MIT Artificial Intelligence Laboratory Vision Grou

Freuder, Eugene C.

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Vision Flash 4December, 1970RevisedFebruary, 1971The Object Partition ProblemEugene C. FreuderVision GroupA.I. LaboratoryM.I.T.I.We provide a theoretical framework for the "objectpartition problem".We begin by considering this problem In a rather abstractcontext. Consider the semantic domain of physical objects, andthe syntactic domain of picture graphs. The object partitionproblem can be viewed In terms of specifying syntacticoperations that yield partitions of the picture graph, whichwhen interpreted In the semantic domaih correspond to possiblepartitions of the scene into physical objects. We term suchsyntactic partitions "physically realizable partitions" orsimply "realizable partitions". The problem has severalaspects. We may seek all possible partitions, the one "best"partition, several plausible partitions, judge or rank proposedpartitions, and so forth.The most Important aspect of the problem, In sone sense, isdetermining the "best" or most likely realizable partition.Guzman's SEE attempts to deal with this problem. Guzman useshis understanding of the semantic physical world of objects tomake local choices on the best partition of a given picturegraph. These choices are somewhat Interrelated. They arecombined to inform global decisions which may also beinterrelated.Guzman's SEE is a remarkable heuristic programmingachievement. However, as might be expected of tho germinalachievement in its field, It lacks a satisfactory theoreticalframework. There Is some difficulty in determining themotivations and implications of Gukian's heuristic decisions.It is not always obvious just what semantic observations informthe selection of the syntactic partitioning choices. As it isnot clear which possible Interpretations are being discarded ateach stage of the procedure, the process cannot producealternative partitions, even where several plausibleinterpretations are preseht in the physical scene. To judge orextend the woek, one must essbntlally rePeat Guzman's experiencewith individual scenes.The program does not function as a gobd element for aheterarchical vision system. There Is no proper framework fromwhich to launch dialogue with other knowledge structuresrelevant to the object partition problem. There Is not theflexibility to provide alternative analysis on the basis ofhigher level dissatisfaction.It may be argued that in order to establish a satisfactorytheoretical base for a property of physical scenes, such asobject partition, one must present a system capable of dealingwith all possible physical interpretations of a given picturegraph, in terms of the relevant property. This would requireour theory of object partition to be "complete" In the sensethat It could deal with any physically realizable partition.One criteria of completeness would be that the theory couldproduce all realizable partitions, though the system would notnecessarily be "generative" oriented.A "complete" characterization of realizable objectpartitions would hopefully provide an organized framework Inwhich choices could be made among possible lbcal partitioningsIn determirlng the "best" possible global partition. Decisionsinvolving these choices would be semantically andheterarchically Informed, and their Implications and motivationswould be clearly understood. Alternative choices could be made,in some plausible order.Such a characterization would cut down our decision spaceby eliminating rncontistent or impossible partioning choices.The nature and range of the remaining heuristic choices would beclarified.Recently Huffman has approached another problem of sceneanalysis, the "configuration" problem, in somewhat this fashionby attempting to identify physically unrealizableconfigurations. The results of his venture recommend this typeof approach.However, Huffman's success has also encouraged speculationthat his theory may contain as well the desired completecharacterization approach to the "object partition" problem.The "physically realizable configuration" and "physicallyrealizable partition" problems are Interrelated andInterdependent. However, they are not Identical, and it wouldbe a mistake to base an approach to one upon the conceptualunits that characterize the other, The treatment of the formerproblem has been begun by Huffman. Much remains to be done,however; some relevant observations will appear in futurepapers.The "characterization" approach to the "object partitionproblem" will be outlined below.II.We take for our basic units of analysis a line predicateand its negation. The predicates are "belong to the same body"and "do not belong to the same body", as applied to the (two)regions bordering a line.This choice of our unit of analysis Is neither as trivialor as circular as It may appear. To begin with the basicpredicate defined Is NOT dquivalent to the predicbte "links".The success of the link predicate is probably a major reasonthat the more fundamental predicate we described was not usedearlier. However, deep problems in partition analysis revealthat "link", while a useful concept, is actually a handicap tooptimal thinking when used as the basic unit of analysis.We note that "belong to the same body" does not even implythat the line referenced corresponds to a physical edge of bothneighboring regions. Consider line AB In the following figure!We begin by enumerating all physically realizableinterpretations or "labellings" of the various types ofvertices, In terms of our two predicates. All physicallyrealizable partlonings of a given picture graph are thenobtained from all consistent combinations of local labellings.(Using the criteria that an interpretation applies to an entireline, i.e. line segment, and thus a line cannot receive oppositelabels from its two vertices.)We will describe this process In some further detail. wedemonstrate first how this approach circumscribes the realizableinterpretations, in achieving a solution to the "all realizablepartitions" aspect of the object partition problem. We then goon to indicate how this approach provides a basis for thedecision making procedure which deals with "best" realizablepartition and related problems. We will gradually shift ourfocus from an abstract theory to a theoretical model embodied ina "partition system" embedded in a heterarchical vision system.We will find it easier perhaps to talk in terms oflabellings and deal with picture graph elements so we Introducethe notation "I" for "belong to same body" and "M" for "do notbelong to same body". (These notations are, If you like,syntactic elements in the picture graph language which areinterpreted as the indicated physical relations.) We will referto "I" labellings as "ties" and "M" labellings as "breaks". Ifwe simply consider any possible syntactic labelling of an n-linevertex we obviously have 2 to the n possible labellings.However, we make the restriction that "tie" is an equivalencerelation, using an observation of the Oropertles' df the "belong"predicate. For an n-element vertex, transitivity thuseliminates all labellings with n-1 tie labellings and one breaklabelling.Consider theee line vertices. The number of physicallyrealizable labellings now corresponds to the number oflabellings with nd "ties", All "breaks", with one tie, twobreaks, and with three ties. I.e.31/013! + 3!/112! + 31/3101=1+3+1= 5A similar analysis can be carried out for n-line vertices.In considering vertices of specific forms, however, we mayfind that different symmetries further reduce the number ofdifferent labellings or group them into classes. We will callthe vertex labellings that have a physically realizableinterpretation simply "realizable labellings".Examine the realizable labellings for the different threeline vertex types, forks, arrows, and T's.ForksArrowsT'sT TTTTWe can easily dispose of the two line vertex "L".Consider also the Interpretable labellings of "K" typevertices.X4 KYhWe find that the number of different realizable labelllngclasses even for the K is surprisingly manageable. Recall thatwe have placed no restrictions on the physical domain (beyondarbitrary planar polyhedra), or on possible Interpretations ofpicture elements. If we wish to do so we may cut down thenumber of realizable labellings 6ven further. For example, wemay Impose restrictions of physical objects to degree threepolyhedra, and certain "general position" requirements.(We might note that this analysis does not pertain tototally disconnected bodies, e.g. joining the two halves of thepartially hidden body in the following figure:This Is a different class of problem.)Applying these realizablb labellings to a scene in allconsistent combinations provides a solution to the "allphysically realizable partitibns' aspsct df 'the pbatitionproblem. We note that finding consistent coribinations alsoinvolves checking foe "global trahsiltivity".. A region R cannotbe asserted to belong to the same body aý region S bytransitivity of the "tie" predicate while at the same time a"break" predicate asserts that R does hot belong to the samebody as S. We have restricted our labellings so that thisinconsistency cannot occur around a single vertex; however, westill have to guard against its occurence across several lineboundaries, as in the following figure:III.A complete characterization of the partition problem hasbeen achieved.Beyond this a foundation has also been laid for dealingwith the "best possible realizable partion" and related aspectsof the partition problen. The objective here is to produce themost plausible partition, with the ability to retrench andproduce alternatives if required.Any realizable partition of a picture graph may be producedby an approPrlatd labelling of..the vertites, At ahy vertex wehave a number of choices of realizable labellings depending onthe type of vertex. We consider factors Which may prohibit ordictate choices, or rate them on a plausibility scale. Thesefactors may be global, in the sense that they affect our choicesof arrow labellings, say, regardless of where the arrows appearin the picture graph, or they may be local, In the sense thatthey affect a decision at a particular vertex. (Or their effectmay fall somewhere in between these extremes.)Our first approach should be to return to the semanticcontext in which our picture graphs are to be Interpreted. Onthe most general level this means combinations of planarpolyhedra. By studying such combinations we can make judgmentsas to the relative plausibility of the various realizabl]labellings for the different vertex types. Even in a grossCstat Ist l CsiI M I C ns 1A1 k #th +t A - 1 1 i I b l ie e, e can o serve a er ginterpretations are more common or likely than others. Theseobservations may be extended to certain combinations ofvertices.Beyond this stage we may recognize that certain restrictedphysical contexts limit the range of Interpretable labellings orotherwise affect the relative likelihood of certainInterpretations. We observed above, for example, that limitingthe physical domain to degree three polyhedra would have thiseffect. Limitation to convex objects, or likelihood of concaveobjects, would also affect labelling choices.The fork possibility, for example, that contains two breaks( and one tie, Is highly unlikely except in a concave object.With the "K" possibilities clearly in hand we mar overcbme theintimidation of K analysis and qulckl obser d -that manypossibilities are highly unlikely or impossible in many commoncontexts. In fact, as the unlikelihood of alternativd choicesrises much faster for K's than for forks say, we may find K's attimes more helpful to our analysis than some "simpler" vertices.And we can afford to rate certain choices as implausible sincethat does not mean we have not dealt with them; they are stillavailable as alternatives In our complete analysis.The point to be made here is not so much that a limitedsystem could be designed that would function well in 6 specificrestricted context. Rather a complete partition decisionstructure, imbedded in a heterarchical vision system, couldemploy Information from a context decision structure to adviseits labelling decisions or alter Its plausibility ratihgs.Of course the context decision structure, in turn, couldbenefit from an understanding of the implications of whichlabelling choices were being made. In patilculpartition process experienced difficulty In pplausible partition within the constraints of tanalysis provided It, the context analyser could bereconsider its findings. In a similar fashion theanalyser may complain that ah iýplausible choice Is.to improper input .data from the peAproceis r; orinformed to expect certain unlikely configuration'of missing lines, shadows, whatever.Aside from what we right call "contitt"' inlabelling choices may be Informed by knowledge anstructules of Many differeht kinds within a hestructure. Background informatloh, for example willdetermine many labellings and circurmscribe or suggThere are a variety of line predicates aside frompartitioning predicate which are relevant to pdecisions (concave and convex fall into this categthere are more global criteria Involved. Future 0ar, if theroviding ahe contektprompted to-partitionperhaps dueIt may beas a resultformation,d knowledgeterarchicalimmediatelyest others.the basicartitioningory). Andapers willdeal with these factors, their relationand to scene analysis in general.Relevant information may restrictto partition analysisor advise ourpartitioning thoites similarly dt other than the vertexlabelling level, of course. Decisions may be Indicated at theIndividual line level, at higher levels InVolving types ofvertex codbinations, whatever.The beauty of our characterization approach Is that allthis Information and all these decisions c6n be done within asystematic, complete framework. All the eealizableihterpretations are available for anblysis and comparison. Thesystem can make decisions With a clear understanding of justWhat the choices ari and what the Impi ca'tfohb d thbe ckoicesare. We know what Possibilities are Jiscbrded at egah d6cisionpoint.These featurbs of the charactetizat6on theory Also Indicatewhy it is a great aid in organtzing, sttmuiating, and clarifyingour thinking, in determining precisely how the relevant factoesdutlined above detertine the ipecific decisioh Procedures usedby a partition system.One of the difficulties in dealing with partitloh analysisis that the problem, like many scene ahalysiS problems, is"potentially global". For any judgment one makes about acertain local configuration implying a certain partidning, onecan usually find an exception by embedding the configuration Ina sufficiently complex environment. Another advantage of ourtheoretical approach is that the basic aspect of this global( determination Is built in to the structure of the consistentlabelling approach, on a network level.Local labelling decisions affect others 16 a potentiallyglobal relationship. Once some (or all) of the lines of avertex have been labelled by neighbbring labelling deciiions,the labelling of that vertex may be determined, or thepossibilities cut down. The labelling 6f one or more tihes of avertex may direct our attention t6 the most plausible labelling,for the vertex type, which agrees with the alkbady labelledlines.The interrelatlonship built into the labelhing systemguarantees that potentially global determining relationshipswill be donsidered. When this n6cessitatei reiotution ofconflicts and discarding decisions obr system will know justwhat decisionS were involved and be able to evaluat6 them. Theappropriate alternatives will be available. And our theoreticalbase thsu.es that our options prOvide a comrplete set df possiblesolutions.Our system, in other words, has the "freedom to fail". Ithas long been an educational cliche that this is a prereouisitefor accomplishment. The very notion of "heuristic" prbgrammingimplies, not algorithms, but sets of principles, some bf whichmay fall at any given application. The most successful conceptsof heuristic programming have dealt with systems that couldrecover from or use these failures in some fashion. Withoutthis capability one is often forced to tightly restlict the16problem domain, or to deal with Ordbietf of hoýel6st complexity.In scene analysis the "p tentiallY global" problmi, inparticuldr, makes It difficUlt to ~ake deo.lsloni that "haVe toWork". Our aroroadh to the "blst pakti6t r.proble tt glVes usthe flexibility to #fil, with the added coh.idente that, sltcewe have characterized the set of realtlzabie artitions, we must,at least in some theoretical sensýe be able to achieve thedesired solution..Ir

The Object Partition Problem

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The Object Partition Problem

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