p ~ ~ T E R N RRECOCNITION U S I N G G E N E R A L I Z E D PORTRA l T S

i V . FJ. V a p n i k a n d A . Ya. L e r n e r

(tdoscow 1 Translated from Avtamarika I Telernckllanjka. VoI. 21. No. 6. pp. 774-780, J u ~ I P , 1963

Origlnai ar t ic le sirhrnltred December 26, 1962

An axlornatlc d e I ~ n r t i o n o I paltern rs Riven. The concepts of "<eneraI~zed portrait*. "distinction", and "recognltionm are ~ n r r o d u c e d . Algor i thms are proposed for learning recognitLon and dlslinction

on the basis of find~ng generalized portraits of patterns.

A numher nf s t ~ l d ~ c s havc appearcd i n recenr years or1 tbe sublect of mode l l ing palttern recopnrrton processes

11-33. A number o f e F f c c t ~ v e algorithms h a v e been proposed for l e a r n ~ n g to drsrrngulsh visual patterns in specral- purpose or Lrnlversal dtgltal computers. The poslng of these problems and the resu!ts obtained cons t i t u t e a n Impor- tant aage In the drvelopment of <elf-nrganiz ~ r ~ g syrtems.

In t h e present papcr a n arrernpt is made to formalize cer ta in concepts connecrcd wtth pattern recagnr t lon. The authors start from the pr~nc ip l e that a partern is defincd by the objective properties oC t h e se t o r objecrr under

mnsidcrar i nn and the s u b ) c c t ~ v c propcrt i c s of IIE n i a d ~ i n e percerv Ing them.

T h ~ s approach has pcrrnltred us t o lntradure t h e concept of "generalized portra t t " and threshold of r ecogn i r~on , the ensemble of w l ~ l c h c l~arac ter~zes tl~c systcrn of machine patterns, and to sl?uw that t h e problem of pattern rccog- n i r ~ n r i In gt-neraI r.rlnrtrtc 7l rhc' two suhs~dla:y problems of rec.~gn!7ing and d i s ~ ! r . g u i s l ~ ~ n p patterns.

Iet us c o n s ~ d c r a n lach~ne. consisting cf a perceDtion dcv~ce PD, a transfnrmatinr~ device T D and a rccognllkon ~ P V I ~ C W ( T I E . 11. We 5'lall term the starc rrf its perception device the trnagcifj, ~i rk ~ r h nhject presenredro the rnacljlnc, artd rhe rwtpurs f of rhc cor~vers~on d e v r c e we tcrm thc d e s c r ~ p r i u n u f the oh,c;. The foLlowing considera tions are based on t h e de fmnl r lon of 111e concept 'pattern".

I et tliere c x ~ s t A rcrtait l set of ohjecrs 11. We sha l l a s f u m ~ that rile F e t O F thelr images for r h c given rnaclrirtc

car1 hc j ~ v i d r d fn to q paft'rns II rhc set of objc'ctt: H c a n be divided ~ n ~ c subsets I i , . . .. . Hn, suc"~ thar after a cerraln s u f j ~ c i e n t numhcr o i objt'crs o I each subset has been shown to t h e nlachine, l t c a n divide rile entlre set I I i n ~ w t h e snnle suhccts H,, . . . , 11,. It is assumed tha t for e a c h sf thcse suhscrs H:. . . . IF, [llcrc exlrts 3 descr~pt ion pzrnlltrlnq an es r jma te ot fhc d e g ~ e e a I correspondence to ~t of t h e d e s c r ~ p r ~ o n s of e a c h object.

The unique asrlpnment of an ohjcct ro 9 s11h~ft 1s then possible w t ~ n . i f the dc$criprioris of rho obj~cr: bclnng- 1ng l u different s ~ ~ h s e t s arc ccn~pared ro the d ~ s c r l p ~ l o n of o n e of the subscrs. a la rger v a l u r rrf t h e csrlrnarlon p o r a m -

eTer 1s ohta~ned krr r he d r r c t i p t ~ o n of the u'bject which beiangs t o that suhet.

1 . D e f ~ n ~ t i n : ~ of p a t t e r n [,et there ex is t a set of Images T. W e .$hall consirler a ccrta In suI7set o l the rrnages Q cT and a c t r t a l n srllgle-

Yalllcd r:a~s inrrnat io i~ .T C- I!. w11ere CI is a given set of single -valued transforman~ons, each of w h i s h pIaccs J n cur - respondencc wit11 each i n ~ a g e @ 6 @ a polnt an the unit s~hcrc It1 Hilhert space.

Wc <hal l ?a;. that rhe set of Images decornpnscs into _n palrcrnl i f r l~e ~ c t Ip can b e d ~ v l c l e d rnro 2 s~rh<e:s $,, . . . , 3, sucli ~ l ~ a r the correspond~ng subscr of pclints on the sphere F1,. . . . T, has t h e followkng property.

-- For each subset Fi I point on the sphere pi can be found such that for any f i 6 Fi, f . 6 pj the lncqualr

1

I

is satisfied.

I I If the set is such that there does lot exist ad image 4 6 T\@, whose corresponding

f. athi the inequalities

we shall say t b r the images 4 divide into defined patrerns 'al, . . . .'4n. In the contrary case we say !har the I.

ages belong to the lndefinire parrerns *GI. . . . 1 , : I ,

1' , ' We shaH say that the images O do not divide into n patterns i f in the given set of transformations u t~. I no transformation on the sphere 9, lor which conditsdn (I) is satisfied*. ' 1 %

8 , It is abvlous hat several transfi)rmariar)s may be found for which conditions (1 ) a r e satisfied. I

I I , we shal l ray that the w~nsformatioo& E UJS similar to the oansfotmation $j t and w r ~ t s 3 ~ - $j,if set b, dividing into n patterns *el, . . . .'@, under the transformation F,, can be decomposed into the same n

pat-. I

terns '9,. . . . . 0, under the transformation 9,. The set of patterns 'el, . . . .'*,, into whlch a glven set of jmaRet decomposes under the rransformation Ti wrl l be termed a system of 9,-homogeneous patterns, and will be de- _

noted by Igt (a) I. 2. R e c o g n i t i o n a n d D i s t i n c t i o n

. I Let us consider a sysrern of 3,- h ~ r n o g e n c o ~ s patterns (@)I. Let the patterns of this system be ,%. ( . - ! the subset of images$i under [he transformation . y i ' l e t there correspond the subsets of paints on the sphere

F1,. .. , Fn. We shall consider a cer ta~n i n a g e E ?' and the poinl f correspond~ng to i t . we call the establlshent of

the membership of f in the subset Fi pattern distinction i f it 1s known In advance rhat the rmage $ belongs to the

set +, the contrary case establishment of the membership of the point f ~n one of the subsen F,, . . . , F, is. termed pattern recognition. The recognition problem can be defined also for a set of indefinite patterns. The fol-

1 % I low in^ geometric interpretations of recagnit ion and distinction can be proposed.

Conditions (1) m a y be written in the form (f, T,) > C,, where C -- min (f,qi). This sig~?illes that In some 1%

, # ' ! functional space i n the sphere with center-at the golnt (pi and radius Ri no description of an object belonging m pt- . '! tern '+j ( j + i ) can occur. ,

In the case of Jistiacalon incidence i n the sphere is sufficient for recognition or the pattern. Howeva the mutual disposlrions of the spheres may be such that certain of them may intersect (Fig. 2) . The portion of the poian belonging s~rnultaneously to two spheres cannot be recognized by inequal i t~es ( 1 ) (i.e.. these points are not images belonging to the wrnple).

li i t is net known in advance ~f the glven irnage belongs ta one of the - n patterns, incidence of its descriprkon w irhjn one of the spheres i s not suffjcient for reccgn it ion. It i s further necessary to de~ermine t f the description 0b the description of the object i s incident within a single sphere only.

Let us consider the conditions (flvi) > ( fJqZ) + (f+Pj) > V ~ V , ) .

We term ki = rain ( f , cp , ) The recognition thresliold at t l ~ e pattern f i

the recognition threshold OF the pattern n & j .

and ~orrcsponding1y.k~ - min ufl$ f i

Let rhe system nf .~ii-homagencous pasterns (0) 1 be such that rhe definlte parterns cornpxng ~t are

* ~ h l s definlrion of pattern can be given in metric space, replactng conduions ( I ) by the condi~lons p(/,r qi) <P(#~,P$

710

2ty . . . ,*9, (a subset of a l l images belonging to the patterns el, - . . , en, and he corresponding suhsct of points on

sphere F1, . . . , En). tl)

~ c c c t d ~ o g to rhc definir~on of pattern. for each subset there exists a certain point o f rhe sphere vk. W e term pojnr on the sphere Q k the generalized portrait o f the pattern Bk .

it F.

F i g . 2 Fig. 3.

It is ohvinl~s tha t ~ h c generalized portraits nf t h e pattcrns Q1, . . . , V n and rl~c scr of t t~c i r rccog~iition t h r e s l~ - aids k l , . . . , k, ful ly characterize such a system of .~i-bomogeneous patterns'.

Cif each I pat- mana t r

" It-

Let k l * 0. . . . . kn 9 0; we put s, = ( l / k i ) *. The inequalities ( I ) may be rewrjcten in the farm

(#is11 > 1 > ( f j s1 l7 > 1 > (jlsJ)

.. ,%. (!IS,) > (JJJ &re

The last raequaliv can be taken as the condition for dirringulsh~ng the patterns *Qi, . . . ,'+, of zllc system

TC [Fi (0) I " , n t of I rho I f a certain object f j does not belong to any of the patterns, F(@) can he estimated itom the quantity '4, ,

and Jt can be ass~gned to that pattern (fj, sj} for which the value ' 4 k 1s maxlmum.

am*

3 pat-

ir ion

ofor z < l , " ' 0 ( z ) =

I 1 for L > 4 .

liere the upper lndex indtcates the index of the system of Fk-homogeneous patterns containing the sample '9i. Then the c o n d ~ t ~ o n for recognizing the pattern '@, ronslsts in determining such x f , rhat

The exjstence o l this requirement Indicates that the operation of r e c o g n i t ~ o n d ~ s not terminare alter the recognirion threshold has been passed, while the aperation of di sr ingursh~n~ terminates w ~ t h passage of the threshold.

Thus, to determine the membership o f some image in a pattern. it is necessary to:

1) compare the descr~pt~onof the object presented t o a l l the generalized portraits of t h e homogeneous sys- t e m ~ i n the memory :

2 ) put o u ~ the s~gnal of passage of the threshold:

'it should be kept i n mind. thar wfth increase of learning (E-e., with Increase in the number of patterns entering kmo rhe 3,- homoqeneous system) patterns can be tedefined. However at tach given Stage there exists ~ t s corres- ponding system o f true Fa- homogeneous patterns.

Thus. IT IS l ike ly that a person ignorant of the Latin alphaher w i l l term the image in Fig. 3 rhc letter 0. A

Person who knows rhe Latin alphabet. on being asked what the drawing represents, w i l l probabIy answer: " I dnn't know."

3 F'! 3) verjfy if requirement (2) has been satisfied in the case of recognlrlon.

? I , ! I 11 I! i s ro introduce one further definition. we term the order oidistlngulshability of TWO pagtem

I (

I and ' a . [he quantity I = 1 - ( w p.), and the order ofdistin~uishsbilltyofthe system of $i-homogeneou3 pi I ,; 1 1 1

li 11 t h e quantlty

; I I I = 1 - mnx I ~ , t p , ) . - 4

JI I t. 1 c I

where v , . qj are the generalized portraits of thc syatenr'lSi (@)I . I? is prsib3e to consider the order of djstingahk biljty as the character~st ic degree o i similarity of the parrerns in t k system of homogeneous patterns,

3 . O n e R e p r e s e n t a t i o n a specific reprcseotarion we shall consider the exc~ta t ion of the receptor f ~ e l d . i.e., a certain

ul, . . . . m,. where 0 s or 5 1,

if. 1 :i [) us consider the set of transformations. U over rhe receptor field. Each transformation Fik 6 Uputs into : I ,

I . i . l h responden= w i t h the sequence al. . . . , t o a certain vnit vector. . I ; '

Assume that by means of the transformation Tik to the sequence . . . , an is placed lnCOrrespoodene the Ill it t~crrnr 19 .. . . - - B d l . and bv means of the rransformarion .?; no the s a n e sequence n.. . - . . w, i e ,i,,,~ ,,

r r ,f i k + i , ( ~ 1 , . . . , a,) = .FIk (aE, . . , , a,) @ $>, (n,. . . . , a,)

W e define the operation "rnu!tipl cation by a number":

,+T (al, . . ., = (c, ,Flk @ c2Fie) (a,, - . . , an)

These operations are neither commutative nor assocj a t i v t . hut any tranrlormation :f t 1 k * l e , l i n can a l w a y s be represented as

.T,ik+iF)+i,, = cITlk @ cZFi, 0 ~35i , .

I

IT is earily shown that this transformarion F* = E,Fi,@ . . . 13 c,yi, is slmrIar ro an arbitrary aansfmormrtil

I n t h e rans sf or mar ion$. let a certain set of images @ d i v i d e Into fhe patterns 'al, ... , '*,, whose generrlii ' k

portraits are q, , . . . , p n .

Let us consider t w c images. lrnage c, belonging to pattern 'el, and image b, - not bel~nging to pattern '9,. l.et,in the transformation Fik,!heir vectors and the generalized p w r r r a ~ t of the pattern "lP1 take the forms

/ - n n r l i ~ t n n c (1 l lnr thece vectors t a k e thc form

1 t

These same images in the aansformation $* w i l l be

Let us cons~der the vector rpl r= (v , . . . ., vh, 0 , . . ., 0). T h e n

1 r n d the inequality u:cp1) > ( f ~ p ' ) f ~ [ ~ ~ w s from inequality ( A ) . Let , r , (a,, . . ., u,,) = (PI, - . ., Pn), , g,(q, . . ., %,)=CO, - ., 0, p s . 01 . * - , 0).

We deflne the operation 'subtraction":

put* Inta c

Let us consider a machine consisring of _n transformation dev~ces each of whlch has the structure deflned by the transformation Fi 6 L!. The rnachlne can real ize one of n rransformarions.

According to the above, such a machine may be replaced by a machine whose entire structrlre is glven by a slrlp,le transformation

,- 5," =c,.F1 2 . . - B C r T n ,

#here 51, . , ., 2,. is the sysrcrn of transformations in the set U,

The11 each set of Images 4. d~viding ~ n t o parterns for the flrst machsne, div ides ~ n r o the same parterns tor he secnrtd machine also.

The strucrntc (system of rransr?vat~onm) of each machine should be cl\oscn according to the purpose o i the nachine and ~ t s Image should be joined in a pattern. can

. , .

The problem nT f ~ n d ~ n g a rransiormat~on provldlne effective patterrr recognition a p p a r e n t l y cannot be solverl rp dlrec~ calculat rnn. Neverthelesr there exlsts a means of flndlnp the required ~srnsformatton, corresponding to

he goal\ ret before thc machine. This is the methnd of directed select~on.

! 4. directed selecrion we understand a progressive improvement of a n arbitrary inltisl structure such that inef- ! licient (with respect t g a given criterion) pot~jons of the s t ruc ture are e l l r n ~ n a r e d and replaced by other random : flrIlctures. 1t IS ohvious that t h ~ s w i l l a lways lead lo progress of rt~e structme.

It should be emphasized rhat this method is not that of random selection over a l l possible structures. and there nrP each succeedlog modllicarion of the machine does not inheri t inefficient components.

I f 2s the ct i ter~on of effjclency of t h e structure we rake rhe order nf pattern distinguishability # ~ n d organlze In

he machine the directed oelecrion of transformations*, then after a certaln number of selection opcratinrrr; the JURlity of the transCormarlons w ~ l l be brou~l~t to the required level of perfect~on.

' 4 - F i n d ~ n g t h e G e n e r a l i z e d P o r t r a i t

According to ~nequa l i~y (I), the gencral~zed portrait "el is the centtt of a sphere in a ce r r a tn functionalspace r l r h ~ which a l l points belonging to a certain subset F, of rhe parrern '9, fa l l , and w ~ t h ~ n which no po~nr of rhe sub 'hFj o f the system contaming the pattern '9, . fa l ls .

1x1 us consider the set TJ of poinrs s,, for which the ~nequal i ty

r ' e - m remove frmn the initial set those rransforrnatiol~s such that in rt~e new strucrttre the order of distinguishahillxy * ' '"eased, and add new fransfvrmations taken a t random from a cerrain set U.

.t

I

I I < i s val id . I 1:

1 ~f t h e set rl i s not empty, the following propositions* hold.

I

1. There exists an unique point r;: 6 q such that

where the coefficienrs a and B are nonvanishing only for rhose vectors f i and f j for which inequality (1) pasres inm equality.

2, The venor s: has minlmurn length among al l vectors s E rl . This signifies that i f for the generalized portrait we t a k the unit vecror q i = s:/ t[ syll , 'the threshold of recognition ki = rnin If (Pi) w i l l be maxjmum,

3, .";; = - - 3 / ~ a , + Ip,,. . . 3 .

., 1 ' 1 Correspondjngly the threshold o f recognition k i for the genetalized portrait

I qi = ": ,I sp I I

, I is equal to I

4. The coefficients ai, & i can be found as t h e coordinates of a stable singular point of the system of equations

A - - E X R + Fi (1 - ('rTfih))v 136 = - SPJ + F , (1 - (vifjd))v

where

IZ for z >,- Cb. ( Z for 1 .< 0 , f', (2) = F 2 ( ; ) --

I0 for : <: 0. \O for L > U.

The generalized portrait can be gendrated i n the following way. I I

A descrip~jon of the ohjecr. which in the first approximation is taken as the gcnsralized portrait. 1s shown to..

the machine. I f when a second object heionging ta the same pattern is shown to the machine, i t 1s not recognized by the machine. the description of this object i s ~ n c l u d e d I n the generalized portrait generated as the second apl mation. It i s clear thar the descriptions of recognized ohjeczs enter wlrh zero weight Into the generation of the eralized portraits.

proxi- gen -

During the learning process i t i s necessary to store in the machine memory not only rht general ized portrait hilt those descriptions o f objects which have taken part i n the generation of this generalized portratt.

Us~ng the concept of "generalized portrait" it i s also possible to solve a number of other problems.

5 . C e r t a i n P r o b l e m s +he problem of autonomous learning may b e formulated rn the f o l l ~ w ~ n g manner.

Let there exlst a certain set o f objects. It is required to divide I T inyo subsets such that i n the i m a g e s f descriptions of the objects satisfy inequality ( 1 ) . If there are several such dtv~slons, lr i s possible ro require that division to be found lor which the order of disrinpuisl~jbilrry is rnaxlmum.

With rhc: discovery of the generalized portrait i t is possible to solve the problem of the "shifting" image.

Let there exisr a certain shifting image w i t h description f (x, t ) . which Far t - can be assigned to one oft

*The proofs o l these propnsitions w i l l be published separately.

her

K i n

, is shown to t recognized second sppmxi- ion of tk gsn-

nagef Uih* :quire thrf

- imaflu.

d to onc off

. -

pAfiern~ **I. - . . , 'en. I t is necessary to determine as soon as possible to which of the patterns rhe shirring Image Hlongs.

TI* solution of t h ~ s problem reduces to the foilowing extrapolation problem: there exist n functions li ( t )

. (f(x, t)u',(x)) ( i = I , 2, . .. , n), defined on t h e fnrerval ID , TI Determine which of the functions first (for rhe msllest value of t * ) passes the recognition threshold (which c a n be solved by exis t ing methods).

L I T E R A T U R E C I T E D

I , F. Rosenblart, Generalized perception of transformation groups. Cybernetics Coll. No. 4 [Russian trans- lation] Izd. inostr. 11r. (1962).

2. E. M. Braverman, EKperiments in teaching a machine to recognize visualpatterns. Avromatika i telemekhan Vol. 23. No. 3 (1962).

3. M.M.&ongard.M~dellrngrecognit~onprocessesonadig~talcompurer. B i o f 1 z t k a . V o l . 4 , N o . 2 ( 1 9 f i l ) .

A l l abbreviations of periodicals in the above bibliography nre itttcr-by-letter transliter- ations or the abbreviations as given in the on~innl Russian jeumsl. S m s or all of lh18 perl- d i c a l literature may well he eveilsble in EngIrsh translation. A complete list of the cover- lo.

cover E n ~ l i s h tmnslationa appears et the back of this iasue.