CRLB-like performance bounds for image deconvolution

_at_(nospam)rmc.ca
Date: 11/05/04 

Date: Fri, 05 Nov 2004 11:12:23 -0500

Hi all,

I am working on a deconvolution algorithm that can be formulated as a
statistical estimation problem. I am looking for some means to quantify
the performance limits of our estimator. I am having difficulties
because the estimation process is really a two-step process, and I don't
know whether the standard CRLB approach is valid.

For clarity, I don't want to go into extensive details, so I will try to
describe the problem in a simplified manner. I don't have a rigorous
background in statistical signal processing, so I may be using the
terminology incorrectly.

We are working with a process to de-blur images. Image formation can be
represented as a convolution, so an image, i(x), is given by a
convolution of an object, o(x), and a blurring function, s(x):

i(x) = o(x)**s(x), ** = convolution

We have a parametric model for s(x), with a parameter set {a(n)}, n =
1...N. Given a noisy measured image i(x), we first attempt to estimate
the parameters a(n) of s(x). Once we have obtained a "good" estimate of
these parameters, we construct an estimate of s(x) and perform a
de-convolution operation to obtain o(x) from i(x) given s(x).

We would like to quantify the uncertainty in the object estimate o(x)
given known noise statistics in the image i(x). We can theoretically
calculate a CRLB on the variance of the estimates for the parameters
{a(n)}, but we then need to relate this variance to the uncertainty in
the object estimate.

A colleague has suggested that this problem is similar to cases in which
a parameter estimation problem contains some parameters of interest and
some nuisance parameters, for which a hybrid or modifed CRLB will be
requried.

Any suggestions on appropriate literature or texts would be greatly
appreciated, as I am having a lot of difficulty in finding the best way
to tackle this problem.

Thanks,
Hal