Simulation and visualization of anisotropic expansion ... - CiteSeerX

5 th AGILE Conference on Geographical Information Science, Palma (Balearic Islands, Spain) April 25 th- 27th 2002

Simulation and visualization of anisotropic expansion phenomena Andrés Molina, M. Dolores Muñoz Dpto. de Informática. Escuela Politécnica Superior Universidad de Jaén Avda. de Madrid 35. 23071 JAÉN. SPAIN Tlf: 953 002 446 Fax: 953 002 420. E-mail: {molina, [email protected]} Abstract. In the isotropic buffer, the generating polygon is a circle, what implies a constant distance from the border of the buffer to the object. The simulation of anisotropic processes, on the contrary requires the use of generating polygons which determine influence areas directionally non uniform. In this article we will describe a method for the design of generating ovals supposing a normal distribution of the surface around a certain direction or dominant direction. Likewise, the problem that presents the simulation and visualization of anisotropic phenomena expansions is approached, analyzing the existent casuistry for the generation of influence areas calculated by means of the operation Sum of Minkowski.

Introduction and previous concepts All the actually customised GIS include the buffer operator, also denominated buffering or influence zone. It is defined as the geometric space of the points that are at a smaller distance or similar to an object (point, poliline or polygon) given [Chow97]. This definition is isotropic or directionally uniform, since the distance of the object to the edge of the buffer is constant in any direction of the plane. One of the disciplines in those that this operator is used is the one of simulation at visualization of environmental processes, such as the study of the contamination that the pesticides and chemical fertilizers produce in the shallow waters of the hydrographic basins, the analysis of the influence of nitrates and silts levels on the growth or the flora, the environmental impact taken place by the installation of new industries in the proximities to urban centers, the determination of areas of high seismic risk … [Odyssey-A][ Odyssey-B ][ Odyssey-C]. All the described applications make use of the operator isotropic buffer. However, there are phenomena like the dispersion or polluting gases emitted by industries that cannot be considered processes directionally uniforms, since the affected area by the gases will depend on the changing direction of the winds. In these cases, it is not possible the use of the operator defined buffer, for that reason it is necessary a new operator anisotropic buffer, able to calculate influence areas directionally non uniform. Two methods for the generation or influence areas exist: triangulation of Voronoi [Córcoles00] and Sum of Minkowski [Okabe00]. In this last, one a secondary polygon or generating polygon is defined as when being located on a point or to move on a poliline or polygon generates a surface formed by the points that this generating polygon finds on its way. In the isotropic buffer, the generating polygon is a circle, what implies a constant distance from the border of the buffer to the object. The simulation of anisotropic processes, on the contrary requires the use of generating polygons which determine influence areas directionally non uniform. In this article we will describe a method for the design of generating ovals supposing a normal distribution of the surface around a certain direction or dominant direction. Likewise, the problem that presents the simulation and visualization of anisotropic phenomena expansions is approached, analyzing the existent casuistry for the generation of influence areas calculated by means of the operation Sum of Minkowski. Distribution of Von Misses We can define the circular variables [Batschelet81] as those that represent directions on the plane, being quantified by means of angles that vary from 0 to 2π. One of the most important differences with regard to the lineal variables is that while these can take values of the whole real straight line ( +∞ ,−∞ ) , the circular variables take cyclic values and consequently, the sum or difference of observations can surpass 360º or even give a negative value, being possible in these cases to find an equivalent value belonging to the interval 0-360º. This characteristic allows to the circular variables a different treatment to the lineal ones, by means of the creation of statistics, methods for correlation analysis and specific distributions for this type of variables.

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Conceptually, a circular distribution can be considered like a bivariated lineal distribution where the total probability (or total mass) is dispersed on the circle unit. Therefore, the same as in the bivariated lineal statistic, it exists in circular statistic an mean vector m of module r and mean angle Φ , in whose tip is the mass center C of the distribution (Fig. 1). Let the random variable Z (Φ ) . If we take a monomodal sample of frequencies n 1 , n 2 , ... , n j in the directions Φ 1 , Φ 2 , ... , Φ j , it is defined the mean vector m ( r , Φ ) as

si x > 0  Arctan[ y x ] Φ =  180 + Arctan[ y x ] si x < 0 where x and y are the projections of m on the axes X and Y respectively: r = x 2 + y2

x=

1 N

j

∑ ni Cos Φ i i =1

y=

1 N

j

∑ ni Sin Φ i i =1

(1)

j

N = ∑ ni

(2)

i =1

Figure 1 If the data are contained in intervals of width λ , r should be corrected, being the correct module rc = r ⋅ c , where λ 2 c= ( λ en radianes ) (3) Sin (λ 2) Among the existent circular distributions [Sigmath], one of the most used for the modelization of circular variables is the Von Misses distribution, whose density function for v-modal and symmetric samples is 1 f (Φ ) = Exp [k Cos v( Φ − θ )] (4) 2π I 0 ( k ) where I 0 is the function Bessel of imaginary pure argument of order 0, v is the number or modes and k it is the concentration parameter [Schow78], that indicates on what measure the distribution around the dominant direction θ is concentrated. The samples v-modal should be considered as having extracted of a distribution generated by the overlapping of v monomodal distributions. When the distances between modes are arbitrary, standard methods do not exist to decompose a v-modal sample in v monomodal samples; in the nature the multimodal samples usually show as bimodal and diametrically opposed. In this case, it is possible to reduce the bimodal sample to a monomodal sample, duplicating the angles. With the new angles, the average vector is calculated m 2 ( r2 , Φ 2 ) using the Eq.(1)(2)(3). To obtain the symmetrical modal angle Φ1 of the original sample, we must cancel the effect of the duplication of the angles, being Φ1 = Φ 2 2 ó Φ1 = Φ 2 2 + 180º . For k = 0 , f (Φ ) degenerates in an uniform distribution. Mardia demonstrates [Mardia75] that the maximum likelihood estimation θˆ and ρˆ for the parameters θ and ρ of a Von Misses distribution is respectively Φ and r. Likewise, it is fulfilled I 1 ( kˆ ) =r I 0 ( kˆ ) For that reason the maximum likelihood kˆ is the solution of the Eq.(5). Minkowski Sum Given two images A and B in R 2 , the sum of Minkowski is defined as [Tuwien]

2

(5)


A ⊕ B := U A + b

(6)

b∈B

Where A is the generating polygon, and B the skeleton or primary element (point, poliline, or polygon. A ⊕ B is generated moving A though each element b∈ B , and adding the result of all the translations later on. The translation of the generating polygon A trough the element b ∈ B is defined as

A + b := {a + b, a ∈ A}

(7)

If we take as generating polygon A, a circle and as primary element the group of points B = {( 2,3), (3, 4), (2,5), (1,5)} :

A ⊕ B = [( A + ( 2,3)) U ( A + (3,4) ) U ( A + (2,5)) U ( A + (1,5))]

(8)

The Fig. 2-a shows the result, as well as A ⊕ L and A ⊕ P , additions in those that have taken as primary elements the poliline L and the polygon P respectively.

Figure 2-a.

Figure 2-b.

Conceptually, the sum of Minkowski is a dilation or expansion of the primary image B, whose form is determined by the generating polygon A. In the previous example we have chosen as generating image a circle. The expansion of the primary image is directionally uniform or isotropic, since the generating image is a symmetrical figure with regard to both axes. If on the contrary, we take as generating polygon an oval, the resulting expansion will be directionally non-uniform or anisotropic, having the front its biggest advance in the direction α where the oval presents its maximum polar radius. The Fig. 2-b shows the sums A ⊕ B , A ⊕ L and A ⊕ P being A an oval with α = 45º.

Modelling of the generating polygon The Fig. 3 represents the “spot” that draws the expansion or a fluid that emanates from the point P. Considering that the external forces ( wind, ocean currents, ...) have acted on it in a constant direction, the steps to follow for the design of a function that simulates its behaviour and it allows us to visualise the expansion process are the following ones: a) Sampling of the spot. Fixed the longitude of the arch λ , we have to sample the surface that covers the spot in the j = 2π λ intervals {[ Φ i − λ 2 , Φ i + λ 2) , i = 1.. j} of mark Φ i = λ (i − 1) . A good approximation to the surface of the sector i is s i = ( λ 2 ) ⋅ d i2 , where d i is the distance of the point P to the front of the spot in the direction Φ i . Table 1 shows the values Φ i , d i and si to λ = 30º , being its histogram the one that the Fig. 4 shows. The observation of the histogram provides a first view of the absence/presence of monomodality and symmetry. In the case of asymmetry, the distribution of Von Misses is not valid, being necessary the adjustment by means or some other skew circular function. Figure 3.

Figure 4.

3


Interval [345, 15)

?i 0º

di 40

si 418.879

Interval [165, 195)

?i 180º

di 20

si 104.720

[ 15, 45)

30º

50

654.498

[195, 225)

210º

10

26.1799

[ 45, 75)

60º

60

942.478

[225, 255)

240º

10

26.1799

[ 75, 105)

90º

50

654.498

[255, 285)

270º

15

58.9049

[105, 135)

120º

40

418.879

[285, 315)

310º

20

104.720

[135, 165)

150º

30

235.619

[315, 345) Table 1.

340º

30

235.619

b) Calculation of the mean vector m . Applying the Eqs. (1)(2)(3) to the sample of the Table 1, we obtain, Φ = 59º and rc = 0. 6 . Once calculated m , it is already possible to verify if the variations of the sample are due to the existence of directedness or to random deviations of an uniform expansion. Among the numerous tests of uniformity, the most used by its simplicity and generality is the Rayleigh test [Batschelet78]. The basic idea of this test is that for directionally uniform samples, rc is small; if on the contrary rc is long enough, the null hypothesis H 0 : “The population is distributed uniformly”, can be rejected in favour of the alternative hypothesis of directedness. The statistical of test is z = N ⋅ r 2 = 135 , obtaining significance for z ≥ z (α ) . Being the critical value z ( 0. 05) = 2. 98 , we find more than enough significance, for that reason we reject H 0 in favour of the alternative hypothesis of directedness. c) Obtaining and validation of the density function f (Φ ) . Once obtained rc , we calculate the concentration parameter k ( k = 0 if we assume uniformity). We observe that with the value k = 1 .5 the Eq.(5) is carried out, being the density function of probability:

f (Φ ) = 0.097 ⋅ Exp [ k Cos(Φ − 59º )] (9) A distribution is a model that helps to interpret and to understand a stochastic phenomenon. Whilst a natural phenomenon is always complex, a model is conceptually simple, and consequently, it will never reproduce in an exact way the behaviour of the phenomenon. This fact outlines the necessity to validate the model by means of the quantification of the deviations between the real behaviour of the phenomenon and the behaviour of the model. Statistically, this is reduced to check if the theoretical distribution f (Φ ) is adjusted to the observed values. There are many tests in order to determine the goodness-of-fit, and all theses are in some way restricted. Cox test [Watson76] is designed to test the goodness-of-fit of a sample drawn from Von Misses distribution. It is based on the second trigonometric moments [Cox74] 1 j 1 j ni Cos(Φ i - Φ ) m s = ∑ ni Sin (Φ i - Φ ) (10) ∑ N i =1 N i =1 Under a Von Misses population of parameters Φ and k, these moments are usually distributed (in an approximate way) with means I 2 (k ) I 0 ( k ) and 0 respectively. Here, I m denotes the modified Bessel function or order m. We reject the assumption of a Von Misses distribution if the second trigonometric moments are too far away from these mean values. For the obtained function (Eq. 9), m c = 0. 0172 and m s = 0.013 . Since both momentum approach to their respective average ( I 2 (1. 5) I 0 (1. 5) = 0. 205 y 0), we can assume that the fit is good. mc =

Starting from the density function of probability f (Φ ) , we obtain the distribution function F (α , β , Φ) : β

F (α , β , Φ ) = ∫α f (Φ ) dΦ

(11)

It returns the probability that the random variable is among the angle α and β , and whose values coincides with the area of the bell which is between both angles (Fig. 5-a).

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Figure 5–a.

Figure 5-b.

d) Modelling of the generating polygon. The generating image must be a polygon that distributes the surface around its center according to the density function f (Φ ) . This means that the surface of the bell between two angles should be equal to the surface of the curvilinear sector of the generating polygon limited by these angles. If the border of the polygon possesses as polar ρ = g( Φ ) , all this is translated to the expression

1

β β 2 ∫α f (Φ ) dΦ = 2 ∫α [g (Φ )] dΦ

(12)

From which is deduced that the generating polygon is the oval of polar radius.

ρ = g (Φ ) = 2 f ( Φ ) =

1 πI 0 ( k )

k  Exp  Cos(Φ − Φ ) 2 

(13)

We can observe that for k = 0 the expansion is uniform, being reduced the oval to a circle of radius 1

π . The Fig. 5-b shows the generating oval of the density function Eq.(9).

During the expansion taken place in the period of time t i , the external forces will have a dominant direction θ i and a velocity v i , to which the front of the spot will move. In consequence, at the end of the period t i , the front of the spot will move forward a distance li = vi ⋅ t i . This makes necessary the modelling of generating ovals G (l ,θ ) whose front is at a distance l of the center. From the Eq. 13 it is deduced that the front of the oval is in the direction Φ , since it is the value of Φ that makes maximum ρ , because of that the polar radius of G (l ,θ ) will be determined by the expression

k  g l ( Φ ) = l ⋅ Exp  [Cos(Φ − Φ ) − 1] 2 

(14)

Visualization of the expansion The process of final expansion E, is the result of the accumulation of the partial expansions Ei that are carried out in the periods of times t1 , t 2 ,...,t n : n

n

i =1

i =1

E = U Ei (l i ,θ i ) = U E i (vi ⋅ t i , θ i )

(15)

Starting from a primary image P, the previous expression is equivalent to

E = [...[[P ⊕ G (l1 ,θ 1 )] ⊕ G (l 2 ,θ 2 )] ⊕ ... ⊕ G (l n ,θ n )]

(16)

For the associative property of the sum of Minkowski, the Eq.(16) can be written as:

E = P ⊕ [G (l1 ,θ 1 ) ⊕ G (l 2 ,θ 2 ) ⊕ ... ⊕ G (l n , θ n )] = P ⊕ G t (17) By which the sum of n partial expansion can be interpreted as one only expansion in which the generating polygon is G t of polar radius

5


n k  g t (Φ ) = ∑ li ⋅ Exp  [Cos( Φ − θ i ) − 1] 2   i =1

(18)

θi

Ei

vi

E1 E2 E3 E4

20

59º

10

120º

30

270º

15

180º

Figure 6. Expansion of the point P. The Table shows the values v i y θ i used in each Ei . The used generating polygon (Eq (13)) possesses value k=1.5. The previous considerations have been exposed assuming that the dominant direction and the speed are constant in every period i. This assumption does not fit reality, since as the size of the spot increases, it is more probable than this includes areas with different external forces in the same period of time. The possible cases when in a period i varies v i and/or θ i , it is the following one: Case 1: The expansion begins with different forces that stay constant during the whole period (the primary image covers areas with different values of v i and/or θ i not modified during the period; for that reason the generating polygons present constant values of v i and/or θ i ). Case 2: The expansion begins with an only force that varies during the period (the primary image possesses fixed values of v i and θ i , and the generating polygon finds areas with different values of v i and/or θ i ). Case 3: The expansion begins with different forces that vary during the period (the primary image included areas with different values of v i and/or θ i and the generating polygon find areas with values of v i and/or θ i ). This case supposes the simultaneous occurrence of the cases 1 and 2.

Figure 7.

Figure 8.

Case 1: The Fig. 7 shows the primary polygon of vertexes P = {(5,15), (15,12), ( 20,7), ( 7,4), (5,15)} . The surface in which this is included is divided in cells, each one of which possesses a value of dominant direction and speed or the external forces that act on it. When at least a side of P included in cells or different values exists, the buffer cannot be calculated applying the sum of Minkowski the way it was made in the previous sections. The generation of the influence area requires:

6


a) Segmentation of each side of the image in so many segments like cells occupies that, so that after the division, each one of the resulting segments possesses an only value of v and θ . After this operation the polygon is described as P = {(5,15 ), (11,13.2 ), (15,12 ), (18,9 ), ( 20,7 ), (11,5 ), (7,4 ), ( 6.09,9 )} , after adding the vertexes (11,13.2 ), (18, 9), (11,5 ) and (6.09,9 ) . b) Sum of Minkowski on each one of the segments s j of the image, using as generating polygons in each segment G (l j , θ j ) , where

l j and θ j are the values of velocity per unit of time and dominant

direction in its containing cell c , the sum should be bounded to this cell.

Ei = U

{ [s j ⊕ G(l j , θ j )] ∩ c}

j

(19)

Case 2: It is frequent that during the course of a period the speed and/or direction of the external forces that determine the form of the spot change. Therefore the generating polygon finds areas with different values of v i and/or θ i , what makes necessary the use of so many generating polygons as different pairs ( v i , θ i ) the expansions finds. The most simple case is shown in the Fig. 8. We observe that in the period i = 1 , the expansion of the point P is the result of the expansions in the cell 1 and in the cell 2:

E1 = E1,1 + E1, 2 = Z1 + Z2 + Z3 (20) The exclusive expansion in the cell 1 (area Z1) is take place from the initial moment t = 0 to the moment t = t c , starting form which the expansion is produced in both cells. Simultaneously the expansion in the cell 1 follows the habitual behaviour during the whole period, for that reason assuming t1 as the unit of time, E1,1 = ( P ⊕ G(l1,1 ,θ1,1 )) ∩ cell 1 = Z1 + Z2

(21)

In the cell 2, the expansion will depend on the moment t c :

E1, 2 = ( P ⊕ G(v1,1 , v1, 2 , θ 1,1 , θ 1, 2 , t c )) ∩ cell 2 = Z3 Being P ⊕ G( v1,1 , v1, 2 ,θ 1,1 , θ1,2 , tc ) the generating polygon of polar radius k k g (Φ ) = v1,1 ⋅ t c ⋅ Exp  [Cos(Φ −θ 1,1 ) − 1] + v1,2 ⋅ (1 − tc ) ⋅ Exp  [Cos (Φ − θ1,2 ) − 1] 2  2  The values of the Fig. 8 are shown in the Table 2.

E1,1 E1,2

0 < t ≤ tc P ⊕ G( 25,90º ) = Z1 ----------

(22)

(23)

tc < t ≤1 ( P ⊕ G( 25,90º ) ) ∩ cell 1 = Z2

( P ⊕ G( 25,40,90 º ,90 º ,0.4 )) ∩ cell 2 = Z3 Table

The Fig. 9 shows different examples of variations in the velocity and/or direction of the forces. The expansions in each cell are those given by the Eqs.(21)(22)(23). Although the procedure has been described by didactic reasons taking as primary image a point, this is expandable to primary images of type poliline and polygon. Case 3: As it was already pointed in the section 3, this is the case in which both previous cases are simultaneously produced. For its resolution it is enough to carry out the segmentation described in the case 1 and to apply the case 2 on the new group of points.

Future works Inside the simulation and visualization of anisotropic processes, the authors have open two complementary investigation lines: particularisation of the expansion phenomena to transport phenomena and analysis of the interactions (continuation, creation, dissipation, bifurcation and coalition) among different objects or spots.

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(a)

( b)

(c)

( d) Figure 9

References Batschelet E. 1978 “Second-Order statistical analysis of directions”. En K. Schmidt-Koening y W. Keeton editores: “Animal, migration, navigation and homing”. Springer, Berlin, pp 3-24. Batschelet E. 1981 “Circular statistics in biology”. Academic Press, London. Córcoles, J.E. 2000 “Una alternativa al cálculo de la zona de Influencia 2D en sistemas de Información geográfica”, Proceedings X Congreso Español de Informática Gráfica CEIG’00, pp 217-231. Chou, Y. 1997 “Exploring Spatial Analysis in GIS”, OnWord Press, Santa Fe. Mardia K.V. 1975 “Statistics of directional data”, Journal of Royal Statistical Society 37, pp 349-393. Odyssey-A http://spatialodyssey.ursus.maine.edu/gisweb/spatdb/gis-lis/gi94036.html Odyssey-B http://spatialodyssey.ursus.maine.edu/gisweb/spatdb/gis-lis/gi94052.html Odyssey-C http://spatialodyssey.ursus.maine.edu/gisweb/spatdb/gis-lis/gi94039.html Okabe A., Boots B. and Sugihara K. 2000 "Spatial Tesellations-Concepts and Applications of Voronoi Diagrams". New York, Wiley. Schow G. 1978 “Estimation of the Concentrator Parameter in Von Misses-Fisher distributions”, Biometrika 65, pp 369-375. Sigmath http://www.sigmath.es.osaka-u.ac.jp/~aki/pdf/pdf.html Tuwien http://h50.iue.tuwien.ac.at:8001/vistaa18/node24.html Watson G. and Williams E. 1956 “On the construction of significance test on the circle and the sphere”, Biometrika 43, pp 344-352. Cox D. and Hinkley D.V. 1975 “Theoretical Statistics”. Chapman and Hall, London.

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