Fast and Reliable Color Region Merging inspired by Decision Tree Pruning Richard Nock Universit´ e Antilles-Guyane D´ ept Scientifique Inter-Facultaire Campus de Schoelcher, B.P. 7209 97275 Schoelcher, France [email protected]Abstract In this paper, we exploit some previous theoretical results about decision tree pruning to derive a color segmentation algorithm which avoids some of the common drawbacks of region merging techniques. The algorithm has both statisti- cal and computational advantages over known approaches. It authorizes the processing of 512 512 images in less than a second on conventional PC computers. Experiments are reported on thirty-five images of various origins, illustrat- ing the quality of the segmentations obtained. 1 Introduction It is established since the Gestalt movement in psy- chology [14] that perceptual grouping plays a fundamental role in human perception. Even though this observation is rooted in the early part of the XX century, the adaptation and automation of the segmentation (and more generally, grouping) task with computers has remained so far a tanta- lizing and central problem for image processing. This is all the more important for computers as grouping authorizes the reduction of the size of data, and, by the way, can re- duce dramatically the space and time complexities for some post-processing tasks, in a field where data are easily avail- able and in huge quantities. Roughly speaking, the problem can be presented as the transformation of the collection of pixels of an image into a meaningful arrangement of regions and objects [8]. But, how can we identify objects ? The Gestalt movement identi- fied some factors leading to the perception of objects : sym- metry, similarity, parallelism, etc. [4]. Though these rules are conceptually satisfying to explain the phenomenon at the human level, their lack of details makes it hard to see them as directly implementable algorithms [4], even if some of them can be extended to rigorous algorithms [15]. So far, image segmentation techniques have tackled the problem by studying mathematical properties of the image seen [16], or more rarely by a direct emphasis on algorithmic and com- putational issues [3]. There are four large categories of approaches to image seg- mentation [16], one of which is of direct interest to us : region growing and merging techniques. In this group of al- gorithms, regions are sets of pixels with homogeneous prop- erties and they are iteratively grown by combining smaller regions or pixels, pixels being taken as elementary regions. Region growing techniques usually work with a statistical test to decide the merging of regions [11, 16]. In the field of image segmentation (or more generally, im- age processing), Machine Learning (ML) techniques might appear tailor-made for high-level tasks, such as in [9]. One might wonder whether applying such techniques or their adaptations to image processing based on low-level cues (typically, image segmentation) might be well worth the try, in a field where the approaches are already nu- merous, of many kinds (local filtering, balloons, snakes, MDL/Bayesian, region growing, etc. [16]), and where heavy mathematical approaches have obtained significant results (see e.g. the results of [16]). We think some clues coming from image processing indicate that ML-derived techniques might be of interest already for low-level pro- cessing tasks. First, vision is widely accepted as an in- ference problem, i.e. the search of what caused the ob- served data [4]. Second, as argued by [12], low-level im- age segmentation should not aim at producing a correct seg- mentation, but rather a good approximation given the post- processing stages of the image. One more reason can give evidence for the possible interest in using ML-related tech- niques. ML and Computational Learning Theory (COLT) works such as [7] have brought a number of useful results, or found a way to use a number of results, in statistics or probability, which remove many undesirable distribution assumptions on the data. The most commonly used theo- retical founding model for ML/COLT is the PAC model of Valiant [13], which removes any such distribution assump- 1
Transcript
Fast and Reliable Color Region Merging inspired by Decision Tree Pruning
It is establishedsince the Gestalt movement in psy-chology[14] thatperceptualgroupingplaysa fundamentalrole in humanperception.Even thoughthis observation isrootedin theearlypartof theXX
���century, theadaptation
and automationof the segmentation(and more generally,grouping)taskwith computershasremainedsofar a tanta-lizing andcentralproblemfor imageprocessing.This is allthe more importantfor computersas groupingauthorizesthe reductionof the sizeof data,and,by the way, canre-ducedramaticallythespaceandtimecomplexitiesfor somepost-processingtasks,in a field wheredataareeasilyavail-ableandin hugequantities.Roughly speaking,the problem can be presentedas thetransformationof the collectionof pixelsof an imageintoa meaningfularrangementof regionsandobjects[8]. But,how canweidentify objects? TheGestaltmovementidenti-fiedsomefactorsleadingto theperceptionof objects: sym-metry, similarity, parallelism,etc. [4]. Thoughtheserulesare conceptuallysatisfyingto explain the phenomenonatthe humanlevel, their lack of detailsmakesit hard to seethemasdirectlyimplementablealgorithms[4], evenif someof themcanbeextendedto rigorousalgorithms[15]. Sofar,imagesegmentationtechniqueshavetackledtheproblemby
studyingmathematicalpropertiesof theimageseen[16], ormorerarelyby a direct emphasison algorithmicandcom-putationalissues[3].Therearefour largecategoriesof approachesto imageseg-mentation[16], one of which is of direct interestto us :regiongrowing andmergingtechniques.In thisgroupof al-gorithms,regionsaresetsof pixelswith homogeneousprop-ertiesandthey areiteratively grown by combiningsmallerregionsor pixels,pixelsbeingtakenaselementaryregions.Region growing techniquesusuallywork with a statisticaltestto decidethemergingof regions[11, 16].In the field of imagesegmentation(or moregenerally, im-ageprocessing),MachineLearning(ML) techniquesmightappeartailor-made for high-level tasks, such as in [9].One might wonder whetherapplying such techniquesortheir adaptationsto imageprocessingbasedon low-levelcues(typically, imagesegmentation)might be well worththe try, in a field where the approachesare alreadynu-merous,of many kinds (local filtering, balloons,snakes,MDL/Bayesian, region growing, etc. [16]), and whereheavy mathematicalapproacheshave obtainedsignificantresults(seee.g. the resultsof [16]). We think somecluescoming from image processingindicate that ML-derivedtechniquesmight be of interestalreadyfor low-level pro-cessingtasks. First, vision is widely acceptedas an in-ferenceproblem, i.e. the searchof what causedthe ob-served data[4]. Second,asarguedby [12], low-level im-agesegmentationshouldnotaimatproducingacorrectseg-mentation,but rathera goodapproximationgiventhepost-processingstagesof the image. Onemorereasoncangiveevidencefor thepossibleinterestin usingML-relatedtech-niques. ML andComputationalLearningTheory(COLT)workssuchas[7] have broughta numberof usefulresults,or found a way to usea numberof results,in statisticsorprobability, which remove many undesirabledistributionassumptionson the data. The mostcommonlyusedtheo-retical foundingmodelfor ML/COLT is thePAC modelofValiant [13], which removesany suchdistribution assump-
1
tions. In imagesegmentation,therehave beena numberofprobabilisticapproachesto theproblem[1, 16] or [4] (Chp.18). Mostof themrely onheavy, penalizingdistributionas-sumptionsonthedata(suchasnormality),andfinally somerecentprominentwork in imagesegmentationhasmadeem-phasison moreconventional,non-probabilisticapproaches,suchasdiscriminantanalysis-type[12, 2].Our aim in this paperis clearlynot to proposea ML algo-rithm to segmentimages,but ratherto exploit someof thecommonpointsbetweenimagesegmentationanddecision-treepruning,andthenadaptthetheoreticalML/COLT ideasand tools of [7] to our setting,to derive a new segmenta-tion algorithm. This paperconsiststhereforeof anoriginal(andnovel) adaptationof previousML/COLT work to low-level imageprocessing,a ratherseldomresult in our con-text. Thenext sectiongoesin depthin thecommonpointsbetweendecision-treepruningandimagesegmentation.Itis followedby asectionpresentingamodelof imagegener-ation. Then,two sectionsdetail respectively the statisticalandalgorithmiccontentsof our approach.The lastsectiondiscussesexperimentsonnumerousimages.
2 From pruning to merging
A decisiontreeis aclassifiermakingrecursivepartitionsof a representationspace,accordingto someclasses.Con-siderfig. 1, showing a 2D representationspacerestrictedto the integer couplesof the domain
� ������� � � ��� ���� forsome
����� ������. Supposethat thereare � classes(not
shown). Thepoint is thata probabilitydistribution is givento domain� classes. Each couple (observation, class) iscalledanexample, andtheproblemis to fit asbestaspos-sible the domainof all examples,usingonly a potentiallysmall subsetof this domain,sampledaccordingto the dis-tribution. Decisiontreesarevery efficient ways to tackletheproblemusinga verysimpleformalism[7]. Thequalityof a decisiontreeis evaluatedby its error probability overthewholedomain,alsocalledstructuralrisk, which we canonly estimateusingthe error frequency over the examplesseen,also called empirical risk [7]. The classificationofanobservationis madeasfollows: eachinternalnode(andtheroot) of thetreeis labeledby a teston a variableof theobservations;eachleaf is labeledby a class(not shown infig. 1). The classificationprocessstartsfrom the root ofthe tree. If the observation satisfiesthe currenttest, thenit follows the right path,otherwiseit follows the left path,until it reachesa leaf,andthereforeis giventhecorrespond-ing class. In fig. 1, an observation for which
�������and ��� ��
would fall in region � , asshown by thedottedpath.Themostpopulardecisiontreelearningalgorithms(CART,C4.5[7]) proceedby growing a very largedecisiontreetofit asbestaspossibletheexamplesseen,andthenpruningitto get(hopefully)agoodfit of thewholedomain.Pruninga
x
y >y2
x > 3xx > 2x
x > 1x
y >y1
00
CAB
y
y1
y2
x 3x2x1 4
y3
x
YesNo
Figure 1. A domain and its recur sive par titionby a decision-tree (see text for details).
treeconsistsin removing iteratively aninternalnodeanditssubtree,therebyreplacingthe nodeby a classlabel. Prun-ing thenodewhosetestis “
� �����” in image1 would boil
down to mergeregions � , ! and " .If we considerthat the color valuesof pixels in an imageare the result of a theoreticalvalue(the oneof the regionthey belongto), combinedwith a randomvariability fac-tor (suchasnoise),thenthe taskof region merging canbelooselyreformulatedastherecognitionof the regionshav-ing thesametheoreticalvaluesin anobservedimage.Thisproblemis quite similar to decisiontreepruning(considerthat theimagehas
��# ��pixelsin fig. 1); moreimportantly,
the tools usedin both problemscanbe the same:the con-centrationof randomvariablesto give confidencebounds,eitherfor thetheoreticalcolorvaluesof groupsof pixels,orfor thestructuralrisksin decision-treepruning.However, our taskappearsto be moredifficult thanprun-ing, sincewe generatemuchmorepotentialconfigurations.Considere.g. fig. 1: the treehasonly 7 possibleprunings(including theoriginal tree),whereasthereare42 possiblesegmentationsof thedomainwith the6 initial regions.Imageprocessinghaspotentiallytwo characteristicswhichmakeit agoodcandidatefor theadaptationof particularMLtechniques.First, the quality of the pruning strongly de-pendson the availablequantityof data,andsmall datasetstypically leadto uncontrollableover-pruning.In imageseg-mentation,thematerialis availablein hugequantities.Sec-ond,imagesegmentationis a field in which fastalgorithmsareobviously crucial with the advent of real-time(video)imageprocessing,but practiceshows that “f ast” is a field-dependentsubjective notion: for example, [1] considera“very fast” algorithmextractinga few dozensof regionsina 512� 512imagein lessthan10 seconds,but on anhigh-endUltraSPARC workstation;[12] giveexecutiontimeson100� 120 imagesof about2 minuteson conventionalma-chines,and thereare many other examples. We show inthatpaperthatusingourML-derivedtoolscanbringhighly
competitive(or better)results,in reducedtimes.
3 Notations and models
The notation $&%'$ standsfor cardinal. The observed im-age, ( , contains $ ()$ pixels, eachcontainingRed-Green-Blue (RGB) values,eachof the threebelongingto the set*�+��-,.� %/%0% � ��1 . We have deliberatelychosennot to usecom-plex formulationsof thecolors,suchasthe 243657398:3 space[1]. ( isanobservationof aperfectscene(.; wedonotknowof, in which pixelsareperfectlyrepresentedby a family ofdistributions, from whicheachof theobservedcolor-level issampled.In (<; , theoptimal (or true)regionsrepresentthe-oreticalobjectssharinga commonhomogeneityproperty:= all pixelsof agiventrueregionhaveidenticalexpecta-
tionsfor eachRGB color-level,= theexpectationsof adjacentregionsaredifferentfor atleastoneRGB color level.( is obtainedfrom (.; by samplingeachtheoreticalpixel
for observedRGB values.Fig. 2 presentsanexampleof acolor-level for onepixel in ( ; andhow to generatethecor-respondingobservedcolor-level of the pixel in ( . In eachpixel of (.; , eachcolor-level is replacedby a setof > in-dependentrandomvariables(r.v.) takingpositivevaluesondomainsboundedby �:?�> , suchthat any possiblesumofoutcomesof these> r.v. with non-zeroprobabilitybelongsto*�+��-,.� %/%0% � ��1 .
Thesamplingof eachpixel andits color levelsaresupposedindependentfrom eachother. Our modelof imagegener-ation doesnot unfortunatelyprevent us to dependon thisusualassumption(alsowidely usedin ML/COLT). Our ex-perimentsshall demonstratethat it doesnot seemto haveanimpacton theresultsof thesegmentation.It is importantto notethat this is theonly assumptionwe make on (<; . Inparticular, we do not make any further assumptionon thenatureof eachdistribution(suchasnormality, homoscedas-ticity), whichcanbethereforedifferentfrom oneanotherineachpixel of a region, aslong asthesumof their expecta-tionsis constantfor eachcolor insidea trueregion.Ourgoalis thenstraightforward:find observedregionsin (approximatingasbestaspossiblethetrueregionsin (.; .> is certainlythelessintuitiveparameterof ourmodel.Wehave chosento introduceit for our model and our analy-sesto be asgeneralaspossible(standardanalyseswouldfix > �@+ ). Practicallyspeaking,> representsthe majoradvantageto beatrade-off parameter, adjustableto obtainacompromisebetweenthepowerof themodelandthequalityof theobservedresults.Indeed,if > is small (say, > �A+ ),thenscarcelynothingcanbeestimatedreliably for smallre-gionsin ( , but our modelis themostgeneralpossible.Ontheotherhand,if > is toolarge,thenourmodelbecomesre-stricted( > � � bringsbacka conventionalBinomial law),
but ourestimationsarethemostreliable.Between,theusercanchoosea convenientvalueto keepa powerful enoughmodelwhile ensuringreliableestimationson ( .4 The merging test and its properties
Theorem 1 (Theindependentboundeddifferenceinequal-ity, [10]) Let X
�CBEDGFH�D � � %/%0% ��DJI<K be a family of inde-pendentr.v. with
DMLtakingvaluesin a set � L for each N .
Supposethat the real-valuedfunction O definedon P L � Lsatisfies$ O B x KRQ O B x S K $<T�U L whenevervectorsx andx S dif-fer only in the N -th coordinate. Let V betheexpectedvalueof ther.v. O B X K . Thenfor any WYX � ,
PrB O B X K6Q V XZW K T [�\ ]_^ ]`ba�c&d a�e ]
(1)
In ourcase,wenow provethatwith highprobability, theobservedaveragefhg of any region f in ( shallnot deviatetoomuchfrom its theoreticalexpectationE g B f K for asinglecolor-level i . By theoreticalexpectation,we meantheav-erageof thesumover eachof its pixelsof theexpectationsof its > distributionsfor color i in (<; .Theorem 2 Let f be a region in ( . Let jMk be the setofregionshaving l pixelsin ( . Fix an iGm * R � G � B 1 . n)opS �q� ,theprobability that thereexistsa region in jsr tur such thatvv f g Q E g B f K vv X � w +, >M$ fJ$6x.y0z�{ , $ jsr tur0$o S | (2)
is no more than opS .(Proof omitted due to the lack of space). Of course,theprobability of occurrenceof the event for someof the R,G, or B levels, is no morethan }�opS . What is interestingintheorem2 is thatregion f is not requiredto belongto atrueregion of (<; . Indeed,if the region merging algorithmhasmergedtogethersubsetsof trueregions ~ Fp� ~ � � %0%/% � ~ L in f ,thentheboundof theorem2 still holds.Theuseof theorem2 will be straightforward. Theprobability that thereexistsa region in ( (regardlessof its size)for which the averageof someof its RGB color deviatesfrom its expectationbymore than the right-hand-sideof inequality 2 is no morethan }�$ ()$ opS , sincethe region canhave size1, 2, ..., or $ ()$ .Then,if we fix o � }�$ ()$ o S (3)
and � B f K�� � w +, >M$ fJ$ x.y0z�{ ,o S6� y0z�{ $ jsr tur'$ | (4)
�region �4�
imageborder��� �region �J�
a pixel of �J�
Figure 2. Generation of a single color -level for one pix el from (<; to ( .thenweknow that,with highprobability(
��+�Q o ), any pos-sibleregion f will have a boundedvariationof f7g aroundEB f K g for all valuesi�m * R � G � B 1 . Becauseof the trian-
gle inequality, weknow thatany two regions f and f7S hav-ing the sameexpectations(E g B f7S K�� E g B f K���� g , n�i�m*
R�G�B 1 ) shallobserve a quantity $ f S g Q f g $ which will
bealsoconcentratedaroundzero,upto radius
� B f K �� B f�S K
actually, since $ f S g Q f g $�T�$ f S g Qq� g $ � $ f g Qq� g $ (tri-angleinequality). If $ f S g Q f g $�T � B f K �� B f�S K for alli�m * R � G � B 1 , thenwe cansupposethat f and f S belong
to the sametrue region in (.; , andmergethem. This givesthe merging predicate� B f � f�S K of our region merging al-gorithm,returningwhetherf and f�S canbemergedor not:� B f � f S K����� � true if f n�iJm * R � G � B 1 �$ f S g Q f g $.T � B f K �
Supposethat the image ( contains ª¤« rows and U�«columns. This represents¬ �@, ª¤«#U�« Q ª#« Q U�« couplesof adjacentpixels (in 4-connexity). Name �« asthe setofthesecouples.DenoteO B % K thefunctionwhich takesa cou-ple of pixels
B&®���® S K , andreturnsthemaximumof the threecolor differences(R, G andB) in absolutevaluebetween®
and® S . Denoteasquicksort( « , O ) the outputof the
quicksortalgorithmfor set « , in increasingorderof func-tion O B % K . For any pixel
®of ( , we denoteas f B&®)K thecur-
rentregion to which®
belongsin ( . Thealgorithmis calledPSIS, for ProbabilisticSortedImageSegmentation.
if Æ�¼¾½.¿ ÂÈǵ�Æ�¼¾½ ± ¿  and Éh¼EÆ�¼¾½.¿  ÀÊÆ˼¾½ ± ¿ Â� =true thenUnion( Æ�¼¾½ ¿  , Æ�¼¾½ ± ¿  );
Onemightwonderwhy wehavechosento orderthecou-plesof pixelsin PSIS. Indeed,sucha requirementis a pri-ori notclearfrom our theory, andit bringsan Ì B $ (�$ y/z�{ $ (�$ Kcomplexity (sometimessmaller, [2]) insteadof the(almost)linear complexity reachablewithout this stage. Therearebasicallythreekind of errorsour algorithmcansuffer withrespectto the optimal segmentation.First, under-mergingrepresentsthecasewhereoneor moreregionsobtainedarestrict subpartsof trueregions.Second,over-mergingrepre-sentsthecasewheresomeregionsobtainedstrictly containmorethanonetrueregion. Third, thereis the“hybrid” (andmostprobable)casewheresomeregionsobtainedcontainmorethanonestrict subpartof true regions. Orderingthepixelstestis away to limit this third kind of error, sinceweapproximatethefollowing property:
(P) All merging testsinsidetrue regionsaremadebeforeany merging testbetweentrueregions.
Obviously, (P) is unreachablein practice;however, from apurely theoreticalpoint of view, (P) hasa key effect whenusedwith our statisticalmaterial:PSIS only suffersover-merging with probability
�Í+JQ o (eq. 3). Indeed,weknow that for any regions f � f7S coming from the sametrue region in ( ; , we have n�i m * R � G � B 1 � $ f S g Q f g $�T$ f S g QÎ� g.$ � $ f�g QÎ� g.$ where
� g � E g B f7S K�� E g B f K . Withprobability
��+:Q o , wealsoknow (theorem2 andeq.3) that$ f S g Q fhg.$�T � B f K �� B f7S K for any regions f � f7S coming
from thesametrue region in (<; . Sincethis is our mergingtest, andsince(P) holds, it follows that the segmentationobtainedis anunder-segmentationof (<; .Therefore,theoreticallyspeaking,(P) allows to eliminate
Image PSIS [1] Image PSIS [1]
Figure 3. Results obtained by PSIS and [1] on four images. Region are white with black bor ders.
under-merging andthe hybrid error casewith high proba-bility; thismotivatedusin fastapproximationsof it. In fact,over-mergingcansomewhatbecontrolledin practicefor re-gionswith sizelargeenough(typically Xb� ) aslong astheboundfor $ j k $ is nottoolarge(seeEq. 4). Thisis thereasonfor our choiceof theupperboundfor $ jMk�$ .6 Experimental results
Dueto thelack of space,we reportexperimentsonly onthirty-fiveimagesamongall onwhichPSISwastested(fig-ures3,4 and5). While lookingattheresults,thereadermaykeepin mind that:
thevaluesof theparametersof PSIS arethesamefor all images: opS ��+ ? B }�$ (�$ � K (theorem2) and> � } , . Furthermore,the imageswereusedasthey are,i.e. withoutany preprocessing.
Therefore, the results of PSIS do not stem from anydomain- or image-dependentpreprocessingor parametertuning.PSIS wasimplementedin C. Fig. 3 displayssomeresultsto becomparedwith [1, 16]. While PSIS obtainedrecord times for processingtheseimages,the resultsarehighly competitivewith theotherapproaches.Considerthewoman image.Herbustis bettersegmentedthan[1]. Whenlooking at [16]’s resultfor this image,our resultappearstobe better: after over a hundrediterations,[16]’s techniqueof regioncompetitiondoesnotmanageto find thewoman’seyes. [16] argue that that the eyes are too small and as-similatedto noise. This is clearlynot a mistakePSIS hasmade.Thehand imagealsodisplaysgoodresults,all thebetterif we considerthat themodelof imagesegmentationof PSIS doesnot explicitly integratetextures.In fig. 4, theleft tabledemonstratesthatPSIS obtainsagainniceresults
on texture segmentation(tennis androck images),allthemoreinterestingif we comparethemto theapproachesof [5, 6], tailor-madefor texture segmentation. The rightimageshows for someimagesa particularregion isolatedbyPSIS. Remarkfromlena andsquirrel thatPSIS isableto isolateregionswith high variability (e.g. thegrass),and obtainsresultseven betterthan [16] on the squir-rel image: their segmentation,althoughtailor-madefortextured images,obtainsa segmentationof the grasswithmany holes,a populardrawback of region-merging tech-niques[16], seealso the result of [2] in fig. 5 (bottomrow, region #2) for the grass. Note also the nice segmen-tation of the truck comparedto [2] (fig. 5, bottom row,region #4). The small imagesin fig. 5 (a-x) display theability of PSIS to handlereasonablegradientsdueto light-ning andslantedsurfaces(imagesb, f, i, j, k, r, s, u,w, x). This correctsanotherdrawbackof many segmenta-tion techniques,relying on the assumptionthat the imageis piecewiseconstant[2]. Note that in imagex, PSIS ob-tainsexactly two regions. Thebowl region (x,#2) approx-imatesnicely thetrueobject,in a betterway than[2], whofind significantlymoreregions,againwith many holescom-paredto PSIS. In imagew, which containsgradientsandlight effects,PSIS findsexactly threeregions,approximat-ing almostperfectlythethreepartsof theimage:theback-ground,thepot, andits lid. Thegradientsandlight effectsof imagej aremuchmoredifficult to handle.In thatcase,PSIS found four regions (all shown), two of which (#2and#3) aregoodapproximationsof two partsof theobject(exterior/interior). More generally, many regionsfoundbyPSIS in thesesmall imagesapproximateconceptuallydis-tinct partsof theobjects:see(a,#4),(b,#3),(c,#4),(e,#3),(f,#3),(n,#4),(s,#3),(t,#3),andmany others.
Image PSIS [5, 6] Image PSIS detail
Figure 4. Results obtained by PSIS and [5, 6] on various images. The “detail” column is a regionof interest in PSIS’s results. PSIS’s segmentations are grey-leveled averaged with white bor ders.Conventions for the results of [5, 6] are various but intuitive .
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Resultsof [2] ÑFigure 5. Some images, and some of the most significant regions obtained by PSIS. The bottom rowsho ws the result of [2] on the street image. The conventions are the same for all region images(everything that is not the region is white), except for region Ò + on the street results, surr oundedby black due to the brightness of the road.
