Image Processing Reference
In-Depth Information
N
M
v
D
1
ω
k
e
i
f
:
Ω
→
(
R
)
,(
f
(ω))
x
=
μ
i
+
φ
i
(
x
).
i
=
1
k
=
The Gaussian distribution at each position
x
, mean function and expectation function
can be derived from here as before.
9.6 Conclusion
We have shown that stochastic processes provide a suitable mathematical foundation
for the definition of uncertain fields in visualization. In the case of given Gaussian
distributions, we have demonstrated how thewell-known interpolationmethods allow
to define Gaussian processes from uncertain field data. We hope that these remarks
will stimulate and simplify research on the visualization of uncertain field data. Of
course, there is much more to say on the topic that would require more space than
available here. For further reading, we recommend the cited literature below.
References
1. Adler, R.: The Geometry of Random Fields. Wiley, Chichester (1981)
2. Adler, R., Taylor, J.: Random Fields and Geometry. Springer, New York (2007)
3. Adler, R., Taylor, J.: Topological complexity of smooth random functions. Lecture Notes in
Mathematics, vol. 2019. Springer, Heidelberg (2011)
4. Doob, J.L.: Stochastic Processes. Wiley, New York (1953)
5. Pöthkow, K., Hege, H.-C: Positional uncertainty of isocontours: condition analysis and proba-
bilistic measures. IEEE Trans. Vis. Comput. Graphics
17
(10), 1393-1406 (2011)
6. Pöthkow, k., Weber, B., Hege, H.-C: Probabilistic marching cubes. Comput. Graphics Forum
30
(3), 931-940 (2011)
7. Rasmussen, C.E., Williams, C.K.I.: Gaussian Processes for Machine Learning. The MIT Press,
Cambridge (2006)
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