Nadeem Salamat
email: nsalam01@univ-lr.fr

Fuzzy Change Detection in a spatial scene

Keywords: spatio-temporal relations, Fuzzy histogram of Allen Relations, Quantitative spatial change, Similarity measure

Spatial relations are used in different fields of image understanding, spatial reasoning, linguistics description of scene and new fields like computing with words. These spatial relations can be used to detect changes in a spatial scene, either it is a change in a topological structure or a change in metric information. Three type of spatial relations can be described by fuzzy Allen histograms. As motion cause changes in a spatial scene, this change can be detected with the help of spatio-temporal relations. Different qualitative methods are used to detect changes in scene, for each type of relation a separate model is used. In this paper a new methods for detecting the change in a spatial scene is proposed. This methods becomes important for detecting change when the metric relations i.e. when directional and distance informations becomes zero in qualitative methods. A histogram of fuzzy Allen relations are used to detect the change in a spatial scene and fuzzy histograms are compared by a similarity measure to detect the change in spatial scene.

I. INTRODUCTION

Advent of digital computers has resultant in a rapid expansion is the use of quantitative methods through most of human knowledge. Imprecise knowledge information give rise to fuzzy methods. Changes are concerned with the movement of objects. In daily life observations, objects change their locations, orientations, shapes and sizes over time such as moving objects, mobile devices, cities, lakes or solid objects in movement due to camera positions. A gradual location change of an object may be interpreted as object movement in the embedded space. Object motion takes place in space and time, therefore behavior of moving objects can be discussed in terms of spatio-temporal relations between the objects in scene. Change in object size by expanding or reducing is correspond to the scaling or deformation of the objects. Mostly the reasoning methods depend upon the object location and shape. Detection of this change is important to take different decisions, categorizing spatio temporal relations or starting new processes. These changes can be observed by multiple observation of scene or by comparing the different frames of a video in video analysis. A spatial scene can be completely described by geometric objects along with their spatial relations such as topological, distance and directional relations. Mostly spatio temporal relations are studied in a separate domains and a variety of researchers such as M. Schneider [START_REF] Güting | Moving objects databases[END_REF], Nico Van de Weghe et al. [START_REF] Delafontaine | Qualitative Relations Between Moving Objects in a Network Changing its Topological Relations[END_REF], [START_REF] Van De Weghe | A qualitative trajectory calculus as a basis for representing moving objects in geographical information systems[END_REF], [START_REF] Van De Weghea | Representing Moving Objects in Computer-based Expert Systems: The Overtake Event Example[END_REF] etc all emphasis on qualitative methods where they always consider the disjoint objects. They used spatio-temporal changes to model the relations between moving objects on road networks. These spatio temporal changes are essential part of artificial intelligence, formation and disappearance of storms and other natural phenomenon [START_REF] Robertson | Wulder, Stamp: spatial-temporal analysis of moving polygons[END_REF] where the some new objects appear and disappear in scene. S. Chaudhary et al. in [START_REF] Archana | Video content description using fuzzy spatio-temporal relations[END_REF] used fuzzy spatio-temporal relations to describe the video contents. They used histogram of forces which represents limited number of topological relations and movement or expansion of spatial objects can cause to change the topological relations, metric relations or both at the same time. Following are the examples of different moving objects, which shows that moving objects can change the shape, size and locations as a result in the scene the topological relations, metric relations or directional relations between the geometric objects are changed.

• According to the ontological point of view, the cities expand due to increase in inhabitants and their expansion is not uniform. A city or residential area expands over time which can change the spatial relations with respect to the other objects over time. This expansion is irregular and may cause only metric relations changes such as distance or directions. Now consider an example of a city and building outside the city, call museum, first of all it is away from city in east, then city gradually expands and covers the land area around it, in this way the city becomes closer and every corner have a different directional relation with the museum. This scene is explained in figure 1 • A log close to a lake may go through a variety of topological relationships. When the water surface is too low, that there is a land between the water surface and log. With increase in water level the log will have imediate access to the water surface with continuous increase in water surface log first partially under water then completely under water at edge and finally it may become an island in lake. These relations are explained in figure 2 Fig. 2. Changes in topological relationship between lake and log. [START_REF] Egenhofer | Reasoning about Gradual Changes of Topological Relationships[END_REF] • Consider that a cat jumps over the wall, then both the objects don't change their size, only directional relation is changed at each instant of time. A pictorial scene looks like figure 3, where circular object is used to represent the cat position. In defining the spatio-temporal relations rigid objects are considered they don't change or split during the movement. Moreover camera motions such as booming, tilting, panning and zooming are not considered. They move continuously and they change their positions to the neighborhood for this purpose concept of gradual changes is introduced. Models for changes of topological relationship are relevant to spatiotemporal reasoning in geographic spaces, as they derive the most like configuration and allow for prediction about significant changes. Such predictions are important methods for reasoning in large scale space and geographic spaces [START_REF] Egenhofer | Reasoning about Gradual Changes of Topological Relationships[END_REF] and point set topology is used to determine the topological relations. Where the topological distances are defined between the topological relations to observe that how much change has occurred and same object pair is considered at two different times. A similar approach is adopted by John Z. in [START_REF] Li | Modeling of Moving Objects in a Video Database[END_REF] where the qualitative directional relations are computed, trajectory of moving objects is represented by 8 directions and distance between these qualitative relations are defined to assess the gradual changes in directional relations. It has been recognized that a qualitative change occur if the deformation of an object effects its topological relations. This method is inspired by Max J. Egenhofer's method [START_REF] Egenhofer | Reasoning about Gradual Changes of Topological Relationships[END_REF] of topological distances by considering the directional relations. Qualitative methods can not detect small changes while the objects are overlapping, contained, contained by or disjoint.

More precisely these methods are qualitative not fuzzy. This paper investigates that how an existing method of combined extraction of topological and directional relations [START_REF] Matsakis | Combined Extraction of Directional and Topological Relationship Information from 2D Concave Objects, in fuzzy modeling with spatial informations for geographic problems[END_REF], [START_REF] Salamat | Spatial relations analysis by using fuzzy operators[END_REF] methods can be used to detect the topological or metric changes between the two spatial objects. Motion can be seen as spatio temporal changes, where spatial relation may change with time. This paper is arranged as follows, section 2 describes the histogram of fuzzy Allen relations, in section 3 effect of spatial change on histogram of fuzzy Allen relations and its measure is described, section 4 consists of experiments and section 5 concludes the paper.

II. HISTOGRAM OF FUZZY ALLEN RELATIONS AND TERMINOLOGY USED

A. Oriented lines, segments and longitudinal sections

Let and be two spatial objects, ( , ) ∈ , where is any real number and ∈ [-, ]. Δ ( ) is an oriented line at orientation angle . ∩ Δ ( ) is the intersection of object and oriented line Δ ( ). It is denoted by ( ), called segment of object and its length is . Similarly for object where ∩ Δ ( ) = ( ) is segment and is its length. is the difference between the maximum value of ∩ Δ ( ) and minimum of ∩ Δ ( )(for details see [START_REF] Matsakis | Combined Extraction of Directional and Topological Relationship Information from 2D Concave Objects, in fuzzy modeling with spatial informations for geographic problems[END_REF]). In case of polygonal object approximations ( , , ) can be calculated from intersecting points of line and object boundary, oriented lines are considered which passes through one vertex of two polygons. If there exist more than one segment then it is called longitudinal section as in case of ( ) in figure 4(a)

B. 1D Allen relations in space

Allen in [START_REF] Allen | Maintaining Knowledge about Temporal Intervals[END_REF], introduced the well known 13 jointly exhaustive and pairwise disjoint (JEPD)interval relations based on temporal interval algebra. These relations are arranged as = {<, , , , , , , , , , , , >}. where {< ,,,,,,} ({ ,,,,,>}

C. Fuzzification of Allen relations

In real applications, small errors in crisp values can change the entire result when gradual changes of topological relations occur over time. To cope with these problems fuzzification was introduced, it comprises the process of transforming crisp values into grades of membership for linguistic terms of fuzzy sets. Fuzzification process of Allen relations do not depend upon particular choice of fuzzy membership function, trapezoidal membership function is used due to flexibility in shape change. Let ( , ) is Allen relation between segments and where ∈ (argument object) and ∈ (reference object), ′ is the distance between ( , ) and its conceptional neighborhood. We consider a fuzzy membership function

: ′ -→ [0, 1].
The fuzzy Allen relations defined by Matsakis [START_REF] Matsakis | Combined Extraction of Directional and Topological Relationship Information from 2D Concave Objects, in fuzzy modeling with spatial informations for geographic problems[END_REF] are

( , ) = (-∞,-∞,--3 /2,--) ( ) ( , ) = (--3 /2,--,--,--/2) ( ) ( , ) = (--,--/2,--/2, ) ( ) ( , ) = ( (-( + )/2,-,-,+∞) ( ) , (-3 /2,-,-,-/2) ( ) (-∞,-∞, /2, ) ( )) ( , ) = ( --/2,-,-,-+ /2 ( ) , (-∞,-∞,-,-( + )/2) ( ) , (-∞,-∞, /2, ) ( )) ( , ) = ( (-( + )/2,-,-,+∞) ( ) , (-3 /2,-,-,-/2) ( ) ( ,2 ,+∞,+∞) ( )) ( , ) = ( --/2,-,-,-+ /2 ( ) , (-∞,-∞,-,-( + )/2) ( ) , ( ,2 ,+∞,+∞) ( )) ( , ) = ( (-,-+ /2,-3 /2,-) ( ) , (-∞,-∞, /2, ) ( )) ( , ) = ( (-,-+ /2,-3 /2,-) ( ) , ( ,2 ,+∞,+∞) ( )) ( , ) = (-,-/2,-/2,0) ( ) ( , ) = (-/2,0,0, /2) ( ) ( , ) = (0, /2,∞,∞) ( ) where = ( , ), = ( , ),
is the length of longitudinal section of argument object , and is the length of longitudinal section of reference object . Most of relations are defined by one membership function and some of them by the minimum value of more than one membership functions like ( ), ( ), ( ℎ), ( ℎ ). In fuzzy set theory, sum of all the relations is one, this gives the definition for fuzzy relation . Fuzzy Allen relations are not Jointly Exhaustive and Pairwise Disjoint (JEPD) because there exist at least two relations between two spatial objects.

D. Treatment of longitudinal sections

During the decomposition process of an object into segments, there can be multiple segments depending on object shape and boundary which is called longitudinal section. Different segments of a longitudinal section are at a certain distance and these distances might effect end results. In polygonal object approximation, instead of applying fuzzification algorithm depicted by Matsakis [START_REF] Matsakis | Combined Extraction of Directional and Topological Relationship Information from 2D Concave Objects, in fuzzy modeling with spatial informations for geographic problems[END_REF] for fuzzification of segments fuzzy operators are used. In this case each fuzzy Allen relation is a member of fuzzy set, fuzzy -and are used for fuzzy integration of available information. ( ) ( ) = ( ( ) ( ), ( ) ( )) When fuzzy operator is used, only one fuzzy value contributes for the resultant value which is . In this case each Allen relation has a fuzzy grade objective is to accumulate the best available information. Suppose that longitudinal section of object has two segments such that = 1 + 2 where 1 is the length of first segment and 2 is the length of second segment and is length of longitudinal section. Let 1 ( 1 ) defines the value of fuzzy Allen relations with the first segment and 2 ( 2 ) represents value of fuzzy Allen relations with the second segment where 1 and 2 are the distances between object and two segments of . Now fuzzy operator is used to get consequent information obtained from two sets of fuzzy Allen relations.

III. CHANGE DETECTION

This section consist of two parts, first the effect of a change on a fuzzy histogram of Allen relations, second section how to measure these changes.

A. Effect of change on fuzzy histogram of Allen relations

To describe the relative position of object pair in an image, there exist three types of relations named as topological, directional and distance relations. These spatial relations are explained in figure 8 and can be arranged as tuple ( , , , ). Where is the topological relation, is the directional relation and denotes the distance relation between objects , and T denotes the time. This representation is already used in [START_REF] Shim | Content-based retrieval using trajectories of moving objects in video databases[END_REF] to represent the spatio-temporal relations between moving objects. When the relations are captured with the help of 1D Allen relations, the distance relations has the importance when the objects are disjoint. For the disjoint relations the distance relations can be estimated from the support of a fuzzy Allen histogram because directional relations are proportional to the visual rang of object [START_REF] Deng | Zalim li, A Statistical Model for Directional Relations Between Spatial Objects[END_REF]. This range is inversely proportional to the distance between the objects.i.e. Closer the distance between objects, larger will be the support for the fuzzy Allen relations where fuzzy allan relations represents the thirteen fuzzy Allen relations and support of histograms represents the directions where histograms have non zero values and it is an interval between [0, 2 ] of a period 2 and is time. To detect the change in a spatial scene topological relations between objects and there histograms can be compared at two different time intervals. When the 13 Allen relations are represented in a histogram, each bin of this histogram consist of a matrix of 13 × where represents the total number of segments treated in the process. Suppose these relations are arranged in a following way

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ < = > ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1,1 1,2
. .

. . . . . . at time 2 when the object moves to the new location with respect to object . Some number of observers in direction 1 claims that the object is before the object for position 1 while no observers claims that object is before object in the given direction for position 2 . but when the object moves towards the new position, the number of observers in new direction 2 who claim that object is before increases and surface area of two objects also changes, by this way histogram translated to the new position. Normalized fuzzy histograms are considered as fuzzy sets, where core and support of a fuzzy set depends directional and distance contents, if core and support is translated, then object changes only directional relation and if shrinks or expands then object changes the distance relations and change detection can be considered as the inverse of assessing similarity between spatial scene, for detecting change in a spatial scene three types of relations, topological, directional and distance relations are compared. When the spatial relations are computed by the fuzzy histogram of Allen relations, the normalized histograms are considered as a fuzzy set whose support is inversely proportional to the distance and each topological relation is corresponds to a histogram in a direction. When the histograms expand or shrinks then the object is changing the distance information. To measure these changes different methods of similarity or dissimilarity can be used, as they are the inverse of each other. A similarity measure is used to access the degree of similarity of two images according to spatial relations between objects. Dissimilarity is the negation of similarity so dissimilarity could be defined as = = 1 -Similarity measures are:

13, ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ Each column
( , ) = | ∩ | | ∪ | = ∑ [ ( ( ), ( ))] ∑ [ ( ( ), ( ))] (1)
(ℎ 1 , ℎ 2 ) = ∑ ( (ℎ 1 ( ), ℎ 2 ( ))) ∑ ( (ℎ 1 ( ), ℎ 2 ( ))) (2)
(ℎ 1 , ℎ 2 ) = ∑ (ℎ 1 ( )ℎ 2 ( )) √ ∑ (ℎ 2 1 ( ) √ ∑ ℎ 2 2 ( )) (3)
is denoted by 6 in Valerie De Witte1 et al. [START_REF] De Witte1 Tom Mlange1 | Image Similarity From Fuzzy Sets to Color Image Applications[END_REF] is histogram comparison method, which is simply an extension of jaccard's coefficient to compare a histogram. stands for Tversky index and is normalized cross correlation for similarity measure. Equation 1 can easily be calculated by using -and -. These measures satisfy :

0 ≤ (ℎ 1 , ℎ 2 ) ≤ 1; ℎ 1 = ℎ 2 ⇒ (ℎ 1 , ℎ 2 ) = 1; (ℎ 1 , ℎ 2 ) = (ℎ 2 , ℎ 1 ); ( ℎ 1 , ℎ 2 ) = (ℎ 1 , ℎ 2 )
Histogram measure represented in equation 3 also has the property:

( 1 ℎ 1 , 2 ℎ 2 ) = (ℎ 1 , ℎ 2 ).
Where normalized cross correlation is invariant to the camera zooming so due to this reason it is preferred to use this similarity measure. As most of the fuzzy Allen relations are reorientation of each other [START_REF] Wawrzyniak-L. Matsakis | Relative positions in words: a system that builds descriptions around Allen relations[END_REF] and for any , this reorientation reflects through the following equations.

( ) = ( + ), ( ) = ( + ), ( ) = ( + ), ( ) = ( + ), ( ) = ( + ), = ( ) = = ( + ), In other words, only eight Allen histograms are independent and ( ), ( ), = ( ) are periodic of period . In this way we have to compare 8 dissimilarity measures corresponds to 8 2D topological relations. A vector or a similarity row or column matrix could be defined. If each element of this matrix is zero then there is no spatial change in the spatial scene.

IV. EXPERIMENTS for the experiment purposes, All the 13 fuzzy Allen histograms are calculated from angle measure [0, 360] with the increment of one degree and similarity between them. In practice 8 fuzzy Allen histograms can be computed from [0,180] using the above cited relations. In the first example (figure 10(a) to 10(d)), we consider two frames of a well known image sequence of Y. Sheikh and M. Shah [START_REF] Sheikh | Bayesian modeling of dynamic scenes for object detection[END_REF] where both objects are moving in opposite directions where the speed of the car is ℎ 1 and speed of man is considered as ℎ 2 . If the one object say reference object (car) is considered at rest and the object is under motion, then net speed of object will be ℎ 1 + ℎ 2 . Here manual segmentation is taken, because here the problem is to detect the change in the scene not the segmentation. As at each frame the distance between the object increases, at the first analysis frame the fuzzy meet relation exists i.e. in terms of Allen relations fuzzy meet and meet by relations exists along with certain directions. When the distance between the objects increase, the histogram for the relations meet and meet by. Then the similarity between the two histograms are calculated. All those histograms which don't exist, the similarity or dissimilarity between them is considered to be zero. Dissimilarity matrix will be (0.25, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0.25)

In second example (For reference figure 11 Dissimilarity matrix will be (0.17, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0.17)

In third example (figure 12(a) to 12(d)) when the object (light gray object )is inside the object (dark gray object), fuzzy Allen histograms are calculated there exist number of histograms due to fuzzification. In the second case object slightly changes its position while remaining inside the object , for experimental purpose object is translated by (-10,20), this results change in the fuzzy histograms. In this example qualitative directional and distance relations don't exist, and there is no qualitative change in spatial scene. But in fact there is a change in spatial scene, that object has changed its location. V. CONCLUSION Most of the existing methods for scene similarity are qualitative, as similarity depends upon topological, directional and distance relations. In these methods, small changes in the spatial relations cannot be detected. A fuzzy method for detecting small changes in scene is introduced and a fuzzy similarity function is used to detect the change in spatial scene structure. This method works when the qualitative methods fail to detect the small changes, particularly when the directional and distance informations become uneffective.i.e. object is inside an object. In dynamic scene analysis or in video analysis these changes have a fundamental importance and can be used to describe the spatial relations between moving objects or for defining different types of spatio temporal relations. These changes may be in topological, directional or distance relations.

  and (a),(b),(c) and (d) are different views of scene over time.

Fig. 1 .

 1 Fig. 1. City changes the size over time and metric relations with museum

Fig. 3 .

 3 Fig. 3. View of cat jump over the wall

Fig. 4 .

 4 Fig. 4. Oriented line Δ ( ), segment as in case of object , longitudinal section as in case of object .( Matsakis [8])

  inverse relations of the cited ones). The relation is the equality spatial relation. All the Allen relations in space are conceptually illustrated in figure5. These relations have a rich support for the topological and directional relations.

Fig. 5 .

 5 Fig. 5. Black segment represents the reference object and gray segment represents argument object. figure extracted from [8]

Fig. 6 .

 6 Fig. 6. Histogram of fuzzy Allen relation in direction

Fig. 7 .Fig. 8 .

 78 Fig. 7. (a) Histogram representation (b) Overlapping objects (c)Un normalized fuzzy histogram of Allen relations (d) Corresponding normalized fuzzy histogram of Allen relations

  corresponds to an observer's arguments. These values are summed up in a single value to form a final opinion, different operators are used. Most values in a column of this matrix are zero. Object under relative motion may undergo the topological structure of scene or topological structure remains same but it changes the order relations or metric relations. When the object changes a topological relation, values changes the rows remaining in the same column, when it changes the directional relation, the values changes from one matrix to other. When objects do not change their topological relation, they only change distance information between them, total number of non zero values in matrix or total number of matrix having same values are changed. This phenomenon is represented by the figure 9(a) and figure figure 9(b) where object is represented by light gray value and object is represented by dark gray value and 1 represents the position of object at time 1 and 2 represents the position of object

Fig. 9 .

 9 Fig. 9. Arguments at different angles where 1 represents position of object at time 1 and 2 represents position of object at time 2

Fig. 10 .

 10 Fig. 10. Two frames of an image sequence of Y. Sheikh and M. Shah [6] and their spatial relations

Fig. 11 .

 11 Fig. 11. (a) and (c) are images where in (c) object A moves towards object B (b) and (d) are their histogram representations.

Fig. 12 .

 12 Fig. 12. (a) and (c) are images where in (c) object A moves towards the center of object B (b) and (d) are their histogram representations.


Nadeem Salamat
email: nadeem.salamat@univ-lr.fr

Fuzzy spatio-temporal relations analysis

Keywords: Spatio-temporal relations, Relative position, Fuzzy Allen relations, Quantitative spatial relations, Fuzzy dissimilarity measure

There are different families of Spatio-temporal relations such as same-place same-time, same-place differenttimes, for road networks like overtake, derive beside and many others. These relations describe the relative positions of objects in a spatial scene. In existing techniques, these relations are defined qualitatively. Due to imprecise knowledge information and compensation power to small errors, fuzzy methods are becoming more important. In this paper, fuzzy spatio-temporal relations same-place differenttime and different-place different time are introduced. To define these relations, histograms of fuzzy Allen relations and fuzzy dissimilarity measure are used.

I. INTRODUCTION

A lot of work has been done in developing the fuzzy spatial relations. These relations include the topological, directional and distance relations. Nowadays the most favorite topic for researchers community in this field is spatio-temporal relations. These relations are mostly depend on their applications domain. In this regard, different examples are considered. A spatial scene can be completely described by geometric objects along with their spatial relations such as topological, distance and directional relations. Mostly spatio temporal relations are studied in a separate domains and a variety of researchers such as M. Schneider [START_REF] Güting | Moving objects databases[END_REF] described the different spatio-temporal relations which involve topological changes. Nico Van de Weghe et al. [START_REF] Delafontaine | Qualitative Relations Between Moving Objects in a Network Changing its Topological Relations[END_REF], [START_REF] Van De Weghe | A qualitative trajectory calculus as a basis for representing moving objects in geographical information systems[END_REF], [START_REF] Van De Weghea | Representing Moving Objects in Computer-based Expert Systems: The Overtake Event Example[END_REF] develop Qualitative Trajectory Calculus (QTC) which deals with moving objects and it always consider the disjoint objects. They used spatio-temporal changes to model the relations between moving objects on road networks. These spatio temporal changes are essential part of artificial intelligence, video content description. In [START_REF] Archana | Video content description using fuzzy spatio-temporal relations[END_REF] fuzzy directional relations are used ( Histogram of forces ) to describe the video contents. Movement or expansion of spatial objects can cause to change the topological, metric or both at the same time. Following are the examples of different moving objects, which shows that moving objects can change the shape, size and locations as a result in the scene the topological, metric or directional relations between the geometric objects are changed.

• A log close to a lake may go through a variety of topological relationships. When the water surface is too low, that there is a land between the water surface and log. With increase in water level, the log will have immediate access to the water surface with continuous increase in water surface log first partially under water then completely under water at edge and finally it may become an island in lake. These relations are explained in figure 1.

Fig. 1. Changes in topological relationship between lake and log. [START_REF] Egenhofer | Reasoning about Gradual Changes of Topological Relationships[END_REF] • A cat jumps over the wall, then both the objects don't change their size, only directional relation is changed at each instant of time. A pictorial scene looks like figure 2, where circular object is used to represent the cat position.

Fig. 2. View of cat jump over the wall

• A boat with in the lake continuously changes its position with respect to an observer standing out side the lake. For example first of all the boat was near the eastern edge of the lake, then it changes continuously its position towards north and finally it reaches near the north edge of lake.

In such a case the boat changes its locations. In figure 3 boat is represented by a small rectangle and big polygon shape object represents the lake. They move continuously and they change their positions to the neighborhood for this purpose the concept of gradual changes in topological relations was introduced. Models for changes of topological relationship are relevant to spatio-temporal Fig. 3. A boat inside a lake continuously changes its position reasoning in geographic spaces, as they derive the most like configuration and allow for prediction about significant changes. Such predictions are important methods for reasoning in large scale space and geographic spaces [START_REF] Egenhofer | Reasoning about Gradual Changes of Topological Relationships[END_REF] and point set topology is used to determine the topological relations and the topological distances are defined between the topological relations to observe that how much change has occurred. In this model object movements which causes changes in directional and distance information are not considered and same object pair is considered at two different times. This model to detect the gradual changes in spatial scene can be used in situation like explained in figure1, where dynamic spatial relations are considered and the major focus is on topological changes of the space. A similar approach is adopted by John Z. [START_REF] Li | Modeling of Moving Objects in a Video Database[END_REF] and the qualitative directional relations are computed, trajectory of moving objects is represented by 8 directions and distance between these qualitative directional relations are defined to assess the gradual changes in directional relations. This model can be used to detect the spatial changes in directional relations where the situation like in figure 2 exists. This method is inspired by Max J. Egenhofer's method [START_REF] Egenhofer | Reasoning about Gradual Changes of Topological Relationships[END_REF] of topological distances by considering the directional relations. In situations like in figure 3, where the distance and directional information becomes invalid and the object does not change the topological relations Sung-Hye et al. [START_REF] Choe | Hierarchical spatial relation based on a contiguity graph: Research articles[END_REF] introduced the method of hierarchical spatial relations based on contiguity graph. This method depends upon the physical properties of objects not on the spatial properties. Another method in qualitative domain, is internal cardinal directions [START_REF] Wang | Qualitative Spatial Reasoning About Internal Cardinal Direction Relations[END_REF] which capture the position of an object inside the other object. This method is based on interval calculus and in this method the central tile or minimum bounding rectangle (MBR)(Also called Minimum Bounding Box (MBB)) for the 9-intersection method is divided into 4 or 9 or 13 subtiles. Qualitative methods can not detect small changes while the objects are overlapping, contained, contained by or disjoint. More precisely the methods are qualitative not fuzzy. In defining the spatio-temporal relations rigid objects are considered they don't change or split during the movement. Moreover camera motions such as booming, tilting, panning and zooming are not considered. This paper investigates that how an existing method of combined extraction of topological and directional relations [START_REF] Matsakis | Combined Extraction of Directional and Topological Relationship Information from 2D Concave Objects, in fuzzy modeling with spatial informations for geographic problems[END_REF], [START_REF] Salamat | Spatial relations analysis by using fuzzy operators[END_REF] can be used to detect the topological or metric changes between the two spatial objects. These changes can be used to define the spatio-temporal relations. In this paper section 2 covers the different aspects of a histogram of fuzzy Allen relations, in section 3 spatio-temporal relations are defined, section 4 covers the effect of spatial scene change and its measure. in section 5 different experiments are taken into account and section 6 concludes the paper.

II. HISTOGRAM OF FUZZY ALLEN RELATIONS AND TERMINOLOGY USED

In this section, the different terms used in this paper and the computation of histogram of fuzzy Allen relations are explained. where ∩ Δ ( ) = ( ) is segment and is its length. is the difference between the maximum value of ∩Δ ( ) and minimum of ∩Δ ( )(for details [START_REF] Matsakis | Combined Extraction of Directional and Topological Relationship Information from 2D Concave Objects, in fuzzy modeling with spatial informations for geographic problems[END_REF]). In case of polygonal object approximations ( , , ) can be calculated from intersecting points of line and object boundary. If there exist more than one segment then it is called longitudinal section as in case of ( ) in figure 4. In this paper all

A. Oriented lines, segments and longitudinal sections

B. 1D Allen relations in space

Allen in [START_REF] Allen | Maintaining Knowledge about Temporal Intervals[END_REF], introduced the well known 13 jointly exhaustive and pairwise disjoint (JEPD)interval relations based on temporal interval algebra. These relations are arranged as = {<, , , , , , , , , , , , >}. where {<,,,,,,} ({ ,,,,,>}

C. Fuzzification of Allen relations

In real applications, small errors in crisp values can change the entire result when gradual changes of topological relations occur over time. To cope with these problems fuzzification was introduced, it comprises the process of transforming crisp values into grades of membership for linguistic terms of fuzzy sets. Fuzzification process of Allen relations do not depend upon particular choice of fuzzy membership function, trapezoidal membership function is used due to flexibility in shape change. Let ( , ) is Allen relation between segments and where ∈ (argument object) and ∈ (reference object), ′ is the distance between ( , ) and its conceptional neighborhood. We consider a fuzzy membership function

: ′ -→ [0, 1]
. Some of the fuzzy Allen relations defined by Matsakis [START_REF] Matsakis | Combined Extraction of Directional and Topological Relationship Information from 2D Concave Objects, in fuzzy modeling with spatial informations for geographic problems[END_REF] are:

• ( , ) = (-∞,-∞,--3 /2,--) ( ), • ( , ) = (0, /2,∞,∞) ( ) • ( , ) = (--3 /2,--,--,--/2) ( ), • ( , ) = (-/2,0,0, /2) ( ) • ( , ) = (--,--/2,--/2, ) ( ), • ( , ) = (-,-/2,-/2,0) ( ) • ( , ) = ( 5 , 3 , 1 ) • ( , ) = ( 6 , 4 , 2 ) • ( , ) = ( 6 , 4 , 1 ) • ( , ) = ( 5 , 3 , 2 ) • ( , ) = ( 7 , 1 ) • ( , ) = ( 7 , 2 ) where 1 = (-∞,-∞, /2, ) ( ), 2 = ( ,2 ,+∞,+∞) ( ), 3 = (-3 /2,-,-,-/2) ( ), 4 = (-∞,-∞,-,-( + )/2) ( ), 5 = (-( + )/2,-,-,+∞) ( ), 6 = --/2,-,-,-+ /2 ( ), 7 = (-,-+ /2,-3 /2,-) ( ), = ( , ), = ( , ),
is the length of longitudinal section of argument object , and is the length of longitudinal section of reference object . Most of relations are defined by one membership function and some of them by the minimum value of more than one membership functions like ( ), (

), ( ℎ), ( ℎ ). In fuzzy set theory, sum of all the relations is one, this gives the definition for fuzzy relation . Fuzzy Allen relations are not Jointly Exhaustive and Pairwise Disjoint (JEPD) because there exist at least two relations between two spatial objects.

D. Treatment of longitudinal sections

During the decomposition process of an object into segments, there can be multiple segments depending on object shape and boundary which is called longitudinal section. Different segments of a longitudinal section are at a certain distance and these distances might effect end results. Fuzzy Allen relations are computed for each segment then fuzzy operators are used for information fusion by considering that each Allen relation is a fuzzy set. Here fuzzy operator ( ( ) ( ) = ( ( ) ( ), ( ) ( )) ) is used. For this operator, only one fuzzy value contributes for the resultant value which is . In this case each Allen relation has a fuzzy grade objective is to accumulate the best available information. Suppose that longitudinal section of object has two segments such that = 1 + 2 where 1 is the length of first segment and 2 is the length of second segment and is length of longitudinal section. Let 1 ( 1 ) defines the value of fuzzy Allen relations with the first segment and 2 ( 2 ) represents value of fuzzy Allen relations with the second segment where 1 and 2 are the distances between object and two segments of . Now fuzzy operator is used to get consequent information obtained from two sets of fuzzy Allen relations.

E. Fuzzy histogram of Allen relations and its normalization

Histogram of fuzzy Allen relations represents the total area of subregions of and that are facing each other in given direction [START_REF] Matsakis | Combined Extraction of Directional and Topological Relationship Information from 2D Concave Objects, in fuzzy modeling with spatial informations for geographic problems[END_REF]. Area of subregions of object and represented by dark gray color in figure 6 represents a histogram of fuzzy Allen relation in given direction . In case of polygonal object approximation which is considered by N. Salamat and El Zahzah [START_REF] Salamat | Spatial relations analysis by using fuzzy operators[END_REF] to meet the problem of temporal complexity, fuzzy histogram of Allen relations can be written mathematically as:

∫ +∞ -∞ ( ∑ ( , ( ), ( )) = ( + ) ∑ =1 ( , )
where is the area of reference object and is area of augmented object in direction , is total number of segments treated and ( , ) is an Allen relation for segments , . Allen histograms can easily be normalized by dividing all Allen relations by the sum of all the Allen relations for every in the given direction. It is represented by ⌈ ( )⌋ where ∈ . ⌈ ( )⌋ = or there is a effect of on due to the presence of both object at same place at same time. In the same way the relation different-place same -time is also logical. This also has the same foundations that could not effect the object because both were at different places at the same time. Moreover these relations can hold in physical domain. Here our interest is to define the spatio-temporal relations same-place differenttime and different-place different-time in the spatial domains or in video analysis. if one object changes its position with respect to the other object in a scene at different time instants, then it is called that object A is at different place with respect to the object and if it doesn't change its relative position then it is called that object has the same-place at different times. An other important fact that the concept of same-place different times can also represent the circular movement of an object around that other object. For such a case object repeats its object location 2 directions. Both of the relations are associated with a change in a spatial scene. In the next section, the occurrence of a spatial change and the method to measure the change is discussed. For change measure a dissimilarity measure is used and if at least one out of 13 dissimilarity measure is non zero then the spatio-temporal relation differentplace different-time exist otherwise the spatio-temporal relation same-place different time exists.

IV. CHANGE DETECTION

This section consist of two parts, first the effect of a change on a fuzzy histogram of Allen relations, second section describes how to measure these changes.

A. Effect of change on fuzzy histogram of Allen relations

To describe the relative position of object pair in an image, there exist three types of relations named as topological, directional and distance relations. These spatial relations are explained in figure 8. When the relations are captured with Fig. 8. Spatial Relations for a static image the help of 1D Allen relations, the distance relations has the importance when the objects are disjoint. For the disjoint relations the distance relations can be estimated from the support of a fuzzy Allen histogram because directional relations are proportional to the visual rang of object [START_REF] Deng | Zalim li, A Statistical Model for Directional Relations Between Spatial Objects[END_REF]. This range is inversely proportional to the distance between the objects.i.e. Closer the distance between objects, larger will be the support for the fuzzy Allen relations histograms and vice versa. Thus the above relations can be described as (Fuzzy Allen Relations, Support of histograms, )

where fuzzy Allan relations represents the thirteen fuzzy Allen relations and support of histograms represents the directions where histograms have non zero values and it is an interval between [0, 2 ] of a period 2 and is time. To detect the change in a spatial scene, a snap shot scene model is considered and histograms of fuzzy Allen relations between same object pair can be compared at two different time instants. Object under relative motion may undergo the changes in topological structure of scene or topological structure remains same but it changes the order or metric relations. Normalized fuzzy histograms are considered as fuzzy sets, where core and support of a fuzzy set depends directional and distance contents, if core and support is translated, then object changes only directional relation and if support shrinks or expands then object changes the distance relations and change detection can be considered as the inverse of assessing similarity between spatial scene, for detecting change in a spatial scene three types of relations, topological, directional and distance relations are compared. When the spatial relations are computed by the fuzzy histogram of Allen relations, the normalized histograms are considered as a fuzzy set whose support is inversely proportional to the distance for each topological relation.

B. Change measure

This change in the spatial scene can exist in three different ways, change in topological, directional and distance relations. These changes can effect the fuzzy Allen histograms in three ways (i) Emerges a new histogram or disappears an existing histogram,(ii) Changes the support of a histogram i.e. histogram support shrinks or expands (iii) Histogram support is translated. When an existing histogram disappears or new histogram appears it means object changes its topological relations. When the histograms expand or shrinks then the object is changing the distance information. To measure these changes different methods of similarity or dissimilarity can be used, as they are the inverse of each other. A similarity measure is used to access the degree of similarity of two images according to spatial relations between objects. Dissimilarity is the negation of similarity so dissimilarity could be defined as = = 1 -and 0 ≤ ≤ 1. Where = 0 shows that there is no change in spatial scene and = 1 shows that there is a complete topological or directional change in the spatial scene.

(ℎ 1 , ℎ 2 ) = ∑ (ℎ 1 ( )ℎ 2 ( )) √ ∑ (ℎ 2 1 ( ) √ ∑ ℎ 2 2 ( )) (1)
is normalized cross correlation for similarity measure. Normalized cross correlation is invariant to the camera zooming so due to this reason it is preferred to use this similarity measure. A vector or a similarity row or column matrix could be defined. If each element of this matrix is zero then there is no spatial change in the spatial scene and the spatio-temporal relation is defined as same-place different-time otherwise the relation will be different-place different-time.

V. EXPERIMENTS

For the experiment purposes, all the 13 fuzzy Allen histograms are calculated from angle measure [0, 360] with the increment of one degree and similarity between them. In the first example (figure 9(a) to 9(d)), we consider two frames of a well known image sequence of Y. Sheikh and M. Shah [START_REF] Sheikh | Bayesian modeling of dynamic scenes for object detection[END_REF] where both objects are moving in opposite directions where the speed of the car is ℎ 1 and speed of man is considered as ℎ 2 . If the one object say reference object (car) is considered at rest and the object is under motion, then net speed of object will be ℎ 1 + ℎ 2 . Here manual segmentation is taken, because here the problem is to detect the change in the scene not the segmentation. As at each frame the distance between the object increases, at the first analysis frame the fuzzy meet relation exists i.e. in terms of Allen relations fuzzy meet and meet by relations exists along with certain directions. When the distance between the objects increase, the histogram for the relations meet and meet by. Then the similarity between the two histograms are calculated. All those histograms which don't exist, the similarity or dissimilarity between them is considered to be zero. Dissimilarity matrix will be (0.3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0.3). This shows that the spatio-temporal relation is different-place different time.

In second example (For reference figure 10 this example two trail objects are considered for experimental purposes, rectangular object represented by dark gray color, and circular object is represented by the light gray color. In fact, the example represents the inverse phenomenon to the example 1. At initial stage the objects are at a certain distance, they have only disjoint topological relation, when the distance between them decreases to a certain extent, emerges a new histogram for the fuzzy meet relation. Although the objects don't meet but fuzzification is object size dependant. As a result, whole topological structure is changed and a new fuzzy topological relations exists in certain directions. Dissimilarity matrix will be (0.2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0.2). This shows that the spatio-temporal relation is different-place different time.

In third example (figure 11(a) to 11(d)) when the object (light gray object )is inside the object (dark gray object), fuzzy Allen histograms are calculated there exist number of histograms due to fuzzification. In the second case object slightly changes its position while remaining inside the object , for experimental purpose object is translated by (-10,20), this results change in the fuzzy histograms. In this example qualitative directional and distance relations don't exist, and there is no qualitative change in spatial scene. But in fact there is a change in spatial scene, that object has changed its location. Dissimilarity matrix will be (0.4, 0, 0, 0.9, 0, 0, 0, 0, 0, 0.9, 0, 0, 0. VI. CONCLUSION Fuzzy techniques are becoming important day by day due to handling the vagueness and uncertainty in every field of image processing, computer vision and database management, defining the fuzzy spatio temporal relations is one of them. In this paper we have proposed a fuzzy method to define the spatio-temporal relations belonging to the family of sameplace different-time and different-place different-time. This method is based on our previous work [START_REF] Salamat | Fuzzy change detection in a spatial scene[END_REF], where we have defined a fuzzy method for detecting a small change in video scene analysis. The fuzzy Allen relations are used for the model and fuzzy dissimilarity measure is used to compare the dissimilarity for fuzzy Allen relations histograms.

  Let and be two spatial objects, ( , ) ∈ , where is any real number and ∈ [-, ]. Δ ( ) is an oriented line at orientation angle . ∩ Δ ( ) is the intersection of object and oriented line Δ ( ). It is denoted by ( ), called segment of object and its length is . Similarly for object

Fig. 4 .

 4 Fig.4. Oriented line Δ ( ), segment as in case of object , longitudinal section as in case of object .( Matsakis[START_REF] Matsakis | Combined Extraction of Directional and Topological Relationship Information from 2D Concave Objects, in fuzzy modeling with spatial informations for geographic problems[END_REF]) 360 directions are considered with an angle increment of one degree and lines are drawn by 2d Bresenham digital line algorithm. A polygonal object approximation is taken and lines passing through at least one vertex of the polygon are taken into account. segments are computed and all pairs of segments can be treated simultaneously. Fuzzy Allen relations are computed for each segment.

  ) are the relation , , , , ℎ, (resp the inverse relations of the cited ones). The relation is the equality spatial relation. All the Allen relations in space are conceptually illustrated in figure 5. These relations have a rich support for the topological and directional relations.

Fig. 5 .

 5 Fig. 5. Black segment represents the reference object and gray segment represents argument object

Fig. 6 .

 6 Fig. 6. Histogram of fuzzy Allen relation in direction

Fig. 7 .

 7 Fig. 7. (a) Histogram representation (b) Overlapping objects (c)Un normalized histogram of fuzzy Allen relations (d) Corresponding normalized histogram of fuzzy Allen relations

Fig. 9 .

 9 Fig. 9. Two frames of an image sequence of Y. Sheikh and M. Shah [7] and their spatial relations

Fig. 10 .

 10 Fig. 10. (a) and (c) are images where in (c) object A (light gray object) moves towards object B, (b),(d) are their histogram representations.

Fig. 11 .

 11 Fig. 11. (a) and (c) are images where in (c) object A moves towards the center of object B (b) and (d) are their histogram representations.

4 ).

 4 This shows that the spatio-temporal relation is different-place different time. All the above cited examples are representing the spatio-temporal relation different-place different-time because each time objects are taken under motion.


Nadeem Salamat
email: nsalamat@univ-lr.fr

El-Hadi Zahzah
email: ezahzah@univ-lr.fr

Fusion of fuzzy spatial relations

Keywords: Topological and metric relations, Fuzzy topological relations, Fusion of spatial relations, Fuzzy directional relations, Matrix of fuzzy relations

Spatial relations are essential for understanding the image configuration and modeling common sense knowledge. In most of existing methods, topological, directional and distance spatial relations are computed separately as they have separate application domains. Introduction of Temporal Geographic Information System (TGIS), spatiotemporal reasoning and study of spatio-temporal relations required the computation of topological and metric spatial relations together. In this paper the fuzzy topological and directional relations are integrated with the help of fuzzy Allen relations and directions are evaluated by specific fuzzy membership functions. A matrix of fuzzy relations is developed where the topological and directional relations are integrated for a 2D scene. Experiments are performed to validate the proposed method. The results are analyzed and interpreted from histograms.

Introduction

Temporal Geographic Information System (TGIS) was introduced in recent years. This enhancement of Geographic Information System (GIS) deeply affects the present studies in GIS and need fusion of different information provided from different sensors given heterogenous data. The fusion of spatial relations will be helpful in defining the spatio-temporal relations, reasoning in many decision processus. In existing methods, topological, directional and distance spatial relations are studied separately. Beside this information, fuzzy spatial relations are also important for fuzzy reasoning, fuzzy image understanding and fuzzy change detection. Fuzzy topological relations are studied by the extended models of 9 -intersections [START_REF] Egenhofer | A Critical Comparison of The 4-Intersection and 9-Intersection Models for Spatial Relations: Formal Analysis[END_REF][START_REF] Du | Fuzzy description of topological relations i: A unified fuzzy 9-intersection model[END_REF] and Region Connected Calculus ( 8) [START_REF] Randell | A spatial logic based on regions and connection[END_REF][START_REF] Rosenfeld | Fuzzy digital topology[END_REF][START_REF] Tang | Spatial object model[l]ing in fuzzy topological spaces : with applications to land cover change[END_REF]. These topological relations are studied through the object model with intermediate boundaries or the well known Egg-Yolk model [START_REF] Clementini | An algebraic model for spatial objects with indeterminate boundaries[END_REF][START_REF] Cohn | The 'egg-yolk' representation of regions with indeterminate boundaries[END_REF]. Fuzzy directional relations can be defined using the mathematical morphology as in [START_REF] Bloch | Directional Relative Position Between Objects in Image Processing: A Comparison Between Fuzzy Approaches[END_REF] and numerical methods as in [START_REF] Miyajima | Spatial Organization in 2D Segmented Images: Representation and Recognition of Primitive Spatial Relations[END_REF][START_REF] Matsakis | F-Histograms and Fuzzy Directional Spatial Relations[END_REF]. Fuzzy directional relations assign a degree to each cardinal direction and different methods are used to assign the degree to a cardinal direction. Another type of metric relation is the distance relations.

These relations provide us the distance information about the closet part of objects and a fuzzy membership function assigns a degree to the distance relation Near, Far and Far off [START_REF] Bloch | Fuzzy distances and image processing[END_REF]. In the approaches for finding the fuzzy directional relations, crisp objects are considered and fuzzy objects are treated by considering the Egg's boundary. Fuzzy methods for directional relations work for the limited set of topological relations. All sort of relations are essential for image understanding, modeling common sense knowledge, natural phenomenon and spatial reasoning [START_REF] Hudelot | Fuzzy Spatial Relation Ontology for Image Interpretation[END_REF]. This requirement of image understanding needs to apply different methods to find the topological, directional and distance relations between the objects and it is needed to have a single method where we can get all the required information about the spatial scene. In this work two types of fuzzy sets are used, 1 fuzzy sets for finding the fuzzy Allen relations and then 1 directional fuzzy sets for evaluating the directions in 2 scene. This method of using the 1 fuzzy sets provide us the integrated topological and directional information of a 2 scene. These integrated information of a 2 scene are represented in a matrix. Each entity of this matrix represents the percentage surface area of two objects having a topological relation in a specific direction. This paper is structured as follows, section 2 discusses in detail the different terms and computations necessary for 1 Allen relations, in section 3 the fusion of topological and directional relations and their interpretation is given. Results for different configurations is given in section 4. Section 5 concludes the paper.

2 Terminology used for computation fuzzy Allen relations ( ) is an oriented line at orientation angle . ∩ ( ) is the intersection of object and oriented line ( ). It is denoted by ( ), called segment of object and its length is . Similarly for object where ∩ ( ) = ( ) is segment and is its length. is the difference between the maximum value of ∩ ( ) and minimum of ∩ ( )(for details [START_REF] Matsakis | Combined Extraction of Directional and Topological Relationship Information from 2D Concave Objects, in fuzzy modeling with spatial informations for geographic problems[END_REF]). In case of polygonal object approximations ( , , ) can be calculated from intersecting points of line and object boundary. If there exist more than one segment, then it is called longitudinal section.In this paper all the 180 directions are considered with an angle increment of one degree and lines are drawn by 2 Bresenham digital line algorithm. A polygonal object approximation is taken and lines passing through the polygon vertices are taken into account.

1 Allen relations in space

Allen [START_REF] Jf | Maintaining Knowledge about Temporal Intervals[END_REF] relations are arranged as = {<, ,,,,,,,,,, , >} with meanings before, meet, overlap, start, finish, during, equal, during_by, fin-ish_by, start_by, overlap_by, meet_by, and after. All the Allen relations in space are conceptually illustrated in figure 1. These relations have a rich support for the topological and directional relations. In the neighborhood graph of Allen relations, three paths can be find (because we assume that objects are monolithic and don't changing size during their movement. Some possible paths are ignored due to this reason ). Depending upon the neighborhood graph of Allen relations, Inverse of these relations can be divided into two categories, object commutativity and reorientation of relations. According to reorientation of a relation, we can write 1 = {<, , , , , , , =} and their inverses as ,,,,,,=}. This shows that relations , =, have their own reorientations. Region Connected Calculus -RCC8 relations are possible combination of 8 independent Allen relations in 1 . These relations and their inverse show that the whole 2 space can be explored with the help of 1 Allen relations using oriented lines varying from (0, ).

2 = {>,

Fuzzification of Allen relations

Fuzzification process of Allen relations do not depend upon particular choice of fuzzy membership function, trapezoidal membership function is used due to flexibility in shape change. Let ( , ) be Allen relation between segments and where ∈ (argument object) and ∈ (reference object), ′ is the distance between ( , ) and its conceptional neighborhood. We consider a fuzzy membership function

: ′ -→ [0, 1].
The fuzzy Allen relations defined by Matsakis [START_REF] Matsakis | Combined Extraction of Directional and Topological Relationship Information from 2D Concave Objects, in fuzzy modeling with spatial informations for geographic problems[END_REF] are

-< ( , ) = (-∞,-∞,--3 /2,--) ( ), -> ( , ) = (0, /2,∞,∞) ( ) - ( , ) = (--3 /2,--,--,--/2) ( ), - ( , ) = (-/2,0,0, /2) ( ) - ( , ) = (-,-/2,-/2,0) ( ) - ( , ) = ( (-( + )/2,-,-,+∞) ( ), (-3 /2,-,-,-/2) ( ), ( ,2 ,+∞,+∞) ( )) -( , ) = ( (-,-+ /2,-3 /2,-) ( ), (-∞,-∞, /2, ) ( ))
where = ( , ), = ( , ) and is the length of longitudinal section of argument object and is the length of longitudinal section of reference object . Most of relations are defined by one membership function like , , , _ and some of them are defined by more than one membership functions like ( ), ( _ ), ( ℎ), ( ℎ _ ). In fuzzy set theory, sum of all the relations is one, this gives the definition for relation fuzzy . During the decomposition process of an object into segments, there can be multiple segments depending on object shape and boundary which is called longitudinal section. Different segments of a longitudinal section are at a certain distance and these distances might effect end results. In polygonal object approximation, fuzzy T-conorms are used for fuzzy integration of available information. Here for simplicity only T-conorm (Fuzzy operator)is used.

( ) ( ) = ( ( ) ( ), ( ) ( ))

Normalized fuzzy histogram of Allen relations

Histogram of fuzzy Allen relation represents total area of subregions of and that are facing each other in given direction . Mathematically it can be written as

[15] ∫ +∞ -∞ ( ∑ ( , ( ), ( )) = ( + ) ∑ =1 ( , )
where is the area of reference object and is area of augmented object in direction , is total number of segments treated and ( , ) is an Allen relation for segments ( , ). These histograms are normalized by dividing all Allen's relations by their sum for every . It is represented by ⌈ ( )⌋ where ∈ . ⌈ ( )⌋ = ( ) ∑ ∈ ( ) . These fuzzy Allen relations are directional fuzzy numbers and can be used to define the quantitative fuzzy directional relations.

Fusion of Topological and directional relations

This section consists of two subsections where in first one it is described how different functions can be used to assess the fuzzy qualitative directions and how the different Allen relations are combined for topological and directional information fusion. In second subsection, the representation method for the fuzzy topological and fuzzy directional relations is described in detail.

Fusion of topological and directional Components

All these equations depicted in section 2.3 assign a numerical value to a topological spatial relation in a direction . To find directional contents of a topological relation, these numerical values are used with directional fuzzy sets. Directions are represented as { , , , , , , , } with meanings East, North_East, North, North_West, West, South_ West, South and South_ East. To assess these fuzzy directional relations, two trigonometric membership functions 2 and 2 are used. As angle distribution is taken to the half plane so opposite Allen relations are used to define the opposite directions except the direction East and West where union of both relations are used. Mathematically these relations can be defined as

- = ∑ 4 =0 2 × 2 (2 ) + ∑ = 3 4 1 × 2 (2 ) - = ∑ 4 =0 1 × 2 (2 ) + ∑ = 3 4 2 × 2 (2 )
- = ∑ 3 4 = 4 2 × 2 (2 )
- = ∑ 3 4 = 4 1 × 2 (2 )
Where ∈ , = 1, 2 given in section 2. The above explanation shows that overlap relation in 2 space is more complex than any other topological relation. In this case all the Allen relations coexist in different directions.

Experiments and discussion

In the following set of examples the objects (argument object) and (reference object) are represented by light and dark grey colors respectively. A matrix of fuzzy topological and directional relations are computed. Figure (2(a)) shows that argument object meets the reference object in direction , their topological and directional relations show that there exist a topological relation meet EC with a maximum degree in direction East along with other neighboring directions and Disjoint topological relation exists due to application of In the last example, the method is applied to a well known sequence of images "water surface sequence" [START_REF] Li | Statistical modeling of complex backgrounds for foreground object detection[END_REF] where a manual segmentation of objects is performed. The tree is taken as a reference object while the shape of the person is taken as an argument object . In figure 3 ples, it is shown that the topological and directional relations can be successfully integrated in a single method. This method will be hopefully used for spatiotemporal reasoning where we need the topological and directional information at each step.

(a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l)

Conclusion

Spatial topological and directional relations are important to understand the scene configuration and spatio-temporal reasoning. Spatio-temporal information fusion is important to have a single method to understand the image configuration completely. In this paper, the key point is that different informations regarding the topological and directional relations are integrated through a single method. These relations are sensitive to the distance between the objects and cardinal directions. Due to the spatio-temporal nature it is possible to construct from the fuzzy Allen relations a set of spatio-temporal relations such as Leave,Enter, Cross, Bypass and others. This method could be used for spatio-temporal reasoning. Our future work will now in defining the spatio-temporal relations of the above mentioned classes between two moving spatial objects.

2. 1

 1 Oriented lines, segments and longitudinal sectionsLet and be two spatial objects and ( , ) ∈ ℝ, where is any real number and ∈ [0, 2 ].

Fig. 1 .

 1 Fig. 1. Black segment represents the reference object and gray segment represents argument object

  mean Disjoint, Externally connected, Partially overlap, Tangent properpart, Non tangent properpart, Tangent properpart inverse, Non Tangent properpart inverse, and Equal.

3. 2 2 . 3 .

 223 Interpretation of topological and directional relationsThe relations are manipulated in (8 × 8) matrix where ( , ) represents the ℎ topological relation in ℎ direction. Rows are arranged in an order of { , Different directions may yield different 1 relations in 2 space and different relations may coexist along the same direction. Below a conclusion about the overall 2 relation is given. 1. Only first row is non zero then objects have fuzzy disjoint (DC ) topological relation . If the first and the second rows are non zero then the overall relation in 2 space is fuzzy EC. If there exist at least one non zero value in third row it means there exist fuzzy topological relation overlap. or If non zero values also exist in TPP along with NTPP (TPPI along with NTPPI ) then the relation will be TPP (NTPP ) in the corresponding direction. 4. Relations PP, PPI, EQ hold if the corresponding relation holds in all directions. A relation will hold if all elements in a row are non zero and all other rows are zero.

Fig. 2 .

 2 Fig. 2. Object pairs with meet topological relation and their directional relations

Fig. 3 .

 3 Fig. 3. Difference frames of image sequence "watersurface" and their spatial relations.


Nadeem Salamat
email: nsalam01@univ-lr.fr

El-Hadi Zahzah
email: 2ezahzah@univ-lr.fr

FUZZY SPATIAL RELATIONS FOR 2D SCENE

Keywords: Fuzzy objects, Fuzzy relations, Fuzzy topological relations, 1D Allen relations, Matrix of fuzzy relations

Different models for computing the spatial relations have been developed in the last decade. Separate methods are used for computing topological, directional and distance relations. These models have also been extended in fuzzy domains based on fuzzy or vague objects and less attention has been paid to fuzzy relations. In this paper 8 fuzzy topological relations are computed with the help of 1D Allen relations and directional relations are evaluated by using fuzzy membership functions. A matrix of fuzzy relations is developed which provides complete set of information about a 2D scene. This method deals with two types of fuzziness (i) fuzziness involved in the objects position (ii) fuzziness about directional relations.

Introduction

The space can be studied through spatial relations between different regions included in it. These spatial relations include topological, directional and distance relations. There is an increasing interest in formalisms that describe properties of space in a qualitative way. Usually such a qualitative representation takes the form of topological relations between regions, there are two famous models for finding topological relations between spatial regions. 9-intersection model [START_REF] Egenhofer | Categorizing Binary Topological Relations Between Regions,Lines and Points in Geographic Databases[END_REF], [START_REF] Egenhofer | A Critical Comparison of The 4-Intersection and 9-Intersection Models for Spatial Relations: Formal Analysis[END_REF], [START_REF]A voronoi-based 9-intersection model for spatial relations[END_REF] and Region Connected Calculus ( 8) [START_REF] Randell | A spatial logic based on regions and connection[END_REF]. Both 9-intersection and models have been extended to the fuzzy domains but each time models are extended for fuzzy objects [START_REF] Shi | A fuzzy topology for computing the interior, boundary, and exterior of spatial objects quantitatively in gis[END_REF], [START_REF] Liu | Quantitative fuzzy topological relations of spatial objects by induced fuzzy topology[END_REF], [START_REF] Palshikar | Fuzzy region connection calculus in finite discrete space domains[END_REF], [START_REF] Schockaert | Fuzzy region connection calculus: An interpretation based on closeness[END_REF], [START_REF] Schockaert | Fuzzy region connection calculus: Representing vague topological information[END_REF] while in most of the qualitative directional relations models objects are taken as points. The 9-intersection model for directional relations [START_REF] Goyal | Similarity of cardinal directions[END_REF] consider the objects by their minimum bounding rectangle (MBR) and then 2D projections are used. To capture internal position of an object internal cardinal directional relations (ICD) model were introduced [START_REF] Tang | Advances in Spatio-Temporal Analysis[END_REF], [START_REF] Liu | On internal cardinal direction relations[END_REF]. All of these models describe the object properties only in one domain and completely ignore the other aspects of the space like 9-intersections for topological relations and 8 models describe well the topological properties of space and ignore the directional and distance relations meanwhile 9-intersection model for directional relations and internal directional relations models describe the objects regarding their global directional contents and ignore the topological properties, as a result one single model can not be used to perfectly describe the object in embedding space. Fuzzy reasoning about spatial relationship can be viewed fuzzy description of object's relative position. In qualitative domain mostly topological relations are studied between vague or fuzzy objects such as in 8 and 9-intersection models for topological relations but less attention has been paid to study the fuzzy spatial relations between crisp objects. Different approaches for directional relations were adopted such as mathematical morphology [START_REF] Bloch | Directional Relative Position Between Objects in Image Processing: A Comparison Between Fuzzy Approaches[END_REF], [START_REF] Bloch | Fuzzy distances and image processing[END_REF], quantitative approaches [START_REF] Miyajima | Spatial Organization in 2D Segmented Images: Representation and Recognition of Primitive Spatial Relations[END_REF], called angle histogram and force histograms [START_REF] Matsakis | F-Histograms and Fuzzy Directional Spatial Relations[END_REF], [START_REF] Laurent Wendling | Représentation de la Position Relative d' Objets 2D au Moyen ′ un Histogramme de Forces[END_REF] where evaluation approach for directional relations was fuzzy and deals only disjoint objects. Another extension of angle histogram was R-histograms [START_REF] Wang | R-histogram: quantitative representation of spatial relations for similarity-based image retrieval[END_REF] which deals with most of the topological relations like disjoint, meet, overlap, contains and contained. In all these approaches, the quantitative or fuzzy directional relations are studied and less attention has been paid to fuzzy topological relations while the topological, directional and distance relations are considered essential to understand the image configuration, modeling common sense knowledge and spatial reasoning [START_REF] Hudelot | Fuzzy Spatial Relation Ontology for Image Interpretation[END_REF]. The idea of combined topological and directional relations was first initiated by J.Malki et al. [START_REF] Malki | Directional relations composition by orientation histogram fusion[END_REF] by using the 1D Allen [START_REF]Maintaining Knowledge about Temporal Intervals[END_REF] relations in spatial domain. This work was revisited by Matsakis and Nikitenko [START_REF] Matsakis | Combined Extraction of Directional and Topological Relationship Information from 2D Concave Objects, in Fuzzy Modeling with Spatial Information for Geographic Problems[END_REF] and fuzziness in the 1D Allen relations was introduced and problem of temporal complexity was resolved by using t-conorms along with polygonal approximation of objects [START_REF] Salamat | Spatial relations analysis by using fuzzy operators[END_REF]. This work is related to fuzzy spatial aspects where the topological and directional relations are evaluated according to fuzzy set theoretical viewpoint in 2D scene and it differs from P.Matsakis work [START_REF] Wawrzyniak | Speaking with spatial relations[END_REF], [START_REF] Matsakis | Relative positions in words: a system that builds descriptions around Allen relations[END_REF] where he used 2D fuzzy sets to explore the limited set of 2D topological relations and 1D fuzzy sets are used to express the 1D fuzzy Allen relations. This paper is structured as follows, section 2 discuss in detail the different terms and computations necessary for 1D Allen relations. In section 3, our method for computing the topological relations along with directional aspects and their interpretation is given, results for different situation is given in section 4 and section 5 concludes the paper.

Terminology used for computation of fuzzy Allen relations

This section describes different terminology used to decompose the space and computation of different terms used in this paper.

Oriented lines, segments and longitudinal sections

and be two spatial objects and ( , ) ∈ ℝ, where is any real number and ∈ [0, 2 ]. Δ ( ) is an oriented line at orientation angle . ∩Δ ( ) is the intersection of object and oriented line Δ ( ). It is denoted by ( ), called segment of object and its length is . Similarly for object where ∩Δ ( ) = ( ) is segment and is its length. is the difference between the maximum value of ∩ Δ ( ) and minimum of ∩ Δ ( )(for details [START_REF] Matsakis | Combined Extraction of Directional and Topological Relationship Information from 2D Concave Objects, in Fuzzy Modeling with Spatial Information for Geographic Problems[END_REF]). In case of polygonal object approximations ( , , ) can be calculated from intersecting points of line and object boundary, oriented lines are considered which passes through at least one vertex of two polygons. If there exist more than one segment then it is called longitudinal section as in case of ( ) in figure 1. In this paper all the 180 directions are considered with Fig. 1: Oriented line Δ ( ), segment as in case of object , longitudinal section as in case of object [START_REF] Matsakis | Combined Extraction of Directional and Topological Relationship Information from 2D Concave Objects, in Fuzzy Modeling with Spatial Information for Geographic Problems[END_REF].

an angle increment of one degree and lines are drawn by 2d Bresenham digital line algorithm. A polygonal object approximation is taken and lines passing through the polygon vertices are taken into account. segments are computed and all pairs of segments can be treated simultaneously. Fuzzy Allen relations are computed for each segment.

1D Allen relations in space

Allen [START_REF]Maintaining Knowledge about Temporal Intervals[END_REF] introduced the well known 13 jointly exhaustive and pairwise disjoint (JEPD) interval relations based on temporal interval algebra. These relations are arranged as = {<, , , , , , , , , , , , >} with meanings before, meet, overlap, start, finish, during, equal, during_by, finish_by, start_by, overlap_by, meet_by, and after. All the Allen relations in space are conceptually illustrated in figure 2. These relations have a rich support for the topological and directional relations. In the neighborhood graph of Allen relations, three paths can be find (because we consider the solid objects and they don't change their size during movement, due to this reason other possible paths are ignored). Depending upon the neighborhood graph of Allen relations, Inverse of a Allen relation can be divided into two categories.

1) Objects Commutativity: There are three continuous paths in this graph. When objects are interchanged, A versed Allen relations and their inverses also change, in this case there are 8 Allen relations and their inverse. These inverses are also called the reorientation of relations. {<, , , , , , , =, , , >} and inverse {>, , , , , , , =, , , , , <} as a result, we can write 1 = {<, , , , , , , =} and their inverses as 2 = {>, , , , , , , =}. This shows that the numeric values , =, have their own reorientations. RCC8 relations are possible combination of 8 independent Allen relations in 1D. These relations and their inverse show that the whole 2D space can be explored with the help of 1D Allen relations using oriented lines varying from (0, ).

Fuzzification of Allen relations

In real applications, small errors in crisp values can change the entire result. To cope these problems fuzzification was introduced, it comprises the process of transforming crisp values into grades of membership for linguistic terms of fuzzy sets. Fuzzification process of Allen relations do not depend upon particular choice of fuzzy membership function, trapezoidal membership function is used due to flexibility in shape change. Let ( , ) is Allen relation between segments and where ∈ (argument object) and ∈ (reference object), ′ is the distance between ( , ) and its conceptional neighborhood. We consider a fuzzy membership function

: ′ -→ [0, 1].
The fuzzy Allen relations defined by Matsakis and Nikitenko [START_REF] Matsakis | Combined Extraction of Directional and Topological Relationship Information from 2D Concave Objects, in Fuzzy Modeling with Spatial Information for Geographic Problems[END_REF] are

• < ( , ) = (-∞,-∞,--3 /2,--) ( ), • > ( , ) = (0, /2,∞,∞) ( ) • ( , ) = (--3 /2,--,--,--/2) ( ), • ( , ) = (-/2,0,0, /2) ( ) • ( , ) = (--,--/2,--/2, ) ( ), • ( , ) = (-,-/2,-/2,0) ( ) • ( , ) = ( 5 , 3 , 1 ) • ( , ) = ( 6 , 4 , 2 ) • ( , ) = ( 6 , 4 , 1 ) • ( , ) = ( 5 , 3 , 2 ) • ( , ) = ( 7 , 1 ) • ( , ) = ( 7 , 2
)

where 1 = (-∞,-∞, /2, ) ( ), 2 = ( ,2 ,+∞,+∞) ( ) 3 = (-3 /2,-,-,-/2) ( ), 4 = (-∞,-∞,-,-( + )/2) ( ), 5 = (-( + )/2,-,-,+∞) ( ), 6 = --/2,-,-,-+ /2 ( ), 7 = (-,-+ /2,-3 /2,-) ( ), = ( , ), =
( , ) and is the length of longitudinal section of argument object and is the length of longitudinal section of reference object . Most of relations are defined by one membership like , , , _ and some of them are defined by more than one membership functions like ( ), ( _ ), ( ℎ), ( ℎ _ ). In fuzzy set theory, sum of all the relations is one. This gives the definition for relation fuzzy . Fuzzy Allen relations are not Jointly Exhaustive and Pairwise Disjoint (JEPD) because there exist at least two relations between two spatial objects. All these equations assign a numeric value to a spatial relation.

Treatment of longitudinal sections

During the decomposition process of an object into segments, there can be multiple segments depending object shape and boundary which is called longitudinal section. Different segments of a longitudinal section are at a certain distance and these distances might effect end results. In polygonal object approximation, fuzzy T-norms and T-conorms are used for fuzzy integration of available information. Here for simplicity only T-conorm (Fuzzy operator)is used.

( ) ( ) = ( ( ) ( ), ( ) ( ))
When fuzzy operator is used, only one fuzzy value contributes for the resultant value which represents the maximum. In this case each Allen relation has a fuzzy grade objective is to accumulate the best available information. Suppose that longitudinal section of object has two segments such that = 1 + 2 where 1 is the length of first segment and 2 is the length of second segment and is length of longitudinal section. Let 1 ( 1 ) defines the value of fuzzy Allen relations with the first segment and 2 ( 2 ) represents value of fuzzy Allen relations with the second segment where 1 and 2 are the distances between object and two segments of . Now fuzzy operator is used to get consequent information obtained from two sets of fuzzy Allen relations.

Normalized fuzzy histogram of Allen relations

Histogram of fuzzy Allen relation represents total area of subregions of and that are facing each other in given direction [START_REF] Matsakis | Combined Extraction of Directional and Topological Relationship Information from 2D Concave Objects, in Fuzzy Modeling with Spatial Information for Geographic Problems[END_REF]. Mathematically it can be written as

[23] ∫ +∞ -∞ ( ∑ ( , ( ), ( )) = ( + ) ∑ =1 ( , )
where is the area of reference object and is area of augmented object in direction , is total number of segments treated and ( , ) is an Allen relation for segments ( , ). These histograms can easily be normalized by dividing the all the Allen relations by the sum of all the Allen relations for every . It is represented by ⌈ ( )⌋ where ∈ . ⌈ ( )⌋ = ( ) ∑ ∈ ( ) . These fuzzy Allen relations are directional fuzzy numbers and can be used to define the quantitative fuzzy directional relations.

Topological and directional relations

This section consists of two subsections where in first subsection, it is described that two different functions can be used to assess the fuzzy qualitative directions and then how the different Allen relations are combined. In the second subsection, the representation method for the fuzzy topological and fuzzy directional relations are described in detail.

Assessment of fuzzy directional Components

All these equations depicted in section 2.3 assign a numeric value to a spatial relation in a direction and it is difficult to analyze the spatial changes in a video scene with the histogram representation. To make the changes visible in a qualitative direction, numeric values are used. The mainstream efforts concentrated on developing computation oriented representation formalisms and their implementations, paying little attention to the process of modeling itself. To cope with the temporal complexity semi interval [START_REF] Freksa | Temporal Reasoning Based on Semi-intervals[END_REF] is used to model the directions. For this purpose 8 compass directions are used to express the directional relations. Directions are represented as { , , , , , , , } with meanings East, North_East, North, North_West, West, South_ West, South and South_ East. To assess these fuzzy directional relations, two trigonometric functions are used. To assess the even directions ({ , , , }), trigonometric function 2 and to assess the odd direction ({ , , , }), trigonometric function 2 is used. As the angle distribution is taken to the half plane so opposite Allen relations are used to define the opposite directions except the direction East and West where union of both relations are used. Mathematically these relations can be defined as

= ⎧  ⎨  ⎩ 2 * 2 (2 ) if ∈ (0, 4 ) 1 * 2 (2 ) if ∈ ( 3 4 , ) 0 otherwise = ⎧  ⎨  ⎩ 1 * 2 (2 ) if ∈ (0, 4 ) 2 * 2 (2 ) if ∈ ( 3 4 , ) 0 otherwise = { 2 * 2 (2 ) if ∈ ( 4 , 3 4 ) 0 otherwise = { 1 * 2 (2 ) if ∈ ( 4 , 3 4 ) 0 otherwise = { 2 * 2 (2 ) if ∈ (0, 2 ) 0 otherwise = { 1 * 2 (2 ) if ∈ (0, 2 ) 0 otherwise = { 2 * 2 (2 ) if ∈ ( 2 , ) 0 otherwise = { 1 * 2 (2 ) if ∈ ( 2 , ) 0

Representation and interpretation of relations

A qualitative direction system will assign a single direction to another object with respect to a reference object and that is the only directional relationship between them. However, in the fuzzy approach, cardinal directions will be have different degrees of truth. This degree of truth value represents the percentage of object which lies in that direction. The relations are manipulated in (8 × 8) matrix where rows show the topological relations and columns show the directional distribution of each topological relation. Each entity of this matrix represents the percentage surface area of two objects having a topological relation in a specific direction. C(i,j) represents the ℎ topological relation in ℎ direction. Rows are arranged in an order of spatial neighboring of RCC8 relations.i.e. { , , , , , , , }. Different directions may yield different 1D relations in 2D space and different relations may coexist along the same direction. When directional relation for complex 2D objects are computed with the help of 1D Allen relations then one relation may exist in multiple directions. In case of complex 2D objects such as concave objects two opposite relations may coexist along the one direction or the one object may have different relations in different directions. Here a method to understand the representation of fuzzy topological and directional relations and deriving the conclusion about the overall 2D relation is given.

1) If all the rows are zero except the first then objects have fuzzy disjoint (DC) topological relation . 2) If non zero values are in first and second row then the overall relation in 2D space is fuzzy meet EC. 3) If there is at least one non zero value in third row it means there exist fuzzy topological relation overlap.

More generally when the two opposite relations exist together i.e TPP and TPPI or NTPP and NTPPI, then topological relation in 2D space will be overlap other relations only explain the situation. 4) Relations PP, PPI, EQ hold if the corresponding relation holds in all directions, a relation will hold if all elements in a row are non zero and all other rows are zero. 5) If the non zero values also exist in TPP along with NTPP (TPPI long with NTPPI) then the relation will be TPP (NTPP) in the corresponding direction. This shows that 1D definition of topological relations could not used directly to define 2D topological relations. Whole above explanation shows that overlap relation in 2D space is more complex than any other topological relation.

Experiments

For the experiment purpose, 180 directions are considered (angle increment is 1 degree) and lines are drawn by 2d Bresenham digital line algorithm. Segments are computed and all pairs of segments can be treated simultaneously. Fuzzy Allen relations are computed for each segment. When the directional relations are evaluated by fuzzy techniques, they can share more than one direction and their directional relations are represented by a degree. Throughout this paper reference object is represented by dark grey color and light grey object represents the argument object . In first example, objects are taken at different distances and it is a well known fact that range of directional relations are inversely proportional to the distance between them.i.e. greater the distance between them, smaller will be their range of directional distributions. When the object is far away from the object represented in figure 3(a), it represents the topological relation disjoint with degree of directional relations E=0.69, NE=0.16, SE=0.15 as represented in figure

Conclusion and future work

In this paper a new method for finding the fuzzy topological and directional relations was studied and all the topological relations are generated by using the fuzzy Allen relations. Directions are evaluated with the help of trigonometric membership functions. This method deals with fuzziness at two levels, fuzziness in the topological relations due to their positions and fuzziness in directional relations. This method differs from P. Matsakis [START_REF] Wawrzyniak | Speaking with spatial relations[END_REF] work, where 2D fuzzy sets to generate a limited number of topological relations are used. Here fuzzy Allen relations are combined to generate the 2D topological relations along with the directional assessment of relations and a new representation is used. It is a numerical description of relative position of objects. Spatio-temporal relations are the emerging issue and hopefully these results will be helpful in extending this work to a spatio-temporal aspects and fuzzy spatio-temporal reasoning and natural language processus. These results will be used in future works to develop the motion verbs and spatio-temporal relations between moving objects.

Fig. 2 :

 2 Fig. 2: Black segment represents the reference object and gray segment represents argument object

Fig. 3 :Fig. 4 :

 34 Fig. 3: Object pairs and its spatial relations when distance between them changes

Fig. 5 :Fig. 6 :Fig. 7 :

 567 Fig. 5: Object pairs when object A lies inside object B and its spatial relations