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Дата изменения: Tue Feb 24 13:18:52 1998 Дата индексирования: Tue Oct 2 19:01:19 2012 Кодировка: Поисковые слова: jupiter |
As discussed in Section 2.4, maximising the posterior probability for
the WF and MEM cases is equivalent to minimising respectively the functions
and
, which are given by (22) and
(23) as
From (19), in each case the standard misfit statistic may
be written in terms of the hidden vector
as
The cross entropy for this
complex image is given by (16) and (17).
Since is a complex vector we may consider
and
to be functions of the real and imaginary parts of the
elements of
. Alternatively, we may consider these functions to
depend upon the complex elements of
, together with their complex
conjugates. While it is clear that the former approach is required in order to
use standard numerical minimisers, a simpler mathematical derivation is
provided by adopting the latter approach. In any case, derivatives with respect
to the real and imaginary parts of
may be easily found using the
relations
Differentiating (31) with respect to and
, we find the gradient of
is given by
and upon differentiating once more we find the Hessian (curvature) matrix of
has the form
Using (32) and (33), the gradient of in
(29) is simply given by
and its Hessian matrix reads
where is the unit matrix of appropriate dimensions. We note that
the Hessian matrix for
is independent of
.
By setting the right-hand side of (34) equal to zero, and remembering
that and
, it is straightforward to obtain the linear relation
(14) for the WF solution. Moreover, from (27), the error
covariance matrix for the reconstructed signal vector is given by
, and using (35) this
is easily shown to be identical to the result (15).
In a similar way, we may calculate the derivatives of defined
in (30). Unfortunately, the form of the cross entropy given in
(16) precludes us from writing its gradient or curvature as a simple
matrix multiplication, and we must instead express them in component form. From
(16) and (17), we find the components of the gradient
vector of the cross entropy are given by
where and a similar
expression exists for
. Differentiating once more we find the
components of the Hessian of the cross entropy to be given by
and equals zero otherwise. We note that these components may be used to define
the (diagonal) metric on the space of images, which is given simply by by
(Skilling (1989); Hobson & Lasenby (1998)).
Using (36) and (37) the gradient and Hessian of
are then easily calculated. In particular, we find that the
Hessian matrix is given by
where is the image space metric and we have used the expression
for the curvature of
given in (33). In contrast to
(35), we see that the Hessian matrix of
depends on
through the metric
.