| Shaihan 2008-01-14, 8:23 am |
|
> Corect me if I'm wrong:
>
> 0. Assume size(A) = [m n], size(b) = [ m 1]
> 1. A and b are known and complex
> 2. A may be rank deficient
> 3. A has no symmetries to exploit
> 4. The solution x is complex
> 5. x may not be unique
> 6. You are unable to quantify the meaning of
> "solutions that are undesirable from a physical point of view"
> 7. You want to find x, possibly nonunique, satisfying
>
> || abs(A*x) - abs(b)|| is a min
>
> You can, at least, get good starting solutions from
> looping over solutions to
>
> A*x = R*b
>
> where
>
> R = diag(exp(2*pi*i*rand(m,1)))
>
> Hope this helps.
>
> Greg
Thank you very much for this; you can indeed do this and I am working
on this now. For anybody else who might be interested, the PhD thesis
of Peter Kassakian (Berkeley 2006) is on just this topic; he calls it
Magnitude Least Squares or MLS.
> 6. You are unable to quantify the meaning of
> "solutions that are undesirable from a physical point of view"
By physicality, I require that i) the spatial phase variation in b is
smooth and ii) ||x||^2 is not too high. I know that condition (ii) can
be dealt with with Tikhonov regularization, however I cannot easily
include (i)
Once again thanks for your help
Shaihan
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