=Paper=
{{Paper
|id=Vol-1132/paper6
|storemode=property
|title=Reasoning about Directions in an Egocentric Spatial Reference Frame
|pdfUrl=https://ceur-ws.org/Vol-1132/paper6.pdf
|volume=Vol-1132
}}
==Reasoning about Directions in an Egocentric Spatial Reference Frame==
https://ceur-ws.org/Vol-1132/paper6.pdf
Reasoning about directions in an egocentric spatial
reference frame
Rupam Baruah Shyamanta M. Hazarika
Biomimetic and Cognitive Robotics Lab Biomimetic and Cognitive Robotics Lab
Computer Science and Engineering, Tezpur University Computer Science and Engineering, Tezpur University
Tezpur - 784028, India Tezpur - 784028, India
rupam.barua.jec@gmail.com shyamanta@ieee.org
Abstract—Within qualitative spatial reasoning, spatial objects moving. For example, vehicles moving on a highway leads
having directions have been abstracted in different ways; directed to different spatio-temporal patterns. In this case, we take the
line segments, oriented points etc. have been used. In certain direction of the vehicle along its front. Each vehicle may find
applications, it becomes necessary to reason about directions of the qualitative direction of other vehicles relative to its own
objects without focusing on the dimensionality of these objects. direction of motion. For example, as shown in Figure 1, we
We present a framework for representation of and reasoning
have different scenario (from top to bottom) such as ’car in
about directions in a qualitative way within an egocentric spatial
reference frame. Qualitative direction has been separated from front of truck’; ’car to left of truck’ and ’car behind truck’. At
spatial location and dimensionality of spatial objects and as such, any instance of time, these directions form a pattern that can
the formalism may be used to represent qualitative direction in a be analyzed to extract higher level semantics. Having seen the
dimension-independent way. The formalism uses fewer numbers above direction relations sequence (from top to bottom) we
of base relations than existing formalisms. Further granularity understand a composite pattern ’truck overtake car’.
can be refined easily. Qualitative direction relations separated
from spatial location and dimensionality can be used to express
spatio-temporal patterns of directional entities using a regular
grammar.
Keywords—QSR; Qualitative direction; Qualitative Direction
Algebra; Composition.
I. I NTRODUCTION
Everyday reasoning involving spatial and temporal at-
tributes is driven through qualitative abstractions rather than
complete quantitative knowledge. Qualitative abstractions are
an integral part of our conceptualization of space and time
[1]. Quantitative information is precise and accurate. However,
such precise information may not be cognitively meaningful
at times. For example, in the domain of traffic analysis, we
can measure the change in direction of a moving car at certain
intervals. This quantitative data is precise, but we will not be
able to extract much high level knowledge from it. Instead,
a qualitative expression like ”the car is approaching me from Fig. 1. Each vehicle may find the qualitative direction of other vehicles
the opposite direction” will convey more information. Further, relative to its own direction of motion. Such qualitative direction relations
space and time are inextricably linked. For any commonsense form the basis of describing motion patterns such as illustrated above.
theory of spatial representation and reasoning, space and
spatial change need to be interwoven! Commonsense theories In QSR literature, different formalisms have been used
of space and spatial change for cognitive agents need to be for representing directions of spatial objects in a qualitative
qualitative rather than quantitative. way. In some works, spatial objects have been abstracted as
directed dimensionless points. In others, directed line segments
Space, in our commonsense knowledge, is characterized by have been used [2]. An interesting issue here is about the
many different attributes. Within Qualitative Spatial Reasoning representation of direction of spatial objects extended in space.
(QSR), spatial characteristics are abstracted in a qualitative If we treat objects in two or three dimensional forms, then
way. Aspects of space that have been treated in a qualitative we would not be able to abstract these as points or lines.
way are spatial orientation, distance, direction, shape and size One needs to separate the issue of representation from the
etc [2]. In this paper, we have taken up one such spatial issue of dimensionality. Definitions of qualitative direction
aspect i.e. qualitative direction. Qualitative direction plays an relations should not be influenced by underlying dimensions.
important role when we think of analyzing patterns formed For example, in dipole relation algebra [3], spatial objects are
by directed objects. Such an object may be stationary or abstracted as directed line segments having a start and an end
37
point. In defining qualitative relations between two dipoles, the egocentric frame of reference and the second corresponds to an
location of end points with respect to a line (whether on the left allocentric frame of reference. When two objects are moving
or on the right side) is considered. Therefore, this formalism in a two dimensional plane, their directions can be specified
will not be suitable if our spatial objects are rectangles instead in different ways. If the plane has a large geographical extent,
of lines. Similarly, in oriented point algebra [4], spatial objects then it is possible that we may use the north-south-east-west
are abstracted as dimensionless points having directions. The reference system. We may say that one is heading north while
direction of a point sets up a coordinate system and orientation the other is heading south-west. If the scope is confined to
labels like F ront, Back etc. can be defined with respect to this a piece of paper, one may use the X-Y coordinate system
coordinate system. As higher dimensional objects are extended for this. Here, we are using an external reference system and
in space, meaning of orientation labels like F ront, Back this type of frame of reference is named as allocentric [7] or
etc. will be quite different. We can cite a formalism called extrinsic [8]. On the other hand, qualitative direction can be
rectangular cardinal directions [5] where spatial orientation represented in a relative way. For this, the direction of one
of rectangles having sides parallel to the axes of projection object is taken as a reference and the direction of others is
is treated. Oriented point algebra will not be suitable here stated with respect to this reference. This direction typically
because the objects under consideration are extended in space depends on factors like topology, size, shape etc. This reference
in two dimensions. In order to have a dimension independent direction is not static; it may change with time as the object
representation of qualitative direction, we should not bring in moves or rotates. Such a frame of reference is termed as
the spatial orientation of any part of the abstracted object into egocentric [7] or intrinsic [8]. In deictic frame of reference [8],
our definition. the concept of direction is defined by an external observer.
Here, the direction in which two or more objects are moving is
For the Qualitative Direction Algebra (QDA) proposed in
specified from the point of view of an external observer. In this
this work, we separate qualitative direction from the issues
paper, we have handled qualitative direction in an egocentric
of spatial location and dimensionality. We propose a direction
spatial reference frame.
model that does not use spatial location labels for representing
qualitative direction. We have used angular measurements
between directions for representing our qualitative direction III. Q UALITATIVE D IRECTION A LGEBRA
relations. Direction relation labels, that we have proposed,
A. Defining the JEPD Set of Direction Relations
are closer to our cognitive perception of object locomotion.
A Jointly Exhaustive and Pairwise Disjoint (JEPD) set of In order to explain the qualitative direction relations, we
binary qualitative relations has been proposed for representing need to introduce a few definitions. The direction in which
and reasoning with qualitative directions. The issue of spatio- an object is headed is specified by a straight line. In a two
temporal continuity of these base relations has also been dimensional plane, the direction of this straight line expresses
addressed. For constraint based reasoning, this set of relations the direction of the object. This straight line, used for specify-
needs to be closed under composition and converse. We have ing the direction of an object, has been termed as a direction
proposed algorithms for finding the converse of a single line. For finding binary qualitative direction relations, at first
relation and also to find the composition of two base relations. we need the direction lines for each object under consideration.
An interesting characteristic of this formalism is that the
granularity can be refined depending on requirement. In QSR, Definition. A Direction Line is a directed line segment in a
continuous input is discretized and qualitative abstractions are 2-D plane having direction dir and magnitude m.
introduced. For example, let us consider distance as a spatial
aspect. Distance of one object from another will have contin-
uous numeric values. From a qualitative view point, we can
discretize these values into three zones and label these as close,
near and f ar. If necessity arises, we can further subdivide
the close range and introduce two labels, namely, veryclose
and close. In doing so, we are refining our granularity in
such a way that finer changes in numerical distance can be
represented. In a similar way, in the formalism proposed in
this paper, it is possible to move to finer granularity so that
smaller changes in angular direction can be represented. The
algorithms proposed for finding converse and composition can Fig. 2. Objects with their intrinsic directions. Direction lines of each object
made to originate at the same point leading to a direction region.
handle refinement of granularity.
For convenience, we assume that the start point for all these
II. E GOCENTRIC R EFERENCE F RAME
direction lines is same. This can be done because the direction
A spatial reference frame is a coordinate system with line represents the direction logically and translation of these
respect to which the qualitative direction labels are defined. lines do not change the direction of the object. Moreover,
Spatial reference frame is an important issue when we want spatial locations of the objects are noway important to us. As
to represent qualitative direction. Tversky advocates that peo- an illustration, in Figure 2, we have shown two objects whose
ple’s spatial mental models use only two basic perspectives - directions are indicated by arrowheads (part A). In part (B) of
locating elements relative to one another from a point of view the same figure, two direction lines are drawn parallel to the
or locating an element to a higher order environmental feature direction of the two objects. We will use the notation dirA
or reference frame [6]. The first of these corresponds to an to mean the direction line of the spatial object A. When we
38
make the direction lines originate at the same point, we have
regions between two consecutive direction lines.
Definition. Direction Region : Let l1 and l2 be two direction
lines having directions dir1 and dir2 respectively and having
a point of intersection o. Let θ be the angle between dir1 and
dir2 in an anticlockwise direction. Then, a direction region
defines a set of direction lines that originate at o and the
direction of any such line is bounded by the angle θ from dir1
in an anticlockwise direction.
The concept of direction regions will be used in algorithms
for finding composition and converse of base relations of the
qualitative direction algebra. We have the task for arriving
at a set of binary qualitative direction relations from the
angles measured between their direction lines. QSR discretizes
the continuous domain and introduces abstractions. We may
tend to think that QSR is same as fuzzy approximations; but Fig. 4. QD8 : Pictorial representation of eight major direction relations.
there is an important difference. Categories in fuzzy approach
are approximations of real values, while categories in QSR region. The + region is direction region with an angular span
depends on application requirement [2]. of 45 degrees measured counterclockwise from one of the
We start with four qualitative abstractions and name these major direction relations introduced earlier. The − region is
as Same , Opposite , LR and RL. These abstractions are direction region with an angular span of 45 degrees measured
derived from the angular displacements between direction clockwise from one of the major direction relations. Since each
lines. Our qualitative relations for direction are binary. So, major direction relation now will give rise to one + region
the direction of one of the objects is taken as the reference and one − region, we will have eight such direction regions in
direction. The direction of the other object is taken as the total. These can be considered as qualitative direction relations,
primary direction and a qualitative relation expresses the because the direction line of the primary may fall along any
relationship of the primary with respect to the reference. of these. For naming these relations, we use a simple notation
where we append a + or − symbol to the name of a major
direction label. For example, the major relation Same results
in two relations, namely, Same+ and Same−. Let A Same
B and let B be the reference object. In the relation Same,
we know that the angle between dirA and dirB is zero.
In Same+, the direction line dirA lies within an angular
span of 45 degrees measured counterclockwise from dirB .
Similarly, in Same−, the direction line dirA lies within an
angular span of 45 degrees measured clockwise from dirB .
Fig. 3. Direction relations based on four qualitative abstractions. These twelve base relations are enumerated in Table I. The
angular spans for all the relations are listed. All these angles
Let A and B be two spatial objects. When we say A Same are measured in anticlockwise direction from the direction line
B, we mean that the angle between dirA and dirB is zero. of the reference object. In Figure 4, these relations are shown
For uniformity, we will measure all angles counterclockwise. pictorially. There are eight equal divisions of the full angular
Meaning of A Opposite B is that the angle between dirA span of 360 degrees. We choose to use this number of divisions
and dirB is 180 degrees. When an object A moves along to express the granularity of the algebra. So, the base relations
dirA in a two dimensional plane, its course of motion divides enumerated in Table I are for a QDA with granularity eight.
the plane equally into two parts. Assume that the direction We denote this as QD8 .
line dirB , corresponding to some object B, intersects dirA
at right angle in a left to right direction. Then, the resulting
QD16 : Refining the Granularity
qualitative relation is named as LR. An identical case in the
right to left direction is termed as RL. We have a set of four The twelve base relations listed in Table I, can record
binary qualitative direction relations here. These relations are change in direction when the direction of the primary object
illustrated in Figure 3. crosses discrete boundaries at integral multiples of 45 degrees.
In many applications, it may be necessary to record change
QD8 : QDA with granularity eight at finer intervals. In our proposed formalism, this can be
done very easily. In this section, we will show one level of
The level of granularity is too coarse and change in
refinement. The same process can be repeated to arrive at even
direction is represented when it crosses a threshold of 90
finer granularity of base relations.
degrees. These are the major direction relations that we are
going to refine further. Naturally, one would like to divide these For refinement, we equally divide the + and − direction
right angles equally so that granularity is refined. An important regions. For example, if we divide the + region for which
aspect is to keep the angular span same for all relations. We the angle range is ]0, 45], we obtain two direction regions of
identify two direction regions, namely, a + region and a − span 22.5 degrees each. The region [0, 22.5] is denoted by
39
TABLE I. D IRECTION RELATIONS OF QD8 .
Sl. Base Angle Converse of Sl. Direction Angle Converse of
No. Relation Range Base Relation No. Relation Range Base Relation
1 Same [0, 0] Same 7 lr [270, 270] rl
2 Same+ ]0, 45] Same− 8 lr+ ]270, 315] rl−
3 Same− ]315, 360[ Same+ 9 lr− ]225, 270[ rl+
4 Opposite [180, 180] Opposite 10 rl [90, 90] lr
5 Opposite+ ]180, 225] Opposite− 11 rl+ ]90, 135] lr−
6 Opposite− ]135, 180[ Opposite+ 12 rl− ]45, 90[ lr+
the symbol +1 and the region ]22.5, 45] is denoted by +2 . We present an algorithm for finding the converse of any
As a result, we get twenty four base relations that are listed qualitative direction relation. Let us assume that A dr B, where
in Table II. This time, change in direction is noticed after a A and B are spatial objects and dr is the direction relation
threshold of 22.5 degrees. The same thing can be done to the holding between their directions. Intuitively, for finding the
− region and the resulting direction regions will be −1 and converse of dr, we should know the number of rotations we
−2 . These refined relations are shown in Figure 5. Now, there should give to the direction line of A to get back to the
are 16 equal divisions of the full span of 360 degrees and direction line of B. This is because of the fact that the converse
accordingly, we have QD16 . expresses the relation of B with respect to A. So, for finding
the converse relation, the direction line of A will be taken
as line 0. Moreover, we should know the relation that results
after these many rotations. An algorithm is presented below
for finding the converse of a qualitative direction relation.
Algorithm Converse(R,m)
R is the relation whose converse has to
be returned and m is the granularity
BEGIN
1.n:= Calc_Rot_Conv(R)
2.If (R==’Same’||R==’Opposite’||R==’LR’
|| R==’RL’) Then
Begin
Conv_Rel := Find_Rel(m-n,m-n)
End
Else
Fig. 5. QD16 : Direction relations one level refined. Begin
Max_rot:= n
In order to apply constraint based reasoning to a set of Min_rot:= n-1
spatial relations, we develop a partition scheme for the objects Conv_Rel:=Find_Rel(m-Max_rot,m-Min_rot)
in the domain under consideration [9] and arrive at a set of
Jointly Exhaustive Pairwise Disjoint (JEPD) base relations. Function Calc Rot Conv returns the number of rotations
General relations are obtained by taking the power set of base needed to align the direction line of the primary object
relations, with top, bottom, union, intersection and complement with that of the reference. The bottom and top lines for the
of relations defined in the set theoretic way [9]. Moreover, an relation are retrieved into local variables p and q. p denotes
identity relation and a converse operation on base relations the index of the bottom line and q denotes the index of the
must be provided. For the set of base relations introduced top line. Since the function returns m − p, we understand
earlier, Same is the identity relation. Each relation is cosed that the maximum required number of rotations is returned
under converse operation. The converses for the base relations by the function. This returned value gets stored in the local
were listed in Table I and in Table II. variable n inside the function Converse. If the relation is
one of Same , Opposite , LR or RL , then we know that
B. Finding Converse and Composition
the direction line of the primary will not fall in a direction
For finding the direction relation that holds between direc- region. It will align with one of the lines (at one of the angles
tions of two spatial objects, one of these objects is considered 90 , 180 , 270 or 360 degrees measured counterclockwise )in
as a reference. A direction line parallel to the direction of the the direction wheel. Then, it is easy to see that we will have
reference object is drawn and this direction line can be desig- to give m − n number of rotations to the direction line of
nated as line 0. The direction relation wheel can now be drawn the primary object. For example, let us consider QD8 and let
with respect to this line according to the granularity level A Opposite B hold. Then, the direction line of the primary
under consideration. Then, the direction line corresponding to aligns with the line at an angle of 180 degrees in the direction
the direction of the primary object is drawn. The direction wheel. The value of < p, q > will be < 4, 4 >. The value
region in which this line falls tells us the direction relation returned by Calc Rot Conv will be 4. Inside the function
of the primary with respect to the reference. All angles are Converse, a call will be made as F ind Rel(8−4, 8−4). The
measured counterclockwise and direction relations in terms of relation whose bottom and top lines are (4, 4) is Opposite.
angle ranges have been listed before. So, the converse of Opposite is computed as Opposite.
40
TABLE II. R EFINED DIRECTION RELATIONS OF QD16 .
Sl. Base Angle Converse of Sl. Base Angle Converse of
No. Relation Range Base Relation No. Relation Range Base Relation
1 Same [0, 0] Same 13 lr [270, 270] rl
2 Same+1 ]0, 22.5] Same−1 14 lr+1 ]270, 292.5] rl−1
3 Same+2 ]22.5, 45] Same−2 15 lr+2 ]292.5, 315] rl−2
4 Same−1 ]337.5, 360[ Same+1 16 lr−1 ]247.5, 270[ lr+1
5 Same−2 ]315, 337.5] Same+2 17 rl−2 ]225, 247.5] lr+2
6 Opposite [180, 180] Opposite 18 rl [90, 90] lr+
7 Opposite+1 ]180, 202.5] Opposite−1 19 rl+1 ]90, 112.5] rl−1
8 Opposite+2 ]202.5, 225] Opposite−2 20 rl+2 ]112.5, 135] rl−2
9 Opposite−1 ]157.5, 180[ Opposite+1 21 rl−1 ]67.5, 90[ rl+1
10 Opposite−2 ]135, 157.5] Opposite+2 22 rl−2 ]45, 67.5] rl+2
8. Begin
9. R := From_Hash(min,max)
For a discussion of other type of relations, let us consider 10. Return
RL+. The bottom and top lines for this relation will be 11.Else If (max==min+2) Then
returned as < p, q > = < 2, 3 >. The value returned from 12.Begin
Calc Rot Conv will be 8 − 2 i.e. 6. Inside the function 13. :=
Converse, M ax rot will be 6 and M in rot will be 5. This 14. If Then
F ind Rel(2, 3). The relation for which dirbottom is 2 and 16. :=
dirtop is 3 is RL+. So, the converse of RL+ is RL+. An 17. :=
outline of the Calc Rot Conv function is given below. 18. Q := Get_From_Hash(a,b)
19. If (c!=-1) Then
Algorithm Calc_Rot_Conv(R,m) 20. Begin
The function Get_Lines gives the bottom 21. S := From_Hash(c,d)
and top direction lines associated with 22. T := From_Hash(e , f)
the relation R 23. End
BEGIN 24. Return
1. < p , q> := Get_Lines(Rel)
2. Return m-p
END For constraint based reasoning, composition of base
relations is an important issue. An algorithm for composition
We assume that the function F ind Rel retrieves the ap- of base relations is presented. Let A , B and C be three
propriate relation from a hash table depending on the pair objects such that A Rel1 B and B Rel2 C hold. We want to
of integers passed to it . Every direction relation can be find Rel1 ◦ Rel2, where ◦ denotes set theoretic composition.
represented by a pair of integers (i, j) where i is the integer For this, direction relation wheel is drawn with respect to
corresponding to dirbottom and j is the integer corresponding the direction of B. Since the relation of A to B is already
to dirtop . For example, when the pair (0, 0) is passed, the known, we can identify the direction line or direction region
retrieved relation is Same , when (0, 1) is passed, the retrieved for A. The remaining task is to fix C in the wheel. Rel2 is
relation is Same+ and so on. An outline of the algorithm for given. But that expresses the relation of B with respect to C.
the F ind Rel function is given below. The algorithm takes We take the converse of Rel2 and identify the direction line
care of the fact that sometimes (while computing composition or region for C with respect to B. To find the composition
of relations) the second argument can be two more than the we need to compute the number of rotations required to align
first and the local variables c and d are used to control this. The direction of C with that of A and find the resulting relation.
algorithm will return a quadruple of relations. Let us assume For any relation Rel, let us denote the lower line of its
that this quadruple is of the form < R, Q, S, T >. In this direction region as RelBottom and the corresponding upper
quadruple, only non null entries are meaningful. For example, line as RelT op . Then, any relation can be expressed as an
if we call F ind Rel(2, 2), then only one relation is returned ordered pairs of the form (RelBottom , RelT op ). For example,
and this is available in R. If the call is like F ind Rel(4, 5), in the Figure 4, the relation Opposite+ can be specified as
then also a single relation is returned in R. The parameters Q (4, 5) and Same as (0, 0).
, S and T become meaningful when max is min + 2.
Algorithm Find_Rel( min , max) Algorithm Compose(Rel1,Rel2,m)
1. c:=d:=-1 ; R:= Q := S := T := Null This algorithms computes composition of
2. If (min==max) Then rel1 and rel2. m is the granularity
3. Begin 1. S:=Converse( Rel2 , m)
4. R := From_Hash(min,min) 2. := Get_Lines(S)
5. Return 3. := Get_Lines(Rel1)
6. End 4. :=Calc_Rot_Comp(,)
7. Else If (max==min+1) Then 5. Find_Rel( min , max )
41
C. Conceptual Neighbourhood the motion sequence of P and Q with respect to R. The
motion sequence of P with respect to R can be expressed as
Conceptual dependency of a spatial relation defines a set Opposite, Opposite−, RL+, RL and that of Q can be stated
of relations that may hold after this relation whenever a as RL, RL−, Same+, Same. Such patterns can be expressed
change is recorded. For example, if at any point of time using a regular grammar and a parser can be used to recognize
Same is the qualitative direction relation that holds between the pattern in an input stream.
directions of two objects, then it is not possible that this
relation will change to Opposite whenever change in direction Elsewhere a general technique has been developed to
is noted. After the relation Same , the possible relations that combine QSR with formal grammars for recognition of motion
may hold can be either Same+ or Same−. The relations events among multiple spatial entities [10]. Use of formal
that may hold after the current relation are termed as its grammars as a recognition technique allows creation of hi-
conceptual neighbour(s). Conceptual neighbours dictate how erarchies of conceptual abstractions; motion events that one
one relation can or cannot change. This gives rise to a notion expects in an input stream are expressed by writing programs
of spatio-temporal continuity which can be exploited in many in this language. Such programs are parsed using context free
applications. Conceptual neighbours are generally expressed grammar and interpreted using regular grammar. Successful in-
by a graph where nodes represent relations and edges are terpretation is equivalent to recognition of the events expressed
drawn from a node to its conceptual neighbours. In Figure 6, in the program.
conceptual dependency of 12 base relations for QD8 is shown.
V. F INAL C OMMENTS
In this paper, a qualitative direction algebra is proposed for
representation and reasoning about directions of spatial objects
in a dimension independent way. Qualitative spatial relations
for QD8 and QD16 have been defined.
Existing formalisms combine dimensionality into defini-
tion of relations and as a result of this, such representation
become unsuitable when dimension scales up. In the proposed
formalism, it is easy to move to a finer granularity. This finer
granularity is realized using fewer base relations than existing
formalisms. A limitation of this approach might be the fact
that it ascertains the directions in a deterministic way. At any
point of time, it is assumed that we know the angles between
the directions with certainty. Future work includes study of
formal properties of these calculi.
R EFERENCES
[1] A. G. Cohn and S. M. Hazarika, “Qualitative spatial representation and
reasoning: an overview,” Fundamenta Informaticae, vol. 46(1-2), pp.
Fig. 6. Conceptual neighbourhood diagram for QD8 . 1–29, 2001.
[2] J. Renz and B. Nebel, “Qualitative spatial reasoning with constraint
calculi,” In Aiello M. and Pratt-Hartmann I. and van Benthem J.(Eds),
IV. QDA AND M OTION PATTERN A NALYSIS Handbook of Spatial Logics, pp. 161–215, 2007.
Qualitative direction relations can be used to represent [3] R. Moratz, J. Renz, and D. Wolter, “Qulatitative spatial reasoning about
line segments,” in In Proceedings of ECAI 2000, 2000, pp. 234–238.
and reason about spatio-temporal patterns of directional en-
[4] T. Mossakowski and R. Moratz, “Qualitative reasoning about relative
tities. One such example is motion pattern analysis. For such direction of oriented points,” Artificial Intelligence, vol. 180-181, pp.
applications, formalisms are required for representation and 34–45, 2012.
recognition of patterns in an input stream. [5] S. Skiadopoulos and M. Koubarakis, “On the consistencyof cardinal
direction constraints,” Artificial Intelligence, vol. 163(1), p. 91135,
2005.
[6] B. Tversky and B. M. Hard, “Embodied and disembodied cognition:
Spatial perspective taking.” Cognition, vol. 110, pp. 124–129, 2009.
[7] R. Klatzky, “Allocentric and egocentric spatial representations,” In
Frank A.U. and Campari I.(Eds):Spatial Information Theory. A The-
oretical Basis for GIS, COSIT93, vol. 716, pp. 14–24, 1998.
[8] E. Clementini, P. D. Felice, and D. Hernandez, “Introduction:frames
Fig. 7. Motion events of a set of directed points. of reference,” In Eilan N. and Mckarthy and Brewer B.(Eds):Spatial
IRepresentation. Problems in Philosophy and Psychology, 1993.
[9] G. Ligozat and J. Renz, “What is a qualitative calculus? a generl
Figure 7 present a synthetic example where three objects framework,” In Zhang C. and Guesgen H.W. and Yeap W.k.(Eds), In
are involved in a motion pattern recognition application. One of Proceedings of PRICAI-04, pp. 53–64, 2004.
the objects is stationary and has a direction along its egocentric [10] R. Baruah and S. M. Hazarika, “Qualitative directions in egocentric and
orientation. The other two objects are non stationery and allocentric spatial reference frames,” International Journal of Computer
changes in their egocentric headings. R is the stationery object Information Systems and Industrial Management Applications., vol. 6,
and it is taken as the reference object. We want to represent pp. 344–354, 2014.
42