Image Processing Reference
In-Depth Information
h
(
x
1
)
h
(
x
2
)
h
(
x
3
)
h
(
x
4
)
f
j
f
j
f
j
f
j
←
x
→
Fig. 3.1
Four probability distributions represented as histograms
h
(
x
i
)
with 32 bins
f
j
(rotated
90
◦
). Maxima (identified with
black circles
) indicate function values with high probability.
Colors
indicate bucketing, the aggregation of bins of the histograms into modes based on the stable mani-
folds of persistent maxima. ©IEEE reprinted, with permission, from Thompson et al. [
10
]
probability given by the cumulative distribution function over that range of function
values. Figure
3.1
illustrates how the distributions have been bucketed by merging
maxima of
h
with their lowest persistence [
2
].
Because our scalar function
f
is represented by probability distribution
h
and we
are interested in identifying regions of high probability, we use a variant on the notion
of persistence. Typically, persistence ranks maxima by the difference in function
value between their value and their paired minima. Instead, we assign a value equal
to the area of the histogram between the pair (we call this ranking
areal persistence
).
By ordering intervals between maxima and minima according to the area underneath
them, peaks in probability density may be eliminated according to the probability
associated with them. The decision of which of the two possible minima (assuming
the maximum is interior) should be merged with the peak is made using regular
persistence: the smaller difference in function value indicates the region to which
corresponds the peak to be eliminated. Buckets can be merged in this fashion until
the probability of the smallest bucket is above some threshold. When the number of
samples is small, this threshold must be close to 1 since our confidence will be low.
Assuming that
f
has a finite variance (so that the central limit theorem holds), the
threshold may be lowered as the number of samples increases. Eventually, each hixel
will have one or more buckets corresponding to probable function values associated
with a peak in the distribution function; each bucket thus corresponds to an estimated
mode
of the distribution.
Figure
3.2
shows the bucket counts for the jet dataset as areal persistence thresh-
olds are varied. At low thresholds, hixels that encompass areas of turbulent behavior
have high bucket counts. As persistence simplification is applied, but increasing
the threshold of areal persistence, buckets are merging indicating the most probable
modes of the dataset.
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