Image Processing Reference
In-Depth Information
the number of dimensions is typically larger than three, the space must be projected
into a 2D or 3D visual space. In the visual space, clusters can be indicated using
enclosing curves or surfaces or by color coding.
Many algorithms for projections exist. The simplest projection of the multi-
dimensional attribute space is to project all data points to a visual space that is
spanned by two of the attribute dimension. This projection leads to a standard 2D
scatter plot. In the context of multivariate volume data visualizations, standard 2D
scatter plots can be extended to continuous scatter plots [
5
,
6
,
15
,
23
,
24
]. Con-
tinuous scatter plots generalize the concept of scatter plots to the visualization of
spatially continuous input data by a continuous and dense plot. It uses the spatial
neighborhood in the object space to allow for applying an interpolation between the
points in the attribute space.
Blaas et al. [
8
] use scatter plots in attribute space, where the multi-dimensional
data is projected into arbitrary planes. In general, one refers to multi-dimensional
scaling for techniques that project high-dimensional data into a low-dimensional
visual space. Given a set of
d
-dimensional data points
, multidimen-
sional projection techniques apply some criterion to generate a representation of the
points in an
m
-dimensional space with
m
{
p
1
,...,
p
n
}
d
. A possible criterion is to preserve
as much as possible the neighborhood relationships amongst the original points. The
projected points are the input to visual representations that convey information about
groups of elements with similar or dissimilar behavior. Several classical techniques
like Sammon Mapping [
34
], FastMap [
12
] or more recent techniques like Nearest-
Neighbor Projection [
37
] or ProjClus [
31
] are described in the literature to handle
different high-dimensional data in different ways. A somewhat outdated survey paper
is given by König [
4
]. Least Square Projection [
32
] is a multidimensional projec-
tion technique that effectively handles large data sets characterized by a sparse data
distribution in the high-dimensional space.
Takanashi et al. [
36
] applied Independent Component Analysis (ICA) on a multi-
dimensional histogram to classify the volume domain. Classification becomes equiv-
alent to interactive clipping in the ICA space. Paulovich et al. [
33
] presented an effi-
cient two-phase mapping approach that allows for fast projection of large data sets
with good projection properties. They applied their approach to multivariate volume
data and were even able to handle time-varying data.
The described projection techniques are based on a set of high-dimensional data
points as input. When pre-clustering the data, one can optimize the projection such
that clusters stay as much separated as possible. Linsen et al. [
25
] are using a respec-
tive projection technique for projecting clusters into a visual space whose layout is
given in form of optimized star coordinates. The main idea of the approach is to
represent the mode clusters of the cluster hierarchy by their barycenter and project
the centers using a linear contracting projection that maximizes the distance between
the clusters' barycenters. Since the method is linear, distances in the projected space
still allow for some interpretations. The contraction property assures that clusters
stay together and do not fall apart. The maximization of the distance between the
barycenters when projecting assures that separated clusters stay separated as much as
possible. Finally, the projection uses a star coordinate approach, where the directions,
≤
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