Jan Caha - Visibility Analysis on Uncertain Surfaces

Visibility Analysis on Uncertain
Surfaces
Jan Caha
InDOG Conference 2013

Department of Geoinformatics
Palacký University Olomouc

Department of Geoinformatics, Palacký University Olomouc, geoinformatics.upol.cz
Introduction

Visibility calculation

Fuzzy surfaces

Possibilistic visibility

Conclusions

Table of Contents

1 Introduction
2 Visibility calculation
3 Fuzzy surfaces
4 Possibilistic visibility
5 Conclusions

InDOG Conference 2013 - 15.10.2013

2/14

Introduction


Fuzzy surfaces


Conclusions

Introduction

∙ visibility analysis sometimes also referred as viewshed

operation
∙ application in landscape planning, acheology, location of

transmitters and receivers, various ecological applications and
obviously determinations of ideal locations for viewing towers
and hiking trails
∙ uncertainty of the surface is very important because the

calculation of visibility is extremely sensitive to any changes
of surface


3/14

Introduction


Fuzzy surfaces


Conclusions

Surfaces with uncertainty

∙ surfaces always contain some amount of uncertainty
∙ uncertainty can have various sources
∙ usually modelled by statistics and consequences on visibility

are estimated by employing Monte Carlo method
∙ such model captures only relatively specific type of

uncertainty, and is not well suited for situations where the
uncertainty is caused by lack of knowledge
∙ fuzzy surfaces provide better framework for assessing impact

of uncertainty on visibility


4/14

Introduction


Fuzzy surfaces


Conclusions

∙ most of the research on visibility calculation in GIS was

performed by Peter Fisher
∙ several aspects that determine the process and may vary

between implementations: approximation of source and
target point and process of inferring elevations from the
grid
∙ the most important part of the algorithm is determination of

so called Line of Sight (LoS)
∙ the line is formed by points Pi = {1, 2, . . . , n}
∙ each point Pi has and elevation e and distance d from the

viewpoint V
∙ the important is an angle 𝛼i , by their comparison visible

points can be identified

5/14

Introduction


Fuzzy surfaces


Conclusions

Visibility calculation - Calculation of 𝛼i
1

∆h
αi

0

V

Pi

∆d

6/14

Introduction


Fuzzy surfaces


Conclusions


Δh
Δd
∙ point Pm on LoS is visible if 𝛼i < 𝛼m for all m < i, otherwise
the point Pm is invisible from the viewpoint V
∙ 𝛼i = arctan


7/14

Introduction


Fuzzy surfaces


Conclusions

Visibility calculation - LoS

1

0

0

1

2


3

4

5

6

7

8

9

10

8/14

Introduction


Fuzzy surfaces


Conclusions

Fuzzy surfaces

∙ fuzzy surface is surface in which value at the position x , y is

not represented by exact number z but by fuzzy number z
˜
∙ contains uncertainty of the input data and in some cases of

uncertainty that arise from the process of interpolation of
the dataset
∙ allows creation of derived characteristics such are slope,

aspect, profile curvatures and visibility with uncertainty of
the surface propagated to it
∙ requires use of fuzzy arithmetic and possibility theory


9/14

Introduction


Fuzzy surfaces


Conclusions

Visibility on fuzzy surfaces
∙ the ΔH will not be a crisp number but a fuzzy number, Δd

remains crisp number
1

∆hmin

αmin

∆hmax

αmax
0

V

Pi

∆d


10/14

Introduction


Fuzzy surfaces


Conclusions


∙ comparison of fuzzy 𝛼i to determine visibility needs to be

done in the framework of possibility theory
∙ possibility and necessity of exceedance are used to determine

possible and necessary visible parts of the LoS
∙ there are several possible outcomes:
∙
∙
∙
∙

Πi = 𝒩i = 0 → invisible
Πi = 𝒩i = 1 → visible
Πi > 0 and 𝒩i = 0 → possibly visible but not necessary
Πi = 1 and 𝒩i > 0 → possibly and necessary visible but not
absolutely sure


11/14

Introduction


Fuzzy surfaces


Conclusions

3

necessary visibility line
2

C
E
B

A

1

possible visibility line

D
0

0

1

2

3

4

5

Comparison of possible and necessary visibility of points C, D, E from viewpoint A
with respect to the point B


12/14

Introduction


Fuzzy surfaces


Conclusions

Conclusions
∙ concept is extension of the classic viewshed operation for

fuzzy surfaces
∙ Monte Carlo is not necessary correct solution
∙ proposed way to handle vagueness and lack of knowledge

about the surface
∙ obtaining two values - possibility and necessity of visibility

instead of just probability of visibility offers more information
∙ future work should focus on comparison of visibility calculated

using proposed approach and classic statistic methods on LoS
and also on presentation of case studies


13/14

Introduction


Fuzzy surfaces


Conclusions

Thank you for your attention.


14/14

Jan Caha - Visibility Analysis on Uncertain Surfaces

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