Image Processing Reference
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distributions, and render uncertain isosurfaces. In all three cases we demonstrate how
using hixels provides the capability to recover prominent features that would other-
wise be either infeasible to compute or ambiguous to infer. We use a collection of
computer tomography data and large scale combustion simulations to illustrate our
techniques.
3.1 Foundations
The concepts presented in this chapter rely on mathematical foundations from both
the topology and statistics communities. A summary of the statistical methods used
in this chapter can be found in [
3
]. The topological tools presented in this paper are
based on Morse theory, a mathematical tool to study how the “shape” of a function is
related to the shape of its domain [
8
,
9
]. Morse theory is a well understood concept in
the context of smooth scalar fields, and has been effectively extended to piecewise-
linear [
2
] and discrete [
4
] domains. Algorithms for computing the Morse-Smale (MS)
complex have been presented in the piecewise linear context [
1
,
2
]aswellasthe
discrete context [
5
-
7
]. In this chapter we summarize the method presented in [
10
]
to extend the use of the MS complex to hixels.
We are interested in characterizing an uncertain scalar field defined at many points
in a metric space,
M
.A
hixel
is a point
x
i
∈ M
with which we associate a histogram
(
x
i
)
(
x
i
)
of scalar values,
h
could either represent a collection
of values in a block of data, collections of values at a location over a set of runs
in an ensemble, or uncertainty about the potential values a location may represent.
Figure
3.1
shows several empirical distributions with maxima identified.
. In our setting, the
h
3.1.1 Bucketing Hixels
When a hixel is defined empirically as a number of samples
n
f
i
on a finite support
{
, we call each entry of the support a
bin
. The probability
distribution (specifically here a probability mass function) is thus given by:
f
j
|
j
∈{
1
,
2
,...,
N
f
}}
n
f
i
N
f
k
=
h
:
f
i
−→
1
n
f
k
for each possible value
f
j
. Whether this distribution is defined empirically or
analytically, for instance as a weighted sum of Gaussians, we are interested in identi-
fying regions of high probability associated with peaks in the probability density. For
that we will perform topological segmentation of the histogram to identify peaks as
well as a range of function values associated with each peak. This range of function
values is called a
bucket
. A bucket aggregates one or more bins and is assigned a
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