Image Processing Reference
In-Depth Information
1run
16 runs
64 runs
256 runs
16384 runs
1
a
j
0
Fig. 3.3
We sample the hixel data for an 8
×
8 blocking of combustion data, and compute the ag-
gregate segmentation for a number of iterations, also varying the level of persistence simplification.
Adjacent white pixels are identified in the interior of the same basin in every single run. The images
converge as the number of iterations increases
left to right
. ©IEEE reprinted, with permission, from
Thompson et al. [
10
]
as a basin boundary. Formally, at each sampled location we compute the aggregate
function
a
j
n
C
i
c
j
. Note that
a
j
can take values between one and zero, where
one indicates it was identified as the boundary of basins in every instance, and zero
meaning it was identified as interior in every instance. In this manner, we visualize
rasterizations of the geometry of the Morse complex.
One point of interest is the amount of sampling required to capture a reasonable
aggregate field. Figure
3.3
shows each aggregate slice for the 8
1
=
×
8 block size, as
number of iterations and topological persistence are varied. The convergence of
these sequences indicates that the distribution represented by the hixels produces
stable modes of segmentation.
3.2.2 Topological Analysis of Statistically Associated Buckets
We next describe a novel statistical technique for recovering prominent topological
features from ensemble data stored in hixel format. This computation is aided by
the fact that ensemble data has a statistical dependence between runs that allows us
to build a structure representing a predictive link between neighboring hixels. Our
algorithm identifies subregions of space and scalar values that are consistent with
positive association and we perform topological segmentation on only those regions.
After bucketing all hixels, we compute a
contingency table
or tabular representation
between each pair of adjacent hixels,
h
i
and
h
j
, of the counts of all observed com-
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