Image Processing Reference
In-Depth Information
namely pairwise similarity/dissimilarity, global alignment, and dependency, are
derived quantities that can provide critical information on the input fields. For
example, when tracking features in fluid flow datasets, it is often desirable to measure
the alignment of a sequence of consecutive time-slices in the data. In Sect.
14.2
we
will review existing work on pairwise derived fields, i.e., the number of input fields
is two. In Sect.
14.3
we will consider global alignment and dependency measures
for the case when there are more than two input fields.
Another scenario of derived fields in the context of multifield visualization is
referred to as decomposition and componentization. In this case, this input may
be considered as a single field. However, its key characteristics are revealed by a
decomposition into multiple derived fields. The behavior of the input field can be
better understood by studying each derived field in the decomposition as well as
the interplay among them. An example of this is the well-known
Hodge-Helmholtz
decomposition
, where an input vector field is decomposed into the sum of three
vector fields: (1)
divergence-free
,(2)
curl-free
, and (3)
harmonic
vector fields. We
will review techniques corresponding to this category in Sect.
14.4
.
14.2 Pairwise Distances and Correlation Measures
A first step towards capturing the relationships between fields in multifield data
is to consider pairwise interactions. In this context we discuss the use of distance
measures, similarity measures, and local correlations between two fields.
14.2.1 Correlation Measures
The correlation coefficient is a standard and popular statistical measure used to
determine if two sets of real values are linearly related by comparing their deviations
from the respective mean values [
4
, Chap. 8]. When two scalar functions are sampled
at discrete points, the correlation coefficient is computed as
i
=
1
(
x
i
−¯
x
)
·
(
y
i
−¯
y
)
ρ
=
i
=
1
(
2
·
i
=
1
(
2
x
i
−¯
x
)
y
i
−¯
y
)
where
x
i
,
y
are the mean
values of the two functions. Two scalar functions have a high correlation coefficient
if they deviate consistently from their respective mean values i.e., if one function
takes a value close to its mean then so does the other function at the same point in the
domain. Note that the correlation coefficient as defined above is a global measure.
However, in the context of two time-varying fields, the correlation coefficient can
be computed at each point resulting in a derived field over the domain. This field
captures the linear relationship between the two time-series data at each point.
y
i
are the corresponding values of the two functions and
x
¯
,
¯
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