Fine-grained Qualitative Spatial Reasoning about Point Positions
UUniversity of BremenDepartment of Computer Science
Diploma Thesis
Fine-grained Qualitative Spatial Reasoningabout Point Positions
Sören SchwertfegerAugust 2005Supervisors:Prof. Christian Freksa, Ph.D.Dr. Reinhard Moratz a r X i v : . [ c s . A I] N ov bstract The ability to persist in the spacial environment is, not only in the robotic context, anessential feature. Positional knowledge is one of the most important aspects of spaceand a number of methods to represent these information have been developed in the inthe research area of spatial cognition. The basic qualitative spatial representation andreasoning techniques are presented in this thesis and several calculi are briefly reviewed.Features and applications of qualitative calculi are summarized. A new calculus forrepresenting and reasoning about qualitative spatial orientation and distances is beingdesigned. It supports an arbitrary level of granularity over ternary relations of points.Ways of improving the complexity of the composition are shown and an implementa-tion of the calculus demonstrates its capabilities. Existing qualitative spatial calculi ofpositional information are compared to the new approach and possibilities for futureresearch are outlined. cknowledgments
First of all, I would like to thank my advisor Dr. Reinhard Moratz for introducingme to the interesting field of qualitative spacial representation and reasoning, for theinspiring discussions and his support. Without him this thesis would not have beenpossible. Furthermore I want to express my gratitude to Prof. Christian Freksa forsupervising this thesis.I am thankful to all members of the spatial cognition group for the creative atmospherethat I enjoyed there. As the five years as a student at the University of Bremen passby I would like to thank my friends and classmates for making these years a wonderfulexperience.Finally I thank my family for their support, especially my daughter for brightening upmy day and my wife for her constant backup. ontents
1. Introduction 10
2. Foundations 13
3. State of the Art 42 ontents
4. The
Fspp
Approach 54
Fspp
Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.1.1.
Fspp
Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.1.2.
Fspp
Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.1.3. The Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.2.
Fspp
Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.2.1. Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.2.2. Unary operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.2.3. Conceptual neighborhood of the
Fspp . . . . . . . . . . . . . . . 634.2.4. Improvement for the composition: Contour tracing . . . . . . . . 634.3.
Fspp
Impelementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.4. Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.4.1. TPCC vs
Fspp . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.4.2. DOI vs
Fspp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.4.3.
Gppc vs Fspp . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.4.4. Classification of the calculi . . . . . . . . . . . . . . . . . . . . . . 79
5. Conclusion 80
A. TPCC definition 83B. DOI composition formula 85C. Topology definition 88D. Bibliography 91 ist of Figures a) location of c wrt. the location of b and the orientation ab ; b) Orienta-tion relations wrt. to the location b and the orientation ab ; c) left/rightand front/back dichotomies in an orientation system . . . . . . . . . . . . 322.3. cone-based representation . . . . . . . . . . . . . . . . . . . . . . . . . . 332.4. The possible locations of c (gray) before and after the unary Inv operation 362.5. Composition - the possible locations of d are shown in gray . . . . . . . . 372.6. The fine lines are the generalized Voronoi diagram of a 2D environment.On the right side is the corresponding GVG. . . . . . . . . . . . . . . . . 413.1. Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2. The 24 atomic relations of the dipole calculus . . . . . . . . . . . . . . . 443.3. The flip-flop partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.4. Names of the TPCC areas . . . . . . . . . . . . . . . . . . . . . . . . . . 463.5. Iconic representation for TPCC-relations . . . . . . . . . . . . . . . . . . 473.6. The iconic representation of the unary TPCC operations . . . . . . . . . 483.7. A DOI and its parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 503.8. Composition of adjacent path segments . . . . . . . . . . . . . . . . . . . 503.9. Resulting DOI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.10. Example configuration in Gppc . . . . . . . . . . . . . . . . . . . . . . . 534.1. Distance from relatum ∆ x and distance range δ x . . . . . . . . . . . . . . 564.2. An example FSPP with 32 orientation distinctions and 13 distances . . . 604.3. Disjunction of 22 Fspp base relations indicating possible locations of thereferent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.4. The temporary
Fspp for the
Inv operation . . . . . . . . . . . . . . . . 624.5. The short cut Sc operation . . . . . . . . . . . . . . . . . . . . . . . . . 624.6. Conceptual neighborhood of an Fspp . . . . . . . . . . . . . . . . . . . . 644.7. The Moore Neighborhood . . . . . . . . . . . . . . . . . . . . . . . . . . 654.8. Pavlidis: check P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678 ist of Figures P (left) and check P (right) . . . . . . . . . . . . . . . 674.10. 8-connected border of an Fspp . . . . . . . . . . . . . . . . . . . . . . . 684.11. Quantitative vs Qualitative calculi . . . . . . . . . . . . . . . . . . . . . . 799 hapter 1.Introduction
Space is a basal part of the existence of all creatures. The ability to cope with the spatialenvironment is thus a primary survival skill of even the smallest insects.All physical entities are located in space and their spatial properties, like location, orien-tation or shape, might change. Because humans are able to deal with spatial knowledgeeffectively we can deliberately and intelligently utilise this environment. We obtain andmodify this knowledge by what we perceive from the external world and by our abilityto derive new knowledge. The communication of spatial knowledge is an essential skillof human beings.The elementary concept of space has thus been a central subject of study in CognitiveScience and Artificial Intelligence. To perceive space, to represent space, to reason aboutspace and to communicate space is an important task in Artificial Intelligence research.Artificial Intelligence is a highly diversified discipline with influences from different clas-sical areas such as Computer Science, Logic, Engineering, Psychology and Philosophy.The two primary methods to deal with space are the quantitative approach and thequalitative approach. The quantitative approach is the classical way which is used inmany traditional disciplines like computational geometry, computer vision and robotics.It uses numerical values within a coordinate system to store the spatial information. Thequalitative methods abstract from the exact numerical values and use a finite vocabularyto handle spatial information. Qualitative representation of spatial knowledge andreasoning about qualitative spatial knowledge is more "human-like" or "cognitivelyadequate" than classical approaches. 10 hapter 1. Introduction
Artificial Intelligence and in particular spatial reasoning have many applications, butan increasingly important one is robotics. Non-static robots most often solve spatialproblems like maintaining information about their position, path-finding, way-planning,map-building or goal-finding. Communicating with humans about spatial knowledge isanother important aspect of robotics. Topological information are often useful, but forrobotics the most important spatial information is the positional information. Qualita-tive approaches are very suited for dealing with spatial knowledge in robotics becausethey can handle imprecise sensor data and omit unnecessary details.The goal of this thesis is to develop a new qualitative calculus for spatial reasoning aboutpositions. Most often it is sufficient to reason about points and to abstract from theextend of the objects. Many existing qualitative spatial representations are too coarsefor some applications, especially for robotics. Thus a calculus for more "Fine-grainedQualitative Spatial Reasoning about Point Positions" is being presented in this thesis.This thesis is written with regard to robotics but the calculus can be used for otherapplications that need to reason about positional information as well.
This thesis will first introduce the reader to the foundations of qualitative spatial reason-ing. State of the art calculi are presented in following chapter. After the new approachis developed, ways of improving it and it’s implementation are explained. A summaryfollowed by the discussion and the outlook is presented after a comparison of the newcalculus with other approaches.The first section of Chapter 2, which introduces the terms and definitions required, isabout spatial representations in general. It presents different important aspects of spacelike topology, orientation, distance and shape. Section 2.2 introduces to spacial reasoningand the unary and binary operations that are needed for it. General constraint basedreasoning is explained as well as the special features of spatial reasoning. The nextSection (2.3) presents the qualitative representation, its properties and it gives reasonswhy to use it. Later in this section qualitative spatial representation and reasoning isintroduced. The last section of Chapter 2 (Section 2.4) is about the applications inwhich qualitative spatial reasoning is used.Chapter sec:StateOfTheArt introduces state of the art calculi. The first calculus inSection 3.1 is about topological aspects of space. The Dipole approach over intrinsic11 hapter 1. Introduction orientation knowledge is presented in Section 3.2. Section 3.3 introduces the TernaryPoint Configuration Calculus from which the calculus developed in this thesis derives.The Distance/orientation-interval propagation is a quantitative approach which is usedfor the computation of the composition (Section 3.4). The Granular Point PositionCalculus introduced in Section 3.5 is recently developed and similar to this thesis’sapproach.The calculus of this diploma thesis is being developed in Chapter 4. The absolutedistance representation is presented in Section 4.1.1 and the relative orientation rep-resentation is introduced in Section 4.1.2. Section 4.1.3 combines those latter two torepresent positional knowledge. The reasoning techniques for the new calculus are ex-plained in Section 4.2 and the conceptual neighborhoods are defined. The algorithmfor the composition as well as the unary operations are developed and improved in thissection, for example by using Pavlidis contour tracing algorithm. Section 4.3 shows codefragments of the implementation as well as the output of an example program whileSection 4.4 compares some important state of the art calculi with the newly developedone.Chapter 5 provides a summary of the results achieved in this thesis as well as a discussionand an outlook for future research possibilities.The appendix contains in depth definitions for the Ternary Point Configuration Cal-culus (A), for the Distance/orientation-interval propagation (B) and for the RegionConnection Calculus (C) as well as the Bibliography.12 hapter 2.Foundations
There are tasks within robotics that can be performed without the need of spatial repre-sentations. Behavior-based robotics does not use an internal model of the environment.The robot rather acts by directly using the input from the sensors and thus reacting tothe changes in its environment. One very early example of behavior-based robotics isWalter’s turtle. There, a light indicates the position of a recharge station. The turtlehas a photo sensor which leads it to the recharge station if the batteries run low onpower and makes it flee from the light once the batteries are full again, [Walter, 1950].Other great contributions to behavior-based robotics have been done by Braitenbergwho developed the famous Braitenberg-vehicles [Braitenberg, 1984] as well as by RonaldArkin [Arkin, 2000].More complex demands on robots require the use of spatial knowledge. In order touse this spatial knowledge and perform tasks like reasoning with this information it isnecessary to have an adequate representation. There are two common ways to representspatial knowledge: the qualitative and the quantitative (it is also known as numerical)approach which are closely introduced in section 2.3.There are different aspects of space that are represented by different kinds of spatialrelationships. The one dimensional Temporal logic will be presented briefly as it hadgreat influence on qualitative spatial reasoning. Topology representations are inherentlyqualitative whereas orientation and distance information can also be expressed quanti-tatively. The latter two combined form the positional information. Those aspects ofspace are often not independent from each other, especially if orientated objects withextension are represented. 13 hapter 2. Foundations
Relation Symbol Inverse symbol ExampleX equal Y = = XXXYYYX before Y < > XXX YYYX meets Y m mi XXXYYYX overlaps Y o oi XXXYYYX during Y d di XXXYYYYYX starts Y s si XXXYYYYYX finishes Y f fi XXXYYYY
Table 2.1.:
Thirteen Relationships
James F. Allen introduced a temporal logic in 1983 based on intervals, [Allen, 1983].Without using the term "qualitative", Allen defined some conditions for his algorithm:It should allow imprecision (relative relations rather than absolute data are used) anduncertainty of information (constraints between two times may exist although the rela-tionship between two times may be unknown). Furthermore the algorithm should varyin its grain of reasoning (years, days, milliseconds etc.) and it should support persistence(facilitate default reasoning).Allen uses intervals of time because he observed that there don’t seem to be any atomsin time. All possible events in time might be broken up into two or more individualevents thus forming an interval of time.In table 2.1 we can see the thirteen possible relationships between intervals that Allenidentified. Convenience allows the collapse of the three during relations (d, s, f) into onerelationship called dur and the three containment relations (di, si, fi) into a relationshipcalled con.The time intervals constitute nodes of a network, while the arcs between those nodesare labeled with one or more of those thirteen relationships (allowing uncertainty fordisjunction of relations). Allen presented an algorithm to calculate the transitive closureof such a network using a "transitivity table" (composition table). He also introduced14 hapter 2. Foundations reference intervals to group clusters of intervals and thus reduced the space requirementsof the representation without greatly affecting the inferential power of his mechanism.[Allen, 1983]A generalization of Allen’s approach was introduced in 1990 by Christian Freksa,[Freksa, 1992a]. Rather than reasoning about intervals of time he used semi-intervals.The start and end-points (which themselves can be seen as semi-intervals on higher levelsof granularity) now define the intervals using the relations before (<), equal (=) andafter (>). Thus all relations of intervals can be uniquely defined by using a maximumof two relations of beginnings and endings. This is possible because of two domain-inherent conditions: beginnings take place before endings and the relations (<, =, >)are transitive.Freksa’s approach uses the "Conceptual Neighborhood" of relations which allows forinferring neighbors of relations between objects about which neighborhoods of relationsto some other object are known. It is more efficient in regards of inferencing while thefull reasoning power is maintained, thus this calculus is cognitively plausible. We see,for example, that having less knowledge corresponds to a simpler representation ratherthan Allen’s method, which adds more relation disjunctions, if less is known. This alsoallows for a drastic compaction of the inferencing knowledge base.[Freksa, 1992a]Efforts have been made to extend temporal logic into spatial dimensions such as using thetemporal distinctions for the two axes of a Cartesian coordinate system [Guesgen, 1989].These approaches lack cognitive plausibility because humans don’t decompose the worldinto two axes nor consider relations on each of them.It must be noted that there are (although both are one dimensional) important differ-ences between time and a one-dimensional space. The most important is, that timealways moves forward whereas space has no fixed direction. Therefore spatial represen-tations differ greatly from temporal logic.
Topological distinctions between spatial entities are a fundamental aspect of spatialknowledge. Because those distinctions are inherently qualitative they are particularlyinteresting for qualitative spatial reasoning.15 hapter 2. Foundations
Topological representations usually describe relationships between spatial regions whichare subsets of some topological space. Most approaches that formalize topological prop-erties of spatial regions use a single primitive relation (the binary connectedness relation)to define many other relations. In topology, formal definitions like neighborhood, inte-rior, exterior, boundary etc. can be used to define relations like disjoint, meet, overlap,contain etc. for two regions.
Figure 2.1.:
Examples for topological relations
Orientation is also a very interesting field for spatial representation and reasoning. Rel-ative orientation of spatial entities is a ternary relationship depending on the referent,the relatum and the frame of reference (reference system) which can be specified eitherby a third object (origin) or by a given direction. The orientation is determined by thedirection in which the referent is located in relation to the frame of reference.Three reference systems can be distinguished: • In the
Intrinsic Reference System the orientation is determined by an inherentfeature of the relatum. • The
Absolute Reference System (also called extrinsic reference system) usesexternal parameters like global coordinate systems to give the relatum an orienta-tion. • A third object (origin) is used in the
Relative Reference System (also calleddeictic reference system) so that the orientation is defined by the "point of view"this third object has to the reference system.16 hapter 2. Foundations
Most spatial representations use points in a two-dimensional space as basic spatial en-tities since developing spatial orientations between extended objects is much more diffi-cult.
The scalar entity distance is, together with topology and orientation, one of the mostimportant aspects of space. In human communication qualitative descriptions of distanceare used. There are absolute distance relations that specify distances between two objectslike "close to", "far" or "very far". There also are relative distance relations that comparethe distance between two objects with the distance to a third object like "closer than" or"further than". Qualitatively described absolute distances depend on the relative scaleof space (for example objects on a desktop, in a home or in a city).Combining distance relations not only depends on distance but also on orientation. If anobject A is far from B and another object C is also far from B we cannot say anythingcertain about the distance between A and C. If the objects were on a line in the sequenceof A B C, A would be very far from C, but if they were on a line in the order of A C Bthen A could be close to C. It is thus appropriate to use distance in combination withorientation which is called positional information.
In this thesis I’ll use the term "positional information" for information structures thatcombine the orientation and distance representation. That means, that the position ofan object is described by a combination of a qualitative orientation and a qualitativedistance. Usually the number of possible relations (atomic relations) is the numberof possible relations of the orientation representation multiplied with the number ofrelations of the distance representation. This may not be true for special cases, forexample if relations are considered where objects are on the same location.Hernández [Hernández, 1994] on contrast uses the term "positional information" forcombinations of orientation information with topological information.17 hapter 2. Foundations
All real world objects have an extension. If spatial representations would avail thisfact improvements could be achieved for some applications. The extension is usuallydescribed by size and shape of the objects. The shape of an object could affect the frameof reference of an relation - in an intrinsic reference system it determines the orientationof this object. The size on the other hand has influence on how humans describe spatialrelations. They prefer large, salient objects as reference objects: Humans rather say"The bicycle is in front of the cathedral" than "The cathedral is behind the bicycle".Problems with extended objects arise if they overlap. Two orientations could thendescribe the same configuration. [Hernández, 1994]
Shape is difficult to describe qualitatively. Using topology one can say whether anobject has holes or whether it is in one piece or not. For finer grained distinctionsshape primitives could be used. Other approaches characterize the boundary of an ob-ject using a sequence of different types of boundary segments or curvature extrema.Furthermore shape could be described with polygons that qualitatively define for eachcorner whether it is convex or concave, whether it is obtuse or right-angled or acuteand with an qualitative description of the direction of the corner. Shape representa-tions can make statements solely about the boundary or about the interior of an object[Cohn and Hazarika, 2001], [Jungert, 1993].
Motion is a spatio-temporal information that can be quite easily expressed qualitatively.An example for such an qualitative representation is given in [Brauer et al., 1997]. Inthis approach, the motion of an object is observed from an deictic point of view with afixed frame rate. The two components orientation and distance, that establish a vector,are used to describe the change of position of the object from one frame to the next.If the object is standing still the orientation and distance are 0. If two subsequentqualitative motion vectors are equal one can increment an index. A motion could be forexample (distance,orientation): ( close,forward ) , ( close,left ) , ( ) , ( far,forward ) .18 hapter 2. Foundations Another method of representation is one with an intrinsic frame of reference given by theorientation of the previous vector (the first vector is always "forward"). Now the framerate is only used for standstills, in other cases a new qualitative value, the velocity, isused. A motion would now look like this: ( short-dist,forward,slow ) , ( medium-dist,left,slow ) , ( ) , ( far,right,fast ) .A quite abstract representation is the propositional one. It identifies a set of movementshapes like "straight-line", "left-turn", "right-turn", "u-turn", "loop", etc. that canhave relations between them that describe differences in magnitude, orientation andvelocity.[Brauer et al., 1997] To make use of their spatial knowledge, most applications have to use reasoning mecha-nisms to derive the knowledge needed. Respectively, in order for a representation to beuseful, we most often have to consider not only its constituents and how they correspondto what is being represented, but also the operations on them. Only if simple search formatching relations is sufficient, one can disregard spatial reasoning. This is for examplethe case, if a robot perceives a configuration of objects and has to search for an objecta user specifies by verbally describing an object configuration (like "go to the box thatis right of the basket").Other applications like navigation or computer vision that are presented in section 2.4make serious use of reasoning algorithms. Spatial reasoning is not possible without thepresence of a spatial representation and the quality of the results the reasoning generatesdepends heavily on the underlying representation.To deal with commonsense knowledge is important for any intelligent system and it hasbeen early recognized as one of the central topics of Artificial Intelligence to representand reason about commonsense knowledge. The question can be raised if the formallogic or the physical space is more fundamental for reasoning processes, [Freksa, 1992b].The aim of research on reasoning is to develop efficient algorithms, since the brute-force"generate-and-test" approach is always available but most often unusable due to it’scomplexity.Reasoning can perform many tasks. Most importantly it can infer knowledge that isimplicit in the knowledge base and make it thus explicit. That way the knowledgebase is extended. Reasoning can also answer queries that are given with only partialknowledge and with a specific context. Various types of consistency can be maintained19 hapter 2. Foundations using reasoning techniques that simplify the use of the knowledge base. In generalreasoning is used to acquire and process new knowledge.
In this section some basic reasoning techniques over spatial representations are presented.In order to do so, an assumption is made which actually holds for many calculi, especiallyfor the calculi relevant for this thesis. The assumption is, that the relations between theobjects that are being manipulated are ternary. Ternary relations are being made overthree objects: the first argument is the origin, the second one is the relatum and thethird one the referent. The relation describes the referent with respect to the frame ofreference (or reference system) determined by the origin and the relatum.In qualitative representations each representation has a finite number of atomic rela-tions corresponding to the finite number of spatial configurations that are distinguishedby the representation. The relation over the three objects corresponds to the spatialconfiguration of those objects. Special cases usually occur when two of the three theobjects are on the same position. In order for the algebra to recognize those cases theyare added to the set of atomic relations.In general the three objects partition the space and the set of atomic relations is givenby this partition.
General relation
A (general) relation is any subset of the set of all atomic relations.The interpretation of such a relation R is as follows: ( ∀ X, Y, Z ) ( R ( X, Y, Z ) ⇔ ∨ r ∈ R r ( X, Y, Z )) .The atomic relations are designed to be Jointly Exhaustive and Pairwise Disjoint (JEPD).That means, given any three objects X, Y, Z , there exists one and only one atomic rela-tion r such that r ( X, Y, Z ) . Unary operations
The ternary relations have three arguments and thus there are 3! = 6 possible permuta-tions for the arrangement of those arguments. Following [Zimmermann and Freksa, 1996]the terminology shown in table 2.2 is used for those transformations.20 hapter 2. Foundations
Table 2.2.:
Transformationsterm symbol permutationidentical Id ( ∀ X, Y, Z )( R ( X, Y, Z ) ⇒ Id ( R ( X, Y, Z ))) inversion
Inv ( ∀ X, Y, Z )( R ( X, Y, Z ) ⇒ Inv ( R ( Y, X, Z ))) short cut Sc ( ∀ X, Y, Z )( R ( X, Y, Z ) ⇒ Sc ( R ( X, Z, Y ))) inverse short cut
Sci ( ∀ X, Y, Z )( R ( X, Y, Z ) ⇒ Sci ( R ( Z, X, Y ))) homing Hm ( ∀ X, Y, Z )( R ( X, Y, Z ) ⇒ Hm ( R ( Y, Z, X ))) inverse homing
Hmi ( ∀ X, Y, Z )( R ( X, Y, Z ) ⇒ Hmi ( R ( Z, Y, X )))
Binary operationsComposition
The composition R ⊗ R of two relations R and R is the most specific relation R j such that: ( ∀ X, Y, Z, W ) ( R ( X, Y, Z ) ∧ R ( X, Z, W ) ⇒ R j ( X, Y, W )) .Given four objects X, Y, Z, W and two atomic relations r and r the conjunction r ( X, Y, Z ) ∧ r ( X, Z, W ) is trivially inconsistent if either of the following holds: ( ∀ X, Y, Z, W ) (( r ( X, Y, Z ) ⇒ ( Z = X )) ∧ ( r ( X, Z, W ) ⇒ ( Z (cid:54) = X )))( ∀ X, Y, Z, W ) (( r ( X, Y, Z ) ⇒ ( Z (cid:54) = X )) ∧ ( r ( X, Z, W ) ⇒ ( Z = X ))) For atomic relations for which the above holds the result of the composition is empty: r ⊗ r = ∅ In general a compositional inference is a deduction from two relational facts of the form R ( a, b ) and R ( a, b ) to a relational fact of the form R ( a, c ) , involving only a and c.The validity of compositional inferences does, in many cases, not depend on the specificelements involved but only on the logical properties on the relations. In such a casethe composition of pairs of relations can be abstracted by table look up as and whenrequired. Given the set of n JEPD atomic relations, one can store in a n × n compositiontable the relationships between x and z for a pair of relations R ( x, y ) and R ( y, z ) . Ingeneral, each entry can be a disjunction of the base relations.For the ternary relations those tables have to be made available for the following twocases: ( ∀ X, Y, Z, W ) (( r ( X, Y, Z ) ⇒ ( Z = X )) ∧ ( r ( X, Z, W ) ⇒ ( Z = X ))) and ( ∀ X, Y, Z, W ) (( r ( X, Y, Z ) ⇒ ( Z (cid:54) = X )) ∧ ( r ( X, Z, W ) ⇒ ( Z (cid:54) = X ))) hapter 2. Foundations For a representation using a limited set of binary relations, the simplicity of the compo-sitional inference makes it an attractive means of effective reasoning.
Intersection
The intersection of two relations R and R is the relation R which consistsof the set-theoretic intersection of the atomic relation sets from R and R : R = R ∩ R .[Isli and Moratz, 1999] The constraint satisfaction problem is identified as an abstract formulation of manydifficult problems in Artificial Intelligence. Given that qualitative representations can beexpressed in form of relations, general constraint satisfaction techniques can be appliedfor various kinds of inference. Constraint satisfaction techniques play an important rolein Computer Science as a whole, and in Artificial Intelligence in particular. Many difficultproblems involving search from areas such as (robot) navigation, temporal reasoning,graph algorithms and machine design and manufacturing can be considered to be specialcases of the constraint satisfaction problem (CSP).Constraint Satisfaction Problems (CSP) generally consist of: • a set of variables X = { X , ..., X n } , • a discrete domain for each variable { D , ..., D n } so that each variable X i has afinite set D i of possible values, • a set { R k } of constraints, defined over some subset of the variable domains, R j ⊆ D i × · · · × D ij , and showing the mutually compatible values for a variablesubset { X i , ..., X ij } . Those constraints are restricting the values the variables cansimultaneously take.The problem is to find an assignment of values to variables such that all constraints aresatisfied. Variants of the problem are to find all such assignments, the best one, if thereexists any at all, etc. 22 hapter 2. Foundations The constraints are restricted to be unary or binary because those are the operationswe defined for the spatial representation. Furthermore it is assumed that all variabledomains have the same cardinality.The straightforward approach to find a satisfying assignment is a backtracking algorithmthat corresponds to an uninformed systematic search. If, after having instantiated allvariables relevant to a set of constraints, any of them is not valid, the algorithm back-tracks to the most recently instantiated variable that still has untried values available.The run-time complexity of this algorithm is exponential, making it useless for realisticinput sizes. This inefficiency arises because the same computations are repeated unnec-essarily many times.Possible solutions to this problem are the modification of the search space, the use ofheuristics to guide the search or the use of the particular problem structure to orientthe search. The latter one is especially interesting for spatial reasoning as we will seelater.
The goal of modifying the search space is to avoid useless computation without missingany of the solutions of the original space. In other words, we are looking for a smallerbut equivalent search space. This can be achieved either prior the search (by improvingthe consistency of the network through constraint propagation) or, in hybrid algorithms,during the search.One way of reducing the number of repeated computations is constraint propagation.There those values are removed from a domain that do not satisfy the correspondingunary predicates, as well as those values for which no matching value can be found in theadjacent domains such that the corresponding binary predicates are satisfied. The formerprocess achieves node consistency, the latter arc consistency. Expressed differently onecan say that this process of constraint relaxation is triggered by incompatible constraints.This concept of local consistency can be generalized to any number of variables. A setof variables is k -consistent if for each set of k − variables with satisfying values, itis possible to find a value for the k th variable such that all constraints among the k variables are satisfied. A set of variables is strong k -consistent if it is j -consistent for all j ≤ k . Of special interest is strong 3-consistency, which is equivalent to arc consistencyplus path consistency. A network is path consistent if any value pair permitted in R ij is also allowed by any other path from i to j . The process of achieving consistency in23 hapter 2. Foundations a network of constraints is called constraint propagation. Several authors give differentdefinitions of constraint propagation, whose equivalence is not obvious.One such definition is that constraint propagation is a way of deriving stronger (i.e.,more restrictive) constraints by analyzing sets of variables and their related constraints.The value elimination consistency procedures mentioned above assume a very generalextensional form of constraints as sets of satisfying value pairs. Whenever a value of thedomain of a variable involved in more than one constraint is removed, a satisfying valuepair might have to be removed from one or more of the other constraints, thus makingthem more restrictive.Another view of constraint propagation is as the process of making implicit constraintsexplicit, where implicit constraints are those not recorded directly in compatible valuepairs, but implied by them. This, however, can be seen as a side effect of the consistencyprocedures. The universal constraint, that allows any value pair, holds implicitly betweentwo variables not explicitly linked together. If the domains of the two variables X i and X j are restricted to a few values, then the implicit constraint R ij can be made explicitas the set of combinations of the two domains.Note that the global full constraint propagation is equivalent to finding the minimalgraph of the CSP, where each value permitted by any explicit constraint belongs to atleast one problem solution. Full constraint propagation is thus as hard as the CSP itself(in general NP-complete). The local constraint propagation techniques used to achievenode-, arc- and path-consistency have polynomial complexities and can be used as pre-processors that substantially reduce the need to backtrack during the search for a globalsolution.Unfortunately, arc- and path-consistency do not eliminate in general the need to back-track during the search because constraints are propagated only locally. Spatial inferencing is concerned with the qualitative spatial analysis of the spatial con-figurations, often in two-dimensional Euclidean space. It uses the reasoning techniquespresented in the previous sections. The composition of spatial relations is very under-determinate. The resulting relation sets tend to contain too many atomic relations. Butoften the structure of the spatial domain inherits certain constraints that allow furtherimprovements for constraint based reasoning.24 hapter 2. Foundations
In section 2.1.1 it is shown how Freksas approach uses constraints of the temporal do-main to improve reasoning about time. An example for these constrains in the spatialdomain is, that distinct solid objects never occupy the same point in space. Hernández[Hernández, 1994], giving another example, uses "abstract maps" that contain for eachobject in a scene a data structure with the same neighborhood structure as the domainrequired for the task at hand. A change in of the point of view can then be easily ac-complished diagrammatically by rotating the labels of the orientation. Hernández alsoobserved, that the composition of pairs of topological and orientation information yieldsmore specific results.In [Freksa, 1991] Freksa shows, that spatial location representations of physical objectsconsist of connected parts (their "neighbors", see section 2.3.3) and that movement inspace is only possible between neighboring locations. Therefore a general representationshould not explicitly check if these constraints hold; the constraints should rather be"build-in" in the representation and reasoning system.
Qualitativeness can be best explained in contrast with it’s counterpart Quantitativeness.First we gonna take a look at the American Heritage Dictionary entries for these twoterms: • quantity: 1. A specified or indefinite number or amount. An exact amount ornumber. 2. The measurable, countable, or comparable property or aspect of athing. • quality: 1. The essential character of something; nature. 2. An inherent ordistinguishing characteristic; property.The most important words for quantity are number and measurable, because spatialquantities are measured which implies that a number is assigned to represent a magni-tude. Usually the assignment can be made by a simple comparison. The magnitude ofthe quantity is compared to a standard quantity, the magnitude of which is arbitrarilychosen to have the measure 1.The term quality is more difficult to explain. The qualitative representations can becharacterized by establishing a correspondence between the abstract entities in the rep-resentation and the actual magnitudes. Quantitative knowledge is obtained whenever a25 hapter 2. Foundations standardized scale is used for anchoring the represented magnitudes. The use of a scaleis also the context in which issues of granularity and resolution are meaningful, since ascale defines a smallest unit of possible distinction below which we are not able to sayanything about a quantity.[Brauer et al., 1997]Qualitative representation provides mechanisms for representing only those features thatare unique or essential, whereas a quantitative representation allows to represent allthose values that can be expressed with respect to a predefined unit. Although qual-itative reasoning allows inferences to be made in absence of complete knowledge, itis not a probabilistic or fuzzy approach but it refuses to differentiate between certainquantities.One primary goal of qualitative spatial representations is to provide a general vocabularyfor performing efficient symbolic spatial inferences and analysis. Although the semanticsof the vocabulary may not be unique, the discrete values for its definition over thecontinuous domains should allow computationally efficient spatial inference and lessambiguous spatial descriptions (i.e., with sufficient precision).A mathematical definition for a qualitative abstraction can look like this:Let x be a quantitative variable, such that x ∈ R , and R ⊆ (cid:60) . If the entire domain R is partioned into a finite set of mutually disjoint subdomains { Q , Q , ..., Q m } , i.e., (cid:83) mi =1 Q i = R , and, furthermore, all numerical values lying within Q i are treated asbeing equivalent and named symbolically by Label ( Q i ) , then the qualitative variable [ x ] corresponding to x is defined as follows: [ x ] ∈ X, X ⊆ m (cid:83) i =1 Label ( Q i ) where Label ( Q i ) is called a primitive qualitative value.[Liu and Daneshmend, 2004] This section is a citation from [Brauer et al., 1997] who pointed out many of the usefulproperties of qualitative representations perfectly.26 hapter 2. Foundations • Qualitative representations make only as many distinctions as necessary to identifyobjects, events, situations, etc. in a given context (recognition task) as opposedto those needed to fully reconstruct a situation (reconstruction task). • All knowledge about the physical world in general, and space in particular, isbased on comparisons between magnitudes. As representations that capture suchcomparisons, qualitative representations reflect the relative arrangement of mag-nitudes, but not absolute information about magnitudes. • The search for distinctive features that characterizes the qualitative approach hasan important side effect: It structures the domain according to the particularviewpoint used. Some of the qualitative distinctions being made are conceptuallycloser to each other than others. This structure is reflected in the set of relationsused to represent the domain. • Qualitative representations are "under-determined" in the sense that they mightcorrespond to many "real" situations. The reason they still can be effectively usedto solve spatial problems is that those problems are always embedded in a partic-ular context. The context, which for simplicity can be taken to be a set of objects,should constrain the relative information enough to allow spatial reasoning, for ex-ample by making it possible to find a unique order along a descriptional dimension.In other words, a representation that can count on being used together with someparticular context does not need to contain as much specific information itself. • Qualitative representations handle vague knowledge by using coarse granularitylevels, which avoid having to commit to specific values on a given dimension. Withother words, the inherent "under-determination" of the representation absorbs thevagueness of our knowledge. • In qualitative representations of space, the structural similarity between the rep-resenting and the represented world prevents us from violating constraints cor-responding to basic properties of the represented world, which in propositionalsystems would have to be restored through revision mechanisms at great cost. • Unlike quantitative representations, which require a scale to be fixed before mea-surements can take place, qualitative representations are independent of fixed gran-ularities. The qualitative distinctions made may correspond to finer or coarser dif-ferences in the represented world, depending on the granularity of the knowledgeavailable and the actual context. 27 hapter 2. Foundations • The informative content of qualitative relations varies. Some describe what wouldcorrespond to a large range of quantitative values of the same quality, while othersmay single out a unique distinctive value. • While the discrimination power of single qualitative relations is kept intentionallylow, the interaction of several relations can lead to arbitrarily fine distinctions. Ifeach relation is considered to represent a set of possible values, the intersectionsof those sets correspond to elements that satisfy all constraints simultaneously.[Brauer et al., 1997]
Of course quantitative representations are very useful for many applications (CAD, 3DGraphics etc), but there are good reasons why quantitative approaches are to be preferredin many other systems: • Advantages in Input and Output – Partial and uncertain information: A value may not be exactly known (deter-mined by the priori fixed scale) due to imprecise sensor data or vague humandescriptions. It then either has to be ignored or assigned to a range of pos-sible values in quantitative approaches. In the latter case there will be morecomputation needed. Qualitative approaches have no problems handling suchinformation. – Transformation: The transformation of a quantitative value to a qualitativeone is done more easily and exactly than in the opposite direction. – Qualitative input: The input for reasoning processes is often qualitative - itis often the result of a comparison rather than a quantitative description. Itis thus better to use qualitative reasoning. – Real world input: Spatial reasoning (in the real world) is in most casesdriven by qualitative abstractions rather than by complete a priori quan-titative knowledge. – Reasoning goal: The goal of reasoning is always qualitative. A decision isbeing processed not a quantitative value. • Interaction with humans 28 hapter 2. Foundations – Missing adequacy: Humans are not very good at determining exact length,orientations ect. but they can easily perform context-dependent comparisons.In quantitative approaches humans would be forced to use quantities to ex-press facts. – Human reasoning: Humans also do qualitative reasoning more easily (andsometimes better). – Communication: Humans are used to communicate spatial facts qualitatively.This is thus their preferred way to interact with computers, too. – Human cognition: Qualitative representations are more transparent and in-tuitive to humans because it is believed that this is the method humansthemselves use to reason about space. • Computation issues – Transformational impedance: As a consequence of the missing adequacy de-scribed above, spatial reasoning systems based on quantitative values mighthave to transform back and forth between their internal representation andthe qualitative one which is used to communicated with humans. Informationmight get lost during these transformations. – Robustness: Qualitative approaches are more robust against errors than nu-merical methods. – Falsifying effects: Quantitative models might falsify the representation byforcing discrete decisions. – Unnecessary details: With qualitative representations unnecessary details canbe omitted which yields to smaller and more transparent representations. – Properties: Quantitative approaches do not have the nice properties thattheir analytical counterparts have. • Complexity – Cheapness: Qualitative knowledge is cheaper than quantitative knowledgebecause it is less informative in a certain sense. – Complexity: The number of values that a descriptional value may take affectsindirectly the complexity of the algorithms operating on them. Not only aremore involved computations required, but the granularity of the representa-tion, as determined by the fixed scale chosen, may make more distinctions29 hapter 2. Foundations than necessary for a given task. For example, the exact positions of all ob-jects in the room are not necessary, if all we need to know is what objects areat a wall that is to be painted.[Freksa, 1992b] [Hernández, 1994]
Quantitative representations usually store the spatial information in a common globalor local coordinate system. Agents might have a local coordinate system derived fromthe inherent orientation of the agent or, in contrast, in the global coordinate system allentities and objects use the same common system which can, for example, be oriented ona building’s floor plan or the cardinal directions. Agents can communicate their spatialknowledge by mapping their local coordinate systems to a global one. That can onlybe done if all agents have access to the global coordinate system. The relevant globalcoordinates can then be exchanged.Using quantitative representations, it is difficult to handle indeterminate or inexactknowledge. For a quantitative representation the precise position and the size of allobjects must be known. Furthermore, detecting certain spatial configurations is quitedifficult but it is often needed to trigger certain actions or behaviors. In the numericalapproach reasoning is done with numerical or geometrical methods like computing atangent to an object or closest points of two objects to each other.[Renz, 2002]The qualitative approach is representing spatial knowledge without exact numericalvalues. It uses a finite vocabulary that describes a finite number of possible relationships.This is, compared to the quantitative approach, closer to how humans represent spatialknowledge. In natural language the human says something like "A is above B" or "Ais next to B" to express spatial information. Those information are usually sufficientto identify an object or follow a route. It is easy to represent indefinite or uncertainknowledge with qualitative methods. For example, one could say "A is left or behindB". Furthermore are rules easy to define "Do something if A is in front of B".Qualitative spatial knowledge is not inexact, even though no numerical values are used.This is because distinctions are only made if necessary and they depend on the levelof granularity that was chosen. Both, the quantitative approach and the qualitativeapproach, have their own right because both have applications where they are bestsuited. Robots that interact with humans usually take advantage of using a qualitative30 hapter 2. Foundations representation. If available, exact coordinate based knowledge should be used. Theparallel use and transformation of both representation approaches is often useful, forexample for a car navigation system by giving directions that guide the way in anunknown city using its street map and data from the Global Positioning System.
Points versus areas
Qualitative spatial knowledge can be represented using spatially extended objects orabstract points. For the one dimensional time Allen [Allen, 1983] pointed out goodreasons why to use intervals of time instead of points. He showed that every event intime (for example "finding a letter") can be decomposited into a time interval ("lookingat a letter" -> "realizing that it is the one that is searched for"). This also holds truefor space since every real life point can be magnified to an area (at least as one staysabove nano-sized structures). Another reason why Allen preferred intervals of time isthe problem of open or closed intervals that occurs if one models intervals of time withpoints of time. Considering a situation where a light is switched off there is a point intime where the light is neither on or off assuming open intervals. With closed intervalsthere would be a point in time where the light is both - on and off. Allan solved thereproblems very elegantly with his Temporal logic 2.1.1.In two dimensional space Allens intervals corresponds to the aspects topology and shape.It is clear that it is needed to consider areas and not points if one wants to reason abouttopology or shape. But problems arise in two dimensional space because there are lots ofpossible classes of shapes which cannot be handled equally well. Freksa [Freksa, 1992b]showed that it is more convenient to use points, especially for other aspects of space likeorientation and distance. One reason is, that the properties of points and their spatialrelations hold for the entire spatial domain. Another, that shapes can be representedusing points at different levels of abstraction. Lastly Freksa explained, that it is desirableto be flexible with respect to the spatial entities and their resolution. That means, thatin one context one might be purely interested in 0-dimensional points such as pointson a map. Other applications might be interested in 1-dimensional information like thewidth of a river or the length of a road. 2-dimensional projections (e.g. area of a lake)or 3-dimensional shapes of objects might be of interest in other contexts.31 hapter 2. Foundations
Orientation
Orientation is a ternary relationship depending on the referent, the relatum and theframe of reference which can be specified either by a third object (origin) or by a givendirection (see section 2.1.3). If the frame of reference is given the orientation can beexpressed using binary relationships. For 2-dimensional space Freksa [Freksa, 1992b]defined orientation as a 1-dimensional feature which is determined by an oriented line.The oriented line is defined by an ordered set of two points. An orientation is thendenoted by an oriented line ab through the two points a and b (see figure 2.2). ba denotes the opposite orientation. The relative orientation in 2-dimensional space is thengiven by two oriented lines (which are represented by two ordered sets of points). Thetwo ordered sets of points can share one point without loss of generality because thefeature orientation is independent of location and vice versa. One can thus describethe orientation of a line bc relative to the orientation line ab . This way the ternaryrelationship is again achieved. Three special cases arise if the locations of (1) a or (2) c or (3) both are identical with the location of b . In the first special case ( a = b ) noorientation information can be represented. In the second ( c = b ) and the third case( a = b = c ) orientation information is unavailable, too, but location information of c isstill available. The point c is called the referent (also primary object or located object), b the relatum (also reference object) and a is called origin (also parent object). Figure 2.2.: a) location of c wrt. the location of b and the orientation ab ; b) Ori-entation relations wrt. to the location b and the orientation ab ; c) left/right andfront/back dichotomies in an orientation system32 hapter 2. Foundations From this point different approaches use diverse methods to determine the qualitativeorientation values. A simple distinction can be made by defining four qualitativelydifferent orientation values labeled same , opposite , left and right . If the point c is onthe line ab on the other side of b than a the orientation is same and if it is on the line ab on the other side of a than b the orientation is opposite . The orientation is left ifthe point c is located on the left semi-plane of the oriented line ab and it is right if c islocated on the right semi-plane of the oriented line ab . Freksa observed that unlike in thecase of linear dimensions, incrementing quantitative orientation leads back to previousorientations. In this sense, orientation is a circular dimension. Figure 2.3.: cone-based representationA substantial gain of information can be achieved by introducing a front/ back di-chotomy. In this so called cardinal direction representation the eight different orien-tation labels straight-front , right-front , right-neutral , right-back , straight-back , left-back , left-neutral and left-front are distinguished (see figure 2.2). Frank [Frank, 1991] calledthis method "projection-based" and Ligozat [Ligozat, 1998] "cardinal algebra". Ligozatalso found that reasoning with this cardinal algebra is NP-complete. Another approachto augment the number of orientation values is called "cone-based" by Frank. Here ninedifferent relations are distinguished: north , north-east , east , south-east , south , south-west , west , north-west and equal (see figure 2.3). In this method the plane is split intoeight slices of 45 ◦ by three lines. All eight segments have the same scope unlike in thecardinal algebra, where the -front and -back relations correspond to an infinite num-33 hapter 2. Foundations ber of angles (as in the cone-based approach) while the straight- and neutral- relationscorrespond to a single angle.Other approaches are presented more detailed in the chapter 3 State of the Art. Distance
Distance is, unlike topology and orientation, a scalar entity. As mentioned in section2.1.4 one has to distinguish between absolute distance relations and relative distancerelations. Absolute distance relations indicate the distance between two points anddivide the real line into a different number of sectors depending on the chosen levelof granularity. They can be represented quantitatively or qualitatively and depend onan uniform global scale. Absolute distance relations name distances whereas relativedistance relations compare the distance of two points with the distance to a thirdpoint. Next to the obvious predicates <, =, > more relations for relative distances canbe defined if needed (e.g. "much shorter", "a little bit less", "much longer").In [Clementini et al., 1997] a general framework for representing qualitative distancesat different levels of granularity has been developed. The space around the relatum RO is partitioned according to a number of totally ordered distance distinctions Q = { q , q , q , ..., q n } , where q is the distance closest to the relatum and q n is the one farthestaway (which can go to infinity). Distance relations are organized in distance systems D defined as: D = ( Q, A, (cid:61) ) where: • Q is the totally ordered set of distance relations; • A is an acceptance function defined as A : Q × O → I , such that, given a referenceobject (relatum) RO and a set of objects O , A ( q i , RO ) returns the geometricinterval δ i ∈ I corresponding to the distance relation q i ; • (cid:61) is an algebraic structure with operations and order relations defined over a setof intervals I . (cid:61) defines the structure relations between intervals.Each distance relation can be associated to an acceptance area surrounding a referenceobject (relatum) which will be circular in isotropic space (that is space which has thesame cost of moving in all directions). 34 hapter 2. FoundationsHomogeneous distance systems are those in which all distance relations have the samestructure relations. That means, the size of the geometric intervals δ i follows a recur-rent pattern. The general type of distance systems where this is not the case is calledaccordingly heterogeneous . More restrictive properties of the structure of these inter-vals include monotonicity (each interval is bigger or equal than its previous) and rangerestriction (any given interval is bigger than the entire range from the origin to theprevious interval).The frame of reference (see section 2.1.3) is important for distance systems, too. In the intrinsic reference system the distance is determined by some inherent characteristics ofthe relatum, like its topology, size or shape (e.g. 50 meters can be considered far awayfrom a small house but they seem to be close if standing next to a skyscraper). Thedistance is determined by some external factor in the extrinsic reference system, like thearrangement of objects, the traveling time or the costs involved. The deictic referencesystem uses an external point of view to determine the distance, like if the objects arevisually perceived by an observer. Conceptual Neighbors
For higher-level reasoning and knowledge abstraction the conceptual neighborhood re-lation provides several advantages. It was originally developed by Freksa for temporalknowledge [Freksa, 1992a], generalizing Allens temporal logic (see section 2.1.1). LaterFreksa successfully applied the conceptual neighborhood principle to spatial knowledge[Freksa, 1992b].In a representation two relations are conceptual neighbors, if there exists an opera-tion in the represented domain that causes a direct transition from one relation to theother. Those operations can be either spatial movement or deformations for the physicalspace. In the cone-based approach described above each relation except equal has threeconceptual neighbors. The conceptual neighbors of north , for example, are north-east , north-west and equal because it is possible to make a direct move there without theneed to traverse any other relation. East on the other hand is not a conceptual neighborof north because there is no direct transition from north to east - possible ways eitherhave to cross north-east , equal or even detour by traversing through north-west . Butthis only holds for the cone-based representation presented here - other representationshave other conceptual neighbors and for more coarse ones there might be a direct wayfrom north to east. The conceptual neighbors for a distance relation q x in the distancerepresentation above are obviously those two distance relations in the totally ordered35 hapter 2. Foundations set Q that are next to q x , viz q x − and q x +1 (there is only one neighbor for the first ( q )and last).A great benefit of conceptual neighborhood structures is, that they intrinsically reflectthe structure of the represented real world with their operations. This makes it possibleto implement reasoning strategies which are strongly biased toward the operations inthe represented domain. Conceptual neighborhoods allow to only consider operationswhich are feasible in the specific domain which can restrict the problem space and thusachieve nice computational advantages. Figure 2.4.:
The possible locations of c (gray) before and after the unary Inv operationIn order to use the reasoning techniques introduced in section 2.2 the unary and binaryoperations have to be defined for the spatial representation. In this section these op-erations are illustrated by using the cardinal direction representation presented in theorientation section above (see figure 2.2 c) ).The first example will be the unary operation of a spatial arrangement where c , thereferent, is at the left-back of ab , where a is the origin and b the relatum. Now wemight need to exchange the origin and the relatum. That is done by using the inverse( Inv ) operation (see table 2.2). B is now the origin and a the relatum. The result ofthat operation is ambiguous, as can be seen in figure 2.4. The space is now partitioned36 hapter 2. Foundations differently because the front/ back dichotomy now divides the space through a . Ofcourse the labels changed, too. For example, what was left is now right and vice versabecause the direction has been changed by 180 ◦ . Because of the qualitative nature of therepresentation we cannot say whether the point c is right-back , right-neutral or right-front of ba . The result of this operation is thus a disjunction of all three possible atomicrelations. Figure 2.5.:
Composition - the possible locations of d are shown in grayThe binary composition operation can have ambiguous results as well. We now have fourpoints a, b, c and d and two spatial relations - one between a, b, and c and the otherbetween b, c and d (always origin, relatum and referent in this order). The compositionof these two relations answers the question where the point d is in relation to a and b (see figure 2.5 - the possible locations of d (for an assumed position of c ) are indicatedgray). In both input-relations the referent ( c respectively d ) is located in the right-front .It is clearly visible that the composition result, which describes the location of d (gray)with a as origin and b as relatum, consist of the atomic relations right-front , right-neutral and right-back . The result of this composition is thus a disjunction of all three possibleatomic relations. 37 hapter 2. Foundations Qualitative representations of space can be used in various application areas in whichspatial knowledge plays a role. It is in particular used in those systems which are char-acterized by uncertainty and incompleteness.This coarse knowledge often occurs in early stages of design projects, in which verballyexpressed spatial requirements and raw sketches are very common. Examples for suchapplication areas include spatial and geographical information systems, computer aidedsystems for architectural design and urban planning, document analysis, computer vi-sion, natural language processing (input and output), visual (programming) languages,qualitative simulations of physical processes and of course navigation, robot-navigationand robot-control.Although so many areas exists, there are quite few existing applications present. Onereason for that is, that most applications require more than just one aspect of space.But since the different aspects of space are not independent of each other it is not pos-sible to simply add different approaches covering single aspects of space to achieve thedesired functionality. Another reason for the lack of existing applications is, that manyapproaches have no or not complete reasoning mechanisms without which the spatialrepresentations are less useful [Renz, 2002].
Geographical Information Systems (GIS)
GIS are these days commonplace, but they have major problems with interaction withthe user. GISs have access to a vast size of vectorized information, without the abilityto sufficiently support intuitive interaction with humans. Users may wish to performqueries that are essentially, or at least largely, qualitative. Egenhofers concept of
NaiveGeography [Egenhofer and Mark, 1995] employs qualitative reasoning techniques to en-hance the human - GIS interface.
Robotics
Robotic navigation can make use of qualitative approaches in high level planning. Be-sides that, robots may navigate using qualitative spatial representations if the robot’smodel of its environment is imperfect, which would lead to an inability to use standardrobot navigation techniques if numerical representations had been chosen. Research in38 hapter 2. Foundations qualitative methods for robotics concentrates in robust qualitative representations andinference machines for exploration, mapping and navigation.In [Freksa et al., 2000] an approach to high level interaction with autonomous robots bymeans of schematic maps was outlined. Schematic Maps are knowledge representationstructures that encode qualitative spatial information about a physical environment. Inthe presented scenario an autonomous mobile robot with rudimentary sensory abilitiesto recognize the presence as well as certain distinguishing features of obstacles had thetask to move to a given location within a structured dynamic spatial environment. In or-der to achieve that task the robot has to implement certain abilities: It has to determinewhere to go to reach the target location which needs spatial knowledge. It further hasto compute what actions are to be made in order to move there. This needs knowledgeabout the relation between motor actions and movements and it also needs knowledgeof the relation between movement and the robots spatial location.In robotics there are no detailed, complete information about the spatial structure ofthe environment available for several reasons. First it is hard to provide detailed spatialknowledge which agrees with the physical space. Secondly there is no persistence - thatmeans that the spatial configurations might change for unpredictable reasons. Lastly theactions of the robots in physical space are typically not fully predictable due to slippingwheels and imprecise sensors and motors.One main reason why autonomous robotics is so difficult is, that the robot lives in twoworlds simultaneously. It exists in the physical world of objects and space as well as inthe abstract world of representation and computation. Those worlds are incommensu-rable: There is no theory that can treat both worlds in the same way. Classical ArtificialIntelligence approaches try to develop formal theories about space that are sufficientlyprecise to describe all that is needed to perform the actions on the level of the represen-tation (successful examples: board games). Qualitative theories on the other hand onlydeal with some aspects of the physical world and leave other aspects to be dealt withseparately.In robotics maps are used for communication between the robot and humans as wellas interaction interfaces between the robot and its environment. The power of mapsas representation media for spatial information stems from the strong correspondencebetween spatial relations in the map and spatial relations in the real world. Thus spatialrelations may be directly read from the map, even if those have not been entered into therepresentation explicitly. Maps distort spatial relations to some extent, most obviouslythey transform the scale.Schematic Maps distort beyond the distortions required for representational reasonsto omit unnecessary details, to simplify shapes and structures, or to make the maps39 hapter 2. Foundations more readable. Typical examples for schematic maps are public transportation mapsor tourist city maps which may severely distort distances and orientations between ob-jects. Schematic maps provide suitable means for communicating navigation instructionsto robots: spatial relationships like neighborhood, connectedness of places, location ofobstacles, etc. can be represented. Those maps can be encoded in terms of qualita-tive spatial relations and qualitative spatial reasoning can be used to infer relationshipsneeded for solving the navigation task. For the correspondence between the map and thespatial environment coarser low-resolution information is more suitable. A large numberof rare or unique configurations can be found this way. If relations from the map andthe environment do not match perfectly, the conceptual neighborhood knowledge (seesection 2.3.3) can be used to determine appropriate matches.Schematic maps may be created in at least three different ways: A human may acquireknowledge about the spatial layout and build the schematic map by entering the rele-vant relationships, perhaps with the help of a computerized design tool. The robot itselfmay explore its environment in its idle time and create a schematic map that reflectslandmarks and spatial relationships between notable entities discovered from the robotsperspective. Furthermore a spatial data base could be used to create schematic maps.The navigation planning and execution starts with the initial schematic map that pro-vides the robot with survey knowledge about its environment. Important features areextracted from the map for identification in the environment. The robot can also enterdiscoveries into the map that it made during its own perceptual explorations. A coarseplan for the route is being produced using global knowledge from the map and localknowledge from its own perception. The resulting plan is a qualitative one. During itsexecution, the robot will change its local environment through locomotion. This enablesit to instantiate the coarse plan by taking into account temporary or unpredicted obsta-cles. Also, the local exploration may unveil serious discrepancies between the map andthe environment that prevent the instantiation of the plan. In this case, the map can beupdated by the newly accumulated knowledge and a revised plan can be generated.
Linguistics
Humans naturally communicate using qualitative expressions. Thus humans can con-struct schematic maps rather easily. Therefore schematic maps can be used for two-waycommunication between humans and robots. The instructor can give a verbal descriptionof the goal. If the robot fails to generate a working plan for some reason it can commu-nicate with the instructor and possibly give an indication about it’s (spatial) problem.40 hapter 2. Foundations
However, in natural language, the use and interpretation of spatial propositions tend tobe ambiguous. There are multiple ways in which natural language spatial relationshipscan be used.[Freksa et al., 2000]
Voronoi graphs
In [Moratz and Wallgrün, 2003b] Moratz and Wallgrün used the Distance/orientation-interval propagation presented in section 3.4 to build an environmental map for mobilerobot navigation. The foundation of this map is a generalized Voronoi graph (GVG),which is the graph corresponding to the generalized Voronoi diagram of the robots freespace that is annotated with additional information (see figure 2.6). The nodes of theGVG correspond to the meet points and the edges correspond to the Voronoi curves.This graph only represents the topology of the GVD and it is annotated with additionalinformation, including DOI information between the neighboring vertices.
Figure 2.6.:
The fine lines are the generalized Voronoi diagram of a 2D environment.On the right side is the corresponding GVG.This representation brings together advantages stemming from the use of topologicalmaps and from the use of Voronoi diagrams in mobile robot applications. Path plan-ning can be done directly by using graph search to search through the GVG. Sinceonly qualitatively different paths are represented path planning becomes very efficient.[Wallgrün, 2002] The DOI information stored in the graph can be used for cycle de-tection. The DOIs are propagated along the path through the GVG that connects thevertices that are checked to be identical. 41 hapter 3.State of the Art
In 1992 Randell, Cui and Cohn developed the Region Connection Calculus [Randell et al., 1992].In this theory topological relations are used as a basis for qualitative spatial representa-tion and reasoning.RCC-8 is a subset of eight relations from RCC. Those eight base relations are: "A isdisconnected from B", "A is externally connected to B", "A partially overlaps B", "A isequal to B", "A is a tangential proper part of B", "A is a non-tangential proper part ofB" and the converse of the latter two relations (A and B are spatial regions). RCC-8 alsocontains all possible unions of these base relations. RCC-8 is the spatial counterpart ofAllen’s temporal interval algebra - the eight base relations correspondent to Allen’s baserelations. Most other relations in RCC are refinements of the RCC-8 base relations.RCC-8 semantics of base relations can be described using propositional logics rather thanfirst-order logic needed in RCC [Bennett, 1996]. Thus reasoning about TCC-8 relationsis decidable.RCC is a fully axiomatized first-order theory for representing topological relations. Allspatial entities are regarded as spatial regions. See appendix C for detailed definitionsin the RCC calculus [Renz, 2002]. 42 hapter 3. State of the Art
An approach for dealing with intrinsic orientation information is presented in[Moratz et al., 2000]. It uses orientated line segments called dipole that are formed bya pair of points - a start point ( s A for a dipole A) and an end point ( e A for a dipole A).These dipoles are used to represent two-dimensional extended spatial objects with anintrinsic orientation (see figure 3.1). The local orientation of the dipoles is even simpleras those relation presented in section 2.3.3. In the dipole representation a point canonly be left (l), right (r) and on the straight line (o) of the referring dipole, merging theatomic relations same and opposite of figure 2.2 b) into one. With the left and right relations between a dipole and the start and end points of another dipole, 24 JEPDatomic relations can be distinguished, given that no more than two points are allowedto be on a line (this is called general position) with the exception that two dipoles mayshare one point. Figure 3.1.:
DipoleGiven two dipoles A and B with their start points s A , s B and their end points e A , e B ,the atomic relations between them can be described as follows: AR s B ∧ AR e B ∧ BR s A ∧ BR e A with R , R , R , R ∈ { r, l, s, e } If R is s or e (meaning that s B is on the same point as s A ( s ) or e A ( e )), R can onlybe r or l and vice versa ( R and R exchanged), since dipoles share maximal one point.This also holds for R and R .A short form of the term above can be written as: AR R R R B (see figure 3.2)43 hapter 3. State of the Art Figure 3.2.:
The 24 atomic relations of the dipole calculus
The Dipole Calculus forms a relation algebra - it is closed under the binary compositionoperation and under the unary intersection, complement and converse operation. It alsosupports an empty relation, an universal relation and an identity relation. Compoundrelations are sets of atomic relations. The composition results of atomic relations areobtained in an composition table. The composition of compound relations can be ob-tained as the union of the compositions of the corresponding atomic relations. DipolesConstraint Satisfaction Problems can be solved using the methods presented in section2.2.2. It has been shown that the Dipole Calculus is NP-hard and in PSPACE.Freksas double-cross calculus describes relations between triples of points, which can beregarded as relationships between a dipole and an isolated point. In contrast to Freksasternary relations, the dipole relations are binary relations which makes reasoning mucheasier. Also, Freksa distinguishes more possible relations between a dipole and a pointthan the dipole.A difficulty of the the Dipole Calculus is, that it presumes intrinsic objects althoughthe intrinsic character of objects may not be existent or invisible for an application’ssensor. 44 hapter 3. State of the Art
The Ternary Point Configuration Calculus (TPCC) is a qualitative spatial reason-ing calculus that uses ternary relations of points developed by Reinhard Moratz[Moratz, 2003]. The distinctions in TPCC are less coarse then in the calculi describedabove and thus TPCC permits more useful differentiations for realistic application sce-narios.
In TPCC we again have point-like objects on a 2D-plane. A relative reference system isgiven by an origin and a relatum. The origin and the relatum define the reference axis.The spatial relation between the reference system and the referent is then describedqualitatively by naming the part of the partition in which the referent lies.
Figure 3.3.:
The flip-flop partitionLigozat [Ligozat, 1993] suggested a system which he called flip-flop calculus. In thiscalculus the reference axis partitions the 2D-plane into two parts - left and right. Thespatial relation between the reference system and the referent is described qualitativelyby naming the part of the partition in which the referent lies. The referent can not onlybe on the left or the right - it can also be on the reference axis itself. In this case Ligozatdistinguishes five configurations. The referent can either be behind the relatum (back -ba), at the same position as the relatum (same as relatum - sr), in front of the relatum(front - fr), at the same point at the origin (same as origin - so) or behind the origin(behind origin - bo). shows this partition. The partition is shown in figure 3.3.45 hapter 3. State of the Art
Points A, B and C can be examples for origin, relatum and referent. Isli and Moratz[Isli, 1999] observed two additional configurations in which origin and relatum haveexactly the same location. In the first configuration the origin and relatum are at thesame point and the referent at some other (double point - dou). Secondly all three pointscan be at the same location (triple point - tri).Infix notation is used to describe configurations. The reference system consisting oforigin and relatum are written in front of the relation symbol - the referent is writtenbehind it.Vorweg et al. [Vorweg et al., 1997] showed empirically that the acceptance regions forfront and back need similar extensions like left and right. In the flip-flop calculus frontand back only have linear acceptance regions.Freksa [Freksa, 1992b] extended the flip-flop calculus by partitioning the 2D-plane witha cross. Therefore the left and right side is respectively divided into a front and back
The TPCC calculus is derived from the cardinal direction calculus and provides finerdifferentiations than the cardinal direction calculus (see Section 2.3.3). Its 2D-plane isdivided into eight slices by adding another cross which is rotated 45 ◦ . Additionally thedistance from the relatum to the referent is compared to the distance from the relatumto the origin to provide a distinction between the two ranges. (See figure 3.4) Figure 3.4.:
Names of the TPCC areas46 hapter 3. State of the Art
The letters f, b, l, r, s, d, c in figure 3.4 stand for front, back, left, right, straight,distant, close. The TPCC has 8 different orientations and 4 precise orientations. With2 distances and three special cases there are 27 possible configurations ( (8 + 4) · A, B clf C ∨ A, B, cfl C would be described by A, B (clf,cfl) C . Figure 3.5.:
Iconic representation for TPCC-relations
The unary and binary operations introduced in sections 2.2 and 2.3.4 are used withTPCC as well. In order to keep the transformation and composition tables small Moratzintroduced an iconic representation for the TPCC-relations seen on table 3.5. Segmentscorresponding to a relation are illustrated as filled segments, sets of base relations haveseveral segments filled. This representation is easier to translate into its semantic contentcompared with a representation that uses the textual relations symbols.The transformation table 3.6 shows that results of transformations may constitute sub-sets of the base relations.
Composition hapter 3. State of the Art Figure 3.6.:
The iconic representation of the unary TPCC operationsTPCC is not closed under strong composition. Therefore a weak composition operationwas introduced with the specific relation such that: ∀ A, B, D : A, B ( r ⊕ r ) D ← ∃ C : A, B ( r ) C ∧ B, C ( r ) D The table for the weak composition can be found in [Moratz, 2003]. More than 700results of the weak composition are represented in this table.
Constraint-based Reasoning
Constraint satisfactory problems are solved as described in section 2.2.2. To express thecalculus in terms of relation algebras is one prerequisite for using this standard constraintalgorithm. But the Ternary Point Configuration Calculus is not closed under transfor-mations and under compositions so this algorithm doesn’t achieve path-consistency.Nevertheless, it is possible to perform simple path-based inferences. This is done whenthe last two relations of a path are composed and the reference system is incrementallymoved toward the beginning of the path in a form of a backward chaining. This can beused to detect cyclic paths. Moratz showed that reasoning with TPCC is in PSPACE.
The Distance/orientation-interval propagation(DOI) by Reinhard Moratz [Moratz and Wallgrün, 2003b]proposes an approach to model the typically imprecise sensor data about orientationsand distances. This approach propagates orientation and distance intervals to produceglobal knowledge. 48 hapter 3. State of the Art
The Distance/orientation-interval propagation(DOI) was designed as a calculus for mo-bile robot indoor exploration. Therefore it reflects the imprecise sensor data robotsusually provide. Furthermore the data delivered by the sensors may be not only impre-cise but also incomplete. This poses serious problems for the integration of local spatialknowledge into survey knowledge which is, for example, required for robot explorationsin unknown environments. Navigation tasks require calculi that handle orientation anddistance information - pure topological information is not sufficient. [Röfer, 1999]The DOI uses a relative reference system which utilizes continuous interval borders formodeling imprecision. Therefore it is not a qualitative calculus but it can be seen as anextension to the qualitative approaches. Benefits of qualitative approaches are combinedwith metric measurements by using DOI. Qualitative calculi can represent imprecise spa-tial knowledge while metric representations are good at distinguishing different spatialentities. The DOI propagation can be combined with a representation based on thegeneralized Voronoi graph of a robots free space developed by Wallgrün. The DOIs areused here to specify the relative positions of the vertices in the Voronoi graph.
The Distance/orientation-interval propagation is based on continuous distance orientation-intervals. There is again, on a 2D-plane, a point-like object. The DOI uses this pointtogether with a reference direction as anchor and it has four additional parameters: r min , r max , φ min and φ max .A DOI d is a set of polar vectors ( r i , φ j ) with: d = { ( r i , φ j ) | r min ≤ r i ≤ r max ∧ φ min ≤ φ j ≤ φ max } There is a special case which represents the spatial arrangement where the goal locationcan be the same as the reference point. This case is represented by the following values: φ max = π , φ min = − π and r min = 0 .In all other cases φ max − φ min ≤ π holds and for convenience it is assumed that − π ≤ φ min ≤ π and − π ≤ φ max ≤ π hapter 3. State of the Art Figure 3.7.:
A DOI and its parameters
The composition between two DOIs is the basic step for propagation along paths (seefigure 3.8).
Figure 3.8.:
Composition of adjacent path segmentsTwo DOIs ( d andd ) can be compositioned into a third ( d ). d = (cid:0) r min , r max , φ min , φ max (cid:1) d = (cid:0) r min , r max , φ min , φ max (cid:1) d = (cid:0) r min , r max , φ min , φ max (cid:1) It holds: d = (cid:16) minr i , r k , φ j , φ l r (cid:80) (cid:0) r i , r k , φ j , φ l (cid:1) , maxr i , r k , φ j , φ l r (cid:80) (cid:0) r i , r k , φ j , φ l (cid:1) minr i , r k , φ j , φ l φ (cid:80) (cid:0) r i , r k , φ j , φ l (cid:1) , maxr i , r k , φ j , φ l φ (cid:80) (cid:0) r i , r k , φ j , φ l (cid:1) (cid:17) hapter 3. State of the Art with: r min ≤ r i ≤ r max , φ min ≤ φ j ≤ φ max , r min ≤ r i ≤ r max , φ min ≤ φ j ≤ φ max The functions r (cid:80) and φ (cid:80) are defined this way: r (cid:80) (cid:0) r i , r k , φ j , φ l (cid:1) = (cid:113)(cid:0) r i sin φ j + r k sin (cid:0) φ j + φ l (cid:1)(cid:1) + (cid:0) r i cos φ j + r k cos (cid:0) φ j + φ l (cid:1)(cid:1) and φ (cid:80) (cid:0) r i , r k , φ j , φ l (cid:1) = tan − r i sin φ j + r k sin ( φ j + φ l ) r i cos φ j + r k cos ( φ j + φ l ) The values for the minimum and maximum of d thus the cases for which r m and φ n havetheir minimum or maximum and can be listed by geometric analysis. These formulasare given in appendix B. Figure 3.9.:
Resulting DOIThe composition is only an approximation in form of an upper bound of the area con-sisting of the vectors ( r m , φ n ) which can be directly composed by vectors ( r i , φ j ) and ( r k , φ l ) from d and d respectively. A typical spatial layout of these areas is shown infigure 3.9.A DOI can be viewed as a set of difference vectors between two points expressed inpolar coordinates relative to the reference direction. A function δ maps two points anda reference direction to the corresponding difference vector. The function φ maps twopoints and a reference direction from the first point to the second. The relation between51 hapter 3. State of the Art points can now be expressed in imprecise relative position and the respective DOIs: ∆ ( φ , P , P ) ∈ d ∧ ∆ ( φ ( P , P ) , P , P ) ∈ d ⇒ ∆ ( φ , P , P ) ∈ d (cid:5) d Now the DOI composition can be used to propagate local, relative spatial knowledgealong a path. There is an anchor point P , a reference direction φ and a sequence ofpoints which determine the path segments P , P , . . . P i , . . . P n . Each Point P i on thepath has an associated DOI d i . A stepwise composition recursively beginning with theend of the path yields the relative position of the end point with respect to the anchorpoint P and the reference direction φ : ∆ ( φ , P , P n ) ∈ d (cid:5) ( d ( . . . ( d n − (cid:5) d n ) . . . )) [Moratz and Wallgrün, 2003a] [Moratz and Wallgrün, 2003b] In [Moratz, 2005] Reinhard Moratz developed a calculus called Granular Point PositionCalculus (
Gppc ). Two points define a relative reference system in two dimensionalspace. The calculus is partitioning the space in several orientations and distances.The special cases for the origin A = ( x A , y A ) , the relatum B = ( x B , y B ) and the referent C = ( x C , y C ) are: A, B dou C := x A = x B ∧ y A = y B ∧ ( x C (cid:54) = x A ∨ y C (cid:54) = y A ) A, B tri C := x A = x B = x C ∧ y A = y B = y C The relative radius for the other cases is defined as: r A,B,C := √ ( x C − x B ) +( y C − y B ) √ ( x B − x A ) +( y B − y A ) A, B sam C := r A,B,C = 0
For A (cid:54) = B (cid:54) = C the relative angle is defined as: φ A,B,C := tan − y C − y B x C − x B − tan − y B − y A x B − x A Gppc allows different levels of granularity and figure 3.10 shows an example configura-tion of an
Gppc with the granularity level m of three. A
Gppc m has (4 m − m ) + 3 hapter 3. State of the Art Figure 3.10.:
Example configuration in
Gppc base relations which are defined as follows: ≤ j ≤ m − ∧ j mod → φ A,B,C = j m π ≤ j ≤ m − ∧ j mod → j − m π < φ A,B,C < j + 14 m π ≤ i ≤ m − ∧ i mod → i − m < r A,B,C < i + 12 m ≤ i ≤ m ∧ i mod → r A,B,C = i m m + 1 ≤ i ≤ m − ∧ i mod → m m − i − < r A,B,C < m m − i +12 m + 2 ≤ i ≤ m − ∧ i mod → r A,B,C = m m − i i = 4 m − → m < r A,B,C
The DOI calculus presented in the section above is used to calculate the compositiontable for the
Gppc . This is possible because the flat segments and their borders aresummarized obtaining a quasi-partition. [Moratz, 2005]53 hapter 4.The
Fspp
Approach
The calculus that is being developed in this thesis designed for usage in indoor robotics,but it can be used for other applications that reason about positional information, aswell.For current robotic navigation and communication systems it is sufficient to work withtwo dimensional space. Reasons for that are, that the movement most often happens onthe ground, that the sensors as well as the maps are two dimensional and also becausehumans most often communicate this way if there is not a good reason to do otherwise.Of course two dimensional reasoning is much easier than three dimensional reasoning,while it is simply not possible to do two dimensional navigation with one dimensionalcalculi.It is also assumed that the space is isometric and homogeneous. Isometric means, thatthe cost of moving is the same in all directions while a homogeneous space is one thathas the same properties at all locations. These constraints are made because otherwiseit would be really difficult to develop an adequate representation.The most important spatial aspects for distinguishing objects are shape, topology andposition. Representing the shape of an object is independent from its position andvice versa if an absolute or relative reference system is used. Therefore it does notinfluence the calculus developed here and it is thus disregarded for it. It is furtherassumed that all considered spatial entities are disjoint. Reasons for that are, thatsensors anyways have problems to distinguish objects that meet each other, that mostobjects actually are disjoint from each other and if they shouldn’t be disjoint it is mostoften not important to recognize that fact. Therefore the most interesting spatial aspectof space for robotics is the position, consisting of orientation information and distanceinformation [Musto et al., 1999]. 54 hapter 4. The
Fspp
Approach
There are already some calculi representing orientation, distance or position information,as it has been shown in chapter 3. Some of them already have been used quite successfullyin robotic context [Berndt et al., 2003]. Nevertheless the new approach for representingand reasoning about positional information presented here should be advantageous overthese calculi.The Dipole calculus premises extended objects with an intrinsic orientation which arequite difficult to recognize for sensors. As discussed in section 2.3.3 it is more convenientto use points for representing orientation and distance.The Ternary Point Configuration Calculus has already been used in some applications.For the given tasks it was rather successful but it became apparent that is was too coarsein many situations. The 8 orientations and two distances generate 16 flat acceptanceareas in TPCC - too less in scenarios with many object or greater distances.The Distance/orientation-interval propagation has a quite accurate representation forsingle objects. It is however not a qualitative approach. It doesn’t support disjunctionsof base relations and has thus difficulties in representing different possible positions -those have to be fitted into the characteristicly shaped DOI acceptance area which maymean a loss of precision.The Granular Point Position Calculus is advantageous over the three latter approaches.It is a qualitative approach that represents point position with an arbitrary level ofgranularity in both distance and orientation. Although it is quite similar to the approachdeveloped in this thesis, it differs in some important points from the
Fspp . A comparisonof these two calculi will be made in section 4.4.The name of the new calculus is "Fine-grained Qualitative Spatial Reasoning about PointPositions" (
Fspp ). It already highlights the major properties of the representation.
Fspp
Representation
Fspp has an arbitrary level of granularity. Depending on the scenario a very fine-grainedrepresentation could be chosen or a quite coarse one which would be less memory andtime consuming. Very little changes need to be done in order to change the granularity.The
Fspp is defined over ternary points: the origin, the relatum and the referent (seesection 2.3.3). The three points are needed for the qualitative orientation, because itwill use a relative frame of reference. The qualitative distance will, however, make notuse of the origin as you will see in the next section.55 hapter 4. The
Fspp
Approach
Fspp
Distance
Hernández distance definitions from [Hernández, 1995] and [Clementini et al., 1997],which already have been summarized in section 2.3.3, will be used for
Fspp , too.
Figure 4.1.:
Distance from relatum ∆ x and distance range δ x The distance system D is defined over a totally ordered set of distance relations Q , anacceptance function A and an algebraic structure (cid:61) : D = ( Q, A, (cid:61) ) (see section 2.3.3). Q = { q , q , q , ..., q n } , given a level of granularity with n + 1 distance distinctions. Thewidth of an acceptance area corresponding to a distance symbol q i is denoted with δ i whereas ∆ i denotes the maximum distance that a relatum can have to the referent forstill falling into the acceptance are of the distance symbol q i (see figure 4.1). Fspp uses an absolute distance representation. The area around the relatum is parti-tioned into a number of circular acceptance areas using a global scale. In the indoorrobotics scenario the action radius is quite low compared to the distance of objects withinthe robots sensor input. It is thus acceptable to use a monotone distance system.The absolute representation was chosen because the quantitative data input (from sen-sors or maps) allows an easy abstraction to an absolute distance representation (opposedto orientation). It is desirable to keep those absolute distance information, because itis very difficult to find out an metric distance for the robot using only relative distancerelations. Absolute distances also yield in better composition results and easier usage inmaps and other high level applications. A drawback is the more difficult human-robotcommunication. 56 hapter 4. The
Fspp
Approach
For the distance part of
Fspp the origin is not needed. The distance is solely definedover the relatum and the referent. That leads to the fact that all unary operationsexcept Id and Sc (see table 2.2) loose all distance information - the resulting Fspp sonly have orientation information, all distances are possible for any possible orientation.But that is no disadvantage. During the constrain propagation process (see section2.2.2) the distance information will be restored if it has been available in the initial setof constraints (relations).In the absolute distance system for length L of the first interval δ has to be specified.It’s value strongly depends on the number of distance distinctions m as well as on thedistance system chosen. The following values seem to be quite plausible and are thusused later on: L = 10 cmm = 24 δ i +1 = δ i · . The last distance interval q has a size of δ = 26 . m ( . · cm ) and the maximumdistance would be at ∆ = 132 m . The Fspp should consider distances greater than themaximum distance to fall into the last distance interval. The size of δ and ∆ is thusinfinite. Besides that exception all distance intervals δ i bigger than ten ( i > ) thanhave about 20% of the size of the total distance ∆ i . This seems to be a good value - butit isn’t suggested by cognitive articles known to the author. Fspp
Orientation
Fspp uses a relative frame of reference for orientation. The advantages of relativepositions are, that no intrinsic characteristics of objects have to be distinguished andthat no global coordinate system has to be used which is often not available in indoorscenarios.The space around the relatum is partitioned according to a number of ordered orientationdistinctions R = { o , o , o , ..., o m } . The acceptance function defines acceptance areasaround the relatum which correspond to the orientation relations ( o j ).Depended on the granularity chosen, different numbers of orientation relations are possi-ble. The number of orientations m is always even. This way the origin is always locatedon the border between the acceptance areas of the orientation relations o m − and o m .57 hapter 4. The Fspp
Approach
These special cases where objects are located directly on the border between two accep-tance areas are very unlikely to happen in real life scenarios. It is thus acceptable todefine, that in such cases all bordering acceptance areas are to be considered as possiblelocations for the referent. This is called quasi-partition.The acceptance function for an angle phi are defined by the following formula: φ in o j → πm j ≤ φ ≤ πm ( j + 1) Now the distance and the orientation are put together. This is done by defining theCartesian product of the set of distance relations Q and the set of orientation relations R : S = Q × R . Thus Fspp has a set S of m · n atomic relations f i,j : S = f , f , · · · f ,j f , f , · · · f ,j ... ... . . . ... f i, f i, · · · f i,j The
Fspp is defined over three points: the origin A with it’s coordinates A = ( x A , y A ) ,the relatum B = ( x B , y B ) and the referent C = ( x C , y C ) . The following definitions aresimilar to those in the Gppc (see section 3.5) because both calculi use similar acceptanceareas.First the three special cases that can occur in all ternary calculi are defined:58 hapter 4. The
Fspp
Approach
A, B dou C := x A = x B ∧ y A = y B ∧ ( x C (cid:54) = x A ∨ y C (cid:54) = y A ) A, B tri C := x A = x B = x C ∧ y A = y B = y C The relative radius for the other cases is defined as: r A,B,C := (cid:112) ( x C − x B ) + ( y C − y B ) (cid:112) ( x B − x A ) + ( y B − y A ) A, B sam C := r A,B,C = 0
For A (cid:54) = B (cid:54) = C the relative angle is defined as: φ A,B,C := tan − y C − y B x C − x B − tan − y B − y A x B − x A The acceptance areas for the
Fspp representation are defined according to the distanceand orientation relations above: C in f i,j → ∆ i − ≤ r A,B,C ≤ ∆ i ∧ πm j ≤ φ A,B,C ≤ πm ( j + 1) For the first and the last distances the following definitions are made: ∆ − = 0∆ n = infiniteIn figure 4.2 an Fspp with 416 (exempt the three special cases) base relations is shown.For display improvements only 13 distances are used rather than the suggested 24, butthe distance system follows the δ i +1 = δ i · . formula. 32 orientations are differentiated.The origin is indicated on the left (it’s distance to relatum and referent is unknown), therelatum is in the middle while the referent is the object that is to be located. In figure4.3 an example for possible locations of the referent are indicated gray. The referentobject has equal probabilities of being in one of the 22 indicated base relations. Thedisplayed Fspp relation thus consists of a disjunction of those 22 base relations.59 hapter 4. The
Fspp
Approach
Figure 4.2.:
An example FSPP with 32 orientation distinctions and 13 distances
Fspp
Reasoning
The unary operations and the binary composition are needed for constraint based rea-soning. But those operation are not trivial in the given representation.
The
Fspp was developed so that it can make use of the Distance/orientation-intervalpropagation . The DOI provides easy access to composition results. The atomic relationsof the
Fspp are exactly shaped like DOIs so that it is no problem to use the DOIcomposition for the atomic relations. This is done on demand and not prior due tothe huge composition table that would have to be created. The composition resultof a general
Fspp relation is then the union of the composition results of the atomicrelations. 60 hapter 4. The
Fspp
Approach
Figure 4.3.:
Disjunction of 22
Fspp base relations indicating possible locations of thereferent
The unary operations are more difficult to develop. The problem is here, that the dis-tance of the origin to the relatum and the referent is unknown which leads to quiteambiguous results. The unary operations are calculated as follows (the identity opera-tion Id is trivial and thus exempted): Inversion
Inv
For the computation of the inversion the composition is used. First a temporary
Fspp iscreated. It represents the possible location of the origin A to the relatum B - A is alwaysat the border in the back and it has an unknown distance. The
Fspp thus looks likein figure 4.4. Now the composition of this temporary
Fspp with the original has theinversion as a result.
Short cut Sc The short cut reorders the
Fspp (ABC) to (ACB). That means, that the distance in-formation is preserved (since the distance BC is the same as CB). But the orientation61 hapter 4. The
Fspp
Approach
Figure 4.4.:
The temporary
Fspp for the
Inv operationis ambiguous after that operation because of the unknown distance of the origin A. The Sc is calculated as follows:The result of Sc ( o i ) is for i < m : all orientations in the range from m to m + i . For i ≥ m : all orientations in the range from m − to i − m . Figure 4.5 shows the original Fspp on the left and the result of the Sc on the right. Figure 4.5.:
The short cut Sc operation62 hapter 4. The Fspp
Approach
Sci , Hm and Hmi
The remaining unary operations can be calculated using those two above:
Sci (ABC) → Inv ( Sc (ABC)) Hm (ABC) → Sc ( Inv (ABC))
Hmi (ABC) → Sc ( Sci (ABC))
Fspp
The definition of the conceptual neighborhood of the
Fspp is quite straight forward.Movement is only possible between the acceptance areas that share an edge since quasi-partition is used. The conceptual neighbors of an
Fspp f i,j are thus f i, ( j +1) mod m , f i, ( j − m ) mod m , f i − ,j and f i +1 . The first two conceptual neighbors are those with thesame distance but neighboring orientations. It is necessary to use the modular operationsince the orientation is circular - the "last" basic orientation relation is next to the"first" one. The latter two conceptual neighbors are those with the same orientationbut with smaller and bigger distance. Those are only existent, of course, if i > respectively r < n . See figure 4.6 for an example. The dark gray atomic relationsare the Fspp relation and the light gray relations are it’s conceptual neighborhood.The geometric neighborhoods introduced in the next section have similarities with theconceptual neighborhood, but those concepts are defined over different basics and fordifferent applications so that they are only partly comparable.
Complexity issues are an important topic not only for Artificial Intelligence but alsofor qualitative spatial reasoning [Dylla and Moratz, 2004]. A quite easy way of reducingthe number of computations needed will be presented in this section.63 hapter 4. The
Fspp
Approach
Figure 4.6.:
Conceptual neighborhood of an
Fspp
A problem in the composition approach described above is, that if the
Fspp have morethan a few basic relations, the number of DOIs that have to be computed is rapidlygrowing. Many of those calculations are unnecessary since they often lead to results(basic
Fspp relations) that already have been found to be possible locations of the ref-erent. A good way to avoid these computations is to only calculate the compositionof those atomic relations that are bordering to atomic relations that are known to notcontain the referent.An efficient way of finding those relations will be shown in this section. The approachis called contour tracing, but it is also known as border or boundary following. In orderto introduce the actual algorithm some definitions have to be made.
Geometrical neighborhood
The geometrical neighborhoods that are interesting for this thesis are defined on a twodimensional plane that is divided with square tessellation. The resulting squares areconsidered to have either a value of 0 or 1. Although the
Fspp basic relations are nosquares because two of their edges are round, they can still be used for these approaches.Since contour tracing is a computer graphics problem the squares are being referred aspixels which can have the color white for the value 0 and black for the value 1. Thatway it is also easier to understand the following terms and algorithm.64 hapter 4. The
Fspp
Approach
Moore NeighborhoodFigure 4.7.:
The Moore NeighborhoodThe Moore neighborhood of a pixel, P , is the set of 8 pixels which share a vertex oredge with that pixel. These pixels are called P , P , P , P , P , P , P and P beginning with the top left one and circling clockwise around the pixel P (see figure4.7). The Moore neighborhood (also known as the 8-neighbors or indirect neighbors) isan important concept that frequently arises in the literature. The 4-neighborhood is a subset of the 8-neighborhood. It only consists of the pixels thatshare an edge with the pixel P . Those are the pixels P , P , P and P of the MooreNeighborhood. Border pixels
Along the lines of neighborhoods the border pixels can be defined as 8-border pixels or4-border pixels. A black pixel is an 8-border pixel if it shares a vertex or edge with atleast one white pixel. Using the neighborhood definitions from above we can say that ablack pixel is an 8-border pixel if at least one of the pixels of its Moore Neighborhoodare white.4-border pixels are thus black pixels which have at least one white pixel in their 4-neighborhood. Those pixels share an edge with one or more white pixels.65 hapter 4. The
Fspp
Approach
Connectivity
A connected component is a set of black pixels, B , such that for every pair of pixels p i and p j in B , there exists a sequence of pixels p i , ..., p j such that:1. all pixels in the sequence are in the set B (they are black) and2. every 2 pixels that are adjacent in the sequence are neighbors.A component is 4-connected if 4-neighborhood used in the second condition and it is8-connected if the Moore Neighborhood is used. Pavlidis contour tracing algorithm
In 1982 Theo Pavlidis introduced his contour tracing algorithm [Pavlidis, 1982]. Withthis algorithm it is possible to extract 4-connected borders as well as 8-connected ones.It is important to keep track of the direction in which you entered the pixel which iscurrently active.The first task that has do be done using this algorithm is to find a black start pixel. Theonly restriction to this pixel is, that its left neighbor ( P in the Moor Neighborhoodmodel) is white. From now on the only important pixels are the ones in front of thecurrent pixel (left, ahead and right front: P , P and P ). Four different cases have tobe considered now:1. If P is black it is declared as our new current pixel and the current direction ischanged by turning left (-90 ◦ ). P is also added to the set of border pixels B . (seefigure 4.8)2. If the first failed ( P is white) and P black the new current pixel is P whichis also added to the set of border pixels B . The current direction is not beingchanged. (see figure 4.9)3. If the first two cases failed ( P and P white) but P is black the new currentpixel is P . P is added to the set of border pixels B and the direction is notbeing changed. (see figure 4.9).4. If first three cases failed (all three pixels P , P and P are white) the currentdirection is changed by turning right (90 ◦ ).66 hapter 4. The Fspp
Approach
These above computations are repeated till one of the following two exit conditions ismet:1. The algorithm will terminate after turning right three times on the same pixel or,2. after reaching the start pixel.
Figure 4.8.:
Pavlidis: check P The above algorithm returns a border that is 8-connected. If a 4-connected border isneeded the first and the third case have to be altered. For the first case pixel P hasto be black, too, or the first case fails. If P is black it also has to be added to B afterproceeding as stated above. In the third case P has to be black, too, or this case fails.If it is black is has to be added to B , too, and the algorithm proceeds as above. Figure 4.9.:
Pavlidis: check P (left) and check P (right)[Pavlidis, 1982] Using Pavlidis algorithm in
Fspp
For
Fspp the 8-connected border is sufficient because it corresponds to the conceptualneighborhood of the calculus. Figure 4.10 shows a random
Fspp on the left. On the67 hapter 4. The
Fspp
Approach right the border atomic relations of this random
Fspp are indicated dark while the inneratomic relations are light gray.
Figure 4.10.:
Fspp
The result of a composition that used at least one
Fspp with border segments is likelyto not be solid but to have holes in it’s representation that need to be filled (i.e. atomicrelations that are needed to be activated). For this task Pavlidis algorithm is used again.This time not only the border relations are stored but also those outside atomic relationsthat are neighbors of the border segments and that do not contain the referent. Thoseoutside segments then enclose the the actual
Fspp relation. Now all atomic relations,that are neighbors of relations that are already activated, are activated, too, if they arenot one of those enclosing segments calculated above. This way a fast filling algorithmfor the
Fspp is implemented.During the constraint propagation it is only needed to fill the
Fspp when doing intersec-tions. For all other operations it is sufficient to use the border relations. It thus mightbe a good idea not to use the contour tracing algorithm after every operation but maybeonly after every second or third. At the end of the inference process the resulting
Fspp sshould be filled. 68 hapter 4. The
Fspp
Approach
Fspp
Impelementation
The
Fspp is implemented in C. It uses a Distance/orientation-interval propagation im-plementation which also has been written in C by the author. The representation of thebasic relations is done by setting bits in an byte array. The naming conventions in theimplementation are different to those in this theses, mainly because the implementationstarted before this document reached the actual
Fspp chapter. The number of orienta-tions m is defined as "ROT_SLICES" (for rotation) and the number of distances n isdefined as "TRANS_SLICES" (for translation). Fspp s are named "Pieces". The vari-ous input and output functions come pairwise - the ones with an A at the end requestthe orientation and the distance index of the atomic relation whereas the ones with anB at the end request the bit-number of the atomic relation. The
BYTE_NUM(x) (x/8) and
BIT_NUM(x) (x%8)) directives are used to access the actual bit in the array. Someshort exemplary functions are shown to demponstrate the
Fspp implementation.The procedures to set a specific atomic relation to an value (either 0 or 1) look likethis: void setA(Pieces *pcs, int rot, int trans, int value){setB(pcs,rot+trans*ROT_SLICES,value);}void setB(Pieces *pcs, int which, int value){if(value==0){pcs->pPcs[BYTE_NUM(which)] =~(1 << BIT_NUM(which)) & pcs->pPcs[BYTE_NUM(which)];}else{pcs->pPcs[BYTE_NUM(which)] =1 << BIT_NUM(which) | pcs->pPcs[BYTE_NUM(which)];}}
The get functions: int getA(Pieces *pcs, int rot, int trans){return getB(pcs, rot+trans*ROT_SLICES);}int getB(Pieces *pcs, int which){ hapter 4. The Fspp
Approach return (((1 << BIT_NUM(which)) & pcs->pPcs[BYTE_NUM(which)]) > 0);}
The union of two FSPPs: void pieceORpiece(Pieces *rtn, Pieces *pcs1, Pieces *pcs2){int i;// setting the last unused bits in the last byte to 0pcs1->pPcs[BYTES_USED-1] &= ~(255<
The composition using DOI: void composition(Pieces *rtn, Pieces *pcs1, Pieces *pcs2){DoiTyp doi1, doi2, comp;Pieces tmp;Searchy search1,search2;int x,y;tmp.pPcs = getPiecesArray();search1.pPiece = pcs1;search1.number = -1; // initializing the serachsearch2.pPiece = pcs2;// searching through the first FSPP for atomic relationswhile((x = getNextTrue(&search1))>= 0){search2.number = -1; // initializing the serach// searching through the second FSPP for atomic relationswhile((y = getNextTrue(&search2))>= 0){singleDoiB(&doi1,x); // generating a DOI with// the atomic relation xsingleDoiB(&doi2,y); // generating a DOI with// the atomic relation y hapter 4. The Fspp
Approach // the compositioncomposition_exact(&doi1,&doi2,&comp,"");clearPiece(&tmp);// now a FPSS is being generated out of// the DOI composition resultdoiToAnder(&tmp,&comp);// the union of this atomic composition result// with the former composition resultspieceORpiece(rtn,rtn,&tmp);}}free(tmp.pPcs);}
The unary operation Short cut Sc : void scCalc(Pieces *rtn, Pieces *in){Searchy search;int x,oldRot,oldTrans,i;search.pPiece = in;search.number = -1;while((x = getNextTrue(&search))>= 0){oldRot = getRot(x);oldTrans = getTrans(x);if(oldRot < ROT_SLICES/2){for(i = ROT_SLICES/2; i <= ROT_SLICES/2+oldRot; i++){setA(rtn,i,oldTrans,1);}}else{for(i = oldRot-ROT_SLICES/2; i < ROT_SLICES/2; i++){setA(rtn,i,oldTrans,1);}}}} Test if two
Fspp s are equal. int testEqual(Pieces *pcs1, Pieces *pcs2){ hapter 4. The Fspp
Approach // set the unused bits at the end to 0pcs1->pPcs[BYTES_USED-1] &= ~(255<
A program that is successfully demonstrating the Sc operation, Pavlidis algorithm andthe the composition. int main(int argc, char* argv[]){ // the FSPPsPieces a,b,c,d,e,f;// testing if the defines are set correctlypretest();// allocate memory for the byte arraysinitPieces(&a);initPieces(&b);initPieces(&c);initPieces(&d);initPieces(&e);initPieces(&f);// initialize an FSPP by spezifing atomic relations// using their orientation and distance indexsetA(&a,6,3,1);setA(&a,6,4,1);setA(&a,5,3,1);// initialize an FSPP using a metric coordinate systemsetPoint(&b,6,39,1);// initialize an FSPP by spezifing atomic relations// using their bit number in the arraysetB(&c,58,1);setB(&c,59,1);// ASCII output of the FSPPs hapter 4. The Fspp
Approach printf(" FSPP a \n");printPieces(&a);printf(" FSPP b \n");printPieces(&b);printf(" FSPP c \n");printPieces(&c);// calculate the short cut of b - store the result in dunary(&d,&b,2);// print the Sc of bprintf("\n\n result of SC b \n");printPieces(&d);// set all atomic relation to 0clearPiece(&d);// composition of a and b - store the result in dcomposition(&d,&a,&b);// print the result of the compositionprintf("\n\n composition result 1\n");printPieces(&d);// composition of a and d - store the result in ecomposition(&e,&a,&d);// do contour tracing and set the FSPP to the contourcalcBoundaryPavlidi(&d);setPcsLikeBoundary(&d);// output of the contourprintf("\n\n contour of composition \n");printPieces(&d);// composition with the contourcomposition(&f,&a,&d);// output to compare the compusition results with// and without contour hapter 4. The Fspp
Approach printf("\n\n composition 2 (calculated without contour)\n");printPieces(&e);printf("\n\n composition 2 (calculated out contour)\n");printPieces(&f);// using a function to test for equalityprintf("\n e and f equal ? %d \n",testEqual(&e,&f));// do contour tracing and set the FSPP to the contourcalcBoundaryPavlidi(&f);setPcsLikeBoundary(&f);// print the contourprintf("\n\n contour of composition \n");printPieces(&f);// freeing the memoryfree(a.pPcs);free(b.pPcs);free(c.pPcs);free(d.pPcs);free(e.pPcs);free(f.pPcs);}
This is the output of the program above (it uses 18 orientation distinctions and 20distances). 74 hapter 4. The
Fspp
Approach
FSPP a TRANS00 10 20| . | . |000 : 00000000000000000000R 001 : 00000000000000000000O 002 : 00000000000000000000T 003 : 00000000000000000000004 : 00000000000000000000005 : 00010000000000000000006 : 00011000000000000000007 : 00000000000000000000008 : 00000000000000000000009 : 00000000000000000000| . | . |010 : 00000000000000000000011 : 00000000000000000000012 : 00000000000000000000013 : 00000000000000000000014 : 00000000000000000000015 : 00000000000000000000016 : 00000000000000000000017 : 00000000000000000000FSPP b TRANS00 10 20| . | . |000 : 00000000000000000000R 001 : 00000000000000000000O 002 : 00000000000000000000T 003 : 00000000000000000000004 : 00000000000000010000005 : 00000000000000000000006 : 00000000000000000000007 : 00000000000000000000008 : 00000000000000000000009 : 00000000000000000000| . | . |010 : 00000000000000000000011 : 00000000000000000000012 : 00000000000000000000013 : 00000000000000000000014 : 00000000000000000000015 : 00000000000000000000016 : 00000000000000000000017 : 00000000000000000000FSPP c TRANS00 10 20| . | . |000 : 00000000000000000000R 001 : 00000000000000000000O 002 : 00000000000000000000T 003 : 00000000000000000000004 : 00010000000000000000005 : 00010000000000000000006 : 00000000000000000000007 : 00000000000000000000008 : 00000000000000000000009 : 00000000000000000000| . | . |010 : 00000000000000000000011 : 00000000000000000000012 : 00000000000000000000013 : 00000000000000000000014 : 00000000000000000000015 : 00000000000000000000016 : 00000000000000000000017 : 00000000000000000000 result of SC bTRANS00 10 20| . | . |000 : 00000000000000000000R 001 : 00000000000000000000O 002 : 00000000000000000000T 003 : 00000000000000000000004 : 00000000000000000000005 : 00000000000000000000006 : 00000000000000000000007 : 00000000000000000000008 : 00000000000000000000009 : 00000000000000010000| . | . |010 : 00000000000000010000011 : 00000000000000010000012 : 00000000000000010000013 : 00000000000000010000014 : 00000000000000000000015 : 00000000000000000000016 : 00000000000000000000017 : 00000000000000000000composition result 1TRANS00 10 20| . | . |000 : 00000000000000000000R 001 : 00000000000000000000O 002 : 00000000000000000000T 003 : 00000000000000000000004 : 00000000000000000000005 : 00000000000000000000006 : 00000000000000000000007 : 00000000000000000000008 : 00000000000000111100009 : 00000000000000111100| . | . |010 : 00000000000000111100011 : 00000000000000111100012 : 00000000000000000000013 : 00000000000000000000014 : 00000000000000000000015 : 00000000000000000000016 : 00000000000000000000017 : 00000000000000000000contour of compositionTRANS00 10 20| . | . |000 : 00000000000000000000R 001 : 00000000000000000000O 002 : 00000000000000000000T 003 : 00000000000000000000004 : 00000000000000000000005 : 00000000000000000000006 : 00000000000000000000007 : 00000000000000000000008 : 00000000000000111100009 : 00000000000000100100| . | . |010 : 00000000000000100100011 : 00000000000000111100012 : 00000000000000000000013 : 00000000000000000000014 : 00000000000000000000015 : 00000000000000000000016 : 00000000000000000000017 : 00000000000000000000 hapter 4. The Fspp
Approach composition 2 (calculated without contour)TRANS00 10 20| . | . |000 : 00000000011111111000R 001 : 00000000001111111000O 002 : 00000000001111000000T 003 : 00000000000000000000004 : 00000000000000000000005 : 00000000000000000000006 : 00000000000000000000007 : 00000000000000000000008 : 00000000000000000000009 : 00000000000000000000| . | . |010 : 00000000000000000000011 : 00000000000000000000012 : 00000000001111110000013 : 00000000011111110000014 : 00000000011111110000015 : 00000000011111110000016 : 00000000011111111000017 : 00000000011111111000composition 2 (calculated out contour)TRANS00 10 20| . | . |000 : 00000000011111111000R 001 : 00000000001111111000O 002 : 00000000001111000000T 003 : 00000000000000000000004 : 00000000000000000000005 : 00000000000000000000006 : 00000000000000000000007 : 00000000000000000000008 : 00000000000000000000009 : 00000000000000000000| . | . |010 : 00000000000000000000011 : 00000000000000000000012 : 00000000001111110000013 : 00000000011111110000014 : 00000000011111110000015 : 00000000011111110000016 : 00000000011111111000017 : 00000000011111111000e and f equal ? 1contour of compositionTRANS00 10 20| . | . |000 : 00000000010000001000R 001 : 00000000001000111000O 002 : 00000000001111000000T 003 : 00000000000000000000004 : 00000000000000000000005 : 00000000000000000000006 : 00000000000000000000007 : 00000000000000000000008 : 00000000000000000000009 : 00000000000000000000| . | . |010 : 00000000000000000000011 : 00000000000000000000012 : 00000000001111110000013 : 00000000010000010000014 : 00000000010000010000015 : 00000000010000010000016 : 00000000010000001000017 : 00000000010000001000 hapter 4. The Fspp
Approach
A comparison and classification of important calculi presented in this thesis is done inthis section. This is done to help to choose the proper approach for spatial problems.It should also show, that the Fine-grained Qualitative Spatial Reasoning about PointPositions calculus developed in this thesis has unique properties which are useful in somecontext.
Fspp
TPCC, which is presented in Section 3.3, has a predetermined set of atomic relations.It differentiates two relative distances and eight orientations. Additionally four lineardirections that are derived from the cardinal direction calculus (see Section 2.3.3) can berepresented (straight front and back, left and right neutral) which give the calculus nicealgebraic properties. For robotic applications these linear relations are less importantbecause objects will almost certainly not be located directly on the straight line. Theconstant number of relatively few atomic relations enables a fast composition usingtable lookup. TCPP has already been used for robotic applications [Berndt et al., 2003].During the test series it became apparent, that, although it is often possible to achievegood results with TPCC, the calculus is too coarse for robust robotic applications.The advantages of
Fspp over TPCC are quite straight forward - it supports more ori-entation distinctions and more distance distinctions than TPCC. In both calculi therelative orientation is defined over three points.
Fspp s absolute distance system is fa-vorably for robotic representations while the relative distance in TPCC is better suitedfor interaction with humans.In applications such as described in [Berndt et al., 2003]
Fspp is expected to performsignificantly better if a high granularity is chosen. Due to it’s finer grain far less am-biguous situations are expected, that are situations in which more than one real lifeobject is located in the acceptance areas of the relation. In situations where no real lifeobject is found the
Fspp , the relation can be expanded by the conceptual neighbors ofthe atomic relations. This would not be possible with TPCC because it instantly wouldlead to very ambiguous results. 77 hapter 4. The
Fspp
Approach
Fspp
The Distance/orientation-interval propagation presented in Section 3.4 is a very math-ematical calculus. The fact that is does not have atomic relations shows that it is not aqualitative approach. DOIs are successfully used in the robotic context, for example inmaps using Voronoi graphs, as mentioned in Section 2.4.The
Fspp uses the Distance/orientation-interval propagation to calculate the composi-tion. It is apparent that the composition algorithm of the
Fspp has a bad complexity,especially compared to the DOI calculus who just needs one (DOI) calculation for it.But in this paper measures have been described to keep even high resolution
Fspp susable for robotic applications. The main advantage of
Fspp over the DOI approach is,that it is, to some extent, a qualitative calculus. Thus the fine properties described inSection 2.3.1 are still available for
Fspp . Gppc vs Fspp
Gppc (see Section 3.5) is quite similar to
Fspp . A minor difference is, that
Gppc hasfixed levels of granularity whereas the number of orientation and distance differentiationsin
Fspp can be chosen freely.
Gppc thus limits the number of different resolution modeswhich might make it more user friendly. The granularity of
Fspp on the other handcan easily be adjusted to the users needs, for example with many distances but feworientations.The most important difference between the both is clearly, that
Gppc uses relativedistances while
Fspp has an absolute distance system. The relative distance systemof
Gppc yields to quite accurate results of the unary operations while it causes moreambiguous composition relations. The situation for the
Fspp is reverse. It has goodcomposition results but the results of the unary operations are quite inexact. The compo-sition is surely the more important operation, but I think that the disadvantages canceleach other out since both unary and binary operations are needed during constraintbased reasoning. The relative distance system of
Gppc is better suited for interactionswith humans while the absolute distances used in
Fspp have advantages for maps.78 hapter 4. The
Fspp
Approach
Only few calculi are absolutely qualitative. Many have more or less quantitative features.The transition from qualitative to quantitative representations is gradual. Figure 4.11shows a classification of different calculi on a line. More qualitative calculi are insertedon the left while the quantitative DOI calculus is on the right. The topological RCC isentirely qualitative, while TPCC can be seen as more number cumbering as the cardinalalgebra because of the relatively higher number of basic relations.This classification also corresponds to the amount of positional information the calculican represent (except of RCC which represents topological knowledge). Approaches tothe left are more coarse than those on the right.
Gppc and even more
Fspp don’t havea fixed position on this line because of the different levels of granularity they offer.
Figure 4.11.:
Quantitative vs Qualitative calculiAn extreme version of
Fspp with just four orientations and one distance is more coarsethan TPCC. It thus would have to be inserted at the X1 mark in the figure. A slightlyfiner
Fspp with, for example, eight orientations and three distances would be locatedat the X2 mark while even higher resolutions can be at the X3 mark. DOIs will alwaysbe more accurate than
Fspp with even the highest level of granularity because they usereal numbers for the orientation and the range. Therefore X4 indicates the rightmostmark possible for
Fspp . In this classification
Gppc can be located in similar regions asthe
Fspp . 79 hapter 5.Conclusion
In this diploma thesis a new calculus for reasoning about qualitative positional informa-tion with the name "Fine-grained Qualitative Spatial Reasoning about Point Positions"(
Fspp ) has been developed. The basic terms and definition have been defined in Section2.1. Representation and reasoning techniques as well as the properties of qualitativespatial reasoning are presented there. State of the art approaches have been outlinedin Section 3. The Ternary Point Configuration Calculus (TPCC) is of special interestbecause not only the
Fspp calculus is derived from it. The Distance/orientation-intervalpropagation is important because it is needed for the reasoning process of
Fspp . TheGranular Point Position Calculus is a recently developed approach that has many simi-larities with the one developed in this thesis.The representation of the
Fspp has been developed in Section 4. The absolute dis-tance system and the relative orientation are outlined and merged into the ternaryposition representation. Unary and binary operations are needed for the reasoning pro-cess. The algorithms for both operations are presented. The composition makes use ofthe Distance/orientation-interval propagation while the unary inversion (
Inv ) operationmakes use of the composition. The algorithm for the unary short cut ( Sc ) operationis presented so that the other operations can be calculated using Inv and Sc . Theconceptual neighborhood concept can be applied to Fspp , too.An important improvement for the calculation of the composition is the use of a contourtrancing algorithm beforehand. The terms that have their origin in computer graphicshave been introduced and the application of the contour tracing algorithm of TheoPavlidis to
Fspp has been explained. The implementation of the
Fspp calculus was80 hapter 5. Conclusion shown to confirm the usability of this approach. The advantages and disadvantages ofthe
Fspp compared to other important calculi have been pointed out and a classificationwas developed over them.
This thesis was written with the aim to develop a new calculus for qualitative spatialreasoning in the robotic context. An arbitrary level of granularity for representingpositional knowledge was intended. I think that the
Fspp approach at hand satisfiesthese expectations. It is defined over three points and offers a free number of distanceand orientation distinctions. The relative orientation and absolute distance offer a goodrepresentation of positional knowledge. The necessary unary operations are defined.Because of the the absolute distance they are quite ambiguous. This is especially thecase for those operations that are defined indirectly over the
Inv and Sc operations.Future research could be done to define the unary Sci , Hm and Hmi operations directlyto get better results during the reasoning process.In
Fspp the composition is done using the DOI composition of the atomic relations.This leads to a huge number of DOI operations for very fine-grained representations.An effective way of reducing this number is presented by only using the border relationsduring the composition. During the extensive tests that have been done with the im-plementation of the
Fspp it was shown that this border-composition is as exact as acomposition with all basic relations. This is however not proved and thus a topic forfuture research. Related to this topic is the question, if 8-connectivity is really suffi-cient for the border-composition or if there are special cases in which 4-connectivity isrequired.Another interesting topic for future research is the question, which level of granularityand what distance systems for the
Fspp representation are advisable for typical applica-tion scenarios. Especially the distance system is interesting and cognitive research andtest series could be done to find representations that are optimal for interaction withhumans as well as for usage in maps and other applications.The comparison with other state of the art calculi revealed that
Fspp has unique prop-erties which are useful in the robotic context. It also became apparent that the recentlydeveloped
Gppc has many similarities with the
Fspp . It could be interesting to find81 hapter 5. Conclusion algorithms that allow to transform knowledge represented with those calculi forth andback. That way the advantages of both approaches could be used.82 ppendix A.TPCC definition
The formal definition of the TPCC relations are described by geometric configurationson the basis of a cartesian coordinate system represented by R .Special Cases for A = ( x A , y A ) , B = ( x B , y B ) , C = ( x c , y c ) : A, B dou C := x A = x B (cid:54) = x C ∧ y A = y B (cid:54) = y C A, B tri C := x A = x B = x C ∧ y A = y B = y C For cases with A (cid:54) = B a relative radius r A,B,C and a relative angle φ A,B,C are defined: r A,B,C := √ ( x C − x B ) +( y C − y B ) √ ( x B − x A ) +( y B − y A ) φ A,B,C := tan − y C − y B x C − x B − tan − y B − y A x B − x A Spatial relations: 83 ppendix A. TPCC definition
A, B sam C := r A,B,C = 0
A, B csb C := 0 < r A,B,C < ∧ φ A,B,C = 0
A, B dsb C := 1 ≤ r A,B,C ∧ φ A,B,C = 0
A, B clb C := 0 < r A,B,C < ∧ < φ A,B,C ≤ π/ A, B dlb C := 1 ≤ r A,B,C ∧ < φ A,B,C ≤ π/ A, B cbl C := 0 < r A,B,C < ∧ π/ < φ A,B,C < π/ A, B dbl C := 1 ≤ r A,B,C ∧ π/ < φ A,B,C < π/ A, B csl C := 0 < r A,B,C < ∧ φ A,B,C = π/ A, B dsl C := 1 ≤ r A,B,C ∧ φ A,B,C = π/ A, B cfl C := 0 < r A,B,C < ∧ π/ < φ A,B,C < / πA, B dfl C := 1 ≤ r A,B,C ∧ π/ < φ A,B,C < / πA, B clf C := 0 < r A,B,C < ∧ / π ≤ φ A,B,C < πA, B dlf C := 1 ≤ r A,B,C ∧ / π ≤ φ A,B,C < πA, B csf C := 0 < r A,B,C < ∧ φ A,B,C = πA, B dsf C := 1 ≤ r A,B,C ∧ φ A,B,C = πA, B crf C := 0 < r A,B,C < ∧ π < φ A,B,C ≤ / πA, B drf C := 1 ≤ r A,B,C ∧ π < φ A,B,C ≤ / πA, B cfr C := 0 < r A,B,C < ∧ / π < φ A,B,C < / πA, B dfr C := 1 ≤ r A,B,C ∧ / π < φ A,B,C < / πA, B csr C := 0 < r A,B,C < ∧ φ A,B,C = 3 / πA, B dsr C := 1 ≤ r A,B,C ∧ φ A,B,C = 3 / πA, B cbr C := 0 < r A,B,C < ∧ / π < φ A,B,C < / πA, B dbr C := 1 ≤ r A,B,C ∧ / π < φ A,B,C < / πA, B crb C := 0 < r A,B,C < ∧ / π ≤ φ A,B,C < πA, B drb C := 1 ≤ r A,B,C ∧ / π ≤ φ A,B,C < π ppendix B.DOI composition formula The resulting DOI d of a composition of the DOIs d and d has its minimum andmaximum radius and angle to be computed: min r i ,r k ,φ j ,φ l r (cid:80) (cid:0) r i , r k , φ j , φ l (cid:1) max r i ,r k ,φ j ,φ l r (cid:80) (cid:0) r i , r k , φ j , φ l (cid:1) min r i ,r k ,φ j ,φ l φ (cid:80) (cid:0) r i , r k , φ j , φ l (cid:1) max r i ,r k ,φ j ,φ l φ (cid:80) (cid:0) r i , r k , φ j , φ l (cid:1) For min r i ,r k ,φ j ,φ l r (cid:80) (cid:0) r i , r k , φ j , φ l (cid:1)
12 geometric cases have to be considered r , . . . , r .The min r i ,r k ,φ j ,φ l r (cid:80) (cid:0) r i , r k , φ j , φ l (cid:1) = min ( r , . . . , r ) ppendix B. DOI composition formula r = r (cid:80) (cid:0) r min , r min , , φ min (cid:1) r = r (cid:80) (cid:0) r min , r min , , φ max (cid:1) r = r (cid:80) (cid:0) r min , r max , , φ min (cid:1) r = r (cid:80) (cid:0) r min , r max , , φ max (cid:1) r = r (cid:80) (cid:0) r max , r min , , φ min (cid:1) r = r (cid:80) (cid:0) r max , r min , , φ max (cid:1) r = r min − r max ⇐ (cid:0) φ min ≤ − π ≤ φ max ∧ r min > r max (cid:1) r = r min − r max ⇐ (cid:0) φ min ≤ − π ≤ φ max ∧ r min > r max (cid:1) r = r (cid:80) (cid:0) r min , r min cos ( π − φ max ) , , φ max (cid:1) ⇐ (cid:16) φ max > π ∧ r min < r min cos ( π − φ max ) < r max (cid:17) r = r (cid:80) (cid:0) r min , r min cos (cid:0) π + φ min (cid:1) , , φ min (cid:1) ⇐ (cid:16) φ min < − π ∧ r min < r min cos (cid:0) π + φ min (cid:1) < r max (cid:17) r = r (cid:80) (cid:0) − cos φ max r min , r min , , φ max (cid:1) ⇐ cos φ maxr min + r min < < cos φ max r min + r max r = r (cid:80) (cid:0) − cos φ min r min , r min , , φ min (cid:1) ⇐ cos φ minr min + r min < < cos φ min r min + r max For max r i ,r k ,φ j ,φ l r (cid:80) (cid:0) r i , r k , φ j , φ l (cid:1) the geometric analysis shows seven distinct cases overwhich the maximum max ( r , . . . , r ) has to be computed: r = r (cid:80) (cid:0) r max , r min , , φ min (cid:1) r = r (cid:80) (cid:0) r max , r min , , φ max (cid:1) r = r (cid:80) (cid:0) r max , r max , , φ min (cid:1) r = r (cid:80) ( r max , r max , , φ max ) r = r (cid:80) (cid:0) r min , r max , , φ min (cid:1) r = r (cid:80) (cid:0) r min , r max , , φ max (cid:1) r = r (cid:80) (cid:0) r min , r max , , (cid:1) ⇐ φ min < < φ max The algorithm for min r i ,r k ,φ j ,φ l φ (cid:80) (cid:0) r i , r k , φ j , φ l (cid:1) and max r i ,r k ,φ j ,φ l φ (cid:80) (cid:0) r i , r k , φ j , φ l (cid:1) havebeen debugged. First the cases are calculated:86 ppendix B. DOI composition formula φ = φ (cid:80) (cid:0) r min , r min , φ min , φ min (cid:1) φ = φ (cid:80) (cid:0) r min , r max , φ min , φ min (cid:1) φ = φ (cid:80) (cid:0) r max , r min , φ min , φ min (cid:1) φ = φ (cid:80) (cid:0) r max , r max , φ min , φ min (cid:1) φ = φ (cid:80) (cid:0) r min , r max , φ min , φ max (cid:1) φ = φ (cid:80) (cid:18) r min , r max , φ min , − π − sin − r max r min (cid:19) ⇐ φ min < − π − sin − r m axr min < φ max φ = φ (cid:80) (cid:0) r min , r min , φ max , φ max (cid:1) φ = φ (cid:80) (cid:0) r min , r max , φ max , φ max (cid:1) φ = φ (cid:80) (cid:0) r max , r min , φ max , φ max (cid:1) φ = φ (cid:80) ( r max , r max , φ max , φ max ) φ = φ (cid:80) (cid:0) r min , r max , φ max , φ min (cid:1) φ = φ (cid:80) (cid:18) r min , r max , φ max , π − r max r min (cid:19) ⇐ φ min < π − r max r min < φ max φ = φ (cid:80) (cid:0) r max , r min , φ max , φ min (cid:1) φ = φ (cid:80) (cid:0) r max , r min , φ min , φ max (cid:1) The values of φ − are modified to be in the range of − π ≤ φ n ≤ . After that φ − are sorted by their value. The difference between the neighboring cases are calculated,with the biggest and the smallest value being neighbors, too, since the values are angleson a circle! The values that have the biggest difference between each other are takeninto account now - they are the new φ min and φ max - with the smaller value being minand the bigger being max.If the difference between the newly calculated φ min and φ max is greater than 180 ◦ thespecial case with φ max = π , φ min = − π and r min = 0 is set because the goal locationcan be at the reference point now! The same holds if the following conditions are met: φ min ≤ − π ≤ φ max or r min ≤ r min ≤ r max or r min ≤ r min ≤ r max .87 ppendix C.Topology definition Definition : topology, topological space: Let U be a non-empty set, the universe. Atopology on U is a family T of subsets of U that satisfies the following axioms:1. U and ∅ belong to T ,2. the union of any number of sets in T belongs to T ,3. the intersection of any two sets of T belongs to T .A topological space is a pair [ U, T ] . The members of T are called open sets.In a topological space [ U, T ] , a subset X of U is called a closed set if its complement X c is an open set, i.e. if X c belongs to T . By applying the DeMorgan laws, we obtain theproperties of closed sets:1. U and ∅ are closed sets,2. the intersection of any number of closed sets is a closed set,3. the union of any two closed sets is a closed set.If the particular topology T on a set U is clear or not important, the U can be referredto as the topological space. Closely related to the concept of an open set is that of aneighborhood. Definition : neighborhood, neighborhood system. Let U be a topological space and p ∈ U be a point in U . • N ⊂ U is said to be a neighborhood of p if there is an open subset O ⊂ U suchthat p ∈ O ⊂ N . 88 ppendix C. Topology definition • The family of all neighborhoods of p is called the neighborhood system of p , denotedas N p .A neighborhood system N p has the property that every finite intersection of members of N p belongs to N p . Based on the notion of neighborhood it is possible to define certainpoints and areas of a region.Def interior, exterior, boundary, closure. Let u be a topological space, X ⊂ U be asubset of U and p ∈ U be a point in U. • p is said to be an interior point of X if there is a neighborhood N of p containedin X . The set of all interior points of X is called the interior of X , denoted i ( X ) . • p is said to be an exterior point of X of there is a neighborhood N of p thatcontains no point of X . The set of all exterior points of X is called the exterior of X , denoted e ( X ) . • p is said to be a boundary point of X if every neighborhood N of p contains atleast one point in X and one point is not in X . The set of all boundary points of X is called the boundary of X , denoted b ( X ) . • The closure of X , denoted c ( X ) , is the smallest closed set which contains X .The closure of a set is equivalent to the union of its interior and its boundary. Everyopen set is equivalent to its interior, every closed set is equivalent to its closure.Def regular open, regular closed. Let x be a subset of a topological space U . • X is said to be regular open if X is equivalent to the interior of its closure, i.e. X = i ( c ( X )) . • X is said to be regular closed if X is equivalent to the closure of its interior, i.e. X = c ( i ( X )) .Two sets of a topological space are called separated if the closure of one set is disjointfrom the other set, and vice-versa. A subset of a topological space is internally connectedif it cannot be written as a union of two separated sets.Topological spaces can be categorized according to how points or closed sets can beseparated by open sets. 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