noise, (1)
Montana State University
www.math.montana.edu/~vogel
Abstract:
The physical problem of interest is too estimate both the phase (or wave-front aberration) and the object (or true image) from atmospheric image data modeled by
noise, (1)
where 
denotes 2-D convolution product and the point spread function 
. The obvious
nonuniqueness issues arising in (1) can be dealt with by taking new image datta generated by a technique known as phase
diversity.
The phase/object estimation problem presents some interesting mathematical and computational challenges. The problem is ill-posed, so regularization must be incorporated to obtain stable, accurate parameter estimates. We employ a penalty approach known as Tikhonov regularization, which requires the minimization of a function of the form
(2)where 
is a nonquadratic fit-to-data function, L is a symmetric positive definite matrix, and 
and 
are small
parameters. After discretization, the number of unknows is quite large 
We apply a pair of algorithms, (i) the limited memory BFGS method (l-BFGS) with line search globalization; and (ii) the Newton/CG algorithm with trust region globalization due to Steihaug. We will present numerical results and point out some interesting parallels between preconditioners for Newton/CG and the initial Hessian for l-BFGS.
Please join us for refreshments before the lecture at 2:30p.m. in room 353 Jabara Hall.
[ Spring 2001]