Volume visualization - The Visualization of Spatial Social Structure - page 258

Box 9.1 Three-dimensional smoothing

The one- and two-dimensional bino-
mial smoothing discussed earlier can
easily be extended to work in three
dimensions. The three matrices shown
here give the smoothing factors
around a single voxel in three dimen-
sions. After a few passes, application
of this filter approximates the trivari-
ate normal distribution.
Below is shown a perspective
view of the bivariate normal kernel
which could be used to smooth a point
distribution. A trivariate function was
used to smooth the cancer cases in
this topic. The width and sharpness
of the kernel is more important than
the actual shape of the function, which
is chosen for its effect on the final
image. These parameters correspond
to the number of passes of binomial
smoothing undertaken. Often it is use-
ful to explore the effects of a range
of parameters (if possible, while the
image changes before your eyes).

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Childhood leukaemia cases in the North of England over the twenty years
prior to 1986 numbered several hundred, with a high level of accuracy involved
in their recording. We can think of these cases as points, sitting geometrically in
a block of two decades of Northern English spacetime (Figure 9.7). If we were
to render these cases as simple points, then, because of their sparsity, we may
well not pick up any slight increase or decrease in the density of cases or some
more subtle spatial and temporal arrangement. To aid visual interpretations here
the points are represented by spheres (Figure 9.8). This process can also create
a truly three-dimensional surface; a single value existing at every point in time
and space is related to the general prevalence of the disease about that time and
around that place.

by a pile of sand, with the height of the pile at the place the person occupies and decreasing away
from him. Suppose there is a similar sandpile around the place of residence of every individual.
Now let all this sand be superimposed. At any point the total height of the sand will be the sum of
the heights of all the individual sandpiles. The total height is a measure of total population influence
at that point, and a contour map or a physical model may be made of the entire surface' (Warntz,
1975, p. 77).