Representing Geometrical Objects by Relational
Spatial Constraints
Debasis Mitra
Department of Computer Sciences
Florida Institute of Technology
Melbourne, Florida, USA
Email: dmitra@zach.fit.edu
Abstract
Qualitative Knowledge Representation and Reasoning encompasses a wealth of
research results accumulated over the last few decades. It primarily deals with
incomplete or disjunctive information in a noisy environment. In this paper we have
introduced a scheme for representing geometrical objects by using one of the spatial
knowledge representation techniques. We have quantified the inaccuracy that entails
such a qualitative representation mechanism. Some important relevant issues are also
being discussed here.
Key Words: Spatio-temporal constraint reasoning, geometrical
representation, spatial information management under noisy input.
knowledge
1. Introduction
Consider a scenario where a military intelligence ground-operative encounters an array of
radars during a reconnaissance mission in an enemy territory. He understands that the shape
of the array is important to determine its purpose, but does not have requisite gadgets to make
any accurate positional assessment about the array. He starts noting down the angular
direction of each radar with respect to the North (say, determined from the Stars in the sky)
from its previous radar position in the array. Thus, he tours around each point in the array and
records this information. Obviously, without an appropriate gadget the angular directions
“estimated” will be very qualitative in nature. In this paper we propose a scheme for, and
discuss the related issues involved in, such a qualitative angular representation of geometrical
objects. Primarily we are concerned here in studying the shape of a polygon that could be
encaptured using the Star-ontology representation scheme developed in the literature. We
define this concept as a kind of shape invariance.
We have first introduced some relevant aspects of the Qualitative Spatio-temporal Knowledge
Representation techniques in a section. Then we have described the particular qualitative
spatial representation scheme, namely the Star-ontology, utilized in this work. Next two
sections introduced our scheme and discussed what remains invariant and to what degree in
the proposed representation. Further discussions on relevant issues follow these two sections.
Lastly we have concluded the paper with a short summary and by mentioning the future
directions of our work in this research.
2. Qualitative Representation of Space
Over the last few decades different schemes for qualitative representation of space and time,
and reasoning in them have been investigated [Chittaro and Montanari, 2000]. Such a
representation (often referred to as an ontology) involves an underlying space S, a well-defined
object p in it, and a scheme B to divide the space S into some equivalent regions when the object
p is placed in the space S. B is a set of “basic qualitative relations” for a second object p’ with
respect to the referred object p in that space S. For example, assume S being the real line R (infinity through +infinity) and the object as any point in it. Then B = {<, >, =}. This is the case
of point-based ontology (Figure 1), first proposed for simple temporal knowledge representation
(with S as a time line) [Vilain and Kautz, 1984].
.
=
<
>
Figure 1: Point-based ontology
A problem instance is (1) a given finite set of objects P={p1, p2, . . ., pn}, (2) a set of qualitative
constraints C between these objects expressed with the elements of B, and (3) a decision
question whether the constraints are satisfiable or not, i.e., can one actually place the objects in
a space S satisfying those constraints in C. This is obviously a constraint satisfaction problem or
a CSP [Bartak, 1999] in the context of spatial knowledge representation.
Example 1 (in the above point-based representation): a set P of three points on a real line R,
P={p1, p2, p3}, a set of three constraints (binary) between the points C = {( p1 < p2), (p2 < p3),
(p3 < p1)} (Figure 2). Obviously, the answer to the relevant decision question is NO, i.e., there
cannot be any satisfiable assignment for the three points following the relational constraints (in
C) between them.
p2
<
<
p1
<
p3
<
Figure 2: The constraint network for Example 1
Often the problem involves making an actual assignment (one or more “solutions” to the
problem) when the answer to the decision question happens to be “yes.” Also, typically the
constraints could be disjunctive (e.g., p1 <= p2) that leads to combinatorial explosion of many
possibilities to be looked into by a reasoning algorithm, which tries to find an answer to the
problem. Such disjunctive reasoning is mostly unimportant for the purpose of this paper.
Another such spatial representation scheme is the RCC-5 (Regional Connection Calculus,
[Randell et al, 1992]) where S is a two-dimensional Euclidean space, objects are closed regions,
and the five possible basic relations between a pair of objects p1 and p2 are defined as B={p1
overlaps p2, p2 non-intersecting p2, p1 included-in p2, p1 includes p2, p1 equal-to p2}, the
terminologies are self-explanatory. This is also a qualitative-knowledge representation scheme
for sets, where S is the universal set U, and the objects are sets in it. A more refined scheme for
representation of regions in 2D space is RCC-8 [Renz, 1998], where each of the three basic
relations non-intersecting, included-in, and includes are divided into two corresponding basic
relations depending on whether the regions p1 and p2 touch each other at their boundary or
not. Thus, “p1 includes p2,” will become (1) “p1 includes-and-touches p2,” and (2) “p1 includeswithout-touching p2.” So, there will be eight basic relations in RCC-8 as opposed to five in RCC5.
As we have mentioned before, there are many such spatial representation schemes that have
been proposed and investigated in the literature so far. The one that is relevant to our work
presented here is the Star-ontology [Mitra, 2002a]. We will introduce it in the following section.
3. The Star-ontology
N
NW
NE
.
East
SW
SE
S
Figure 3: Cardinal-direction ontology
3
2
.
1 (East)
2
Figure 4: A Star-ontology()
In “Star-ontology” the space S (refer to the previous section) is the 2D-Euclidean space, and the
objects are points. The basic relations between a pair of points are relative angular directions
between them. For example, in a special case of Star-ontology (Cardinal-directions, [Ligozat,
1998]) B is {Equal, East, Northeast, North, Northwest, West, Southwest, South, Southeast},
where East, West, North, South are linear directions, the other four corner relations are the
four Cartesian quadrants (Figure 3). The angular zoning here is with 90-degrees angle. Starontology [Mitra, 2002b] generalizes this over any arbitrary angle , where  = 360/ degrees,
for any even integer . Surprisingly, odd value of  is useless for any representation or
reasoning purpose ([Mitra, 2000b], the reason behind this property is yet unknown to us and is
under current investigation). In the generalized Star-ontology the set of basic relations has 2
+1 number of elements. Mitra has designated them [Mitra, 2002b] by the numbers, B =
{0Equal, 1East, 2, 3, . . ., 2}, even numbered relations represent 2D conic-section regions,
while the odd numbered relations are the lines radiating like spokes of a wheel from the
referred point (Figure 4).
4. Representing Polygons Qualitatively
We propose to represent a polygon by using qualitative description of relative positions between
the points forming the polygon using some Star-ontology (for some even integer value of ).
Exhaustively, we could express all possible binary relations between every pair of points, n(n1)/2 of them, for n points. However, the following assertion shows that we need only n number
of such binary relations.
Assertion 1: In order to represent an n-point polygon it is necessary and sufficient to have n
number of binary relations between every ordered pair of consecutive points of the polygon in
either clockwise or counter-clockwise sense.
Example 2: A quadrilateral is described in =12 ( =30-degrees) as {(p1 2 p2), (p2 22 p3), (p3 14
p4), (p4 8 p1)}.
Justification for the assertion 1: The necessary part of the assertion could be justified by
observing that (n-1) or less number of binary relations would leave the description open without
completing the loop between the points forming the polygon or the network remains
unconstrained. Since, we do not specify the length of each side of the polygon, without the
“triangulation” with the last constraint between pn and p1 the shape of the polygon will be
completely undetermined. The sufficient part of the assertion could be justified by the
observation that any additional binary relation between any other pair of points could be
derived in terms of some of these n binary relations, and so, is redundant.
An obvious point to note about the proposed representation scheme is that it is inaccurate from
a strict quantitative point of view. This means that a polygon recovered from a given
description may be distorted. In the next section we will investigate what remains invariant,
and to what degree, in such a qualitative description of a polygon.
5. Shape-orientation Invariance
Definition 1: P or the -description of a polygon P using a Star-ontology with -degree
angular zoning scheme is a set of binary constraints between each adjacent pair of points of the
Polygon. See example 2 for such a -description of a quadrilateral.
Note that a -description not only describes the shape but also the orientation of the polygon
with respect to the frame of reference of the underlying space S. This is because the qualitative
relations (in each element of the -description) between the points are with respect to the same
orientation of the space.
Definition 2: Two polygons P1 and P2 are said to be -shape-orientation-equivalent or -osequivalent when they have the same number of points and their respective -descriptions
match, or P1 = P2.
Definition 3: When a group of polygons are -os-equivalent to each other their corresponding P
is called the -os-invariant for the group.
The following assertion and the two theorems quantify how much information do we loose in a
-description.
Assertion 2: -description does not preserve any scale information.
The justification for this assertion is quite trivial.
Theorem 1: For a pair of -os-equivalent polygons P1 and P2 the maximum distortion between
any corresponding internal angle is 2, i.e., if an internal angle is x1 in P1, then the
corresponding angle x2 in P2 can be expressed as x1- < x2 < x1+.
Proof of Theorem 1: Consider any internal angle x1 of polygon P1 at the point p2, between the
three consecutive points p1, p2, and p3 (Figure 5). Note that x1 is a quantitative measure and
not a qualitative description, i.e., not an angular zone in a Star-ontology. Let d1 and d2 be two
directed lines (vectors) parallel to the linear zone 1 (directional frame of reference in the space)
for any Star-ontology, passing respectively through p1 and p2 (Figure 5). Assume 1 is the angle
at the point p2 from the line d1 to the line p1p2 measured counterclockwise (following the
scheme of Star-ontology), and 2 is that at p3 from d2 to the line p2p3. The relation between x1,
1, and 2 could be expressed as x1 =   ( 1 - 2), + or – depending on whether the three angles
are less than or greater than .
p2
p1
d1 (Zone 1)
p3
d2 (1)
Figure 5: Two lines of a polygon
When 1 and 2 are expressed as an angular region in a Star-ontology within a -description of
the polygon (instead of being quantitative values of angles), then their accuracy could be
expressed as 1’<1<1’+, and 2’<2<2’+ (the primed parameters are the starting point of the
respective zone). Thus, if 1 is expressed as zone 2, then that would mean 0<1<. In that
representation the inner angle x1 would be expressed as   (1 - 2) - < x1 <   (1 - 2) +.
This is true for any internal angle of the polygon. Hence the relative discrepancy between the
corresponding angles of any two polygons having the same –description would be maximum
2, or the theorem is proved. 
Theorem 2: For a pair of -os-equivalent polygons P1 and P2 the maximum distortion in their
relative orientation (between the two whole polygons, with respect to the frame of reference)
that can occur is .
Proof of Theorem 2: Every pair of corresponding arms of the two respective polygons are
described by an angular zone with respect to a fixed direction in space (say, the line represented
by the zone 1 or the basic-relation 1) in the anti-clockwise direction. A zone can have maximum
angular width (if the zone is even numbered or a 2D-zone) as . Hence, the maximum possible
relative angular separation between any pair of corresponding lines of P1 and P2 in the space
could be . 
Note that the inaccuracy of a -description captured by the above two theorems is maximal. In
case the description happens to use a 1D region or a linear zone (odd numbered, e.g., 1, 3, etc.)
rather than a 2D region, then the description is practically quantitative and the inaccuracy is
zero.
While recovering from a -description of a polygon a rigid constraint is imposed by the
geometry, namely, that the inner angles must add up to (n-2), for an n point-polygon. For
example, if the original object is a rectangle, and the description is with =15-degree, then the
distortion in recovering could cause the internal angles range from 74-degree through 104degree. However, all the internal angles of the recovered quadrilateral must add up to 360degrees, e.g., they could be 95, 95, 94, and 76. This constraint will make a recovery
algorithm non-trivial and possibly having some amount of backtracking, although (we
conjecture) the recovery problem is unlikely to be NP-hard. Although n relations are necessary
and sufficient for a –description of a polygon with n points, any recovery algorithm could be
made faster with some redundant relations between pairs of additional points. How much
redundancy is good is a point of our future investigation.
6. Discussion
In a noisy environment it is possible that the user could make a mistake in a zoning description
between two points. Such a mistake could be off by one 2D zone on either side. Thus, a relation
given as (x1 22 x2) or x2 is at zone 22 with respect to x1, could actually be (x1 20 x2) or (x1 24
x2). Hence, the inaccuracy in the description will be 3 instead of only . In that case the
inaccuracy of inner angles of a polygon will be 6 instead of 2. However, such errors in the description may lead to inconsistent description that cannot triangulate to a closed polygon.
Hence, if such errors are expected in an application, then a consistency-checking algorithm has
to be run, before a polygon is attempted to be recovered from its -description. An interesting
point in this regard is that such a consistency-checking algorithm would be polynomial in its
time complexity even for >4 cases. This is because the contiguous 3 zone disjunctive-relations
would fall under what is called the tractable sub-problem of the otherwise NP-hard consistency
checking-problem in Star-ontology [Mitra, 2002b].
An advantage of the -description of a polygon is that one could do some type of efficient
machine-learning by generalization. Suppose a set of possibly non-identical polygons with the
same number of points is provided as input corresponding to the same object in space. One can
efficiently detect a maximal value of  for which their -description is the same, i.e., if such a
maximal  value exists. GCM of the half of all the internal angles could be a starting point for
finding that  value, and the -description of every polygon in the set has to be generated in
order to check if they are same or not. Then, later a new input polygon could be easily checked
whether it represents the same object as that for the set of polygons, by matching the description of the new polygon for that particular value of maximal . This possibility can have
application in image analysis for images taken in a noisy environment, e.g., distorted satellite
images.
The representation is good for any geometrical object described by an ordered set of points,
instead of only for polygons. For example, a path description, or a robotic arms-movement could
be represented using the constraints from a Star-ontology as described here. However, open
loops may set new challenges than addressed in this paper for the closed loop polygons.
Another possibility is to represent multiple objects rather than single geometrical object as
addressed here. Such a representation may need unification of many special representation
schemes. For example, Gerivini and Renz [2002] have recently attempted such a unification
between the RCC-8 (see above) and a size-constraining (of the spatial-region objects) schemes in
two dimensional Euclidean space. A further unification with the Star-ontology in that scheme
will provide additional directional information. Subsequently, that scheme will be rich enough
for handling qualitative representations of multiple geometrical objects in 2D spaces, in a
similar way we addressed the representation of single objects comprised of multiple points.
7. Conclusion and Future Directions
In this work we have presented a mechanism for representing geometrical objects by borrowing
a scheme from the area of Qualitative Spatial Knowledge Representation and Reasoning. Such
a representation is very useful in a noise-prone application where a quantitative measurement
is either not possible or is not warranted for some reasons. We have studied the level of
accuracy in the proposed representation for the case of polygons. Some of the possibilities and
issues with this representation are also being discussed here.
A natural future direction of this work is to develop and implement some of the algorithms
touched upon in this article, e.g., for recovery from a -description. Our future work will also
investigate how useful the proposed representation is in the applications suggested above,
particularly in image analyses. We plan to set up an experiment for testing the machine
learning aspect mentioned before. We are looking for some data set involving distorted pictures
of some known objects, possibly some satellite images, for this purpose. Also, an interesting
dimension of this work is to attempt redefining geometry or trigonometry in a qualitative
fashion (using the Star-ontology) as opposed to the traditional analytical way (personal
communication with Jochen renz).
8. Acknowledgement
This work is supported by a grant (IIS-0296xxx) from the US National Science Foundation.
Unknown reviewers’ comments were helpful and may set new future directions for this work.
The author acknowledges discussions with Jochen Renz during a Summer Visit partially
supported by the Vienna University of Technology, and some fruitful comments from Hyoung
Rae-Kim, Gerard Ligozat, William D. Shoaff, Philip K. Chan and Nilanjana Bhattacharyya.
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