CR Bound for Poisson Statistics

We model the data, d, as a set of independent, Poisson random variables with rate vector g given by

where P is a distortion matrix. The PDF of d, given a, is

and the F matrix has elements

If we define a diagonal matrix Y with diagonal elements , F is

with inverse

where is a diagonal matrix with elements and Q is the inverse of P.

To evaluate Q, and , we assume that P is a block circulant matrix, as would be the case if P represents a convolution of the object with a point response function and if we neglect the end effects. In this case Q is also a block circulant matrix and can be evaluated with Fourier techniques.

We are working towards the evaluation of Eq. (4). In the case where â is an unbiased estimate, b is zero and the matrix C is an identity. Then and the CR bound on â is a vector with elements

This is also a convolution-like operation, but with the sum over index , it is actually a correlation operation.

This equation provides a simple prescription for obtaining the bound associated with an unbiased estimator such as the inverse filter. Construct the inverse filter, square its elements in the spatial domain, and convolve the transpose of the resulting function with g. The only problem with this approach is that g is unknown. However, reasonable approximations can be obtained by using an estimate based on â.

If the estimate â is biased and is given by

where W is any block circulant matrix, then we can show that

The recipe for is modified by the addition of the squared bias term and by the substitution of the filter for the inverse filter .

In our examples we use a Wiener filter described in the spatial frequency domain by

where is the Fourier transform of the point spread function, normalized to unit area, and a noise control parameter.