Caha - Visibility on a fuzzy surface: A case study

Visibility on a fuzzy surface:A case study

Jan [email protected]

InDOG Conference 2014

Katedra GeoinformatikyUniverzita Palackého v Olomouci

Katedra Geoinformatiky, Univerzita Palackého v Olomouci, geoinformatics.upol.cz

Introduction Fuzzy surface Possibilistic visibility References

Motivation

Far better an approximate answer to the right question,which is often vague, than an exact answer to the wrongquestion, which can always be made precise.

John W. Tukey

InDOG Conference 2014 - 14.10.2014 2/18



Introduction

∙ calculation of visibility on fuzzy surface (representation ofsurface and its uncertainty)

∙ the calculations of visibility with uncertain surfaces were donebefore – Fisher (1994) and Anile et al. (2003), however thesetwo examples use different approaches

∙ focus only on calculation of visibility on the line of sight,the process of inferring the line of sight is described elsewhere(Caha, 2014)

∙ theoretical background for the presented approach are set inmy PhD thesis (Caha, 2014) and presentation from last year’sconference (Caha, 2013)




Fuzzy surface

∙ fuzzy surface is a surface in which value at the position x , yis not represented by exact number z but by fuzzy number z

∙ contains the height of the surface and its uncertainty∙ this uncertainty of the surface is directly propagated to the

derivatives of such surface∙ requires use of fuzzy arithmetic (Kaufmann and Gupta,

1985) and possibility theory (Dubois and Prade, 1986)




Fuzzy surface - case study

∙ for the purpose of the case study used surface based onartificially generated set of points

∙ the surface interpolated by method proposed by Loquin andDubois (2010) that accounts for user’s uncertainty aboutparameters of kriging

∙ based on uncertain values of range, sill and nugget themethod produces fuzzy surface

∙ complete information about the dataset and the process isprovided in Caha (2014)





surface elevation182.8

121.3

km0 0.5 1

The modal value of a fuzzy surface.





1.750

-2.35 0

km0 0.5 1

The differences of a fuzzy surface from the modal value. The difference between theminimal and modal value (left) and the maximal and modal value (right).




Visibility

∙ method to determine which areas of the surface are visiblefrom the given viewpoint

∙ very sensitive on quality of the input data, relatively smalluncertainty can have large impact on result

∙ Fisher (1994) used statistical methods (Monte Carlo) toestimate visibility on surface with uncertainty

∙ Anile et al. (2003) presented visibility calculation on fuzzysurface but the method was actually optimization of visibilityand thus provided far too optimistic estimates of visibility




Possibilistic visibility - terminology

∙ fuzzy visibility - originally used by Fisher (1992)∙ later recognized as incorrect use of the term and described as

probable visibility (Fisher, 1994)∙ in the same article described fuzzy visibility as being

dependant on the distance from viewpoint∙ Anile et al. (2003) used the term fuzzy visibility to describe

visibility on fuzzy surface∙ to avoid the collision with existing terms the term

possibilistic visibility was chosen because the algorithmutilizes possibility theory




Visibility on fuzzy surfaces∙ the difference is in calculation of vertical angle between

viewpoint and points on the line of sight∙ the ΔH will not be a crisp number but a fuzzy number (Δd

is a crisp number)∙ the highest angle is propagated through the line of sight

V Pi

Pid

Pie− V ePiαmin

Piαmax

distance

elev

atio

n




Possibilistic visibility

∙ if a point Pi should be visible then its vertical angle (Pi𝛼)must be higher than vertical angles of all points between thepoint and the viewpoint

∙ simple for crisp numbers, complex problem for fuzzynumbers

∙ with utilization of possibility theory the comparison can bedone with the usage of four indices

∙ indices are taken from the possibility theory (Dubois andPrade, 1986) and they are used for comparison of fuzzynumbers (vertical angles)

∙ indices: possibility of visibility, necessity of visibility,possibility of strict visibility and necessity of strictvisibility




Example - comparison of fuzzy nubmers

0 1 2 30

0.5

1

µA(x)

x

X1 Y

Π X1 ≥ Y

N X1 ≥ Y

X2

Π X2 > Y

N X2 > Y

Comparison of fuzzy numbers X1 and X2 to Y with the four indices visualized.




Example - propagation of maximal angle

distance

elev

atio

n

V P1

P LOS1

N LOS1

P2

P LOS2

P3

P LOS3

P4

The necessary line of sight N LOS1 and the possible lines of sight P LOS1, P LOS2, PLOS3. The example shows how the propagation of a maximal angle affects thepossible line of sight.




Case study results

1 1

00

km0 0.5 1

The possibilistic visibility from the viewpoint (1.8 meter above the surface). Thepossibility (left) and necessity (right) of visibility.




Case study results

1 1

00

km0 0.5 1

The possibilistic visibility from the viewpoint (1.8 meter above the surface). The strictpossibility (left) and strict necessity (right) of visibility.




Conclusions∙ possibility of visibility results in rather large visible area

(optimistic estimate), strict necessity of visibility identifiesrelatively small area (pessimistic estimate)

∙ two remaining indices can be used as a supportiveinformation for reasoning about the possibilistic visibility

∙ outcome provides user with complex information regardingthe visibility by providing four graduated indices instead ofone Boolean value

∙ four indices allow the inconsistency of data to be consideredbecause uncertain data can provide contradictionaryinformation

∙ the possibilistic visibility is better in providing complexassessment of uncertainty in the visibility analysis thanexisting methods




References IANILE, A. M., FURNO, P., GALLO, G., MASSOLO, A. A fuzzy approach to visibility

maps creation over digital terrains. Fuzzy Sets and Systems, 135, 1, s. 63–80, 2003.CAHA, J. Visibility analysis on uncertain surfaces. In: Second InDOG Doctoral

Conference. Proceedings. Univerzita Palackého v Olomouci, Katedrageoinformatiky., 2013.

CAHA, J. Uncertainty Propagation in Fuzzy Surface Analysis. Phd, Palacký Universityin Olomouc, 2014.

DUBOIS, D., PRADE, H. Possibility Theory: An approach to ComputerizedProcessing of Uncertainty. New York : Plenum Press, 1986. ISBN 0-306-42520-3.

FISHER, P. F. 1st Experiments in Viewshed Uncertainty - Simulating FuzzyViewsheds. Photogrammetric Engineering and Remote Sensing, 58, 3, s. 345–352,1992.

FISHER, P. F. Probable and fuzzy models of the viewshed operation. In: WORBOYS,M. (Ed.), Innovations in GIS 1, s. 161–175. Taylor & Francis, 1994.

KAUFMANN, A., GUPTA, M. M. Introduction to Fuzzy Arithmetic. New York : VanNostrand Reinhold Company, 1985. ISBN 044230079.

LOQUIN, K., DUBOIS, D. Kriging with Ill-Known Variogram and Data. In:DESHPANDE, A., HUNTER, A. (Eds.), Scalable Uncertainty Management SE - 5l,6379 / Lecture Notes in Computer Science, s. 219–235. Springer Berlin /Heidelberg, 2010.




Thank you for your attention.


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Caha - Visibility on a fuzzy surface: A case study

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