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A Proximity Compatibility Function among 3-D Surfaces for
Environment Modelling
Liscano, Ramiro; Elgazzar, Shadia; Wong, A.K.C.
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3
3
3-D SURFACES FOR ENVIRONMENT MODELLING
RAMIRO LISCANO SHADIA ELGAZZAR ANDREW. K. C. WONG
Abstract
1 Introduction
2 Determining adjacent surfaces Ramiro.Liscano@nrc.ca elgazzar@iit.nrc.ca akcwong@watnow.uwaterlo o.ca
Institute forInformation Technology Departmentof SystemsDesign
National ResearchCouncil Universityof Waterlo o
Ottawa, Ont. K1A 0R6 Waterlo o, Ont. N2K3G1
CANADA CANADA
This article de nes a metho d for computing a
proximitycompatibilityfunctionamongfragmented
3-D surfaces for environment mo delling. Fragmented
surfaces are a commono ccurrence after the
segmen-tation pro cess has b een appliedto 3-D sensory data,
inparticularfordatatakenfromlargeindo or
environ-ments. This proximitycompatibilityfunction among
surfaces gives anindicationonhow closethesurfaces
aretoeach otherbased onacommongapde nedb
e-tween theb oundariesof thesurfaces. This particular
approach p erforms most of the computations in the
2-D image plane and when required will usethe 3-D
informationinthedata. This issimplerthantackling
thewholeprobleminEuclideanspaceandasanadded
b onus,sections ofthealgorithmp ertainingtothe2-D
imageplane can b e applied to classical2-D intensity
images.
Keywords: Surface proximity, Surface grouping,
Environmentmo delling
Research in the domain of computer mo delling
from 3-D data has primarilyfo cussed on the
extrac-tionof3-Dsurfaces andvolumetricprimitivesforthe
purp ose of either object recognition or creating more
precise mo dels from 3-D sensory data of machined
parts [7, 13]. These typ e of objects can easily b e
carried and placed in a controlled environment and
scanned usingahighresolution active sensor. Thisis
signi cantly di erent from the mo delling of large
in-do orenvironmentswhere it is necessary to bring the
sensor to the environment, changing the
characteris-ticsofthesenseddatadramatically.Theresultisthat
nearlyallscanstakenintheseenvironmentsconsistof
fragmenteddata anditb ecomes necessary to develop
algorithmsthatcanreasonamongthefragmented
sur-faces.
Researchinthemo dellingofindo orenvironments
has primarily fo cussed on the incremental synthesis
ofsensorviews and/orp ositionestimationof the
sen-sor[15,14,1]. Forthesesystemstob ecomeviableto ols
forComputer AidedDesign (CAD)it is necessary to
developapproachesthathyp othesizetheformationof
morecomp ositefeaturesfromthesurfaces.
Recently, there hasb een interest inthe
interpre-tationofscenes takenfrom3-Ddataprimarilyforthe
creation of mo dels of indo or environments [9, 12, 5].
Alloftheseapproachesrelyontheabilitytodetermine
theconnectivityamongthesurfaces. Insome
circum-stancesthesensorydataapp earsrelativelycleanwith
littleornogaps[12]andconnectivitycaneasilyb e
de-termined. Whileintheotherexamples[9,5]gapswere
presentbutlittleissaidab outthealgorithmsusedfor
determiningsurface adjacencyandproximity.
Thisarticlepresentsanapproachforcomputinga
proximitycompatibilityfunctionamong3-Dsurfaces.
Thisproximitycompatibilityfunctioncanb e usedby
otheralgorithmsasameasureofcon dencein
group-ing the surfaces. In particular to this research, the
proximitycompatibility functionwas used inthe
de-sign of a Bayesian Network for the grouping of 3-D
surfaces [10]. The approach presented can b e
sepa-rated into 2distinctop erations; thedeterminationof
adjacentsurfaces alongwith acommonarea of
inter-est b etween them,and thecalculationof aproximity
factorthatcan b eusedfordetermininghowclosethe
surfaceb oundariesare toeachother.
Itis necessary to de nethecontext inwhichthe
proximitymeasure istob eappliedsince itisp ossible
to have severaldi erent typ es of proximitymeasures
mea-S 2.1 Extracting a surface'sb oundary
2.2 Determining adjacent surfaces the sensor's view p oint. This implies that the data
p ointsareplacedinaparticularorderdetermined
pri-marilyfromthescanningcharacteristicsofthesensor,
i.e. they are not a cloud of 3-D p oints. When the
data isviewedwith resp ect to the camerathere
can-notb e anyoverlap amongthesurfaces, andtherefore
anyproximitymeasuresthatrelyonsurfacetosurface
overlap cannot b e applied. For this typ e of sensory
dataaproximitymeasurebasedonthesurfaceb
ound-aries is more appropriate. The result of this is that
the majority of the computations required to
deter-mine adjacent surfaces and their resp ective gaps are
p erformedinthe2-Dimageplane,andonlyattheend
of thepro cess arethe 3-D values usedfor computing
theproximitymeasure.
The approach is rather simple in theory but in
practice smo othing op erators and approximations to
theb oundariesmustb eappliedtoreducethee ectof
noise ontheimage. Thepro cedure fordetermininga
commongapisasfollows:Foranysurfaceinanimage
extractab oundary thatrepresentstheb order ofthat
surface. For each pixel on the b oundary determine
itsadjacentsurface. Collecttheb oundaryp ointsthat
share acommonadjacency to each otherand
reorga-nizethemso as tode ne acommonp olygonb etween
thesurfaces.
Thedatawasacquiredusingacompactlaser
cam-era called BIRIS [2]. Figure 1 is an intensity image
asso ciatedwiththe3-Dscanofacorner ofthelab
ora-tory. Planarsurfacesareextractedfromthe3-Dp oints
using analgorithmbased onahierarchical
segmenta-tionpro ceduredevelop edbyBoulanger[3].Theresult
of that algorithmisan imageof data p ointsgroup ed
intoasetofplanarsurfacesasshownin gure 2.
An edge trackingalgorithmis appliedto each of
thesurfacesdepictedinthelab eledimageshownin
g-ure2. Theedgetrackingalgorithmisanextensionof
theonedevelop edbyGaoetal[8]inthatitcomputes
anestimateforthecurvatureoftheedgeasitis
track-ingtheb oundaryofasurface. Thecurvaturealongthe
edgeiscomputedas adi erence inarunningaverage
of the gray level gradientvalues alongthe b oundary.
Currentlythis lteruses theaverageof 3pixel
gradi-entvaluesandapp earstob eableto lterthemajority
of largechanges inthegradientvalues which are due
primarilytothediscretizationoftheimage. Thehigh
curvature p oints are used to de ne a p olygon which
can b e used as another form of representing the 3-D
p olygons,inparticularforvirtualrealitymo delling.
Not everyhigh curvatureb oundaryp ointisused
inde ningthep olygonrepresentationforthesurfaces.
Figure1: Intensityimageofascan fromalab oratory.
Anysequentialsetofhighcurvaturep ointsisreplaced
by a straight line de ned by the rst and last high
curvaturep ointsinthatsegment. Theresultofthisisa
p olygon,asdepictedforthesurfacesin gure3,whose
cornersarehighcurvaturep oints(represente daswhite
pixels) connected by straight lines or low curvature
edges(represe nted as graypixels). This pro cedure of
by-passing aset of consecutive high curvature p oints
canb ejusti edbythefactthatinmostcasestheseset
ofp ointsareasso ciatedwithvirtualb oundariescaused
byweaksensorvaluesandsolittleinformationis lost
replacingonevirtualb oundarywithanotherstraighter
one.
Thisreducedsetofhighcurvaturep ointsareused
tode nea3-Dp olygonwhichcanb eusedtorepresent
thesurfaces forvirtualrealitymo delling. Figure 4is
aview of the surfaces, taken from aVRML (Virtual
RealityMarkupLanguage)browser,thatwerede ned
using the reduced set of high curvature p oints. For
thisdisplaythenumb erofp ointsusedinde ningthe
p olygonswas reduced byafactorof 10over usingall
the high curvature p oints. This is a signi cant
im-provementinthep erformanceoftheVRMLbrowser.
Beingabletojumpacross thegaps results in
de-terminingmorep ossibleadjacentsegmentsthanwere
originally computed using direct contact as a
crite-rion. This addedinformationisimp ortantwhen
deal-ing with fragmented surfaces. For example, surface
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1 1 S i S m S S S S S S P P S S S 4 5 10 13 16 17 2 1 22.3 Determining the commongap Figure 2: Surface lab elmapof ascan fromalab
ora-tory.
projection algorithm resulted in 5other p ossible
ad-jacentsurfaces , , , ,and andtherefore
agreater p ossibilityofb eingabletojointhesurfaces.
Adjacent surfaces can b e determined by
project-ing fromeach of the b oundary's pixels ina direction
away from the surface. At a globalp ersp ective, the
b oundaries of a surface that were formed due to a
weaksensorsignalmaintainsomedirectional
informa-tionab outthecontinuationofthat surface. This can
b e observed in the top and b ottom edges of surface
in gure 3. This direction of theb oundary is
af-fectedbythep ersp ective transformationofthesensor
but their e ect is fairly minimal since their
projec-tiontendsnottoresultinanintersectionwithanother
surface. Boundariesthat arereal b oundaries, formed
by intersections or jumpedges, maintaintheir
direc-tionalinformationandapp earasedgeswithlessnoise.
Atalo calp ersp ective, direction ofthe segmentsthat
makeupab oundarycanvarydramatically. Thise ect
is reduced substantially by cho osing to represe nt the
b oundaryasap olygonsincethelargedeviationsinthe
gradientthato ccurb etweensequentialhighcurvature
p ointsareremoved.
The p ointsat which the normalprojections ofa
b oundary intersect with another surface's b oundary
arelab eledastheneighb ourp ointsofthe rstb
ound-ary. These are recorded and are used to computea
commongap b etween the2 adjacentsurfaces. While
thep ointsalongaparticularb oundaryarerepresente d
Figure3: Surfaceb oundariesofascanfromalab
ora-tory.
ingp ointshavenoparticularordering. Thisis
demon-stratedin gure5wherealltheorderedp ointsb etween
to haveneighb ouringpixelvalues onsurface
,buttheorderofthosecorresp ondingneighb ouring
p ointsisnotapparentandmustb e determined.
An approximation for the shap e of the gap b
e-tweenthesurfacesisdonebycollectingtheneighb our
p oints and ordering them such that a p olygonal
sur-face is de ned common to b oth surfaces and .
This isdone inthe followingmanner: Determinethe
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(b)
(a)
S13
S17
s s 1 1 1 1 1 1 i j i j i j i j ss s s ss s s 1 2 2 1 2 1 2 1 2 2 17 13 17 133 Computing a proximitymeasure S i S m S q S u S m S q pr ox i j g g i j i j pr ox i j S S S S S S S S S P P P P P P S f S ;S A A A ; A A A S S f S;S S S S S
Figure5: Neighb ourp ointsof with .
extremap oints(blackp ointsin gure 5)ofthe
neigh-b our p oints on that corresp ond to . Determine
the segment of theb oundary on that containsall
the neighb our p oints. Extract fromthis segmentthe
corner p oints, since all that is required to de ne the
segmentaretheextremap ointsandthecorner p oints
withinthesegment. Thegapisde nedbyjoiningthis
segment with anothersegmentfromthe b oundary of
thatcontainedtheneighb ourp ointsthatarewithin
thesegmentfrom ,see gure 5.
Theredo es notexistone uniquegap b etween the
surfaces,infactatleast2commongapscanb ede ned;
One with resp ect to the rst surface connecting it to
anothersecond surface and thereverse, one fromthe
secondsurfacebacktothe rst. Thisisdemonstrated
inthegapsde nedin gure6(a)and(b)where gure6
(a)arethegapsde nedwithresp ec ttosurface and
gure6(b)thosewithresp ect tosurface . Thiswill
result in2valuesofproximityas de nedinsection3.
As well as these 2 p ossible cases multiple gaps
canb ede nedwithresp ecttoonesurfacetowardsthe
other,asshownin gure6(a)asthehatchedp olygons.
One gap is de ned fromthe continuous set of p oints
startingat to and anotherfrom to .
Thissplitisaresultofthedirectionofthelinede ned
from to whose projectiondo es notintersect
with the . In this situation one can sum up the
e ect of these 2gaps when computing the proximity
measure.
The objective is to de ne a compatibility
func-tion among the surfaces that quanti es the distance
b etweentheb oundariesof2surfaces. Ourapproachis
similartotheproximityapproachusedbyFanetal.[6]
fordeterminingtheproximityb etween theendp oints
the gap b etween the 2closest endp ointsof the lines
to the sumof thelengths of the2 lines. Inthis
par-ticular casethe3-Dsurfacesreplace thelinesandthe
distance b etween the2 closest end p ointsis replaced
byacommonp olygonshap edgap. Insteadofusinga
lineardistanceasameasurementoftheproximityitis
moreappropriate to use thearea of the surfaces and
of the gap. This results in the followingmeasure of
proximity,
( )=
+
(1)
where istheareaofthegapb etween surfaces
and ., and and are thearea of thesurfaces
and . The closer the surfaces are to each other
thesmallerthevalueof ( ).
Up to this p oint all computations where p
er-formedutilizing the 2-D image represe ntation of the
3-D sensoryp oints. Togeta prop er estimate forthe
proximity measure b etween the surfaces it is
neces-sary to compute the area of the 3-D surfaces. Note
that the approach could b e used on 2-D surfaces by
simplycomputingtheareaofthe2-Dp olygon,inthis
mannerthisapproachcaneasilyb eappliedtosurfaces
ina2-D image plane. The areas forthe surfaces are
computedusing an algorithmfor computingthe area
ofap olygon. Anestimateoftheareaofthegapisp
er-formedbyapproximatingitas atriangularmesh [11]
and summingthe areasof thetriangles in3-Dspace.
This triangular mesh approximation is necessary b
e-causetheshap eofthegapcanb e anirregularshap ed
surfaceandnotplanar. Anexampleofthisisshownin
gure7(a)wherethegapde nedb etweensurfaces
and hasb eentriangulatedinthe2-Dimage. The
result ofthistriangulationin3-Disshownin gure7
(b).
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4 Discussion i j i j i j i j i i j j 1 2 2 1 2 3 7 10 13 12 6 S S S S S S S S S S S ;S S S S S S S S S S S SFigure6: Proximitygapsb etween surface andsurface (a)andb etween surface and surface .
Pr. Pr. Pr. 1 6 0.23 9 12 0.23 13 10 0.09 1 9 0.04 10 17 0.03 13 5 0.01 4 17 0.12 10 9 0.03 14 8 0.31 4 5 0.39 10 12 0.15 14 11 3.47 5 4 0.24 10 15 0.52 15 18 0.85 5 17 0.23 10 5 0.03 15 10 0.55 5 10 0.05 10 13 0.10 15 12 1.46 6 12 11.3 11 14 2.29 16 11 1.10 6 9 0.00 11 8 13.5 16 17 0.01 6 1 0.27 11 16 6.5 17 16 0.00 8 11 13.4 12 15 1.51 17 10 0.01 8 14 0.30 12 10 0.51 17 13 0.05 9 1 0.02 12 9 0.60 18 15 0.86 9 6 0.17 12 6 3.44 18 17 0.48 9 12 0.00 13 17 0.03 18 10 0.48
Table1: Proximitymeasuresamongadjacentsurfaces.
Table 1 presents calculated proximity values
amongtheadjacentsurfaces. Notallthesurfaceshave
b eenincludedinthecalculations,anumb erofsurfaces
b elowacertainareawereeliminatedattheb eginning
of the pro cess. Inparticular surfaces , and
were allconsidered to osmallinsizeto pro cess. Note,
a smallvalue for proximityimplies that the surfaces
arecloser.
Adjacent surfaces willnothave proximityvalues
of0for several reasons. First,the pro cess of de ning
asurface b oundary results inthe creation of a small
gap b etween the surfaces. Even if the surfaces were
originallyconnected there willalwaysb e asmallgap
ondly, irregular b oundary shap es can result in gaps
that extend b eyond thelengthof theedges originally
shared b etween the adjacent surfaces. For example,
surfaces and in gure 1(a) share acommon
edge. Smallgapswillb e detectedat thetopandb
ot-tom of the commonedge due to the irregular shap e
of this edge. Thirdly, the commonedgeb etween the
surfacesmayb eajumpedge,sothesizeofthegapcan
b eratherlargeindepth. Thisisthesituationdepicted
b etweenthesurfaces and wheretheyapp earas
joined surfaces in the imagebut are at di erent
dis-tances fromthesensor.
Atmost2proximitymeasuresmayexistb etween
adjacent surfaces and should b e nearly of the same
value. Table 1(b)shouldb e interpreted asthe
prox-imitymeasureb etween and relativeto ,
there-fore another mayexist b etween and relative to
. In some circumstances only one proximity
mea-sure willb eavailable. Thismayo ccurwhenthep
oly-gonthatde nesthegapb etweenthesurfaceshas
self-intersectingedges. Auniquealgorithmwasdevelop ed
to automaticallytry to correct for thisbut will
even-tuallyfailifto omanyself-intersectionsarediscovered.
Self-intersectionmayseemtheoreticallyimp ossiblebut
in practice can o ccur when the b oundaries are very
closetoeach other.
Theproximitymeasurecomputesonevaluetob e
usedfordeterminingifadjacentsurfacescanb e
consid-ered proximaltoeach other. Theproblemstill exists
ofhowonewouldcho oseareasonablethreshold value
forproximity.
Equation1isaratioofsurfaceareas,thoseofthe
area of the gap to the sum of the areas of the
5 Conclusions
Arti cialvisionformobilerobots: stereo visionandmultisensoryperception.
Proceedingsofthe 1991IEEE International Con-ferenceonRoboticsandAutomation
Proceedingsof the FifteenthInternationalWorkshop
onMaximum EntropyandBayesianMethods
Thefuzzysystemshandbook
The2ndWorkshoponSensorFusionand
EnvironmentModelling
IEEE Trans. onPatternAnalysisandMachineIntelligence
Machine vision for three-dimensional scenes
Pattern Recogni-tion
IEICE Trans. on Communications, Electronics,
In-formation, and Systems
Proceedingsof Vision Interface 97
ComputationalGeometry: Theory
andApplications
Proceedings
of the SPIE: Videometrics IV
Three-dimensional object recognition from range images
Proceedingsofthe SPIE: Sensor Fusion andAerospaceApplications
3D Dynamic Scene Analysis: A StereoBased Approach
[1] N.Ayache.
(Cambridge: MIT
Press,1991).
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[3] P.Boulanger. Hierarchicalsegmentationofrangeand color images based on bayesian decision theory. In
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[5] M.Devy,J.Colly,P.Grandjean,andT.Baron. Envi-ronment mo dellin gfrom laser/cameramultisens ory system. In
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forReview,Kelowna,Canada,May19-231997. [11] R.Seidel.Asimpleandfastrandomizedalgorithmfor
computingtrap ezoidal decomp ositio ns andfor trian-gulatin gp olygons.
,1(1),July1991,pages51{64. [12] V.Sequeira,J.G.M.Goncalves,andM.I.Rib eiro.3d
scene mo dellin gfrom multiple views. In
, volume 2598, pages 114{127,Philad ephi a, Pa.,Octob er25-261995. [13] M. Suk and S. M. Bhandarkar.
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. (Berlin:
Springer-Verla g,1992). bypropagatingasurface's b oundary, ina2-D image,
untilitcollideswithanothersurface. Reasonable
prox-imityvalues have moreto dowith the psychology of
p erceptualgroupinganddonotfollowanyhardrules.
Ingeneral it app ears that surfaces separated bygaps
thatareab outthesamesizeasthemselvesdonotgive
the app earance of b eing proximal. If the 2 surfaces
are ab outthesamesize then thisresults ina
thresh-old value of 0.5, so that any ratio ab ove 0.5 can b e
consideredasnon-proximalsurfaces.
Insteadofusingtheab ovethresholdruleasacrisp
value it is also p ossible to mapthe proximityvalues
to a fuzzy set by using adeclining S-curve [4]. This
typ eofmappingwasusedbyLiscanoetal.[10]inthe
designofaBayesian Networkforthegroupingof3-D
surfaces. Theuse of aBayesian Network allowedthe
proximityvaluetob e propagatedand combinedwith
othermeasures ofevidencewithouthavingtomakea
decisiononproximityatanearlystageofthegrouping
op eration.
Thisarticlepresentedanapproachforcomputing
aproximitycompatibilityfunctionb etween2surfaces
in3-Dspacebasedonaratiooftheareaofacommon
gap b etween the surfaces to the sum of the area of
the surfaces. The approach taken was to determine
adjacency b etween fragmentedsurfaces and compute
ameasureofproximityb etween2adjacentsurfacesby
computing a common gap b etween their b oundaries.
The diculty is in estimating the shap e of the gap
b etween the 2 surfaces. The presented metho dology
computes this shap e using the image plane acquired
from a sensor view, thus reducing the complexity of
theproblemdownto 2-Dimageanalysisinsteadof
3-D geometrical analysis. After the shap e of the gap
is determined the actual surface areas are estimated
in 3-D space and the gap is triangulated so that an
estimateofitssurfaceareacan b ep erformed.
The proximity measure was applied to 3-D
sen-sory data acquired froman indo or environment that
was segmented into planar surfaces. The approach
couldb eappliedtoanyothertyp eofordered3-D
sen-sory p oints, but itis b etter suitedfor data that
con-tainsfragmentedsurfaces. Ifthesensorydataisdense
(withrelativelysmallgaps),thensurfaceadjacencyis
easilydetermined andit issimplertoextend the
sur-faces until they join. The surfaces donot necessarily
have tob e planar,b ecause thealgorithmop erates on
