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Consider an image characterized by its intensity
distribution I(x,y), corresponding to the observation of an
object O(x,y) through an optical system. If the
imaging system is linear and shiftinvariant, the relation between
the object and the image in the same coordinate frame is a
convolution:
I(x,y)= O(x,y) * P(x,y) + N(x,y) 


(14.99) 
P(x,y) is the point spread function (PSF) of the imaging system, and
N(x,y) is an additive noise. In Fourier space we have:



(14.100) 
We want to determine O(x,y) knowing I(x,y) and P(x,y). This
inverse problem has led to a large amount of work, the main difficulties
being the existence of: (i) a cutoff frequency of the
PSF, and (ii) an intensity noise (see for example [6]).
Equation 14.99 is always an illposed problem.
This means that there is not a unique leastsquares solution of minimal norm
and a regularization is
necessary.
The best restoration algorithms are generally iterative [24].
Van Cittert [41] proposed the following iteration:



(14.101) 
where
is a converging parameter generally taken as 1. In
this equation, the object distribution is modified by adding a term
proportional to the residual. But this algorithm diverges when we
have noise [12]. Another iterative algorithm is provided by
the minimization of the norm
[21] and leads to:



(14.102) 
where
P_{s}(x,y)=P(x,y).
Tikhonov's regularization [40] consists of minimizing the term:



(14.103) 
where H corresponds to a highpass filter.
This criterion contains two terms;
the first one,
,
expresses
fidelity to the data I(x,y) and the second one,
,
smoothness of the restored image.
is the
regularization parameter and represents the tradeoff between
fidelity to the data and the restored image smoothness. Finding
the optimal value
necessitates use of numeric techniques such as
CrossValidation [15] [14].
This method works well, but it is relatively long
and produces smoothed images. This second point can be a real problem
when we seek compact structures as is the case in astronomical imaging.
An iterative approach for computing maximum likelihood estimates may be used.
The Lucy method [#lucy<#15258,#katsaggelos<#15259,#adorf<#15260] uses such
an iterative approach:



(14.104) 
and



(14.105) 
where P^{*} is the conjugate of the PSF.
Next: Regularization in the wavelet
Up: Deconvolution
Previous: Deconvolution
Petra Nass
19990615