Visual Exploration of Multivariate Volume Data Based on Clustering - Scientific Visualization: Uncertainty, Multifield, Biomedical, and Scalable Visualization - page 166

16.2 Automatic Clustering of Attribute Space
Cluster analysis divides data into meaningful or useful groups (clusters). Clustering
algorithms can be categorized with respect to their properties of being based on
partitioning, hierarchical, based on density, or based on grids. In partitioningmethods,
data sets are divided into an number of clusters and each object must belong to exactly
one cluster. In hierarchical methods, data sets are represented using similarity trees
and clusters are extracted from this hierarchical tree. In density-based methods,
clusters are a dense region of points separated by low-density regions. In grid-based
methods, the data space is divided into a finite number of cells that form a grid
structure and all of the clustering operations are performed on the cells. A complete
review of existing methods is beyond the scope of this chapter, but we refer to
respective survey papers [ 13 , 18 ].
In the context of multivariate volume data visualization, Maciejewski et al. [ 29 ]
proposed to apply a clustering of the attribute space to the 2D histogram obtained
by the range of a scalar function and the magnitude of its gradients. The clustering
is being visualized as a segmented image of the 2D histogram using color coding of
the segments. This segmented image serves as a user interface to select clusters and
display them in a volume visualization. This approach is limited to two-dimensional
attribute spaces.
When dealing with higher-dimensional attribute spaces, it is favorable to use
a hierarchical clustering method, as the cluster hierarchy can be used for visual
encoding of the clustering result. Moreover, a density-based approach is desirable, as
the computed density values can be assigned to the respective sample points in object
space leading to a volumetric density field. This density field can be exploited for
volume visualization methods. Density-based clustering methods are either kernel-
based or grid-based. Kernel-based methods convolve each point with some kernel
function and sum up the contributions of all kernels to compute the density. Grid-
based methods subdivide the space into cells, count the number of points that fall
into each cell, and compute the density as the number of points within a cell divided
by its area. Typically, the grid is a uniform, rectilinear one, i.e., it represents a multi-
dimensional histogram. In multivariate volume data, attribute values may be given at
different scales. Hence, when using kernel-based methods one would need to apply
anisotropic kernels. How to choose the scaling in the individual dimensions becomes
an issue. Thus, grid-based methods are favorable, as they split each dimension into
a number of cells independent of the scales the attributes are given in.
Linsen et al. [ 25 - 28 ] presented an approach for multivariate volume data visual-
ization that uses a hierarchical density-based approach where densities are computed
over a grid. The main advantage of that approach over other techniques with similar
properties is the direct identification of clusters without any threshold parameter of
density level sets. This property is achieved by the observation that density is pro-
portional to the number of points per cell when assuming cells of equal size. Hence,
density becomes an integer value and one can iterate over the density values. To
estimate all non-empty cells, a partitioning algorithm is used that iterates through all
dimensions. Given the multi-dimensional histogram, clusters are defined as largest
sets of neighboring non-empty cells, where neighboring refers to sharing a common