Image Processing Reference
In-Depth Information
h
1
h
1
-
h
2
efg
a
h
3
420
h
2
a
b
231
h
e
c
051
d
600
b
f
i
h
1
-
h
3
h ij
a
c
g
j
f
510
b
141
d
c
240
d
015
y
x
Fig. 3.4
Shown is a 2-dimensional spatial domain (
x
and
y
) and a function,
f
, on which the data
is hixelated (
vertical axis
). Hixel
h
1
has buckets
a
,
b
,
c
,and
d
;
h
2
has
e
,
f
,and
g
;and
h
3
has
h
,
i
,and
j
.Onthe
right
, two contingency tables are shown tabulating the simultaneously observed
samples for
h
1
−
h
2
and
h
1
−
h
3
. ©IEEE reprinted, with permission, from Thompson et al. [
10
]
binations of values as shown in Fig.
3.4
. By considering simultaneously observed
samples of
h
i
and
h
j
, it is possible to identify pairs of buckets that co-occur more
frequently than if they were statistically independent by identifying those whose
pointwise mutual information
(pmi) is greater than zero. Pointwise mutual informa-
tion is a statistical measure of association between realizations of discrete random
variables. The pmi of a realization
(
x
,
y
)
of a pair of discrete random variables
(
X
,
Y
)
is defined as:
p
)
(
x
,
y
)
(
X
,
Y
pmi
(
x
,
y
)
:=
log
)
,
p
X
(
x
)
p
Y
(
y
where
p
X
,
p
Y
, and
p
respectively denote the probability density functions of
X
,
Y
, and the joint probability
(
X
,
Y
)
, for all possible outcomes of
X
and
Y
. When the
joint probability vanishes the pmi is set to
(
X
,
Y
)
. Note that if
X
and
Y
are independent,
then the pointwise mutual information vanishes everywhere the joint probability does
not. Naturally, as this is a pointwise quantity, a zero value of the pmi does not indicate
mutual independence of the random variables.
Pairs of buckets in neighboring hixels, with a pmi greater then some
−∞
0, can
be treated as edges in a graph connecting buckets We call these connected compo-
nents
sheets
, illustrated in Fig.
3.5
. Sheets are geometrically like lower-dimensional
surfaces in the product space of the spatial variables and the scalar data. Once we
have selected sheets, we compute topological basins of minima and maxima on each
sheet individually. We examine sheets on a mixture of 2 stochastic processes shown
in Fig.
3.6
a. This data highlights the fact that individual hixels can be multi-modal
and can behave as both a minimum and maximum. A naive analysis that computes
the mean or median followed by standard topological segmentation would fail to
incorporate the multi-modal nature of the data. To addresses this issue, topologi-
cal analysis is performed on sheets of the domain that have likely simultaneously
observable sets of behavior.
ε
≥
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