Image Processing Reference
In-Depth Information
={
f
1
,
f
2
,
f
3
,
where
x
represents the Euclidean norm of vector
x
[
9
]. Let
F
...,
f
m
}
M
be a set of smooth scalar fields defined on a manifold
. The derivative at
a point
p
∈ M
is written as a matrix of partial derivatives,
⎡
⎣
⎤
⎦
∂
f
1
)...
∂
f
1
∂
x
1
(
p
x
n
(
p
)
∂
.
.
.
.
.
dF
(
p
)
=
∂
f
m
)...
∂
f
m
∂
x
1
(
x
n
(
)
p
p
∂
p
at point
p
is defined as the norm of the matrix
The
multifield comparison measure
η
p
dF
(
p
)
,
η
=
dF
(
p
)
.
14.3.3.1 Properties and Computation
F
The measure
p
satisfies three important properties: symmetry, coordinate system
independence and stability.
•
η
Symmetry
. The measure is independent of the permutation of the functions in
F
.
•
Coordinate system independence
. The norm of the matrix
dF
at a point
p
does
not depend on the coordinate system used to represent
p
.
•
Stability
. A finite change in the functions results in a bounded change in the
multifield comparison measure. The amount of change additionally depends on
the size of the triangle.
Evaluating the multifield comparison measure at a point requires the solution to a
maximization problem. Nagaraj et al. show that this computation can be reduced to
the faster evaluation of the maximum eigenvalue of a positive semi-definite matrix
Λ
=
(
T
dF
(
p
))
(
dF
(
p
))
x
2
p
x
T
η
=
Λ
max
n
x
∈R
,
x
=
1
√
=
max
{
λ
:
λ
is a diagonal element of
Λ
}
√
T
=
max
{
λ
:
λ
is an eigenvalue of
(
dF
(
p
))
(
dF
(
p
))
}
.
The derivative matrix
dF
is constant within each mesh element if the scalar field
is available as a sample and linearly interpolated within elements of a triangulation.
(
p
)
14.3.3.2 Applications
The multifield comparison measure has been applied to study various real-world
data from weather modeling, climate simulations, and combustion simulations. In
particular, it was used to study a simulation of the hurricane Isabel and the analysis
of a global wind pattern data set.
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