Presentation from Third InDOG Doctoral Conference in Olomouc, Czech Republic. 13. - 16. October 2014

1. Visibility on a fuzzy surface:
A case study
Jan Caha
jan.caha1@vsb.cz
InDOG Conference 2014
Katedra Geoinformatiky
Univerzita Palackého v Olomouci

2. 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 wrong
question, which can always be made precise.
John W. Tukey
InDOG Conference 2014 - 14.10.2014 2/18

3. Katedra Geoinformatiky, Univerzita Palackého v Olomouci, geoinformatics.upol.cz
Introduction Fuzzy surface Possibilistic visibility References
Introduction
∙ calculation of visibility on fuzzy surface (representation of
surface and its uncertainty)
∙ the calculations of visibility with uncertain surfaces were done
before – Fisher (1994) and Anile et al. (2003), however these
two 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 in
my PhD thesis (Caha, 2014) and presentation from last year’s
conference (Caha, 2013)
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4. Katedra Geoinformatiky, Univerzita Palackého v Olomouci, geoinformatics.upol.cz
Introduction Fuzzy surface Possibilistic visibility References
Fuzzy surface
∙ fuzzy surface is a surface in which value at the position x, y
is 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)
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5. Katedra Geoinformatiky, Univerzita Palackého v Olomouci, geoinformatics.upol.cz
Introduction Fuzzy surface Possibilistic visibility References
Fuzzy surface - case study
∙ for the purpose of the case study used surface based on
artificially generated set of points
∙ the surface interpolated by method proposed by Loquin and
Dubois (2010) that accounts for user’s uncertainty about
parameters of kriging
∙ based on uncertain values of range, sill and nugget the
method produces fuzzy surface
∙ complete information about the dataset and the process is
provided in Caha (2014)
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6. Katedra Geoinformatiky, Univerzita Palackého v Olomouci, geoinformatics.upol.cz
Introduction Fuzzy surface Possibilistic visibility References
Fuzzy surface - case study
surface elevation
182.8
121.3
0 0.5 1 km
The modal value of a fuzzy surface.
InDOG Conference 2014 - 14.10.2014 6/18

7. Katedra Geoinformatiky, Univerzita Palackého v Olomouci, geoinformatics.upol.cz
Introduction Fuzzy surface Possibilistic visibility References
Fuzzy surface - case study
0 0.5 1 km
0 1.75
-2.35 0
The differences of a fuzzy surface from the modal value. The difference between the
minimal and modal value (left) and the maximal and modal value (right).
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8. Katedra Geoinformatiky, Univerzita Palackého v Olomouci, geoinformatics.upol.cz
Introduction Fuzzy surface Possibilistic visibility References
Visibility
∙ method to determine which areas of the surface are visible
from the given viewpoint
∙ very sensitive on quality of the input data, relatively small
uncertainty can have large impact on result
∙ Fisher (1994) used statistical methods (Monte Carlo) to
estimate visibility on surface with uncertainty
∙ Anile et al. (2003) presented visibility calculation on fuzzy
surface but the method was actually optimization of visibility
and thus provided far too optimistic estimates of visibility
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9. Katedra Geoinformatiky, Univerzita Palackého v Olomouci, geoinformatics.upol.cz
Introduction Fuzzy surface Possibilistic visibility References
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 algorithm
utilizes possibility theory
InDOG Conference 2014 - 14.10.2014 9/18

10. Katedra Geoinformatiky, Univerzita Palackého v Olomouci, geoinformatics.upol.cz
Introduction Fuzzy surface Possibilistic visibility References
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
˜ P˜ie − V˜ e Pi
min
˜ Pi
max
distance
elevation
InDOG Conference 2014 - 14.10.2014 10/18

11. Katedra Geoinformatiky, Univerzita Palackého v Olomouci, geoinformatics.upol.cz
Introduction Fuzzy surface Possibilistic visibility References
Possibilistic visibility
∙ if a point Pi should be visible then its vertical angle (Pi훼)
must be higher than vertical angles of all points between the
point and the viewpoint
∙ simple for crisp numbers, complex problem for fuzzy
numbers
∙ with utilization of possibility theory the comparison can be
done with the usage of four indices
∙ indices are taken from the possibility theory (Dubois and
Prade, 1986) and they are used for comparison of fuzzy
numbers (vertical angles)
∙ indices: possibility of visibility, necessity of visibility,
possibility of strict visibility and necessity of strict
visibility
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12. Katedra Geoinformatiky, Univerzita Palackého v Olomouci, geoinformatics.upol.cz
Introduction Fuzzy surface Possibilistic visibility References
Example - comparison of fuzzy nubmers
X˜1 ˜ Y
X˜1 ˜ Y
0 1 2 3
1
0.5
0
μA˜(x)
x
N X˜1 ˜ Y
X˜2
X˜2 Y˜
N X˜2 Y˜
Comparison of fuzzy numbers ˜X
1 and ˜X
2 to ˜Y
with the four indices visualized.
InDOG Conference 2014 - 14.10.2014 12/18

13. Katedra Geoinformatiky, Univerzita Palackého v Olomouci, geoinformatics.upol.cz
Introduction Fuzzy surface Possibilistic visibility References
Example - propagation of maximal angle
distance
elevation
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, P
LOS3. The example shows how the propagation of a maximal angle affects the
possible line of sight.
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14. Katedra Geoinformatiky, Univerzita Palackého v Olomouci, geoinformatics.upol.cz
Introduction Fuzzy surface Possibilistic visibility References
Case study results
0 0.5 1 km
1 1
0 0
The possibilistic visibility from the viewpoint (1.8 meter above the surface). The
possibility (left) and necessity (right) of visibility.
InDOG Conference 2014 - 14.10.2014 14/18

15. Katedra Geoinformatiky, Univerzita Palackého v Olomouci, geoinformatics.upol.cz
Introduction Fuzzy surface Possibilistic visibility References
Case study results
0 0.5 1 km
1 1
0 0
The possibilistic visibility from the viewpoint (1.8 meter above the surface). The strict
possibility (left) and strict necessity (right) of visibility.
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16. Katedra Geoinformatiky, Univerzita Palackého v Olomouci, geoinformatics.upol.cz
Introduction Fuzzy surface Possibilistic visibility References
Conclusions
∙ possibility of visibility results in rather large visible area
(optimistic estimate), strict necessity of visibility identifies
relatively small area (pessimistic estimate)
∙ two remaining indices can be used as a supportive
information for reasoning about the possibilistic visibility
∙ outcome provides user with complex information regarding
the visibility by providing four graduated indices instead of
one Boolean value
∙ four indices allow the inconsistency of data to be considered
because uncertain data can provide contradictionary
information
∙ the possibilistic visibility is better in providing complex
assessment of uncertainty in the visibility analysis than
existing methods
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17. Katedra Geoinformatiky, Univerzita Palackého v Olomouci, geoinformatics.upol.cz
Introduction Fuzzy surface Possibilistic visibility References
References I
ANILE, 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, Katedra
geoinformatiky., 2013.
CAHA, J. Uncertainty Propagation in Fuzzy Surface Analysis. Phd, Palacký University
in Olomouc, 2014.
DUBOIS, D., PRADE, H. Possibility Theory: An approach to Computerized
Processing of Uncertainty. New York : Plenum Press, 1986. ISBN 0-306-42520-3.
FISHER, P. F. 1st Experiments in Viewshed Uncertainty - Simulating Fuzzy
Viewsheds. 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 : Van
Nostrand 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.
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18. Katedra Geoinformatiky, Univerzita Palackého v Olomouci, geoinformatics.upol.cz
Introduction Fuzzy surface Possibilistic visibility References
Thank you for your attention.
InDOG Conference 2014 - 14.10.2014 18/18