Geoscience Reference
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MaxEnt algorithm, which does not suffer from the limitations of Capon's method and which
possesses a number of other desirable features as a consequence of the incorporation of rather
informative prior information.
4. MaxEnt formulation
The algorithm described below derives from the MaxEnt spectral analysis method, a Bayesian
method based on maximizing the Shannon entropy of the spectrum (Shannon & Weaver,
1949). The method should not be confused with the maximum entropy method (MEM or ME)
or similar autoregressive models, with which it has only a remote connection (Jaynes, 1982).
MaxEnt was originally applied to spaced-receiver imaging by Gull & Daniell (1978) and also
by Wernecke & D'Addario (1977) in a more generalized way. Variations on the technique were
published later by Wu (1984), Skilling & Bryan (1984), and Cornwell & Evans (1985). MaxEnt
is now applied to a wide array of problems, including natural language processing (NLP),
quantum physics, and climate science, to name a few.
Expansive rationales for MaxEnt have been given by Ables (1974), Jaynes (1982; 1985), Skilling
(1991), and Daniell (1991), among others. MaxEnt is a Bayesian optimization technique that
maximizes the MAP (maximum a posteriori) probability of an image given prior probability
rooted in Shannon's entropy and constraints related to the model prediction error, error
bounds, image support, certain normalization, and other factors. As entropy admits only
globally positive brightness distributions, it rejects the vast majority of candidate solutions
in favor of a small, allowable solution subspace. The entropy metric favors uniform images
in the absence of contrary information but is nevertheless edge preserving. Moreover, the
use of entropy for prior probability makes the algorithm minimally dependent on unknown
quantities and, in that sense, bias and artifact free. It is a formalization of Occam's razor.
The algorithm described here is based on one developed by Wilczek & Drapatz (1985) (WD85)
for radio astronomy. The real valued brightness will be represented by the symbol
f
i
=
f
(
θ
i
)
.
The visibility data come from normalized cross-correlation estimates
v
1
v
2
(
k
d
j
)=
N
1
V
(5)
|
v
1
|
2
−
|
v
2
|
2
−
N
2
where the
v
1,2
represent quadrature voltage samples from a pair of receivers spaced by a
distance
d
j
and
N
1,2
are the corresponding noise estimates. The angle brackets above are
the expectation. We will represent the visibility data by the symbol
g
j
=
g
and assign
two real values for each baseline; one each for the real and imaginary part of (5). Given
M
interferometry baselines with nonzero length, there will be a total of 2
M
(
k
d
j
)
+
1 distinct visibility
data. (The visibility for the zero baseline is identically unity.)
In matrix notation, (2) may be expressed as
g
t
e
t
f
t
h
+
=
(6)
where
g
,
e
, and
f
are column vectors and
t
represents the transpose. Here, the elements of
the matrix
h
(
h
ij
) are the real or the imaginary part of the point spread function exp
(
ik
d
j
·
σ
i
)
, depending on whether
g
j
is the real or imaginary part of (5), and
e
j
represents the
corresponding random error arising from the finite number of samples used to estimate (5).
The elements
f
i
of vector
f
represent the brightness distribution evaluated across the defined
image space.
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