Citation |

- Permanent Link:
- https://ufdc.ufl.edu/UF00097624/00001
## Material Information- Title:
- Object extraction and identification in picture processing
- Creator:
- Lin, Peter Pie-teh, 1945-
- Publication Date:
- 1972
- Copyright Date:
- 1972
- Language:
- English
- Physical Description:
- xiii, 140 leaves. : ; 28 cm.
## Subjects- Subjects / Keywords:
- Blood cells ( jstor )
Centroids ( jstor ) Connected regions ( jstor ) Diagrams ( jstor ) Extraction ( jstor ) Graph theory ( jstor ) Pictorial representation ( jstor ) Skin ( jstor ) Surface contours ( jstor ) Vertices ( jstor ) Dissertations, Academic -- Electrical Engineering -- UF Electrical Engineering thesis Ph. D Electronic digital computers ( lcsh ) Optical pattern recognition ( lcsh ) Photographic interpretation ( lcsh ) Scanning systems ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- 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- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 022660234 ( AlephBibNum )
13948416 ( OCLC ) ADA4971 ( NOTIS )
## UFDC Membership |

Downloads |

## This item has the following downloads: |

Full Text |

,C IECT E.:'.P CTI,: I r rrir i T ii- T, '- I N ir' TU'.E ''. UE3 I.',I By 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 UNTVIVrPR IT'i C,'F rLOP.iD\ 1972 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; . TABLE OF CCONTENT.:I ACK:L;'.'L rcE: iEF T;:TS . .. LIST OF TA L.LFS . . . LIST OF FI UiPES . . . 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 . . . . . . II OCiJECT F.YTPF_.TI I: FY THE PC.R. A IEi T 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 . . . . III 3OBJECT E:'.TF'LiACT rIO. E:' THE :O:IlTOLiR 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 . IV GrP PiH 1 THtinpOR .PPF'.OACHI T', i'ICTUPE PROC' SS . . . . ....... . . .1. Some Graph Theiory F.ackgroiundls and the Proporties cf i nlT . . Fa2e ill vi Sii :.: 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 LIST I F TA.i L'.E 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 LISI OF FIGI-i::Ei 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 viii 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 CHAPT Ei; i INJTF [i'LICT I '.: 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 11111 11111 1 1 1 1 1 11111 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- ject. 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 2 112_ 1 1 : 11 1 -1 : 1 -11- 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 nucleus. 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' a' /- "r!*,1:ch inc T' s" 'N, 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, b- a a b a a bb+ b b a a b (a-bV) a 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 elsi if S = -(, then ffS = flg(S) f else if S = (-(S1 S )), h then fl'S) = f((-,f( )) ffl-s)). else if S = (-(S -S~, .l, then f(S) = (f(f-S ) )xf((--S ) i) else if = 1' '51 0 i 0 *,-,.,' the n fl.- f(= ifl ( i 1 1 i f(i* 5. else 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 a 1L step 4 c 1 step S i Figure 1.10. The graph of a vPDL. "1 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 dp/' 1a ,,) L I i , T I c. FI- ':i : dm 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. 3 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 OP)iECT EXTF:.T CTIO. J E:' THE CPADIE iT rIETHiHDi 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- -1 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. 31 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 path, 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. Inpr.u 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 I! 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 n*o 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 2001 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 I,: theP wrI n P t r ~p I,. .-- rr~,-, kit vap~cn to in .2th Ith i--- tr .111 point -i 3 id newtcnIt b.'uri I- ILi 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. (3 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 boundary. 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 areas. 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,. S0 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. D 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 ~-t [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 50CI 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. CHAPTER III OESJ CCT ['.:TRACT TiO' F THE fC ITOLr .',i.1 LY: II 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 51 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 .5 (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: th 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 PCL 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 nnn -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 ) the V C. c lo ed r c n - P,-,, ion re- inter- i n t r PC,- 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 12 13 Fipuriat3.S. *1he Wssegmq representing Ohw iniclusioin I e 1 L n i'rja c irtturs in Fi 'irTC I.S. 71 9 'N! 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.(:,. / / I 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. CHAPT F: IV GPPH THE,-' ,\AF rl,O H TOI 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 A1 4', 4 -. 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 1 '. 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. Lil SV, ,FI 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 t :ro.re- ion. Foi = 1,.. ,t-1, v i n int r: C i ..i t e 1 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 11100I 111IO A -= 11 10011 i0 0 011 00011 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 1 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. 1- \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( . 1- 1(. 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 1 the pio.p: t '- (" ,l'V ) < l'(V. ,''-V.) for all partitions 11 \6 (-- ---- ---" V .' Figure .-. An ex.namplr : ,f a 'c iihted rnph. 1 3 F, 0 M of. v i '4 Figure .1.4. A\r; .15T of the iscihted grahnl shcwii in Finure 4.3. ( ,\. > 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 graph. 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. .1 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 Input 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 , K! A.- .. L2 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 |

Full Text |

PAGE 1 OBJECT EXTRACTION AND IDENTIFICATION IN PICTURE PROCESSING By Peter Pei-teh Lin A Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy UNIVERSITY OF FLORIDA 1972 PAGE 2 To my parents PAGE 3 ACKNOWLEDGEMENTS This work was supported in part by the National Science Foundation Grant GK-2786, the Office of Naval Research Contract N00014-68-A-0173-0001, the Center for Informatics Research, and the Graduate School of the University of Florida. This financial assistance is gratefully acknowledged. The author would like to thank the members of his supervisory committee and departmental faculty representatives for the fine job they did in reading and therefore improving the presentation made in this dissertation. Dr. Raymond L. Hackett from the Medical School should be acknowledged for the information on medical aspects. For his time and effort in directing the research, the author would especially like to thank Dr. Julius T. Tou. Without his stimulating ideas, his invaluable suggestions, his honest criticism, and his enlightening guidance, this dissertation would not have been possible. The author's utmost thanks must go to him. Miss Jean Roman should be acknowledged for proofreading. 111 PAGE 4 TABLE OF CONTENTS Page ACKNOWLEDGEMENTS iii LIST OF TABLES vi LIST OF FIGURES vii ABSTRACT xi Chapter I INTRODUCTION 1 1.1. A Survey of Research in the Area of Picture Processing .... 3 1.2. Summary of the Remaining Chapters 24 II OBJECT EXTRACTION BY THE GRADIENT METHOD 2 7 2.1. Boundary Segments Finding 29 2.2. Combining Boundary Segments to Form the Boundary Contours of Objects . . . 38 III OBJECT EXTRACTION BY THE CONTOUR ANALYSIS 51 3.1. Some Fundamental Concepts in Binary Pictures 55 3.2. Contours Finding in Multilevel Picture 57 3.3. Inclusion Relation Among Contours 65 3.4. Object Extraction by Comparison . . 67 IV GRAPH THEORY APPROACH TO PICTURE PROCESSING 71 4.1. Some Graph Theory Backgrounds and the Properties of anMST .... 71 IV PAGE 5 TABLE OF CONTENTS (continued) Page 4.2. The Representation of a Digitized Picture by a Weighted Graph 87 V FEATURE EXTRACTION 9 5.1. Some Fundamental Local Features . . 91 5.2. Global Features 102 VI EXPERIMENTS AND CONCLUSIONS 106 6.1. Experiments with Chromosome Pictures 106 6.2. Experiments on Skin Cell Pictures 119 6.3. Experiments on Blood Cell Pictures 126 6.4. Conclusions and Further Research. . 132 1IBLIOGRAPHY 137 HOGRAPHICAL SKETCH 140 PAGE 6 LIST OF TABLES Table Page 5.1 Table of area, x-increment and y-increment 92 5.2 Table of moment 93 5.3 Table of moment of inertia and product of inertia 99 6.1 Local features of nuclei in Figure 6.12 124 VI PAGE 7 LIST OF FIGURES Figure Page 1.1 A scene of two cubes 1 1.2 A block diagram of a general picture processing system 3 1.3 Examples of (a) hexagonal grid and (b) rectangular grid 4 1.4 A binary picture of numeral "6" 6 1.5 A digitized picture of a portion of a skin cell picture 8 1.6 Linkage among faces 11 1.7 Several inhibited links 11 1.8 An example of the decomposition of three-dimensional objects 13 1.9 Local completeness of {+, x, -, *} ... 17 1.10 The graph of a vPDL 20 1.11 An example of the structure descriptions of a picture 22 1.12 Grammar for the class of pictures of chromosomes and three examples of chromosomes 23 2.1 The 8-neighboring points of a picture point p and the octal chain codes .... 28 2.2 The enhanced picture of Figure 1.4 ... 33 2.3 Flow diagram for searching for tentative boundary paths 34 2.4 Histogram of b-values of Figure 2.3 .. . 35 VII PAGE 8 LIST OF FIGURES (continued) Figure Page 2.5 Flow diagram for filling gaps between tentative boundary paths 37 2.6 Examples of degeneration from (a) a nonjunction boundary point and (b) a junction point on the boundary 39 2.7 Flow diagram for ordering nonsingular paths around junction areas 41 2.8 Flow diagram for finding all the smallest contours and the exterior boundary in an isolated picture 44 2.9 Examples of (a) overlapping, (b) selffolding and (c) touching 45 2.10 Examples of looping nodes 45 2.11 Flow diagram for combining paths in forming the boundary contours of objects 47 3.1 The binary pictures transformed from Figure 1.5 52 3.2 Flow diagram for finding contours in a multi-level picture 63 3.3 The labeled picture obtained from Figure 1.5 64 3.4 State diagram for finding the inclusion relation among contours 66 3.5 The Hasse graph representing the inclusion relation among contours in Figure 1.5 68 3.6 The Hasse graph obtained through the deletion of small contours 69 3.7 The Hasse graph obtained through the deletion of similar contours 70 4.1 An example of an undirected finite graph 72 viii PAGE 9 LIST OF FIGURES (continued) Figure p age 4.2 An example of a simple graph 73 4.3 An example of a weighted graph 79 4.4 An MST of the weighted graoh shown in Figure 4.3 79 4.5 Flow diagram for finding an MST of a weighted graph 81 4.6 Flow diagram for finding radial paths and centers in a tree 85 4.7 Three methods for connecting picture points 88 5.1 X-y coordinate system and the eight possible line segments with the corresponding octal chain codes 91 5.2 Principal axis direction 101 6.1 An 8-level picture of a human chromosome 10 7 6.2 The enhanced picture of Figure 6.1 ... 108 6.3 The boundary picture obtained from Figure 6 . 2 by the gradient method .... 109 6.4 The smoothed difference function of the boundary shown in Figure 6.4 .... Ill 6.5 The labeled picture obtained from Figure 6.1 112 6.6 The boundary picture obtained from Figure 6 . 1 by the contour analysis . . . 113 6.7 The smoothed difference function of the contour shown in Figure 6.6 114 6.8 An MST obtained from Figure 6.1 116 6.9 The four radial paths in Figure 6.8 obtained by deleting small branches . . . 117 PAGE 10 LIST OF FIGURES (continued) Figure Page 6.10 An 8-level picture of a portion of a skin cell picture 120 6.11 The labeled picture obtained from Figure 6.10 121 6.12 The boundary picture obtained from Figure 6.10 by the contour analysis . . . 123 6.13 Flow diagram of finding the distribution of objects in Figure 6.10 125 6.14 An MST used to describe the distribution of objects in Figure 6.10 127 6.15 The clusters of objects in Figure 6.10 128 6.16 An 8-level picture of a portion of a blood cell picture 129 6.17 Flow diagram for counting overlapping blood cells 131 6.18 The histogram of the intensities of a blood cell picture 133 PAGE 11 Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy OBJECT EXTRACTION AND IDENTIFICATION IN PICTURE PROCESSING By Peter Pei-teh Lin March, 19 72 Chairman: Dr. Julius T. Tou Major Department: Electrical Engineering Any computerized picture processing system can generally be divided into four major units: a picture digitizer, an object extractor, a feature extractor and a classifier. This dissertation is concerned mainly with new approaches to object extraction and feature extraction. Three information handling methods have been developed which may be used to mechanize the extraction of objects from multi-level pictures. These methods are those of the gradient analysis, the contour analysis and the graph theory approach. In the gradient analysis method, a locally optimal threshold is used to find the boundary points. In this new approach, high efficiency is achieved because the tentative boundary paths are simultaneously found. Then, after filling the gaps along the boundaries or at the intersections and removing the tails, boundary segments are found. Rules are xi PAGE 12 set to combine the boundary segments in order to decompose the overlapping, self-folding and touching objects in an area picture. The contour analysis method is developed on the assumption that the threshold used for transfering a multi-level picture to a binary picture is approximately constant in a window. This method permits very successful object extraction for a multi-level picture with selective deletion of nonboundary contours. In the graph theory approach, a multi-level picture is transfered to a weighted graph. An MSF (Minimal Spanning Forest) of the weighted graph is then found. By finding the principal paths of a tree in the MSF, the skeleton of the object corresponding to this tree can then be found. The boundary contour of an object has been encoded by a sequence of octal chain codes. A local feature extractor has been designed to find the area, centroid, shape, principal axis direction and the elongation index of an object with the knowledge of the sequence of octal chain codes of the boundary contour. A global feature extractor has been designed to find the inclusion relationship among objects and the distribution of objects in a picture. The inclusion relationship is represented by a Kasse graph. The distribution of objects may be represented by an MST (Minimal Spanning Tree) . The newly designed object extractor and the feature extractor methods have been tested by analysis of the XII PAGE 13 information in the pictures of chromosomes, skin cells and blood cells. In evaluation of chromosome pictures, the major tasks are to identify and catagorize all chromosomes. In analysis of the histological skin cell photomicrographs, the problem is to find the structure of cells in epidermis in order to detect the degree of the malignancy of possible tumors. In evaluation of the blood cell photomicrographs, the goal is to obtain the histogram of the blood cell photointensities in order to reveal critical diagnostic information. In each of these three evaluation tests very promising results were achieved by the use of combinations of the new techniques. A more complete computerized picture proces sing system is suggested, as an extension of the newly developed techniques. xm PAGE 14 CHAPTER I INTRODUCTION Picture processing is a process which transfers scenes to descriptions. For example, when a picture processor "sees" a writing "Q", it should be able to tell that it is the Chinese character for "sun." When a picture processor "sees" the scene as shown in Figure 1.1, it should tell that there are two cubes, A and B, in the scene, where cube A is in front of cube B. Figure 1.1. A scene of two cubes The processor performs two main functions: the first is "to see" and the second is "to give the descriptions" PAGE 15 from what is "seen." "To see" is the process usually called object extraction from the scene. "To give the descriptions" from what has been "seen" includes feature extraction and identification. In general, human beings are the best picture processors up to the present. One drawback of man's ability as a picture processor is that his visual system is easily tired. Mechanization of the picture processing became possible after the invention of the modern computer. This mechanization is very desirable as it frees manpower from routine visual tasks. There are two principal types of pictures encountered in everyday life. One is the picture of three-dimensional objects. This type of picture is the projection of the three-dimensional object on a picture plane. The projection is supposed to exhibit the depth information. Several researchers^ J have conducted research dealing with this type of three-dimensional picture. The other type of picture is two-dimensional. Two-dimensional pictures are either artificial pictures, such as characters^ ' ' and maps, ' or natural images whose depth information is not important and almost cannot be seen in the picture planes, such as pictures of particle tracks in the bubble chamber/ ' ' fingerprints *Â• ^ and cell images.** ^ From here on, "picture processing" means the mechanization of picture processor unless otherwise specified. PAGE 16 A picture processing system can generally be divided into four parts: a picture digitizer, an object extractor, a feature extractor and a classifier. The picture digitizer and the object extractor perform the function of "seeing." The feature extractor and the classifier perform the function of "giving the descriptions." Figure 1.2 is a block diagram of a general picture processing system. optical picture picture digitizer PAGE 17 Â• i * * (12) . . are two principal ways to quantize v * a picture plane: the hexagonal grid and the rectangular grid. Figure 1.3 shows the two types of grids. Hexagonal grids have the advantage of having six neighboring picture points, which are nearest to p, for every picture point p. They have the drawback of being based on an uncommon, non-orthogonal coordinate system. The rectangular grids contain only four neighboring picture points, which are nearest to p, for every picture point p, but it is very easy to access every picture point. (a) (b) Figure 1.3. Examples of (a) hexagonal grid and (b) rectangular grid. The rectangular grid forms an orthogonal coordinate system. Hence the picture plane becomes I =1X1, where I and I are subsets of the integer set. From here on, all digitized pictures use rectangular grids. The intensities of a quantized picture are quantized into n levels. Usually n is set equal to 2 because this maximizes PAGE 18 storage efficiency within the bit-oriented digital computers. A 2 -level picture is called a k-bit picture. If k = 1, the digitized picture is called a binary picture. Ann-level digitized picture is a mapping g: I->-N, where N = (0,1,..., n-1) is the set of quantized intensity values. A digitized picture can also be represented as a matrix. The location of a picture point is specified by the location of the element in the matrix. The intensity of a picture point is indicated by the value of the corresponding element in the matrix. Figure 1.4 is a binary picture of a numeral "6" represented as this matrix form. A picture digitizer performs a transformation from an f mapping to a new mapping g. A complete picture digitizing system, PIDAC (Pictorial Data Acquisition Computer) , has been implemented at the CIR (Center for Informatics Research) in the University of Florida. The PIDAC, which is a modification of the FIDAC^ J (Film Input to Digital Automatic Computer) system, is one of the better picture digitizers available today. It consists of a CRT, two lenses, a photomultiplier, an a-d converter and a scan control unit. The digitized pictures are stored on a magnetic tape. The PIDAC can alternately be interfaced with a digital computer to store the digitized form from a picture. The maximum spatial resolution of the PIDAC is 1,240 spots along the long axis and 800 lines per 35 mm film. The maximum digitized level of the PIDAC is 2 4 = 64. Very good 8-level PAGE 19 000000000000000000000000 000000001110000000000000 000000011110000000000000 000000011110000000000000 000000111110000000000000 000000111100000000000000 000001111100000000000000 000001111000000000000000 000011111000000000000000 000011111000000000000000 000011110000000000000000 000111110000000000000000 000111111111100000000000 000111111111111000000000 000111111111111000000000 000011111111111100000000 000011111000111100000000 000011111000111100000000 000001111111111100000000 000000111111111100000000 000000011111111000000000 000000000111100000000000 000000000 000000000000000 000000000000000000000000 Figure 1.4. A binary picture of numeral M 6 . " PAGE 20 pictures can be achieved by the PIDAC. The scanning speed of the PIDAC is .3 sec/picture. Figure 1.5 is a portion of an 8-level skin cell picture obtained from PIDAC. Once a digitized picture is obtained, the picture data are then accessible by digital computers. The next process is the extraction of objects from the picture. 1.1.2. Object Extraction There are mainly three methods used in extracting objects from the scene. The first method finds the boundaries of objects and then decomposes objects from boundaries. The second method finds the thresholds to transfer a multilevel picture to a binary picture and then finds the contours of the binary pictures. The third method finds the clusters in a picture and considers each cluster as an object. There are two main approaches used to find the boundaries of objects. One approach finds the enhanced picture first and then finds the boundaries. The enhanced pictures can be found either from the spatial domain or from the spatial frequency domain. To find the enhanced pictures directly from the picture plane (spatial domain) , the most frequently used methods are the gradient method^ ' and the Laplacian method. * ' By the gradient method, each picture point in an enhanced picture is set to have a value equal to the gradient of intensity at that picture point. PAGE 21 3 PAGE 22 Since the data are digitized, approximation of gradient is f 11 2 used, such as h(i,j) = [ (g(i , j ) -g(i + l , j +1) ) + 2 1/2 (g(i+l,jlg(i,j+l)) ] , which is a very good approximation except that square root calculation is involved. Approximation of the Laplacian function is needed for digitized data, such as ^ J L(i,j) = g(i-l , j ) +g (i+1 , j ) + g(i , j -1) +g(i , j + 1) -4g (i , j ) . The function L then represents the enhanced picture. The enhanced picture will normally have high values at the boundaries. To find the enhanced picture from the spatial frequency domain, ** the picture f is transformed to a Fourier spectrum F first. A highpass filter H is applied to enhance values of F at high frequencies relative to those at low frequencies. The inverse Fourier transformation of FH is the corresponding enhanced picture. After the enhanced picture is found, a threshold is then set to find the boundary points. Boundary points are connected by a multi-step process. ' The other approach finds the boundaries by use of a matched filter which can extract the boundaries directly from the picture g. The purpose of the edge operator used by Huechel^ ' is to fit an ideal edge element to any empirically obtained edge element. In scanning the picture, when an edge is found by the edge operator, scanning is interrupted and the edge is traced until lost. After the boundaries in a picture are found, objects f 21 are to be extracted. Guzman^ J did the work on extracting PAGE 23 10 three-dimensional objects. The main idea used to extract the three-dimensional objects was based on the a priori knowledge of the possibility of two faces belonging to an object. A vertex is in general a point of intersection of two or more boundaries of regions. A program SEE has been built to examine the configuration of lines meeting at the vertex to obtain evidence relevant to whether the .regions involved belong to some object. Two types of links, strong links and weak links, are used. Figure 1.6 shows the linkage of faces at several vertices. A solid line implies a strong link and a dotted line implies a weak link. Figure 1.7 shows the links which are inhibited. A region is defined as a surface bounded by simply closed curves. A nucleus is a set of regions. Two nuclei, A and B, are linked if the regions a and b are linked where a e A and b e B. Three rules are set to link the nuclei. First rule : If two nuclei are linked by two or more strong links, they are merged into a larger nucleus . Second rule : If nuclei A and B are joined by a strong and a weak link, they are merged into a new nucleus . Third rule : If nucleus A consists of a single region, has one link with nucleus B and no links with any other nucleus, A and B are merged. PAGE 24 11 "Fork" "Arrow" Leg" "Matching T's" Figure 1.6. Linkage among faces Figure 1.7. Several inhibited links PAGE 25 12 The first rule is applied on the picture repeatedly until it is no longer possible to combine nuclei. The second and third rules are then applied successively. Figure 1.8 is an example of the decomposition of threedimensional objects. In step 1, every nucleus corresponds to a region; for example, nuclei A, B and C correspond to regions a, b and c, respectively. There are two strong links connecting nuclei A and B. One link comes from the Y intersection of regions a, b and c. The other link comes from the arrow intersection of regions a and b . All other links are derived in the same way. Step 2 is the straightforward application of the three rules to combine nuclei. The regions correspond to nuclei in a group on an object, for example, regions a, b and c form an object. It is obvious that Guzman's method can be applied only to the pictures of three-dimensional objects. The second method uses thresholds to transfer a multilevel picture and then finds the contours of the binary pictures as object boundaries. Prewitt^ 11 ^ used the local minima of the optical density^f requency distribution of a picture as the thresholds to find the background levels, cytoplasm levels and nucleus levels. The third method is the clustering method. Zahn fl6) proposed a method to group points into objects by the clustering method, which is graph theory oriented. This clustering method is motivated by the perception of two- PAGE 26 13 Figure 1.8. An example of the decomposition of three dimensional objects. PAGE 27 14 dimensional point sets as separate "gestalts." The principle of grouping used is "proximity" as described by Wertf 171 heimer. v The proposed method is applicable to binary pictures. For a binary picture, every picture point with grey value 1 is a vertex. Picture points having grey values 1 are called object points. The connection between object points is called an edge. A weight is assigned to every edge. It is equal to the Euclidean distance between the corresponding object points. An MST (Minimal Spanning Tree) T is defined as a spanning tree of G whose weight is minimum among all spanning trees of G. Some edges in the MST can be deleted by using a factor as the measure of the significant edge inconsistency. The MST is then clustered to a forest. Every tree in the forest clusters together all the points in one object. 1.1.3. Feature Extraction Feature extraction strongly depends on the type of pictures handled. How big the feature set should be depends on the purpose of handling the picture. There are two main types of features. One is the local feature which depends on individual objects in the picture. Area and centroid were presented by Freeman. J Eden^ ' has proposed the fundamental strokes as the features of handwritten English characters. Topological features are proposed by Tcu and Gonzalez^ * for characterizing handwritten characters. Topological features have been used PAGE 28 15 for automatic fingerprint interpretation. ' A skeleton^ ' has been proposed to describe indirectly the shape of objects. A skeleton can be thought of as a generalized axis of symmetry of an object. At first the concept was applied f 201 f 211 to the binary pictures. Rosenfeld, " ; Montanari, } Phil(22) brick^ -* and others have developed algorithms to find the SKeletons in binary pictures. Levi v J generalized the concept to the multi-level pictures by defining a new distance function which took the grey level intensities into consideration. Ledley^ ' used the ratio of the number of concavities to the number of segments of the boundary as the only feature in detecting the mitotic cells. The other type of feature is the global feature. Global features are the ones which reveal the interrelationship among objects in the picture. Sometimes an object can be described in terms of fundamental components. Global features can also be used to describe the interrelationship among fundamental components of an object. Inclusion relar 2 41 tion among regions can be found by MANS. Several linguistic descriptions have been used to describe the global (7) features. Narasimhan v * used syntax-directed hierarchy labeling to describe the particle tracks in the bubble f251 chamber. Shaw proposed a PDL^~ * (Picture Description Language) which may be the most formal and useful linguistic approach in picture processing up to date. Linguistic approach has the advantage of describing the picture formally. PAGE 29 16 The PDL (Picture Description Language) is a picture or graph algebra over the set of primitive structural descriptions under the operations +, -, x, *, and /. Figure 1.9 shows the local completeness of the operations [+, -, x, *]. Elements in the PDL are considered equal if they are equivalent. The equivalent relation is defined as: 1. S.^ is weakly equivalent to S 2 if there exists an isomorphism between graphs of S, and S 2 such that the corresponding edges have identical names. 2. Sj is equivalent to S if (a) S, is weakly equivalent to S 2> and (b) tailCS^ = tail(S 2 ) and head(S ;[ ) = head(S 2 ). A number of useful algebraic properties are given below: 1. Each of the binary operators is associative. 2. * is the only commutative operator; x and are "weakly" commutative . 3. The unary operator ~ (a) acts as complementation in a Boolean algebra. (~(s 1+ s 2 )) = CC~s 2 )+(~s 1 )3 (~(s 1 *s 2 )) = CC~s 2 )*(~s 1 )) (b) obeys a "de Morgan's law" with respect to x and -. (~cs 1 xs 2 )) = ((~s 2 )-(~s 1 )) C~(s 1 -s 2 )) = ((~s 2 )x(-s 1 )) (c) Involution: (~C~S)) = S. PAGE 30 17 Concatenation Description a+b axb a-b b + a a*b (a+b)* A Figure 1.9. Local completeness of {+, x, -, *} PAGE 31 4. The / operator. (a) (/(/s)) = S. (b) C/(S 1 b S 2 )) = (/S 1 )j2( b (/S 2 ), where b is a binary operator . 5. The null point primitive X. (a) S + X = X + S. (b) S + X = S,S-X = S, XxS = S. (c) ~A = X. (d) X0 b X = X. By using some of the algebraic properties of PDL to move unary operators and label designators as far as possible within an expression, a standard form f(S) PDL of an expression S can be obtained. f(S) is defined by: if (S = s 1 V S = C/Sj) V S = OS^ V S (-(/Sj))) primitive (S^ , then f(S) = S else if S = (S 1 b S 2 ), b e{+,x,-,*}, then f(S) = (f (S^ fe f (S 2 ) ) else if S = S 1 , then f(S) = f(g(S)) else if S = (-(S 1 0S 2 )), 0e{+,*}, then f(S) = (f ( (~S 2 ) ) 0f ( (-S^ ) ) else if S = (-(SjXS^), then f(S) = (f ( (~S 2 ) ) -f ( (-S^ ) ) else if S = C-CSj-S^), then f(S) = (f ( (~s 2 ) ) xf ( (-S^ ) ) PAGE 32 19 else if S = (/(S 1 0S 2 )), 0e{+,-,x,*}, then f(S) = (f((/S 1 ))0f((/S 2 ))) else if S = C~C~S.jP), then f(S) = ffS^ else if S = C~C/S 1 )) V S = C/C-Sj)), then f(S) = f ( (~f ( (/S^ ) ) ) else if S = C/C/S^)), then f(S) = fCC/Sp) A valid PDL expression (vPDL) is the one whose standard form is such that if (/p ) appears in it one or more Â£ times for some primitive p and label Â£ , then p also appears once and only once outside the scope of a /. The graph described by a vPDL S is defined by the following algorithm: 1. Transform S into standard form by applying the function f Â£ 2. Replace each expression of the form (/p ) by a new primitive p/ . This removes all / operators. 3. Generate the connectivity graph of the resulting expression. 4. Connect the tail and head nodes of each edge p/ to the Â£ corresponding nodes of p . Â£ 5. Eliminate all edges of the form p/ . The above algorithm formally defines the meaning of labeled expressions and the / operator. Figure 1.10 shows the graph of a vPDL. PAGE 33 20 step 2 ((((a 1 + b) * (b + a))* c) + (/a 1 )) ((((a 1 + b) * (b + a))* c) + a 1 /) step 3 step 4 step 5 Figure 1.10. The graph of a vPDL, PAGE 34 21 It has been proved that any vPDL describes a unique primitive connectivity and any connected set of primitives can be effectively described by a vPDL. It has also been shown that the origin (tail) of a picture can be at any convenient place. The set of rules or grammar G that generates (describes) the class of pictures PÂ„ will be a type 2 (context-free) phrase structure grammar with the following restrictions. Each production is of the form: S + pdl 1 |pdl 2 |pdl 3 | ... |pdl n , n > 1, where S is a nonterminal symbol and pdl. is any PDL expression with the addition that non-terminal symbols are allowable replacements for primitive class names. Sentences of L(G) will consist of PDL expressions; thus, the class of terminal symbols of G will be a subset of {+ ,x, -,*, ~ ,/,(,) } V{primitive class names} V{label designators} Each grammar G will have one distinguished nonterminal symbol from which L(G) may be generated; the symbol on the left part of the first production of G will be the distinguished symbol. The hierarchic structural description HÂ„(C) of a picture CePp having primitive structural description To(C)eL(G) is defined as the parse of T<-.(C) according to G; H~(C) is conveniently represented as a parenthesisfree tree. A simple example of PDL description of a house is given in PAGE 35 G: 22 (a) House -*> C(vm + (h+ (Â»vm) ) ) * Triangle) Triangle > ((dp + dm) * h) L(G) =[((vm + (h+(-vm))) * ((dp + dm)* h))] dp //> dm\ h (a) G,L(G), and primitives vm Cb) A T s (c i ) = ((vm + (h+Â£vm)))* ((dp + dm) * h)) H (c) : s v i J House (b) Examples and parse of a "house' Figure 1.11. An example of the structure descriptions of a picture. PAGE 36 23 Figure 1.11. Note that all three pictures of houses in Figure 1.11 have the same primitive structural descriptions, which can be accepted by the grammar G. PDL can describe very well the interrelationship between primitives, but it does not have the ability to find the primitives. Figure 1.12 shows three pictures of chromosomes and the accompanying grammar to describe the pictures. Chromosome * Kl * K2 K2 -> v + Kl + vl v + Kl| Kl + vl Kl Kl -> p+v+p primitives: P /^ v aj a/1 T(C X ) PAGE 37 24 1.1.4. Classification Once a good set of features has been extracted, many f 26) classification techniques^ * are available. If the set of features of different categories are linearly separable, linear classification^ '' can be used; otherwise non-linear r ?6~) classification^" * should be used. Multi-level classification^ J is sometimes used. For linguistic description of f 2 5 2 8) features, a grammar^ ' J can be designed to accept a sentence S only if S describes a picture of some specific category. A grammar can then serve the purpose of classifying objects. For example, any picture having a PDL expression which is acceptable by the grammar shown in Figure 1.12 is classified as a chromosome. Note that all three pictures shown in Figure 1.12 will be accepted as chromosomes. The main problem of designing a grammar is that it has to be complete in the sense that it should be able to accept all pictures in a category. Here we like to emphasize that in order to have a good result on the classification, a good set of features is required. If the features set is poor, no matter how good the classification technique is the result will be of poor quality. 1.2. Summary of the Remaining Chapters Chapters II and III present two different methods of object extraction. Chapter II uses the gradient method to find the enhanced picture. Boundary points are found by PAGE 38 25 adaptively thresholding the gradients. Boundary paths will be found in the process of finding the boundary points. Gaps will then be filled in and the boundary segments are then found. Special laws are used to combine the boundary segments to form the boundaries of individual objects. Overlapping, self-folding and touching objects are decomposed. The gradients used are integers rather than real numbers, such as those used by Roberts. ' Hence less storage is required. The threshold is adaptive rather than fixed. It is then less sensitive to the noise. Chapter III presents the contour analysis method. This method was motivated in experimenting with area picture data by showing different levels of a picture in a display unit. The main idea is that in a small window section in the picture, the threshold for transferring the picture to a binary picture is approximately constant. The thresholds are adaptive rather than fixed, such as those used by Prewitt. The result of this method is very successful, especially for the area pictures. Chapter IV discusses the graph theory approach to the picture processing. Skeletons of objects can be found by this approach. Further theoretical research should be done in this area. Chapter V presents the extraction of several important features for area pictures. Area, centroid, shape, principal axis direction and elongation index of an object are the local features discussed in this chapter. Inclusion PAGE 39 26 relation among objects and the distribution of objects in a picture are the global features presented in this chapter. Chapter VI discusses the experiments with biomedical images by the above methods and suggests further research. Chromosome, skin cell and blood cell pictures are analyzed. In experimenting with chromosome pictures, the main problem is finding chromosomes in a picture. In experimenting with skin cell pictures, the main problem is to detect the tumors In experimenting with blood cell pictures, the main problem is to find the histogram of the intensities. All the experiments show very promising results. It is hoped that further research can produce a more sophisticated image analyzing system. PAGE 40 CHAPTER II OBJECT EXTRACTION BY THE GRADIENT METHOD When human beings look at any scene, the impact information we get is the shapes of the objects in the scene. The information revealing the shapes of the objects are the boundaries. If the objects are overlapped, it is possible to use the information of boundaries and grey intensities to decompose objects. As we can very easily imagine, one way to find the boundaries is to use the fact that usually the boundaries consist of those points having very high change of intensities from their neighbors. Using this property to find the boundaries is called the gradient method, because the change of intensities is measured by the gradient. The gradient method is very good if there are high contrasts in the boundaries, even if there exist some nonuniform distribution of the intensities in the objects. Some definitions will be introduced before getting into the problem. Definition 2.1 . --A point p. in the picture plane I is an 8-neighboring point of the picture point p in I if and only if 0 PAGE 41 28 8-neighboring points of p are labeled as shown in Figure 2.1, The set of all 8-neighboring points of p is denoted as N(p). The octal chain code which encodes the line segment from p to p. is i. Because the i's values range from to 7, the code is an octal code. It is obvious that any curve in the digitized picture is approximated by a sequence of line segments which join their points to their 8-neighboring points. Hence any curve in the digitized picture can be represented by a sequence of octal chain codes and the start point of the curve. Let C be a curve represented by the chain codes c, ... c and the r J In start point o. The reverse of the curve C can then be represented by the sequence C" = (c -4) R ... (c,-4)Â„ and the start point o* which is the end point of the curve C. Because of the small storage required to store the chain code and because it is easy to manipulate, this technique is used throughout the dissertation to encode the curve in the digitized picture. Figure 2.1. The 8-neighboring points of a picture point p and the octal chain codes. PAGE 42 29 2.1. Boundary Segments Finding Physically, a boundary point is a picture point which had a high increase of intensity from its neighbors. The rate of change of intensity from p. to p, where p. is an 8neighboring point of p, can be evaluated by a differentiator, which is defined as h*(p,p i ) = [g(p) -g (p i ) ] /d (p ,p . ) , where g is the picture function and d is the Euclidean distance. It is easily seen from Figure 2.1 that d(p,p.) 1 if i is even and d(p,p.) = /I if i is odd. The h* function can then be redefined as h*(p,p i ) = â€¢* [g(p) -gCp^ ,E(i) ] , where E(i) = if i is odd and E(i) = 1 if i is even. The V* function is defined by the following mapping table. PAGE 43 30 of the h** function is discrete, there is a one-to-one mapping h*** from R = range (h**) onto I b , where I fe is a subset of the integer numbers, and h*** preserves the ordering of the elements. h*** serves the purpose of quantizing the range of h**. A new function h can then be defined as the composition of h*** and h**, that is, h = h***.h**. This function is a measure of the quantized rate of increase of intensities. We can write the h function in terms of a ? function, hCp, Pi ) = ng(p)-g( Pi ),E(i)]. For n = 8, the function can be represented by the following mapping table: V E(i) 1 g(p)-g(P i ) -7 -6 -5 4 -5 -2 -10 12 3 4 1 3 5 6 8 10 11 002479 12 13 14 The magnitude of the gradient at a picture point p is defined as the maximal increase of intensity from the neighboring points to the picture point. The following definition is then yielded. Definition 2.2. --The gradient b (or sometimes will be called b-value) at a picture point p of the picture g is defined as b(p) = max (h(p,p.)}, ie[0,l,...,7] 1 where the h function is defined earlier. PAGE 44 31 The enhanced picture of a digitized picture g is a digitized picture with the intensity at every picture point equal to the gradient at the corresponding picture point in the g picture. The enhanced picture ofann-level picture is of 2n-l level. It will be seen later that one extra value is required to identify the boundary points. It is then obvious that a k bit picture will yield a k+1 bit enhanced picture. The enhanced picture of Figure 1.4, which is a binary picture of numeral "6", is shown in Figure 2.2. Because of the unavoidable noise appearing in the picture, there is no way to find the real boundaries in one step. We can break the processof finding boundaries into several steps. The first step is to find all those points which can quite possibly be boundary points. These points are called tentative boundary points. The fact that the boundary points of a boundary path are connected can be used in the process of finding tentative boundaries. The following section details the scheme of finding the tentative boundary paths. Octal chain codes are used to encode the paths. 2.1.1. Tentative Boundary Path Searching The method of finding the tentative boundary paths is based on the principle that a tentative boundary point is a point which has a gradient greater than the gradients of the neighboring nonboundary points and is connected to some PAGE 45 32 other tentative boundary points. Figure 2.3 shows the flow chart used to search the tentative boundary paths. The input to this tentative boundary path finder is the raw digitized picture. The output is: 1. The tentative boundary uicture which is a binary picture having value -1 at the tentative boundary point and elsewhere . 2. A list of tentative boundary paths. For each tentative boundary path, it has 2a. the start point of the tentative boundary path, 2b. the length of the tentative boundary path, 2c. a sequence of octal chain codes which encode the path , 2d. an indicator which denotes whether the tentative boundary path is closed or open, and 2e. the end point of the tentative boundary path if the indicator denotes that it is an open path. The threshold 6 in the flow chart, which is used to pick up the first point in a tentative boundary path, is usually decided by the following method. First, find the histogram of the b values in the picture, which is a plot of number of picture points whose gradients are greater than or equal to a b-value. Figure 2.4 is the histogram of the enhanced picture shown in Figure 2.3. To determine which is 6 from this histogram, find PAGE 46 33 PAGE 47 34 Input digitized picture g | Find the enhanced picture"] Note p* as PAGE 48 35 the greatest drop in the histogram from 0-1 to example G will be set as 1. In this n, PAGE 49 36 path 2 : start point = (16,10), length = 10 , octal chain codes 4432100765, closed contour. In this special example, the tentative boundary paths are the final boundary contours. In most of the practical cases, because of the existence of noise, overlapping, selffolding and touching, there will exist gaps between tentative boundary paths and tails of tentative boundary paths. The following sections discuss the strategies of solving these problems . 2.1.2. Procedure of Filling Gaps and Determining Line Segments To fill the gaps between tentative boundary paths, one has to examine the extreme points of tentative boundary paths. For an extreme point p of a tentative boundary path C, let p' be the nearest tentative boundary point which is not on C or is on C and has more than five points from p along C. If the Euclidean distance between p and p' is less than 3, p is then connected to p' through the shortest path. If p' is an extreme point of a tentative boundary path C , tentative boundaries C and C will then be combined. If p' is not an extreme point, p' will then be an intersection node. Figure 2.5 shows the flow diagram for filling gaps. PAGE 50 37 Input the enhanced picture and the list of tentative boundary paths Figure 2.5. Flow diagram for filling gaps between tentative boundary paths. PAGE 51 38 A boundary segment is defined as a boundary path between successive junction nodes or a boundary path which does not have junction nodes on it. A tail of a boundary is the boundary path between an extreme point, which is not a junction node, and a junction node. Hence the main procedure in determining boundary segments is to order the junction nodes along the boundary paths. Once boundary segments are determined, the boundary segments joined at a junction node can easily be noted. Because only contours of objects are of interest, all tails will be erased. 2.2. Combining Boundary Segments to Form the Boundary Contours of Obiects Sometimes a point p on the boundary will degenerate into several junction nodes after applying the process stated in the previous section to the digitized pictures. It happens most often when p is a real junction point. Let p be a boundary point which degenerates into k junction nodes n..,...,n,. N = {n, , ...,n, } is the complete set of junction nodes degenerated from the point p. The ideal cases (i.e., no degeneration) are: 1. point p is not a junction point and N is empty, and 2. point p is a junction point and the cardinal number of N is 1. P Figure 2.6 shows some degeneration cases. In the degeneration cases, there must exist singular paths connectinj PAGE 52 59 the degenerated junction nodes. Let N be the set of all junction nodes in the picture. The degenerate relation R is defined on N such that nRn', where n, n'eN, if there exists a sequence of intersection nodes n, = n,n, . . . ,n-. = n' satisfying the condition that there is a singular segment between junction nodes n. and n,, i = l,...,k-l. The dei l+l generate relation R is obviously an equivalence relation. The equivalence relation R can thus partition N into a collection of equivalence classes. Every equivalence class is then a complete set of junction nodes degenerated from some point. In the processing, we have to find the singular segments first, then \ PAGE 53 40 A threshold y is assigned such that if a boundary segment is of length less than u, it is classified as a singular segment; otherwise it is nonsingular. Let N be an equivalence class induced by the relation R and let S be the set of all singular segments connecting to the junction nodes in N. (N,S) forms a junction area. Let E be the set of all nonsingular segments connecting to the junction nodes in N. The ordering of the elements in E is very useful in combining boundary segments. It is important to point out that only the chain codes, which encode the line segments connecting the junction nodes, are used to detect the ordering of the nonsingular boundary segments around the junction area. The ordering of segments around the junction area can be either clockwise or counterclockwise. Because of the line encoding scheme (octal chain code) we used, there are at most eight boundary segments joining at a junction node. Figure 2.7 shows the flow diagram for ordering the nonsingular segments around junction areas . Every equivalence class of junction nodes can be thought of as a single junction node. Any nonsingular segment which connects to some other nonsingular segment through the equivalence class of junction nodes is considered as through the corresponding single junction node. We can then imagine the boundary segments picture as an ideal one in the sense that no singular path exists. PAGE 54 41 Input list of boundary segments and list of junction nodes -J Return ) Note p as a junction node having not been processed yet . Set k = 0. Note S is the nonsingu lar path connecting to p yes Set S = S'. p = extreme point of S which is not p. Figure 2.7. Flow diagram for ordering nonsingular paths around junction areas. PAGE 55 42 A nonsingular segment S. is connected to a nonsingular segment S. if and only if there exists an equivalence class N of junction nodes such that both Sand S. contain some junction nodes in N. This relation is denoted by E. The relation E" of connectivity of two nonsingular paths is a transitive closure of the relation E. The picture consisting only of the nonsingular segments (and the singular paths which connect them) of the equivalence class induced by the relation E' is called an isolated picture. A contour, is defined as a simply closed curve. We can partition a contour into a sequence of successive adjacent boundary segments. Hence the concatenation of successive adjacent boundary segments can form a contour, if the condition of being a contour is satisfied. A picture point p is included by a contour if p is a point on the contour or if every ray initiated from p will meet an odd number of times with the contour. If the above condition is not satisfied, the picture point is said to be excluded by the contour. The set of all picture points which are included by a contour is called the region enclosed by the contour. Definition 2. 3 . --In an isolated picture, an elementary region is defined as a region enclosed by a minimum contour in the sense that the region does not include any region which is enclosed by a contour in the picture. A region in an isolated picture, which is the union of all elementary regions, is called the whole region of the picture. The PAGE 56 43 contour in an isolated picture which encloses the whole region is called the exterior boundary of the isolated picture. Any nonsingular segment in the exterior boundary is called an exterior segment. Let m be the number of junction areas and k be the number of nonsingular segments in an isolated picture; f 29) there are k-m+1 elementary regions and one exterior boundary. The above facts are useful in terminating the searching process. Figure 2.8 shows the flow diagram for finding all the minimum contours and the exterior boundary in an isolated picture. From here on, the terms "path" and "node" are used to imply "nonsingular segment" and "junction area," respectively, unless otherwise specified. Let us look at the different examples shown in figure 2.9 to get a feeling of "how our visual systems combine paths into object boundary contours." It is amazing that we don't have to know the intensity in each elementary region to find out that in (a) there are two objects: one is enclosed by the boundary contour consisting of paths S n and S 2 , the other is enclosed by the boundary contour consisting of paths S-, and S,; in (b) there is a self-folding object enclosed by the contour consisting of paths S n and S, with the region enclosed by the contour consisting of paths S, and S ? as the folded part; and in (c) there are two touching objects: one is enclosed by the PAGE 57 44 Input isolated picture Set k=0, k'=m-n+l and i=0. m=nuTnber of paths n=nuniber of nodes Note p as an unprocessed node Note S-: has not combined with Sj+i (the index is around p) . Set S = 3.j and k-k+1. I i = i+1 Set S'=Sj+i, p=the extreme point of S' which is not p, Put S* in the ith list. yes A counterclockwise contour is the exterior boundary. A clockwise contour is a smallest contour. Set j = index of S' around the node p. Figure 2.8. Flow diagram for finding all the smallest contours and the exterior boundary in an isolated picture. PAGE 58 45 boundary contour consisting of paths S,, and S, , the other is enclosed by the boundary contour consisting of paths S, and S . (a) overlapping (b) self -folding Figure 2.9. Examples of (a) overlapping, (b) self-folding and (c) touching. Sometimes objects touching at a point may occur such as the examples shown in Figure 2.10. We call this kind of node the looping node. Figure 2.10. Examples of looping nodes. PAGE 59 46 If there is an odd number of paths connecting to a junction node, self -folding or touching may occur. Usually the path which belongs to two objects (here we consider self-folding as a special case of touching) is quite "straight." The straightness of a path can be determined by the filtered sequence of differences of successive chain codes, which can be obtained by the digital filtering method. If each filtered difference of successive chain codes of a path has absolute value less than 1, the path is then considered as "straight." Figure 2.11 is a flow diagram for combining the paths in forming the boundary contours of objects. Now we have to find out which objects are overlapped, which objects are touched and which objects are self-folded. Let C, and C^ be the boundaries of two objects 0-, and 0~ , respectively. If there does not exist a common node between C. and Cy , 0, and 0are separate objects. If there exists a common node between C, and Cand if there does not exist a common path between C, and C~, 0, and 0,, are overlapped objects, such as Figure 2.9(a). If there exists a common path between C, and C~ , 0, and 0~ are either touching or self -folding . Assume that both C. and C~ have a common path S. Trace C, and C^ so that S is traced in the same direction. If both the contours C, and C ? are traced in the same direction (that is, either both are clockwise or both are counterclockwise), 0, and 2 form a self-folding object. PAGE 60 47 Input the list of paths and the list of junction nodes 1 Order paths around junction areas Find all the smallest contours and the exterior boundary of every isolated picture. Note an extreme point p of a path C which has not been processed yet. Set C*, C S =C; p*,p s =p. Count(p')=0 for all junction nodes in the oicture. |count (p) = count (p)+l| Â©-* Erase C from the list Set C=C*, p=p*. Figure 2.11. Flow diagram for combining paths in forming the boundary contours of objects. PAGE 61 48 Note C = S-; and C'=S, PAGE 62 49 Erase C from the junction PErase C from the junction Pyes Â© Note C be the first exterior path connecting to p such that there are interior paths between them. Erase C and C from the junction p Note C be the interior path such that it is in the same interior region as C and the number of paths between C and the corresponding nearest exterior path is equal to the number of paths between C' and the corresponding nearest exterior path J Figure 2.11. Continued PAGE 63 50 If C-, and C~ are traced in different directions, 1 and ? are touching each other. For example, in Figure 2.9(b), the common path is S, . Trace C, = S,Sand C~ = S,S~ so that S, is traced upward in both cases. Both C, and C ? are traced in the clockwise direction. Hence CL and ? are self-folding. In Figure 2.9(c), trace C 1 = S-S, and C 2 = S-S., so that the common path S 1 is traced upward in both contours. C, is then traced in counterclockwise direction, while C ? is traced in clockwise direction. Hence C, and C ? are touching each other. PAGE 64 CHAPTER III OBJECT EXTRACTION BY THE CONTOUR ANALYSIS It is known ^ ' that boundary paths are very easily extracted from a binary picture. Ann-level picture g can be transformed into n-1 binary pictures g. = PAGE 65 52 1111111111111111110 1111111111111111110 111' 1111111111111110 1111111111111111 110 1111111111111111100 111111 1111 111111100 1111111111111111100 1111111111111111100 1111111111111111000 1111111111111111000 1111111111111111000 1 PAGE 66 53 1 PAGE 67 54 0000000000000000000 0000000000000000000 0000000000000000000 0000000000000000000 0000000000000000000 0000000000000000000 0000000000000000000 0000000000000000000 0000000000000000000 0000000000000000000 0000000000000000000 0000000001000000000 0000000000000000000 0000000000000000000 0000000000000000000 0000000000000000000 0000000000000000000 (e) B 5 Figure 3.1. Continued PAGE 68 55 Figure 1.5. Since the highest level in Figure 1.5 is 5, both g 6 and S_ are trivial pictures, having in every picture point, and are not shown in Figure 3.1. Boundary contours can be extracted from $ , j=l,...,n-l. Inclusion relations can be set up among boundary contours. A boundary contour C-. in 6 will be included in a boundary contour C ? in 3-, where i PAGE 69 56 p. such that d(p,p-)<2. A contour is a sequence of picture points (p Q ,. . . ,P m _ 1 ) such that 0 PAGE 70 5 7 3.2. Contours Finding in Multi-level Picture In a multi-level picture g, a picture point p is a boundary point if there exists a point p. such that d(p,p.)<2 and g(p)>g(p-)> P-: is called an adjacent background point, with respect to the contours passing through the boundary point p in binary pictures S Â• , g(p ) PAGE 71 58 A rotator is a vector initiating from an object point to its eight-neighboring points, and hence can be represented by an octal chain code. A rotator rotating in counterclockwise direction is needed in the process of finding contours. If a rotator pointed to the picture point p if it will next point top r .,,~. __ * Â•'modS' rotator is the index of the neighboring point. In finding the contours in a picture g, first scan the picture g in forward direction. Let p be the first picture point satisfying the conditions that g(p)>g(p ) and either (* (W the picture point p has not been labeled before, or the picture point p is on some contour C , which has been found, and g(C') PAGE 72 39 reduced to c^,...c ,. If the reduced sequence is empty, then the contour is one which can be broken into two paths S, = c, ...c and S~ = c .^...c such that S= S ~. 11 m, l m. +i m 21 A contour having an empty reduced sequence is considered to be an exterior contour. If the reduced sequence is net empty, then the sum of the differences of adjacent octal chain code is the parameter to indicate whether the contour is encoded clockwise or counterclockwise. If the sum is -8, the contour is traced in a clockwise direction. Hence the contour is an interior one. If the sum is +8, the contour is traced in a counterclockwise direction. Hence the contour is an exterior one. Figure 3.2 shows the flow diagram used to find contours in a multi-level picture. Figure 3.3 is the labeled picture of the 8-level picture shown in Figure 1.5. There are 14 contours in Figure 1.5. Contour 1: start point = (1,1), octal chain codes = 22222222222222224444444444 4456 56 5666 566656660000 000 000000000, length = 61, intensity = 1, background intensity = 0, exterior contour. Contour 2: start point = (1,1), octal chain codes = 22222222222222224444534454 5 56 566 56 766 76 5 500 0000 00 000 000 PAGE 73 60 length = 55 , intensity = 2 , background intensity = 1, exterior contour. Contour 3: start point = (1,1), octal chain codes = 22222222666656660 length = 17, intensity = 3, background intensity = 2, exterior contour. Contour 4: start point = (1,6), octal chain codes = 44344540000000 length = 14, intensity = 3, background intensity = 2, exterior contour. Contour 5: start point = (1,10), length = 0, intensity = 4 , background intensity = 3, exterior contour. Contour 6: start point = (2,1), octal chain codes = 22650, length = 5 , intensity = 4, background intensity = 3, exterior contour. PAGE 74 61 Contour 7: start point = (2,6), octal chain codes = 4444322102111007666656653, length = 25, intensity = 2, background intensity = 1, interior contour. Contour 8: start point = (5,12), octal chain codes = 1210211022134345456655676660, length = 28, intensity = 3, background intensity = 2, exterior contour. Contour 9: start point = (9,11), octal chain codes = 2171222245556660, length = 16 , intensity = 4, background intensity = 3, exterior contour. Contour 10: start point = (11,4), octal chain codes = 2221776553, length = 10, intensity = 2 , background intensity = 1, interior contour. Contour 11: start point = (12,6), length = , -> dUJ. PAGE 75 62 intensity = 4, background intensity = 3, exterior contour. Contour 12: start point = (12,10), length 0, intensity = 5, background intensity = 4, exterior contour. Contour 13: start point = (14,1), octal chain codes = 222451766, length = 9, intensity = 3, background intensity = 2, exterior contour. Contour 14: start point = (16,6), octal chain codes = 1753, length = 4, intensity = 2, background intensity = 1, interior contour. PAGE 76 63 Note the i t " contour as interior contour, if it is clockwise. Otherwise the i.th contour is noted as the exterior contour. Input digitized picture g Set start point as p. i=i+l, j=j+2, k=k+2, and L(p)=j . Initialize the rotator bv setting r=0. Find the first p_, satisfyinj the conditions of being a boundary point and having g(p r ) >Â§'(Â£*), if C* exists. Set L(o r )=L. Store r in the i tn contour sequence . Set p=p r and r=fr+4) , Â„. 'mod 8 Label the corresponding adjacent background points p's as k if L(p) PAGE 77 64 ->-.? 13131111 19191221 19131221 19131221 13122120 13122120 13122120 13122120 13122120 11122721 11272627 27262627 27262627 33272627 33322722 33323335 33331111 21151 20212 20202 20 1 20 1 1 1 1 12 1202 20202 21212 22232 22292 22232 23 32 35232 34352 35 91 51515 12121 02020 111 12020 12021 02021 02123 12223 22325 32530 42530 42530 425 4 32525 22323 11114 17151515 21151414 20211411 20212211 20212323 2123 323 2223 323 23242423 24252524 24252524 25302523 31302522 30252311 25242311 23231110 11111010 9 9 9 9 1111 9 411010 1010 1 1010 1 1110 1 1110 1 111010 221110 2311 9 2310 1110 1110 10 9 9 9 9 8 9 8 8 8 8 8 .-K 8 8 8 8 8 8 Figure 3.3. The labeled picture obtained from Figure 1.5 PAGE 78 65 3.3. Inclusion Relation Among Contours As stated in the beginning of the chapter, inclusion relation must be set among contours to extract objects. Inclusion relation can be easily found from the labeled picture L and the original n-level picture g. The labels and intensities of two successive picture points p and p (in the forward raster direction, i.e., p is at the right-hand side of the picture point p ) are required. The information from the label of the picture point p is stored as a state. Three kinds of labels exist. State 1 is that L(p ) PAGE 79 66 L(p)>n 5 is even / no action L(p)>n $ is even / no action Up) PAGE 80 67 contours C, ,. . . ,C . Let C, ..... ,C , , where the indices are 1 ' ' m 1 ' m ' ordered according to the ascending order of the corresponding label values, be the contours satisfying the condition of entering the region at the picture point p. C, ,,..., C , will be put in the array in the order. Applying this inclusion relation finding process to the contours in Figure 1.5, the Hasse graph will turn out to be the one shown in Figure 3.5. 3.4. Object Extraction by Comparison Now we are in the final stage of extracting objects. Let H be the Hasse graph representing the inclusion relation among contours in a picture. Let the area of the region enclosed by a contour C be denoted as A (which can be found by the method presented in Section 5.1.1.). A threshold cu is set such that if A < a-. , the node corresponding to C is deleted from Hj It is obvious that if a contour C satisfies the above condition, all its descendants will satisfy the condition and will thus be deleted. A reasonable value for a, would be 9, because it is usually impossible to filter out the noise disturbing the shape of a contour if the area enclosed by the contour is less than 9. Let H' be the subgraph of H obtained by this deletion process. The H' obtained from the Hasse graph H shown in Figure 3.5 is shown in Figure 3.6. PAGE 81 68 1 Figure 3.5. The Hasse graph representing the inclusion relation among contours in Figure 1.5. PAGE 82 69 Figure 3.6. The Hasse graph obtained through the deletion of small contours. A dissimilarity measurement between two contours C, and C is defined as dc Ci ,c 2 ) . Im^-m^Hm^-m^Mh^-mÂ™! where M^ 1 -' , M^ 1 -' and M*1 -' are the number of critical points, c ' p v the number of peak points and the number of valley points on the contour C, i = l, 2. A threshold a 2 is set to extract objects from H' . If a node corresponding to a contour C is the only son of a node corresponding to a contour C* , and if D(C,C*) < a^ , the node corresponding to the contour C is deleted from H' . First the levels of the nodes in H* are assigned. The level of the root in H' is assigned as 1 and the levels of PAGE 83 70 all sons or nodes of level k are assigned as k+1. The deletion procedure is then applied to H' from the nodes with largest level assignment to the nodes with level 1. Let the resulting graph be denoted as H". Every node in H" can possibly correspond to the contour of an object. Figure 3.7 shows H" obtained from Figure 3.6. Figure 3.7. The Hasse graph obtained through the deletion of similar contours. Contours 7 and 10 are not under consideration because they are interior contours. Contour 1, which touches the picture frame, is also not under consideration. Hence only one object, which is enclosed by contour 8, is extracted. PAGE 84 CHAPTER IV GRAPH THEORY APPROACH TO PICTURE PROCESSING In this chapter we are proposing a method to extract objects in a multi-level picture by the clustering method. This approach can detect the gestalt clusters, which are objects, in the picture and can presumably give the "skeletons" of objects. We first transfer the n-level picture into a weighted graph G and then f ind an MST (Minimal Spanning Tree) of every isolated weighted graph G. of G. Based on the statistics of an MST. we can cluster an MST. Every cluster is an object in the picture. Some major paths of an MST restricted to a cluster form a "skeleton" of an object. Seme graph theory backgrounds and properties of the MST will be discussed in the following section. (29) 4.1. Some Graph Theory^ Backgrounds and the Properties of an MST An undirected finite graph G = {V,E,F} consists of a set V of m vertices, where V = {v, ,...,v }, a set E of k ' 1 ' ' m ' edges, where E = {e, , . . . ,e, } , and a function F, a mapping from E into V and V, the set of all unordered pairs of members of V. Figure 4.1 shows an example of an undirected 71 PAGE 85 72 Figure 4.1. An example of an undirected finite graph graph. In Figure 4.1, V = {v-^...^} and E = {e 1 ,...,e g } If e. is in E, then F(e.) = (v. Â§v. ), for some vert V x 2 ices 3 ' v 3 v. and v. in V, such as F(e.) = (v 9 Â§v 7 ) in Figure 4.1. An edge e. is incident with vertices v. and v. , if Ffe-1 3 i x i 2 ' j J = Cvqv). For example, in Figure 4.1, e, is incident x l x 2 i with v ? and v . If F(e.) = (v. Â§v. ), then e. is a loop, ^ j j i 1 i 2 j such as e g in Figure 4.1. The number n(v.) of edges, which are incident with a vertex v. , is called the degree of the vertex v.. For example, n(v ? ) = 4 in Figure 4.1. v. and i 1 v. are adjacent vertices, if there exists an edge e. such 2 3 that F(e.) = (v. %v. ). For example, v. and v. are adjacent J 12 L Â£ vertices in Figure 4.1. Let e. and e. be two distinct J l J 2 edges. If F(e. ) = (v Â§v. ), and if F(e. ) = (v. Â§v. ), J l X l 1 2 ^2 x 2 x 3 then e. and e. are adjacent edges. Furthermore, if J l J 2 Vj = v, , then e. and e. are parallel edges. For example, 3 1 J l J 2 PAGE 86 73 in Figure 4.1, e, and e, are adjacent edges, and e, and e ? are parallel edges. A simple graph is a graph having no loop and no pair of parallel edges. The graph, shown in Figure 4.1, is not a simple graph, because e., and e~ are parallel edges, and eÂ„ is a loop. Figure 4.2 shows an example of a simple graph. v, -o < v 3 v 5 Figure 4.2. An example of a simple graph, A graph G' = {V'jE'jF'} is a subgraph of a graph G = {V,E,F} 1. if V is a subset of V and E' is a subset of E, 2. if for every e. in E', F'(e.) = F(e.)> and 3. if for every e. in E', F(e.) = (v. $v. ), v. and v. j j i 1 i 2 x 1 i 2 are in V . A finite sequence of edges, e. ,...,e. , is an edge J l J t progression (or edge sequence) of length t if there is a sequence of vertices, v. ,v. ,...,v. , such that for each x x l x t y = l,...,t, F(e. ) = (v. Â§v. ). If v. f v. , the edge J y x y-l x y 1 t progression is open (or non-cyclic), such as e,, e., e,, e in Figure 4.1. If v. = v. , the edge progression is 1 X t PAGE 87 74 closed (or cyclic), such as e.,, e, , e ? , e, in Figure 4.1. An edge progression is said to be from v. to v. ; v. is the initial vertex and vis the terminal vertex of the t progression. For y = l,...,t-l, v. is an intermediate V vertex of the progression. A chain progression (or noncyclic path) is an open edge progression in which no edge is repeated in the sequence, such as e,, e~, e, in Figure 4.1. A circuit progression (or cyclic path) is a closed edge progression in which no edge is repeated in the sequence, such as e.,, e., e,, e fi , e_, e q in Figure 4.1. A simple chain progression (or simple path or arc) is a chain progression in which no vertex is repeated in the vertex sequence, such as e.-, e ? in Figure 4.1. A simple circuit progression (or circuit) is a circuit progression in which v. = v. but there is no other duplication of any vertex 1 X t in the vertex sequence, such as e ., , e_, e q in Figure 4.1. Let v. and v. be two vertices of a graph G, v. and 1 t 1 v. are connected vertices if v. = v. or if there exists X t x X t an edge progression, e. ,...,e. with vertex sequence v. , J l J t x v. ,...,v. . The existence of an edge progression from X l X t v. to v. implies the existence of an arc from vto v. , X \ x V so a pair of distinct vertices is connected if and only if there is an arc joining them. G is a connected graph if for any vertices v. and v. in V, v. and v. are connected. 7 l j ' l 3 A mxm matrix A = (a) can be defined such that 1 j (1 if i=j or if v. and v. are adjacent vertices otherwise. PAGE 88 75 For example, the A matrix for the graph shown in Figure 4.2 is A = fmool 11000 10100 00011 00011 For some integer S such that A s = A S+ , the matrix A s = (s) (s) _ (a| J ) is called the connection matrix in that a^ J = 1 if and only if the vertices v. and v. are connected. For example, the connection matrix of the graph shown in Figure 4.2 is PAGE 89 76 of a connected subgraph G' of G, then G. = G'; that is, G. is a maximal connected subgraph of G and is called an isolated component of G. Definition 4.1 . --A tree is a connected graph having no circuit. A circuit-free graph having q connected components is a forest of q trees. If T = {V,E,F} is a tree and e is an edge of T, then the subgraph G = {V,E-(e), Fp_f >} of T is disconnected, where F, -, implies the function F restricted on the domain E-Cej E-(e). Hence no subgraph derived from a tree, which has all the vertices and lesser number of edges, is connected. Thus a tree is a minimal connected graph. Definition 4.2 . --Let G = {V,E,F} be a connected graph, and let v. and vbe two distinct vertices in V. The disi 3 tance d(v.,v.) between v. and v. is defined as the minimum i' j J i j length of the arcs from v. to v.. If v= v., dfv.,v.) is defined equal to 0. The distance function defined above satisfies the metric axioms: 1. d(v i ,v i ] = 0, 2. d(v i ,v.) = d(v. ,v i ), and 3. d(v i ,v t ) <_ dO-^v.) + d(v.,v t ) V v i ,v.,v t e V. Definition 4. 3 . --Let T = {V,E,F} be a tree and v. is a vertex in T. If n(v-) = 1, vertex \. is termed as a leaf of the tree T. An arc a(v.,v.) from v. to v. is called a i r i J diametral path when its length I is maximal among the PAGE 90 77 distances between any two vertices; I is the diameter of the tree T. A vertex c in V is a center of T if r (c) = min {r (v. ) } = r n v Â± eV x U where r(v.) is defined as max{d (v. , v . ) } . rÂ„ is called the i VjeV i 3 radius of T. Let v. be a leaf of T. The longest arc from l to v. is called a major path from v.. l J l The following theorem reveals the properties of the centers of a tree. Theorem 4.1. --Let T be a tree of diameter I and afv. , V v. ) be a diametral path, having the corresponding sequence x l of vertices v. ,v. ,...,v. . When I is even T has a single 1 1 I center c = vand has a radius r n = 1/2. All major 1 (Â£/2) U paths go through c. When I is odd T has two centers, c, = v. and c = v. , and has a radius r~ = 1 1 (i-l)/2 L 1 (Â£+l)/2 U (Â£+l)/2. All major paths pass through both centers. Definition 4. 4 . --Let G = {V,E,F} be a connected graph, and T = {VÂ„,E T ,F T } be a tree and a subgraph of G. If VÂ„ = V, then T spans G, T is termed a spanning tree of G. Definition 4. 5 . --A weighted graph G = {V,E,F,W} is a graph {V,E,F} with the assignment of a weight to each edge in E. W is the weight function which maps E into real numbers, that is, the weight of an edge e. is W(e.). The weight of an edge progression e. ,...,e. is defined equal 3 1 J t to W(e. )+...+W(e). The weight of G is defined equal to J l J t PAGE 91 78 the sum of the weights of all edges in G. Let T = {V ,E F T ,1\ T T }, where U r T is the restriction of W on EÂ™, be a spanning tree of G, which is also connected. T is said to be a Minimal Spanning Tree (MST) of G if the weight of T is minimal among all spanning trees of G. Figure 4.3 shows an example of a weighted graph G. Figure 4.4 shows the corresponding MST of G. Definition 4.6 . --Let (V-^V^ be a partition of the vertex set V of a weighted graph G = {V,E,F,W}. The weight W( V 1' V 2^ across the partition is defined as the smallest weight among all edges which join one vertex in V and the other in V-j. The set of edges ECV^V^ which span a partition will be referred to as the cut set of {V V ? } and a link is any edge in E(V lf V 2 ) whose weight is equal to the weight WCV-^V^. The set of all links in ECV^V^ is called link set LCV-^V^ of {V^V,}. The following theorem allows us to find an MST of a weighted graph from the link sets. Theorem 4. 2 . --An MST contains at least one edge from the link set LfV^V^ of every partition {V^V }. Every edge of an MST is a link of some partition of V. Theorem 4.3 reveals that the appropriate clusters can be found as subtrees of any MST. Theorem 4. 3 . --If Vj is a non-empty subset of V with the property W(V. ,V. ) < W(V V-V.) for all partitions 12 PAGE 92 79 Figure 4.3. An example of a weighted graph, Figure 4.4. An MST of the weighted graoh shown in Figure 4.3. PAGE 93 80 {V, ,V. } of V. , then the restriction of any MST to V. i x i 2 1 x forms a subtree of the MST. 4.1.1. Finding an MST of a Weighted Graph From Theorem 4.2, it is clear that an MST can be found from a connected graph by building up a subtree T*, to which a link of {V T ,,V G -V T ,} is added. Let m be the total number of vertices of G. We can set V Q = {v , ...,v }. Three arrays are required to achieve the purpose of finding an MST from a weighted graph G. L J 1. Vertex array X: It indicates which vertices are in VÂ„ , , that is, if X(i) = 1, then v Â± e V while if X(i) = 0, then v. t VÂ„, . 2. Reference array R: If X(i) = 1, R(i) specifies the index of the vertex v. in V T , , if v, is adjacent to v. in T*. If X(i) = 0, R(i) specifies the index of vertex v. in V T , , such that W(v,,v ) = min {W(v. ,vj } , where W(v. ,v ) is the x J v q e V-pt 1 q i q weight of the edge joining vertices v. and v . 1 q 3. Weight array Z: Z(i) is equal to the weight of the edge incident with v. and v (i) , that is, Z(i) = WCv if v R(i) ) The link of {V T ,,V G -V T ,} can be found from the vertex array X and the weight array Z by noting that the edge connecting a vertex v Â± and v R ,.. , where v i is not in the subtree I" of MST (i.e., X(i) = 0), has a weight Z(i) equal to min ^V(j)|X(j) = 0}. The edge is then added to T' by j={l, . . . ,m} setting X(R(i)) = 1. Figure 4.5 shows the flow diagram for finding an MST of a connected weighted graph. PAGE 94 81 Input a weighted graph Initialization : X(l) = 1, X(i) = 0, i = 2,...,m. R(i) = 1 if Vj is adjacent to V!, = 0, otherwise. Z(i) = W(vi,v R m) if R(i) f 6/ = w n if R(i) = 0, where w^ > all weights in the graph. k = 1. Find j such that Z(j) = iuin{Z(i) |X(i) = 0}. Set X(j) = 1. If X(i) = and Z(i) > W(vi,Vj] set R(i) = j and Z(i) = (vi,V-;) k = k+1. Figure 4.5. Flow diagram for finding an MST of a weighted graph. PAGE 95 82 4.1.2. Finding Major Paths As is easily seen, the most straightforward clustering can be done by setting a threshold w, such that if an edge in an MST is of weight greater than w, the edge is deleted from the MST. The deletion of heavily weighted edges from an MST will yield a forest of subtrees. Every subtree corresponds to a cluster of the connected weighted graph G. It should be pointed out that an MST of a connected weighted graph is not unique. From Theorem 4.3, it is known that this non-uniqueness of the MSTs does not restrict the formation of the same clusters from different MSTs of a graph. In many cases, more sophisticated consideration should be taken to cluster the graph. Statistics of the weights of edges on major paths should be taken into consideration. For a tree, usually there exist many major paths. A systematic method should be set to find all major paths. Welldistinguished major paths are of interest. Two major paths can be considered as well distinguished if they have only a small portion of paths in common. Also, the branches from a diametral 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 following are useful definitions. Definition 4. 7 . --Let T, be a tree of diameter I, which is even. There is only one center c in T, . The arc a(v,c) from a leaf v to the center c is called a radial path. If d(v,c) = 1/2, a(v,c) is called a maximal radial path. Let PAGE 96 83 T 2 be a tree of diameter I, which is odd. There are two centers, c, and c 2 , in T . For any leaf v, if d(v,c) > d(v,c.)> where i f j, the arc from v to c., a(v,c)> is called a radial path. If d(v,c.) = (Â£+l)/2, a(v,c.) is called a maximal radial path. Let a(v.,v.) and afv^.v.) r ^ 1 >^ Â». |.Â» y be two arcs, where both v. and v are leaves, and v. be the only common vertex of the two arcs. If d(v.,v.) <_ d(v ,v.)> then a(v.,v.) is called a branch and any arc containing a(v ,v.) is called a stem. v. is called a branching vertex. A relation R is defined on all radial paths in a tree T. Let s, and s~ be two radial paths in T. If s, and s, contain a common subsequence of more than one vertex, then s,Rs . It is obvious that R is an equivalence relation. Let T be a tree having only one center c and v. ,...,v. be X l X t all vertices adjacent to c. Every radial path has a subsequence of vertices (v. ,c) for some vertex v. ,j=l,...,t. Hence there are t equivalence classes of radial paths induced bv the equivalent relation R. Let S. denote all radial paths having a subsequence of vertices (vj , c). Any j radial path of S. can be combined with any maximal-radial path of S . , to form a major path, where j f j ' e{l , . . . , t} . Let s^ J = fv^ , v^ ,...,v. , c) be a maximal -radial path u *Â• u, ' u ? ' ' i, J r in S.,. (v^,v^ t ...,v. , c ,v. ,... ,v( ] ,'\v(J'h then 3' v -u 1 'u ? ' ' i . ' ' i Â» * u' 2 * U'j J forms a major path. If T is a tree having two centers, c, and c 7 , every radial path S in T either contains the subsequence of vertices (c,,c ? ), or contains the subsequence PAGE 97 84 of vertices (c^c-^. Hence there are two equivalence classes induced bv the relation R. Let S = fv c . , c . 1 be a u u i x J radial path and S , = (v , ,...,v , ,c,c) be a 1 U C*-l)/2 J X maximal radial path. S u and S , can be combined into a major path (v , . . . , c . ,c . , v , ,...,v , ). 1 3 U CA-l)/2 1 Because of less storage required and easy combination into major paths, the storage structure of radial paths would be that only one radial path, which has maximum length among all radial paths in the same equivalence class, of every equivalence class is stored in the full sequence. Any other radial paths are stored as branches. Every leaf in a tree initializes a sequence. Trim all the leaves from the tree. If the adjacent vertex v. of a leaf v. does not turn out to be a leaf after v. is trimmed, the corresponding sequence will represent a branch. The sequence having v^ as a leaf will be the corresponding stem. The procedure is iterated until either there are only two vertices left or there is only one vertex left. The vertices finally left are the centers of the tree. Figure 4.6 shows the flow diagram for finding radial paths and centers in a tree. The tree shown in Figure 4.4 has only one center v.. The diameter of the tree is 4. The set of radial paths, having full sequences, is { (Vj , v 3 , v 4 ) , (v ? ,v 5 , v 4 ) } . 6 ,v 5 ) is a branch depending on the maximal radial path (v 7 ,v,,v.). 2 ,v 4 ) is a branch having (v^v^v^ and (v ? ,v r,v.) as PAGE 98 85 Input a nontrivial tree For every leaf in the tree, initialize a stack which has the leaf as the only clement, Let K be the total number of leaves in the tree. PAGE 99 86 Figure 4.6. Continued PAGE 100 87 stems. (v.,,v_,v.) and (v_,v 5 ,v.) can be combined into a diametral path (v, , v., , v. , v~ , v_) . From the branch (v,,v<-) we can find a maximal radial path ( v c> v r> v ,i) which is dependent on the maximal radial path (v.,\' ,v.). (v v v.) and ( v 6Â» v c> v 4) can be combined into a diametral path (v, , v_ , v. ,v. , v,} . ( v 2' v a) ^s independent oÂ£ any other radial paths in the tree, hence we can combine 1. ( v 2 ' v 4^ with ^ v i' v 3Â» v 4^ t0 a ma J or P ath Cv 2 ,v 4 ,v 3 ,v 1 ) , 2. (v 2 ,v 4 ) with ( v 7Â»v 5> v-] to a major path (v_ , v. , v 5 , v_) , and 3. (v 2 >v 4 ) with (v 6 ,v 5 ,v 4 ) to a major path (v 2 ,v 4 , v 5 ,v 6 ) . 4.2. The Representation of a Digitized Picture by a Weighted Graph A digitized picture g can be represented by a weighted graph G in the following manner. Every picture point is considered as a vertex. Several possible methods are used to define the connection of vertices and the weight of the corresponding edges. 1. Method 1 is that every picture point p is connected to any of the four-neighboring picture points p' with a weight l/(g(p) + g(p')) except that g(p)+g(p') = 0. 2. Method 2 is that every picture point p is connected to any of the eight-neighboring picture points p' with a weight of d(p,p')/(g(p)+g(p')) except that g(p)+g(p') = 0. 3. Method 3 is that every picture point p is connected to the 16 -neighboring picture points p' with a weight of PAGE 101 d(p,p*)/fg(p) + g(p')) except that g(p)+g(p f ) = 0. Figure 4.7 shows the three methods for connecting nodes. method 1 method 2 method 3 Figure 4.7. Three methods for connecting picture points Because the MST will he invariant under the monotone transformation of the weights, a quantizer can be used to transfer monotonically the weights assigned by the previous paragraph to a subset W of the natural numbers. Let g be a n-level picture. It is easily seen that for method 1, the cardinal number of W is 2n-3; for method 2, the cardinal number of W is 4n-5; and for method 3, the cardinal number of W is 6n-7. The weighted graph G, representing a multi-level picture, is not necessarily a connected graph. Once the weighted graph G of a digitized picture is found, the flow diagram shown in Figure 4.1 can be applied to G to find an MST. The edges in the MST found, having weights equal to PAGE 102 89 the initial weight w are deleted. Hence the result would be a minimal spanning forest. For the same reason of encoding a curve by the chain code (discussed in detail in Chapter II) , chain codes are used to represent edges, instead of using a pair of vertices or assigning a name to an edge. In method 1, from a vertex to another vertex, there are only four possible edges, hence a 2-bit chain code is used to encode edges. In method 2, from a vertex to another vertex, there are only eight possible edges, hence an octal chain code is used to encode the edges. In method 3, there are only 16 possible edges from a vertex to another vertex, hence a hexadecimal chain code is used to encode the edges. PAGE 103 CHAPTER V FEATURE EXTRACTION As stated in Chapter I, feature extraction strongly depends on the type of pictures handled. Here in this chapter, we are discussing only area pictures. There are mainly two types of features: local features and global features. Local features are the ones which are dependent on the individual objects and are independent of any other objects, for example, area, centroid, shape of object, principal axis direction, elongation index, etc. Global features are the ones which describe the interrelationship of objects in the picture, for example, the inclusion relation of objects in the picture, overlapping, touching of objects in the picture, the distribution of objects in the picture, etc. Assume that the objects are described by the boundary contours. The boundary contours are described by the octal chain codes. An object which has one exterior contour C Q and q interior contours C, ,...,C can be deq q scribed by R r U R where R r is the region enclosed L j=l L j C j by the contour C, j = 0,...,q, where is the substraction in the set theory and U is the union. 90 PAGE 104 91 5.1. Some Fundamental Local Features The most important and useful local features are areas centroids, shapes, major principal axis directions and the corresponding elongation index of objects. We are attacking the problems one by one. Remember that the boundarycontours are described by the start points and the associated sequences of octal chain codes 5.1.1. Area Since the contours are encoded into a sequence of straight lines, the area inside the contour is the algebraic sum of areas of the strips between the line segments and the x axis. Figure 5.1 shows the x-y coordinate system and the eight possible line segments with the corresponding octal chain codes. PAGE 105 92 The following is a table of the area a. of the strip r. between a segment, corresponding to the octal chain code s i which starts at (x^y^, and the x axis, and the increment Ax_ L and Ay i in the x and y coordinates, respectively. From Table S.l we can get a. Â± = Ax i (y i + Ay./2). The sign of the area enclosed by a contour is positive if the contour is encoded in the clockwise direction and is negative otherwise. The area of an object which is represented by q q C " ._ R C. 1S then eq . ual to iarea(.R c )| I |area(R )|. The ratio |area(R )j/|area(R r ) | is a normalized measurement of the size of the hole enclosed by the interior contour C. , j=l,.. . ,q. Table 5.1 Table of area, x-increment and y-increment Octal chain code Area x-increment y-increment S i a i A *i Ay i 1 2 1 yÂ± -i o (y^l/2) -i i 3 ' Y Â± +l/2 1 i 4 110 5 7-1/2 1 -i 6 -1 7 -C/i-1/2) -1 -1 PAGE 106 93 5.1.2. Centroid The centroid of a region enclosed by a contour C can be found by the mean value technique. The y coordinate of m the centroid is v = ( I a. y-)/A, where a. is the area of -i l J i J l i= 1 the strip between the corresponding segment and the x axis, y. is the mean y coordinate of the strip, m is the length m of the sequence representing C and A = I ais the area 1-1 of the region. Table 5.2 gives the a. y. values for the & Â° i J i eight possible segments. A formula can be derived, ay. = Ax^y? + Ay^'i + A yi /3))/2. Table 5.2 Table of moment Octal chain Moment of a^ Moment of a^t code about the x-axis about the y-axis s a y Â• a ! x ! 1 1 ; 1 1 1 -v?/2 1 -(7+ Y i + l/3)/2 Cx? x Â± * l/3)/2 2 x?/2 3 (y+ Y Â± + l/3)/2 [x? + x i + l/3)/2 4 y^/2 5 (yyi + l/3)/2 -(x? + x Â± + l/3)/2 6 -x?/2 7 -CyYi + l/3)/2 -(x? x i + l/3)/2 PAGE 107 94 m wh The x coordinate of the centroid is x = E ax!/A' ere a! is the area of the strip between the corresponding segment and the y axis, x.' is the mean x coordinate of the m 1 strio and A' = E a.' is the area of the region enclosed bv the contour C which has opposite sign from that of A. Bysymmetry a formula can be derived, a! x! = Ay.fx? + Ax. (x. + Ax./3))/2. 11 ' l v l i^i l ' Â•* J ' For an object which has holes, the net centroid is x = C|A C |x c E |A |x )/A j = l j j and y = (|A C | y Q E | A | y )/A, j=l j j where AÂ„ is the area, (xÂ„ , y_. ) is the centroid of the j j j region enclosed by the contour C., j=0,l,...,q and A is the net area of the objects. 5.1.3. Shape The shape of an object can be described by the shape of the exterior boundary contour of the object and the shapes of interior boundary contours of the object if they exist. A method to describe the shape of a contour is then required. Let r.s consider a contour in the real plane first. Second order differentiation can very well describe the shape of the contour because it can partition the contour into concave and convex portions. In the digitized picture, the PAGE 108 95 octal chain code describes the first order differentiation of the curve. The difference c 'c' of two successive octal chain codes c and c 1 , which is defined in Section 3.2, i.e., c 1 c' =c-c'+8p, where p is an integer such that the absolute value of the difference is less than or equal to 4, can describe the second order differentiation. Let Cq ,c, , . . . , c , be the sequence of octal chain codes representing the contour C and X = xÂ„,...,x -, , where x= r s ' ' m-1' i c-., c , i = 0, l,...,m-l and c = c n be the correspondl+l i ' ' ' ' m U r ing difference sequence. Because of the quantization error and noise in the picture, high frequency noise appears in f 321 the X sequence. A digital filter^ ' is then required to filter out this high frequency noise. A Manning window function W, which is defined as w k = 1 cos(2^k/K) , < k < K, is used here to convolve the input sequence X to filter out the high frequency noise. The output Y of the digital filter W is then K ^ ' ^mod m j = J * J -^mod m i = 0,...,m-l. The output sequence Y is called the smoothed difference sequence. The change of signs between successive y's in the Y sequence is important to note. If the value of y. is positive, the i * point in the contour is in the concave portion PAGE 109 96 of the contour. If the value of y. is negative, the i point in the contour is in the convex portion of the contour. The Y sequence is periodic because the last element in Y is one ahead of the first element in Y. We can thus group the Y sequence into a sequence of t subsequences YÂ„,...,Y , such that Y = YÂ„ ... Y. _, , The signs of all y. values in a subsequence are the same and the signs of y. values in a subsequence are different from those of y. values in an adjacent subsequence. Let N = (n n ,...,n -, ) be the sequence such that the n. point in the contour is the start point of the Y. portion of the contour, i = 0,...,t-l. We call these points the critical points. It is noted that Y i " () 'n i ' 1 '(V 1 Vod m -"" y Cn (i + i;| 1} ), v mod e mod m i = 0,... ,t-l. A sign specification a should be accompanied with the N sequence such that a = sgn(y ). From the value of a we n o th can tell that the portion of the contour from the n. point to the n^. ... point, i = 0,...,t-l, is a concave por(i+l) ft > > > f v Â•'mod t tion or a convex portion. If a = +1 and if i is even (odd), it implies that sgn(y ) = +1 (-1). Hence the portion of the contour from the n. point to the n,... point v J mod t is a concave (convex) one. If a = -1, the situation is just the opposite to that of a = +1. PAGE 110 97 The window size K of the Harming filter is chosen such that if K 1 ^ K is used as the window size, then for every contour C in the picture, any two successive critical points are within 6 length along the contour 5, where 4 < 6 < 9 is a threshold, and if K' < K is used as the window size, then there exists a contour C having the N sequence (if t f 1) such that (n. n. ,) , < 5 for some 1 < i < t, where m i l-l mod m Â— Â— Â— ' is the length of the contour C. Let the i contour point have a local maximum |y. | value and |y|>a. The i contour point is said to be an extreme point. If y. > 0, then the i contour point is on a concave portion. Hence the i contour point is also called a valley point. If y. < 0, the i contour point is also called a peak point. If there is one or less extreme point between two sue*t* V* + Vi cessive critical points, the n. -, and the n. points, the curvature of the portion of the contour between the successive critical points can well be defined as sgnfy )((n. n,] , /dfn. , n. , ) -1) , where 5 w n. -, l i-l^mod m r i ' i-l y '' i-l th d(n. n. -, ) is the Euclidean distance between n. -, and *Â• l Â±-l J i-l the npoints. It should be pointed out that the Euclidean distance between the starting point and the end point of a curve can be found without the specification of the locations of the starting point and the end point, if the octal chain code sequence of the curve is known. Let c, ... c , be the n 1 m PAGE 111 98 octal chain code sequence of a curve C. We can calculate the x(y) -projection of the vector from the start point of C to the end point of C by summing up the x(y) increments contributed by the octal chain codes as defined in Table m' m' 5.1, i.e., AX = E Ax-, AY = E Ay.. The Euclidean disi=l x i=l x tance between the start point and the end point of the curve 7 m ' m' ~ C is then AX Z + AY Z = ( S Ax.) l + ( E Ay . ) . i=l x i=l x 5.1.4. Principal Axis Direction and Elongation Index If a contour C has no very deep concavity, we can assume that the region enclosed is an elliptical one. Generally there are two principal axes of the region through the centroid. In order to find the principal axes, we have to find 1,1, the moments of inertia of the region about the x' y ' b x and y axes, respectively, and P , the product of inertia of the region. Here the x-y coordinate system is the translation of the old x-y coordinate system obtained by setting the centroid of the region as the new origin. Note that both I and I are positive and are given by n Â— jY n Â— j Â— j I = E av. and I = E a! x! , where ay. is the i = l 7 i = l Â— j moment of inertia of aabout the y-axis and a! x! is the moment of inertia of a. about the x-axis. P = sgn(A) l xy b ^ J n ' E ax.y., where a. x.y. is the product of inertia of a-. . , l l'l* ii^i v l 1 = 1 T oTable 5.3 shows the av. , a! x! and ax.y. values 3-1' 11 11 7 1 Formulas for those calculations can be derived: PAGE 112 99 3 o X PAGE 113 100 a i y i 2 Ax i (4y i 3 + A yi (6 yi 2 + UAy-^y. + 1))/12, a!x! 2 = Ay. (4x. 3 + AX-C6X. + 4(Ax i )x i + 1))/12, and a i X i y i = Ax i(Xi ( 2x 4 Ax i ))/ 4 + (Ax i Ay i JX i y i /2 ^j(Ay i )y i /3 ^AxJx^G + 1/8. Let 9 be the angle oÂ£ rotation from the x,y axes to the two principal axes x' and y', respectively, as shown in Figure 5.2. f 33") A formula 1 J has been derived such that tan 2 8 = -2P /{I -I ") . Hence we can find this angle of rotation xy v y x6 very easily. A major principal axis is defined as the axis about which the moment of inertia is minimum. Hence, in order to find the major principal axis, we have to calculate I., and I ,,and to find out which has less value. The formula for I , and I , are: x y I x , = Cl x + I y )/2 + ((I x -l y )/2)* cos20 P xy sin26 and I , = (I x +I y )/2 C(I x -I y )/2)* cos29 + ? xy sin 29. The major principal direction can be represented by 8 or 0+^/2 depending on x' or y' being the major principal axis, respectively. The ratio of the moment of inertia about the major principal axis to the moment of inertia about the minor principal axis is called the elongation index. The elongation index ranges from to i. It indicates how PAGE 114 101 -** x (x,y) Figure 5.2. Principal axis direction, PAGE 115 102 sharp the region is. The less the elongation index, the sharper the region is. For example, if the elongation index is 1, the region is a circle, x^hile if the elongation index is almost 0, the region is almost approaching a line. 5.2. Global Features The global features are the ones which describe the interrelationship of objects in the picture. Since sometimes an object may contain several holes, we can consider the regions enclosed by the boundary contours as individual objects and find the global features among the regions to describe the object itself. To find the global features among objects, only the regions enclosed by the exterior boundary contours need to be considered. The following are several nontrivial global features. 5.2.1. Inclusion Relation Among Objects Inclusion relations among objects can be derived directly from the inclusion relation among contours. Let H be the Hasse graph representing the inclusion relation among contours and N' be the set of all nodes in H which correspond to exterior contours of objects in the picture. Let H' be the graph which represents the transitive reduction of the reachability among nodes in N' revealed by the graph H. H" is then a Hasse graph which represents the inclusion relation among objects. Since there is a one-to-one PAGE 116 103 correspondence between exterior contours of objects and objects, the nodes in N' are in one-to-one correspondence with the objects. 5.2.2. Distribution of Objects in the Picture (With Consideration to ~tne Distances Between Objects) The distance between two objects and 0' is defined as the Euclidean distance between the corresponding centroids o and o'. One method to describe the distribution of objects in the picture is using the Minimal Spanning Tree (MST) technique, which is discussed in detail in Chapter IV. A K-nearest-neighbor graph G = (N, E, W) can be set by the following rules: 1. For any object in the picture, there exists one and only one node n in N. 2. There exists an edge connecting nodes n and n' if either 0' is a K-nearest-neighbor of object or is a K-nearestneighbor of object 0', where objects and 0' are corresponding to nodes n and n' , respectively. 3. Let e be an edge connecting nodes n and n', the weight W(e) assigned to edge e is defined as the distance between objects and 0', where and 0' are the objects corresponding to nodes n and n 1 , respectively. Let T be an MST of the weighted graph G. T can be found by the method presented by the flow diagram 4.1. T can reveal the information describing gestalt clusters of objects in the picture. The philosophy is the same as PAGE 117 104 stated in Chapter IV, except we consider every object in the picture as an individual vertex. The definition of the distance between objects stated in the last paragraph is a good one if the object size and shape are quite uniform through the picture. If the objects' sizes and shapes are not uniform through the picture, the distance between two objects and 0' may better be defined as min{dCp,p') [peO and p'eO 1 }. 5.2.3. Distribution of Objects in the Picture (With Consideration to Both the Distances Between Objects and Relative Principal Axes' Directions .Among Objects) If all objects in the picture are quite elliptical in shape, to our visual systems the reasonable gestalt clusters of objects should take not only the distance between objects into consideration, but also the angle between the principal axes of objects into consideration. One practical example is the cluster of cells in the epidermis, which will be discussed in detail in Chapter VI. The method used to describe the distribution of objects by considering not only the distance between objects, but also the relative principal axes' directions among objects is the same as the one presented in the last section, except in setting the K-nearest-neighbor weighted graph, the weights assignment should be modified (i.e., rule 3 should be changed). Let e be an edge connecting nodes n and n', the weight W(e) assigned to edge e is defined as d(0,0'). + PAGE 118 105 Y9(0,0'), where d(0,0') is the distance between objects and 0', 3(0,0') is the angle between the major principal axes of objects and 0' and and 0' are the objects corresponding to nodes n and n', respectively. The weighted graph in the last section is a special case of the weighted graph in this section under the assumption that y = 0. PAGE 119 CHAPTER VI EXPERIMENTS AND CONCLUSIONS All techniques stated in the previous chapters have been implemented by FORTRAN programs. Biomedical images are chosen as the experiment data. It should be pointed out that the techniques are not restricted to biomedical images. The reason that the author chose this special type of pictures is due to the great need for the handling of biomedical images. Chromosome pictures, skin cell pictures and blood cell pictures are the main pictures we work with in the experiments. 6.1. Experiments With Chromosome Pictures Figure 6.1 is an 8-level digitized chromosome pic(2 3^ ture. J Figure 6.2 is the enhanced picture and Figure 6.3 is the boundary picture obtained by the programs based on the gradient technique. There is only one boundary in the picture with a length equal to 207. The corresponding sequence of octal chain codes is 4443312011010101010 0011010121210121345454555545 5 4545 445454456544332111010 0000101110 001012110112223 32 34 33 4 334 33321000 70 7 7 70 77 76 70 700132 3332 33334 333 32100 70 7 77 106 PAGE 120 107 00 I) 00 00 000 0900 0000000 0000000012221100000'"! 00 n 1 3 6 6 6 5 3 I 00000 000000000 00000012666 6 Â£6410000 000 0000900000000 0000156666 6 66 4 10O0CO0 0000000000000 00 00000 1366666666 3 10000 0000000 0000 000 12 4 6 6666666420000 000 00000000 00 0000 12346 6 66666 6 3 111100 00 00 0000000') 00000 135 6 6666666432113321000 0000000000)01112366666 6 664221112566 3 100 00 00000012345:. 666666643111112 4 666 6 310 0000000001246666 66666431111112466666520 00000024 6 66666 6 6 5452111112 4 5666666420 00 000014666666653211111335656656666 3 10 000000136666666421111134666666 6 66665200 000001366666 6 43111123466666666666551000 00 0015 6 6 66642111135666'6666655543210000 000002665666211123666666653321111000000 000014666 663111346666665311100000000000 00001566564 111366666664 20 000 00000000 0001256654211356666653200000000000000 000146664322366666422000000000000000 00 0003566 53335666643210000 0000000 000000 01466644666666421000000000000000000000 0035666666666652 000000000 600 00000000 00256666666664200000 OOOOCOOOOOOOOOO 9 00 02 666666666320000000000000000000000000 2S6666666310000000000000000000 00000 00366666664200000000000000 00 00 013666 6 6664100000000000 000000000000 00 003566666662000000000000000000000000000 002355666664200000000000000000000000000 002443 4666664100 0000000000000 0000000 013454356666631000000000000000 OOQOOOO 013553 2 35666664210000000000000000000900 00346 4 32246666653100000 000000000000000 00256632 12466666653100000000000 00009 002566531 136666666652100000000000000 900 002466662123 6 66666665321000000000000 9 00 001366665211256666666642000000000900 9 00 000366665 4 21124666666663100000000 9 00900 00025666 6 63111235666666 4 2000000000009 000136666652111125666664100000000000 00 0000256660 553111115 4 664200000 000 00 9 00 000002666666642111112211000000000000000 00000136666666521000000 9 000000 00 9 9 00 000000135666666631000000000000000000 00 000 0000236666666310000 00000000009 900 000 0000013666666 4 10000 9000000 il 00 900 900 00000 00001246666310 00900000000000 9 00 00000 00 000125 3 320 00 0000 90 000:00 00000 00000000000000 0900000000000 9 00 Figure 6.1. An 8 -level picture of a human chromosome PAGE 121 108 PAGE 122 109 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *** * * * * * * * * * * * * * * * * * * * * * * * * * ****** * * * * * * * * * * * * * * * *Â•***. * * * * * * * * * * * * * * *. * * * * * * * * * * * *" / * * . * * * * * * * * * * * * * * * * * * * * * . * * * * * * * * * * * * * * * * * * * * Figure 6.3. The boundary picture obtained from Figure 6.2 by'the gradient method. PAGE 123 110 077 76 7 75 766 7666665 7566 756556656556544554445 545545555. The optimal Hanning window filter, used to filter out the noise in the sequence of differences of successive octal chain codes, has a window size equal to 26. The filtered sequence of differences of successive octal chain codes in shown in Figure 6.4. There are six critical points, the 17 th , 45 th , 77 th , 104 th , 122 nd and 140 th points on the contour. The 5 th , 65 th , 113 th and 148 th points on the contour are the peak points. The 33 rd , 97 th and 131 st points in the contour are the valley points. The parameter is +1, which implies that the portions (17, 44), (77, 103) and (122,139) are the concave portions and the portions (45, 76), (104, 121) and (140, 16) are the convex portions. The portion (i, j) denotes the portion of the contour from the i th point to the j point in the contour. Figure 6.5 is the labeled output picture of Figure 6.1 using the contour analysis. Figure 6.6 is the contour picture obtained by the method stated in Chapter III. There is only one object contour in the final picture with a length equal to 201. The corresponding sequence of octal chain codes is 44444 32 2111001101010010110112112 345555 45545454544454 55 564 34322 21101000010010110011011111122 53333434 334 33 2210000 770 7 7 70 76 77012 32 3332 34 3334 322100 70 70 770 76 776 7 6666 766 56 766 566 7565 5665665 5 545 54 445545 45 5 5456. PAGE 124 Ill bo o ^ U ed 13 PI 3 o X) CD +-> m o pj o Â•H P O PI m 0) u pi a> <1) m Â•H Â•ti T3 0J -P o o 6 3 bo >. PAGE 125 112 s S 3 3 8 8 8 3 000000000088 9111111 99 38000000 3 8 9131919191713 9 8 3 3 8 91119 6 6 6191915 9 S 8 S 91319 6 6 6 6 61915 98000 8 S 3 9 1319 6 6 6 6 6 61913 9 8 8 8 3 91115L9 6 6 6 6 6 6191311 3 8 8 0000000 00000 OSS 911131319 6 6 6 6 61919151010 9 9 8 8 8 3 3 8 8 8 913171919 6 6 6 619191513111010131311 9 8 8 00000000088 8 99 911131919 6 6 6 6 1 9191511 1 1 1 10101117 19 1913 9 8 8 0000000 38 9111315171719 6 6 6 6 191915151010101010111519 6 61913 9 8 0000000 8 3 0111519191919 6 619191915 1 31010101010 LO 111519 6 6 6191711 8 8 811151919 6 6 6 6 1919 1715 151 ! 1010101011151719 6 6 6 6191511 8 8 8 91519 6 6 6 6 6191713 L1101010 1010131317191919 6 6 6 6 61913 9 8 8 8 91319 6 6 6 61919151110101010101315191919 6 6 6 5 6 619191311 8 3 8 9131* 6 6 6 6191513101010101113151919 6 6 6 619191919191713 9 8 8. 8 91719 6 6 619151110:0101013171919 6 6 6191919171717151311 9 S 8 8 81119 6 6 6 6191110101011131919 6 6 1919171313 L 1 9 9 9 9 S 3 8 8 91519 6 6 61913101010131519 6 6 6 (,191713 9 9 9 3 S 3 8 8 8 8 8 91719 6 61915101010131919 6 6 619191511 3 8 S 8 8 8 9111710 61915111010131719 6 61919171311 3300000000000000 3 91519 61915131211131919 61919151111 888000000000000000 8 813171919171414131719 6 619151311 9 S 8 8 91519 6191615191919 6 6191511 988 8 00000000000 0000000 8131719 6 61019 6 6 6 6191711 88800000000000000000000 8111719 6 6 6 6 6 6 6191611 880000000 00 0000000000000 81119 6 6 6 6 6 6 6191311 8800000000000000000000000 3111719 6 6 6 6 61913 938000000000000000000000000 8 8151Q 6 6 6 6 6191511 S 8 8 91319 6 6 6 6 61915 9800000000000000090000000000 8 813171919 6 6 6 61911 830000000000000000000 00000 81113171719 6 6 6191511 88000000000000 00000000000 8 8111516161519 6 6 61915 98800000000000000000000000 8 913152115141719 6 6 61915 9 888000000000000000000000 8 915212114141-1719 6 6 6191511 98SOOOOOOO00O0OOOOOOOOO 8 81^157515151^251519 6 6 6191713 9388000000000000000000 811212525152512251519 6 6 619191713 9 8 8 8 8112125252115121215H) 6 6 6 6 619191311 9 3 3 S 8111523 62323251225131919 6 6 6 6 619171311 9800000000000000 S 9152' 6 62521251010111719 6 6 6 6 619191511 SSOOOOOOOOCOOOO 8 81525 6 6 625151110101115 1919 6 6 6 6 61915 930000000000000 8117173 6 6 6231510101011131719 6 6 6 6191511 8 0000000 0000 8 91375 6 6 62521111010101011171919 6 61915 9 30000000000000 3 3112123 6 6 62521151910 101010151519191511 880000000000000 8 81125 6 6 6 6252315111010 9 9 91111 9 9 3 S 9132325 6 6 6 6252111 9 8 8 8 S 8 8 8 8 8 8 01 -,ii'3 6 6 6 6252513 9 8 8 3 8 111525 6 6 6 6 62515 930000000 00000000000 8 3 9132525 6 6 62315 980000000000000000000 8 8 911152523252315 9 8 8 8 3 91113 151311 3 8 000000OO0008S83SS8SOO000 000000000000000 Figure 6.5. The labeled picture obtained from Figure 6.1. PAGE 126 113 l l l l l l 1 1 ll 1 1 1 l 1 1 l 1 11 ill ll l l ill llll ll 1 1 l ll ll 11 ill ll 1 l ll llll l I ill 1 1 11 ll 11l i ll lllll 1 1 l ill 1 ill ll 111 1 1 1 1 ill I'll l 1 11 ll 1 l 1 l 1 l 1 l 1 l 1 1 1 1 1 1 1 1 1 l 1. l l 11 l 1 1 1 11 ill ll ill 1 1 1 11 11 ill l 111 l I ill 1 II ll l ll lllll l l 1 l II ll 1 l 1 1 l ill ill Figure 6.6. The boundary picture obtained from Figure 6.1 by. the contour analysis. PAGE 127 114 PAGE 128 115 The optimal Harming window filter, used to filter out the noise in the sequence of differences of successive octal chain codes, has a window size equal to 28. The filtered sequence of differences of the successive octal chain codes is shown in Figure 6.7. There are six critical points, the nr .th rfrth -74th no th ,,,th . ,__rd ., 19 ,45 ,74 ,98 , 116 and 133 points in the contour. The 6 , 62 , 107 and 141 points in the contour are the peak points. The 31 , 90 and 125 points in the contour are the valley points. The a parameter is +1, which implies that the portions (19, 44), (74, 97) and (116, 132) are the concave portions and the portions (45, 73), (98, 115) and (133, 18) are the convex portions. It should be pointed out that the dissimilarity of the contours obtained by the gradient method and the contour analysis is equal to as is expected. Method 2 as stated in Section 4.2 is used to find the weighted graph of Figure 6.1. An MST has been found as shown in Figure 6.8. The radius of the MST is r n = 51. There are two centers (23, 7) and (24, 7) in the MST. a is set as 1/4, hence any branch of length less than [otr n ] = 12 is removed. Only four radial paths (including two maximal radial paths) are remaining, which are shown in Figure 6.9. Radial path 1: start point = (6, 34), chain code sequence = 00066006002020202020020020200 00200202020220222222, PAGE 129 116 Figure 6.8. An MST obtained from Figure 6.1 PAGE 130 117 Figure 6;9. The four radial paths in Figure 6.8 obtained by deleting small branches. PAGE 131 118 length = 51 (i.e., it is a maximal radial path). Radial path 2: start point = (15,39), chain code sequence = 7006660002020200020002002002 00202002200202000 200066, length = 51 (i.e., it is a maximal radial path). Radial path 3: start point = (46,25), chain code sequence = 7766606006060060060606060660 66666600066, length = 39. Radial path 4: start point (50, 20) , chain code sequence = 7006606006060060660606600660 6666666666666066022, length = 47. Radial path 3 is dependent on maximal radial path 2, while radial path 4 is dependent on maximal radial path 1. (22, 7) and (25, 10) are the only two branching vertices in the MST with the removal of small branches. The octal chain code sequence of the path from (22, 7) to (25, 10) is 222444 which is of length 6. Since this length is much smaller than the length of any radial path, the skeleton of the picture can be considered as consisting of four arms. Each arm is a subpath of a relative diametral path from a leaf to the nearest branching vertex. The branching vertices are shown by e> , and the centers are shown by Â• in Figure 6.9. PAGE 132 119 6.2. Experiments on Skin Cell Pictures It is known ^ s that there are two types of cells in the epidermis, basal cells and squamous cells. If the epidermis is normal, the basal cells' nuclei are generally perpendicular to the dermis-epidermal junction, and the squamous cells' nuclei are parallel to the epidermal-dermal junction. The basal cell carcinoma will appear in an island The main features of the tumor cells in the island are: 1. the nuclei are crowded, and 2. there is no uniform pattern of the alteration of the major principal axis direction of the nuclei. The hair follicle shaft can be distinguished from the basal cell carcinoma island because the hair follicle has an acellular protein part which makes the hair follicle look like a doughnut instead of a solid island. The sebaceous glands can be easily distinguished from the basal cells carcinoma in that the sabaceous cells' nuclei are quite widely separated. All other structures in the skin are characterized by having scattered nuclei or being in the form of a small island. In our experiment we first extract nuclei from the picture. Figure 6.10 is a portion of the skin cell picture obtained through the quantization of the PIDAC. Contour analysis is used to extract objects from the picture. Figure 6.11 shows the intermediate result, which is the PAGE 133 120 2 2 2 2 2 2 2 2 2 2 2 2 2 12 2 11. 222222222. PAGE 134 121 in 1 n 1 1 1 1 u m 1 1 1 1 1 1 1 u l u 3 1 : y 9 o d o : ) i u rc 1 1 vt r ? a s s : 2 2 2 2 J 212121110 I 110 iUOlOlll. 0101010 1 9 3 2 2 2 : 2 2 2 2 21 no ii : '.p.; :,mu l l l l g 3 2 2 2 2 2 2 2 2111010 L i 01 11 6 I6l\l0 I 1 1 1 1 1 9 2 2 2 2 2 2 2111121 1101017! 111119 2 2252S 211111021l021191U7232517l\lO1010 1119 !252 12425 10J . 2 1 11 1 72 3262 i 17 17 11 10 till 22S242S1110 1212 M2110U1727272222171J11031 1 1 1. . :::;t517."3^3233 72317H110313f363733 J . .'11,1010 137 1 9 3 " L71616N10 1 1 I 9 S 11716111010 I I 1 9 i 317161110 1 1 I 1 1 9 111413 11 2 11141414 2 11 2 2 2 : 111S1S 2 2 11191S 2 2 1119181313 192918191ft 292833; M8! 525101 I 2111172327 113S343S191819181110 12120 01021111727 3 = : 1155313:) 5191S181110 121202010111722232727 19 3395339 19 1S1 110102 1202 1202110172 5262 72 1940 404140 3191311112120202021112 3272727 26; 194041414ni019l81S111021 11252 2623 194041454141 1019111010 1212020211117252322171643111010 I 1 I 1 19414S44454S4 119181 11010 12120 21111617171745164746111010 1 1 1 L94-0414949494S1119181U0 1 12110101110111 . M746111010 1 9 19404M9 5494140191110101010 1 11010101011 S75050514746U10 1 1 19404149 3 5 494119181110111010 1 1 11010114-50515047461110 1 9 19414149 5494S41191S1110101110 1 1 11011 1 7S0S154S1 171 11010 9 8 19104019 5494SS353191811111010 1 11010114751545554514711 9 8 8 194041485749484848411918111019101010114751 45 15451471110 9 194041S7S65749494948411S1S11S9S 10114 5051 4 151504711 9 3 11194041574 849494948 44119191 'Â• 947 = 5151 4515047461110 1119191940 U414S4948 441 101959596 L4747475051"504747461110 1 191S131S19404041484S41404141 136 14661616*14747 1" 4646 11 10 10 1 1113 2181819404041 4414 HI -. 616060604646466145111010 1 1 1162 2 21S19 340414141404119136160 1 1 1 1606164111010 1 1 631S1S1S1S19 367404041401918616060 1 160t>0(>06 165651110 I 9 1118691S19 367666740404>H9l5 , filoO 1 16060616165 365111010 9 11696871 3 37573 3 519 3101S6160 16060616565747465641110 9 19 3777031797272S1191819191S6160606061646S747575746511 9 3 19777677797879S3808H81SIS 1885616 161646575747575746510 19867786797SS5S2S31918 2 2S5848S64656S75907 59075651110 19868786 379 383 S191818858484S564S97475 ".3641110 878737S6 3 3 3 3 3 319183518843564657475909075651110 9 8786S6S686 3 '3 3 3 3 3191985848565 37475 475-4651110 9 199387S786 3 3 3 394949419188513A4656575756S6S1110 9 8 939293S6869494949494959-19181913641161656S11111010 9 S 879386869494959595959594971918131110111111101010 9 S 8 198786949495939893989S9796191S11111010101010 19SS0000 8786S69S9S9S99**99959494191S1110i010 1 1 1 1** 1 9 198694959898******989594941918111110************ 9 ****94949S99****989S989S191S18**1U010**** !** 3 S ****94959899****99999594 19 13****18111010******** 9 19**94959899******999893191318181819111010****** 8 19 394959S9S****** 599939119**1918**191110****** 9 19**i9495989l)******999594.94************19111 )** I 9 19**9.1949599 5 S99989S9S94 3**********19101010 9 3 ***Â»**94919999 599993S9594 3************ i9H10 19 3 Â»*****9495989S999S989.5 " : 9 t* ************* i? 1 110 9 8 8 ****** v 4.9 1959 S 1595 *9594**************** 151 1 10 1 9 8 19******r)49.ig.i**g49 5 q, ! 9. 1 **********t*****. giiioio 9 3 8 Â• *****************g.|9.| 3***********s****iyi%niQ 19 3 19**************** 3ft***s************* lg i 81ll o 19 8 ig********ftft3*ft************;.***ftft*****ig].{l 12 QlQl g g lg ****** 5******************************19] iiiimo g 9 3 9 S 9 3 3 3 3 8 3 8 s 8 S 8 8 3 8 9 S S 8 8 3 8 S 3 9 9 1 3 9 S 3 a a oig**** c .-**>*********. 19************** ^ ^ a a !g******* ********* 3**3 1 g A ***.> *. * : * * * ft******* : lg*A*ft**A*AAAft*A****AA: 1 g * * * * ft * r. A S ft ft ft A * >. ft ft A * ft : lg**A* 5 t A*** ****** AAA: 19*AAA*A**A*jAAAAA*AA*: ig********************; 1 g A A * A A A A * A ft A .-: A * ^ :: ft ft A : igAAAAAAAAAAAAftft 3 A ft A A i 19 3 3***4************: 9 ^ 1 9 aaa 3ft********* 1 g lSl3 ****** 1 | lnl010 1 1 1 5 3*******A*A 5 1 9i S xA**********i 1IO ioiO 1 ********* uiQjgxgi 3** ******** ** U i ** 10 i ******* 1919 131 8** lS A *******************tO ***** 19 is 13 IS** ****** **************** ft ft * * ft ft ft I g 1 O* *** ft* i fc* *********** I 1 in 1 Q fe* * ** * 1 nig***** ** * * fc ************* *** 1 1 10 5******19**! 3**********************10] a ******** ***191 3************************10 a 5******19! -*********** 2 2**1110 A/IAA.-AAAAA-ft*********** ~ J J 1 I 1 ************* 3191 ,s is* *******-Â»* 211101010 111919**19*** ************ 191111191 1191911****3 1 19: * 3 3 3l919'S******* f HIllOlllO i&19191919191!>lD191Ulll**ll 91111 9 9 Figure 6.11. The labeled picture obtained from Figure 6.10 PAGE 135 122 labeled picture of Figure 6.10. Since this is only a portion of the epidermis, the cytoplasma part touches the picture frame, which is then not under consideration. It should also be pointed out that the cytoplasma parts of the cells are merged together, hence it is very hard to tell the cytoplasma boundary of each individual cell. The objects extracted are then only the nuclei as is expected. Figure 6.12 shows (nuclei) object boundaries. Picture points which have the value i = 1,...,9 are the boundary points of object i. There are in total 9 nuclei in the picture . Local features of individual nuclei are then found by the methods stated in Chapter V. Table 6.1 shows the results of the local features of the 9 nuclei. XBAR and YBAR specify the location of the centroid of an object. AREA is the area of an object. THETA is the angle (in radians) of the major principal axis of an object with respect to the positive y axis. The elongation index of an object is what is defined in Section 5.1.4. The distribution of nuclei in the picture is described by the MST as stated in Section 5.2.3. The weight assigned to the edge connecting two nodes n and n' is D(0,0') + Y8(O,0'), where and 0' are the nuclei corresponding to the nodes n and n', respectively. The appropriate MST (i.e., the appropriate y value) is found through the flow diagram, shown in Figure 6.13, by changing y value until proper PAGE 136 123 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 5 5 5 5 5 5 5 5 5 5 5 5 5 5 S 5 5 S 5 5 5 5 5 1 1 111 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 7 7 7 7 7 7 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 6 9 9 9 g 9 9 .9 9 9 9 9 Figure 6.12. The boundary picture obtained from Figure 6.10 by the contour analysis. PAGE 137 124 Table 6.1 Local features of nuclei in Figure 6.12 Nucleus PAGE 138 125 Input centroids, major principal axes directions of nuclei y = ay = 10 Find the weighted graph with D(0,0*)+y6(0,0') assigned as weight to the edge connecting tA^o nodes corresponding to the two nuclei and 0* Y+Ay Figure 6.13. Flow diagram of finding the distribution of objects in Figure 6.10. PAGE 139 126 clusters are obtained. The clustering method used in Figure 6.13 is that if W(e) > w+aa , then the edge e is erased, where w is the mean weight in the tree and a is the standard & w deviation of the weights in the tree and a is an adjustable parameter and is set equal to 3 in the experiment. Figure 6.13 is then a flow diagram which not only gives the result of the appropriate MST but also gives the result of the clusters of the MST. Figure 6.14 is the MST obtained by this method and the corresponding y value is 20. Figure 6.15 shows the clusters of the MST. From Figure 6.15 we can tell that the nuclei in the same cluster are of the same type. Nucleus 1 is a basal nucleus. Nuclei 3 and 4 are basal nuclei. Nuclei 9 and 3 are basal nuclei. Nucleus 2 is a squamous nucleus. Nuclei 5, 6 and 7 are squamous nuclei. 6.3. Experiments on Blood Cell Pictures The histogram of blood cell density via the cell intensity reveals information for some disease diagnosis. Blood cell pictures are different from skin cell pictures in that there are some blood cells whose intensities are lower than their corresponding background intensities, while every skin cell intensity is higher than its corresponding background intensity. Hence a hole can be a cell in the blood cell pictures. Figure 6.16 is a portion of the blood cell pictures in our experiment. PAGE 140 127 35.46158 14.93248 33.55661 Figure 6.14. An MST used to describe the distribution of objects in Figure 6.10. PAGE 141 128 o 1 2o 9 4 6 3 76 Figure 6.15. The clusters of objects in Figure 6.10 PAGE 142 129 PAGE 143 130 The intensity of a cell is defined as the average of the intensities of picture points inside the cell. Let H be the Hasse graph of all the contours in the picture which represent the inclusion relation. Let C be the object contour of a cell and H c be the subtree of H with C as the root. A level assignment to the subtree H c is very easily done from the level assignment of H. Assume that the level of C in H is L(C), the level of any contour C in H f is then L(C') L(C) + 1, where L(C') is the level of C in H. The intensity of the cell is then n m, m 1 o .2 * CACC|) E J ACcf 1 '^)) g(ch g (o) = i=Li=i Â— i w : l A(C) where n Q is the highest level of the subtree H c , m. is the number of level i nodes in H c , A(C') is the area enclosed by the contour C, C 1 . is the contour corresponding to a level i node in H c , mj is the number of sons of the node corresponding to the contour c] and c/ 1 '^ is a contour corresponding to a son of the node corresponding to the contour C^. . In this special experiment, the intensity of the cell is assigned as the integer which is closest to the value g(0) calculated from the above equation. The counting of overlapping cells is done by the flow diagram shown in Figure 6.17. K in the flow diagram is the number of cells which jointly form a single object boundary found by the contour analysis. PAGE 144 131 Input ordered valley points on a contour |Set K=0 , J=Q and M=m M<1 I K=K+i| ^ M=l M>1 [Set i = | K = K+l. Delete vand v,.. ,.. ( 1+1) mod N the v's sequence and rearrange the index of v's. M = M-2. M<1 M=l v J mod +M>1 (i-1) T~T7 mod M 1k=k+jI Figure 6.17. Flow diagram for counting overlapping blood cells. PAGE 145 132 Figure 6.18 shows the histogram of intensities of a blood cell picture in our experiment. In Figure 6.18, n represents the total number of cells. 6.4. Conclusions and Further Research From the experiments, it is found that the two methods of object extraction are good in different situations: 1. Object extraction by the gradient method can work verywell if the boundary points have very high gradients. That is, it works well if the boundaries are sharp. 2. Object extraction by contour analysis works well under the condition that the objects in a picture occupy quite uniform intensity regions. This method is especially good for area pictures. By using the graph-theory approach to picture processing, a clustering method can then be applied. A multilevel picture g can be transferred into a weighted graph G. From the weighted graph G, a minimal spanning forest of G can then be found. The clustering method is then applied to every MST. Let r Q be the radius of an MST T. Small branches in T are removed if their lengths are less than ar Q , where o PAGE 146 133 PAGE 147 134 length t and (w. ,...,w. ) be the corresponding weight sex l x t quence, that is, w. is the weight of the edge connecting X j vand v. Significant local minima weights are I (j-D J noted from the histograms of the weights along the major paths. Several cluster ranges of weights can be found between significant local minima. If w. and w. are in 1 . 1 , . , -, 3 (3-1) different cluster ranges and if w. < w. , then the edge X j 1 (j-D connecting vertices vand v. are broken. After this clustering method is applied to all major paths in T, T will be clustered into several subtrees. Hence the minimal spanning forest will be clustered into several trees. Let two clusters be represented by two trees T n and TÂ„ . T-, and 1 2 1 T~ will be linked together if (1) the average weights w, and w~ of T. and Tare very close, that is, |w, -w~| < w , and (2) if a leaf v, in T., and a leaf v~ in T~ are adjacent in G T , and if the weight w' of the edge incident with v, and w l +w 2 |Wl +w 2 i v^ are very close to Â— ^ Â— > "that is | Â— Â— w' | < w . After this linking process, every tree will represent an object in the picture. The skeleton of an object can be represented by the significant major paths of the tree representing the object. From the experiments on the skin cell picture, it is found that object extraction by contour analysis is a very promising method to handle cell pictures. The global feature used in the experiment also shows that the description of the distribution of nuclei in the picture is a very reasonable global feature. PAGE 148 155 It is suggested that a further development of the system is needed in which a slide is taken as a roll of films and the PIDAC can handle a roll of films. It is possible to work on the individual film and to store the local features of objects in the individual films. In describing the global features, it is possible to consider all films at one time by knowing the location of each film in the slide. Once such a system is set, the distribution of the nuclei in the picture can be found more completely. The completion of the clustering stated in Section 6.2 can be done by linking together the nuclei which definitely belong to the same type. When one scans through the picture, if the object contour point of nucleus is always just ahead of a contour point of a high level contour C and if another nucleus 0' satisfies the same condition for the same contour C, nuclei and 0' are said to be the same type. Let each cluster obtained by the method stated in Section 6.2 be considered as a vertex. Clusters S. and S. are connected if there exists the definite same type nuclei 0, , and 0^ , in S. and S., respectively. A weight W(S,S.) is then assigned to the edge connecting Sand S. by W(S i ,S j ) = D (Â°k'Â°i ) = min{D(oj,,oj t )|oÂ£ t e S Â± , o] , e S. and 0, 1 , and 0^, are definitely Â£' j k' I ' ' the same type}. PAGE 149 136 A weighted graph is then set. Find the corresponding minimal spanning forest. If the clusters Sand S. are connected in the minimal spanning forest, connect 0, and Oi in the graph obtained through the cluster shown in Section 6.2. After this process, it is easily seen that for a normal epidermis, there should be two large clusters. An island of nuclei can be found to be a cluster through the distribution of the objects in a picture by considering only the distance between objects (i.e., y = 0) . A tumor detection system can then be set by observing the resulting clusters of nuclei in an island. The blood cell extraction is more complicated than the skin cell extraction because some blood cells appear as holes in the picture. While the result we get in the blood cell picture analysis is quite reasonable, a more detailed research should be conducted to achieve a more complete object extraction system in dealing with the pictures which have holes as objects. Again the PIDAC should be developed to be able to handle a rool of films which can then find more reasonable data about the density of the cells, because the result from one film is too restricted. The average of the density of a roll of films can give a very satisfactory result. PAGE 150 BIBLIOGRAPHY L. G. Roberts, "Machine Perception of Three-dimensional Solids," in Optical and Electro -optical Informatio n Processing (edited by J. T7 lippett et al.), M.I.TT Press, Cambridge, Mass., 1965. A. Guzman , "Computer Recognition of Three-dimensional Objects in a Visual Scene," Ph.D. Dissertation, M.I.T. , 196 8. B. Raphael, "Programming a Robot," Proc. IFIPS, Edinburgh, 1968. M. Eden and M. Halle, "Characterization of Cursive Handwriting," Proceedings of the Fourth Symposium on Information Theory (edited by C. Cherry) , Butterwcrth Scientific Publications, London, 1961. J. T. Tou and R. C. Gonzalez, "A New Approach to Automatic Recognition of Handwritten Characters," Technical Report No. 70-101, CIR, University of Florida, 1970. R. H. Cofer,"The Automation of Map Reading," Ph.D. Dissertation, University of Florida, 1971. R. Narasimhan, "Syntax-directed Interpretation of Classes of Pictures," Co mm. ACM , Vol.* 9, No. 3, March 1966. H. S. White, "DAPR Digital Automatic Pattern Recognition for Bubble Chambers," in Pictorial Pattern Recognitio n (edited by G. C. Cheng et al .), Thompson Book' Co . , Washington , ' D . C . , 1968. W. J. Hankley and J. T. Tou, "Automatic Fingerprint Interpretation and Classification via Contextual Analysis and Topological Coding," in Pictorial Pa t tern Recognition (edited by G. C. Cheng et al . J ", Thompson Book Co. , Washington, D.C., 1968. 137 PAGE 151 V 138 10. R. S. Ledlev, "Automatic Pattern Recognition for Clinical Medicine," Proc. of the IEEE , Vol. 57, No. 11, November 1969. 11. J. M. S. Prewitt and W. S. Mendelsohn, "The Analysis of Cell Images," Annals of the New York Academy of Sciences , Vol. 128, 1966. 12. H. Freeman, "On the Encoding of Arbitrary Geometric Configurations," IRE Trans, on Electronic Computers , Vol. EC-10, No. 2, June 1961. 13. R. S. Ledley et ai . , "FIDAC: Film Input to Digital Automatic Computer and Associated Syntax-directed Pattern Recognition Programming System," in Optical and Electrooptical Information Processing , M . I . T . Press , Cambridge, Mass., 1965 . -14. J. K. Hawkins, "Image Processing Principles and Techniques," in Advances in Information Systems Science , Vol. 3 (edited by J. T. Tou) , Plenum Publishing Co., N. Y. , 1970. 15. M. H. Huechel, "An Operator Which Locates Edges in Digitized Pictures," J . ACM , Vol. 18, No. 1, January 1971. 16. C. T. Zahn, "Graph-theoretical Methods for Detecting and Describing Gestalt Clusters," IEEE Trans, on Com puters , Vol. C-20, No. 1, January 1971. 17. M. Wertheimer, "Principles of Perceptual Organization," in Readings in Perception (edited by D. Beardsley and M. Wertheimer) , Van No strand, Princeton, N. J., 19.5 8. 18. H. Freeman, "Techniques for the Digital Computer Analysis of Chain-coded Arbitrary Plane Curves," Proc . National Electron Conference, Chicago, 111., Vol. 17, T9"6X 19. H. Blum, "A Transformation for Extracting New Descriptors of Shape," in Models for the Perception of Speech and Visual Form (edited by W. Wathen et al .j, M.I . T . Press , Cambridge , Mass., 196 7. 20. A. Rosenfeld and J. L. Pfaltz, "Sequential Operations in Digital Picture Processing," J. ACM , Vol. 13, October 1966. 21. U. Montanari, "A Method for Obtaining Skeletons Using a Quasi-Euclidean Distance," J. ACM, Vol. 15, No. 4, 1968. PAGE 152 139 22. 0. Philbrick, "Shape Description with the Medial Axis Transformation," in Pictorial Pattern Recognition (edited by G. C. Cheng et al .), Thompson book Co.. Washington, D. C, 1968. 23. G. levi and U. Montanari, "A Grey-weighted Skeleton," Information and Control , Vol. 1?', No. 1, August 19 70. 24. J. L. Pfaltz, J. W. Snively, Jr., and A. Rosenfeld, "Local and Global Picture Processing by Computer," in Pictorial Pattern Recognition (edited by G. C. Cheng et al .), Thompson Book Co., Washington ,' D. C. , 1968. 25. A. C. Shaw, "A Formal Picture Description Scheme as a Basis for Picture Processing Systems'," Informatio n ard Control , Vol. 14, No. 1, January 1969. 26. J. T. Tou, "Engineering Principles of Pattern Recognition," in Advances in Information Systems Science, Vol. 1 (edited by J. T. Tou) , Plenum Publishing Co., N. Y. , 1968. 27. N. J. Nilsson, Learn ing Machines, McGraw-Hill Book Co., N.Y., 1965. = 28. S. K. Chang, "The Analysis and Synthesis of Twodimensional Patterns Using Picture -processing Grammars, Ph.D. Dissertation, University of California, Berkeley, 1969. 29. 0. Ore, Theory of Graphs , American Mathematical Society, Providence, 1962. 30. F. Hohn, "Algebraic Foundation for Automata Theory," in Applied Automata Theory (edited by J. T. Tou), Academic Press, N. Y. , 1968. 31. J. C. Gower and G. J. S. Ross, "Minimum Spanning Trees and Single Linkage Cluster Analysis," Appli ed Statistics , Vol. 18, No. 1, 1969. ~^ 32. B. Gold and C. M. Rader, Digital Processing of Signals , McGraw-Hill Book Co., N. Y. , 1969. " 33. F. P. Beer and E. R. Johnston, Mechanics for Engineers , McGraw-Hill Book Co., N. Y. , 19S~6~: ' 34. R. L. Hackett , Private communication, Medical School, University of Florida, 1971. PAGE 153 BIOGRAPHICAL SKETCH Peter Pei-teh Lin was born December 8, 1945, in Chekiang, China. In July, 1962, he graduated from ChienKuo High School. In July, 1966, he received the degree of Bachelor of Science in Electrical Engineering from National Taiwan University. In 1967 he enrolled in the Graduate School of the University of Florida. He worked as a graduate assistant in the Center for Informatics Research until August, 1969, when he received the degree of Master of Science in Engineering. From September, 1969, until the present time he has pursued his work toward the degree of Doctor of Philosophy. 140 PAGE 154 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly' presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. JuliusTT' Tou, Chairman Graduate Research Professor of Electrical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Arnold H. Nevis Professor of Medicine, Electrical Engineering and Biophysics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. (R^\ \&S^A^ c Zoran R. Pop-SÂ£o j anarvic Associate Professor of Mathematics This dissertation was submitted to the Dean of the College of Engineering and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. March, 1972 e~a n , College of Engineering Dean, Graduate School xml version 1.0 encoding UTF-8 REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID ELS6EJ9AD_YHUAFV INGEST_TIME 2017-07-14T23:33:49Z PACKAGE UF00097624_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES |