Belief networks for the perceptual grouping of 3-D surfaces

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Belief networks for the perceptual grouping of 3-D surfaces

Liscano, Ramiro; Elgazzar, Shadia; Wong, A.K.C.

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Abstract

1 Introduction

Surfaces

R.Liscano S.Elgazzar A. K.C.Wong

Ramiro.Liscano@nrc.ca elgazzar@iit.nrc.ca akcwong@watnow.uwaterlo o.ca InstituteforInformationTechnology Dept. ofSystemsDesign

NationalResearchCouncil UniversityofWaterlo o Ottawa,Ont. K1A0R6 Waterlo o,Ont. N2K 3G1

CANADA CANADA

Research in the domain of mo deling using 3-D data has primarily fo cussed on the extraction of 3-D surfaces and volumetric primitives for the purp ose of either object recognition or creating moreprecise mo dels from3-D sensorydataof machinedparts. Thesetyp eof objectscan easily b e carried and placed in a controlled environment and scanned using a high resolution active sensor. Forthemo delingoflargeindo orenvironmentsitisnecessarytobringthesensortotheenvironment, Thisarticle introduces anapproach, basedon BayesianNetworks,for the grouping of 3-D surfaces extracted fromdata obtained by a laserranging sensor. A methodology based on the decomposition of an object into its sub-parts is used for specifying the structure of the network. Conditional probabilities are computed using a setof compatibilityfunctionsthat measurethe area measure in the quality of t of the data to a model that the features may havecome from. These compatibility functions are akin to measures that are used for the perceptual organization of features in the computer vision domain except that they have been developed for 3-D range data. An approach is presented for the mapping of the compatibility functions to conditional probabilities that are required by the Bayesian network. An example of a Bayesiannetwork is presentedthat modelsthe detectionof cornersandcontinuityamongplanar surfaces andusesbothrangeandintensityvalues asfeaturessets. TheBayesiannetworkisusedtocomputeabeliefvaluein theformationofcorners andcontinuity among the surfaces which inturn canbeused to decide if surfaces shouldbejoined. Resultsandanalysisarepresentedforanactual setofintensityandrangeimagestaken ofa typical indoor scene of a robotics laboratory.

2 Constructing the Bayesian Network

scans taken in these environments consist of missingdata and o ccluded objects. Recent attempts in the mo deling of larger environments [1, 2, 3, 4] from 3-D sensory data have demonstrated it to b e a dicult task, due mainly to the amount of missing, sparse, and obscured data. This requires algorithms to b e develop ed that can manageseveral sources of evidence for determining how surfaces should b e group ed and should also maintain b elief values in the formation of these groupings.

Research in the mo deling of indo or environments has primarily fo cussed on the incremental synthesis of sensor views and/or p osition estimation of the sensor [5, 6, 7, 8]. For these systems to b ecome viable to ols for Computer Aided Design (CAD)it is necessary to develop approaches thatcanextractgeometricsurfacesfromthedataandhyp othesize theformationofmorecomp osite featuresfromthesurfaces,i.ejunctionsamongthesurfaces. Inmostcircumstancesasetofheuristics have had to b e declared to decide what and how the surfaces should b e joined. Unfortunately these heuristics tend to b e emb edded into the algorithms and no formal approaches are used for maintaining a b elief value in the grouping pro cess. This leads to systems that are not easily extendable and theuser hasno abilityto sense ofthequalityin the group edsurfaces.

Thisarticlepresentsanapproachtothegroupingofsurfacesbasedontheformalismsdevelop ed for Bayesian networks. A Bayesian network oers a cohesive approach to the sp ecication of relationships among feature sets as well as a metho dology for the computation of b elief values in the existence of a formation among the features. Bayesian networks have b een used in the domain of computer vision forobject recognition [9, 10, 11, 12], multi-agent vision systems [13], road scene recognition [14], tracking of dynamic objects in images [15], and p erceptual grouping for2-D images [16]. A common thread among those systems in the useof constraints amongthe featurestohelpeither recognize ortrackobjectsin a scene. Theseconstraints areamathematical expression of p erceptual organization and the Bayesian network enforces a structure that denes intermediateformationsamongthosefeatures. TheBayesiannetworkalsoenforcesacomputational mo delforb eliefvaluesin theexistenceofthoseformations. Thisissimilarinconcepttothesystem develop edbyLevittetal.[9]inthattheb elief valuesoftheno desintheBayesianNetworkrepresent a probability of the existence of formations among the features. This is a well sound mo del and thechallengestill remainsofdeterminingapproachesforconstructingthenetworkanddetermining theconditional probabilitie s.

Thisparticularimplementation ofa BayesianNetwork isbasedon themo deling of comp osite geo-metricalformationsfromotherfundamentalfeaturescomputedfromthesensorydata. Acomp osite geometrical formation is created from the aggregation and combination of other geometrical for-mationsuntil nally it is comp osed of indivi dual comp onents that can b e detected by thesensor.

2.1 The Network's Structure

Surfaces Vertices 3-D Points Edges

Edges

Cube Surfaces 3-D Points

Surfaces 3-D Points

toaconditional probabilityforthe Bayesian network. The Bayesian Networkallows the enco ding of the exp ected geometricalformations that thesensor maydetect and infer a b elief value in the existenceof thatformation.

A Bayesian Network is a directed acyclic graph with the no des representing a hyp othesis of the existence of a prop osition and the arcs signifying the causal relationship from one prop osition to thenext. Thisresults in a simple mo delof theuniverse but one that can b euseful forp erceptual grouping. Deciding on the structure and variables of a Bayesian network is a mutual exercise. The approach prop osed in this article for guiding the design of the network's structure is the theory of hierarchical decomp osition by parts that has b een previously used for the mo deling of objects[17,18]. Decomp osition bypartsis agenericconceptthat cannotb euseddirectlybut asa guide thedesign ofthe network. It wasalso primarily a concept develop ed for thedecomp osition of objects forthe representation in Computer Aided Design (CAD)thatdo nottake into account thefeaturesandcomp onentsthatcan actuallyb edetectedbya sensor. Figure 1illustratesa cub e decomp osedintoits comp onents thatdirectlymaps tothe Bayesian networkshowntotheright of thedecomp osition hierarchy.

The pro cess ofpartdecomp osition can leadtomanysolutionsand mustb eguided bythetyp e ofsensorydataand anyintermediategroupings thatcan b eextractedfrom thatdata ingathering evidencefortheformationoftheobject. Forexample,ingure1thecub ecanb edecomp osedinto and thatareformationsarising fromthedetectionof and . Ifit wasnotp ossible todetect thenthisstructurewouldcollapse tosimply thelefthand branch mo deling the formation of a from and . The decomp osition pro cedure mustalsoconsiderthetyp eofcompatibilityfunctionthatwill determinehowwellthedatatsthe mo del represented bya no de. Each directed edgeorset ofdirected edges can b econsidered asan application of a constraint to thegrouping pro cedure and therefore implies anothercompatibility function applied to thedata. Later in section 2.2a pro cedure will b e outlined on how toconvert compatibility functionstoconditional probabilities, in deciding the structureitis simply amatter ofknowing conceptuallyhowthisgroupingmightb ep erformed. Forexample thedecomp osition of to ,in gure1, canb e p erformedbyrelaxingan imp osed constrainton tting a set of p oints toa planar surface. The compatibility function would corresp ond toa measure of t ofthe3-D p ointstoa planar surface.

Each no de in the network is a hyp othesized formation representing a b o olean variable that re ects the existence of that particular formation from the sensorydata. It is p ossible to create a no de that represents several formations [12] but this leads to fairly complicated compatibility functionsandanetworkthatisdiculttoextendwhenitisdiscoveredthatoneparticularformation within theno deis dep endent on anothercomp osite formation.

Surfaces

Vertices

3D Points

Edges

Cube

Bayesian Network

Decomposition by Parts

Figure 1: Decomp osition byparts ofa cub e.

Theleaf no desin thenetworktendtocorresp ondtoreadily detectable 2-Dor3-Dfeatures,for example,edges,3-Dp oints,intensity,and/orcolor. Itisatthisp ointthatanyhardevidencewillb e entered intothenetwork. Allother intermediate no des corresp ondtohyp othesizedformationsand groupings among the feature sets. Forenvironment mo deling the purp ose is to detectparticular generalsurfaceformations thatcould b eused forextracting overall shap eof a ro om.

The stepsforcreating thestructureof thenetworkare thefollowing:

Deneasetof variablesthatrepresentthenalcomp ositeformationsthataredesired to b e hyp othesized aswell as the variables of the features that are readily detected by the sensor.

Using the decomp osition by parts metho dology, decomp ose the comp osite formations into sub-formationskeepingin mind thefollowing p oints:

Eachsub-formationsshouldcorresp ondtotherelaxationofageometricconstraint(these will translatedirectlyto compatibilityfunctions).

Eventually these sub-formations will connect to the leaf no des that corresp ond to the variables representing thefeatures thatare readily detectedbythesensor.

  PG PG Cube f PG f ; S compatibili tyfunction A A A

2.2 The Conditional Probabilities

edge. Thesearecausallinksrepresentingthecauseeectrelationb etweentheparentgrouping tothe childsub-grouping.

Conditional probabilities arederived from compatibility functions that measure how well a set of features match a particular p erceptual grouping. This p erceptual grouping is representative of a particular geometric constraint amongthe set of features that imp osed on the child formation leads to the parent formation. They are not direct measures of how well the features represent a particular formation represented by the no de but are simply one measure among several that lead to that desired formation. This is an imp ortant p oint, in that the compatibility function is not representative of the formation of a complicated object but primarily consists of simple p erceptualgroupingmeasuresleadingtowardsthatformation. Soforexamplein termsofthe formationdepicted in gure 1 thecompatibility functionsmeasure theplanar qualityof a surface, proximityamongedges,proximityamongsurfaces,angularrelationshipsamongedges,andangular relationships amongsurfaces. Later in section 3 an actual example is presented forthe detection of corners and continuity amongplanar surfaces. Compatibilityfunctions thenare measures that conformcloser tothosefundamentalsetsdened originally bytheGestaltlaws oforganization,i.e. symmetry, proximity, continuity, .... The dierence is that they can b e develop ed for dierent typ es of sensory data b esides the original 2-D human p erception world envisioned in the Gestalt laws. Withthis in mind thefollowing denition forcompatibilityfunctions ispresented.

A isafunction( ( ))basedonageometricconstraintapplied toa setofattributes ofa sensoryfeaturethatmeasurehowwellthese featuresmatchthep erceptual grouping .

Fromthis p oint on,compatibilityfunctionswillb ereferredsolelyas withouttheargument .

Inordertoplacecompatibilityfunctionsonasamelevelofreferenceamappingfunctionisused to map the results from a compatibili ty function to a certainty value that exhibits the following prop erties,

Itis b oundedb etween theinterval(0 1).

It is a decreasing monotonic function, so that a compatibility value equal to 0 signies a certainty value of 1. This is equivalent to a p erfect match b etween the features and the p erceptualgrouping.

There areseveral of these typ e ofmappings and in this particular implementation we prop ose thedeclin ing -Curve function,shownin gure 2and representedmathematically as,

0

0.5

1

0

0

β

1

γ

f

_PG

S

(

f

PG

,0

,

α

,

β

)

Figure2: Adeclini ng -curve.

This mappingoersthefollowing advantagestotheuser:

It maps a physical measure toa subjective condence value and therefore separatesthese 2 conceptually dierent environments. The user can op erate with physical formsof measure-ments and thenapply a subjective measure that represents how well that physical measure comesclose toa particularp erceptual grouping;

The -Curve function allows the user to dene the resolution of a match of the geometric featureswith thep erceived formation. It is p ossible touse thesamecompatibilityfunctions but dierent -Curvesfor2 dierent geometricalformations;

Compatibility functionsandconstraint functionshave thedrawbackthat theyare usedmostly as a means of determining how well the evidence supp orts a hyp othesized formation. Therefore they cannot b e used as a mechanism for determining a-priori probabilitie s that are required for

Surfaces

Vertices

3D Points

Edges

Cube

Bayesian Network Modelling Consequences

Sub-Parts Composition

as = ( ), where and resp ectively signify the existence and non-existence of the formation . The compatibility functions can b e used directly for computing the conditional probabilitie s when the parent'sstates are but no such mapping exists for when they are . This do es not completely prevent the use of Bayesian networks but requires that the directional edges b e reversed so as to represent consequential knowledge, i.e. from evidence to hyp othesis. For example,the cub e decomp ositionrepresented in gure 1mustnowb e inverted as shownin gure 3.

Figure 3: Formation ofcub e asa consequence of theevidence.

InthisdirectionofcomputationthenetworkresemblesthePerceptualInferenceNetwork(PIN) develop ed bySarkarandBoyer[16]withthesignicant dierencethatthestructureofthenetwork is develop ed using causal theory and the p erceptual groupingconstraints areasso ciated with the edges of the network. The advantages of using causality over consequences are the removal of multiple instantiationsof similar no des that can o ccurwhen the structure of thenetwork is built usingconsequences. Forexample,theexistence ofasetof parallellines mayb ecaused bymultiple formations and is re ected in a causal networkby a no de representing parallel lines and multiple parentformationsthathavecausedtheparallelformation. Whenthenetworkisdesigned tore ect consequencesitisfairlyeasytointro duceanotherno derepresentingaparallel formationanywhere alongthenetwork. Thisofcourseis adesignmetho dologythana limitation imp osedbytheactual

Y Y ( ) ( ) TRUE FALSE j i j i j i j i j i j S : j 0S j S : j 0 S j : j i i i i j PG i j PG i j A ;A j i i i A pa A PG A ;A i i A pa A PG A ;A i i i i i pa A A pa A S P a a f P a a f pa A A e A A S P a pa A f e P a pa A f e pa A P a pa A P a pa A

Let ( )b eafunctionthatreturnstheparentno desof . Itisonlynecessarythentosp ecify theconditional probabilities forthe caseswhen thestates of ( ) are . Forsingle parent no des the -curve mappingscan b e mapp eddirectlyintoconditional probabilities resultingin the following conditional probabilities forserial connections,

( ) = ( )

( ) = 1 ( )

(2)

where ( ) = .

Forno deswithmultipleparents,i.e. convergentno des,theapproachmustaccountfortheedges to the other parent no des. There are 2 p ossible situations that o ccur; either the formation is a consequence of similar features b eing group ed orevidence has tob e combined from the grouping of dissimilar feature sets. A simple example of this is p ortrayed in gure 3, where edges can b e usedtosupp orttheformationof vertices whileb oth 3-Dp ointsand edgesare usedtosupp ort the formationofsurfaces.

If the comp osite formation represents the grouping of similar features then a compatibility functioncanb edevelop edthatmeasuresthequalityoftofthefeaturestoamo delofthecomp osite formation. This is typically what has b een develop ed in the computer vision domain for the grouping of edges. For these typ e of formations the conditional probabiliti es can b e computed usingequation2.

Forthesituations wherea comp osite formationcan come ab out fromseveral dierent typ es of featuresetsseparate compatibility functions arerequired for eachdierent set offeatures and the results ofthese mustb ecombined. Forthese cases,each directed edge, ,from aparentno de toa child no de hasasso ciated withita compatibilityfunction and acorresp onding -Curve mapping. Withthis in mind thefollowing calculation forcomputing theconditional probabilities isprop osed,

( ( )) = ( ( ))

( ( ))=1 ( ( ))

(3)

These2 conditions can o ccurtogetherwhichcan b emanagedbyconsideringone compatibility functionforsimilarfeaturesetsandcombiningthecompatibilityfunctionsofdissimil arfeaturesets usingequation 3.

Forthesituations whenanyof thestatesof ( )are thenthefollowing conditionals apply,

( ( )) = 0 ( ( )) = 1

Corner

(Cor )

Continuity

(Con )

CoPlanar

(Cop )

Edge

(

_{Edg i )}

3D Points

₍

3DPt i )

Polygon

(

_{Pol j )}

Vertex

(Ver )

Edge

(

_{Edg j )}

3D Points

(

_{3DPt j )}

Parallel

(Pll )

Polygon

(

_{Pol i )}

Cop Pll Ver Pol

3DPt Edg

This section will present an example of a Bayesian network for the detection of corners and con-tinuity among planar surfaces. This network is depicted in gure 4 and mo dels the grouping of surfacesand their resp ective edges intocorners and continuoussurfaces.

Figure 4: A Bayesian networkforthe detectionof corners andcontinuity amongplanar surfaces.

ThisparticularexampleisaBayesiannetworkdepictingthecomputation ofb eliefvalues inthe direction from evidence tohyp otheses, orin the direction of consequences. Figure 4is a grouping mo del depicting theexistence of corners( )and continuoussurfaces ( ) from the existence of coplanar surfaces ( ), parallel surfaces ( ), vertices ( ), p olygons ( ), and a set of 3-D p oints ( ) and their resp ective edges ( ). The 3-D p oints and edges are the only instantiated no des in the network since they represent actual sensory data, the other no des are hyp othesized formationsthat containb elief values computedfrom theinstantiated evidence.

Thisparticularstructuredecomp osestheformationofcornersandcontinuoussurfacesby relax-ingoneparticularconstraintamongthegroupingforeachlevelinthenetworkasonetraversesthe network from toptob ottom. Thisis a decomp ositionbypartsop erationwith intermediate no des

X 2  j 2 j i j i j f f f S f f d N d ; d j N f f f S;S ; S S 3.1 Compatibility Functions pr ll plan edg e plan plan j j j j pr ll copl pr ll i j S S S S i j Ver Pll 3.1.1 Planar Surfaces ( )

3.1.2 Parallel ( ) and Coplanar ( )

groupingcompatibility functions areb eing applied as one movesup the network. In this network one cansee thegroupingof similarfeatures;forexample,theformationof parallel surfacesamong 2 p olygons using one compatibili ty function ( ), and the grouping of dissimil ar features, like theformation of p olygons froma set of 3-D p oints and their corresp onding b oundary that uses 2 compatibility functions( and ).

Also similarcompatibili tyfunctionsareusedfortheformationofvertices ( )aswellasthat forparallel surfaces ( ). These 2 formations are contrary to each other and a dierent -curve function isused toactuallycomputetheconditionals.

Averybrief listing ofthecompatibility functionsarepresentedwith appropriatereferences toany detail information. Some of these functions, in particular proximity among surfaces, are fairly unique and require moreexplanationthancan b ep erformedin this article.

Acompatibilityfunction fordetermining thequalityof t forasetof p ointstoa planar surfaceis totakethemeanofthesquareddistanceofthep ointsfromthesurface. Theresultisthefollowing compatibility function forplanar surfacesfrom 3-D p oints,

( )= 1

(5)

where isthedistanceofp oint fromthesurfaceand arethenumb erofp ointsthatdene the surface.

Ameasureofparallelismb etweentwoplanarsurfacescanb ederivedbyevaluatingthelengthofthe vectorcomputedfrom thecrosspro duct ofthenormalstothesurfaces. Thisleads tothefollowing geometricalcompatibility function,

( )= N N (6)

whereN and N arethenormalscorresp onding tothe planar surfaces and .

Coplanarsurfaces are a restricted case of parallel surfaces with the added constraint that the angleb etween thenormalsofthesurfacesand theline joiningthecenter p ointsofthetwosurfaces isapproximately90 . This leads tothe following equation,

0 i j i j i i i i i 3.1.3 Proximity ( ) 3.1.4 Surface Boundary ( ) copl i j S S S S S j i pr ox pr ox i j i j i j i j i j edg e edg e i S i S i i S S i C C S S <a;b> a b f f S;S Ar ea Gap S;S Ar ea S S ; Gap S ;S S S f f S;P Ar ea S Ar ea P Ar ea S ; Ar ea S Ar ea P P A

where and corresp ond to the lo cation of the centers of the planar surfaces and resp ectively and the op eratoris adotpro ductop eration b etween vectors and . This is afairlylo oseconstraint thatactuallymeasureshowwellthecentroidsamong2surfacesarealigned thatalong withequation 6 do eslead tocoplanarsurfaces.

Dening a proximitycompatibili ty function for 3-Dsurfaces is challenging, b ecause thedenition of proximity is not unique. The prop osed approach denes a compatibility function among the surfacesbymeasuringthedistanceb etween theb oundaryoftwosurfacesandissummarized inthe following equation,

( )=

( ( ))

( + )

(8)

where ( )computesap olygonwithab oundarycommonwiththesurfaces and based ontheirresp ectiveb oundaries. Thecommonb oundary isdeterminedbyapproximatingthesurface bya p olygon and projectingtheverticesof thep olygon awayfromthe surfaceuntilthey intersect with another adjacent surface's b oundary. This op eration can b e p erformed in the 2-D image plane representative of an ordered set of 3-D p oints. This orderedset of 3-D p oints isa common representation forscanningtyp e ofrange sensors. Thecommonb oundaries b etweenthe2surfaces areusedtodenea3-Dsurfacecomp osedoftrianglesofwhosesurfaceareacanb eapproximatedby addingup thesurfacearea ofthetriangles. Thispro cedure isratherchallengin g and theapproach oersa unique metho dfordetermining adjacent surfaces [19].

This compatibility function measures how well the edge of a planar surface denes a p olygon representing that surface. The compatibility function involves a ratio of the area dened by the p olygon tothe areaof thesurfaceitself and iscomputedusing the followingequation,

( )=

( ) ( )

( )

(9)

where ( )computes thesurfaceareabased onthe p oints deningthesurfaceand ( ) computesthesurfaceareabasedthep olygon, asso ciatedwiththesurface . Thesetwovalues inmostsituationsarenotequivalentsincetheedgesextractedforthesurfacearestraightwhilethe originalsurfacewillconsistofirregularlyshap ed edges. Theestimationofthesurfacebyap olygon isbased on determininghigh curvature p ointsalong thesurfaces b oundary[19]

N N

aq aq

The Bayesian network and compatibili ty functions describ ed in section 3 were applied toa set of planarsurfacesextractedfromrangep ointsacquiredusingacompactlasercameracalledBIRIS[20]. The output of the camera consists of one acquisition of 256 range and intensity values along a projected plane of light. To acquire more datathan that of a single acquisition theBIRIS sensor was mounted onto a pan and tilt unit. The ability to tilt the sensor is crucial in b eing able to acquire more data, since the eld of view of the sensor is fairly limited. When the Biris Laser Scanner is panned at a constant sp eed for a xed tilt angle the result is a rectangular image of dimension 256x ,where isthenumb er ofacquisitions takenduring thepanning sequence. Thecharacteristicsof this particularversionof thesensorare shownin table 1.

Laserp ower 24mW, (two12mWlaser projectors) wavelength 680nm(visible red)

Fieldof view 19deg. Fo cal Length 20mm

Accuracy 3 mm @1m, 14mm@2 m, 42mm @3 m Range 0.5 m- 5.0m

Table 1: Sp ecications forthe BIRIS rangesensor.

Figure 5 shows the extracted planar surfaces from range data taken of a ro om with a typical layoutapproximatelytheshap eofthat showningure6. Theindivi dual segmentsintheintensity image are shown using separate grayvalues to represent each segment. The advantage of main-tainingthe p oints in an image isthat one can take advantage of the alreadyestablished scanning orderinginherent in thesensor.

The Bayesian network depicted in gure 4 was applied to each set of adjacent surfaces in gure5. Inthisexample therewere25totalsurfacegroupings ofwhichthecomputedb elief values arepresentedintable2. Table2alsolistsasmallcommentforeachgroupingthatmentionsreasons why the b elief values for particular surface formations are low. Low values for b oth corner and continuitydo not necessarily signify bad results they can re ectsituations where theedge shared by the 2 adjacent surfaces is in fact a jump edge and the surfaces cannot form either a corner or continuous surface.

The results are consistent in that the formation of a corner is counter to the formation of continuous surfaces. The b elief values re ect this in thefollowing manner. When theb elie f of the formation of a corner is high the b elief in the formation of a continuous surface is low, and the converse is true. The exception are cases where b oth values are close to 0.0 which in some cases arecharacteristicof surfaces with\Jump"edgesand in other caseslowcertaintyvalues causedby

(a)

(b)

S14

S8

S11

S16

S3

S4

S17

S13

S10

S12

S15

S5

S18

S9

S2

S7

S6

S1

Figure5: Surfaces extractedfrom therangedata;intensityimage(a) andisometric view(b).

highcompatibilityfunctions. UsingtheBayesian networkitisrathersimpletoinferwhythese low values in b oth corners and continuityexists, it is simply a matterof searching throughthe graph lo oking foro ccurrences in low compatibility values.

Figure 7 depicts the results in table 2 as an image where the surface pairs that have b elief values (0.7 orover) of b eing continuous or corners have b een displayed using the same intensity value. Surfaces that had low b elief values in either b eing planar or had bad edge representations aredepicted in adark colorwithonly their b oundaries showing.

Theseresultssuggestthatanothersurfacegrouping,knownasa\JumpSurface",shouldp erhaps b eintro duced tothemo del and will b econsidered forfuture networks. These typ eof surfaces are also presented in gure 7. Discontinuities, like \Jump Surfaces", are generally not considered a p erceptualgroupingformationbut certainly doadd knowledgethat canb e usedforinterpretinga scene.

Some results are notso p ositive but do re ectthe reality of thesituation. It wouldhave b een desirable tohavesurfaces and toresultin ahighb elief valueofb eing acontinuous surface. Thiswasnotso,thereasonwasin thelowvaluethatsurface wasaplanarsurface. Atleastthis low certaintyvalue in the surface'splanar quality is re ected throughout other formations where surface is part of and a low b elie f value is computed that can hint to a further investigation

x

y

z

60 °

Robot

Poles

Wall

Figure 6: Aschematic of thetopview of therob otlabshowing sensorp osition.

intothereasonb ehind thesmall value.

It should b e notes thatthe results obtained are biased bythe parametersused in the -curve mapping. Table3 lists thevalues usedforthearguments and for -curve mapping.

These values were subjectively selected but the selection follows a certain reasoning. The proximity value of 0.25represents the desire toconsider surfaces withgaps smaller than 1/4 the sum oftheareasof the2 neighb ouring surfaces asproximal,anythingb eyond 0.5 isnotproximal. Thissameargumentcan b eapplied totheedgecompatibility value sinceitis alsoaratioof areas. Surfacesareconsideredparallel ifthesurfacenormalsarewithin10radswithresp ecttoeachother, and they are not parallel if b eyond 20 rads. Coplanarity is similar toparallel surfaces and share the same arguments. The parameter values for the quality of the surface b eing considered as a plane was determined by computing the average in the variance of the data p oints of a typical set of p oints gathered from the BIRIS sensor. Using the actual variances in the data as a guide fordetermining theplanarity arguments for and is imp ortant since thesensor itself is notan accuratesensoratlongranges. Ifanarbitrarylowabsolutevalueisusedtheresultwouldb esmall b elief values in theexistenceofallplanar surfaces. Thesecurrentsettingsshowthattherob otand p ole surfaces are reasonable planar surfaces which is not the case but b ecause they are closer to thesensorhave smallervariances in therange thenthebackgroundwalls.

        4 4 5 5 5 6 8 8 16 5 Conclusions OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK i j i j pr ox plan plan plan plan plan edg e plan plan plan S f S f S : S f S : S f S : S f S : S f S : S f S S f S : S f S : S f S :  

1 6 0.32 0.0 ( low 10 12 0.32 0.0 Jump Edge

1 9 0.87 0.0 10 13 0.01 0.72 4 5 0.0 0.0 ( ( )) 00 10 15 0.0 0.0 Jump Edge 4 17 0.0 0.0 ( ( )) 00 10 17 0.9 0.0 5 10 0.0 0.0 ( ( )) 00 10 18 0.0 0.0 Jump Edge 5 13 0.0 0.0 ( ( )) 00 11 14 0.0 0.0 Jump Edge 5 17 0.0 0.0 ( ( )) 00 11 16 0.0 0.0 Jump Edge 6 9 0.63 0.0 ( ( ))low 12 15 0.0 0.0 Jump Edge 6 12 0.0 0.0 Jump Edge 13 17 0.77 0.0

8 11 0.0 0.0 ( ( )) 00 15 18 0.0 0.0 Jump Edge 8 14 0.0 0.0 ( ( )) 00 16 17 0.02 0.27 ( ( )) 00

9 10 0.87 0.0 17 18 0.0 0.0 Jump Edge

9 12 0.22 0.0 Jump Edge

Table 2: Belief values for the formationof corners and continuity among the surfaces in gure 5 (a). Compatibility Units proximity 0.25 0.50 unitless coplanarity 10 20 Rads parallel 10 20 Rads planarity 0.578 1.156 cm edges 0.25 0.50 unitless

Table3: Values for thearguments and forparticularcompatibilityfunctions.

This article presented an approach in the use of Bayesian networks for the grouping of features and the detection of higher order complex structures. An example was presented of a Bayesian Network for the grouping of 3-D surfaces into either corners or continuous planar surfaces. An approachforthesp ecication oftheBayesianNetworkwaspresentedthatusedtheconcept ofpart decomp ositionfordeterminingthestructureofthenetworkandp erceptualgroupingcompatibility functions that can b e used todetermine the conditional probabilitie s required by thenetwork. It was shown that the design of the network can b e done by considering causal relationships from

Continuous and Corner Surfaces

Jump Surfaces

Isolated Surface

Surfaces with Low Belief Values

FALSE

Figure7: Imageof thesurfaces depictingthose that arecontinuous,corners, orjumpsurfaces.

the mo del to its sub-parts but the actual implementation and computations must b e p erformed in a b ottom up fashion from the sub-parts to the mo del. This is imp osed by the inabili ty to sp ecify compatibility functions for theconditions when the parent no des are .A network was presented that group ed surfacedata into corners and continuous surfaces that wastested on 3-Dsensorydatacaptured froma typical indo orenvironment.

Thegroupingresultsapp eartob econsistentbutwhatismoreb enecialisthedeclarativenature ofpro cedure implied bytheuseof a Bayesiannetwork. Thisfacilitates thesearchforreasonswhy b elief values mayb elow. The pro cedural knowledgerequired top erformthegrouping isalso well encapsulated ascompatibili tyfunctions, facilitating the useand development ofthenetwork.

To reduce the inherent computational complexity asso ciated with grouping op erations the grouping pro cess was only applied to adjacent surfaces. The next challenge is to determine an approach forcomparingsurfaces that arenotdirectly adjacent without having tocompare all the surfaces. Once this issue is resolved it will b e p ossible to apply the same Bayesian network pro-cedure to the detection of o ccluded surfaces. Clearly there are many other examples that need to b e investigated like, 3 surface corners, jump surfaces, and textured surfaces like brick walls. Development has b een o ccurring into a Bayesian Attributed Hyp ergraph (BAHG) that combines the b enets of Bayesian networks and attributed hyp ergraphs. This representation will facilitate thecomputationofthecompatibilityfunctionsbyintegratingtherelationshipsandattributeswith the creation of the Bayesian network. Currently multiple separate networks are created as the

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