrd thr, iu.nctirn area can be either clockwuise or counter-

lock.i se. because ,of the lin, crncodirji scheme l oc.tal chain

code) ue -us--d, there .are at most eight boundary: seoements

joininu:a at a junction node. Ficurc 2.7 sh .ows. the flow. dia-

graii foir ordic ring the nons inC.ular s-1a mnents around junction


Ever:.. equivalence class of junction nodes can be

thought of as a single junction node. An.. nonsingular seg-

rent i.hich connects to scme other nons inul ar segment through

the equivalence class of junction nodes is considered as

through the corresponding single junction node. We can then

imagine the boundary segmerints plturi c as an ideal one in the

sense- that no singular path exists.

Ir, p u
L, : .- 1 1 1' 1. Jr IC
in 1 t C.t .rc t 1 r, Ode

Figpuire 27. Flow di3arcri for ord.:rink, nunl.inguilr poth-
3rcUlcim Jukct ion .arcis.~

A nonr inrig l] r .c ter ent i is connected tu a ncns insular

ieement I. if Jiid onCi. if there exists an equivalence class

Nj ,:f jirunctioun nodes such that both 3. and -. contain sofre

lniir tion nodes irn N. This relation is ,denoted b" E. The

relation E' :f con ect f of tio ncin r ir, Tg lar paths is a

tranris t i e closure of the rel ation E. The picture coinsisting

*-i ly of the ncr F'in:kcu.i 1 r segTene ts I'Ind the sirgul r p.n t h

which ,co>irinect tl h ir of the equ i lernce class ind'ice d b', the

relation E' i cIailleJ an i solateJ picture.

A co rnt:',r_ is defined as a si -1 cpl closed curved We can

partition contour into i sequence or successive adjacent

bounJjdar' se.-.ents. Hence the c.:.ncatenation of successive

ad j1.ia nt bounJary s ec.rentsj can form a contour, if the con-

dition o:f being a contour is sati-fied. A picture point p

is included by a contour if p is a point .:n the contour or

if every ray initiated from p ill meet an odd nu rber of

time: u..;ith the contour. If the above condition is nr t satis-

fied, the picture point is said to be excl Jided b\ the con-

tour. Tic iet of all picture points -.hich are included by

a contour is called the region enclosed .b the contour.

Definit ion 2.3. --In n isolated picture, an elementary

re icn is defined as a region enclosed by a ri:ninuin contour

in the sen:e that the rcpion doe- not include any region

which is enclosed bv a contour in the picture. A region in

an isolated picture, which i; the union of all elementary

regions, is called the whole iepion of the picture. The

co)ntot:r in in is.lat d picture i.iich ericlosc til, CiJ'ole

region is called the e:.terior boundary o.f the isol-.iated pic-

ture. An:. nonsingular segpment in the exterior boundary. is

called an e':terior sceicL-nt.

Let im be the number of junction area .and k be t'h

numniber of nonsinguil r scerents in an is.iat--d pic:turc;

thcr-' ir a e k-nmi :leITientar\, rec irons and .ne, eX.t r ir:r

boundary. The above facts are useful in termniriat in the

searching process. Ficure 2 ?. hoi. the fl ti. dia'. ri for

finding all the minimumn ,contours and the e:..terio'r b.oundar,

in an isolated picture.

From here on, the terrmsr; "pa lth" 'nd "njde'" ire isid tc

imply,' "nonslncular secnent" and "junction r ar' respectively ,

unless o)thcri.ise specified.

Let us look at the different e:xampnles shco:.n in figure

.9 to et a feeling of "hoi-i our visual s-'stens c.:mbine

paths into object boundary. contours."

It is ama:ine that we don' t have to Lnr'.n m the irn .en iti

in each elementary' recionr to find cut that in I'a) there are

tir. object : one is cncioses d b;. the bound.ar'.' cont.our corn-

sisting of paths So and -,, the other is enclor d bh tl,-

boundary. contour consistin.; of paths 5i nd '.3: in fb' there

is a self- folding object enclo.'sed by the contour consisting

of pat his q a.d r' with the rgjion enclosed by the contour

consisting of p:iths S1 and C, as the folded part, T an. in

(c) there are two touching objects: one is enclosed by th-

r, L Ih
i~:1:rJ P LC I r E

r.0 r,,. 1 r r. in

Figure 2.5.

Flol diacrmr for finding all rhe smallest
coiitoirz arnd the exterior h1Ou~ndar'- in an
ijsIlat'd picture

bour,Jar' c.ontou.r con .;is- tli of paths :ind : 1, the i otht r
is e closed b.. the bou.,njd r:. c.rntio r c.:n.E t in. f p ith 51
ind 5,.


3) I

(! ) Ibi I(c
overla ppin i .el f folIdin t:. uchir,,

Fig re 2.9. E:. ample- c of li o Vi rla pping itl sel f-fold ng
and (ci tonchinc.

Scn rctir'-sh objects touching r! t a~ poirt nay. occur -uich
.is the c: ,mir.ples sho:.n in FiIrc :. 1'. 1 W call this kind of
node the looping nod'.

0 o)

Figure 2.1 E.amp3m l of ioopin! nodcs.

If there is an *'_dd r unb i- r of paths co-nnect inc to a junction

node J clf foldin or touch irn ma:' occur. .isuall ,' the path

which bel. -ngs r to .wo objects here i.e consider self-fold. ing

as a special case of tou:chlin i' is quite str a ht. The

strai htrn,:ss of ia path can be d.le teirriined b- the filtered

:-equerce ,:.f differ enc, es of suc excessive cha rin codes-, 1.lliich

,an be obtained b'.' the digital filterinr r method. If eich

filtered difference of successive chain codes of a path

has absolute calue less than 1. the path is then considered

a "straight." Ficure 2.11 is a flow dia. Cram for conmbinine

tile- paths ir, formiiin the bounder. contours of object, .

It. u -e have to find out uhich obeiects are overlapped,

.hiich objects are touched and which objects are self-folded.

L.ct C1 'ad C., bc the boundaries of tu-.o object 06 and ,.1

respect i-el.:. If there dc es not rcyit a comnilon node beti.cen

a and C,, '1 and -.1 are separate objects. if there exists

a corinrion node betiueen C] and C, and if there does nor exist

a conlarion path bet ,een .1 and. Ca, 01 and 0, are overlapped

objects such as F ,tiure .9(a) If there ex:is s a commiinon

path bei. een C. and C., 0.1 ad 0:, are either touching or

ilf- fol din. .ssane that both Ci and C, have a common path

S. Trace C1 .ind C: so: that S is traced in the same direc-

tion. If both the contours C1 and C, are traced in the

same direction (that is, either both arc clockwi is or both

are counterclockwise . 01 and OC form a self-folding object.

th-:- lI t ..f p.at r,.j
t h. L ..f 1 ricr l.ir r :.,,d _

s r u il i d
ulni r L:on rc .

Fird i thi-.
rm a! 1l.': .: ontour-
and tL : EL.: rior F

N--.: ,il 3 L i. '- *- L 11'ir.
h.:.und r', of : .p r '

S r :: 1. c.

1 ----- I /.,' ln'
I I.: I . : tr, ., p,:' , o f ..
p[.th L 'i, 1 ch l.ri not t., en
prc ....." _- =: ". ' ": p I

n J ir th i t

l t jr l
[i:,:,, 7 : I l = ii: i: ,r n I p 'I-*

c"CC _I
C /' .:':u rin r e

r n .L ]'/ ': n -i 1, -

SL" P ,p
^ .)

Flow' dJiagr'im for c.railbinin- paths
the boundary cJnr. ours oi objects.

in forming

Figure 11i.

L r c rr ~i ;
C. I L

Fipgur 2.11. Continued

Figure 2.11. Con t iniued


If 1ll and .,r tr.i.:cd in d i ffr: rent dir ::t ion '.1 and 0,

are tou.:hin. e:ich other. For e:-airple, in Figure '. .b ,

the cornnnn path is 51. Tracei an = 11 2

1 2
tl, '.at S"1 i 1 r',ce.1J ,pr," rdJ lr, b,:,lh c:-, s 5. B .th C, and ,C, atre

t r2,:,:,j in. thi c Ic l I, Ji .;t in.:n Hierncc I, nd 0-, ar .

;self -foldinr, In ri pjrc 2.9(1 ) trace i = '0 1 and C = S-,S

5.o that the :anm on patli 51 is traced Lupard in both :contour .

C1 is t.hec i 3c a.:cJd in c: -nt r'It ce l.:'c l.i.se iirecti:n, .rhile C

is tra.c,:d in cl .:cki.e Jire :ti fH, :e oI, and C, are touch-

inr each *other.



It is Lnoun 11 that boundir. paitlh- iar, ver., c~:-l.

et.xtr.iLct from a bin air picture. A' n n-lev:l p i c: u ; z c n

be transformed into n-1 bin ry, pictures f = :. i -'. 1=1,

...,n-1 suchI that

^ 'p.i. = 1 if '; !.i f 1
c 1 -
II otih rw i se

.Vpi i, E heCre I is the picture plane 3nd i is tl-e threshold

to trinrsf)rmi grey' picture into 3 binir.. picture. Ihe

transfornrm ticon frn rn n-level picture intc: ai collect in of

(n-l) bin r'y pictures (t . 1 is denote. bt =

{ '" -1 The r ascn fcr e:xcludin E-n picture e from

con ide- rat ion is th'3t it is a trivin l picture -iith l's every-

:.here in the picture plane. A binary picture 1- i;s i id to

be a subplcture of a binary picture derii.ted by ,

if and E. have the --arie picture pl ane I and p : I
1 1
i(p) = 0 implies E ( p = 0. It is eas. il seen that :

.. c. Obviiou sl'.' *I is a one-t.o-,one t r.il f.-t mp tlion frnom

an n-level picture into i collection cf In-1) binary pict..ire-s

I 2 ,...,Bn- l . cE ). Figure 3.1 shows the bin.-r:.

picttJrs t r-ansl.rnmed fr'-'ri th- S-level picture .shorlun inL


1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ,:
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ,)
1 1 ] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 l ] ". ,
1 1 1 1 1 1 1 1 1 I 1 1 1 1 0 ,
1 1 1 i 1 1 1 i i i 1 1 1 1 1 1 1 1 o
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11
1 1 1 1 1 1 i 1 1 1 1 1 1 1 1 1:- 1 o (i
1 1 1 1 1 l 1 1 1 1 ] 1 1 1 1 1 (, 0 0
11111111 1111 1 I
1 1 1 1 1 1 1 1 1 1 1 1 1 l i 1 i I:
1 1 1 1 1 1 1 1 1 1 1 1 1 1 '1 i 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 ) i0 iI
1 1 1 1 1 1 1 1 1 1 1 1 1 1 i i.
1 1 1 1 1 1 1 1 1 1 1 l 0 I, ,
-1 1 1 1 I 1 1 1 ] l 0 0 i I' i0i

1 1 1 l 1 1 1 1 1 1 1 1 1 1 1 o i1 I P:
1 1 1 1 i 1 1 1 1 1 1 1 1 1 ,) 0 i Ii

1 1 1 1 I0 i 0 1 i 1 1 1 1 0 1.1 "
1 1 1 1 0 0 ii 0 (i 0 1 1 1 1 i ii ( 1 0 0
i 1 1 ,i 0 i' i [i 0 1 1 1 1 ii ,) ii i
1 1 1 o 0 1-0 i i I I 1 1 1 ii'. 0 n
1 '1 I : 1 i 0 0 1 1 1 1 1 1 IU 0 IJ I
1 1 1 I.i 0 0 1 1 1 1 1 1 1 1 i 0 'i
1 1 1 i i i 1 1 1 i 1 1 1 1 1 ii i .i 0ii
1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 i)
1 1 1 1 1 1 1 1 1 1 1 1 0 Ci I 0 I:i 0
1 I 1 1 1 1 1 1 1 1 1 1 I ii ii i. 0
1 I0 ii 1 1 1 1 1 1 1 1 1 1 0 R'IC (i 1: iI
1 1 i'C I 1 1 1 i i 1 1 1 1 0 o" 1: 0 1' 1
1 1 1 1 1 1 1 1 1 1 1 II II II 1: I 1
1 1 1 1 i 1 1 1 ] 1 1 0 ii 0 0 0 0 0
1 1 1 1 1 II 1 1 1 ,) 0 i ,, I-i 0 f .i i o o ii

Firu re 3.1. The bir:i r/ pictures transformed froni
Firu re 1.5.

Ic) ,-3

(d) C

Figure 3.1.

Cori t iiu.: J

Fi eure 2.1.

Con tinued

i(l ci

Figure 1.5. .Sincc the hi,:hest level i l Fi ure 1. is 5,

both E.6 and n ? a re trivial pictIIure hi. irg I In ever:1 p -

tu.ir,- r, int and are nor.t shol n 1I, Figure 3. 1.

Bund.ar-. contours can b:. extracted from : i = ,...,n-1.

Inclusicr, relat iois c.ian be :et up :'..- bundir, c.:nt:'ui .

A brundar,- contour C. I in ill be i ncl .de. in .1 b.un, ar

c:'nt lr .r in i here i-i. If the shapes *:f L. d C, ar,

?imilar, C, is noie lii. l'.- to inciu-le the- uh.:ole .).;i.:ct. This -

fact is quite obvio usz. fror, the -::xpe rirrments. l ing this f.Ict

to'i extract cobjectsz from the s:ce-ne i: -'er:. eff cti- c especi-

ally" for area pictures, thosc hlnavin.: onli.c objects .:consisint g

*:tf 1reas.

5.1. Som e FunLpd'aicnt:' 1l Conce e t
in inar P ictui r '

A picture point in tlhe- picture plane :'f bin.ar p:ict,.ire

. is said to be an obicct point if fBip) = i, o.theriis it

is a bac!:gi'round p,:.in \ picture point pa; is said t,: be.

directly co r nect-d to p' I if ...lp,p' Il, ".here d i3 the

Euclidean list. -nc function. -n obj.1ict point p is said tc.

be connected to an object point p' if thereC e 1sts :1a .se u1iI'C

of object points (pD" . ,I such that p = p and p' =

and. p is directly connected to p i l i= 1,... m. A ma:i:il

set r.f connected object poirnts- in 3 binary picture r. is an

element in that picture. An object point p in a binary. pic-

ture 6 is a boundary point if thli-re c::ists a backgrounJ point

p sU'ch that dtp, p... .' c nt ur is a sequence: oTf picture

points (pn, .. p i such that 0 I (p i .2, ihe re i = 0

. .m- 1., =p and mr iS trh leneith of the contour. *l ; stated

in the, i ist chapter, it cain also be represented b- cqLqu,-nce

of c'c il clain .co.-'des A picture point p is inclui.d by a

con toulur if p i a point in the sequ ence, cr ever', rn', initi-

nted froi r ip ill meet an -dd number of t irrles with the con-

toul. I he set of all picture point_ which are included by,

.a cont,-,ur C is called th- region enclosed by C. An exterior

contour c; an element B in a picture is a contour such that

when the c. onto .ur 1 traced in a c locb ie d i r e .ct i,,on r t, then

.ll point ; in the eiiment will be -o-n the right-h.nd side.

An interior contour of .n eilcn-nt in ai picture is a contour

ci.ih that L.hen thIe contour s1 traced in a cl-cl, %.1is e direction,

then all Fpi -iur points in P. are in the left-hand side. It

is --a ily s'-n r that for in-' element B, there is only one ex-

teri'. r contour ind there iz i finite number of interior

control Theie is one e:-tei io contour and one interior

contour in the bir-nary picture ihon in Figure 1.1.

Let -. be a subpicture .of the binary picture 2.. For

any e::terior contour in there exists an e:.:t rior

contour C(i in E such that the region enclosed b: is

.A subset of the region enclosed b'- C(i) For any interior

contour C in there imay)' e:.ist an interior contour

C(i ill ii such that the region enclosed by ()' is a

u bt set f the region enclnoed by C( '

3.2. ContouLir Findind in 'n l l t-lt v 1' F'lcti lr

In a mnulti-leiel picture c a pictur.- point p i

boundaj rv point if there: i:-:i:st 3 point pi iuch that ifp,pi '

and pi 'l lp pi 13 c.:a l l J Ian .ad: c-nt h ac e r r-i j pci nt

iith r pc t t thce co:int:.i.ur r: i ass ino through thc ho.:.u dar.,

point p in binary pictur-s i. i p iin :-rdc r t.

find the conto:urs, a labeling_ scheme is used. i.ct a be a

n-level picture. As'lsume that n is even .hi ch is ric- ral

the case as n=i for thic picture' : 'oi.n in Ficure 1.5. The

rll-s; of the- !abelin: s';hcnEr. are:

1. 3all b c I rilar', point in thc i t' c.nt,-: r .ir' l abc lc d a

2. all d.-ij.cicnt biac!ground p ints i.ith respect to the ith

contour .re labeled da n-2+2i.

3. the lab elin; o.:f the- odd numbers has pr;i.rity :,vicer that

of the even numbers. and

4. for -odd numbers, the lab l in: .:f large, numbers h.a prior -

ity over the small numbers. For ecten n.mibrs the 1tI eling

of small number rs has priority over th.: la r:c nunh rs:.

It is c:sil,' se-. n th.t a contour C in a multi- lei 1

picture i. a contour in the binar.. pictures L bil :i lClI,

I.-here 'C) = ni n : p) nd b(C ax g(j q i;s n adiace-nt
background j o rin.t %.ith respect t.o C). glC) and b iCl are

called the int-ensitv ind the back round intensity, of the

contour C, respect ively.

A JotiatCor i.: .1 Vector initiating from .n object point

to: its eiLclit nei i r'oring-, p:oi.'t'-i and I.h nce can be represr nte.d

by an octal chain code. .3 rtator rotating in couunterclock.-

mise direction is need-:d in the process o-f finding: c:,int' 'iurs..

if .2 r:-tati:,r pointed to L the picture point p it ''ill n t

p..ir tt... i 'i ,,S. The r:otat.r i the i nJ o-f thc nei lh-

bor i pn oin .

In finding th cont.ouirs in a picture first c.ian tne

picture c in forward dircctior. l.ct p be the first picture

point at i f ,'in the conditi ons that ci p 'cgl., I land either '

the picture Foirit p ha. no;. been labeled before .or the

pictui.r point rP is on so-mie contour which has been found,

and i( I *lp'. p is then the -tart point of a contour to

b f [Iunid. t t the initial p:.-: ition c.f the rotator in the

direction indicated b- nct-il chain code 1. Find thC first

object rp:oint point d b., thY rotator t.'hich rotates in a

con unte r loCil 1.s direction. afterr the second cintc ur point

ha. been found the initial posit ion is c t at the direction

indicated by the oct l] chain co.d li i i where i is the

octal chair code indicating the position of lat rotator.

By thi netlie-d of finding rhe contours, an interior ccntour

will b-c tncoded in a clocki.si e direction, i-.hile .an ct terior

contour .11l be encoded in a counter locki:ise direction.

Let cl. ..c bc the se.qluencc of the octal chain codes

of a contour ci i- is deI uted if Jci-c. = e,

tiii proce-.lure, the sequence of ti.e chain code rill be

reduced tr, .. .cm,. If the rJeducLJ sequence is ei-Lt',.

then the con tur is one whichh can he bro r into tr.,:, paths

S1 = 1...c 1 andJ = c ... i uch tlh.t = '1

A contour bhavin an e .in.ty r .,t >.lrced sequence i: ci n.:J i rc-_d to

be an -'-terior contour. If the redui:d equence is n.:t

empty, th,n th.c Suil. off the Jiff. r:nce*3 of adjacent c-til

chain code is thi- p3raim-ter tc- inJlic't i.h'.t'-r th' cuntour

is cncc.ded clock.t.is or cci.Iun ercloc:l-i i I f thc Su-ii is -

the : .:ntour is trac: d i n a clo.i.i;.se di re:tion. He rn:L th.e

conturmi, iS= an interior c In If the sum is *S, the C:ntur

is traced in a countcrclocl. iS e direction. Hercn: the co n-

tour iE in exterior one. FieurF 5. 2 .-ho.s.: the fIowL d.ia Iri'm

usr-.ed to find contours in a riul ti-lev .l picture.

Figure 3.3 is the lbeled picture iof rhi x.-le\ el pic-

ture shiun ian F ic r.c 1 .3.

There are 1-1 contours in F iure 1.5.

Contour 1 t..art point = (1, ,

octal chain cod-s = 222' 2::2: 44J44 J4
.1 S 6 6 6 6 c r r.i 0 ) c 1. I 'I 1.1
0 r.I o o i l n jI ,

length = 61,

inrt nsity = 1,

bacl.ronnd intensity = 0,

exterior contour.

Contour 2: start point = (1,11,

octal chain codes = 222::22222: 22: 4 4-1 -4154
I)E n oS 766 7 3 0 00n0 C"
S S o cr, So 5. '00 0 .ii 0
i-i iJ

Co n t .1 r 7. :

Co C Iit .ir :

Contour S5:

Contour 6:

length = 55,

intensive =

hackcr'jound interim. 1,

C :
start point r = I .1 ,

octal chain codes= 22222222666t.56,)

length = 1",

intensity ,' = ,

ba..i::c uriou inten itt = ?.

exterio:,r cnt-ur.

start p. int = 1,6)

octal chain codes = 143 14341J 00li0lIO

length = 14,

ir ternsi = 7.,

bacl.gr, und intense i ty = 2

ex:tie T r contour.

start point = ( 1,10 ,

length = 0

intensity = 1

backcgrouncd intensity = 3,

exterior coritour.

start po:.int = (2,1l ,

octal chain codes = 22hI0,

length = 5,

intens ~ y = 4

backL round intensity = 3,

exterior contour.

C on tou r -

C.'ntour 7:

Contour 9:

Contour 10:

Contour 11:

S tart point = 2 i -

octal chain codes = -JJJ4 :,'lO 1 1007 ,:: . r, r,?

length = 25,

intensi t = ,

backeriounJ irnt nsit,' 1,

in tL rC, vr cornto,:'ur.

start point = 5,125 ,

octal chain codes = 12'10_i 11102 13-13-13 51 55 r66 0

length -= :,

intensity, =

background intensit,' = 2,

exterior contour.

start point = 9,11 l,

octal chain coJcs 1= 21"-1: 245S r.'. ,
length : i6. --1-'- '*-
lcneth = 16, '.

int, nsit = 4,

backg~i cund intens;it = 3,

e::terior co.nt our.

star t p in = (i'1 I ,

octal chain .:.-Jes = 17"'6.53,

length = 10.

intensity = 2,

background intensity = 1,

interior contour.

start point = 1.12,6),

length = 0,

Cont tour 12

i ontour 13

Contour 1 1

intern it, = ,

ba:l.c round internsit = :

C tC rc or Cion cO:u.ir .

start point = i ,10l,

lenr th = 0,

intensity = 5

bac I r S,:n nd interns it = -,

e:yterior contour.

start point = 114. i,

octal chain c:d0es = -:-22 5106

len.2th = -1

intensity =

baclgc rou'lnd interns t',' = ,

, t c r I r c O t o r .

start point = 116,6).

octal chain code-3 = 1753,

lencth' = .1

intensity':. = _,

badc ground int en sit = 1.

interior Con touT.

* -'. -t

p t1 i r.

r: I I J

NICA to, Al~ ,n conwor~
of Int'T101 OlcoujiriT
if it is W OU'll e.
c.rti'ur 1. r wnt1.I

Figire 3.2.

F md

p UNJ.Th tr n. 1

has nr c v o n 1 Wo lom V`
-i1- Iii

'I .I.I

or th: invcr ic, f th

o n t o r c k
PI 1 I I 1he lea 1
Q -.r hI in I P i
E SiLAiTE l't
1- r1,~ =) 1

-F ier N5 . -

E t 1. r r 0 "-.A -, r

L It Ir*. Ii.

F I ii Aiagimr ri fiII finj(lIin, contours in a !,,] ltI-
ie~ei lctl:

131'1 l i5 i5 51"151' 1i 9 9
1 9]9L: 12'i0 12 1 : 1 115 14111.11 1' 1 9 .
19 131 2::1 1 1 2.iU -0 0:' 11111 010) 1 1 9 .
91I1 : 1: 1 1 1l.'2 l21 1111 1 1 'i 99
17.1 2 i"i 1 1 O2 .' 1 : 1110 1 9 ,.
1 .1 1.'i1 0 1 1 :'1 2--.- ; 111 1 9 ... P
151221. : 1 r 1 :11 '011 i i :11111 9 i
131 .- 1 -1 1' l :1 '' 3.'42 111 9 S ci
15 1 1 1 ) 1 1'-172. 4 52 2-. 9 9. ii
11 2-2'21.' 1 II: r7 u.i'- '42,1.10 9 I I)

2 -.; ,.,27 42-'4 4 4 111 9 C i; 1 i,
-2 -I ,1 y
37. 7 -.3 43:22.'21 111-'.11l[' 9 0 i 0 L0
73". 311-1i1i '. 9 11111 9 9 9 9 3. ,, u, 11 i

Figure 5.3. The labele.ld ictir hr o:btinrd from FiEure 1...

3. 3. Inc li.- ion P : 1 at i n Aimn-7 l.',*:ntour s

As st' 1: ed : n tihe beginning n f tlie :h:ipter inc:i ion

relation irusl t Vl. sI-et amon'-l c.nt:Iour to ext rac t :.biec:t .

Inclusion rcltin .n can be rasily, found from; the iab-led rpic-

ture L and th, o rigin -l n-ilevel picture n. Th la.itels and

intcnsiti.:- s of t 1 success. pi ctu.re points p. r l.j r_1 ( in

the f:r; uird raster dir e tion, i.e.. p i: at the r i.iht- i'htId

Side c.f the pi ctirIe- p:'-int ,. I are rcqui r-d -1 in fjr,.'rnmat lio

front the label Cf tli p 1ict .ure- point p, is stored ar a .tate.

ThrT:- ilnds; c.f label l e::-ist; rate 1 is that i.lp, ."n, that

is, p. i5 not labeled in th: labI.liric r,:ce .tte it'

tliit L.(p ', n an.d is ev\ n thatt is. P is .3 aidi '- iit b i.-

Q round pc.nt. t tate 3 i ; that Ll.j) .n and is .:dd, that is;.

P is a bound ir,' point. Figiur. 3..-1 is the -tatit- iacr-in

for finding the incluii rn rlItion amioir c ontours. The ac-

tions in Fi, re 3 .4 should be e-.pl n in- .d. .in array is ir iti-

ated e\veCry time a line in the uictI ure i sca'nne.- d. _-'.t.te 1 -5S

the initi.ji s .ita e. if the action c:insi; s c. f enter rine tihe

regio.r, the rc gion nramr ic put ini the arra:. If the ictijon

consist.- of leaving the ri 2io. n, tlh region name is tai.en

out of tli a rray. Thi- inclusion rc ltlron is partiailv

ordered relation, i.hici can bce represented b., a Hasse

graph. 13 Every tire a contour C is put in the array, the

contour C is included in the cont'lur i-hich is ne.,:t to C in

the arr31ia. At a picture point p, there na; pass several

Li pl>n i i ri no act in

Lip in i en Lip ln i odd, if the-
no acti n ,-. contour rat p is 'in e:-
i L,, pi''. te 'l ,or contour, if p 1 f
S" .. 'I1 p, rio act ion, I f
L p .I "- e nter the
nac cti-:n ", cgi r if the cont I :ur at
/ ..' p' i an interior contour,
if ,-,. 'l enter the
're i-n. it p,:,'l p
no acti t n.

S Li nr .! 1. even if the
/ :'nt.' r p,:. is in e. -
/ te or c.rontour, i r,,'l
I*pl leave the re i'oi, \
if 'p, l 'in) no act icr::
S if the contrur at Po i. i 7
interim r c:ntoL r if ,t ip.)'
/ i,. | 1 n,:, ac t i:n if ,_i ,
lea.e the re i n.

Li p n \
r, ,c .i Lipl;n a ic odJ i .rter \.

1 the rr -7ici. 3

Llpl)n if-..t'he c,:o- Li p)-n > is odd
tour at p,, is.. i
cXt i- i:.r ,._,ntoi'-- ir the c-t< u our If the coitO LI
no action. if the at p i; an ev- p is in inter
contour at p, 1i: ln tEcri or con tcur, count ,.our, Itf
in teri-:.r contour if gip.-.l g(p.i p enter
le: : tih,- r 1 gi:on. enter t.e region, region and 1'
i f g(pol lp the region en
no action if ,by an interior,
ip,,') i p) tour passing:
letav the re ion if aFir,- .g rp)
enclo:-cd I.v an enter r the reg
exter i.or ccntour and le.ave the
passing p,:,. ion; enclosed
either by an
ior contour o
exterior cont
vith intensity
:,g .p) passing

r at

Po )
V C.
c lo ed
r c n -


i n t r


Figure 3. 4.

Stat;. di i ram foi finding the inclusion
relation i aong contours .

co tour .... L i.. , . -her- the i rn.i -: : ire

orJc ijed a -dcc.c:lin c rto tI a s:cen iine order of th-- co r re 5;:rcn -

ine label vcillues, be the contCoujr sa ti : i rl t h ccnditicn

of entering the re ion at the rpictue :point p. C1 m

iill be pu.it in tlc array in the order.

Arpi ',in t l i _- incluri.-i n ri laticon t'in.in. proc:,. to

the c 'rinto ri.is in Figur': 1. ., thi- H'. s .:ri t 1.ill turn o t.it

to be the one shor.n in Fieure -..

3. Object E:.tr ctiri n b O:',m arson

No:. i.e 3le in the final sta-e o f :i : ti .: .,:t i. ri .h e : ;ts.

Let H be the Has?.e riph iepres.4entin7 the inclusion :, el ti:on

amIr contcu s in a pic tur. Let the areai cf t he r-cicn

r~icl,.s.-d by a contour. r C b dcnrot d i: as A whiih c'in e fo rind

by tli h- m L t ;ioJ Fpr :nt Ed in S'r: ti:. n .1.1 . A t hre should

.:, is set suchi that if ** :'1' the n:ode .:c: re5spo.:ndinr to C

is deleted from H.' It is obviOLu that if .] contour ati s-

fies the above condition.: all its descend -nt '.ill s tz f'

the condition and i.ill thus he delert -A re on b 1 .lue

for -:'1 I.tCuld b,:- 9', b c-c aisi i t is usually impossible to

filter out the noise istuiirb in'. the shape iof a c orntour if

the area enclosed b\- the contour is lir-s thai 9. Let H'

be the subraph of H obtain'.d- by thi di:lA:liction pioces-.

The H' obtained from rhc Ha-S. cr.ph H sh.:.i:n in Fipire 3.-

is shc-i.n in F :Ilrc 3.1'.

S1.1 `~. 10

6 5 911



Fipuriat3.S. *1he Wssegmq representing Ohw iniclusioin
I e 1 L n i'rja c irtturs in Fi 'irTC I.S.




Figure ..t 1he Hass? e gr .apl obtained through thhe d, letluii
of small cc ntour .

A di s ii a ri ,t ME r 3 j ur,- : Lit 1, r,-- t.',:, cont,: rs T

Rnd C, is d- fi ned as

D(C1. 9 = ( 1) A1 l 1O 1 1-. ]

where n) i and( are the FUmber of critical rcirns,

the numbc-r of peak: points and the n0il.,Tibr of valle- poinl t

on tic contour C. i=l, 2.

.\ threshold d .:, is set ti a:.:t i art objects from ii'. If

a node correspondinS to a co.'ntour C is the onil son of a

node corresponding to .1 contour C', and if FiIC,C'J ,

the node corresponding to the contour C is deleted from I-H'

First the levels of tho nodes in H' ar? assr nied. ThI,

level of the root in H' is ai.sicned as 1 and the levels of

all sons ofc nodes of level aire assigned as K+i. The

Sdlet ion procedure is then applied to H' from the nodes
with 1 arcest leiE l .issi nrirTent to the nodes La;th level 1.

Let the r s -iiltin' -rraphl be- denot d .-1s H". Every node in H"

can possibly correspond to the conroi. tr of ra obi.j t. Figure

3.' shoc, H" obtained from i.i ure 3.(:,.




FlIure 3.". The Has.e raiph obtained through the deletion
f s similar contours.

Contours .nd 10 are not under consideration because they

are interior contours. Contour 1, which touches the pic-

ture frame, is also not under consideration. Hence only

one object ..hich is enclosed by contour S, is extracted.


PI Tlu r.E P r.- F i 1: ; -'

In this chii Fpt r t c ire propo'?sir,n n i thod to e- tr.ac

objects in a multi-le I el pictu,,re bL. the cl .Isterin- methodo.

This appror ach c'in dJetect the o":-stalt cl.: t.:t-rs i.hich ia e ,t-

jects, ir. tIhe' 1.ictur. ard can;r press in; i'.: the "s eiet.:r. ns

of objects. We fir-t tran sfcr the n -level picture int: 3

rweighted graph C and t'ien find sari 'I T i'li i ril p -.nni ir: rcT e

of evere isolated 5,.eighted gra.peh 3. *f g. F. Si d on tiL e

statistics of an IT. ie c n cluster an ';'T. Evcrv .- clu- r,

is an object in the picture. S.me maj)r path of an T i -

stricted to: a cluster fcormi a ske i e rn" of ain 'bl ect C:,me

graph theor.- bacgl ro.urnds and prope rt i e. of the M!ST Till be

di scus se d in the follet. in C ;ct iion.

( "- I
4. 1 ra Cr.ph T :or '* r T.ct c rojurids
a.-r t ic 'r.: ertr ic i I .

An undirected finite ?raph G = (V,E,F) consist ; f .

set V of m vert ces, o .here V = (t1 m, a set E ,of

edges, iwhcre E = 1 .. ,en, and fn tio function F, a rippi n

from E into V anj \, the set :.f all 1 unorde red pairs c.f !meri-

bct s of V. Figure 4.1 shi.s an e:a- iile of an u;idirc cti:d



-. t*---,
4 5

FiguLre 4.1. .an -x:samrpl. of an undirected finite graiph.

graph. In Figur- .I V = {\1 .. v } and F = { 1 . e } "

If e. is in I, then Fle .1 = (v i for some vertices
'. arnd Vi. in V, such s F i = v..' ) in Figure 4.1.
\ 1 -
Arn 'dce- ev. is incident with vertices i and v. if F(c.)

= C 4i i. F r exmnple in Figure 4.1, e is incident
1 i;2
with v., nd v. If F e ) = I'\. i then e. is l1oop
':' 1 '
such a. <' in Figure J. The nu fiber r,(v ) of edgcs, which

are incident with 3 verte.: vi., is called the dteree of the

l er te:. For cx-irtmple, rn v ,' = 4 in Figure 4.1. v. and
1 i1
vi, are adjacent .vertic's, if three exists an edge e such

th'ia F(. ) = ([ 4iV ) For example, v1 and ', are adjacent

V trticcs in Figure 4.1. Let e and e. be two distinct

edges. If F(C .*Ii. ') and if F (v . )
.11 -1 i, 1, 1
then n. and are adiacenIt edges rurthermo're, if
3i 1
tv = v: 1, th n c .l and e. cre parallel vedgcec. For e a3mp le,
3 1 1 B I

in Figure 4.1, 1 arid :.- are adi c. nt c:"J- rand c ind r ,

are parallel dge. A Sim[:,l,: r Fiph i: .i graph havir,, r,,:

l:p and ri p..ir of parallel ed es. The iraph, i sh:hi n in

Fig r. I1.1, i. not a i iiiple raT ph, becai e and er a Arc

pa i ral le ed es, i .1 io,':.p. F i ur' 1." h :i -

ar.ple *f 'a si.plc gi ph.



1. if r is a subset ,-f T h t f i nd is T ub t of E,

=, ,F } I n '
2. if f.or ever. e. in F', F' ic = he ), and
1 I
3. if for ever-'. in E', Fl'c j = tL 1 v'

art in n .'

A fii :c- sequence :t ed e: e ... is an ,dge

progre-;s ion for eJi'e -equence)j -of length t if th. re is a

sequence of \'rti e. v. ,. ,...,v. x uch that for -ach
0 1 ;t
S= .. ,t F i = i ) if v. / v the edec
SwI 1 1 T.
-u i-1 i 0 t
pro ress--ion i- opin Ior nrin-c:. lic), such as 3 5, eC

in Figure- 4.1. If v. = v. th edgJ e pro r-:re sion i
i0 t

cl sed for cvclic i such : e., e e2, in Figure 4.1.

An edce pro: res si: is said to be from to '. i s

the ini r i l ,- r te.-. and vi is the termini al vei te.. :of the
:ro.re- ion. Foi = 1,.. ,t-1, v i n int r: C i ..i t e
v\er t. \ :1 tl pro res sior,. A chain pr:.g: i.ss n i .:r non-

c' lic path i is an open edge pro: e .:irn in which n... ed_,-

is repeated in the sequence, such as _, c,' e, in Figure

I.]. A circuit [procressi:.n Ior c- clic path) is a cl,.sed

ed.e pro:-cress ion in whichh no: ede- is r-epeated in the 5e-

lquenc. such 1s eC. ', et. , e. in F iur' 4.1. .

simple chain pr''o essic.rl or simple path o:r arcil is ch -in

pro re ''ss" ic.r in -.whichl no ,ert.e is rep- itc-d in the vrrtcx e -

luernce, such as e e-, in Ficur, -.1. .A simple circuit

p:ro,.ressi n oI) r circuits is a c circuit pr.:,grc 'sis..r in which

. = v. hut there is n c. other du.pl ication:r of any '.c.rtce

in the vertex sequence, such as -. c 0, in Figu re .1.

Let v. and i. he t.'o T. v rtices f a g aph ., v. and
iI it 10
. are connected vertices if v. = \v. or if tlei-ie c2'ists

in edcge pr:o ression, e. .,e i th vertex sequ.ncc \-
It I
The e:.istenc, c.f an e cc j progresss ion from

v. t.' v. implies the eiftence of an arc from v. to v.
'0 t 0 t
so: a pair ,,f distinct vertices is connected if and onrly if

there is an aic ji:,ining ther.. C is a connected graph if

for an,- v.ertic s v. and v. in V v. and v. are connected.

A mrim matrix. A = (3. i can be defined such that

S1 if i=j or if 'i and v. are adjacent vertices
ai = o1 t
i 0 ot her.'iso.

For e.am[le, the A rma rix for the graph hl,,lio n in FiLuire 4.2

i t

Fo. ii:.T I.: nteg:r . *. .:I-h th. t .' i = ..: 1, the 'tr =

(. I is ,;lle.j the connec, t i on rmatri:.-: in that = 1

3nd only irf the \errices v 'iid v ire :onnectej. For x-

ample, the connecztior. m-tri i: c-f the .jraph ihoi.n in ri,,u, .

4.2 is

A -= 11 10011
i0 0 011

The connectivity rele tionri iC :n the \V-. rtic.ie of : graph

is an eq.lUivaler.cc rc l tion. Let lthe p'rtitiip n 1 :f V, tb. the

co,,nnectivit.: r 1 at ion, be q equ .iv le. nc,: cl- : :i I .

V ) For e..ample, V1 = v } :and = ( ,v' in

Fiure 4.1. Let F. be the subset of E .ach of wihiclh i

incident iith xerticcs. in i. For .any i = j, there d.c,-

not exist an edge e in E. su:h that it joins a vLerte:.. in ',

and a verte. in Thus, if e i is in E in. F c = iv. .v
1 F1 t
S), then both v. and v. are in V.. !ienc : E ,. ,F 1

is .3 partition cf E. For example, = rel,e,) and F

{e-} in Figure 4.2. T heref, re, '. = (V ,E ,F)j where F

isF h restriction to E ., 1defi cs a su.1 ,r.iph of G.

Each 31uch GC is obvi os siv connectc-d If G. is a s'ibgr ph

of a connnected isub r iph C.' of C, then C = G'; th t is, I.

is a ma:y:ximal connected stiib r.iph of C and is called an iso-

lated com ponent ccf r ,.

Definition 1.1.--A tree :s a conncr ctc J gr:aph having no

circuit. A circuit -free 2raph having q conncricted components

is a torest c:f .: trees.

If T = IV,E,F) is a tree .and i .an edc e o:f T. then

the subgr-ph C = (V,E-le?, F ., r :f I is disconrnected,

,here- F- iimplies the fun.ti :on F restricted o:n the domain

E-(eI. Henci nor suibgraph dFriv'ed froin a tree, ihich has all

the ver t ices ian lesser n riml- r of cdn .:s,is connected. Thus

a tree is a minimal connected crarh.

Definition 4. .--.et C = fV',E,F} be a connected graph,

and let v. and v be tr,:o distinct vertices in V. The dis-
1 3
tance dlv ,i j beti.een v. and v is defined as the minimum
1 1 1
length .of the arcs from v. to v If v. = v., d Iv.,v j is
i 1 3 *
defined eqijal to 0 .

The distance functi': n defined above satisfies the

in tric a~ ici s:

1. di(I .i = r ,

2. d(v.i,v. = d(vi,v ),\ and
d i I J
3. d'(v ,v l d(v ,v ) dfv v V v ,v.,v V.
i t 1 t t- I *1
Definition .3. --Let T = (V,E,F} be a trce anid v. is
a verte: in I. I f rni .) = 1, verte:x: is. termed as a Icaf

of the tree i. An arc a(v. from v. to v. ir called a

diametral path when its length '. is maximal amonc the

dist.nces bet e.-? n a rn'. t ,:' c rtic s:-; i s the di.I an.t, r ,f

the tree T. A veite:: c in V is a center of if

r(c; = min I'r( ) = r.

\wh,-re r '' ) is defined s ,a'.I l' ) r. i. ,- 1 t l

radius nrf T. Let v b: a; I-ca :a f f ihe 1 .r e t arc.: tr iiw

v is cal led a major path fr-m ., .
1 1
The following the,.: rem rc ieal s the- riro re : i e t:f the

centers of 3 rree.

Iheorem .1.--Let T -be a tree .f di xiieter and a( .

I be a di ametr al path, havji. ng the I corr-,pon -. .equ ._-

af ver tices v ,v. ,... ,v h n 2. is e rn T h,. a sinr le

center c = v. and has a radius I. = '.' All i-ma :
1 I ,' 2 ) '
paths Fc thruch c hen is .:dd T ha ti:c cin tcr ,

c = i and c, = 3nd has a radius L -L

1), 2. All :a r path pass thrcou h bc th centers.

Definr i on I.J.--Let C = (',E.Fi be a connected rranh,

and T = '' E F, be a tree aid a s ubgraph *f C. If =

V, then I spans C. is tcrmTed d spahin rin tree ,: f I .

De finit i it n 4.5.--A wi. gi hted gr'-ph G = ( ",E,F.W) is 2

graph l'V, ,F} '..ith the assi inment of a ueie ht to each eige:

in F. 15 the V,' hight funictio:n which .Japs E int: real

numbers, that is, the weight of an edge e. iT ( j). The

tight of an ed- e pro'ricss .:'.n J. ... 1i defined equal

to W (e I .. .*\'e ). The weight of G is defiin d .qual t.
1 .1 t

the sum i:of the weights of all J1 dg in G. Let T = l'V ,E ,

FT ) .her'; l iT r the r'- traction :lc-f 1. cn T, be .i span-

nine tret- of C. i.hich is also connc cte T 1. 5 31,J to b-

a :liniriu l panrninn g Tree i.IST'I o'f G if the v eiS ht of 1 is

minimal uimo1 ing all s rpannirn' tr.:ees .:f C. Figure 1.. -,,s an

example of a i'eighti J graph G. Figure 4.4 sho.ws: the c rr es-

ponJdin ,'liST o:f i1.

Definit on 6.--Let V\l'"' 2 be a partition o:f the

\erte:: s: t V :. f .Ii ihted gir'iph G = f(V,EF, w The i:w eight

WI'\ IV, ]i cross; tli, partition is define-d a the smallest

e'eight .among all --:dgc i.hich join cone \vertex in 'i and the

other irn V 'he s t o:'f edgcs E('V1\ \' which l .pan a parti-

tion ;,ill be referred to s the cut set of iV ,V., and a

linl is .ino edge in [(1 ,',) whose :'eight is -equi3al to the

weight l('V,1',j. The set of all lin.s in E(V1,V,) is called

link set ,, of V

The fol lio;in- theli 'rem allows us to find an '1.T of a

ie ghted.j graph from the link sets.

The:rem -1. .--An ;1ST contain at least one edve frcm

the lini set i' V ,V l of ever' partition (V ,\ Ever

edge of anilST is a link. of some partition oi f V.

Thteren 1.3 revealc that the appropriate clusters can

be found as subtree- of an,; HST.

Tihir remi 4.3.--I V. is a non-empty subset of V r ith
the pio.p: t '- (" ,l'V ) < l'(V. ,''-V.) for all partitions

\6 (-- ---- ---" V .'

Figure .-. An ex.namplr : ,f a 'c iihted rnph.

1 3

F, 0 M of. v i


Figure .1.4. A\r; .15T of the iscihted grahnl shcwii in Finure

( ,\. > of V'' th rln the restriction. n :f any. M-.T to V.
i1. i i 1
ft.-rme a uibtree of the '15'.

J.1.1. fi tin ; n IHT of a Weicht'd Graph

Fron Theo. rei7 1.2. it is clear that ain '1:. can be found

fromii a conncc i d raiph I build ina 'i.p :a subtr-ee T', to -.,hich

a. link of (f\', V-\ T is added. Let a he the total number

of verticei-s of G. wle can set Q\., = I\ ... ,v Three ar-

raisc ar- r.eqlured to achiL 'e the purrpose o'f finding an ST

fr m a u..'-i .:htCed gra ph G. 31

1. Verte:: array: .: It indicates ihich vertices are in ,

that ; if i i = 1. thcn -. VT ,. uhiie if X(i) = 0,

then \i d V ,.

2. Reference array R.: If X(i) = 1, R(i) specifies the in-

Jd.x of the vertex v. in \' if is adjacent to v. in T'

If X i) = 0, 1Rll s pecifin the inJey of verte.: v in V ,,

such that i ,' i. = in ( l '\'v. here i' vi ) is the
1 3 VqEVl i C q
ieiglt of the ed.ce joining vertices v and v .
t q
3. Weh'i t array -: Z(i) is equal to the weight of the edge

inciJent u i h v. and \'v that i;, "(i) = Wir(v -, 1.

The link of iV ,,V-G can be fc.unr from the vertex

arrayv i and the i.eilht array by noting that the ege con-

necting a \erte:x v and vn R il, where vi is not in the sub-

tree T' of PIST (i. ., Xli) = 0) has a eeiaht Z(i) equal to

minn fh'li) |' ij 0). The edge is then added to T' by
j={1 . ,m) '
letting .'(PFI ) = 1. Figure 4.5 shows the filo. diagram for

finding in 1iST of a connected weighted graph.

I eihted 1 t
a '.14i i criph

In i i i a :t 1on .
v(1) = 1, i l n 0.
1 = . .. ,nI .
F(i = 1 if ij i ad :ernt
tco \'1
= 0, c.her.i e.
7.(1 = b l. Vit,'F.|iI ) it
F: i l I 0,
= w it" Fiil = 0,
.,here i.r, l el i..' i hts in
the grap.
the : r1: n

= .-1 n:h l ,

Find j such Th '.i
t (i = in il| = 0 I
. :t ( )1 = 1.
I f i: ; ) = 0 a d j.: I Wi| v ,l
set F.i = arid 1i = I.a ,- \ i)
: = }1+1.

Figure 4.3. Flo dcli:gr i for findrin gan 15T of .a ~eghtLed

4. 1. Fin.j in ,d Ili .r Parth-s

A.: is easily s eenr, the r-:,t straw i -ht f.or,'. Ird cl'ste r in

can be done b. settin._b a thr;esl hoi.l t., such that ir an edge

in an V.ST i :f .eight greater [lhan 1., the edige is deleted

fr.: the 11 The elet ion cf heav il ,':-ighti e e Jgc from.

an f il l ie ld a c fores: t .-.f sub tr.' : s. Ev ri .- subtree cor-

resprinds to a cluster e .f the c.onre-cted i.e ighted graph C.

it -sh:.uld be p.:'inted o.ut that a-n :15T of c.:.nrn-ctcJ w. i tghted

graph is not r n niq ue. F rOn rhe orcn 4. it is l.no)i n that

this n''on-uriquene -s of the ilT ; does ncot re trict the forria-

tion of the same clusters fro.-rT different l'SiT. tf a graph.

In many cases, IT.ore : .ophist i:ated cons i crat iorn should be

taken to cluster the r.iaph. Stat istics o:f the w'Ci ht.s of

Cedes *-r, major paths -should be taken inrto consideration.

For a tiiC uisu ally there exist manv major paths. s --

ter.itic m.etho..l shc:.luld be set to find all majo r paths. Nell-

d stinguisheh d major paths are- of interest. Tir.' i mnai r paths

can be co:nsiderc-d as .ull distinguished if the-, have only a

small po-rtion of paths in c.:oimm-onii Also, the- branches from

a diame. trl path are of interest. Since a tree is a simple

graph, any arc in a tree can be represented by a sequence

of vertices. The. follow ing are useful definitions.

Definition r 4.".--Let Ti be a tree ,of diameter ., i.hich

is even. There- is only one center c in T1. The arc a(v,c)

front a leaf v to the center c is called radial path. If

d(v,c) = .'/2, S(v,c) is called a an..:imal radial path. L t

T.. be a tree of diameter '. hichl iS .-- .d. There are t..

centers, ci and c.. in T For an:, lif v. if d(i. ci

d( vc 1 here i ij. the arc f'ro.:m v to:, c aI'\ i
i i 1
called a radial path. If dI' c = ('.* 1 ' vc i

called a, ma::imai radial path. Let a(n ,v ,ia d i v ,v.

be to .arc: i.herc bh.-th a .and t are leaves, e rd v. tb tIh
1 t )
onl ccrom on verte:. of the cto arci If dlv .v d 'v ,.

then a I'.,v .1 is ca1iled a branch and an. ar: ,r.ntainine

Sv jt ) is called s rcn. is c.'ll:d a br3nnchin verte:.

A relation P is defined on all radial paths i], a tree

T. Let s1 and s. be ti.o radial paths in r. if .1 and ,

contain a co rnii:e n subseqluence .f imore than or ne,:' rt t rher.

s1 s It is obvious that R is an equiiva lenc, relation.

Let i be a tree hax inr g ,.rnly' on center c and . be:-

su-qucnce of vertices (. ,c) for sone er te: v. = ..., t.
Hence there arc t equivalence classes of radial paths ir.-

diic--d b.' the equivalent r l'.tio: n r:. L-t 5; denote all

radio al path: having a ubsequence of vrtici s fv1 ,)L ..n',

radial path ,of can be: c' om b incd o th i n. mi.l... I l-r.dial

path of .' to forr., major path, here j j': l,....t .

Let 5 uJ = (v .Iv.. .... .v. ,C ce a maximal-radial path
1 u i i
in u ,cv . 2 the-

forms a nlajor path. if T i a tree having ti%,. centers, cl

and c,, ever,'c radial path S in T ci-Ther conta ins the siub-

sequence of vertices (c1 ,c-. or contains the subsequence

of .vertices (c ,,c ). lience there izre t1io eqluiva ol nce cla-sses

induced b. the rel i oi.n F.. Let S = iv .... c .c ) be

1 1
radix l p1 th :ind I = (v J... v ,.n 1 bn J -
u' u 1 "u'l .i 1
i:.:imal radial p th. c u c nd i can be combinej into

l,.:r p ath ( l ,... u ',",.- ^ ,..., 1).

ec., use of less sto:ra c requi red J in.ld easy coriL nation

into. n im j r pati s, the stor 3a structure cf. radial p.atiLi

ui.iuld he thlit onli. one ri dl.ll path, i.hicli h -ir naxirium li'neth

3mrin; all r iJd l paths in the iame equivalence class. of

eerv-- equivilonce class is stored in the full sequence.

An\',- other rr i i.al piths are stored a3 brinches .

Every lIeaf in i tree initiali:es a sequence. Trim all

the le, ies fror. the tree. if the adiacent \verte:: c. jf a

leaf does not turn out to be 9 le-f after v. is trimmed,

the corre spondi nr, sequence uill represent a branch. The
sequence having v\ as a l t f i.ill be the correspondine stem.

The procedure is iterated until either there ire only two

Services left or there is only one vertex letIt. The v\er-

tices finally cl ft ire the centers :of the trec Figute 4.6

shows the f'lcl. dia ram for finding radial paths and center

in 3 tree.

The tree shoun in Figure -1..1 has only one center \ .

The d i.neter of the trcc is J. The set of radial paths,

having full s quences, is ((v v 4) ,-' v5,v '}. (v6, 5)

is a bi anch dcpcnding on the m ximial radial .path (v,v \ ).

( 7 ,v41i is a ranch having ( ,1' 3 4) and f( ,, 5 ,\'4) a

3 r r t r 1 1 i l Tr

lo r -.: ',' 1 :' r in El,: r, ... ,
Lir l I 1 1 *i . l .' h i
th ,- 1 ', i : 1 f : -.I I',l l- : r, .
Let r tI, t, r: ; :,tot l n'_r[j'T ." r o f
l..3 ',:5 i n ;r -.." r. i ,-. t = p 'i.

. -, - .*.-I "Thr'

,, i u" l t' C1 --" [r, =

L r h l r, ,, r

mtil 1,'.-1,

(' E = -i r .l 1 .

Put in th i- l ,
A.- ..

Figure 4.0. Fl c, diai r. m ra f.or findJin a ra.. lial p ith .nn.l
celtc r in trfec.

.7 -~ 4

Figuin 41. 6 Comit irned

.tte-r [I' ,'-,3 4) irnd l.-,v ,v car be combined into a

diamctral path (VI v v 4, V ,v r Frmir the branch iv ,t

w:e can finrd a maximal radial ,p th 'l:, v 1 which i -

pendent on the maximal radial path (vi.,c ,-,v it'. v, ,

and Iv.. 5 \ 1 can be comrrbinr d int t a di imetr al pIath

l1, V4. V , 41_ i indc p ndc t o:f an: ot her
radial paths in the tree. hence i e ca-in c':Fbir e

1. i ; i\ ui th i', v v I.1 to a n.aji. r path I' , -, .

3. ['v ,.. v. ii'th v( v, 4i t: ira major path I >., C ,*.4. ,,I. d
t h t v 1 t u m 3 o r F 3 t h f 4 '-

4.2. lThe F:-prc-scrnt ti: n t.f a i .o-i t :ed-
Pictuiire b.- a 'e l:htcd Lr rph

A digitized pictir' r c c n cb rcprcscnted b. a veijhtdcj

graph G in the following manner. Evcry picture puint

ccnsidcred .as a verte:. Several por' sibic: n t- c tld re us-:d

to. dcfi ne the .connection of \c-:r tic aind the -1 tc iht of th.

corr i sponrdi n? edge..

1. lMethod 1 is that every picture point p is connected to

any of the four ne ihb r ing picture points D' ; with a i.eit ht

1, "(g p)+ fp')) e:-:cept that pI (p '. = 0.

2. M-ethod 2 is that i' cry picture point p is connected to

:any of th. c ight-ncighbo:l in picture points p' .;with a weight

of d p,p'. (gI(p) g(p')) e:..cept that g '(p) (p = 0.

3. Hlethod 3 is that every picture point p is connected to

the I16-neighboring picture points p' with a i\eight of

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