Image Processing Reference
In-Depth Information
Sauber et al. [
22
] address this issue by introducing the multifield-graph. Nodes of
this graph correspond to each subset of input fields and are displayed with icons that
graphically represent the similarity between the fields. Nodes are laid out in layers
corresponding to the number of fields in the subset. Two nodes in adjacent layers
are connected by an edge if the fields in lower layer node are also compared in the
upper layer node. The correlation/similarity and the size of domain with high cor-
relation/similarity are represented by the size and color of a disk displayed within
each node. A selective display of nodes enables focusing on nodes that represent
high correlations.
The derived field
κ
also extends to multiple fields. Given
k
scalar fields,
κ
is
defined as the norm of the wedge product between the 1-forms d
f
i
,
κ
=
d
f
1
∧
d
f
2
∧···∧
d
f
k
.
Each one form d
f
i
corresponds to the gradient
f
i
. The wedge product is a nat-
ural extension of the cross product of two gradient vectors and represents the
k
-
dimensional volume of the parallelpiped spanned by the
k
gradients [
3
]. While the
comparison measure
∇
does not satisfy the triangle inequality, it satisfies a number
of useful algebraic properties.
κ
1. Symmetry:
κ(...,
f
i
,...,
f
j
,...)
=
κ(...,
f
j
,...,
f
i
,...)
.
2. Degeneracy:
κ(
F
)
=
0ifd
f
i
=
d
f
j
for 1
≤
i
=
j
≤
k
.
3. Scaling:
κ(α
f
1
+
β,
f
2
,...,
f
k
)
=|
α
|·
κ(
f
1
,
f
2
,...,
f
k
)
, with
α, β
∈ R
.
κ(
f
1
+
g
1
,
f
2
,...,
f
k
)
≤
κ(
f
1
,
f
2
,...,
f
k
)
+
κ(
g
1
,
f
2
,...,
f
k
)
4. Sub-additivity:
.
5. Sub-multiplicativity:
κ(
f
1
,...,
f
i
,
f
i
+
1
,...,
f
k
)
vol
(
M
)
≤
κ(
f
1
,...,
f
i
)
·
κ(
f
i
+
1
,...,
f
k
)
.
14.3.1.2 Computation and Applications
In practice, the scalar fields are measured at discrete points in the domain and linearly
interpolated within elements in a triangulation of the manifold. In such a setting,
GSIMand
can be computed in a loop over the
d
-simplices in the triangulation. Since
all functions are linear over a
d
-simplex, their gradients/differentials are constant
within each mesh element. The norm of the
k
-form is evaluated at a point within the
d
-simplex directly from the formula and weighted by the volume of the
d
-simplex.
In a typical application of the multifield-graph, the user selects a particular node
using a visual interface and analyzes the derived field corresponding to that particular
node. Figure
14.5
shows the multifield-graph for the hurricane Isabel data set with
six scalar fields. Selected nodes are displayed within three of the five possible layers.
The extension of
κ
to
k
fields is directly visualized for two- and three-dimensional
domains to study the relationship between the fields.
κ
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