In Proceedings of DAFQA lmage Understandinq
Workshop 1989, pp. 1076-1088, May 1989.
Computer Analysis of Regular Repetitive Textures
Leonard G. C. h e y '
Take0 Kanade
School for ComputaScience
CZlllbegkMelloaUniveniry
PiUsburgh PA 15213
Abstract
Regular rcpttitive texam arc common in rcal-workl scaux, occurring in both n a n d and man-made environments.
Their analysis is important for image segmentatim and fos shape recovery from surf= texture. There are two
fundamental problems in analyzing reguiar repetitive texture. F d y , rbc frcsucncy interpretation of any regular
t e x m is ambiguous since that arc many altanative imqmatiap tbatccmspmdto the same texture. Secondiy,
the v a y definition of regular repetitionis c i m h since the
elanencard the repetitive hzquency are defined
in tamsof each otber. In this papa, we address tbcsc t w o p o b l a n s a n d ~ a n a u s w c rto each. To address the
ambiguity of finqpmcy htmpmaa'on we p~lto
ll the lattice theory and choose sucassiivc minima as the most
fpndamental f n q u a q vccuxs of the t~topre.To deal with thedcfidkm OfregularnpetitiOn, we compare
t h e S t C U C t U- d topnmimtntfcaMesinthe~Tbesttbearmcal
~
concepsarehrporatcdintoa
working systrm, capable of d y z i a g and scgmauing regular repshive tcxtlacs in real-world images. In contrast
with pevioruwark, ora tcchniqueinvolvcsentirely localanalysisaadistherebyrobusttotcxturedistonion.
Regular repetitive exaxes are CQI~IIY)EIin real-worfd scenes. Tbey occur both as arcSulL of natural pmcesses (e-g.
the repetitive mom of reptile skin) and the eff.mof man (eg.man's aukiag of a city scc11c). Understanding these
textures is imponam not only as abasis for image SCgmGntation butaiso because regular repetitive textures can
provi&valuabIcinfonnationfor~gsrtrfactopieatatioa
A fundamental problem in analyzing regular textures, howeva, is that the &finition of regular repetitive texture is
circular. The fnsoencr of the texture is Mined as the spatial dispiacancnt becwccn elantnts of the texture. but the
element of the texture is definedas that portion of the image that is ngulariy rcpeatcd This circular dependency is
usually handXedby obtaining informationabout the reptitivc iiesuenCy without considaingthe natllre of the texture
element or vice vasa. In both applloaches a global adysis of the textme is pufkxmcd, restricting the applicability
oftheaIglxithstoundistoro#i samplesdasin@crepetitivetcxture
In comas to these approaches,omwurkanploys apmdyfocalaaaiysis toidentify tbe repetitive structure of the
mos~
(&minant) fin nsplarnqeitive ocrmres i n d - d images. In this way, we identify the
resnlarnpecitive=hti-m * between textme elemenowithoatirtntifvinp the texMeeianeau themselves.
A second problem in aaaIyzing regular repetitive textmw is that: cvtll whea we know the locations of textw
eluxmts, tbue are many alternative pairs of fhqwncy vccmrs hat equally dcscribc the pattcnr of the texture
I
c
elements. This problem has been haadled in the past by choosing either the shortest frequency vectors [8,14,151 or
those vectors which provide the simplest grammatical suucture for the texture region [IO]. The former approach has
the heuristic advantage that the frequency vectors are independent of the shape of the texture region. In this paper,
we develop additional reasons for preferring the shonest frequency vectors (which we call the fundamental
frequency vectors based on fesultsfiom lattice tbeory. Theseresults paovide additional properties of the fundamental
frequency vectols that are UsGful for extracting the freqpeacy of d - w d d kxllms.
2 Existing Frequency-Based Approaches
Both of tbcsc paoblans imply that an analysisncais tocuuidersmail samphofthe image (of the order of a few
texmre elements) ata time. Becapse tbe size of the tcxrmcrJanrnh is afnnnim of the fnquency of the repetition,
the appropriate image sample sizevaries as a frmction oftk hypmks&
'
npecitinfrequency. It therefore becomes
~ ~ C C C S Sto
~ Cp~~ c t 3 a
s large nambaofsampies of differem sizes i m u d of simply applying a global analysis t~ a
Single image. It is nasarpriSingthen that these t d m i p c s have yero be applied to reai-wmid images.
3 Existing Element-Based Approaches
from each other p v i d e d that their texture elements m sufficiently diffennt 'Ihis capability is very useful when
dealing with real-world images where it is common to encounter more than one regular repetitive texture in the same
image.
.-
The element-based approaches do have one major
the reptcitive firesuency is determined by a global
y
within the texture. Frequency
analysis of the tcxtue. Such a global analysis will fail if the i k q u a ~ ~varies
varhthns commonly occur in textures in real-world8 imagts as a rcsult of ptrspective imaging, and existing
techniquesare Unabletoanal~suchimages.
4 The Dominant Feature Assumption
Ratha than working with the raw image intensities on the one hand or tcxplre dements on the other, we work with
fmaves of the image in somearbitrary featrae space. Weconsider the imagc to consist of a set of these features,
appropriately located (Conceptually. we convat the h a p into its fcature-space rtpresentation). We define a
texture element T as a set of feaopcs (FI,
F2,.J',) axh of which has a location x(FJ relative to the origin of T. We
assume that a feature dots not span two or more t c x m elancny but allow atCxture elanent to span any number of
feaaves
We now defiw a hrnction P, called thepromtrcncc functh, wbae P(Fi) is a scalar value. The function P is
amtinuous, in thatlmilnfCgtmtShavesimilar~ofP.Acoavemmt
*
famforPisafuactionthatmeasures
-
infonnacioa coaoemdthe fcatare. It is highly d l d y thuaq two fcatlnes in Twill have the samc pmminence
value P. In taq f
aanradomly p a a t a i T,all values o f P will be distina with probabii 1. It follows that we can
idtnafy any partiEularftanrrc by its pmhmce valw P. In paticular, wc can uniquely ida~tifyone of the features
of T,sayFpa3 ltre m o s t p o m b c n t f ~in T. Condczthefolbwing:
vi PCFr) LPCF,)
We call Fp thedomtranrfeaure of the texture clement T. Notice thatifP isan infamation measure. then F,,is that
feaaue in Twhich comains the most information. Forthisrcason, tbe feature with tbe largest d u e of P is selected
asthedominantfeaane.
Consider now a regular repetitive tcxtrnc consisting of texture ekmcnfs TI,T2,....Tm that art identical copies of T. If
we amiyze this tcxm and extract the feaauw FI1, FuE
,'
we will fmd that thac is a group of features FpI.
Fpz,...,Fh that are all equally prominent and are morc prominexu than any orha feaaucs in the texture. The
&mbuuu feonvc assumption starea that such a set of dominant features always exists
Dominant Feature Assumption: Every regular repetitive pattern exhibits some feaaae that occurs once in
each texture clement and is more prodneat thanany abafeenneoccnrringintbe same mumclement
6
Undathedorniaantf~asSMlpaon
*
* i t i s p o J s i b l e * t othe
~ strrrctm~ofqetitive tcxplrts from
the ft!anlK+space rqxcsaltatimofthe t
e
x
m sincc thae ismdnninnnt fcaam f a eacb t e x 4 the smlcaae of the
Qminantfeomres~the~0fthetexeb,as~infiigrmc1.
5 Fundamental Frequency Vectors
The fimdmentaI ficqmcy vectors are defined= thcshatestpairof fnsllrerrcy v c c t ~ 3in the texture. They have a
number of useful propaties that am be derived from lattice theory. In this theory, wtassannc that we arc dealing
I
with an undistorted regular repetitive textme. Undistort#i regular @ve
We define an lmdistorted regular repetition as follows.
textures are closely related to lattices.
Defmition 1: A regular repetitionR is a set of points (x&u+p: i.j integers) in the plane where ~0 is an
arbiaary point and u and v are lineariy independent vectors. R is the transhe of a latrice. The vectors u and
v ate a basis for R.
We ckfm tbe fmvlamrntnlfreqpency vectors a’and v’ofaregular repctitionR as the sbaroesr and second-shortest
0
lineatlyindependcntf . i q m c y vccfors dR.
Dcfinitb. 2: M a e M y , kt V = ( x i - % : x i a € &xi#%) be the 9 c of
~ all possible frequency ve~tQS
of R. Define the first fdamaml1llect01
u’ to be any one dtbt shortesr v t c t ~ n
(me3surtd with
Euclidean length) in V. Let U = (id: iintega). Define the s a m d -tal
firesucncy vcctoz v’ to be
any one d t h e shortesr vcc&xs in V-U. Tbe V~EOPJa ’ d #are known as suaxssh minima in lattice
tbyaretbefrmdamentplfreqmcy M c t a s o f R .
*.
TbisdefmiIion of tb fudmWalvccto~3has saneambiguity in the choice of u’and v’. lhis ambiguity
has thrte @bIc solt~cts.Fdy.thae is the ambiguity between u’ and -u’and between v’ and -4.Secondly, if
la’l=lvl, ambiguity can occur in the labeling of the funrlwnclltal fnquarcy vcctocs. However, both cases are not
saiousproblemsdwiunot~w~.
1.-
fmulanrmtai fnqparcy vcctcxs u’and v‘ f a m a basis f a R ; Le. the set (+,+h’+lv:
k, 1 integers)
is
idtntical toR
2 Tbcredoes notexist foaR apairofbasis vccoonaand b for which IrJdr7aodIbklv’l; is.thac is no basis
forR Consistingof Veaonshorotrthanthe baga f d a m a m l hqucacy vcum.
3. Thae &anotexist apair of basis VCC~QJa d b f a R fa which IaMbkbaWfi Le. u’and v’ describe
the minimnm--
!slmcad Inlit
OfR.
4. The fundamanal f i l q m c y vectors u’ aad v’ are tbe most p c q d *d a r basis for R; ie. l u ’ d / lu’llvl is
minimai among aIl bases of R.
related to the Relative Neighburhood Graph (RNG) of tbe texture elements. The RNG [16] is a bidirectional graph
on a set of points P = (p , , h ....,pn) in the plane. The RNG connects two points pi and pj if and only if there exists
no other element pit€ P such that Ipl-pj<lp,-pj and Ih-p)dp,pJ. The RNG has been proposed as a model of
human perceptual grouping of dot pattern (12). We have shown in [s] that the RNG of an undistorted regular
repetitive texture capexactly the hdamental fresuency nlatiaaships between the texture elements. More
specifically, if W is the set Of all possible
kqtmlcy vecms of R (induding all forms of ambiguity)
then the RNG o f R is wraaly that graph rhatconnectseach pointxleR to a l l points in the comsponding set
[x;ewxve w)).
6 Analyzing Real-World Textures
In analyzing real-world tex~llrw.our aim is to discma the f d a r m m i frequency vectors betsvecn the dominant
fcaturcs of the texture elements. The applicarh of the thearwcal cmccpts of dominant features and fundamental
frequency vccoors is MIL direct, however, because of the variations that can o c ~ in
m real-world textures. F
irstly,the
texture elements of r e a l - d textures often vary from cach othet,giving rise to variarions in the prominence of the
dominant feaanes. Secondly, r e a l - d d regular npetitive textures am often distorted as a d t of effects such as
surface curva~eandpaspective
imaging, causing theresub obtainedfroa3 lattice theory to be violated.
-
la ordu to deal with tbe Variations real-worldregalartnanes, w w tbsnlative dominance of the features
ratha than starching far absolutdy rLminant
Rathex than computing the Relative Nughbourhood Graph,
we dcvtlop asQuctlnaj dcsaiption in which miouspropadtsofthe f o d u n a d frequency vectcm are combined
to produce agraph with weighted edges. Successive dinanem steps lead to ambust algorithm that can reliably
. extract the
frequency desaiptioa
Tbc system that we have implcmenwr
illl algaidnn cxl&?bg O f f O r p m a j Q ~
Thc first phase is feature
detectioa. It extram blob-like fhm the i n p u t i m a g e a n d ~ a p o m i # n c evalue and image location
with each feattm. 'Ibe amnaphaseestablishesbesic sullcad retationshrpr betwcenthefeatuns,basedonthe
dominant feature assIIIllptioll and PropatitJ of the fundamental fnsparcy &Tbe suucplral relationships are
rcpresarted as weighted links berwtca the fThc links and theirssoch@d weights are passed to the third
phase which collstrtlcts l d y regular repetitive sutmmes and aoachs evaluaMns to each of them. Multiple
candidate repetitive sauuures are evaluated far cach iomgc feamre. 'Ihe ~ phase
I I decides up on a locally
canSiStcnt repetitive stracture intaprctatiOn based upon the canpetingrepecicivestnmms. A relaxation algorithm
is used to obtain c0nsistcncy by wnstraintppgaha
7 Smooth Thresbolding
m u g h w the algolihms described below, we express evahaom
*
00 rbnmgs0.0 to 1.0. Thesc evaluations have
'
a similar role to fuzzy reasolling or probability vahm~
Namrl
ofbm EniDes the fam of thresholding
applied to a mcasmui value. Instcad of directly thresholding the dam, m e emplay "smooth tksholding" functions
that convat measured values into cvahations on the mge 0.0 to 1.0. 'Ibesmooth thdwlding functions Tand TL
arc based on the tunh function. In the following equations. r is the nominal threshold d u e such that 77% T. a)is 0.5.
and CJ is the spread such that T(.r+a,r,a)is 0.75. T is a particular parameterization of the Sigmoid dismbution: T,
is a logarithmic version which is mOct appropriate for evaluating ratios.
8 Feature Extradon
- 9 Basic Structural Relationships
1. Prejudice urincblc Fattrrres prefer to be Iinkcd to otha fames which are equally or m m prominent than
themselvts.
b
'
I
dirtctional link between Fi and Fi. The evaluation is suppressive, in tbat large values of S
, weaken the link. The
constants 8 and 1.5 are the values used in our experiments.
2InterferenceRinciuk The link buwcut two fuulrres is only suong if them is no other equally or more prominent
featurewithin thedominated region d t b e twofeann#.
TheintafcrmapinciplecaptunscancepUdaind~bothtbeQminant feaopre assumption and the relationship
between tbe f i m h n a d kqumcy vcctm a d theRNG. R a d thattbeRNG linlrs mgethetpairs of points where
tbae is no otbapointpoint dasa to both points rmlercnnirlantwL'Ibis is eqnivalart to linking pats of points
when thac is no other point within the "Irme" of the rwo points rmdcr&ckath
(the "lune" is the intersection of
two circles as shown by the doatdlinein figure 3).
In adapting the RNG ztsult to nal-worjd image data, we must allow fasiightdisaepaacles
*
in the positions of the
dominant feaarres, panimWiy as a FGsult of distorrioa dtbe textare. The 'lune" is much too sensitive to small
Variatiwu in tbe pasitions oftbe featllns and tbe "true" timAamenlnllkquency v c c t ~ swould ohen be disallowed,
so we use anthatlieswithin tbe "tune". We call thiarcgbu thedomhtcdngbn of the two features. As with
the application ofthe &diceprincipiewe do not simply rnakcabinary decision about whethaaparticuiar feature
lies inside ocoutsidt the dombcd region ofotha feanrns Ratha, the i n t d a m c c ofa feaauc incrtases from 0
towanis 0.5 as the feamn appaacbes the cam dthe Itmirraml region. Figum 3 shows some contours of the
dominatedrtgion-
what
k
.
11Relaxation
After phase three of the algorithm, we have computed rnany competing regular repetitive smcture hypotheses for
each feaarre in the image. If a particuiar feature is a dominant feature of a regular repetitive texture. we expect that
neighboring dominant featurcs will also exist aad that thesc neishbaiDg farapnes will have compatible hypotheses
~~g~
'
shouldbeconsistentis
about tbe local repuitivc sansmre 'Ibis coasmmc
*
implcmenrcdinadaxamn alguithmtbatdeDcnruaes
'
atmastnpecitivc s&manrc f a cach texture clement In
the e vent that a fcamre is not pmicipm
. g in a m r e p e r i t i v t smame, thac is a lack of supprt fkom
neighboringfqandtbenlrutatlon aleorithm*
thatnorepecitivestrPctmcisplesent
-
-
l2 Results
Our systan pafornu region segmentation on xcgu&r repetitiw tcxmres p l y as a SidGeffCa of the detailed
analysis. The results ipc srpprisingly good ascan be scen in both figures 7 and 9. In addition, the smctural
description produced by our system is very detailed. It may thucfore be useful as a basis for a detailed shape-hmtexture analysis. For example, the oneeyed stereo approach of [13] may be directly applicable m the structural
grids extracted by our system.
W Conclusion
Figure 1: Repetitive strucnue of dopninruufeatures ofa tatwe.
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0
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