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 .