Geoscience Reference
In-Depth Information
MaxEnt explicitly associates the prior probability of a candidate image with the Shannon
entropy of the brightness distribution,
S
=
. Here,
F
=
f
t
1=
g
0
=
g
is the
total image brightness (“1” being a column with unity elements). Of all distributions, the
uniform one has the highest entropy. In that sense, entropy is a smoothness metric. The
entropy of an image is also related to the likelihood of occurrence in a random assembly
process. All things being equal, a high-entropy distribution should be favored over a low
entropy one. The former represents a broadly accessible class of solutions, while the latter
represents an unlikely outcome that should only be considered if the data demand it. Finally,
only non-negative brightness distributions are allowed by
S
. In incorporating it, we reject the
vast majority of candidate images in favor of a small subclass of physically obtainable ones.
−
∑
i
f
i
ln
(
f
i
/
F
)
(
0
)
Neglecting error bounds for the moment, the brightness distribution that maximizes
S
while
being constrained by (6) is the extremum of the functional:
g
t
f
t
h
f
t
1
E
(
f
(
λ
,
L
)) =
S
+(
−
)
λ
+
L
(
−
F
)
(7)
where the
is a column vector of Lagrange multipliers introduced to enforce the constraints
by the principles of variational mechanics and
L
is another Lagrange multiplier enforcing the
normalization of the brightness. Maximizing (7) with respect to the
f
i
and to
L
yields a model
for the brightness, parametrized by the
λ
λ
j
:
F
e
−
[
h
λ
]
i
Z
f
i
=
(8)
=
∑
i
e
−
[
h
λ
]
i
Z
(9)
where we note how
Z
plays the role of Gibbs' partition function here.
Statistical errors are accounted for in WD85 by adapting (7) to enforce a constraint on the
expectation of
2
.
χ
The constraint is incorporated with the addition of another Lagrange
multiplier (
). The constraint regarding the normalization of the brightness is enforced by
the form of
f
resulting from (8) and need not be enforced further.
Λ
E
(
f
(
e
,
λ
,
Λ
))
)
λ
+ Λ
e
t
Ce
−
Σ
g
t
e
t
f
t
h
=
+(
+
−
S
+
Λ
e
t
Ce
−
Σ
g
t
e
t
=(
+
)
λ
+
F
ln
Z
(10)
where the last step was accomplished by substituting (8) and (9) into
S
. The
term constrains
the error norm, calculated in terms of theoretical error covariance matrix
C
, which we take
to be diagonal. Rather than finding the brightness with the smallest model prediction error
which also has a high entropy, WD85 finds the brightness which deviates from the data in
a prescribed way so as to have the highest possible entropy consistent with experimental
uncertainties.
Σ
Maximizing (10) with respect to the Lagrange multipliers yields 2
M
+
1 algebraic equations:
g
t
e
t
f
t
h
+
−
=
0
(11)
which merely restates (6). Maximizing with respect to the error terms in
e
yields equations
relating them to the elements of
λ
:
C
−
1
e
λ
+
=
2
Λ
0
(12)
