Region-based approximation of probability distributions (forvisibility between imprecise points among obstacles)Citation for published version (APA):Buchin, K., Kostitsyna, I., Löffler, M., & Silveira, R. I. (2015). Region-based approximation of probabilitydistributions (for visibility between imprecise points among obstacles). In U. Brandes, & D. Eppstein (Eds.), Proc.17th Workshop on Algorithm Engineering and Experiments (ALENEX) (pp. 94-103). Society for Industrial andApplied Mathematics (SIAM). https://doi.org/10.1137/1.9781611973754.9

DOI:10.1137/1.9781611973754.9

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Region-based Approximation Algorithms

for Visibility between Imprecise Locations

Kevin Buchin∗ Irina Kostitsyna∗ Maarten Loffler† Rodrigo I. Silveira‡

Abstract

In this paper we present new geometric algorithms for

approximating the visibility between two imprecise locations

amidst a set of obstacles, where the imprecise locations

are modeled by continuous probability distributions. Our

techniques are based on approximating distributions by a

set of regions rather than on approximating by a discrete

point sample. In this way we obtain guaranteed error bounds,

and the results are more robust than similar results based

on discrete point sets. We implemented our techniques and

present an experimental evaluation. The experiments show

that the actual error of our region-based approximation

scheme converges quickly when increasing the complexity

of the regions.

1 Introduction

Visibility has attracted much attention in geographicinformation science due to its many applications. Thequestion of whether two objects can see each otheramidst a group of obstacles arises naturally in plentyof situations. A basic operation in visibility analysis iscomputing the space visible from a certain location in alandscape, where the obstacles occluding vision originatefrom the terrain topography, and sometimes also fromthe vegetation and buildings in the surroundings [3, 8, 19,23, 31, 32]. The portion of a terrain that is visible froman observation point is known as the viewshed of thepoint. The equivalent concept for urban environments isusually called isovist [30]. For an overview on algorithmsto compute viewsheds and other visibility structures, seethe survey by De Floriani and Magillo [11]. Visibilityand viewsheds also play a role in many other areas, suchas animal behaviour studies. For instance, Aspbury andGibson [1] use viewsheds to study grouse behavior in alek, the area in which they perform their mating displays.

∗Dept. of Mathematics and Computer Science, TU Eindhoven;

{k.a.buchin, i.kostitsyna}@tue.nl.†Department of Computing and Information Sciences, Utrecht

University; m.loffler@uu.nl.‡Dept. de Matematica & CIDMA, Universidade de Aveiro,

and Dept. Matematica Aplicada II, Universitat Politecnica deCatalunya; rodrigo.silveira@ua.pt.

Camp et al. [28] use viewsheds of eagle nest locations todetermine human disturbance of nesting activities.

The focus of the research presented in this paper isthe study of visibility under location uncertainty.

In the GIS literature, it has long been recognizedthat data acquired with physical devices are subjectto uncertainty (from measurement and sampling errors,among several others) [10, 22], and that this is particu-larly relevant for location data of moving objects [27, 29].In this context, a widespread way to model an uncer-tain location—especially in the area of spatiotemporaldatabases—is by assuming that the exact location ofan object is at distance at most d from its estimatedlocation. This is equivalent to representing the object’slocation with a disk of radius d, centered at the esti-mated location: the uncertainty region. In addition,it is often assumed that the exact object location fol-lows a probability density function, defined as 0 for anypoint outside of the uncertainty region. Even thoughthe choice of the probability density function dependson the application, most previous work uses uniform andGaussian distributions (e.g., [6, 7, 27, 29]). In particular,Gaussian distributions have been shown to be appro-priate to model the error in locations obtained fromGPS [34] and to model the error in location betweenmeasurements [15]. To the best of our knowledge, previ-ous work on uncertain visibility has focused on modelingthe uncertainty present in the terrain model, usuallyby using some type of error propagation analysis basedon Monte Carlo simulation (e.g., [9, 24]). While bothvisibility and uncertainty have been extensively studiedin GIS, we are not aware of research combining visibilityand location uncertainty.

Another field where both visibility and data uncer-tainty have been studied extensively is computationalgeometry. A lot of attention has been devoted to thedesign of efficient algorithms for computing visibilityregions of points in environments like polygons [16] orterrains [18]. Multiple variants and extensions of visibil-ity concepts have also been studied. For surveys of thevast literature on this topic, we refer to Ghosh [12] andDe Floriani and Magillo [11].

In the computational geometry literature, various

Figure 1: Two pairs of similar point sets on opposite sidesof a collection of obstacles. The green points can all see eachother, whereas none of the blue points can.

models to deal with (spatial) data uncertainty have beensuggested. Motivated by finite coordinate precision,Guibas et al. [13] have considered uncertainty regionsfor points. Similar concepts have been later studiedin a variety of settings [2, 14, 20, 25, 26]. Most of theprevious work on uncertainty in computational geometrydoes not consider uncertainty modeled by probabilitydistributions.

Performing geometric algorithms on probability dis-tributions directly is often computationally intractable.Instead, the distributions can be approximated. A popu-lar approach is to describe a distribution by a point set.For instance, for tracking imprecise objects a particlefilter uses a discrete set of locations to model uncer-tainty [33]. Loffler and Phillips [21] and Jørgenson etal. [17] discuss several geometric problems on pointswith probability distributions, and show how to solvethem using discrete point sets that have guaranteed errorbounds.

However, when combining visibility queries withpoint-based approximations of probability distributions,we may lose any theoretical guarantees on the computedprobabilities. The choice of points may greatly influencethe resulting probability, as illustrated in Figure 1.Instead, we propose to approximate distributions byregions.

Contribution In this paper we present a newmethod to approximate the probability that two impre-cise points can see each other. We consider an abstractsetting in which the imprecise points are modeled byprobability distributions, and the visibility is constrainedby a set of polygonal obstacles. Our techniques are basedon using regions—rather than point samples—to approx-imate this probability, allowing to compute probabilitiesguaranteed to any desired accuracy. We have imple-mented and experimentally evaluated our method. Weshow experimentally that, in the settings considered, ourtechniques are more robust and reliable than those basedon point sampling, and that the empirical error of ourapproximation reduces quickly with the complexity ofthe regions used in the approximation.

Finally, even though in this paper we use theterm visibility, our results also apply to other contexts.

For example, in sensor networks visibility problemsare formulated as coverage problems. This type ofmonitoring can be performed by radars, antennas,routers and basically any device that is able to sendor receive some sort of wireless signal. Each of thesedevices can be placed almost anywhere on a terrain, socoverage is the unit that measures the quality of thechosen device placement scheme. Numerous problemsrelated to visibility arise in this area. We refer to thesurvey by Yick et al. [35] for a more detailed treatmentof the subject.

2 Modeling

In this section we explain how we model the situation oftwo imprecise points amid a collection of obstacles. Wewant to approximate the probability distributions thatgovern the locations of the imprecise points by sets ofuniform distributions. Here we only give an overview ofthe technical details; we refer the reader to the technicalsupplement of this paper [5] for more details and proofs.

Assume that an imprecise point p in the plane isgiven by a two-dimensional probability distribution µ.Let M be a set of weighted regions in the plane, andlet w(M) denote the uniform weight of a region M ∈M.Let M(p) = {M ∈ M | p ∈ M} be the subset of Mcontaining a point p ∈ R2. A set M defines a functionm(p) =

∑M∈M(p) w(M) that sums the weights of all

regions containing p. We say that M ε-approximatesµ if the symmetric difference of the volumes under mand µ is at most ε; that is, if

∫p∈R2 |µ(p)−m(p)| ≤ ε.

Figure 2(a) illustrates the concept.Any probability distribution µ can be approximated

in this way, but the total complexity ofM, i.e., the sumof the complexities of each of its regions, depends onvarious factors: the shape of µ, the shape of allowedregions in M, and the error parameter ε. To focusthe discussion, in this work we limit our attention toGaussian distributions, since they are natural and, asmentioned in the introduction, have been shown to beappropriate for modeling the uncertainty in commonly-used types of location data [15, 34].

2.1 Approximation with Disks A natural way toapproximate a Gaussian distribution by using a set ofregions is by using concentric disks. Thus, given aGaussian probability distribution µ, and a maximumallowed error ε, we would like to compute a set M ofk disks that ε-approximate µ. We may assume µ iscentered at the origin, leaving only a parameter σ thatgoverns the shape of µ, that is,

µ(x, y) =1

2πσ2e−

x2+y2

2σ2 ,

(a)

r1 r2 rkρ2 ρkρ1=0

(b)

Figure 2: (a) A probability density function µ (yellow) can be approximated by a set of weighted regions M, representinga function m (purple). (b) A Gaussian distribution (green) is ε-approximated by a step function (red), which is representedby k weighted disks.

or in polar coordinates,

µ(r, θ) =1

2πσ2e−

r2

2σ2 .

Function µ does not depend on θ, therefore, in thefollowing, we will omit it and write µ(r) for brevity.

We are looking for a set of radii r1, . . . , rk andcorresponding weights w1, . . . , wk such that the set ofdisks centered at the origin with radii ri and weights wi ε-approximate µ. We use these disks to define a cylindricalstep function. Figure 2(b) shows a 2-dimensional cross-section of the situation. Minimizing the volume betweenthe step function an µ, we obtain the following lemma:

Lemma 2.1. Let µ be a Gaussian distribution withstandard deviation σ. Let k be a given integer. Then theminimum-error approximation of µ by k disks is givenby

(2.1)

ri =

√2σ2 log

k(k+1)

(k− i+1)2,

wi =1

2πσ2

(k− i+1)(k− i+2)

k(k+1),

where i ∈ (1, . . . , k).

Proof. Let V (ρ) be the volume under the probabilitydistribution µ outside the disk of radius ρ:

V (ρ) =

∫∫D

µ(x, y)dxdy =

2π∫0

∞∫ρ

µ(r, θ)rdrdθ

=

∞∫ρ

r

σ2e−

r2

2σ2 dr = e−ρ2

2σ2 .

Then the symmetric difference between the functionµ and the cylindrical step function defined by k disksof radii r1, r2, . . . , rk and k weights w1, w2, . . . , wk is afunction given by the following formula:

F = w1(πr21−πρ21)−(V (0)−V (r1))

+V (r1)−V (ρ2)−w2(πρ22−πr21)

+w2(πr22−πρ22)−(V (ρ2)−V (r2))

+ ...

+wk(πr2k−πρ2k)−(V (ρk)−V (rk))

+V (rk) ,

(2.2)

where 0< r1 < r2 < · · ·< rk are the radii of the disks ap-proximating the distribution function µ, w1, w2, . . . , wkare the corresponding weights, and 0 = ρ1<ρ2< · · ·<ρkare the intersections of the step function with µ (refer toFigure 2(b)). After minimizing function F over all ri’sand wi’s we attain Formulas 2.1. For the details refer tothe technical supplement of this article [5]. �

Since the error allowed ε is given, we can solveEquation 2.2 for ε. This leads to the following result.

Theorem 2.1. Let µ be a Gaussian distribution withstandard deviation σ. Given ε> 0, we can ε-approximateµ by a set of

k =

⌈1

eε−1

⌉=O(1/ε)

weighted disks.

Proof. When minimizing function F given by For-mula 2.2, we obtain the following dependency for ρi

Figure 3: A Gaussian distribution, given by isolines at εlevels (red), and a set of polygons that can be used to obtainan additive approximation (blue).

and ri:

2ρ2i = r2i−1 +r2i ,

2e−r2i2σ2 = e−

ρ2i2σ2 +e−

ρ2i+1

2σ2 .

Using these equations and Lemma 2.1, the formula 2.2for function F can be simplified to

F =r21

2σ2= log

k+1

k.

Function F gives the error of approximating the distri-bution function µ by the set of disks:

ε= F = logk+1

k,

and thus,

k =

⌈1

eε−1

⌉=O

(1

ε

).

�

2.2 Approximation with Polygons The curvedboundaries of disks make geometric computations morecomplicated. Therefore, next we consider approximatingµ by a set of polygons. Computing a set of polygons ofminimum total complexity is a challenging mathematicalproblem that we leave to future investigation. However,we can easily obtain a set of polygons at most twiceas large as the minimum, by first computing a set ofk disks with guaranteed error ε and then choosing 2kregular polygons that stay within the annuli definedby those disks. Figure 3 illustrates this idea; since therelative widths of the annuli change, polygons of differentcomplexity are used for different annuli.

Knowing the widths of the annuli we can calculatethe total complexity of the approximation.

Theorem 2.2. A Gaussian distribution with standarddeviation σ can be ε-approximated by O(1/ε) polygonsof complexity O(1/

√ε) each.

Proof. First, we compute a set of k=

⌈1

eε−1

⌉concentric

disks by Formulas 2.1 that approximate the distributionfunction µ with guaranteed error ε. For each disk withradius ri we find two radii r′i and r′′i from the followingequations:

µ(r′i) =1

2(µ(ρi)+µ(ri)) ,

µ(r′′i ) =1

2(µ(ri)+µ(ρi+1)) .

Then we choose 2k regular polygons that stay withinannuli defined by pairs of radii {r′i, ri} and {ri, r′′i } withweights wi and (wi+wi+1)/2 correspondingly. These2k polygons ε-approximate the probability distributionfunction µ. Refer to the technical supplement of thisarticle [5] for the detailed proof of that fact.

The complexity of a regular polygon inscribed in anannulus depends only on the ratio of the radii. Thatis, given an annulus with inner radius r′ and outerradius r, we can fit a regular dπ/ arccos r

′

r e-gon in it.Similarly, given an annulus with inner radius r and outerradius r′′, we can fit a regular dπ/ arccos r

r′′ e-gon. Refer

to [5] for the proof that dπ/ arccos r′

r e = O( 1√ε) and

dπ/ arccos rr′′ e=O( 1√

ε). �

2.3 The environment Now consider a set of obsta-cles R in the plane. We assume that the obstacles aredisjoint convex polygons with m vertices in total (non-convex obstacles can always be first decomposed intoconvex pieces). We approximate two imprecise pointswith probability distributions µ1 and µ2 with two setsof weighted regions M1 and M2, each consisting of reg-ular polygons. For every pair of polygons P1 ∈M1 andP2 ∈M2, we compute the probability that a point p1chosen uniformly at random from P1 can see a point p2chosen uniformly at random from P2. We say that twopoints can see each other if and only if the straight linesegment connecting them does not intersect any obstaclefrom R.

The probability that two points p1 = (x1, y1) andp2 = (x2, y2) chosen uniformly at random from P1 andP2, respectively, can see each other can be computed bythe formula:

(2.3) prob=

∫∫∫∫v(x1, y1, x2, y2)dx1dy1dx2dy2∫∫∫∫

dx1dy1dx2dy2,

where v(x1, y1, x2, y2) is 1 if the points see each other,and 0 otherwise.

ℓ2

ℓ1

ℓ3

ℓ5ℓ4

s1

s2s3

s4

(a)

ℓ*

ℓ*

ℓ*

ℓ2

ℓ3

ℓ* ℓ 4ℓ 5

ℓ*ℓ1

(b)

Figure 4: (a) Two polygons P1 and P2 in primal space. The orange region represents the set of lines intersecting P1

and P2 through s1, s2, s3, s4. (b) Partition L∗ in dual space. The orange cell corresponds to all lines in the primal spaceintersecting s1, s2, s3, and s4.

ℓ1ℓ2

ℓ3

ℓ4

(a)

ℓ*ℓ1

ℓ*ℓ2

ℓ*ℓ3

ℓ*ℓ4

(b)

Figure 5: (a) Primal space: polygons P1 and P2, and an obstacle between them. (b) Dual space: the “hourglass” shapeH∗ (shown gray) in the dual space that corresponds to a set H of all lines in the primal space that intersect the obstacle.

To compute prob we consider a dual space wherea point with coordinates (α, β) corresponds to a liney = αx−β in the primal space (see [4] for a detailedexposition of geometric point-line duality). Our firstgoal is to decompose this dual space into a set of simpleregions (called cells in the following) with the propertythat, for all lines in primal space represented in a cell,we can compute their contribution to prob in constanttime.

For this, we first construct a region L∗ in the dualspace that corresponds to the set L of lines that stab(i.e., intersect) both P1 and P2. This region can bepartitioned into cells, each one corresponding to the setof all lines that cross the same four segments of P1 andP2 (refer to Figure 4(a)). The complexity of the resultingsubdivision is summarized in the following lemma:

Lemma 2.2. Given two convex polygons P1 and P2 oftotal size n, the complexity of the partition L∗ in thedual space, which corresponds to the set of lines L thatstab P1 and P2, is O(n2).

Proof. Each vertex of L∗ corresponds to a line in primalspace through two vertices of P1 and P2. Therefore, thetotal complexity of L∗ is O(n2). �

Next we need to extend the subdivision in order totake obstacles into account. To that end, we constructfor each obstacle h∈R a region H∗ in dual space, whichcorresponds to the set of all lines that intersect h. Sinceobstacles are convex, H∗ has an “hourglass” shape (seeFigure 5). We now compute the subdivision of the dualplane resulting from overlaying the partition L∗ and theregions H∗. The cells of this overlay now provide thedecomposition of the dual space that we need, allowingto distinguish the (primal space) lines that stab P1 andP2, but no obstacle.

Lemma 2.3. Given two polygons P1 and P2 of total sizen and obstacles of total complexity m, we can computethe probability that a pair of points drawn uniformly atrandom from P1×P2 can see each other in O((m+n)2)time, assuming that we can compute the visibility withineach cell of the overlay of L∗ and H∗ in constant time.

Proof. The polygons P1 and P2, and the obstacles arebounded by a total of O(m+n) line segments in primalspace, thus the overlay has complexity O((m+n)2). Ifthe visibility can be computed in constant time withineach cell of the arrangement in the dual space, it takesO((m+n)2) total time to compute the probability of

(a) (b)

Figure 6: (a) An example instance in the forest setting. (b) An example instance in the urban setting with the buildingsof the old centre of Utrecht as obstacles.

visibility between a pair of points chosen uniformly atrandom from P1×P2. �

It remains to show that we can compute the visibilitywithin every cell of the overlay in constant time. In thefollowing we show that this is indeed the case, that is, weprovide the steps necessary to calculate how much thelines of a given cell contribute to prob. For simplicity ofpresentation, we assume that P1 and P2 are separable bya vertical line, and that P1 and P2 are disjoint from R.This will allow us to write the solution in a more conciseway without loss of generality. In case that P1 and P2

cannot be separated by a vertical line, the convexity of P1

and P2, together with their disjointness, guarantee thatwe can find a non-vertical separating line, and rotate thescene to make the line vertical.

Consider line `, given by the formula y = αx−β,which goes through points p1 = (x1, y1) ∈ P1 and p2 =(x2, y2) ∈ P2. In dual space, point `∗, corresponding toline `, has coordinates (α, β). Substitute variables y1and y2 in Formula 2.3 with α and β: (x1, y1, x2, y2)←(x1, α, x2, β), where α(x1, y1, x2, y2) = y2−y1/x2−x1and β(x1, y1, x2, y2) = (x1y2−x2y1)/(x2−x1). We canexpress the probability of two points, distributed uni-formly at random in P1 and P2, seeing each other as

(2.4) prob=

∫∫∫∫v(x1, α, x2, β)|J |dx1dx2dαdβ∫∫∫∫

|J |dx1dx2dαdβ,

where

J = det

[dy1dα

dy1dβ

dy2dα

dy2dβ

]=

1

det

[dαdy1

dβdy1

dαdy2

dβdy2

] = x2−x1 .

The expression in Equation 2.4 can be solved analytically.We defer the tedious details to the technical supplementof this article [5].

Theorem 2.3. Given two convex polygons P1 and P2 oftotal size n and a set of obstacles of total size m, we cancompute the probability that a point p1 chosen uniformlyat random in P1 sees a point p2 chosen uniformly atrandom in P2 in O((m+n)2) time.

Combining Theorems 2.2 and 2.3, the main resultof this section follows:

Theorem 2.4. Given two imprecise points, modeledas Gaussian distributions µ1 and µ2 with standarddeviations σ1 and σ2, and obstacles of total complexitym, we can approximate the probability that p and q seeeach other with an error at most ε in O(1/ε3 +m2/ε2)time.

3 Experiments

We implemented our method in C++ using the Com-putational Geometry Algorithms Library (CGAL), andtested it on synthetically generated and real data.

3.1 Setup We generated several random scenarios,each consisting of two imprecise points modeled asGaussian distributions and approximated by sets oflayered regions, as explained in Section 2. Thesetwo Gaussian distributions are parameterized by σ,their standard deviation, while the approximationsare parameterized by ε, the desired accuracy of theapproximations.

●●●●●●●●●●

●●●●●●●●●●

●

0 1000 2000 3000 4000n0.00

0.05

0.10

0.15

0.20

ε

Figure 7: The error (ε) that our method guaranteesas a function of the total complexity of the approxi-mation (n) in a general setting.

●

●

●●

●● ●

0 100 200 300 400 500 600n0.15

0.16

0.17

0.18

0.19

0.20p

Figure 8: The probability of visibility as computedby our method (for the forest setting presented inFigure 6(a)), as a function of the complexity of theapproximation (n).

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2·105 4·105 6·105 8·105(n+m)2

5·10-610-5

1.5·10-52·10-5

2.5·10-53·10-5

p

(a)

2·104 4·104 6·104 8·104 105n

2·10-54·10-56·10-58·10-510-4

p

(b)

Figure 9: The probability of visibility in the configuration of Figure 6(b). (a) As calculated by our method. The probabilityis shown as function of the value (n+m)2. Different runs result in slightly different measurements, due to randomnessin the orientation. (b) As estimated by sampling uniformly at random pairs of points from the two distributions. Theprobability is shown as function of the number of pairs sampled. The plot shows the function for three independent runs ofthe algorithm.

For the obstacles, we distinguish two settings.

forest: A set of 300 randomly generated obstacles, eachrepresenting the trunk of a tree, modeled as a regularpentagon, in a 100m×100m domain. Figure 6(a)shows an example.

urban: A set of 24 footprints of real buildings, repre-senting a section of the old city center of Utrecht,spanning a 550m×350m region. Figure 6(b) showsan example.

The two settings were chosen as to illustrate twonatural, but clearly distinct, environments. The forestsetting showcases a large number of obstacles of smallsize, whereas the urban setting contains just a few largerobstacles, which create a maze-like domain.

We carried out a number of different experiments to,firstly, investigate how the region-based model changeswhen the parameters ε and σ vary, and secondly, tocompare the region-based model to the point-basedapproach.

3.1.1 Goals of the experiments We begin our ex-perimental evaluation with an analysis of the fundamen-tal properties of our region-based approximation scheme.For this we first study the effect of changing the parame-ter of the approximation, specifically ε. We expect thatthe true value of the error of the visibility, calculatedby our method, will be significantly smaller than ourtheoretical upper bounds. Thus, in applications wherea provable worst-case error bound is not required but

an empirical error bound suffices, the value ε of theprovable error bound does not need to be very smalland can be adjusted accordingly. We also evaluate thedependency of the visibility on the standard deviation σof the distributions.

With a suitable ε in place, next we want to comparethe point-based and region-based approximations of theprobability distributions of the imprecise points. Forthis we study the convergence behavior of point-basedmethods. We also investigate the dependency of thevisibility on σ. We want to know how our method worksat different levels of precision of the coordinates of ourentities.

In summary, our experiments address the followingquestions:

1. How do the results depend on various parameters?Our parameters are

• ε: the approximation quality, and

• σ: the standard deviation of the probabilitydistributions.

2. How does the region-based approach compare to asimple point-based approach, where we approximatethe probability by sampling a large number of pairsof points from the two distributions? Particularly,we wish to know

• whether the region-based and point-basedapproaches give consistent results, that is,whether they yield similar probabilities in thelimit case, and

• how many samples in the point-based approachare needed before we get a reasonably stableapproximation.

3.2 Results In the first experiment we evaluated thetotal size n of the approximation regions depending on ε.Figure 7 shows the guaranteed error as a function of thetotal complexity of the approximation. The plot exhibitsthe inverse behavior of Theorem 2.2, as expected, andshows the constants as they occur in practice.

Secondly, knowing the dependency between n, thesize of approximation, and ε, we plot the computedvisibility as a function of n (for the forest settingpresented in Figure 6(a)). We can see in Figure 8 thatthe visibility seems to stabilise around ≈ 0.185. Thisindicates that the true error, as expected, is much lowerthan the theoretically guaranteed error that our methodprovides. More specifically, we observe that the empiricalerror is already less than 0.03 for n= 26.

Figure 10 shows how the computed visibility varieswith σ in the same forest setting. We observe that the

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■■ ■ ■ ■

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◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆

0 1 2 3 4 5σ

0.10

0.12

0.14

0.16

0.18

0.20

p

Figure 10: The probability of visibility as computed byour method, for various values of the standard deviationof the distributions (σ) in the forest setting (presented inFigure 6(a)). Three lines represent three different ε values,0.05 (green), 0.1 (orange), and 0.2 (purple).

probability of visibility grows fast while σ is less than 2.5,and then stabilizes with a slow steady grow, which canbe explained by the fact that in the particular settingof the Figure 6(a) there are a number of trees blockingthe visibility between the centers of the distributions.Furthermore, we again observe that the dependency ofthe result on ε is very low, due to the fact that theempirical error is much lower than the provable errorbound.

To compare our method with the point-basedmethod we iteratively sampled the distributions in theurban setting (Figure 6(b)). Figure 9(a) shows theprobability of visibility as a function of (n+m)2, whichis in the order of the complexity of the partition in thedual space. Figure 9(b) shows the estimated probabilityof the point-based method as a function of the numberof samples. Since the probability of visibility in thisparticular setting is very small, it takes about 20,000samples before the point-based method even detects thatthe probability is non-zero. In comparison, our methodestimates a non-zero probability even for the smallestcomplexity (ε=0.5; n=44). Furthermore, we see thatthe probability converges very slowly for the point-basedmethod. Additionally, the experiments provide empiricalevidence of the fact that the point-based approach notonly results in a relatively slow convergence rate forthe error, but also that any guarantee on the errorwould need to come with a considerable probability offailure. In contrast, our region-based approach producesresults with error guaranteed to be smaller than theinput parameter ε. Again the empirical error of theregion-based approach is significantly lower than theguaranteed error: for ε=0.05 the approach computesa probability of 0.000032, which seems to only slightlyunderestimate the actual probability.

4 Conclusions

In this paper we presented geometric techniques thatprovide a sound estimation of the probability of visibilitybetween imprecise points amidst a group of obstacles.These techniques are based on approximating theseprobability distributions by sets of polygons. Thistype of approximation allowed us to give guarantees onhow closely the computed probabilities approximate theactual probabilities. Our experiments provide evidencethat these techniques provide strong bounds, and inthe settings tested give better results than point-basedapproximation techniques.

Given that the empirically obtained error is muchsmaller than the analytically obtained upper bound onthe error, an interesting open problem is whether wecan formulate conditions under which tighter theoreticalguarantees on the error can be proven. Furthermore,with the region-based approximation proven successfulin the setting of visibility, we plan to investigate othergeometric problems on imprecise points for which theregion-based approximation of probability distributionsresults in faster geometric approximation algorithms.

Acknowledgments

Kevin Buchin, Irina Kostitsyna, and Maarten Loffler aresupported by the Netherlands Organisation for ScientificResearch (NWO) under grant no. 612.001.106, 612.001.207,and 639.021.123, respectively. Rodrigo Silveira was fundedby Portuguese funds through CIDMA and FCT, withinproject PEst-OE/MAT/UI4106/2014, and by FCT grantSFRH/BPD/ 88455/2012. In addition, Rodrigo Silveirawas partially supported by projects MINECO MTM2012-30951/FEDER, Gen. Cat. DGR2014SGR46, and by ESFEUROCORES program EuroGIGA-ComPoSe IP04-MICINNproject EUI-EURC-2011-4306.

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